Perturbation Methods
Perturbation Methods
ALI HASAN NAYFEH
WILEYVCH
WileyVCH Verlag GmbH & Co. KGaA
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Bibliographic informatioa published by Die Deutscbe Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at 0 the above problem is a wellposed problem and admits a unique solution. However, if E = 0, (2.3.1) reduces to
4z*  Al = 0
(2.3.6)
which is parabolic (diffusion equation). The solution of (2.3.6) cannot in general satisfy all the boundary conditions (2.3.2) through (2.3.5) and one of them must be dropped. As discussed in Section 4.1.2, (2.3.5) must be dropped, and thus the resulting solution is not valid near y = 1. For small E the solution of the reduced equation is expected to be close to the exact solution except in a narrow region near y = 1 where the latter changes rapidly so that it satisfies the boundary condition which is about to be lost. It should be noted that the singular nature of the problem depends not only on the change in type of equation but also on the given region in which the solution is obtained. Although the solution of (2.3.1) in the region 0 5 5, y I 1 does not tend to that of (2.3.6) uniformly, the solution of (2.3.1) tends uniformly to the solution of (2.3.6) in the upper halfplane. Next we discuss an example in which the change in type of equation does not lead to nonuniformities.
LONG WAVES O N LlQUlDS FLOWlNG DOWN INCLINED PLANES In this section we consider the characteristics of waves on the surface of a liquid film flowing down an inclined plane (Figure 26). This is a rather involved example, and it is discussed here because it illustrates a general technique for long nonlinear dispersive waves. The flow is governed by the 2.3.2.
2.3.
TYPE CHANGE OF EQUATION
39
Figure 26
NavierStokes equations
ali+  =ae o a? ag aa + aaa + u"afi 1 a$ =  + gsine + ~ Q ' l i a?
ag a5 afi a/;  + u ,ao + 8 ae =  1 _
ai
(2.3.7)
(2.3.8)
+
 g c o s e v ~ 2 ~(2.3.9) ai a., ag P a?l where li and fi are the components of the velocity in the x and y directions, 8 is liquid pressure, t is time, and p and Y are the liquid's density and kinematic viscosity, respectively. At the liquidlsolid interface, both components of velocity vanish; that is
a=fi=o
at $ = O
(2.3.10)
If the liquid surface is flat (i.e., no waves), the following laminar steady solution exists g sin 0 P=0 (2h$  g2),
o=
P
2v
= 0,
(2.3.1 1)
 pg cos 8(g  h,)
In this solution we used the boundary condition ati/a$ = 0 at g = h, (i.e., no shear). We next consider fluctuations in this steadyflow configuration. We introduce dimensionless quantities according to a = ho/l,
= OIUL,
h = h/h,, u
y = $/h,,
iu = t/U,,
x = $11 u = fi/aU,
(2.3.12)
40 STRAIGHTFORWARD EXPANSIONS AND NONUNIFORMITY where U, = gh,2 sin 13/2v, 1is a characteristic length of the waves, and a is a dimensionless quantity measuring the shallowness of the liquid. Substituting (2.3.12) into (2.3.7) through (2.3.9) and using (2.3.11), we obtain the following equations for the dimensionless fluctuations u, Ut
+ u, = 0
(2.3.13)
+ (u + U)U, + u(u, + U ’ ) = Ut
4 (u
R
+ uR1
+ a2u,,)
(tlvv
1 2 + V)U, + uu, =  p +  (uvv + a2u,,) Ra2 ’ aR
(2.3.14) (2.3.15)
where R = ULh$v is a Reynolds number for the liquid film, U = 2y  y2, and primes denote differentiation with respect to y. Equation (2.3.13) can be solved by introducing a stream function ~ ( zY,, t ) such that ( u , 4 = (Yv9 YJ Then (2.3.14) and (2.3.15) can be combined into Yw
+ w + Y J Y ~ , , (u” + Y ~ J Y , I 2 a 2 ~ ~ ~ + a’R[~,,t + ( U + YJY,,,  v,v,,I  a 4 ~ x z z z (2.3.16)
= aRrywt
Equation (2.3.16) must be supplemented by boundary conditions. At the solid/liquid interface, (2.3.10) gives Yv(% 0 9 0 =
[email protected],0,t ) = 0
(2.3.17)
At the free surface the nornial velocity of the liquid is equal to the velocity of the interface; that is
h,
+ ( U + yv)hZ+ yZ = 0
a t y = h(s)
(2.3.18)
Moreover, the condition that the tangential stress vanishes at this free surface gives
(U’
+ yw  a2y,,)(l
 a2hZ2) 4a2yzvh,= 0
at y = h
(2.3.19)
Finally, the continuity of the normal stress across this free surface demands that  p t (h
 1)cot 0 
Th,, cosec 8 (1
+ u2h,8)”’
2a
1  u2h:
 2uYzu 1
+ u8h,
+ uzh:
Q2Yzs)hz
(YV”
1
=C
at y = h (2.3.20)
~ ~
2.3. TYPE
CHANGE OF EQUATION
41
where from (2.3.14)
P x = kalYgYY  &R"f',t 4 (u I Y,)Yzy  (u' and
+ Y,,)W,I + 8 u ~ , , ~(2.3.21)
T=U
Pd2
with CT the liquid surface tension. To determine an equation describing the dimensionless elevation of the disturbed surface, we follow Benney (1 966a) by first finding a perturbation solution for (2.3.161, (2.3.17), and (2.3.19) through (2.3.21) in powers of u and substituting the expansion for y into (2.3.18). Thus we let Y = yo
+ ay, +
P = r*'p1
+ po + up1
(2.3.22)
in the former equations and equate coefficients of like powers of u to obtain Order a" YOVYVV
=0
yo = yoy= 0 at
yoyy= 2(y Order
y =0
(2.3.24)
at y = h
(2.3.25)
ywVy= 0 at y = h dc
Ylvvvv
where
 1)
(2.3.23)
= R"P'ovvt
+ ( u + y o c / ) ~ o ~ v v (u" +
~ovyy)yo~I
(2.3.27)
y1 = yly= 0 at y = 0
(2.3.28)
ylVV = 0 at y = h
(2.3.29)
po = (h  1) cot 8  Tcosec 8 h,,
PO== &"~vvu
(2.3.26)
 &R"Povr
at y =
/I
(2.3.30)
+ ( u + Wou)Ylo~Y  (u' + W o v v ) ~ o ~ l (2.3.31)
The solution of (2.3.23) through (2.3.26) is Then (2.3.27) becomes
Yo = (h  l)Y2
Y l Y V V U = 2R(h, whose solution subject to (2.3.28) is
(2.3.32)
+ 2hh,Y)
Substituting for y1 into (2.3.29) through (2.3.31) and solving the resulting
42
STRAIGHTFORWARD EXPANSIONS AND NONUNlFORMITY
equations for A and B, we obtain
B = &(ha cot 8  T cosec e h,,) A
=
h(h, cot 8  T cosec 0 h,,,)
Since
a
 EY(Z, Y, f)lu=hl ax
 &Rh2(h,+ Qh'h,)
= (Yuh,
(2.3.34)
+ Yz)lv=n
we can rewrite (2.3.18) as
h,
a + (2h  h2)h, + ax [y(x,y,
f)ly=hl
=0
+ O(a) into this equation, we have h, + 2h2h2 = 0 Then y+, h , t ) = $h3(h, cot 8  T cosec 8 h,,,) + h ~ h ~ Letting y = yo + uyl + O(a2)in (2.3.35), we obtain h, + 2h2h2 + u [Qh3(c0t 8  +Rh3)h, + Q T cosec 8 h3h,,  2h2(cot 8  3 h 3 ) h , 2 + 2Tcosec 8 h2h2,,,] + 0(u2) = 0
(2.3.35 )
Substituting y = yo
(2.3.36) h ,
(2.3.37)
Let us summarize what we have accomplished in this section. We started with the elliptic equation (2.3.16) which was replaced by (2.3.23) and (2.3.27) which are clearly not elliptic. From these perturbation equations we arrived at (2.3.37) which is clearly hyperbolic. Thus nonuniformities due to the type change of equation did not arise because of the unboundedness of the domain. 2.4.
The Presence of Singularities
In this class the expansions have singularities within the region of interest, which are not part of the exact solution. Moreover, in the higherorder terms, the singularities are not only preserved, but they become even more pronounced. 2.4.1. SHIFT IN SINGULARITY As a first example in this class, we consider the problem discussed by Lighthill (1949a)
dY dx
(x + q )
+ ( 2 + x)y
=0
with y(1)
= e'
(2.4.1)
2.4.
THE PRESENCE OF SINGULARITIES
43
This equation is singular along the h e x = EY; the boundary condition makes the exact solution ~(z) positive for x 2 0, hence y(x) is regular in O<X c;' both break down near r = cz'. An expansion valid near this singularity is determined in Section 4.1.6 by using the method of matched asymptotic expansions. 2.4.4. TURNING POINT PROBLEMS As a last example, we consider the asymptotic expansions of the solutions
of
y"
+
 z')y
A2(1
(2.4.43)
=0
for large A. For Izi < 1, the solutions of this equation are oscillatory, while for Is1 > 1 they behave as exponentials. This suggests an expansion of the form = eA4(s;A) (2.4.44) where d = d o ( 4 A1+1(4 * . (2.4.45)
+
+
Substituting this expansion into (2.4.43) and equating coefficients of like powers of A, we obtain + 2;
2+;4;
= (1
+ 4;
The solutions of these equations are
 22)
(2.4.46)
=0
(2.4.47)
f i Y d 5 d . r f
IsdrP_l d7
9, = 4 In 4;
for 1x1 < 1
for 1.1
+ a constant
>1
(2.4.48)
(2.4.49)
Hence
for 1x1 < 1 (2.4.50)
2.5.
' T H E ROLE OF COORDINATE SYSTEMS
49
and = i.2
 1)1/4
[
a2 exp
(AJ'JZ
d7)
+ b, exp (A/'JFZ
dT)
]
for 1x1 > 1 (2.4.51) where ai and bi are constants. The expansions (2.4.50) and (2.4.51) are called the LiouvilleGreen or WKB approximation (Section 7.1.3). These expansions are singular at z = k l , hence they are not uniformly valid. The points z = & l are called turning points. The nonuniformity in this example arose because of the representation of the solutions in terms of elementary functions (namely, exponential and circular functions). The expansions show that the behavior of the solution changes across 1x1 = 1 from oscillatory to exponential, hence we need to represent the solutions by functions that exhibit this qualitative behavior. In this case the appropriate functions are the Airy functions (Section 7.3.1). 2.5. The Role of Coordinate Systems
In obtaining a parameter perturbation for a quantity such as u ( z ; E), we first choose an independent variable which need not be the physical independent variable z but a function 5 of z and the small parameter E. Then we assume
2 dm(~)um[5(z; 4,
u =
E)]
m=O
as
E + 0
(2.5.1)
where is an asymptotic sequence. We substitute this expansion into the governing equations, expand for small E keeping now 5 fixed, and then equate the coefficient of each 6, to zero. Thus
(2.5.2)
It is clear that, for a given sequence d,, u, depends on the choice of ((s;6). , Some of these choices 5 lead to nonuniform expansions, whereas others lead to uniform expansions. For example, choosing 5 = f in (2.1.1), we obtain the nonuniform expansion u ( t ; E) = a cost
+
EU~[$
sin t
+ &(cos 3t  cos t ) ] + 0(c2)
(2.5.3)
50 STRAIGHTFORWARD EXPANSIONS
AND NONUNIFORMITY
for Duffing's equation (Section 2.1.1). Had we chosen we would have obtained u ( t ; €) = 0
cos 5
+ €a3 +os 32
35  cos
5 = [l + (3/8)cu2]t,
g+ O(E2)
(2.5.4)
+
which is uniformly valid. Coordinates such as 5 = [l (3/8)€u2]t, which lead to uniform expansions, are called optimal coordinates (Kaplun, 1954). As a second example, we consider the model for the weak nonlinear instability problem of Section 2.1.2. Equation (2.1.18) gives the following expansion u = E cos u,t cos kx
+
1
+
~ t sin [ a,t cos ~ kx terms bounded as t + co (2.5.5) 320, This nonuniform expansion corresponds to the choice 5 = t. Had we chosen 5 = o,[l  (9/32012)]t,we would have obtained E
u = E cos 5 cos k x
+ O(e3)
(2.5.6)
which is uniformly valid for all 1. Hence this latter 5 is an optimal coordinate. As a third example, we consider supersonic flow past a thin airfoil (Section 2.1.3). Equation (2.1.36) gives the following expansion for the axial velocity component
1+
_
M4 yT'T"  TT" O(e3) (2.5.7) 2 B3 where T = T ( x  B y ) ; y = eT(x) (airfoil shape). This nonuniform expansion was obtained holding x and y fixed. Had we chosen y + 1 ~ 4 t = y and x  B y = (  c  ET'(5) (2.5.8) 2 B2 to be fixed, we would have obtained
1+
O ( E ~ )(2.5.9)  = I  €  "(') + E 2 [ $ ( 1  M4(y '))T"  TT" U B 4B2 which is uniformly valid. Hence (2.5.8) are optimal coordinates. It should be mentioned that a coordinate may be optimal for O(E) but not optimal for O(e2). For example u
5 = (1
+ &a2)t
(2.5.10)
2.5.
THE ROLE OF COORDINATE SYSTEMS
51
is an optimal coordinate for O ( 0 for the Duffing equation (2.1.1), whereas it is not optimal for any higher order. However,
1; = (1
+
* M ) t
(2.5.1 1)
$€a2
is optimal for O(c2) for (2.1.1). Since most straightforward perturbation expansions (obtained by holding the physical coordinates fixed) are nonuniform, perturbation techniques have been devised to render these expansions uniformly valid. In the method of strained coordinates (Chapter 3), some of these expansions are rendered uniformly valid by determination of the optimal coordinates as nearidentity transformations. In some of the problems considered, such as (2.2.1) and (2.2.2), a uniformly valid expansion was given by (2.2.1 1) as y = b exp (1  x)
+ ( a  be) exp (x  1)  + O(r)
(2.5.12)
+ O(e)
(2.5.13)
This expansion could not be obtained by keeping x fixed or by keeping x / e fixed. Had we kept x fixed, we would have obtained y = b exp (1  x)
= a # be in general. However, which is not valid near x = 0 because ~(0) had we kept x/c fixed, we would have obtained
y = be
+ ( a  be) exp (
3
(2.514)
which is not uniformly valid because y(1) = b # be. Thus the solution is represented by two different expansions using the coordinates (scales) x and x / e . Since they are different asymptotic representations of the same function, they can be related to each other by what is called the matching principle (Chapter 4). This suggests that uniformly valid expansions can be obtained by determining different expansions using different scales, relating these expansions by matching, and then combining the expansions. This is the method of matched asymptotic expansions described in Chapter 4. Rather than obtaining, say, two expansions using two different scales to represent the asymptotic solutions of (2.2.1), we keep both x and x / e or some functions of them fixed in carrying out the expansions. This means that we increase the number of independent variables to two and transform our original ordinary differential equation into a partial differential equation. This is the method of multiple scales described in Chapter 6. In oscillation problems such as ii
+ u = Ef(u, li)
52
STRAIGHTFORWARD EXPANSIONS AND NONUNIFORMITY
the unperturbed solution (i.e.,
E
= 0) is
u=acos+,
(2.5.15)
+=t+e
where Q and 0 are constants. If E # 0 the solution can still be expressed in the above form provided that a and 0 are taken to be timedependent. The method of variation of parameters (Section 5.1.1) can be used to find the following equations for (I and
+
da  E
dt
sin + f [ a cos
+, a
d4 €  = 1   cos + f [ a cos dt
U
(2.5.16)
sin +]
+, a
(2.5.17)
sin +]
To determine a uniformly valid expansion of the solution of these equations, we can introduce nearidentity transformations for both dependent variables a and rather than a nearidentity transformation for the independent variable as in the method of strained coordinates. This is the method of averaging described in Chapter 5.
+
Exercises 2.1. Determine a threeterm expansion for the solution near unity of (5
 1)(Z  T) +
= 0,
€
* = &+2.4 (3.1.14) Therefore u = a cos (wt
+ 4) + 32 u3 cos 3(wf + 4) + O(c2)
(3.1.15)
60
THE METHOD OF STRAINED COORDINATES
where a and 8 are constants of integration, and 0
 $a24
= (1 =1
+ &$&z
+
+ $a26  L 2 566d
2
+
* *
)I
0(E3)
(3.1.16)
3.1.2. TRANSITION CURVES FOR THE MATHIEU EQUATION As a second example, we determine the transition curves that separate stable from unstable solutions of the Mathieu equation
ii
+ (6 + c cos 2t)u = 0
(3.1.17)
which has been studied extensively. This equation is a special case of Hill's equation which is a linear differential equation with periodic coefficients. Equations similar to this appear in many problems in applied mathematics such as stability of a transverse column subjected to a periodic longitudinal load, stability of periodic solutions of nonlinear conservative systems, electromagnetic wave propagation in a medium with periodic structure, the lunar motion, and the excitation of certain electrical systems. The qualitative nature of the solutions of (3.1.17) can be described by using Floquet theory (see, e.g., Coddington and Levinson, 1955, Chapter 3). This equation has normal solutions of the form (3. I. 18) where 4 is a periodic function o f t having a period 7r or 27r, and y may be real or complex depending on the values of the parameters 6 and E . Floquet theory shows that the transition curves, in the de plane, separating stable from unstable solutions, correspond to periodic solutions of (3.1.17). Some of these curves are determined below by  both 6 and II as functions  expanding of E . Thus we let 6 = n2 cd, + e2d2 + * (3.1 .19) u = eY'+(t)
+ +
u ( t ) = 240
EUI
+
E2U2
+
 *
..
(3.1.20)
where n is an integer including zero, and u,/uo is bounded for all m in order that (3.1.20) be a uniformly valid asymptotic expansion. Substituting (3.1.19) and (3.1.20) into (3.1.17), expanding, and equating coefficients of equal powers of E , we obtain
+ ii, + ii2 + ii,
n2uo
=0
(3.1.21)
n2u1
=
(3.1.22)
n2u2
= (6,
(al + cos 2t)u0
+ cos 2t)u,  6,u0
(3.1.23)
The solution of the zerothorder equation is cos nt sin nt
n=0,1,2,
...
(3.1.24)
THE METHOD OF STRAINED PARAMETERS
3.1.
61
Next, we determine the higher approximations for the cases n = 0, 1, and 2. The Case of n = 0 In this case uo = 1 , and (3.1.22) becomes
 cos 2t
ii, = 6,
(3. I .25)
In order for (3.1.20) to be a uniformly valid asymptotic expansion, 6, must vanish; and 24,
+c
= Q cos 2t
(3. I .26)
where c is a constant. With uo and u1 known (3.1.23) becomes
ii, = 6,  t  c cos 2t  Q cos 4t
(3.1.27)
In order for uz/uoto be bounded, 6, must be equal to  l / S , hence 6
Q
=
€2
+
(3. I .28)
0(€3)
The Case of n = 1 I n this case uo = cos t or sin t. Taking uo = cos t, we find that (3.1.22) becomes ii, u, = (& 4)cos I  g cos 3t (3.1.29)
+
+
In order that ul/uobe bounded, 6, must be equal to 1/2, and then u, = r'&cos 3t
(3.1.30)
Equation (3.1.23) then becomes ii,
+
= (&
+ &)COSt +
cos 3t 
cos 5t
(3.1.31)
The condition that u,/uo be bounded demands that 6, =  1/32, hence
~
6=I 
€
+
2 o(€3)
(3.1.32)
Had we used uo = sin t, we would have obtained the transition curve 6=I
+ ic 
~
€
+
20 ( € 3 )
(3.1.33)
The Case of n = 2 In this case uo = cos 2t or sin 2t, and (3.1.22) becomes, in the former case, ii,
+ 4u,
=
+
 6, cos 2t  g cos 4t
(3.1.34)
Since ul/uo must be bounded, 6, must vanish, hence u, =
Q
+ & cos 4r
(3.1.35)
62
THE METHOD OF STRAINED COORDINATES
Substituting for u, and ul into (3.1.23) yields ii2
+ 4242 = (a2  A)cos 2t  2s cos 6t
(3.1.36)
Because u2/uomust be bounded, we require that 6, = 5/48, hence
6 =4
+
+
&€2
Taking u, = sin 22 leads to
(3.1.37)
O(E3)
+O(4
6 = 4  &€2
(3.1.38)
3.1.3. CHARACTERISTIC EXPONENTS FOR THE MATHIEU EQUATION (WHITTAKERS METHOD) Floquet theory (see, e.g., Coddington and Levinson 1955, Chapter 3) shows that (3.1.17) has solutions of the form indicated in (3.1.18), where is periodic (having a period 57 or 257) and y is a real or a complex constant. Substituting (3.1.18) into (3.1.17), we transform the latter into
+
6 + 2y4 + (6 + y2 +
f
cos 2t)+ = 0
(3.1 .39)
Since the transition curves correspond to y = 0, Whittaker (1914) obtained an approximation to near the transition curves by assuming the following expansions for 4, 6, and y
+
4 = +o 6 = 6,
+
.+l
+
+ €dl +
y = ‘y1
+
e2yz
+ 
e2+2 €26,
+ *
+
*
*
*
(3.1.40) (3.1.41) (3.1.42)
Next we carry out the solution for the case 6, = 1. Substituting (3.1.40) through (3.1.42) into (3.1.39) and equating like powers of z , we obtain (3.1.43) $0 40 =0
+
61 + 41 = 2y14,  (6, + cos 2t)bo 4 2 + 42 = 2~141  2y2&  (?I2 + &)+o  (6, + cos 2t)+1
(3.1.44) (3.1.45)
The general solution of (3.1.43) is +o = a cos 2
+ b sin t
(3.1.46)
where a and b are constants. Equation (3.1.44) then becomes $1
+ 4, = ~ y , a+ (+ W]
sin t  [2y1b
+ (4 + 6,)al cos 2
 $a cos 3t  &bsin 32 (3.1.47)
3.1.
THE METHOD OF STRAINED PARAMETERS
63
Since I#J is periodic, the terms that produce secular terms must vanish; that is
+ (t  6,)b = 0
2y1a
(3.1.48)
(S + &)a + 2y1b = 0
(3.1.49)
For a nontrivial solution for a and 6, the determinant of the coefficients of u and b in (3.1.48) and (3.1.49) must be zero. Hence (3.1.50)
Then
(3.1.51)
The solution of (3.1.47) subject to the conditions (3.1.48) and (3.1.49) becomes (3.1.52) I#J1 = &a cos 3t &b sin 3t
+
With the above results (3.1.45) becomes
BZ + I#Jz = [2yza  (6,
+ y12+ & P I
+ +
sin  [(d, y12 &)a 2 ~ &cos ] t NST
+
+
(3.1.53)
Terms that produce secular terms will be eliminated if
+ y12 + &)b = 0 (6, + + &)u + 2yzb = 0 2y2a  (6,
(3.1.54) (3.1.55)
Y12
Since b is related to a by (3.1.51), (3.1.54) and (3.1.55) are satisfied simultaneously if and only if y z = 0 and
6 , = ylz&
(3.1.56)
Therefore, to first approximation 14
= ae
(cos t
+
cos 3t)
+612Yl t (sin t +
6=1
If we let then
+ €a1 + iE2(d:
&c
sin 3t)
 $1 + 0 ( € 3 )
1+
6, = 4 cos 20
y1 = t sin 2a,
O(c2)
(3.1.57) (3.1.58) (3.1.59)
6 , = a c o s 40  2)
b  sin2a = cot0 a cos20 1
(3.1.60)
64
THE METHOD OF STRAINED COORDINATES
Hence (3.1.57) and (3.1.58) become = a"ee(l/4)(sin
6 =1
lokt
+ .Q
[sin (t
cos 2a
 a) + &c
+ +z(cos
sin (3t  4 1 4a  2)
+
+ O(r2)
0(€3)
(3.1.61) (3.1.62)
where a" is a constant.
3.1.4. THE STABILITY OF THE TRIANGULAR POINTS IN THE ELLIPTIC RESTRICTED PROBLEM OF THREE BODIES We consider next a fourthorder system with periodic coefficients involving the stability of the triangular points in the elliptic restricted problem of three bodies. The problem is governed mathematically by 2 ' '
 2y' 
y"
+ zx' 
1
h2" ecosf
+
=o
(3.1.63)
+hly
=o
(3.1.64)
 y)]
(3.1.65)
1 ecosf where primes denote differentiation with respect to f,and h1.z = $[I f J1  3p(l
Equations (3.1.63) through (3.1.65) describe the linearized motion of a particle near the triangular points in the restricted problem of three bodies. Here e is the eccentricity of the orbit of the two primaries and p is the ratio of the smaller primary to the sum of the two primaries. If e = 0, it is known that the motion is stable if 0 p < ji m 0.03852 and unstable for p 2 p. Therefore F is the intersection of a transition curve (from stable to unstable motion) with the y axis in the ey plane. Also, it is known from Floquet theory that periodic solutions with periods 277 and 47r correspond to transition curves. In the interval 0 y < p, the period 2a corresponds to y = 0, while 47r corresponds to po = (1  2&/3)/2; y = 0 corresponds to a transition curve that coincides with the e axis. In the following discussion we present an analysis following Nayfeh and Kame1 (1970a) for the determination of the transition curves that intersect the y axis at yo. We assume that
; (ct
+ y2u1 = o + y2u3 = 2coc2u~ + Su,"
(3.1.186) (3.1.187)
We take the solution of (3.1.186) to be u1 = cos k0,
coz = a2
+ yakz
(3.1.188)
3.2.
LIGHTHILL’S TECHNIQUE
77
so that (3.1.185) coincides with (3.1.183) to O(a).Then (3.1.187) becomes
+ y2u3 = (2c0c2k2 + $/I) cos k0 + $/Icos 3k0
(cO2 u’)u~,
(3.1.189)
Secular terms are eliminated if c2 = 3/I/8q,ka. Then
’
us=
32y2
cos3kB
Therefore a3i3 cos 3kO u = a cos k0  32y2
c = vIu 2
+ y2k’
[
1
+
*
]
343 + ... 8(a2k2 y2)
+
(3.1.190)
The technique used in this section was formalized by Stoker (1957) for the treatment of surface waves in liquids. This technique was used to treat the interaction of capillary and gravity waves in deep and finitedepth water by Pierson and Fife (1961) and Barakat and Houston (1968), respectively. It was also used by Maslowe and Kelly (1970) to treat waves in KelvinHelmholtz flow. Instead of expanding the phase speed, we could have expanded the wavenumber to determine the wave number shift or the frequency to determine the frequency shift. Variations of this technique were applied to a variety of problems. For example, Malkus and Veronis (1958) treated the Binard convection problem. Pedlowsky (1967) determined the response of a bounded ocean to a surface wind oscillating near one of the Rossby wave frequencies. Keller and Ting (1966) and Millman and Keller (1969) obtained periodic solutions for various systems governed by nonlinear partial differential equations, while Keller and Millman (1969) treated nonlinear electromagnetic and acoustic wave propagation. Rajappa (1970) investigated nonlinear RayleighTaylor instability. 3.2. Lighthill’s Technique The essence of Lighthill’s technique is t o expand not only the dependent variable u(z,, zl, . . , xn; c) in powers of the small parameter E, but also to expand one of the independent variables, say zl,in powers of E . Lighthill (1949a, 1961) introduced a new independent variable and then expanded both u and x1 in powers of E with coefficients depending on s. To a first approximation he assumed that x1 and s are identical. Thus Lighthill assumed
.
78
THE METHOD OF STRAINED COORDINATES
the following expansions for u and xI m
u = 2 (Emum(& 22,5 3 , . . . , xn) m=O
m
XI
= s 4
Z : E r n t r n ( S , x.2, x3,. m=l
. . , xn)
(3.2.1) (3.2.2)
It is clear that the straightforward expansion (Poincark type) consists of (3.2.1) alone with s replaced by zl.Since this straightforward expansion is not uniformly valid, Lighthill introduced (3.2.2) and chose Em (called straining functions) so as to make both of the above expansions uniformly valid; that is, he chose 5, so that the resulting approximation is uniformly valid. In some cases this is accomplished by requiring that and Urn1
Em
be bounded
(3.2.3)
tm1
In other words, higher approximations shall be no more singular than the Jirst. Comparing (3.2.1) and (3.2.2) with (3.1.2)and (3.1.4), we see that Lighthill’s technique is an extension of the method of strained parameters. This technique was modified by Kuo (1953, 1956) to apply to viscous flows. For this reason Tsien (1956) called it the PLK method, that is, the Poincarb LighthillKuo method. This method was applied to a variety of problems, especially wave propagation in nondispersive media. Lighthill (1949b) treated conical shock waves in steady supersonic flow. Whitham (1952) determined the pattern of shock waves on an axisymmetric projectile in steady supersonic flow and treated the propagation of spherical shocks in stars (1953). Legras (1951, 1953) and Lee and Sheppard (1966) applied it to steady supersonic flow past a thin airfoil, while Rao (1956) applied it to sonic booms. Holt and Schwartz (1963), Sakurai (1965), Holt (1967), and Akinsete and Lee (1969) investigated nonsimilar effects in the collapsing of an empty spherical cavity, while Jahsman (1968) treated the collapse of a gasfilled spherical cavity. Sirignano and Crocco (1964) analyzed combustion instability which is driven by chemical kinetics. Savage and Hasegawa (1967) studied the attenuation of pulses in metals, while Sakurai (1968) discussed the effect of plasma impedance in the problem of inverse pinch. Einaudi (1969, 1970) applied it to the propagation of acoustic gravity waves. Lewak (1969) and Zawadzki and Lewak (1971) solved Vlasov’s equation. Espedal (1971) used a combination of this technique and the method of matched asymptotic expansions to determine the effect of ionion collision on an ionacoustic plasma pulse. Asano and Taniuti (1969, 1970) and Asano (1970) extended this technique to nondispersive wave propagation in slightly inhomogeneous media.
3.2.
LIGHTHILL'S TECHNIQUE
79
Melnik (1965) applied it to the entropy layer in the vicinity of a conical symmetry plane. Mclntyre (1966) treated optimal control with discontinuous forcing functions. Ross (1970) applied it to diffusioncoupled biochemical reaction kinetics. Barua (1954) analyzed secondary flows due to rotation in an unheated tube, Morton (1959) treated laminar convection in a heated tube, and Morris (1965) investigated the case of laminar convection in a vertical tube. Chang, Akins, and Bankoff (1966) analyzed the free convection of a liquid metal from a uniformly heated plate. Crane (1959) rendered an asymptotic expansion for boundary layers uniformly valid. Goldburg and Cheng (1961) discussed the anomaly arising from the application of this technique and parabolic coordinates to the trailing edge boundary layer. Ockendon (1966) investigated the separation points in the Newtonian theory of hypersonic flow. Since Lighthill's technique is a generalization of the method of strained parameters, the first technique gives results identical to those obtained using the latter technique whenever it is applicable. Therefore the examples discussed next are problems that cannot be treated by the method of strained parameters. 3.2.1. A FlRSTORDER DIFFERENTIAL EQUATION The first example treated by Lighthill is the firstorder differential equation (3.2.4) where y(z) and Y(.c) are regular functions for all 5 of interest. Wasow (1955) determined the necessary conditions for the convergence of Lighthill's expansion for this problem; the proof had an error which was corrected by Lighthill. Usher (1971) investigated the necessary conditions for the applicability of this technique to equations of the form y'
=fk, Y> + .g(%
Y)
+
*
..
Comstock ( 1 968) showed that Lighthill's technique may lead to an erroneous expansion (see Exercise 3.28) for ( 2 " + 6y)y' + nx"ly = rnz'"l , Y(1) = a > 1
while Burnside ( 1 970) investigated the uniformity of the expansion obtained by straining z = x n rather than 2. It is clear that the region of nonuniformity is in the neighborhood of x = 0. For E = 0, (3.2.4) has the solution y = [exp
jz
[J' y ( e x p J ' v
d7)
+ c]
(3.2.5)
80
THE METHOD OF STRAINED COORDINATES
Let q(0) = qo, then expf
4O dt = xaoR(x)
(3.2.6)
t
where R(x) stands for a function regular at x = 0. Since r(x) is regular at
x=o
y = R(x)
+ O(x*o)
as
2 +
(3.2.7)
0
except when qo is a negative integer. In the latter case y = R(z)
+ O(x90 In x)
as x + 0
(3.2.8)
Equations (3.2.7) and (3.2.8) show that the zerothorder solution of (3.2.4) is bounded or unbounded as x + 0 depending on whether qo < 0 or qo 2 0. In order to show the details of the method, we apply it to a specific example for which qo = 2. In this case we consider the following problem treated by Lighthill (1949a) and Tsien (1956)
+ ey) dY + (2 + x)y = 0, dx
(z
y(1) = Ael
(3.2.9)
where A is a constant. Following Lighthill, we assume that (3.2.10) x =s
+ 2 ErnXm(S) m
(3.2.11)
m=1
Then
(3.2.12) In order to apply the boundary condition, we need to deErmine the value of s, denote it by S, corresponding to x = 1 ; that is, we must solve m
s = 1  2 Pxm(S)
(3.2.13)
m=l
We expand 2 in powers of
E
according to
s = 1 + ES1
+ €2s2+  
(3.2.14)
*
Substituting (3.2.14) into (3.2.13), expanding, and equating coefficients of equal powers of E lead to
s = 1  q ( 1 )  €2[X2(1)  xl(l)x;(l)]
+
*
.
*
(3.2.15)
3.2.
LIGHTHILL'S TECHNIQUE
81
The boundary condition can now be written as
+
Ael = ~ ~ ( 1 4y,(l) )
or
 yX(l)z,(l)]
+ ...
(3.2.16)
yo(I) = Ael
(3.2.17)
Ydl) = Y6(1)21(1)
(3.2.18)
Substituting (3.2.10) through (3.2.12) into (3.2.9), expanding, and equating the coefficients of qo and E to zero yield sy;,
+ (2 + S)Yo = 0 + ( 2 + S>Yi = (2 + s)Yoz;  (yo + Y;)Z~  yoy;
S Y ~
(3.2.19) (3.2.20)
The solution for yo is
yo = A e  Y 2
(3.2.21)
With this solution (3.2.20) becomes
If x1 = 0, (3.2.22) reduces to the equation for the firstorder term in the straightforward expansion, where y1 is more singular at z = 0 than yo. In fact, yo = 0(r2), while y1 = 0 ( c 5as ) x + 0. To render the above expansion uniformly valid, x1 can be chosen so that y1 is no more singular than yo by eliminating the righthand side of (3.2.22). However, Lighthill found that a uniformly valid expansion can be obtained by choosing z1to eliminate the worst singularity. Thus he put (3.2.23)
or A 3s2
xl=
Then (3.2.22) becomes
Hence
3s3
(3.2.24)
+ Ae"($ +
s4

s,'
e6 t
5)
 +  dE 4
(3.2.25)
1
(3.2.26)
The straining function x2 can be found from the elimination of the worst singularity in yz to be 3A2 x2=(3.2.27) 1oS4
82
THE METHOD OF STRAINED COORDINATES
Therefore
where x = s    eA 3s2
3e2A2 los*
(3.2.29)
The roughest approximation that is uniformly valid near the origin is y = A~'s~
where s is the root of
(3.2.30) (3.2.31)
which is approximately x when x 2 0 and E 0, and stops at x w (3/2)(2Ae/3)ll8 if A < 0. If A = 1 then x = 0 corresponds to
+
S =
Hence at x = 0
Y=
yr3+ (7/a y y 3 +
(;)""+
10 3 10
U(r)
(3.2.32)
O(1)
(3.2.33)
E
3.2.2. THE ONEDIMENSIONAL EARTHMOONSPACESHIP PROBLEM The onedimensional earthmoonspaceship problem was studied by Nayfeh (1965a) and can be reduced to (see Section 2.4.2)
+1 P 2 We assume that
t(0) = 0
+ Ptl(4 + 0(P2) =s + + 0(P2)
t = tds) 2
,
Pl(S)
(3.2.34) (3.2.3 5 ) (3.2.36)
3.2.
LIGHTHILL'S TECHNIQUE
83
Substituting (3.2.35) and (3.2.36) into (3.2.34) and equating coefficients of equal powers of p , we have 202 = s,
The solution of (3.2.37) is
JZ
to =
t,(O) = 0
(3.2.37)
p
(3.2.39)
If z1= 0, I, = 0 t h (1  z)] as z +1. The singularity in t , can be removed if the righthand side of (3.2.38) is eliminated; that is (3.2.40) (3.2.41) Therefore, to first approximation where s is the root of
J2
t = gs3/2
+~
( p )
(3.2.42)
3.2.3. A SOLID CYLINDER EXPANDING UNIFORMLY I N STILL AIR Let us next solve the problem of a cylindrical shock wave produced by a cylindrical solid body expanding uniformly from zero in inviscid nonconducting still air. This problem was also studied by Lighthill. The radial expansion velocity is assumed to be ca, where a, is the speed of sound in still air and is a small quantity. The shock propagates with a uniform radial velocity Ma, where M is the shock Mach number. The flow between the cylinder and the shock is adiabatic and isentropic, hence it can be represented by a potential function q ( r , t ) (the radial velocity q = qJ given (3.2.44) where u is the local speed of sound and is related to a, by Bernoulli's equation; namely (3.2.45)
84 THE METHOD OF STRAINED COORDINATES
where y is the ratio of the gas specific heats. The gas is assumed perfect with constant specific heats. Three boundary conditions must be satisfied by pl. The velocity of the air at the cylinder's surface is equal to the velocity of its expansion, that is (3.2.46) The second condition is the continuity of pl across the shock. Since pl = 0 in still air pl(Ma,t) = 0 (3.2.47) The third condition is the RankineHugonoit relation between the shock velocity and the velocity of the air behind it; that is
2a,(M2  1)
 (Maor) = apl
(3.2.48) ar M(Y 1) Since there is no fundamental length in this problem, all flow quantities are functions only of r/uot.Thus we let ql
+
= a02tf(x),
Then the problem becomes
r x =a0t
(3.2.49)
subject to the boundary conditions df dx
=€
(3.2.51)
f ( M )= 0
(3.2.52)
(€)
(3.2.53) Since there are three boundary conditions imposed on a secondorder differential equation, a relationship must exist between M and E. Since E is small, f is small, hence the zerothorder term in the straightforward expansion is the solution of the linearized form of (3.2.50); that is (1  5 2 ) d y dx2
1df =0 + xdx
(3.3
3.2.
LIGHTHILL'S TECHNIQUE
85
Using the above boundary conditions, we find that .
f = e2[,/l
2 2
 1 dx,
(3.2.55)
M =1
This approximate solution has no physical meaning for x > 1, and the shock Mach number must be greater than 1 for there to be any propagation. In order to determine a valid solution beyond z = 1 , and in order to determine by how much M exceeds 1 , we find it convenient to transform the secondorder equation (3.2.50) into a system of two firstorder equations by letting df= g
(3.2.56)
dx
Then we assume the following expansions
+ €"fi+ . . g = E 2 g o + e4g, + x = s + 2x1(s) + e4x2(s)+ . . . M = 1 + C ~ M+, r4M, + . . f = €yo
(3.2.57)
*
(3.2.58)
*
(3.2.59) (3.2.60)
*
The zerothorder term is given by (3.2.54) if x is replaced by s; that is
(3.2.61)
The firstorder problem is (1  s2)g;
+
S
 (1  sZ)g;x; + [2sx, S
As s t 1, go + J 2 ( 1  s);fO
+ ( y + 1)sgo  ( y  1)folg; goxl = 0
 4f; = g ,
+go4
2 (1
(3.2.62) (3.2.63)
 s ) ~ ' ~hence , (3.2.62) becomes

xl + y + 1 + O ( , / G ) as s 1 (3.2.64) J2(1  S) Thus gl will have a singularity at s = 1 unless x1 = 0, and gl(l) = y 1 as a consequence. g, =
+
86
THE METHOD OF STRAINED COORDINATES
In order to determine M,, we use the boundary conditions at the shock. If S corresponds to the position of the shock x = M, then to order e2 d= 1
+ €2Ml+
(3.2.65)
 * .
and the boundary condition (3.2.53) gives 2g0(1
+ €2M,+   *) +
* * *
+
4
= c2M1 Y + l
Substituting for go from (3.2.61) and equating the coefficients of sides, we obtain MI = 0. Hence S= 1
+ e4[[M,
 s,(l)]
+
* *
.
Then (3.2.52) and (3.2.57) give
(3.2.66)
* *
c2
on both (3.2.67) (3.2.68)
h(1) = 0 The secondorder equation gives for g2
In order to remove the singularity from g,, we set 5 2
Hence d = 1 + e4[[M2  ( y condition (3.2.53) gives J 2nyn  l)!,
n
>2
(3.5.15)
Therefore the expansion (3.5.13) is divergent; in fact, it is “more” divergent than (3.5.10), and the expansion breaks down as z approaches O ( E ~ ’ ~All ). that has been achieved is the exchange of an invalid expansion of one variable for an invalid expansion of the other. The reason for the breakdown of the expansion is the dropping of the highest derivative whose effect is small for large z but becomes significant as z approaches the region O ( P ) . 3.5.3. THE EARTHMOONSPACESHIP PROBLEM Next we show that the application of the method of strained coordinates to the twodimensional earthmoonspace problem (introduced in Section 2.4.2) leads to an invalid expansion. To render the expansion (2.4.17) and (2.4.1 8) uniformly valid using the method of strained coordinates, we substitute (3.5.16) x =s pl(s) .*
+
+ 
into this expansion and obtain
J2
1 t = 3 sin’ pJs P
1 7 Js(l P
 p2s)
(3.5.17) The straining function x1 is chosen so that the term between the brackets is no more singular than the first term as s 1 ; that is XI =

1
2(1  $1
ln(1  s)
(3.5.18)
The straining function could have been chosen to eliminate the term in the brackets. The resulting firstorder expansion is then
y = #us
+ O(F2)
(3.5.20)
EXERCISES
103
where
(3 S . 2 1) Comparing this expansion with the exact solution, Nayfeh (1965a) showed that this expansion gives a divergent trajectory near the moon although the expansion obtained in the onedimensional case agrees fairly well with the exact solution. In the onedimensional case there is a singularity at x = 1 p/(l  p z ) O(p2) which is outside the range of interest 0 5 x 5 1. In the straightforward expansion the singularity is shifted to 5 = 1, a n d the straining of z moves the singularity from 5 = 1 toward its right position. However, in the twodimensional case there is a sharp change in the direction of the spacecraft in the neighborhood of the moon, and the straining which is O ( p ) cannot cope with such a sharp change.
+
+
Exercises 3.1. Consider the problem
ii
+ u = €2,
u(0) = a,
ri(0) = 0
(a) Determine a secondorder (threeterm) straightforward expansion and discuss its uniformity. (b) Render this expansion uniformly valid using the method of renormalization. (c) Determine a firstorder (twoterm) uniformly valid expansion using the method of strained parameters and compare the result with (b). 3.2. (a) Show that the motion of a point mass that moves freely along a parabola z2 = 2pz rotating about its axis with an angular velocity w is given by
(b) Determine a twoterm straightforward expansion for small amplitude and discuss its uniformity. (c) Render this expansion uniformly valid using the method of renormalization. (d) Determine a oneterm uniformly valid expansion using the method of strained parameters and compare the result with (c). 3.3. Determine a twoterm uniformly valid expansion for small amplitudes for the solution of
which describes the oscillations of a pendulum.
104
THE METHOD OF STRAINED COORDINATES
3.4. Determine a secondorder uniformly valid expansion for the periodic solution of ii u = r(1  u2)U
+
Note that the amplitude is not arbitrary. 3.5. Determine a firstorder uniformly valid expansion for the periodic solution of ii u = €(1  z)U
+
Ti
where
is a constant. 3.6. Consider the equation
+z =
u2
7
ii
+ wO2u= €2+ k cos w t
Determine firstorder uniformly valid expansions for the periodic solutions when (a) wo M 2w [Hint: let wo = 2w PU and u = uo cul . . . where uo = a cos (2wt B) ( 1 / 3 ) k ~ cos  ~ w t , then determine a and 3/ from the equation for u,], and (b) wo M w / 2 (a is arbitrary in this case).
+
+ +
3.7. Consider the equation ii
+
+
+ wo2u = cu3 + k cos w t
Determine firstorder uniformly valid expansions for the periodic solutions when (a) wo M 3 0 and (b) wo M 4 3 . 3.8. Determine secondorder expansions for the odd solutions corresponding to the transition curves of ii (6 € cos 2t)u = 0 when 6 is near 1 and 4. 3.9. Consider the equation
+ +
ii
+
1
+ 6u
€COS2t
=o
(a) Determine secondorder expansions for the transition curves near
6 = 0, 1, and 4 (Shen, 1959). (b) Use Whittaker’s technique to determine secondorder expansions for
u near these curves. 3.10. Consider the equation
ii
+
6 1
 ccos2t  ECOS2t u = o
(a) Determine secondorderexpansions for the first three transition curves (Rand and Tseng, 1969) (Le., near 6 = 0, 1, and 4). (b) Use Whittaker’s technique to determine u near these curves. 3.11. Consider the equation ii
+ (6 + rcos3t)u = o
EXERCISES
105
Determine secondorder expansions for the first three transition curves using both the method of strained parameters and Whittaker’s technique. 3.12. Determine a periodic solution to O ( E )of the problem (Mulholland, 1971) ...
u
+ ii + U + u
= (1  u2
2
 ii2)(ii
+ U)
3.13. Determine firstorder expansions for ii
+ lu = EU3
subject to (a) u(0) = u ( n ) = 0, and (b) u ( t ) = u(t 3.14. Determine a firstorder expansion for ii
+ 277).
+ lu = r(sin 2t + uz)u u ( r ) = u(t + 2a)
3.15. Determine firstorder expansions for
+ ilu = €tu U O ) = U ( f + 277) (b) ii + Au = .(a cos 2t + u ( t ) = n ( t + 277) (a) ii
sin 2r)u
3.16. Determine a firstorder expansion for small amplitudes of the periodic solution of k t  x + g ( i  c0s e)  ( I z)~2 = o m
+
+
e + l + x
sine
+
xe
=o
which describe the oscillation of a swinging spring of length I and constant k when oI2= k / m m 4oz2 = 4gll. 3.17. The free vibrations of a simply supported beam on an elastic foundation are given by u,,,, yu EYU3 Ut* = 0
+ +
u(0, t ) =
U(T,
+
t ) = u,,(O, t ) = u Z z ( a ,t ) = 0
u(x, 0) = u sin 2,
ut(z, 0 ) = 0
where >’, a , and t are constants. Determine an expansion to O(E)of the frequency of oscillation (Han, 1965). 3.18. Carry out the expansions of Sections 3.1.4 and 3.1.5 to second order. 3.19. Consider uniform traveling wave solutions of Utt
 uzz
+ u = €U3
106 THE METHOD OF STRAINED COORDINATES of the form u = a exp i(k.l:  w t ) and wave number shifts. 3.20. Consider the problem
+ higher harmonics and determine the frequency
+ u,, + u,x2r = €2
Utt
U ( X ,0) = a cos kx,
u~(z, 0) = 0
(a) Determine a firstorder straightforward expansion. (b) Render this expansion valid using the method of renormalization. (c) Determine an expansion valid for t = O(el) using the method of strained parameters. (d) Show that the frequency is invalid near k = 1. (e) Apply the method of renormalization to this frequency to remove the singularity. (f) Show that the resulting expansion is erroneous. 3.21. Consider the eigenvalue problem +xx
+ +YV + A+
= ex2+
+ ( x , 0 ) = C(x, r) = C(0, Y) =
+(X,
Y)
=0
Determine firstorder expansions when 1 is near 2 and 5 . 3.22. Consider the problem
v2+ + A+
+
=
E
f ( x , Y,
z)+
with vanishing on the surfaces of a cube of length X. Determine firstorder expansions when I = 3 and 6 if (a)f = z2and (b)f = z2y. 3.23. The free transverse vibrations of a simply supported beam are given by ElwrZZz

T=
+ =0 ’( w , ) ~d.z
Tw,,
ES
w ( 0 , I ) = w(1, t ) = W,,(O, w(x,O)
=
TX
asin
I
,
PW~(
I) =
w,,(l,
t) =0
wt(x,O) = 0
where E, I, p. S, a, and lare constant. Determine a firstorder expansion for small amplitude (Evensen, 1968). 3.24. Consider the problem (.1:
+ €Y)Y’ + y = 0 ,
y(1) = 1
(a) Determine a secondorder straightforward expansion. What is its region of nonuniformity ? (b) Render this expansion uniformly valid using the method of renorrnalization.
EXERCISES
107
(c) Determine a firstorder expansion (two terms in y and three terms in z) using Lighthill's technique and compare it with (b). (d) Determine the exact solution by interchanging the roles of dependent and independent variables and compare it with (b) and (c). 3.25. Show that a uniformly valid expansion of
is where z = t  ~ ( Inb t + b + 1) 3.26. Consider the problem
+ O(c2).
(z + .y)y'  *y = I + 2 2 , y(1) = 1 (a) Determine a secondorder straightforward expansion and discuss its uniformity. (b) Render this expansion uniformly valid using the method of renormalization. (c) Obtain a firstorder expansion using Lighthill's technique and compare the result with (b). 3.27. Use the method of renormalization to render the expansion of Exercise 2.12 uniformly valid. 3.28. Consider the problem (sn
+ q ) y ' + nxnly Y(l) = a
 mxm1
=O
>1
(a) Show that its exact solution is given by zny
+
&!.2 I
= xm
+ (a +
*€a2
 1)
(b) Show that application of Lighthill's technique gives y "Yo
= (5"
+ a  ?)I
(c) Show that (Comstock, 1968) theapproximate solution is erroneous near (x = 0) except for special values of m and n. (d) Introduce a new variable z = xn in the original problem and then strain the variable z to determine an approximate solution to y. Determine the conditions under which this new expansion is valid near the origin (Burnside, 1970), hence determine the role of changing the independent variable on rendering the approximate solution uniformly valid. 3.29. Consider the problem (1
+
EU)
au
ax
au += 0, ay
u(z,0) = E+(Z)
108 THE METHOD OF
STRAINED COORDINATES
(a) Determine a firstorder straightforward expansion for E da
+ (1 
+ 2A2 sin y + a
3
128 5a5 +cos 5y 128
2 +
128
8, cos 3y
(5.4.24)
For there to be no secular terms in u2
Hence 5a5
Me=
3072
cos 5 y 
(5.4.26)
1024
Therefore the solution to second order is given by
€2
u = a cos y   sin 3y
32
2U3
where
1024
da EU  = (I dt 2
[$a2cos 5 y
 *a2),
+ (a2 + 8) cos 3y] + O(e3) 4
a2 =
1
(5
+
(5.4.27)
(5.4.28)
l ) F
+
dY 2 A1 dA1 (5.4.29)  1 $a2) dt= [2a(do 256 where a, is a constant. Using (5.4.22) and (5.4.28), we can write (5.4.29) as
+'
2   ( EI dY 1  Hence
16
dt
2
8a
E
y=ttlnu+a 16 8 where y o is a constant.
A]
da dt
1 4a 21
7r 64
2
+yo
(5.4.30)
178
VARIATION OF PARAMETERS AND AVERAGING
5.4.3. THE KLEINGORDON EQUATION As a third example, we consider nonlinear waves governed by Utt
 c2u,,
+ A2u = € j  ( U , ut, us)
(5.4.3 1)
after Montgomery and Tidman (1964). If E = 0, (5.4.31) admits solutions of the form u = a cos ( k ~mot 4) (5.4.32)
+
where a and 4 are constants, and ko and oosatisfy the dispersion relationship
+ A2
wo2 = c2k02
(5.4.3 3)
For small but finite E , we seek an expansion of the form
+ €#,(a,y ) +  
u = a cos y
(5.4.34)
*
where a is a slowly varying function of both time and position according to
 = cA,(a) + c2A2(a)+ . . .
(5.4.35)
+ e2&(a) + . . .
(5.4.36)
aa at
i3U  &,(a)
ax
and y is a new phase variable which coincides with the phase of (5.4.32) for r = O
aw  = wo + at 'a  k,
ax
Ec,(u)
+ c2C,(a) + . . .
(5.4.37)
+ E D , ( ~+) e2D,(a) + . . .
(5.4.38)
In this case also, no u, contains the fundamental cos y. Substituting (5.4.34) through (5.4.38) into (5.4.31), using (5.4.33), and equating the coefficients of E on both sides, we obtain 12
(z:2
+
1 ul) = 2(w0A1
+ c2koB,)sin y
 2a(w0C1 + C2koDl) cos y
+ f [ a cos y, amo sin y , ako
sin py]
(5.4.39)
We now Fourier analyze f in terms of y as f [ a cos y, am, sin y , ako sin y ]
+ 2 [f,(a) sin n y + gJa) cos n y ] W
= go(a)
Tl=l
(5.4.40)
5.5.
AVERAGING USING CANONICAL VARIABLES
179
Eliminating secular terms, we have 2w0A, Sa(woC1
+ 2c%,B1 =fl(u)
+ c2koD,) = g1(a)
(5.4.41) (5.4.42)
Then the solution of (5.4.39) is
and D1from (5.4.35) through (5.4.38) into (5.4.41) Substituting for A,, B,, C1, and (5.4.42), we obtain
 + woaa aa
I
ax
at
!
at
h(a)
= E
2w,
&(a) ’ + w o, ag  = f
ax
2aw0
(5.4.44) (5.4.45)
where 0; = dwo/dko,the group velocity, and /? =
Iffl = 0
a
B = E(Z
I+J
 k0x + mot
=
h,(s
 wit)
+ wit)* 4aw0wh + hz(s  w i t )
(5.4.46) (5.4.47) (5.4.48)
where h, and hz are determined from the initial or boundary conditions. Equations (5.4.44) and (5.4.45) can be solved easily if a and fl are functions of either time or position only. 5.5.
The Method of Averaging by Using Canonical Variables
Let us consider a conservative dynamic system governed by the following Lagrange equations (5.5.1)
where q = ( q l , q 2 ,. . . , q N } is the generalized coordinate vector, t is the independent variable, overdots denote differentiation with respect to I , L(Q, q, t ) = T  V is the Lagrangian, and T and V are the kinetic and potential energies. Let us define the generalized momentum vector p = {Pl?PZ, . P N ) by aL (5.5.2) Pi = aqa ’
’ 7
180 VARIATION
OF PARAMETERS AND AVERAGING
and the Hamiltonian H by H = pT4  L
(5.5.3)
where pT denotes the transpose of p (if p is a column vector, pT is a row vector). Considering H as a function of p, q, and t only, we can write aH dH = 2 dqi i=l 6’4, Also, from (5.5.3)
aH + 2aH dp, +  d t api at
(5.5.4)
i=l
The second and third terms on the righthand side of (5.5.5) cancel according to (5.5.2). Moreover, since pi = aL/aq, from (5.5.1) and (5.5.2), we can write (5.5.5) as (5.5.6)
Comparing (5.5.4) and (5.5.6), we obtain the following canonical equations of Hamilton . = aH q. (5.5.7) ’ api aH (5.5.8) pi=aqi aL aH    (5.5.9) at at These equations replace Lagrange’s equations. Under a transformation from q and p to Q(q, p, t) and P ( q , p. f ) , (5.5.7) and (5.5.8) are transformed into
Q, t )
(5.5.10)
Pi = gi(P, Q, t ) If there exists a function K ( P , Q, t) such that
(5.5.1 1)
Qi
f i = a K
=fi(P,
and gi =
api then (5.5.10) and (5.5.11) assume the form
K  a
(5.5.12)
aQi
(5.5.13)
5.5.
AVERAGING USING CANONICAL VARIABLES
181
and Q and P are called canonical variables, and the transformation from q and p to Q and P is said to be a canonical transformation with respect to the function K . Canonical transformations can be generated using a socalled generating function S(P, q, t ) according to (e.g., Goldstein, 1965, Chapter 8; Meirovitch, 1970, Chapter 9)
p. =as,
as (5.5.14) ap, Once these equations are solved for q = q(P, Q, t ) and p = p(P, Q, t ) , K is related to H by '
K(P,
Q, t )
Qi=
841
= fWP,
Q, t ) , W, Q, 0, t l
+ as
(5.5.15)
If a canonical transformation can be found such that K = 0, then P is a constant vector according to the second relationship in (5.5.13), hence S is a function of q and t only. Since p i = from the first relationship in (5.5.14), S must satisfy the following socalled HamiltonJacobi equation
If S is a complete solution of (5.5.16), then (5.5.14) furnishes a complete set of integrals of the equations (5.5.17) Complete solutions of (5.5.16) are not available for a general H . However, if H = H , fi where is small compared to Ho and a complete solution S,,(P,, . . . , P N ,q l , . . . ,.qN,t ) is available for
+
H,(flSP, . . . , (5.5.18) a41 then an approximate solution of (5.5.17) can be obtained by using the method of averaging in conjunction with the method of variation of parameters. Thus we use a generating function
s = S,(P,, . . . , PN, 41, . . . ,qhr, t )
(5.5.19)
where P is timevarying rather than being a constant. Hence P and Q are given by aK aK p . = (5.5.20) ' aQi ap, 9
ea=
182
VARIATION OF PARAMETERS AND AVERAGING
where
K = H o + A +as0  A
(5.5.21)
at
as a result of (5.5.18). If the solution qo(P,Q, t) and po(P, Q , t) of (5.5.17) with H = Ho has a period T with respect to t , then an approximate solution to (5.5.17) is still given by qo and po but P and Q are given by (5.5.20) with K replaced by its average over T; that is, by
In (5.5.22), P and Q are held constant. We next illustrate this technique by its application to three specific examples. 5.5.1. THE DUFFING EQUATION We consider again the equation q
+
oo2q
+ €93 = 0
(5.5.23)
The Hamiltonian Corresponding to this equation is
+ wo8q2)+ &q4
H = $(p2
(5.5.24)
The HamiltonJacobi equation corresponding to the case e = 0 is (5.5.25)
This equation can be solved by separation of variables; that is, by letting
s = wl) +dt)
(5.5.26)
Equation (5.5.25) separates into or a =  u t
3=u
and
(27+
wo2q2
= 2a
Hence
or
s
s = ut +
s
(5.5.27)
1I
J2a
 ~ 2 a wzq2 dq
(5.5.28)
 w:q2
(5.5.29)
dq
with u the new momentum. Consequently, the new coordinate 2
2 112
(2aw0q)
1
d q = t+arcsin0 0
Ooq
Jz;;
is given by (5.5.30)
5.5.
AVERAGING USING CANONICAL VARIABLES
183
so that
J%.
q = sin
wo(t
+ 8)
(5.5.31)
0 0
We could have written this solution for (5.5.23) by inspection if E = 0. However, the canonical variables CL and ,6 were obtained naturally by solving the HamiltonJacobi equation (5.5.25). Since E? = (1/4)q4 = (ca2/wO4) sin4 wo(t p), the variational equations (5.5.20) become aaf@ i ’ a = 8. = a f i (5.5.32a)
+
aa
Now
so that (5.5.32b)
3ca and ,6 = 4t + Po 4w0 where Po is a constant. Therefore, to first approximation ct
= a constant
(5.5.33)
(5.5.34) in agreement with the expansions obtained in Sections 5.4.1 and 5.3 by using the KrylovBogoliubovMitropolski and Struble techniques if we identify &/w, by uo. 5.5.2. THE MATHIEU EQUATION As a second example, we consider for positive then
w.
If we let
q
+
( 0 2
+
E
cos 299 = 0
9=P
p
=
 (WZ + E cos 2t)q
(5.5.35) (5.5.36) (5.5.37)
from (5.5.35). These equations can be written in the form (5.5.38)
184
VARIATION OF PARAMETERS AND AVERAGING
where
H = )(p2
+ w2q2) + &42 cos 2t
As in Section 5.5.1, the solution of (5.5.38) for
4%
q = cos w(t W
E
(5.5.39)
= 0 is
+ g)
Hence & = (1/2)q2 cos 2t = (ca/w2) cos 2t cos2 w ( t equations (5.5.20)become
(5.5.40)
+ g)and thevariationaI (5.5.41)
Since
0 if
(I?)=
i
wisawayfrom 1
cos 2[(w  1)t + wp]
L:2
if w  1 = O(E)
(5 5 4 2 )
In the former case M and g are constants to first approximation. In the second case we inttoduce the new canonical variables a* and p* using the generating function so that
s* = cc*[(w  1)t + wj?]
(5.5.43) (5.5.44)
p * =  as*  (w  1)t a%*
+0p
(5.5.45)
Consequently, a* and ,8* are canonical variables with respect to the Hamiltonian K = ( ~ ) +  =as*  c o s€a* 2/3*+(wl)a* (5.5.46) at 4w Hence (5.5.47) (5.5.48)
5.5.
AVERAGING USING CANONICAL VARIABLES
185
Elimination of t from (5.5.47) and (5.5.48) gives c
 d(cos 2p*) 40
du* = a* ~
w
 1+
Hence In a* = ln
[
w
1
f_
40
cos 2p*
1
+ 420 cos 28* + a constant
(5.5.49)
Thus the motion is unstable (a* unbounded) if
That is, to first approximation o < l + f e
or w > l  $ f s
u) = 1 f ) c
or
The curves w2 = 1 f
&
(5.5.50) (5.5.51)
separate the stable from the unstable regions in the w  E plane. These curves are in agreement with those obtained in Section 3.1.2 by using the LindstedtPoincarC method and in Section 3.1.3 by using Whittaker's method.
5.5.3. A SWINGING SPRING Following Kane and Kahn (1968), we consider the nonlinear oscillations of a spring swinging in a vertical plane as shown in Figure 51. This problem
Figure 51
was introduced by Gorelik and Witt (1933) to illustrate internal resonances. The kinetic and potential energies of the mass m are
+ (I + v = tkz2 + mg(1 + z)(l  cos 0)
.T = t r n [ i 2
.)"2]
(5.5.52) (5.5.53)
186
VARIATION OF PARAMETERS AND AVERAGING
where x is the stretch in the spring beyond its equilibrium. Therefore L =T
V
= $rn[i2
+ ( 1 + x)'%']  mg(l + x)(1  cos 0)  $kx2 (5.5.54)
Since
i3L p,=,,
H =ips
 mi,
aL pe =  = rn(1
ae
+ x)%
(5.5.55)
+ ep,  L
For small x and 8 and x = O(e), H can be expanded into
3x2p; XPe2 2:rng164 + + 0 ( 0 5 ) (5.5.57)
rn 13 2m14 If we keep the quadratic terms in H, then a complete solution of the corresponding HamiltonJacobi equation can be obtained as follows. The HamiltonJacobi equation in this case is
(5.5.58) where S = S(x, 6 , t ) . T o solve this equation we let
(5.5.59)
hence
(5.5.60) (5.5.61) Therefore p z = as  = Jm(2u,
ax
 kx2)
(5.5.62) (5.5.63)
S
= (ul
+ a2)t+
s
dm(2a,
s
 kx') dx +
Jrn~(2u, mg18') $0
(5.5.64)
5.5.
AVERAGING USING CANONICAL VARIABLES
187
Consequently
p1   = 
m dx = m(2a1  kx')
2
+ &arcsin
B2=G=r+/ as
m12 de = t JmP(2a,  m g W )
Hence
x =
(5.5.65)
Z J ;
+ Jiarcsin
Odz
J"k sin Bl
e=J%
(5.5.66)
2%
(5.5.67)
sin B,
(5.5.68)
p , = J2ma1 cos B,
(5.5.69)
pe = 1J2mu, cos B,
(5.5.70)

where
,
I
To first approximation the variational equations correspond to
Ei
2
"P,2
= arngxe  
rn 13
.

azJo(l {sin B, lJ2k
+ % sin [(wl
+ 3 sin (Bl + 2Bz)
 2wdt
+ olB1 2w28,1)
Thus is fast varying unless w ,  2w2 = E , where the latter case the slowly varying part of A is
E
(5.5.71)
is a small quantity. In
To eliminate the explicit dependence of (fl)on t , we make a further canonical transformation from u1 and B1 to al* and PI* according to S*(al*, pl,
t) =
€al* I 2w,
+3 al*B1 2w,
(5.5.73)
188 VARIATION OF PARAMETERS AND AVERAGING Thus
(5.5.74)
Therefore K = a constant because aK/at = 0. The variational equations become al* =

aK  202cu,Ju1* cos y
(5.5.76)
881* & , =    a K  202cl%,Ju1* cos y
(5.5.77)
aB2
pl*

E = aK = 
aEl*
ZK
202
 ~ C U , E ~  "sin' y 
@z =  =  c J a , *
aE2
where
(5.5.78)
sin y
(5.5.79)
(5.5.80)
Equations similar to (5.5.76) through (5.5.80) were obtained by Mettler (1959) and Sethna (1965) using the method of averaging. Adding (5.5.76) and (5.5.77) and integrating, we obtain a,*
+ u2 = E = a constant
(5.5.8 1)
Hence the motion is completely bounded. Elimination of y from (5.5.75) and (5.5.77) gives
(2J
= C'E;(E
where
 Ut) 
p2i2a2)  K]'
= C 2 [ F 2 ( ~2 ) G'(E~)]
F = fa2JE  tcZ,
G =~
~
(5.5.82)
2
~
2
E K 2]
(5.5.83) )
The functions F(ctZ)and G(E,) are shown schematically in Figure 52. For
5.6.
VON ZEIPEL’S PROCEDURE
189
F. G
Figure 52
real motion, F2must be greater than or equal to G2.The points where G meets F correspond to the vanishing of both u, and al*.A curve such as GIwhich meets both branches of F or one branch of F at two different points corresponds to a periodic motion for the amplitudes and the phases, hence it corresponds to an aperiodic motion. The solution for the amplitudes and the phases can be written in terms of Jacobi elliptic functions. However, the points where G2 touches the branches of F represent periodic motions where the nonlinearity adjusts the frequencies o1and o2to produce perfect resonance. 5.6.
Von Zeipel’s Procedme
To determine a first approximation to Hamiltonian systems, the method of variation of parameters in conjunction with the method of averaging was used in Section 5.5. To determine higher approximations, von Zeipel (1916) devised a technique which is described and applied in this section to the first two examples discussed in the previous section. The essence of this technique is to expand the generating function S in powers of a small parameter E as
190 m
VARIATION OF PARAMETERS AND AVERAGING
2 ens,,,
and then determine S,, recursively by solving a chain of partial
n=O
differential equations. Let the system under consideration be described by the Hamiltonian m
WP, q, t ) = 2enHn(p, q, t ) , n=O
E
and
at
FF,"
(5.6.13)
Thus K,, contains longperiod terms only, while S, contains shortperiod terms only. The functions S, can be obtained by solving successively the chain of partial differential equations in (5.6.13). The basic idea underlying this technique is the same as that underlying the generalized method of averaging of Section 5.2.3. Stern (1971~)showed that, for Hamiltonian systems, Kruskal's technique is equivalent to von Zeipel's technique. In both techniques we introduce nearidentity transformations from the old dependent variables, which contain long as well as shortperiod terms, into new dependent variables which contain longperiod terms. The basic difference between the two techniques is that the transformation in the von Zeipel method must be canonical, while the transformation in the generalized method of averaging need not be canonical and the system need not be described by a Hamiltonian. Morrison (1966b) showed that up to second order the von Zeipel procedure is a particular case of the generalized method of averaging. Giacaglia (1964) carried
192
VARIATION OF PARAMETERS AND AVERAGING
out the expansion to any order, while Barrar (1970) investigated the convergence of the von Zeipel procedure. Musen (1965) showed that the equations governing the expansion take a concise form if written in terms of Faa de Bruno operators (1857). We next determine secondorder expansions for the first two problems discussed in the previous section. 5.6.1. THE DUFFING EQUATION As a first example, we consider the Duffing equation (5.5.23) which corresponds to the Hamiltonain (5.5.24). The solution of the problem corresponding to H, is given by (5.5.31) as found in Section 5.5.1. Hence
+ b)
(5.6.14)
B. =  a f i
(5.6.15)
2
A = 4E M sin‘
wo(r
0 0
aA ag’
&=
aM
To determine an approximate solution to (5.6.15), we introduce a nearidentity transformation from u and ,8 to u* and B* using the generating function (5.6.16) (5.6.17)
and the Hamiltonian
is transformed into
m
K = 2enKn(a*,@, t ) n=l
 ‘,sin4wO(r + b) 0 0
[ as,
a*
+ as + a@
as, +€+8+... at at Equating coefficients of equal powers of K1
=
Me2 0 0
[:
E
E
2
as
l2
+ . .. ~
(5.6.18)
on both sides, we have
as  4 cos 2w0(t + ,!I)+ Q cos 4w0(t + @)I + at‘
(5.6.19) (5.6.20)
Equating Kl to the longperiod term on the righthand side of (5.6.19), we obtain 3M*2 K, =(5.6.21) 8wO4
5.6.
VON ZEIPEL'S PROCEDURE
193
Hence (5.6.19) becomes
as,

at
a*2 +7 [4 oo
cos 2w0(t
+ ,8) +
cos 4w0(t
+ @)I = 0
(5.6.22)
+ p)]
(5.6.23)
The solution of (5.6.22) is
+
a*2
S1 = 5[sin 2w0(t ,8)  $ sin 4w,(t 40, With this value of S , , (5.6.20) becomes a*3 K , = * [cos2wo(t 0 0
+ @) /cos4wo(t + /?)I sin4 wo(t + 8) + asat
(5.6.24)
Equating K , to the longperiod term on the righthand side of this equation, we have 17a*3 (5.6.25) K z =  z Hence, to second order (5.6.26)
and u* 
aK
=0
or a* = aconstant
(5.6.27)
ag*
or (5.6.28)
where gois a constant. Having determined S , , we obtain a = a*
+ as, + . . . E 
= a* 4
p*
ag a
*'
E d2WO
[cos 2w0(t
=g
+ as, aa* + . . .
=
+
+ @)  & cos 4w0(t + #?)I + O(E') (5.6.29)
E 
a* E 5
200
[sin 2w0(t
+ p)  isin 4w0(t + p)] + O(e2)
(5.6.30)
194
VARIATION OF PARAMETERS AND AVERAGING
Solving (5.6.29) and (5.6.30) for u and @ in terms of a* and u = u*
+
€U*2
EU*
2w0
gives
[COS 2%(t
+ p*) 
[sin 2w0(l
+ p*)  & sin 4w0(t + b*)] + O(E')
2w0
B = p*  5
p*
COS 400(t
+ p*)] 4 O(E') (5.6.31) (5.6.32)
To compare this expansion with those obtained by other methods, we substitute for u and /?from (5.6.31) and (5.6.32) into (5.5.31), expand for small E , keeping a* and p* fixed, and obtain q =
J2a* (I 0 0

8.5)
sin(wt
a* + pol  J2a* sin 3(wt + Po) + O ( 2 ) 16w: E 
(5.6.33)
where w = wo(l
If we let
+ I*)
(5.6.34)
then J2u* 
a
&€
2
a3
+ o(€2)
(5.6.35)
w0
(00
Hence
+
where (5.6.37)
This expansion is in agreement with those obtained in Section 5.3 by using Struble's method and in Section 5.4.1 by using the KrylovBogoliubovMitropolski method. 5.6.2. THE MATHIEU EQUATION Next we determine a secondorder expansion for Mathieu's equation (5.5.35) which corresponds to the Hamiltonian (5.5.39). The solution corresponding to Ifo can be written as (see Section 5.5.1)
JZ
q = cos o ( t w
+ p),
p = J20(
sin w(t
+ p)
(5.6.38a)
5.6. Hence a and
Transforming from a and (5.6.16), we obtain =
rK,
195
are canonical variables with respect to
E7 =
K
VON ZEIPEL'S PROCEDURE
= w2
/? to
CO2
o(2
a* and
+ g) cos 2t g* using
(5.6.38b)
the generating function
+ e2K2 + . . .
(5.6.39) since a is given by (5.6.17). Equating coefficients of equal powers of E in (5.6.39) yields K,
a* +  {cos 2t + * cos 2[(w + 1)t + w g ] at 2w2 + B cos 2[(0  1)t + w p ] )
as
= l
(5.6.40) (5.6.41)
Two cases arise depending on whether w is near 1 (resonance) or w is away from 1. We consider both cases next starting with the latter case.
The Case of w Away from I . In this case all the terms on the righthand side of (5.6.40) are fast varying. Hence Kl = 0, and
1
S , =   sin 2t u* 4cu2
+ sin 2[(w + 1)t + w p ] + sin 2[(w2(w1)t1)+ w g ] 2(w
I
+ 1)
(5.6.42)
Substituting for S , into (5.6.41), we obtain K,
as,
a*
at
4w3
=    cos'
w(t
+ ,8)
+ cos 4Kw2(w++1)t1)+ wg1 + cos 4[(w2(w1)t1)+ Og] + w
cos 4w(t
+ g) + cos 4t + 2 cos 2w(t + p) w2 
1
1
(5.6.43)
196 VARIATION OF If
w
PARAMETERS AND AVERAGING
is also away from 2
K,= 1
a*
16 UP(^
 1)
(5.6.44)
because the rest of the terms on the righthand side of (5.6.43) are fast varying and must be set equal to as@ In this case (5.6.45) Hence
b * =  a K   1
16 ~
aa*
Consequently
/3*=
a* = a constant
(5.6.46)
 1) + o(4
(5.6.47)
€2 ~
(
j
o
~
1 €9 + /3Q 16 o ~ ( o 1) ~

(5.6.48)
and q is given by (6.2.95) if we make the following replacement of variables u + q , q,+w , a +o
+
w/30 and w + w[l

+J2u*/w
 ~ ~ / 1 6 w ~( wl ) ]~
(5.6.49)
Thus q is bounded and the motion in this case is stable. If w is near 2, however, cos2[(w  2)t wp] is slowly varying, and it has to be included in K,; otherwise S, will contain secular terms or a small divisor depending on whether w is exactly equal to 2 or not. Equating K2 to the longperiod terms on the righthand side of (5.6.43), we obtain
+
K2=
w cos 2[(w  2)t + wp] “ * ( + ] 16w3 1 w1 0’
(5.6.50)
To an error of O(E), /?can be replaced by /3* in (5.6.50). To analyze the motion in this case, we remove the explicit dependence of K on t by transusing ’ the generating function forming from a* and @*to a‘ and ,!I Thus
S’ = a‘[(w  2)t
+
0/3*]
(5.6.5 1)
(5.6.52)
as (w  2)t
P I = % 
+
0/3*
(5.6.53)
5.6. VON ZEIPEL’S PROCEDURE 197 and
K‘ = K
+ (w  2)a’ = (w  2)a’ 
E’
Hence a.,=   a K’
aP‘

€
2
“[* 160~
0 
1
+
‘“1
w2  1
a’ sin 28’ 8w2(w  1)
(5.6.54)
(5.6.55)
(5.6.56)
The solution of (5.6.55) and (5.6.56) can be obtained as in Section 5.5.2 to be €2
160(0’  1)
 16w2(w2  1) cos 28‘1
+ a constant
(5.6.57)
Therefore, for instability
2
E2
16w(w2  1)
16w2(w  1)
or
5t2 w2192
t2
192
+ o(4
(5.6.58)
Consequently, the transition curves that separate stable from unstable motions in the w2  E plane and emanating from w = 2 are given by w2 = 4
52 + 48  + O(t3)
and w2 = 4
E2  48 + o ( € ~ ) (5.6.59)
in agreement with those obtained in Section 3.1.2.
+
The Case of w Near I. In this case cos2[(w  1)t w p ] is slowly varying, and therefore it should remain in K , ; otherwise S, is singular at w = 1 as evident from (5.6.42). Equating Kl to the longperiod terms in (5.6.40), we have K , = cos 2 [ ( 0  1)t U*
4w2
Hence
+ wp]
a*     {COS 2t + 4 cos 2[(w + 1)t + w P ] } as,
at
2w2
(5.6.60)
(5.6.61)
198
VARIATION OF PARAMETERS AND AVERAGING
The solution of (5.6.61) is sin 22
0 + 1)t + w,B] + sin 2 [ (2(w + 1) )
(5.6.62)
Substituting for S1into (5.6.41) gives K z = as2 
U*
+ +
+
+ 8)cos 2t
cos 2[(w l ) t wP3 cos2 w(t 403(w 1) Equating K2 to the longperiod terms in (5.6.63), we have at
Kz =

Therefore, to second order K =  cos 2[(0 4w2 €a*
U*
32w3(o
+ 1)
 1)t + COB]
€'a* 32w3(o
Moreover
 a*

(5.6.64)
+ 1) + O(P)
+ 1) cos 2[(0 + 1)t + wp] + O(e2)
€U*
440
(5.6.63)
(5.6.65)
(5.6.66)
p* = #? + E as, + . . . aa*
=B
E
 ,(sin
+ sin 2I(w + 1)t + wB]) + O(@)
+
(5.6.67) 40 2(w 1) Solving (5.6.66) and (5.6.67) for u and B in terms of a* and ,8* gives
2t
Substituting for ,Iinto ? (5.6.65) gives K = cos 2[(w 4m2 €U*
 1)t + wP*] 
6%
*
32w3(w
+ 1)
5.6.
VON ZEIPEL’S PROCJZXJRE
199
The presence of the last term on the righthand side of (5.6.70) exhibits a shortcoming of the von Zeipel procedure in its present form, in which the fast and slowly varying terms in (5.6.40) were determined using mixed variables (new momenta but old coordinates). Had we expressed (5.6.39) in terms of the new variables a* and /3*, this last term would have been absorbed into S,, which might contribute to the slowly varying part of K3. In fact, such a representation in mixed variables was recognized by Breakwell (see Schechter, 1968) to lead to invalid results for the motion of a particle near the triangular points in the restricted problem of three bodies (Breakwell and Pringle, 1966). Using this suggestion, Schechter obtained a valid expansion by expressing the Hamiltonian in terms of the new variables before averaging to determine the slowly varying part (long period). Musen (1 965) developed algorithms for effecting the transformation, to any order, of variables and arbitrary functions from the old to the new variables and vice versa. Lacina (1969a, b) and Stern (1970a, 1971a) obtained expressions for general nearidentity canonical transformations from the old variables to the new variables. With these transformations, they modified the HamiltonJacobi equation. The resulting perturbation schemes may be related to other perturbation schemes which use canonical variables by the proper choice of certain expressions entering into these transformations. A technique of determining integrals of motion of a system governed by a Hamiltonian was developed by Whittaker (1916, 1937), Cherry (1927), Contopoulos (1963), McNamara and Whiteman (1967), and Coffey (1969). The technique is based on the fact that the equation
is satisfied by any integral of the canonical equations of motion
aH . Q=,p=
aP
aH aP
An effective and powerful technique of effecting the transformation of variables and arbitrary functions to new variables has been developed by Hori (1966, 1967) using the Lie series, and Deprit (1969) and Kame1 (1969, 1970) using the Lie transforms. This technique is described in Section 5.7. Had we expressed (5.6.39) in terms of the new variables, we would have found that EU*
K = cos Z[(W  l)t 4w2
+ o#?*] 3203(w + 1) + O(P) E2a*
(5.6.71)
We remove the explicit dependence of K on t by changing from u* and ,6*
200
VARIATION OF
PARAMETERS AND AVERAGING
to u1 and /?' by using the generating function
+ w/3*]
S' = d [ ( w  1)t
Hence
(5.6.72)
a s = wu' a* = 
(5.6.73)
a@*
as  /3'=z(w
and
K' = K
as' = COS +at 40 Eu~
28'

+ w/3*
1)t
e2a'
32w2(w
Therefore
+ 1) +
(5.6.74)
(W
 1 ) ~ ' (5.6.75) (5.6.76)
As in Section 5.5.2, the solution of (5.6.76) and (5.6.77) is
In u' = In[&

cos 28'
2 32w2(w
1
+ 1) + w  1 + a constant
Hence the transition curves are given by €2
Consequently
or
4w w = 1 f *€ w2 = 1 f 4.
32w2(w
+ 1)
+ O(2) # +

(5.6.78)
&€2
O(E3)
(5.6.79) (5.6.80)
These curves are in agreement with those obtained in Section 3.1.2 by using the LindstedtPoincark technique, and in Section 3.1.3 by using the Whittaker technique. 5.7. Averaging by Using the Lie Series and Transforms In analyzing the oscillations of a weakly nonlinear system, the method of variation of parameters is usually used to transform the equations governing these oscillations into the standard form * k = f(x; €) = 2
m=O
em ,f,(X)
m.
(5.7.1)
5.7.
AVERAGING BY USING LIE SERIES A N D TRANSFORMS
where f,(x) =
201
amf 1
a E m c=o
Here x and f a r e vectors with N components. The vector x may represent, for example, the amplitudes and the phases of the system, or the orbital parameters of the unperturbed twobody problem. If we denote the components of the vector f, by fmn, then a component xk of the vector x is said to be a rapidly rotating phase iff,, f 0. To analyze this standard system, we found it to be useful (see Section 5.2.3) to introduce a nearidentity transformation
x = X(y;
€)
=y
+ .X,(y) + cZX,(y) + .
* *
(5.7.2a)
from x to y such that the system (5.7.1) is transformed into En
(5.7.2b) n! where g, contains longperiod terms only. In Section 5.2.3, X, and g, were determined by substituting (5.7.2) into (5.7.1) and separating the short and the longperiod terms assuming that X, contains shortperiod terms only. Y = g ( y ; 4 = 2 gg,(y) n=o
5.7.1. THE LIE SERIES AND TRANSFORMS In this section we define the transformation (5.7.2a) as the solution of the N differential equations dx
 = W(x; E ) , de
X(E
= 0) = y
(5.7.3)
The vector W is called the generating vector. It seems at first that we are turning in circles because we are proposing to simplify the original system of differential equations by solving a system of N differential equations. This is not the case, because we are interested in the solution of (5.7.1) for large 1 , whereas we need the solution of (5.7.3) for small E ; which is a significant simplification. Equation (5.7.3) generates the socalled Lie transforms (Kamel, 1970), which are invertible because they are close to the identity. If W does not depend on C , (5.7.3) generates the socalled Lie series. For a canonical system, Hori (1966, 1967) and Deprit (1969) took (5.7.4a)
202
VARIATION OF PARAMETERS AND AVERAGING
with q the system’s coordinates,, p the conjugate momenta, and t the time, and defined
[9
w = s,
,
s = S(q,p, t ;
€)
(5.7.4b)
with S the generating function. Hori (1966) constructed a nonrecursive algorithm using the Lie series to m
determine the transform K = 2: (P/n!)K,(Q, P,t ) of a Hamiltonian m
n=o
H = 2 ( P / n ! ) H , ( q ,p, t). Deprit (1969) constructed another algorithm to ,=O
generate K recursively using the Lie transforms. Kamel (1969b), Campbell and Jefferys (1970), and Mersman (1970) showed the equivalence of Hori’s and Deprit’s theories, while Kamel (1969a) simplified Deprit’s algorithm. Hori (1970) showed that, to second order, Lie transforms are equivalent to von Zeipel’s technique. Shniad (1970) proved that the von Zeipel transformation is equivalent to the Deprit transformation, while Mersman (1971) established the equivalence of the Hori, Deprit and von Zeipel transformations. It should be mentioned that a perturbation theory based on the Lie series and transforms has several advantages over von Zeipel’s procedure. The generating function is not a mixed function of the old and new variables, the theory is canonically invariant, and it is possible to give a direct expansion of any function of the old variables in terms of the new variables. Kamel (1970) introduced the transformation (5.7.3) and constructed an algorithm to transform the standard system (5.7.1) into (5.7.2b). Moreover, he constructed algorithms (1) to transform any vector function from the old to the new variables and (2) to determine (5.7.2a) and its inverse. Henrard (1970) and Kamel (1971) investigated more deeply the mathematical and operational significance of these algorithms. We next construct these generalized algorithms and then specialize them to the canonical case in Section 5.7.5. 5.7.2. GENERALIZED ALGORITHMS Let the solution of (5.7.3) be x = X(y; e ) and let its inverse be y = Y(x; c) so that dx = X, dy and dy = Y,dx (5.7.5) where and
xY  ax. * a Y;
(Jacobian matrix)
5.7.
AVERAGING BY USING LIE SFBIES AND TRANSFORMS 203
From (5.7.5) we have
dx = XYYxdx
so that
XyYx = Z
Hence
(identity matrix) (inverse of X,)
Y, = (XY)l
(5.7.6)
The second relationship in (5.7.5) gives = Yxi
jl
which can be rewritten using (5.7.1) as
i = g(y; ). of
= Yxf
IXSXW)
(5.7.7)
We are interested in developing the righthand side of (5.7.7) in powers E so that (5.7.8)
From (5.7.7)
=
[az
(Yxf)
Now
dx
a
+
a
(YXO]xXw€)
(5.7.9)
0"
(5.7.10) (Yxf) = Yx + f ae af ax ae Since y = Y(x; E ) is the inverse of x = X(y; e) [the solution of dx/de =W(x; e) subject to x(c = 0) = y]
ae
ay + Yxdx Q =0=so that
aE
dc
ay
=
a€
de
=
ay + YXW ae
YxW
Hence (5.7.10) can be rewritten as
a
 (Yxf)
a€
= Yx
ar a    (Y,w)f a€ ax
(5.7.11)
Using this expression and noting that dXldE = W , we rewrite (5.7.9) as d
9 =  t(Yxf)x=xcY;€)l dc dc
204
VARIATION OF PARAMETERS AND AVERAGING
which can be simplified to (5.7.12) where (5.7.13)
By recurrence, (5.7.12) gives d"g = mxDnflx=x(Y;€)
Since X(E = 0) = y
(5.7.14)
den
Y&=, = I
(identity matrix)
so that (5.7.15) To determine D"f, we assume that W can be expanded as
(5.7.16)
so that the transformation (5.7.3) can be generated, successively, to any m
order. I f f = 2 (rn/n!)fn, (5.7.13) becomes n0
+
Letting n = k 1 in the first term, and n = k  m in the second term, we can rewrite this expression as
or
2
€k
Df=zk . +zk! OD
k=o
Yfk+l
m
k=O
k
%km
m=o z c m k [ z w m + l 
Df = k=O
where
2  ff) k!
ax (5.7.17) (5.7.18a) (5.7.18b)
cmk =
k! (k  m)! m !
5.7.
AVERAGING BY USING LIE SERIES AND TRANSFORMS
By recurrence, (5.7.17) and (5.7.18) with fk = f:)
205
give (5.7.19)
where k
f y = f&l’
+ 2 CmkLm+lf:,l’
(5.7.20)
m0
Hence (5.7.21) The recurrence relationship (5.7.20) can be best visualized using the Lie triangle introduced by Deprit (1969); it is somewhat reminiscent of the Pascal triangle and shown in Figure 53.
Figure 53. Lie triangle. (ftn) = &(”), f, = fn(0))
(5.7.22)
In carrying out perturbation solutions, we often need to express a vector 2
F(x; E) = 2  F,(x), n=O n !
F,(x) = !??!
acn
1
c=o
(5.7.23a)
206
VARIATION OF PARAMETERS AND AVERAGING
in terms of the new variable y in the form
= 2  F'n'(y), n=o n! a,
En
F'n'(y) = d"F dan
1
X=Y.
(5.7.23b) C=O
where (5.7.23~) Using (5.7.16) and (5.7.23a), we express this [as we expressed (5.7.17)] in the form (5.7.24) where
By recurrence (5.7.24) and (5.7.25) give (5.7.27) where m0
with FLo' = Fk. Hence
* En G(y; 6 ) = 2  Ftn), n=O n !
F(n) = Fc)[
(5.7.29)
XY
Equations (5.7.27) through (5.7.29) have the same form as (5.7.19) through (5.7.21) except for the different operator 9, so that (5.7.28) can also be visualized using the Lie triangle. 5.7.3. SIMPLIFIED GENERAL ALGORITHMS To simpiify an algorithm such as (5.7.20), Kame1 (1969, 1970) wrote it first as
5.7.
AVERAGING BY USING LIE SERIES AND TRANspoRMs
207
He then eliminated successively the functions on the righthand side to obtain eventually f r ) as a linear functional of ftn+,), f(n+*l),. . . ,f(n).Thus we assume that
fp)=
f(n+K)
 2k CjkGjf(n+ki)
(5.7.3 1)
j=l
where Gj is a linear operator which is a functional of 4,JC.+~, . . . ,L1. Substituting (5.7.31) into (5.7.30) yields the following recursion relationship j1
G, = Li For example
m=l 2 C~21LmGim,
15j
In
G, = L, G, = L,  L1L1 G , = L3  Ll(L.2  L1L1)  2L&1
(5.7.32)
(5.7.33)
For n = 0 and n = 1, (5.7.31) gives fCk)= f,
+ zcjvj,kj k
i=l
k
f p = f'k+k+l' IcjYj,kj+l
(5.7.34) (5.7.35)
j=1
where
51
This is the simplified algorithm of Kamel in which fj., = 0 if fti) = 0 because the second index i, in the recursion relationship, is fixed. A more convenient form of this algorithm was obtained by Kamel by writing (5.7.35) as (5.7 * 37) However, from (5.7.18a)
j=l
so that (5.7.37) can be rewritten as
Since dx/dc = W from (5.7.3) (5.7.39)
208
VARIATION OF PARAMETERS AND AVERAGING
where
I
x(n+l)(y) = 
dnW den
Hence from (5.7.16)and (5.7.34)
for n 2
E=O.X=Y
k1
= wk k zC;'Xj,p+j,
X"'
k 2
(5.7.40)
j=1
where = gjX(i'
XjSi
 z.C;21=gmXjm.a j1
(5.7.41)
m1
To determine the inverse transform y =x
* +z  y'"'(x) n! En
(5.7.42)
n=l
we eliminate x  y from (5.7.39)and (5.7.42)to obtain
However
2  y'"'(x) O0
En
n=l
n!
u=
u =2
n=l
=
En 2 x'"'(y) O0
n=l
n!
En co E n un(x) = 2 r u'"'(y)
n!
n1
n.
(5.7.43a)
(5.7.43b)
(5.7.43c)
(5.7.44) where
is defined by (5.7.41).
5.7.4. PROCEDURE OUTLINE Consider a system of differential equations written in the standard form n=o
n.
(5.7.45)
The essence of the algorithms of the previous section is to introduce a transformation from x to y so that (5.7.45)becomes
i = 2: 7 gn(Y) n. O0
En
(5.7.46)
n=O
where g, does not contain shortperiod terms. To do this we construct a
5.7.
AVERAGING BY USING LIE SERIES A N D TRANSFORMS
209
nearidentity transformation
x =y
+2 x'"'(y) n! En
n=l
(5.7.47)
Under such a transformation a vector F(x; E) = 2  Fn(x) n=O n ! En
becomes
2
F(x; E ) = 2  Ftn'(y) n=o n!
(5.7.48)
(5.7.49)
The algorithms described in the previous section to generate these transformations can be implemented on a computer because they can be effected by the recursive application of elementary operations. We describe below the procedure to second order. It is initiated by putting (5.7.50)
Then we begin the firstorder expansion by writing the linear partial differential equation (5.7.51) gl(Y) = fdY) + LlfO We choose g, to equal the longperiod terms in f,, solve the resulting equation for W1, and compute x") = w,
Fl,o= 6P1F'" F'" = F1
+ Fl,o
(5.7.52)
To prepare for the secondorder expansion, we compute g1.1 = L1g1 Then we set up the differential equation
+
+
+
(5.7.53) g2 = f, Llfl g1.1 b f o and choose g, to be equal to the longperiod part of the righthand side. This completes the expansion to second order. We illustrate this procedure by applying it to van der Pol's equation q
whose solution for
E
=0
+ q = E(1  q")4
(5.7.54)
can be written as
q=acos+,
+ = t + p
(5.7.5 5)
210
VARIATION OF PARAMETERS AND AVERAGING
By the method of variation of parameters, (5.7.54) can be replaced by (see Section 5.2.3) ci = *€[U(l  tu2)  U C , ta3C,] (5.7.56) where
+ = 1 + &[(I
C , = cos n+
+
(5.7.57)
 &zz)S,  ta2S4]
and S, = sin n+
LI
Equations (5.7.56) and (5.7.57) have the same form as (5.7.1) with
x= fo=
[
&z(l  @)
f1=
C]
(5.7.58)
(5.7.59)
 *uc,
+
+(1  ifu2)Sz &l2S4
We transform now from x = we obtain
>1
[i] [I].
f, = 0 for n
to y =
go=
1
Qu3C4
(5.7.60) (5.7.61)
From (5.7.50) and (5.7.59)
c3
(5.7.62)
From (5.7.18b) and (5.7.59), we find that awn
L,fo =  
(5.7.63)
awl g, = fl  
(5.7.64)
ad*
Hence (5.7.51) becomes
a+*
Choosing W, to remove the shortperiod terms in fi gives &2*(1
g,=[
Solving the resultant equation gives
w1=
[
)u*St*
 &*a) 0
+
t(l  *U*2)C2*
where S,* = sin n+* and C,* = cos n+*.
1
&U*3S4*
+
(5.7.65)
I
&2*2c4*
(5.7.66)
AVERAGING BY USING LIE SERIES A N D TRANSFORMS
5.7.
21 1
With (5.7.61) and (5.7.63), (5.7.53) becomes (5.7.67)
Choosing Wz to remove the shortperiod terms, we obtain gz = (Llfl)
+ (L1g1)
(5.7.68)
Since g, consists of longperiod terms only, and W1consists of shortperiod terms only, (Llgl) = 0. Hence (5.7.69)
Therefore
in agreement with the expansion obtained in Section 5.2.3 using the generalized method of averaging. To compare the expansion obtained in this section with that obtained in Section 5.4.2 by using the KrylovBogoliubovMitropolski technique, we need to express (5.7.55) in terms of the new variables. Here En
27 n=O n.
4 =
with From (5.7.50)
Fo = acos(6,
(5.7.71)
Fn = 0 for n 2 1
P o )= a* cos $*
(5.7.72) (5.7.73)
Then (5.7.52) and (5.7.26) give
Hence
Fl,o= 2Z1(a* cos $*) = [cos +*, a* = $a*(l  $ u * ~ sin ) #* 
q = a* cos d*  $ea*[(l
sin +*]W, sin 3+*
 &a*z)sin $* + +a*2sin 34*] + O(e2)
(5.7.74)
which can be rewritten as q = a* cos p
where dy=lrz dt
*3  ra sin 3 y
32
(I8   8 + ?*) 256 a*2
+ O(2) + O(P)
(5.7.75a)
(5.7.75b)
212 VARIATION OF
PARAMETERS AND AVERAGING
in agreement with the expansion obtained in Section 5.4.2 using the KrylovBogoliubovMitropolski technique. 5.7.5. ALGORITHMS FOR CANONICAL SYSTEMS Hori (1966, 1967) and Deprit (1969) used the Lie series and transforms, respectively, to transform a Hamiltonian
into a new Hamiltonian K(P, Q, t ; c) = 2  U P , Q, 2) n=o n! En
If we let
(5.7.77)
(5.7.78)
X =
(5.7.79) then g can be generated from a Hamiltonian K as g=
[
Kq
(5.7.80)
In this case the algorithm of Section 5.7.3 reduces to the scalar form (Kame), 1969a) Ko = Ho(Q, P, 2 )
where
af as, af as, L;f = aQ ap apaa
(5.7.82a) (5.7.82 b)
and
5.7.
AVERAGING BY USING LIE SERIES AND TRANSFORMS
213
Under the above transformation the old variables are given in terms of the new variables as (5.7.83) where (5.7.84a) and (5.7.84b)
31
p3.. 1. = L'.3"'
 Z: C21LAPjm,i m=l
To third order, the above algorithms become
K O = Ho
(5.7.85) (5.7.86)
K2 = H, K3 = H3
+ L;H1 + L;Ki
9s2
+ L;H2 + 2LLH1 + 2LiK2 + L;Kl
(5.7.87)
9 t
 Liz&
9s3 (5.7.88)
9 t
214 VARIATION OF PARAMETERS AND AVERAGING We next illustrate this procedure by its application to the swinging spring represented by the Hamiltonian (5.5.57). Using the solution (5.5.67) through (5.5.70), we transform this Hamiltonian into H = H,
where
+ 3H2+
(5.7.91)
*..
I
H, = %J3 [sin2B,  2 cos2B,] sin B1 l k
(5.7.92)
1 a: 12a,a, . 2 H , = s i n 4 ~ , sin B, cos2B, 3 mgl k12
(5.7.93)
+
+
and Bi = w i ( t pi). We transform a, p, and H into a*, @*,and K = KO Kl &K2 . . . using the algorithm defined by (5.7.85) through (5.7.87). Since H , = 0, KO = 0 by (5.7.85), and 9 S n / 9 r = as,/& by (5.7.82b). With (5.7.92), (5.7.86) becomes
+ +

K 1  
",/$ 1
+
(sin B1*
+ + sin (Bl* + 2B2*)
sin [(ol 20,)t
+ wlpl*  2w2p2*])
3
+
(5.7.94)
All the terms in K, have short periods unless w1 w 20,. In the latter case sin [(ol 2022 olpl*  20,/?,*] has a long period (slowly varying). Choosing S, to eliminate the shortperiod terms results in
+
*
+
1
3 cos(Bl* 2B2*) (5.7.96) awl 202) If we choose Szto eliminate the shortperiod terms in (5.7.87), we obtain
+
+
K , = (H,) (GHd The averaged values in K , are given by
+ (&Kl)
(5.7.97)
(5.7.98)
*=
3al*a2* k12
a2 +w(H,)=  
8mgl
a$ 2k12
30(1*Ue*
+
k2"
(5.7.99)
AVERAGING BY USING LIE SERIES AND TRANSFORMS
5.7. (LiH,) =
215
&H asl\ + /aH, as,\ ''\ag, a d \Z ZJ 
= [I+
4(W1
+ 20.4
]
*=  9a1*a2*
a2
4kZ2
o2
+ 2w2
4k12 o1
(5.7.100)
In (5.7.99) and (5.7.100) use has been made of the fact that o1M 2w2 (i.e., kl M 4mg from the definitions of o1and 02). Hence K2 
and
33af 32k12
+39al*a2* 16k12
(5.7.101)
K = Ko+K1+hK2+..* I
2
39a1*a2* ++ ... 64k12 32kl'
33a: 
(5.7.102)
To remove the explicit dependence of K on t , we transform from a * and u' and B' using the generating function
#?*to
S' = a;
[(w, o12w2)t+ A*] + adSz*
(5.7.103)
so that (5.7.104) (5.7.105) (5.7.106) (5.7.107)
and
asi
K'=K+=at
3a;Ja;  sin (wl/li  2w2,9i) 21J2k a 33a; 3944
+64k12, + 32kl

o1 202 , 0 1
a1 (5.7.108)
216
VARIATION OF PARAMETERS AND AVERAGING
(5.7.109) U'2 

1J2k
sin y
pz=.I
where
3JU;  sin y 21,/2k
cos y
394 ++ 32k1'
w1
334 394 + , 32k12
y = 0lt%  2w,B;,
(5.7.110)
32kl
2w,
0 1
(5.7.11 1) (5.7.112) (5.7.113)
5.8. Averaging by Using Lagrangians
Instead of using canonical variables, which require the use of Hamiltonians, Sturrock (1958, 1962) developed a technique that does not require canonical variables. It consists of averaging the Lagrangian and then writing down the corresponding EulerLagrange equations. Whitham (1965a, 1967a, b, 1970) developed a similar technique for waves in which the frequency and wave numberas well as the amplitude are slowly varying functions of space and time. Bisshop (1 969) supplied a more rigorous justification of this technique. Although this technique is not as elegant as those using canonical variables, it has the advantage of being directly applicable to partial as well as ordinary differential equations. Kawakami (1970) and Kawakami and Yagishita (1971) used canonical variables in conjunction with Hamiltonians to solve the nonlinear Vlasov equation. This technique was applied to a variety of problems of wave propagation in fluids and plasmas. Lighthill (1965, 1967) applied Whitham's theory to moderate waves in deep water where pseudofrequencies are absent, while Karpman and Krushkal' (1 969) used Whitham's theory to study the decay of a plane wave into separate wave packets. Howe (1967) studied openchannel steady flow past a solid surface of finite wave group shape. Bretherton (1968) treated linear wave propagation in slowly varying wave guides, while Bretherton and Garrett (1968) investigated slowly varying waves in inhomogeneous media. Garrett (1968), Drazin (1969), and Rarity (1969) analyzed nonlinear internal gravity waves; the effects of shear and slight atmospheric stratification were determined by Garrett and Drazin, respectively. Simmons (1969) studied the interaction of capillary and gravity waves, while Grimshaw
5.8.
AVERAGING BY USING LAGRANGIANS
217
(1970) discussed solitary waves in water of variable depth. Crapper (1970) investigated the generation of capillary waves by gravity waves. Dougherty (1970), Galloway and Crawford (1970), and Galloway and Kim (1971) treated nonlinear waves in plasmas. Dewar (1970) investigated the interaction between hydromagnetic waves and an inhomogeneous medium, while Tang and Sivasubramanian (1 971) studied the nonlinear instability of modulated waves in a magnetoplasma. Lowell (1970) analyzed wave propagation in lattices with an anharmonic potential. We describe this technique and its application to three examples. 5.8.1. A MODEL FOR DISPERSIVE WAVES As a first example, we analyze slowly varying wave train solutions for Bretherton’s (1 964) model equation
ht
+
4222,
+ 422+ 4 =
€43
(5.8.1)
with the nonlinear term e d 3 rather than €4” If we neglect the nonlinear term ed3, (5.8.1) admits the uniform traveling wave solution
+ = a cos 0,
where
(I)
0 = k x  cot
(5.8.2)
and k satisfy the dispersion relationship 0)’
= k4  k2
+1
(5.8.3)
To determine a slowly varying wave train solution by using the variational approach, we first write the Lagrangian corresponding to (5.8.1); that is L =
b4t2  44:2
+ 4402  BV +
(5.8.4)
It can be easily verified that (5.8.1) is the EulerLagrange equation corresponding to this Lagrangian. We assume an expansion of the form
4 = a c o s e + E p A , c o s n+e cqr2) 4,
n=2
where
k = 0=,
=
et
(5.8.5) (5.8.6)
and a , o,k , and A i are slowly varying functions of x and t . If 0 is twice continuously differentiable, (5.8.6) gives the compatability relationship kt
+
W,
=0
(5.8.7)
Since secular terms appear first at O(€) in the straightforward expansion,
218
VARIATION OF PARAMETERS AND AVERAGING
we assume that a,, a,, w,, w , , k,, and k , are O(E).Thus
4,
+ a, cos 8 + cw 2 nAn sin n8 + O ( 2 ) sin 8 + a, cos 8  E k 2 nAn sin n8 + O(e2) 03
= aw sin 8
+, = ak
n=2
m
n=2
$ ,, = ak2 cos 8  2a,k sin 8
..
 c k 2 z n2Ancos n8 + 0 ( e 2 ) m
n=2
Substituting these expressions into (5.8.4), we find that the resulting Lagrangian depends implicitly on z and r through 8 , a, o,k , and A j . Its variation with z and t through 8 is much faster than that through the other parameters, because when 6 varies over [0,211], the period of the 8 terms in L, the other parameters hardly change. As in the other versions of the method of averaging, we average L over 8 from 0 to 2rr, keeping a, o,k, and A t constant. To do this we first average each term in L separately. Thus
$ ,: = 4a2k4

Hence
$4
= $a4
+ O(c2)
+ O(c)

9 = L = f ( 0 2 k4 + k2  l)a2 + 2 a4 + O(2)
(5.8.8) 32 The averaged Lagrangian 2 is an explicit function of a and an implicit function of 8 through w and k. Now we write the EulerLagrange equations with respect to the variables a and 8. The EulerLagrange equation with respect to a is a9’EpIaa = 0; it gives the dispersion relationship w2 = k4  k2
+ 1 + $€a2+ O(r2)
(5.8.9)
Note that this dispersion relationship can be obtained by using the principle of harmonic balance; that is, we let = a cos 8 in (5.8.1) and equate the coefficients of cos 8 on both sides. Since w = 8, and k = 8,, the EulerLagrange equation with respect to 8 is
+
or
”(”)ae, at
+
d(?z)= i&E= 0 ax ae, ae
(5.8.10) (5.8.11)
5.8.
AVERAGING BY USING LAGRANGIANS
219
However,
Hence
aw
= $ma2 + O(E'),
a
 (OM') at
ak
= $(2k3
a [(2k3  k)a'] +ax
 k)a' + O(e2) =0
(5.8.12)
To simplify (5.8.12), we differentiate (5.8.9) with respect to k to obtain WW'
= 2k3  k
+ O(e2)
where a)' = dwjdk is the group velocity. Hence we rewrite (5.8.12) as
or
(5.8.13) Since w = ~ ( k ) , = o'kt, and the second term in (5.8.13) vanishes according to (5.8.7). Hence (5.8.13) simplifies into (5.8.14)
Moreover, since o = w(k), (5.8.7) can be rewritten as
ak + 0' ak =0 at
ax
(5.8.15)
Therefore the spatial and temporal variations of the amplitude a, frequency a), and wave number k are given by (5.8.9), (5.8.14), and (5.8.15). 5.8.2. A MODEL FOR WAVEWAVE INTERACTION Had we carried out the expansion of the previous section to second order, we would have obtained (b = a cos 0
€a3 cos 30 + 0 ( e 2 ) + 32(9k4  1)
(5.8.16)
Although valid for a wide range of values Df k , this expansion breaks down near k2 = 1/3. This case is referred to as the thirdharmonic resonance case in which both cos 0 and cos 38 satisfy the same dispersion relationship; that is, both the fundamental and its third harmonic have the same phase speed wlk.
220
VARIATION OF PARAMETERS AND AVERAGING
To determine an expansion valid near k2 = 113 for 5.8.1), we assume the expansion to be
4 = a, COS 6, where
+ a3 cos + 2 A,
8, = k,x  o,t
with
a,
63
E
n+1.3
+ pn,
cos (6, f vn)
on2= kn4  kn2
+ O(E’)
(5.8.17)
+1
(5.8.18)
k , w 3 k , and w 3 m 3 0 1
(5.8.1 9)
Note that the firstorder term contains the fundamental cos 8 , and its third harmonic 8,. Since we are interested in the case k12 M 113, we take w i and ki to be constants, whereas we take p i, v i , ai, and A j to be slowly varying functions of x and t. We next substitute this expansion into (5.8.4) and average the resulting Lagrangian over ei, keeping pi, vi, ui, and Ad constants. In performing the averaging we note that, although Oi are fast varying, 6 3  38, is slowly varying. Thus
#zz = 1 2 (k:


2 k1.3
+ 4ki3piZ)a: + O(E’)
With these expressions 9becomes
where 6
= 63
 38,
34 + 32  (a: + 4al2a; + a:) + u13u3cos 6 8
(5.8.20)
+ p3  38,
(5.8.21)
= (k,
 3kl)x  (w3  3w,)t
In arriving at (5.8.20) we used the dispersion relationship (5.8.18) and the definition of the group velocity o:= (2ki3  ki)/wj. With wi and ki constants, us and pi are governed by the EulerLagrange equations
a =9o aai
(5.8.22)
5.8.
AVERAGING BY USING LAGRANGIANS
221
and (5.8.23) Substituting for 2 from (5.8.20) into these equations and using (5.8.21), we obtain
B3t
+
~ 4 8 3 ,=
a01

at
8w3
[6ai2
+ 3a3' + a13ag1cos 61
3€ + w118%  =  a,a, sin 6 9
ax
801
aa3 + w3, aa, = E 3 a , sin 6
at
8w3
ax
(5.8.25) (5.8.26) (5.8.27)
These equations are in agreement with those obtained in Section 6.2.9 by using the method of multiple scales. 5.8.3. THE NONLINEAR KLEINGORDON EQUATION As a last example, we consider after Whitham (1965a) the nonlinear KleinGordon equation (5.8.28) utt  u,, V'(u) = 0 where V ( u ) is any nonlinear potential function which yields oscillatory solutions. Scott (1970) treated the special case V(u) = cos u , which represents the propagation of a magnetic flux on a long Josephson tunnel junction. The Lagrangian corresponding to this equation is
+
L = $Ut2  $11;
 V(u)
(5.8.29)
For a uniform wave train solution u ( @ , (5.8.28) becomes where
(0'

+ V ' ( U )= 0
k2)U,,
and
w = Ot
k = 0,
(5.8.30) (5.8.31)
A first integral of (5.8.30) is
4(0'  k 2 ) U ;
+ V ( U )= E
(5.8.32)
Integrating this equation gives w2 
k2
du
(5.8.33)
222 VARIATION OF PARAMETERS AND AVERAGING We assume that u is periodic with a period that can be normalized to unity so that
(5.8.34) To determine approximate equations for slowly varying E, o,and k, we substitute u(O) for u in (5.8.29) and average the resulting Lagrangian over the interval [0, 11. Thus L = $(w2  k2)U;  V(U)= (0’  k2)u;  E
as a result of (5.8.32), hence L[u(B)]dO = (w2  k2)J
1
1482
d8 
(5.8.35)
EP (5.8.36)
The variation of 9 with respect to E gives the dispersion relation (5.8.34). Since w = Ot and k = O,, the EulerLagrange equation corresponding to the variable 8 is
(5.8.37) Substituting for 2’ into (5.8.37), we obtain a( w W ) at
where
+a (kW) = 0 ax
(5.8.38) (5.8.39)
The problem formulation is completed by augmenting (5.8.34) and (5.8.38) by the cornpatability relation ak  +  =aoo at
ax
(5.8.40)
EXERCISES
223
Exercises 5.1. Use the StuartWatsonEckhaus solutions to (a) ii
technique to determine approximate
+ Iu = €12
u(0) = u(n) = 0 (b)
~
t
+
t ~~,, u =E U
u ( x , 0) = a cos 2 ,
~
Ut(Z,
0) = 0
5.2. Use Struble’s technique to determine uniform secondorder expansions for (Struble, 1962)
+ u = r ( l  u2)li (b) ii + ( h + cos 2r)u = 0 (a) ii
E
5.3. Use the KrylovBogoliubov technique to determine approximate solutions for (a) ii + W o % =  E l i (li( (b) ii
(c) ii
+ wo2u = r ( l  u2)u + cu3 + (8 + c cos 2t)u = 0
5.4. Consider Mathieu’s equation
ii
+ (6 + rcos2r)u = 0
Determine uniform secondorder expansions using (a) the KrylovBogoliubovMitropolski technique, (b) the generalized method of averaging, and (c) the Lie transforms. 5.5. Consider the equation
+ fao2q =
tq3
+ K cOS
or
(a) Show that it corresponds to the Hamiltonian
H
=
&(pz + wo2y2)  t c @  K y cos wf
(b:) Determine a firstorder expansion when K = O(1) and w is away from 3 0 0 , 00, (c) K = 0 ( 1 ) and w is near 3w0. (d) K = 0(1) and w is near 4 3 . (e) K = O ( E )and w is near w0. 5.6. Consider the equation
7 + w02q = ~ ( 1 q2)q + K cos wf
Use the KrylovBogoliubov method to determine firstorder expansions for the cases enumerated in Exercise 5.5.
224 VARIATION
OF PARAMETERS AND AVERAGING
5.7. Use the generalized method of averaging, the KrylovBogoliubovMitre polski method, and Kamel's method to determine secondorder expansions for ii
+ u = E(l
 u2)U
+
€113
Compare the results of the three methods. Which of these techniques would you recommend for such problems? 5.8. Consider the problem
+ 2pli + v2u =  € f ( u , li)
ii
AS E 0
u = uoept cos (mot
(a) For
by letting u
c
+ (bo),
wo =
.\/v2
 pz
# 0 determine a uniform expansion following Mendelson (1970)
= u(a, v),
where
u=u,+ru,+". du    p u eA,(a) dt
+
+ ...
9 = wo + '&(a) + . . . dt (b) Determine an expansion using the KrylovBogoliubovMitropolski technique. (c) Which of these expansions is more accurate? 5.9. Consider the problem 12
+ 6~ + Eblurn + EbzUnl
 cb.&' cos At
=0
where 6, E , bi, and L are constants and m is an odd while n is an even integer with m > n. (a) For small E find a solution of the form u = a(t)cos 0 ,
0 = wt
 C(t),
o =
A/n
and use the method of averaging to determine equations for a and (b (Tso and Caughey, 1965). (b) Determine a Hamiltonian corresponding to the above equation and then determine a firstorder expansion using canonical variables for 6 near 02. (c) Compare the results of the two techniques. 5.10. The problem of a spherical pendulum (i.e., a particle moving under the action of gravity on the surface of a smooth fixed sphere) is represented by the Hamiltonian (Johansen and Kane, 1969)
where qi andpi are the coordinates and momenta of the particle, m is its mass, g is the gravitational acceleration, and I is the radius of the sphere.
EXERCISES
22 5
(a) Determine a firstorder solution for small amplitudes using the method of averaging with canonical variables. (b) Determine a secondorder expansion using Lie transforms. (c) Determine a secondorder expansion using the generalized method of averaging. (d) Compare the three resulting expansions. 5.11. Consider the problem of a swinging spring with damping j.
+ f+ + ik
3'
m
+g(i
 cos e)  ( I
+ +I2 = o
2 xe = o e + s,e + g sine + I+x l + x :
If w12 = k / m and w2' = g / l , determine secondorder uniform expansions using the generalized method of averaging, and the Lie transforms for the cases (a) w1 w 20, and (b) (0, * 3w,. 5.12. Consider (3.1.63) through (3.1.65). (a) Show that the corresponding Hamiltonian is H =
+ p,') + 7 2 ~ 1 7 1 ~ + 2 &Q12 + 72')
 +(h1722
+ hzy12)(1 + e cosf)'
(b) Use the method of averaging with canonical variables to determine a firstorder expansion near the transition curves starting at (p,,, e) where p,,= (1  2\'2/3)/2. (c) Use the Lie transforms to determine a secondorder expansion near these transition curves.
5.13. Consider the motion of a particle described by the Hamiltonian
+
+
+
(6 + &)7? 413 + 29i7z2 H = +(pi2 + pz2) 4(42p1  qlpz) + where 6 is a constant parameter. (a) Show that the circular frequencies of the linearized problem are 1 and 2 when 6 = 1 , (b) use the method of averaging with canonical variables to determine a firstorder expansion for small amplitudes when 6 * 1 , (c) determine a firstorder expansion using the method of averaging, and (d) which of these techniques would you recommend for such problems? 5.14. Consider the problem (Sethna, 1965) i. !J
+ = e(3b12' + 2bZzy + b&)  €612 + K ~ C OAtS + w 2 y = e(b,x2 + 26,xy + 3b4y2)  eS,$ + K , sin I t W
~
~
T
If internal resonance means w1 w 202 or w, = 2w1, resonant excitation means I, m w1 or w 2 , soft excitation is denoted by K i = EkE with ki = O (l), and hard excitation is denoted by K i = 0(1), use the method of averaging to determine firstorder expansions for the following cases:
(a) Hard nonresonant excitation in the absence of internal resonance. (b) hard nonresonant excitation in the presence of internal resonance. (c) soft resonant excitation in the absence of internal resonance. (d) soft resonant excitation in the presence of internal resonance.
226
VARIATION OF PARAMETERS AND AVERAGING
5.15. Traveling waves in a cold plasma are governed by
a 2 +(pu) = 0 at ax au
au
at
ax
+u+E=O
+
Let p = 1 O ( E ) ,u = O ( E ) ,and E = O ( E ) . Use the method of averaging to determine the temporal as well as the spatial variation of the amplitude and phase of a monochromatic traveling wave. 5.16. Consider the problem $ t t  4zx 4 4 = 0 (a) Show that e = kx  wt 4 = a cos e,
+
is a solution of this equation if w2 = k2 1. (b) Show that the above equation can be written in the conservation forms (Whitham, 1965b)
a
a
at M4," + 4x2 + 491 + ax (  + 2 4 t )
=0
(c) Let 4 = a cos 8 with a = a ( x , t ) , k = ,e, and w = Bt in the conservation equations. Assume that a, w , and k are slowly varying functions of x and t. Hence average these equations over @ = 0 to 2 ~ keeping , a, w , and k constants, and obtain
ak + o r ak at
aa2
5.17. Consider the equation
at
Utt
=0
ax
a +(Wh2) ax
 c2ux,
=0
+ u = €U3
(a) Write down the Lagrangian corresponding to this equation, (b) determine a firstorder expansion for traveling waves with constant wave number and frequency but both spatially and temporally varying amplitude and phase. 5.18. The problem of nonlinear transverse oscillations in a homogeneous freefree beam with a nonlinear momentcurvature relationship is described by the Lagrangian
EXERCISES
227
where p , ,8, and t are constants. Determine a firstorder expansion for traveling waves with slowly varying amplitudes and phases using (a) the variational approach, and (b) writing down the governing equation and then using the method of averaging. 5.19. Consider Bretherton's (1964) model equation 4tt
+ 4,xx + h x +
+
=
P
(a) Show that the linear problem has the dispersion relationship W'
= k 4  k2 + 1
(b) Determine the wave number corresponding to nthharmonic resonance. (c) Use the method of averaging to determine a firstorder expansion near
the secondharmonic resonance condition (let the amplitudes and phases be functions of z and t). (d) Write down the corresponding Lagrangian, and then use the variational approach to determine a firstorder expansion near the secondharmonic resonance condition.
Perturbation Methods ALI HASAN NAYFEH Copyright Q 2004 WILEYVCH Valag GmbH & Co. KGaA
CHAPTER 6
The Method of Multiple Scales
6.1. Description of the Method There are three variants of the method of multiple scales. We describe them by discussing the linear damped oscillator x+x=
2d
(6.1.1)
We chose this example because its exact solution is available for comparison with the approximate solution obtained, and because we will be able to display the different variants of the method more clearly without involving ourselves in algebra. To start, let us determine a straightforward asymptotic expansion for small 4. Thus we assume that x = xo + EX1
+ GX, + . . .
(6.1.2)
Substituting (6.1.2) into (6.1 .I) and equating coefficients of equal powers of E to zero lead to (6.1.3) xo xo = 0
+ XI + x1 = 2x o xz + 2.2 = 2x1
The general solution of (6.1.3) is 20
= a cos ( t
+ 4)
(6.1.4) (6.1.5)
(6.1.6)
where a and (b are arbitrary constants. Substituting for xo into (6.1.4) and solving the resulting equation, we obtain 21
= at cos ( t
+ 4)
(6.1.7)
Substituting for x1 in (6.1.5) and solving for x,, we obtain xZ = +at2cos ( t
+ 4) + Qat sin ( t + 9) 228
(6.1.8)
6.1.
DESCRIPTION OF THE METHOD
229
Therefore 2
= a cos ( 2
+ 4) Ear cos ( r + 4) + +r2a[t2cos (t + 4) + t sin (t + I#)] + O ( 2 )
(6.1.9)
It is obvious that (6.1.9) is a poor approximation to 5 when t is as large as rl, because then the second (a1) and the third (c2z2) terms are not small compared to G, and rzl, respectively (xl and z2 contain secular terms), as was assumed when we carried out the above expansion. Thus the straightforward expansion is not valid when t increases to O(cl), and the source of the difficulty is the infinite domain as discussed in Section 2.1. The failure of the above straightfonvard expansion can be seen by investigating the exact solution of (6.1.1), which is given by 2
= ae"cos
141 e2 t + 41
(6.1.10)
Equation (6.1.9) can be obtained by expanding (6.1.10) for small E with t kept fixed. Thus the exponent and cosine factors are represented by
+ gc2t2 + . . c o s ( J 1  r 2 t + 4)= cos(t + 4) + +2tsin exp (d) = 1  r t
(6.1.11) (t
+ 4) i...
(6.1 .12)
It is clear that exp (d) can be approximated by a finite number of terms only if the combination d is small. Since E is small, this means that t = O(1). When t is as large as rl, et is not small and the truncated expansion breaks down. The above truncated series is satisfactory up to a certain value of t after which exp ( rf) and the truncated series differ from each other by a quantity that exceeds the prescribed limit of accuracy. Adding more terms to the truncated series increases the value of r to a new value 1' for which this truncated series is satisfactory. However, for t > t', the difference between exp ( r t ) and the new truncated series again exceeds the prescribed limit of accuracy. All terms of the series are needed to give a satisfactory expansion for exp ( d) for all 1. Thus to determine an expansion valid for times as large as cl, the combination rt should be considered a single variable TI = O(1). Then any truncated expansion for exp (d) valid for times as large as rl is of the form exp ( e t ) = exp ( T I )
(6.1.13)
Similarly, the truncated expansion (6.1.12) is not satisfactory when t is as large as O(E~). To obtain a truncated asymptotic expansion for cos [Jl  c2 t 41 valid for t = O ( c 2 ) , r2t should be considered a single
+
230
THE METHOD OF MULTIPLE SCALES
variable T2 = O(1). With this condition
cos [JI  e2 t
+ 41 = cos [t  iT2 + 4  $e4t + ‘1 = cos (t  i T z + 4) + &*t sin (t  $T2 + $) + . * .
*
.
(6.1.14) Expansion (6.1.14) is valid when t = O(zc2) because the correction term (second term) is O(e2) or less for all times up to O(63. However, this expansion breaks down when t = O(E~)because the second term ceases to be small compared to the first. To obtain an expansion valid for times as large as O(c4), another variable, T4= e4t = O(1), should be introduced. The above discussion suggests that z(t; e) depends explicitly on t , d , e 2 t , . . . , as well as E itself. This can also be seen from the exact solution. Thus, in order to determine a truncated expansion valid for all f up to O<E”‘>,where M is a positive integer, we must determine the dependence of z on the M 1 different time scales To, T I ,. . . , T M , where
+
T , = emt
(6.1.1S)
The time scale Tl is slower than To,while the time scale T, is slower than Tl. In general, T,, is slower than Tnl. Thus we assume that s(t; E ) = Z(T0, TI,
=
. . . , TIM;E )
2 emsm(To,Ti, . . . , T M )+ O(eTfi1)
M1
m=O
(6.1.16)
The error in (6.1.16) is stated O(eTM)to remind the reader that this expansion is valid for times up to O ( E  ~ ) .Beyond these times, we must use other time scales to keep the expansion uniformly valid. Equations (6.1.15) and (6.1.16) show that the problem has been transformed from an ordinary differential equation to a partial differential equation. If the original problem is a partial differential equation, then the introduction of different time scales increases the number of independent variables. By using the chain rule, the time derivative is transformed according to (6.1.17) Equations (6.1.15) through (6.1.17) formulate one version of the method of multiple scales ; namely, the manyvariable version. This technique has been developed by Sturrock (1957, 1963), Frieman (1963), Nayfeh (196% d , 1968), and Sandri (1965, 1967). Equations (6.1.16) and (6.1.17) show that a uniformly valid expansion is obtained by expanding the derivatives as well as the dependent variables in powers of the small parameter. Hence Sturrock and Nayfeh called this technique the derivativeexpansion method.
6.1.
DESCRIPTION OF THE METHOD
231
Substitution of (6.1.16) and (6.1.17) into (6.1.1) and equating coefficients of like powers of E , we obtain equations for determining xo,xl, . . . ,xM. The solutions of these equations contain arbitrary functions of the time scales T I ,T,, . . . , Ti?,. In order to determine these functions, additional conditions need to be imposed. If (6.1.16) is to be valid for times as large as EM, P x , should be a small correction to which in turn should be a small correction to E ~  ~ X ,  ~ .Thus we require that
2,< oo for all T,,T,, . . . , T~ 2,1
This condition does not mean that each x, is bounded. In fact, each x, may be unbounded. However, this condition requires, as in Lighthill's technique (Section 3.2), that higher approximations be no more singular than the first term. This condition is equivalent to the elimination of secular terms. The second version of the method of multiple scales was introduced by Cole and Kevorkian (1963) and applied by Kevorkian (1966a) and Cole (1968) to several examples. Morrison (1966a) showed that this procedure is equivalent to the method of averaging to second order, while Perko (1969) established their equivalence to nth order. Kevorkian (1966b) showed the equivalence of this procedure and von Zeipel's method to first order. If we investigate the exact solution (6.1.10), we find that t appears in either of the combinations E? or 41  3f.Hence to determine an expansion valid for large times, one introduces the two time scales f =
E?
and
7 = 41
 G t = (1  +cz  4c4 +
. . a )
t
(6.1.18)
Therefore Cole and Kevorkian (1963) assumed that x(t; 4 = q f , 17; 4
2 E m X m O ; 7)+ 0 ( c M )
M1
=
where
(6.1.19)
m=O
t =Et,
q = (1
with constant w,. In this case transformed according to
+
+
E ~ C O ~ c3w3
+    +c
~ c o ~ (6.1.20) ) ~
is slower than 7,and the time derivative is
(6.1.21) These two versions can be generalized considerably. Thus the manyvariable version can be generalized (Nayfeh, 1967b) by using an asymptotic
232
THE METHOD OF MULTIPLE SCALES
sequence d , ( ~ ) rather than powers of
E.
Thus
T,, = B n ( E ) f
(6.1.22)
n = 2 dn(E) a dt
qr=o
aTn
(6.1.23)
Equations (6.1.22) and (6.1.23) can be generalized further by letting Tn
= 4I(E)ga[rU,,(4fl
(6.1.24) (6.1.25)
where ,u,,(E) is another asymptotic sequence. Thus (6.1.24) allows for linear as well as nonlinear time scales. Similarly the twovariable expansion procedure can also be generalized. Thus (6.1.20) and (6.1.21) can be generalized to
(6.1.27) This general form was developed by several investigators including Kuzmak (1959), Cochran (1962), Mahony (1962), and Nayfeh (1964, 1965b). Klimas, Ramnath, and Sandri (1970) investigated the role of gauge transformations for uniformization of asymptotic expansions. The method of multiple scales is so popular that it is being rediscovered just about every 6 months. It has been applied to a wide variety of problems in physics, engineering, and applied mathematics. Cole and Kevorkian (1963), Nayfeh (1965c, 1967b, 1968), Kevorkian (1966a), Davis and Alfriend (1967), Schwertassek (1969), and Musa (1967)Rasmussen (1970), and Reiss (1971) analyzed weakly linear and nonlinear vibrations governed by second or thirdorder ordinary differential equations. Kuzmak (1959) studied nonlinear oscillations in secondorder differential equations with slowly varying coefficients. Cochran (1962), Nayfeh (1964, 1965b), and Fowkes (19681) used the generalized version to analyze turning point problems for secondorder linear differential equations. Cochran (1962), Nayfeh (1964, 1965b), and Ramnath and Sandri (1969) used the generalized method to study linear equations with variable coefficients, while Cheng and Wu (1970) analyzed the effect of the scales on the problem of an aging spring. Noerdlinger and Petrosian (1971) discussed a linear inhomogeneous equation with slowly varying coefficients which describes the
6.1.
DESCRIPTION OF THE METHOD
233
effect of cosmological expansion on selfgravitating ensembles of particles, while Kevorkian (1971) investigated the problem of passage through resonance for a onedimensional oscillator with a slowly varying frequency. Cochran (1962), O’Malley (1968a, b), and Sear1 (1971) applied the generalized method to boundary value problems for certain nonlinear secondorder differential equations, while Cochran (1 962) and Ackerberg and O’Malley (1970) applied this method to secondorder equations that exhibit turning points and boundary layers. Tam (1968) used the generalized version to solve the OrrSommerfeld equation. In orbital mechanics, Nayfeh (1965a) used the generalized version to analyze the earthmoonspaceship problem. Ting and Brofman (1964) and Nayfeh (1966) analyzed the problem of takeoff of a satellite from a circular orbit with a small thrust, Shi and Eckstein (1966) investigated takeoff from an elliptical orbit with a small thrust, Kevorkian (1966a) and Brofman (1967) studied the motion of a satellite subjected to a small thrust or drag, and Eckstein and Shi (1967) analyzed the motion of a satellite with variable mass and low thrust. Eckstein, Shi, and Kevorkian (1966a) determined the motion of a satellite around the primary in the restricted problem of three bodies, while Alfriend and Rand (1969) determined the stability of the triangular points in the elliptic restricted problem of three bodies. Eckstein, Shi, and Kevorkian (1966~)evaluated higherorder terms in the motion of a satellite using the energy integral and evaluated the effects of eccentricity and inclination (1966b). Shi and Eckstein (1968) analyzed the motion of an artificial satellite having a period commensurable with the rotation period of the primary. Alfriend (1970) and Nayfeh (1971b) studied the twotoone resonances, while Nayfeh and Kame1 (1970b) and Alfriend (1971b) studied the threetoone resonances near the equilateral libration points. Alfriend (1971a) analyzed twotoone resonances in twodegreeoffreedom Hamiltonian systems. In flight mechanics, Ashley (1967) discussed the role of different time scales in flight mechanics, while Nayfeh and Saric (1971b) analyzed nonlinear resonances in the motion of a missile with slight asymmetries. Nayfeh (1969a) used the generalized version to study the motion of a rolling missile with variable roll rate and dynamic pressure but linear aerodynamics, while Nayfeh and Saric (1972a) studied the motion of a missile with nonlinear aerodynamics and variable roll rate and dynamic pressure. Ramnath (1970b) studied the transition dynamics of VTOL aircraft. In solid mechanics, Amazigo, Budiansky and Carrier (1970) analyzed the nonlinear buckling of imperfect columns, while Reiss and Matkowsky (1971) investigated the nonlinear dynamic buckling of a compressed elastic column. Mortell (1968) analyzed the problem of a traveling wave on a cylindrical shell and the propagation of waves on a spherical shell (1969). Kelly (1965)
234
THE METHOD OF MULTIPLE SCALES
and Morino (1969) studied nonlinear panel flutter, while Sprigs, Messiter, and Anderson (1969) discussed membrane flutter. In partial differential equations, Cochran (1962), Nayfeh (1965b), and Comstock (1971) treated elliptic equations. Fowkes (1968, Part 11) obtained uniformly valid expansions for caustic problems. Neubert (1970) obtained solutions for the Helmholtz equation for turbulent water. Wingate and Davis (1970) discussed the propagation of waves in an inhomogeneous rod. Keller and Kogelman (1970) treated a nonlinear initial boundary value problem for a partial differential equation. Luke (1966) studied the KleinGordon equation and general variational equations of second order, while Emery (1970) treated the case of several dependent variables and several rapidly rotating phases. Ablowitz and Benney (1970) investigated the evolution of multiphase modes for the KleinGordon equation. Nayfeh and Hassan (1971) and Nayfeh and Saric (1972b) discussed nonlinear dispersive waves on the interface of two fluids and in a hot electron plasma. Parker (1969) analyzed the effects of relaxation and diffusive damping on dispersive waves. In wave interactions, Benney and Saffman (1966), Benney (1967), Davidson (1967), Benney and Newell (1967), Hoult (1968), Newell (1968), and Benney and Newell (1969) investigated the nonlinear interaction of random waves in a dispersive medium. Davidson (1969) studied the time evolution of wave correlations in a uniformly turbulent ensemble of weakly nonlinear dispersive systems. In water waves, Carrier (1966) analyzed gravity waves in water of variable depth, while Hoogstraten (1968) and Freeman and Johnson (1970) studied shallow water waves in shear flows. Jacobs (1967) solved the tidal equations. Murray (1968) treated free surface oscillations in a tank resulting from drainage. Chu and Mei (1970) studied slowly varying Stokes’ waves. McGoldrick (1970) and Nayfeh (1 970b) treated the secondharmonic resonance case, while Nayfeh (1970d, 1971a) investigated the thirdharmonic resonance case in the interaction of capillary and gravity waves. In atmospheric science, Newell (1969) treated the resonant interaction of Rossby wave packets, while Stone (1969) analyzed the problem of baroclinic waves. Shabbar (1971) discussed the sideband resonance mechanism in the atmosphere supporting Rossby waves, while Lindzen (1971) studied the propagation of equatorial Tanai and Kelvin waves through shear. In plasma physics, Ball (1964), Taussig (1969), and Tam (1969, 1970) analyzed the propagation of nonlinear waves in a cold plasma, while Nayfeh (1965d) and Das (1971) investigated nonlinear oscillations in a hot electron plasma. Davidson (1968) treated nonlinear oscillations in a VlasovMaxwell plasma. Peyret (1966) analyzed plasma waves in an accelerator, while Butler and Gribben (1968) discussed nonlinear waves in a nonuniform plasma.
6.1.
DESCRIPTION OF THE METHOD
235
Maroli and Pozzoli (1969) studied the penetration of highfrequency electromagnetic waves into a slightly ionized plasma. AbrahamShrauner (1970a, b) investigated the suppression of runaway of electrons in a Lorentz gas. Chen and Lewak (1970), Chen (1971), and Prasad (1971) studied parametric excitation in a plasma, while Lewak (1971) discussed the interaction of electrostatic waves in a plasma. Dobrowolny and Rogister (1971) and Rogister (1971) analyzed the propagation of hydromagnetic waves in a highbeta plasma. In hydrodynamic and plasma stability, Frieman and Rutherford (1964) developed a kinetic theory for weakly unstable plasmas, while Albright (1970) analyzed the stabilization of transverse plasma instability. Kelly (1967) investigated the stability of an inviscid shear layer. Benney and Roskes (1969) analyzed the instability of gravity waves. Kiang (1969) and Nayfeh (1969b) studied RayleighTaylor instability, while Newel1 and Whitehead (1969) analyzed postcritical RayleighBinard convection. Nayfeh (1 970c) investigated the nonlinear stability of a liquid jet. Nayfeh and Saric (1971a) studied nonlinear KelvinHelmholtz instability, while Puri (197 1) analyzed the effects of viscosity and membrane on the oscillation of two superposed fluids. Stewartson and Stuart (1971) treated the nonlinear stability of plane Poiseuille flow. Mitchell (1971) applied this technique to combustion instability. In fluid mechanics, Germain (1967) and Lick (1970) reviewed recent developments in aerodynamics and nonlinear wave propagation in fluids including the methods of matched asymptotic expansions, strained coordinates, and multiple scales. Benney (1965) analyzed the flow field produced by finiteamplitude oscillation of a disk about a steady state of rotation, while Barcilon (1970) studied the linear viscous theory of steady rotating fluid flows. Rubbert and Landahl (1967) discussed the transonic airfoil problem. Peyret (1970) treated the problem of steady flow of a conducting perfectly compressible fluid in a channel. Chong and Sirovich (1971) studied the problem of steady supersonic dissipative gas dynamics. Cheng, Kirsch, and Lee (1971) analyzed the behavior of a strong shock wave initiated by a point explosion and driven continuously outward by an inner contact surface. In general physics, Caughey and Payne (1967) used a combination of the method of multiple scales and the method of matched asymptotic expansions to solve the FokkerPlanck equation arising from the response of selfexcitated oscillators to random excitations. Brau (1967) obtained a stochastic theory for the dissociation and recombination of diatomic molecules. Ramnath (1 970a) obtained an approximation for the ThomasFermi model in atomic physics and treated a class of nonlinear differential equations arising in astrophysics (1971). Meyer (1971) investigated Rayleigh scattering
236
THE METHOD OF MULTIPLE SCALES
of a laser beam from a massive relativistic twolevel atom, while Nienhuis (1970) studied Brownian motion with a rotational degree of freedom. In statistical mechanics, Maroli (1966) solved Boltznann's equation to obtain a kinetic theory of highfrequency resonance gas discharge breakdown, while Caldirola, De Barbieri, and Maroli (1966) solved Boltzmann's equation for the electronic distribution fiinction. De Barbieri and Maroli (1967) solved the Liouville equation to analyze the dynamics of weakly ionized gases, while Goldberg and Sandri (1967) and Ramanathan and Sandri (1969) derived sets of hierarchical equations. In the remainder of this section, we describe the three versions of the method of multiple scales and their application to the simple linear damped oscillator given by (6.1.1). In the following sections we apply these techniques to different problems in mathematical physics. MANYVARIABLE VERSION (THE DERIVATIVEEXPANSION PROCEDURE) Substituting (6.1.16) and (6. I. 17) into (6.1.1) and equating coefficients of like powers of E, we obtain the following equations for xo, x l , and x2 6.1 . I .
azx0 +xo=o
(6.1.28)
aT,"
azxl + XI
= 2
aT,"
azX2 8x1  2 + x2 = 2 
aT,"
aTo
aZX, ~
aToaTl
ax0
aTo
 2
aaXo
azx0 azx0 2 ___
aTlz
(6.1.29)
aTo aTl aToaTz
8x0
 2  (6.1.30)
aT
The general solution of (6.1.28) is
+
xo = Ao(Tl, T2)eiT0 AO(Tl,T2)eiT0
(6.1.31)
where A, is the complex conjugate of A,. This solution is simply equivalent to (6.1.6) where a and #J are taken to be functions of the slow time scales T I and T , rather than being constants. Substituting for xo from (6.1.3 1) into (6.1.29), we obtain
The general solution of (6. I .32a) is
+ xl(Tl,T2)eiT0
x1 = A,(T,, T2)eiT0
6.1. DESCRIPTION
OF THE METHOD
237
Comparing (6.1.32b) with (6.1.31) shows that EX, is a small correction to x only when €To = cf is small. In order to obtain an expansion valid for times as large as O(E~),the secular terms, Toexp (&iT,,), in (6.1.32b) must vanish; that is (6.1.33) or A = a,( Tz)eT1
(6.1.34)
Then (6.1.32b) becomes
+
x1 = A,(T,, Tz)etTo Al(T,, T2)e'To
(6.1.35)
 + x2 = Q(Tl, T2)eiTo Q(Tl, Tz)e'To
(6.1.36)
Using xo and x1 in (6.1.30), we obtain aZx,
where
a T,'
Q(T,, T2)= 2iA1
aA + 2i ,
+
aoeT1 2i e T I (6.1.37) aT1 The terms on the righthand side of (6.1.36) produce secular terms because the particular solution is x2 = $iQ(Tl, Tz)ToeaT0 ,+i&(T,,T2)Toe'To (6.1.38) These secular terms make e2x2 the same order as €5, when t is as large as O(cl). In order to eliminate these secular terms, Q must vanish; that is
(6.1.39) In general, one does not need to solve for x2 in order to arrive at (6.1.39). One needs only to inspect (6.1.36) and eliminate terms that produce secular terms. The general solution of (6.1.39) is
[
A, = al(Tz)
1
Tl + ti (ao + 2i a TZ
Substituting for A, into (6.1.35), we obtain a,
1
eaT1
+ 2iaT* T, eT1eiTD+ CC
(6.1.40)
(6.1.41)
where CC stands for the complex conjugate of the preceding expression. However xo = [aoe'To + go,eiT~]eT1 (6.1.42)
238
THE METHOD OF MULTIPLE SCALES
Therefore, as TI+ co, although z, and 5, + O , E Z , becomes O(z,) as t increases to O(ce). Thus the expansion z, €5, breaks down for t as large ) the coefficients of TI in the brackets in (6.1.41) vanish; as O ( E  ~unless that is, unless
+
(6.1.43)
or
 a 00eiT,/2
(6.1.44)
a,
where a,, is a constant. Then (6.1.40) becomes A, = al(T2)eT1
(6.1.45)
Therefore
The function a,(T,) can be determined by carrying out the expansion to third order al(T2) = alle"~'2
(6.1.47)
where a,, is a constant. If we assume that the initial conditions are such that z(0) = a cos 4 and i ( 0 ) = (sin $J1  + E cos 4) and replace T, by €9,we obtain z = aeFtcos i t  +e2t 4) R (6.1.48)
+ +
where R is the remainder. From (6.1.10) and (6.1.48), we find that
For linear equations such as (6.1.1), we may introduce the different time scales without expanding z.Thus using (6.1.17) in (6.1.1), we obtain a2
a2
[%+2.
aT, aTl
+
+2) az 4aT;
a2
aT, aT,
a +
+
.)z
(6.1.50)
6.1.
DESCRIPTION OF THE METHOD
Equating the coefficients of like powers of
E
239
to zero yields (6.1.51) (6.1.52) (6.1.53)
The general solution of (6.1.51) is
+ x(Tl, T2)ePiTO
x = A(T,, Tz)e'TO
(6.1.54)
Substituting into (6.1.52), we obtain (6.1.55)
Since (6.1.55) is valid for all To, the coefficients of exp (iTo)and exp (iTo) must vanish; that is aA +A=O (6.1.56) aTI or A = a(Tz)eT1 (6.1.57) Substituting (6.1.54) into (6.1.53) yields (6.1.58)
Thus A aA + 2JaTl + 2i =0 aT2
a2A aT12
(6.1.59)
Substituting for A from (6.1.57) into (6.1.59) gives
aa
2iu=O
Hence
(6.1.60)
a T,
a = aoe'T*'2
(6.1.61)
where a, is a constant. Therefore (6.1.54) becomes =a
eTiei(T.T,12)
+
cc
(6.1.62)
240
THE METHOD OF MULTIPLE SCALES
Expressing (6.1.62) in terms of t yields x = aeEtcos( t  fr2t
+ 4)
(6.1.63)
where a, = (1/2)a exp (+I. This result is in full agreement with (6.1.48). 6.1.2. THE TWOVARIABLE EXPANSION PROCEDURE Changing the independent variable from r to 6 and q as defined by (6.1.21), we transform (6.1.1) into
 241 We assume that 5
= xo(5, q)
+2wz+**.)
ax  2 2  ax
a7
at
+ 4 5 , q ) + e2x*(t,7)+  
(6.1.64)
(6.1.65)
Substituting (6.1.65) into (6.1.64) and equating coefficients of equal powers of E on both sides, we obtain o+x,=o a2
(6.1.66)
atx
atxz
 + XZ + 2 arlz
azxl ,+x,+22=2
8x0
817
%
azx0 w , 7 arl
a t aq
zx ' = 2 ax,  2ax, + 2ataq + aat2 aq at agXl
(6.1.67) (6.1.68)
The general solution of (6.1.66) is xo = A,(E)e'q
+ A,(t)e'q
(6.1.69)
With this solution, (6.1.67) becomes (6.1.70)
Eliminating the terms that produce secular terms in (6.1.70) gives
Hence
dAo +A,=O dt
(6.1.71)
6.1.
DESCRIPTION OF T H E METHOD
241
The solution of (6.1.71) is
A , = a,ee (6.1.73) where a, is a constant. Substituting the above solutions for xo and z1into (6.1.68) gives
_a Z x ,
aq
+ x2 = [2i(%
+ A') + ( 2 0 , + l)a,ed
1
es
+ CC
(6.1.74)
Eliminating the terms that produce secular terms in (6.1.74) yields
dA1 + whose solution is
= +i(2w2
AE
+ l)a,ec
(6.1.75)
+
A , = alept  f i ( 2 0 , l)a,Ee+ (6.1.76) Substituting for A , into (6.1.72) and comparing the result with (6.1.69) show that x l / z 0 is unbounded as 6 + 00 unless
$
(6.1.77)
02=
Therefore in terms of t (6.1.65) becomes
+ +
x = ae" cos ( t  f 2 t 4) O ( 2 ) (6. I . 78) where Q, €al = (1/2)a exp ti+). This expression is in full agreement with that obtained using the manyvariable version (derivativeexpansion method).
+
6.1.3. GENERALIZED METHODNONLINEAR SCALES We first introduce a new variable T = ct to transform (6.1.1) into
c ( ;d'x ;,+2$)+z=o
(6.1.79)
In order to determine a uniformly valid expansion, we let E = 7 ,
r / =  g1(7) e
+
&(T)
+
€g1(T)
+
' ' '
,
gi(0) = 0 (6.1.80)
where g iis determined in the course of analysis. The derivatives with respect to T are then transformed according to (6.1.81)
+ [+g
+ g&) + cgi(E) +
* *
1'5
(6.1.82)
242 THE METHOD OF MULTIPLE SCALES We assume that x possesses a uniformly valid expansion of the form x = xo(E, q)
+ q ( E , 7) +
E2X2(E, q)
+
* *
.
(6.1.83)
Substituting (6.1.81) through (6.1.83) into (6.1.79) and equating coefficients of like powers of E, we obtain (6.1.84)
(6.1.85)
The general solution of (6.1.84) is
(6.1.86)
ie +
Substituting for xo into (6.1.85) gives r2
a2x,
+ zl = [( % + g1
2i)A,
g1
The terms on the righthand side of (6.1.87) produce secular terms. In order to eliminate secular terms
( + +i9 + 2i g1 g1
7Ao = 0 (6.1.88)
Since (6.1.88) must be valid for all q , and A, # 0 for a nontrivial solution, we require that g,: = 0 or g, = CE since q(0) = 0 (6.1.89) where c is an arbitrary constant which can be taken to be unity without loss of generality. Then (6.1.88) becomes A;
whose solution is
+ (1 + ig;)Ao
=0
A 0  aoeSim(QJ
(6.1.90) (6.1.91)
where a, is a constant. With A, and g, known 2,
= aoerei(r/~)
+ (igerei(r/c)
(6.1.92)
Equation (6.1.92) shows that gocancels out, hence the solution is independent
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
243
of the value of go. Therefore we might as well set it equal to zero without loss of generality. Hence A, becomes A, = a,e&
(6.1.93)
As a result of (6.1.88), the solution for x, is
+
(6.1.94) z, = Al(t)eiq A,(t)ea'J With g, = and go = 0, the equation for x 2 can be determined by substituting (6.1.81) through (6.1.83) into (6.1.79) and equating the coefficient of cz to zero. Thus
azxz +s,+2
a2x, az, ,aZx, azx, ax + 2+2 g 1 2 +7+2 ata7 arl arl a t at O
=0
(6.1.95)
Substituting for zo and z, into (6.1.95) yields
 + r2 = [2 i(A;
ar2
+ A,)  (2g; 4 l)~,e*]e*~+ CC
(6.1.96)
Eliminating the terms that produce secular terms in (6.1.96), we obtain A;
+ A , = 4i(2g; + l)a,ec
(6.1.97)
+ t)e[
(6.1.98)
The solution of (6.1.97) is A, = ale(  +iao(2g,
where a, is a constant. Equation (6.1.98) shows that xl/xo is unbounded as t 4 00 unless g1= *t (6.1.99) In terms of t = T / C , the expansion becomes
+
L
= ue'tcos ( t
 +c2t + +) + O(2)
(6.1.100)
where a, al= (lj2)a exp ($). This expansion is again in agreement with those obtained using the derivativeexpansion and the twovariable expansion versions of the method of multiple scales. 6.2.
Applications of the DerivativeExpansion Method
6.2.1. THE DUFFING EQUATION The second example to which we apply the derivativeexpansion method is the Duffing equation
 + w,"u d2u
dt2
+
Ell3
=0
(6.2.1)
244
THE METHOD OF MULTIPLE SCALES
We assume that
2
a
Then
d (6.2.3) =Do+~Dl+~2D2+..., D =dt aT, Substituting (6.2.2) and (6.2.3) into (6.2.1) and equating coefficients of each power of E to zero, we have
+ oo2uo= 0 Do%, + oo2u1= 2DoDluo  uO3
Do%,
(6.2.4) (6.2.5)
+
DO2u2 wO2u2= 2DoD1~1  2DoDZuO  Dl2U0  3Uo2u1 (6.2.6) The solution of (6.2.4) is
+
uo = A(Tl,T2)eiWoTa A(Tl,T2)eaWoTa Equation (6.2.5) then becomes
+
+
(6.2.7)
+
D,2ul o 2 u 1 = [2iw,D1A 3A2A]eiWoTo  A I e 3 i W o T o CC (6.2.8) In order that u,/uo be bounded for all To, terms that produce secular terms must be eliminated. Hence
+
2 i o , ~ , A 3A2K = O
(6.2.9)
and the solution for u1 becomes (6.2.10)
To solve (6.2.9), we let A = (1/2)uei6 with real a and $, separate real and imaginary parts, and obtain (6.2.11)
Hence a = u(T2),
4 = a 2 T + $o(T2) 3
8WO Substituting for uo and u1 into (6.2.6) yields
(6.2.12)
 Q(T,, TZ)eiWaTo + CC (6.2.13)
where Q
= 2i0oD1B
+ 3A2B + 6 A A B+ 2ioODzA  8w:
(6.2.14)
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
245
Secular terms are eliminated if B=O
and
(6.2.15)
15A3A2 2iwoD2A = 
(6.2.16)
8W,2
With Q = 0, the solution of u2, disregarding the homogeneous solution, is (6.2.17) Letting A = (l/2)aei+ in (6.2.16) and separating real and imaginary parts, we obtain
w
aa= o , aT2
15
ma=
aT2
256~0,
(6.2.18)
(I4
Equations (6.2.12) and (6.2.18) lead to a = a constant, hence (6.2.19a) where
x
is a constant. Therefore
(6.2.19b) Substituting for uo, ul, and u, into (6.2.2), keeping in mind that A = (1/2)aexp (i4) and expressing the result in terms of t , we obtain u =a
cos ( O l t
+ x) +
2
32w0 Ea3
(1 
320,~
2a +10240:
where w =
W"
3a2 +80,
E
cos 5(wt
15a4  2
2560,~
+ x)
cos 3(wt
€=)
+ x) + O(c3)
+ O(E3)
In the last two terms of (6.2.20a), w0 is replaced by
w
(6.2.20a)
(6.2.20b)
with an error O(e3).
6.2.2. T H E VAN DER POL OSCILLATOR As a second example, we consider the van der Pol oscillator
+ u d2u dt2
= E(l 
du
2)
dt
(6.2.21)
246
THE METHOD OF MULTIPLE SCALES
Substituting (6.2.2) and (6.2.3) into (6.2.21) and equating the coefficients of like powers of e, we obtain
+ u, = 0 Do%, + = 2D,D1~0 + (1  U;)D,U, Do%, + = 2DoD1~1  Dl2uo  2DoDZuo + (1  U;)D,U,
Do2uo
~1
UZ
+ (1  U ~ ) D l U O 2u,u,D,u,
(6.2.22) (6.2.23) (6.2.24)
The solution of (6.2.22) is
+
uo = A(Tl, T2)eiTa J(Tl, Tz)eiTa
Substituting for D:u,
uo into
(6.2.25)
(6.2.23) gives
+ u1 = i(2D1A
A
+ A2x)eaTo iA3e3iT0+ CC
(6.2.26)
To eliminate terms that produce secular terms, we require the vanishing of the coefficients of exp ( f i T , ) ; that is 2D1A = A  A Z K
(6.2.27)
Then the solution of (6.2.26) is
+ iiA3e3'To+ CC
u1 = B(Tl, Tz)eiTo In order to solve (6.2.27), we let
A = M T l , T2) exp i4V1, T,)
(6.2.28) (6.2.29)
Separating real and imaginary parts in (6.2.27), we obtain (6.2.30)
Hence (6.2.31)
If we are interested in the first approximation to u , then we consider B, and c as constants. Mofeover, if u(0) = a, and du(O)/dt = 0, then where
u = a cos t
+ O(E)
4,
(6.2.32) (6.2.33)
which is in agreement with the expansion obtained in Section 5.4.2 using the KrylovBogoliubovMitropolski technique.
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
247
To determine the second approximation, we need to determine the functions B, 4, and c. Thus we substitute for u, and u1 into (6.2.24) and obtain where
Do%,
+ u, = Q(Tl, Tz)eiTo+ &(Tl, T2)eCiT0+ NST
+ i(l  2AX)B  iA2B 2iD,A
Q = 2iD,B
(6.2.34a)
 D12A
+ (1  2AK)DIA  A2D1K+ &A3K2 (6.2.34b)
Secular terms will be eliminated if Q = 0. To solve (6.2.34b) with Q = 0, we let B = (1/2)ib exp i$ with b real and $ defined in (6.2.29). We substitute for A and B into (6.2.34b) with Q = 0, separate real and imaginary parts, and obtain
aa a T2
 = 0,
ab 2  ( I  &')b a TI
=
or
a = a(Tl)
,
d4 d2a  2 ~ +  (1 dT, dTl
(6.2.35a) 2
)da
dTl

128
a5
(6.2.35b)
With the help of (6.2.30), (6.2.35b) can be expressed in the form
(:I
d 
  +2: : (
6 :)
dT
+ (&a

k)
du
(6.2.36b)
Integrating, we obtain h = a($
+ ;ITl + %%a3 &u In a + ab,(T,)
(6.2.36~)
In order that u,/uo be bounded for all TI, the coefficient of Tl in the above expression for h must vanish. This condition gives
4 = AT2 where tl
+
(6.2.37)
40
is a constant. The expansion of u to second approximation is then
= a cos [(l
 r'G.2)t + $,I
 e { ( ( ; t a 3
In a
+ ab,)
+ &u3
sin [(l
sin 3[(1  &;).'t
 i%c2)f + $,I
+ do])+ O(r2)
(6.2.38)
where a is defined by (6.2.33) and b, is considered a constant to within the order of error indicated. To an error of O(rz), this expression can be written
248
THE METHOD OF MULTIPLE SCALES
as
u = a cos (t
where
 6) 
6 = &E2t
sin 3(t  6)
+ O(8)
+ 8. In a  +a2 + 6 ,
(6.2.39a) (6.2.39b)
and 6, = 4,  ~ b = , a constant. This last form of the solution is in full agreement with that obtained in Section 5.4.2 using the KrylovBogoliubovMitropolski method. 6.2.3. FORCED OSCILLATIONS OF THE VAN DER POL
EQUATION We consider next the response of the van der Pol oscillator, discussed in the previous section, to a periodic external force; that is, the oscillations of the equation du d2U w t u = ~ ( 1 u2)K cos I t (6.2.40) dt2
+
dt
+
where K and 1 are real constants. Four cases arise depending on whether the excitation (external force) is "soft" [i.e., K = O ( E ) ]or "hard" [i.e., K = O(l)], and whether the excitation is resonant [i.e., I  w, = O ( E ) ]or nonresonant [i.e., 1  o, = O(l)]. Soft Nonresonant Excitation. In this case K = ck, where k = 0(1),and we express cos I t in the form cos IT,. To determine a first approximation to u , we let (6.2.41) = u,(T,, TI) %(TO, Tl) O(E2)
+
+
with To = t and TI = ct. Substituting (6.2.41) into (6.2.40) and equating the coefficients of eo and ,E on both sides, we obtain D,Zu, Do2u,
+
wo2uo =
0
+ wo2u1 = 2D,Dlu, + (1  u,~)D,u,+ k cos IT ,
(6.2.42) (6.2.43)
The solution of (6.2.42) is u, =
A(T,)e'"OTO
+ A(T,)ei"oTo
(6.2,44)
Substituting for u, into (6.2.43) gives Dtu,
+ w2u1 = iw0(2A' + A  A A) e + 3ke'"'  iwnA3e3i00T0+ CC 2
iwoTo
(6.2.45)
For there to be no secular terms, we require that 2A' = A  A 2 K
(6.2.46)
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
6.2.
249
where the prime denotes differentiation with respect to T I .Then the solution of (6.2.45) is u1 = B(Tl)eimOTO+
A
!
eiATo
2 coo  I 2
+iA3 e3iooTo
+ CC
80,
(6.2.47)
Letting A = (l/2)a exp i# in (6.2.46), separating real and imaginary parts, and solving the resulting equations, we find that # is a constant, while a is given by (6.2.33). Therefore, to first approximation u = a cos mot
+ O(E)
(6.2.48)
where a is given by (6.2.33). Equations (6.2.33) and (6.2.48) show that, to first approximation, neither the phase nor the amplitude is affected by the presence of a soft nonresonant excitation. Moreover, the natural response (i.e., the case with k = 0) dominates the forced response, as expected, since the forcing function is soft. However, as the forcing frequency I approaches the natural frequency coo, the forced response becomes more significant and approaches infinity as can be seen from (6.2.47), and the above expansion is no longer valid. Hard Nonresonant Excitation. (6.2.43) are modified to
+ Do2u1 +
Do2u0
In this case K = O(1) and (6.2.42) and (6.2.49)
= K cos IT,
W ~ U , ~
0
= ~2DoD,uO ~ 1
+ (I
 UO~)D~UO
(6.2.50)
The solution of (6.2.49) is
+
uo = A(Tl)eiooTo A(Tl)ei"oTo
Substituting for
uo into
+ w: ~
 I2
cos IT,
(6.2.51)
(6.2.50) gives
+
D ~ u , w:ul = ioo[2A'
where = 1  K2/2(wO2require that
+ A7  A2A]eimoTo + CC + NST
(6.2.52)
. In order to eliminate the secular terms, we
2A' = A17  A2K
(6.2.53)
To solve (6.2.53), we let A = (1/2)aexp i+, separate real and imaginary
250 THE METHOD OF MULTIPLE SCALES parts, and obtain
4 = a constant and (6.2.54)
The solution of (6.2.54) can be obtained by separation of variables to be In u2  In (q  &z2) = qT,
If u(0) = a, u is given by
+ a constant
+ [K/(wO2 A2)3 and du(O)/dt = 0, the first approximation to u = a cos w,t
where 2
a =
+
1
+ w,2  A2 cos At + O(E) ~
411,
(2
(6.2.55)
(6.2.56)
The steadystate solution (i.e., t co) depends on whether q is positive or negative [i.e., K 2is less than or greater than 2(oO2For negative q, exp (  q f ) + co as t co, hence a + 0 as f co, and the steadystate solution is f
f
us =
f
~
m:
 A2
cos I t
+ O(e)
(6.2.57)
However, for positive q , exp (  c q f ) + O as t + 00, and a+ 2 4 % as 1 a.Consequently, the steadystate solution is f
U, =

2Jq cos mot
+ cos I t + O(E)  A2 6J02
(6.2.5 8)
Therefore, if 11, is negative, the natural response fades away and the steadystate solution consists of the forced response only. However, if q is positive, the steadystate solution is a combination of the natural and forced responses, with the amplitude of the natural response modified by the presence of the hard excitation. Soft Resonant Excitation. In this case K = ck with k = O(1), and A CT = O(1). In order to determine a valid asymptotic expansion in this case, we express the excitation in terms of To and TI according to w, = o c with the detuning
K C O SA t = ck cos (mot
+ oct) = ck cos (w,T, + aT,)
(6.2.59)
With this expression for the excitation, the equations for u, and u1 of (6.2.41)
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
are Di2u1
+
+ wo2uo= 0 + (1  uo2)D,u0 + k cos Do%,
0 0 ~ ~= 1
2DODluo
The general solution of (6.2.60) is
(wOTO
+
uo = A(T,)eiUoTO K(Tl)e*wOTO Hence (6.2.61) becomes Dtu,
+ w;u1
= [io0(2A'
+ oT,)
25 1
(6.2.60) (6.2.61) (6.2.62)
+ A  A2K)+ gkeiuT1]eimoTo  io0A3e3iw0T0 + CC
(6.2.63) The terms proportional to exp ( &iwoTo)in (6.2.63) produce secular terms with respect to the time scale To because the terms in the brackets are functions of TI only. In order that u,/uo be bounded for all To 2A' = A  A2K  1 ike'"T1 (6.2.64) 200 To solve (6.2.64), we let A = (1/2)a exp i4, separate real and imaginary parts, and obtain (6.2.65) (6.2.66) To eliminate the explicit time dependence of the righthand sides of (6.2.65) and (6.2.66), we let (6.2.67) Hence (6.2.65) and (6.2.66) become da
 = ga(1
dT,
k .  $2)+ sin y
200
(6.2.68)
k cos y (6.2.69) d TI 20,a Periodic solutions of the externally excited oscillator (6.2.40) correspond to the stationary solutions of (6.2.68) and (6.2.69); that is, da/dTl = d y / d T l = 0 , or k . (6.2.70) $41  42) sin Q = O
d v  a + 
+
o+
k 2w06
200
cos
4=0
(6.2.71)
252 THE =HOD
OF MULTIPLE SCALES
where the tilde refers to the stationary solution. Elimination of Q from these equations leads to the following frequency response equation p(1
k2
 P )+ ~ 4a2p = 7 = F2, 400
a"2 p =
4
(6.2.72)
+
For a given excitation amplitude rk and frequency 1 = wo e a , (6.2.72) furnishes p , hence the amplitude of harmonic oscillations. To first approximation the harmonic oscillation is given by
+ 6)+ O ( € )
u = a"C0S (w,t
while the frequency of oscillation is
(6.2.73)
(6.2.74) Therefore, as 1 approaches wo, the natural response is entrained by the forced response. The result is a synchronization of the output at the excitation frequency. The stability of these harmonic oscillations can be obtained by letting a = a"+ Aa,
y
=4
+ Ay
(6.2.75)
Developing the righthand sides of (6.2.68) and (6.2.69) in powers of Aa and Ay and keeping only linear terms, we have = &(l $r?")Aa
k +cos @ A y
(6.2.76) dT1 200 k d(Aw)    k (6.2.77) cos @ ha sin $7 A y dT1 2w0h 2w0a If we let Aa a exp mT, and A y a exp mT,,then rn must satisfy the equation

m2  a m
where =1
 2p,
+A =0
A = t(l  4p
(6.2.78)
+ 3p2) +
(6.2.79)
where use has been made of (6.2.70) and (6.2.71). The discriminant of (6.2.78) is D = p2  40' (6.2.80) The loci iR = A = D = 0 are called separatrices and shown in Figure 61. The locus A = 0 is an ellipse whose center is p = 213, 0 = 0, while the locus D = 0 is the two straight lines p = f2a. The interior points of the ellipse correspond to saddle points, hence the corresponding harmonic oscillations are unstable. The points exterior to the ellipse are nodes if
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
253
P
t
10
0.5
0
0 5
0
I .o
Figure 61
D 2 0 and foci if D < 0. The harmonic oscillations corresponding to these points are stable or unstable according to whether p is greater or less than 1/2. Hard Resonant Excitation. The analysis for this case can be obtained as a special case of the previous case, with k = K ~ Ewhere , the amplitude of excitation K = O(1). Hence k is very large because c is small. Thus, for B near zero, (6.2.72) shows that there exists only one amplitude p for harmonic oscillation and it is stable. As k increases without bound the amplitude increases without bound also.
6.2.4. PARAMETRIC RESONANCEMATHIEU EQUATION Let us return to the Mathieu equation discussed in Section 3.1.2, namely ii
+ (6 +
f
(6.2.81)
cos 2t)u = 0
According to the Floquet theory of linear differential equations with periodic coefficients, the 6r plane is divided into regions of stability and instability which are separated by transition curves along which u is periodic with a period of either 7r or 2 ~ In. Section 3.1.2, we determined approximations to the transition curves using the LindstedtPoincare method. In this section we find not only the transition curves but also the solutions, hence the degree of stability or instability, as we found in Section 3.1.3 using Whittaker's method. To this end we let 6
and assume that u = uo(To, Ti,
Tz)
= coo2
with positive wo
+ fui(To, Ti, + 7'2)
f2u2(T0,
(6.2.82) Ti, Tz)
+
*
. (6.2.83)
254
THE METHOD OF MULTIPLE SCALES
Different cases have to be distinguished depending on whether far away from an integer n.
near or
coo is
Solution f o r wo Far Awayfrom an Integer. We express cos 2t in terms of the time scale To as cos 2T0. Substituting (6.2.83) into (6.2.81) and equating the coefficients of zo, z, and 8 to zero, we obtain
Do2u2
+
Do2u1 ~
0
+
Do%, 0 0 ~ ~= 1
+ wo2uo= 0
(6.2.84)
~DOD,U,  U O cos 2To
(6.2.85)
= ~2DoD1~1 ~ 2  (01'
+ 2DODz)uo 
The solution of (6.2.84) is u,, = A(T', T2)eiooTo 2(Tl, T&
+
~1
cos 2To (6.2.86)
i o o To
(6.2.87)
Substituting for uo into (6.2.85) yields
D;ul
+ oopul= 2io

+ cc
 4Aei(oo2)To
0D1A ~ ~ W O T ~ OA ~ ~ ( w o + ~ ) T o
(6.2.88) Since wo is far away from 1, secular terms will be eliminated if D,A = 0 or A = A(T,). Then the solution of u1 is
Substituting for uo and u1 into (6.2.86) yields
D;u2
+ w:u2
= 2
1qo:  1)

+ 1)
+l6(oO 1)
~~i(go+r)TO
l6(oO
+ cc
A~~(QIo~)To
(6.2.90) Since wo is far away from 1 or 2, secular terms will be eliminated if i
D2A = 
A (6.2.91) 1600(w:  1) If we let A = (1/2)a exp iq5 and separate real and imaginary parts, we obtain
da =o, Therefore
dT2
a = a constant,
1 16w0(w2 1)
(6.2.92)
4 =  l 6 ~ 0 ( 0 0~ 1) T2 + $0
(6.2.93)
d 4_
dT2
and

1
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
255
where $o is a constant. With (6.2.91) the solution of (6.2.90) is u2
=
Aei(oo+ltTo
1
W w O
+ l)(wo + 2) + 128(00  1l)(oo 2) Aei(oo4)To+ CC
(6.2.94)
Summarizing, to O(e2), the solution for u is u = a cos (cot
+ +o)
where (6.2.96) We emphasize again that this expansion is valid only when coo is away from 1 and 2. As wo + 1 or 2, u + 03. An expansion valid near oo= 1 is obtained next. Solution for wo Near 1. In this case we let
6=1
+
Ed1
+ €26, + 
(6.2.97)
* *
with 6, and 6, = O(1). Equation (6.2.97) modifies (6.2.84) through (6.2.86) into (6.2.98) Do2uo uo = 0
+
+ = 2DoDl~o  61~0 cos 2To Do%, + u2 = 2DoD1ul  (D12 + 2D0D2)u0 Do2u,
(6.2.99)
UO
UI
The solution of (6.2.98) is
6 1 ~ 1
+ K(T,, ~
uo = A(T,, Tz)eiTO
62u0  ~1 cos 2To (6.2.100)
~ ) e  ~ (6.2.101) ~ ~
Substituting for uo into (6.2.99) gives D ~ u+ , u1 = (  2 i D , A  6,A  &A)eiTo gAe3*To+ CC (6.2.102) Secular terms with respect to the time scale To will be eliminated if DIA = fi(6,A
+ $2)
(6.2.103)
256
THE METHOD OF MULTIPLE SCALES
Then the solution of (6.2.102) is =116 (
u,
1
~ ~ 3 i T +o ~  ~  3 i T o
To solve (6.2.103), we assume that A = A,
+ iA,
(6.2.104) (6.2.105)
where A, and A ; are real, and separate real and imaginary parts to obtain (6.2.106) (6.2.107) The solution of these equations is
A, = ul(T2)eYITI
+ uz(T2)eY1T1
(6.2.108)
A, = 2y1 [ U ~ ( T , ) ~~uz(7'.z)ey1T'] '~' t  61
where
(6.2.109)
Y12 = H i  d l 7
(6.2.110) Here a, and u2 are real valued functions of the time scale T,; however, to first approximation, u, and a, are constant. Equations (6.2.108) through (6.2.1 10) show that A grows exponentially with TI(i.e., with t) if y1 is real or I6,l 5 1/2, and A oscillates with Tl if y1 is imagi&ry or IS,l 2 1/2 (in this case the solution is written in terms of cos ylTl and sin ylT; to keep A, and Ai real). Hence the boundaries (transition curves) that separate the stable from the unstable domains emanating from 6 = 1, E = 0 are given to a first approximation by 6, = f 1/2 or
+ O(3)
(6.2.111)
+ DlzA + (6, + &)AleiTo + CC + NST
(6.2.112)
6 = 1 f 4.
To determine a second approximation to u and the transition curves, we substitute for u, and u1 into (6.2.100) and obtain
+
D ~ u , u2 = [2iD2A
The condition that must be satisfied for there to be no secular terms is
+
2iD2A
+ D12A + (6, + &)A
=0
(6.2.1 13)
Since A = A, iA,, (6.2.113) gives the following equations for A, and A i upon separation of real and imaginary parts 'A 2 2 +aA;=O (6.2.114) aT2 2 *a A . aA, = 0 (6.2.115) aT2
+
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
where
~ 1 +2
257
+
(6.2.116) a = 6, Replacing A, and A i by their expressions from (6.2.108) and (6.2.109) and equating the coefficients of exp (fylTl) to zero because they are functions of T,, we obtain da1 dT2 da2
2+
2Yl
3  61 2Yl
dT,  g  6,
aa, = 0, Ma2
= 0,
These equations lead to da, _ da,  0 dT2
 4y1 d a l + aa, = 0 (6.2.117) 4  61 dT, 4y1 da2 + aa, = 0 (6.2.118) 4  61dTz
or a, = a constant and a2 = a constant (6.2.119)
dT,
(6.2.120) Therefore, to second approximation, the solution is given by (3.1.57) through (3.1.62) which were obtained by Whittaker's method. 6.2.5. THE VAN DER POL OSCILLATOR WITH DELAYED AMPLITUDE LIMITING The next example is a thirdorder problem as opposed to the previous examples which are of the second order. It is given in dimensionless quantities by de (6.2.121) 7dZ
+z
= u2
(6.2.122) dr where v is voltage, t is time, e is excitation, coo is natural frequency, T is delay time, Z is output of the lowpass filter, and p is a measure of the servoloop gain. This oscillator was first studied by Golay (1964) and then by Scott (1966) and Nayfeh (1967b, 1968). We consider here free oscillations (i.e., e = 0) and refer the reader to Nayfeh (1968) for the forced oscillation case. To determine a first approximation to the above equations, we assume that = uo(To, Ti) poi(Tm Ti) * . . (6.2.123)
+
+
(6.2.124) (6.2.125)
258
THE METHOD OF MULTIPLE SCALES
Substituting (6.2.123) through (6.2.125) into (6.2.121) and (6.2.122) and equating coefficients of like powers of p , we obtain
+ T D J ~+ Zo = uo2
(6.2.126)
D O 2 ~ , w ~ =~0 u ~ Do2u,
(6.2.127)
+ wo2u1 = 20,[( 1  Z,)U,  D p , ] +
TDJ1
2 1
= TDlzo
(6.2.128)
+ 2UoU1
(6.2.129)
The solution of (6.2.126) is
+
(6.2.130)
+ Z o = A K + A2e2iwoTo+ CC
(6.2.131)
uo = A(Tl)eiwoTo ,4(Tl)eieoTo
Substituting for u, into (6.2.127) gives .rD,Zo Its solution is Z , = B(T,)eTo"
Ape2imoTo
+ 2 A R + 1 + 2ioO7 + 1  2iw07 ~2~2i~~DoTo
(6.2.132)
With u, and Z , known (6.2.128) becomes
 6i00 where
+ 2iw07 + CC
A3e3ia0To
1
(6.2.133)
A2K (6.2.134)  DIA 1 2ioO7 Secular terms will be eliminated if Q = 0. Then the solution for u1 is Q =A2A2K
+
+
+
u1 = 2AB7 1  iOOT e[ioo(llr)]To 3i A3e3ie0T0 CC (6.2.135) 1  2iwO7 40, 1 2iwor To solve the equation Q = 0, we let A = (1/2)aei* with real a and separate real and imaginary parts in (6.2.134), and obtain
+
4,
(6.2.136)
d 4 &u$a2
(6.2.137)
d TI
where a, =
3 1
+ 8w,2r2 + 4W02T2'
a.=
 1 +2WoT 40,272
(6.2.13 8)
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
259
The solutions of (6.2.136) and (6.2.137) are (6.2.139)
where a, is the initial amplitude and 4, is a constant. To determine B, we substitute for u,, Z,, and u1 into (6.2.129); that is
In order that Zl/Zo be bounded for all To, the coefficient of exp (  T , / T ) must vanish; hence (4.2.142)
where we made use of A = (1/2)a exp i4. Substituting for u2 from (6.2.139) into (6.2.142) and solving the resulting equation, we obtain (6.2.143)
where
Therefore, to first approximation u = a cos (mot
z
=
+
a'
2J1
+ 4) + 001)
cos [ h o t
+ 4W,2T2
+ 2 4  tan'
(6.2.144)
+ * a z + O(p)
~WOT]
(6.2.145)
with a, 4,and B given, respectively, by (6.2.139), (6.2.140), and (6.2.143). 6.2.6. THE STABILITY OF THE TRIANGULAR POINTS I N THE ELLIPTIC RESTRICTED PROBLEM OF THREE BODIES The next two examples are fourthorder problemsone is linear and the second is nonlinear. Consider first the stability of the triangular points in the restricted problem of three bodies treated in Sections 3.1.4 and 3.1.5 using the LindstedtPoincart and Whittaker techniques. This problem was
260
THE METHOD OF MULTIPLE SCALES
treated first using the method of multiple scales by Alfriend and Rand (1969). The problem reduces mathematically to the stability of solutions of (3.1.63) through (3.1.65). In this section we use the method of multiple scales to determine the transition curves intersecting the p axis at po = (1  2d5/3)/2 and determine the behavior of x and y near these transition curves. We set cosf = cos Toand assume that 2
+
= xo(To,Tl) exl(To,Tl)
+ 
+ eYl(To9 Tl) +  p = po + epl +   
Y = YO(T0, Tl) where
To =f and
Hence
(6.2.146)
*
(6.2.147)
*
(6.2.148)
Tl = ef
d  Do+eDl+..,
D,=
(6.2.149)
a
(6.2.150) aT, Substituting (6.2.146) through (6.2.150) into (3.1.63) through (3.1.65) and equating coefficients of eO and e to zero, we obtain
df
Order e0
Order e
 2Doy1
Do2x1
+
Do2xo  2Doyo  bgo = 0
(6.2.15 1)
Dozy,
(6.2.152)
+ 2D0x0  a d o = 0
box1 = 220081~0
Do2Y1 2Dg1 a091 = 2DoD1yo
+ 2Dlyo + b1xo  bso cos To
(6.2.153)
 20120  b1yo  U
(6.2.154)
~ ~ C O ToS
where ui and b, are given by (3.1.71) and (3.1.72). The solution of (6.2.151) and (6.2.152) is
+
where
xo = A(Tl) cos &To B(T,) sin &To
(6.2.155)
yo = uB(Tl) cos *To uA(T,) sin *To
(6.2.156)

+ &)l = b, + & = 4(7  433)
a = (ao
(6.2.157)
The zerothorder solution determines the righthand sides of (6.2.153) and (6.2.154). Thus they become
+ Qlsin *To+ NST Dozyl + 2Dg1  aoyl = P, cos &To+ Q2sin &To+ NST DO2q 2D0yl  bgl = P, cos gTo
(6.2.158) (6.2.1 59)
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
6.2. where
261
P, = (20:  1)B’ + (b,  $b,)A
(6.2.160)
Pz = (a  2)A’  a(b,
(6.2.161)
Qi
(6.2.162)
+ &z,)B =  ( 2 ~ 1)A’ + (b, + 4bo)B Qz = (a  2)B‘ + a(b,  Bao)A
(6.2.163)
To determine a first approximation, we need not solve for x, and y1 but only insure that xl/zoand yl/yoare bounded for all To. This is the reason why we spelled out the terms that produce secular terms. To eliminate the secular terms, we can find the particular secular solution and then determine the condition for its vanishing. The resulting particular solution is of the form x = 0, (6.2.164) y = R1cos &To S, sin 4To or (6.2.165) x = R, cos &To S2sin *To y = 0,
+
+
Thus we can determine the conditions for the elimination of the secular terms by assuming a particular solution of the form (6.2.164) or (6.2.165). The results of using either form are the same. Substituting (6.2.164) into (6.2.158) and (6.2.159) and equating the coefficients of cos (T0/2) and sin (T0/2) on both sides, we obtain
Ri = Q i ,
s, = P1
(6.2.166)
R, = aP,,
S , = aQ2
(6.2.167)
Elimination of R, and S, from (6.2.166) and (6.2.167) leads to the required conditions; that is PI = aQz, Ql = UP, (6.2.168) Substituting for P,,P,,Q,, and Q, from (6.2.160) through (6.2.163) into (6.2.168) leads to the following two equations for A and B (1  4a (1
+ a2)A’ + [(l  a2)bl + $(bo aauo)]B= 0
 4a + a2)B’ . [(l  a2)b,  $(b,  a2u,)]A = 0
(6.2.169) (6.2.170)
To solve these equations, we let A = a exp y,T,
B = b exp y,T,
(6.2.17 1)
and obtain
+ a2)yla+ [(l  a2)bl + +(bo a2ao)]b= 0  [(l  a2)b,  i(bo  a2uo)]u+ (1  4a + aX2)byl =0 (1  4a
(6.2.172) (6.2.173)
262
THE METHOD OF MULTIPLE SCALES
These equations are the same as (3.1.105) and (3.1.106) obtained using Whittaker's method. Hence y l and b/a are given by (3.1.107) and (3.1.108), while x and y are given by (3.1.109). This expansion was continued to second order by Alfriend and Rand (1969). 6.2.7. A SWINGING SPRING We consider next the nonlinear swinging spring discussed in Sections 5.5.3 and 5.7.5 and described by the Lagrangian (5.5.54). The equations of motion corresponding to (5.5.54) are 2
k ++ g(i  C O e)~  ( I + x)P = o m
e+
sine+l + X
2
I+%
ke = 0
(6.2.174) (6.2.175)
We seek an asymptotic solution of these equations for small but finite z and 8 of the form
+ +e ( t ) = .el(r0, rl)+ w T 0 , + .  .
x ( r ) = E X ~ ( Tl) T ~ , E ~ ~ Tl) ~ ( T .~. ,
(6.2.176) (6.2.177)
where T, = E n t and E is of the order of the amplitudes of oscillation. Substituting (6.2.176) and (6.2.177) into (6.2.174) and (6.2.175) and equating coefficients of like powers of E, we obtain Order
e
D:x,
+ w;x1
= 0,
+ wz28, = 0,
Do201 Order
k m
o12 =
(6.2.178)
oz2=
(6.2.179)
1
€2
D : ~ , + w:x2 = 20,01x1
o;e,
 ige12+ i(ooe,)2
(6.2.180)
w2 2 + wz2e2=  2 ~ , 0 ~ e +, 2 x,81   ( ~ o z l ) ( ~ , e l (6.2.181) ) 1 1
The solution of the firstorder equations is
+ ,4(Tl)eimiTn = B ( T l ) e i m ~+ T ~B(T,)e'+'o
z1= A(Tl)eimiTn
(6.2.182)
8,
(6.2.183)
6.2.
APPLICATIONS OF THE DERNATIVEEXPANSION METHOD
263
Then (6.2.180) and (6.2.181) become
+ +gBB + CC
Do2x,+ q 2 x 2= 2iqD1Aeim1To  +g Bze2i*,To
(6.2.184)
I f A and B are constants, the particular solutions of (6.2.184) and (6.2.185)
are
8, =

wz(w2 h ( W 1

0z(w2
4 ( 0 1
+ 201)
+ 2%)
ABei(Ol+O2)T0
 2w1) ABei(ol02)To + cc  2%)
(6.2.187)
which tend to co as w1 20,. Consequently, the expansions (6.2.176) and (6.2.177) break down when o1R!20,. To obtain an expansion valid when w1 2wz,we let f
w,
 2w, = fU,
U
(6.2.188)
= O(1)
and let A and B be functions of TI rather than being constants. Moreover, using (6.2.188) we express exp (2iozTo)and exp [i(wl  03T0]in (6.2.184) and (6.2.185) as exp (2iw2T0)= exp (iwlTo iaT,) exp [i(wl  w2)To]= exp (iwzTo iaTl) to obtain
+
Dtx2
+ wI2x2= (2iw,D,A + +gB2eiuT1)eiOlT0 + CC + N S T
028,
+ wz28, =  [2iwzDlB wz(w*
 2w1)
I
1
~ j ~ i u T eimo,To , +
Eliminating secular terms, we have
(6.2.189)
cc + NST (6.2.190)
2iw1D,A = $gB2 exp (iaT,) 2iwzDlB = wz(w2  20,) A B exp (iaT,) I
(6.2.191)
264 THE METHOD OF
MULTIPLE SCALES
Letting A = (1/2)ia1 exp (iolB1) and B = (1/2)ia2 exp (io&, with real a, and pi,and separating real and imaginary parts, we obtain (6.2.192) Liz=
3EWz
41
(6.2.193)
a1a2cos y
(6.2.194)
A =  &3P a, sin y where
(6.2.195)
Y = 4 3 1  2wzBz + (01  2w2P
(6.2.196) If we let aI2 = co,a,*/o,k and a22= 2a2/rngl, (6.2.192) through (6.2.196) go over into (5.5.76) through (5.5.80) which were obtained using the method of averaging in conjunction with canonical variables. 6.2.8. A MODEL FOR WEAK NONLINEAR INSTABILITY We next consider the model problem us
 u,,  u = u3
16.2.197) cos kx, u ~ ( z0) , =0 for weak nonlinear instability of standing waves, which was discussed in Sections 2.1.2, 3.4.2, and 3.5.1. Away from k = 1, a uniformly valid solution for standing waves is (Section 3.4.2) U(X, 0) = E
where
u = c cos ot cos kx
+ O(E~)
9Ez
I+
32(k2  1)
(6.2.198)
...
It is clear that this expansion breaks down when k  1 = O ( 2 ) . To determine an expansion valid near k = 1, we introduce the new variable 5' = kx in (6.2.197) to obtain Utt
 k 2 U c 5  u = u3
u ( l , 0) = E cos E,
Moreover, we let k =1
+ C2k,
ut(t, 0) = 0
with k z = 0(1)
(6.2.199)
(6.2.200)
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
265
and assume that
~ ( 51 ,; E )
( 5To, , T l , T,;E )
=~
=
+ e2u2+ e3u3 + . . .
(6.2.201)
E U ~
where T , = E n t . Substituting (6.2.200) and (6.2.201) into (6.2.199) and equating coefficients of equal powers of B , we obtain Order
E
a2ul   azUl aT:
u1 = cos 5,
Order
B,
u2
(6.2.202)
8% 0
at
a TO
azU3 at2
u3
at2
= 0,
Order e3 aT:
=0
aZu2   a z U z  u2 = 2aT:
aZu3
u1
at2
aZu,
aToaTl
8%   au1 at TO T l
a
T,=O
a
T, = 0
azUl    2
= u13 + 2k2
at2
= 0,
aT:
(6.2.203)
a2u2
aToaTl
 2
a%   aul  au2 at T,, = 0 a TO
aT,
aZu
a~~
aTo (6.2.204)
aTl
The solution of the firstorder problem is u1 = u(T1,T2) cos
E,
u(0,O) = 1
(6.2.205)
Then (6.2.203) becomes
The solution of (6.2.206) will contain a term proportional to To making u2/ul unbounded as To,co unless aa/aTl = 0 at TI = Tz= 0. Then U, =
b(T,, T2)cos E ,
b(0,O) = 0
(6.2.207)
With the first and secondorder solutions known, the equation for ug
266
THE METHOD OF MULTIPLE SCALES
becomes
aZu3 ar,2
azU3
2a3  2k2a   cos 5
aq2
ap
+ &a3cos 3[
(6.2.208)
Secular terms will be eliminated if
a% + (2k2  $a2)a = 0 aT,2
(6.2.209)
The'initial conditions for a were obtained above as
aa and  = 0 at T,=O (6.2.210) a TI To determine b(T,, T,) and the dependence of a on T,, we need to carry out the expansion to higher order. If we limit the analysis to O ( 4 , we can regard a as a function of TI within an error of O(E2f). A first integral for (6.2.209) and (6.2.210) is a = l
Since a(T,) is real, the righthand side of (6.2.211) must be positive, hence p. Since a(0) = 1, a2 increases without bound if j3 < 1 and oscillates between 0 and 1 if j3 > 1. Therefore /? = 1 or k, = 3/8 separates the stable from the unstable regions. Hence the condition of neutral stability is
a2 must be outside the interval whose ends are 1 and
k =1
+ $e2
(6.2.212)
in agreement with (3.5.6). The solution for a is a Jacobian elliptic function.
6.2.9. A MODEL FOR WAVEWAVE INTERACTION We again consider Bretherton's (1964) model equation
+
++
A t 42222 + 4Zz =C V (6.2.213) which was treated in Sections 5.8.1 and 5.8.2 using the variational approach. The linear problem admits the uniform traveling wave solution
4 = a cos ( k z  wt
+ j3)
(6.2.214)
where a, k, w , and j3 are constants and w and k satisfy the dispersion relationship w2 = k4  k2 1 (6.2.215)
+
Harmonic resonance may occur whenever ( w , k) and (nw,nk) for some integer n 2 2 satisfy (6.2.215). This occurs at all k2 = 1/n for n 2 2. At
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
267
these wave numbers the fundamental and its nth harmonic have the same phase speed. Since the nonlinearity is cubic in our equation, the fundamental corresponding to k2 = 1/3 interacts to O(E)only with its third harmonic (k2 = 3). If the nonlinearity is ~ r j 5for ~ some integer m , then the fundamental ( k 2 = l/m) interacts to O(E)with its mth harmonic (k2= m).If we consider interactions to orders higher than E, harmonic resonances other than the third can occur even for a cubic nonlinearity. To determine a firstorder expansion valid near k2 = 1/3, we let where
4 = +o(To, Ti,xo,xi) + ~+i(To,Ti, xo,x i ) +
* * *
(6.2.21 6)
X,,= cnx
T, = c n t ,
Substituting this expansion into (6.2.213) and equating coefficients of like powers of c , we obtain
a v o+ a'+, + a v o + 4, = 0
L(4,) =2aT, L(+J
ax: ax, a",  4 8'4,  2  a240 = 40" 2a ~ a, ~ , ax: ax, ax, ax, 2
(6.2.217) (6.2.218)
The solution of (6.2.217) is taken to be
+, = A,(Tl, Xl)eiel + A,(Tl, Xl)eies + CC
where
(6.2.219)
8, = k,X,  wnTo = k:
 k:
w 3 M 3w1,
k,
0 , '
+,
+1
M
(6.2.220)
3k1
Note that is assumed to contain the two interacting harmonics. Had we assumed it to contain exp (&), we would have found it t o be invalid (see Sections 5.8.2 and 6.4.8). Substituting for 4, into (6.2.218), we obtain
+ 3(2A1X1+ A3X3)A,eie3+ A13e3ie1+ A,3e3ies +
3,4 ZA ei(ea+zel)
+
3A 1A 3 zei(ze3+e1) + 3A1A 3 zei(ze381)
1
3
+
3 ~ 2~ 1
where w , = dw,/dk: group velocity.
ei(B~2h) 3
+ cc
(6.2.221)
268
THE METHOD OF MULTIPLE SCALES
Because of the interaction between the two modes, terms that produce secular terms other than the usual exp (id,) occur in (6.2.221). To recognize these terms we consider the perfect resonance case in which d3 = 30, so that exp (id,) and exp (3i0,) produce secular terms. We immediately see that exp ( 3 i Q and exp [i(d,  20,)] produce secular terms. For the nearresonance case, we indicate their secular behavior by expressing them in terms of exp (id,) and exp (id3). To do this we observe that
e3  30, = r = (k,  3k,)x0  (W3  301)~0 Although X o and To are 0(1), I' becomes slowly varying as k3 w3 + 30,, hence we express this slow variation by rewriting I' as
f
3kl and
(6.2.222) With this function we express exp (3i0,) and exp [i(0,  20,)] as exp (3i0,) = exp [i(0,  r)],
exp [i(0,
 20,)l
= exp [i(0,
+ r)].
Eliminating the terms that produce secular terms on the righthand side of (6.2.221), we have
(6.2.224) Letting A, = (1/2)u, exp (i/?,) with real a, and ?/, in (6.2.223) and (6.2.224) and separating real and imaginary parts, we have (6.2.225)
at% aT1
3 + olax, = 801 (a,
2
+ 2a3 + alu3cos 6) 2
aa3 = a,3sinS 1 + wjaT, ax, 803
(6.2.227)
& 3
at% + o3fa/?, 1 2  = ( 6 ~ 1+ 3a: where
a
ax,
gW3
6 =r
+ p3  38,
(6.2.226)
+ a 1 3 u ~cos ' S)
(6.2.228) (6.2.229)
6.2.
APPLICATIONS OF THE DERIVATIVEEXPANSION METHOD
269
Equations (6.2.225) through (6.2.228) are in full agreement with (5.8.24) through (5.8.27) obtained using the variational approach. 6.2.10. LIMITATIONS OF THE DERIVATIVEEXPANSION METHOD This method applies to wavetype problems only. It does not apply to unstable cases except when the instability is weak, such as the nonlinear stability problem discussed in Section 6.2.8. For k > 1, u is bounded and the expansion (6.2.198) is valid for times as large as e2 if k is away from 1. This expansion is valid only for small times for k < 1 and away from 1. Near k = 1 , the instability is weak and a valid solution is given by (6.2.211) for times as large as el. In the case of hyperbolic equations, this method applies to dispersive waves only when the initial conditions can be represented by the superposition of a finite number of sinusoidal functions. For a linearly nondispersive wave problem such as (6.2.230) u ( 5 , 0) =f(z),
au ( 5 , O ) at
=0
(6.2.231)
this method does not provide a solution even iff(z) is a sinusoidal function such as cos 5. To see this we let u = uo(5, To, Ti)
where
+
To = t ,
E U ~ ( Z ,To, Z'1)
+
* * *
(6.2.232)
Tl = et
Substituting (6.2.232) into (6.2.230) and (6.2.231) and equating like powers of E, we obtain Order
EO
(6.2.233) uo(z, 0,O)= cos
Order
5,
E
au0 aT0
(5,
0,O) = 0
(6.2.234) (6.2.235)
u(5, 0,O) = 0,
au, (x,0,O)=
aT0
 (x,0,O) aT1
(6.2.236)
270 THE METHOD OF MULTIPLE SCALES
The solution of (6.2.233) and (6.2.234) is uo = ,4(Tl)ei(zTo)+ A( Tl)ei(ZTO)
where
(6.2.237)
A(0) = 3 Substituting this zerothorder solution into (6.2.235) gives
+
2AA + ,42e2”’2TO’
+ pe2i(zTo)
(6.2.238) The righthand side of this equation contains terms that produce secular To)]in addition to terms. They are the terms proportional to exp [f2i(z exp [fi(z  To)].In order that ul/uobe bounded for all To,all these terms must be eliminated. However, there is no way in which this can be done. In the previous examples such terms were proportional to exp [fi(z  To)], hence A was chosen to eliminate them. In this case, if a nontrivial solution is desired, A can be chosen in such a way as to eliminate the terms proportional to exp [fi(z  To)].The resulting expansion contains secular terms, hence it is not valid for large times. Expansions valid for large times and general initial conditions for nondispersive waves were obtained in Sections 3.2.4 and 3.2.5 using the method of strained coordinates.

6.3. The TwoVariable Expansion Procedure
6.3.1. THE DUFFING EQUATION Let us consider again the equation
+ €U3 = 0
+4
+
d2u dt2
We assume that where
 + w,”u u = uo(5, 17)
5 = €f,
5 , 17)
17 = (1
+
E2U2(E,
+
(6.3.1)
17)
+
* * *
+ .**)t
(6.3.2)
(6.3.3) Substituting (6.3.2) and (6.3.3) into (6.3.1) and equating like powers of E, we obtain t2W2
€304
(6.3.4) (6.3.5)
6.3.
THE TWOVARIABLE EXPANSION PROCEDURE
The general solution of (6.3.4) is ~0
= A d E ) cos woq
Then (6.3.5) becomes azUl
y a7
+ w;u1
+ B,(E) sin w,q
271
(6.3.7)
+ :(A: + A,B,2)] cos woq + [ 2 w , ~ ; %(B: + A ~ B , )sin ] woq
= [2w,BA

MOB:) cos 30,q Secular terms will be eliminated if
+ $(B:
 ~4,213,)sin 30,q
+ $(A: + A , B t ) = 0 2w,A;  $(B: + A:Bo) = 0 2woBA
(6.3.8)
(6.3.9) (6.3.10)
Adding B, times (6.3.9) to A , times (6.3.10) gives
+
+
d (A,' B:) = 0 or A; B: = u 2 = a constant dE Using (6.3.11), we can express (6.3.9) and (6.3.10) in the form
B; + w I A O
where
Ah  O J ~ B=, 0
= 0,
(6.3.11)
(6.3.12)
3 8WO
w1 =  a'.
Hence
A , = a cos (wlE 4+),
B, = a sin (wlE
+ $)
(6.3.13)
where 4 is a constant. With the secular terms eliminated, the solution of u1 becomes
 I
(B:  ~ A Z B , )sin 3w0q (6.3.14)
320, Substituting for A , and Bo into u, and ul, we get u,, =
cos e
u1 = A,(@ cos 8
where
(6.3.15) u3 + B,(E) sin 0 + 320: 
0 = woq
cOS
+ olt + 4
313
(6.3.16) (6.3.17)
272
THE METHOD OF MULTIPLE SCALES
Replacing uo and u1 in (6.3.6) by their expressions from (6.3.15) through (6.3.17), we obtain
aZu2+ o;u2 
=
+ 3a')
cos 8  (2w0A; 2w0w1B1
128~:

(2w0B1' +
pu2A,
 200w1Al  awl2  2uo2wo2
+ $a2Bl)sin 8 + NST
(6.3.18)
Secular terms are eliminated if A, = B, = 0 and
or 02=
15
a4
(6.3.19)
2560;
Therefore, to second approximation
u = u cos (of where
d dt
=  (w0q
or
€a3 + +) + 32w0 cos 3(wt + (6) + O(E') 2
d + colt) = [(ao + dt
+E
~
+ .w. . ) t ] ~
(6.3.20)
~
(6.3.21)
This expansion is in full agreement with those obtained in Section 3.1.1 by using the LindstedtPoincarC methods, in Section 5.4.1 by using methods of averaging, and in Section 6.2.1 by using the derivativeexpansion method. 6.3.2. THE VAN DER POL OSCILLATOR The second example we consider is the van der Pol oscillator
d2u du  + 11 = E ( l  u2)(6.3.22) dt2 dt discussed in Sections 5.4.2, 5.7.4, and 6.2.2. We assume that u possesses the following uniformly valid expansion (Cole and Kevorkian, 1963 ;Kevorkian, 1966a) u = Uo(5,V)
+ % ( 5 , 7) + c2u,(5, rl) +
'
..
(6.3.23)
where 5 and '7 are defined in (6.3.3). Substituting (6.3.3) and (6.3.23) into
~
6.3.
THE TWOVARIABLE EXPANSION PROCEDURE
(6.3.22), and equating like powers of
E,
a2u,
we obtain
+u,=o
(6.3.24)
ar12
 + u1=  2 2
+ (1  u i ) au,
aZu, + u 2 = 2
aZu,  aZu,   20,
a2u
ar2
at aq
a7
star
aZu
273
(6.3.25)
all
at2
ar12
+ (1  u:)( aU, + 2 au , 2u,u, au, all
The general solution of (6.3.24) is uo = A o ( t ) cos q
at
arl
+ B,(5) sin q
(6.3.26)
(6.3.27)
Hence (6.3.25) becomes
aZu + u 1 = [2BA
f (I  A,2
+
4
+ [ 2 4 ,  (1  A'
7 B . I cos q B ' ) ~ , ] sinq
+ $(A03  3A0B,2)sin 377 + #(~,3 3A02B,,) cos 317
(6.2.28)
Elimination of secular terms necessitates satisfying the following conditions (6.3.29)
4
(6.3.30) Subtracting B, times (6.3.29) from A, times (6.3.30) gives p'
where
p is
 p(l
 gp) = 0
(6.3.31)
the square of the amplitude of the zerothorder solution; that is
+ Bo2
p = u2 = A,'
(6.3.3 2)
By separation of variables we integrate (6.3.31) to obtain
(5 4
a' =
1
+
 1)c'
(6.3.3 3)
274 THE METHOD OF MULTIPLE
SCALES
where a, is the initial amplitude. Expressing A, and B, in terms of the phase 4 and the amplitude a, we obtain A,, = a cos
+,
B, = a sin 4
(6.3.34)
Substituting into either (6.3.29) o r (6.3.30) and using (6.3.31), we find that
+' = 0
+ = $,
or
=a
Hence u, can be expressed as u, = a c o s
+
(q
(6.3.3 5)
constant
(6.3.36)
$0)
With (6.3.29) and (6.3.30) satisfied, the solution of (6.3.28) is
u1 = ME) cos ( q
+ do) + B1(t) sin ( q + 4,)  a3 sin 3(q + 4,) 32
(6.3.37)
Substituting for u, and u1 into (6.3.26) gives aZU + uz =
[2B;
1
+ (1  &a2)B, a" + 2wza + (1  &?)a' + us 128
(6.3.38)
To eliminate secular terms we require that 2B;
 (1  &a2)Bl= 2w,a  a" + (1
a5  $a2)ar+ 128
2A;  (1  &?)A, = 0
(6.3.39) (6.3.40)
Using (6.3.31) and (6.3.32), we express the above two equations as
A'
('a"' 1
(6.3.42)
lA,=O
The solutions of these equations are B, = a(oz
+ &)l  b,a + &a In a  &a3
(6.3.43) (6.3.44)
A, = ~ , a ~ e  ~
where a, and bl are constants. Since as t unbounded as E 03 unless
+
+
oz= 2. 16
co, 6
+
co and a
f
2, ul/uo is (6.3.45)
6.3.
THE TWOVARIABLE EXPANSION PROCEDURE
275
Therefore, to second approximation u = (a
+ ~ a ~ a ~ e cos  ' ~[(I )  &e2)t + +,I  J ( ~ a 3 &a In a + bla) sin [(I  +z))t + +,I \ a3 +sin 3[(1 32
where
a=
h 2 ) t
+
+0]
I+
2
,
O(E') (6.3.46) (6.3.47)
This expansion is in full agreement with (6.2.38) obtained using the derivativeexpansion method if we identify a, with (5, 4 , 3 a l .
+
6.3.3. THE STABILITY OF THE TRIANGULAR POINTS IN THE ELLIPTIC RESTRICTED PROBLEM OF THREE BODIES Let us consider again the parametric resonance problem treated in Section 6.2.6 using the derivativeexpansion method. The problem is described mathematically by (3.1.63) through (3.1.65). To determine a uniformly valid expansion near the transition curves using the twovariable expansion procedure, we need to use different time scales from those given by (6.3.3). The appropriate time scales are
E
= (e
+ wze2+
. .)t,
(6.3.48)
7 =r We assume that z and y possess expansions of the form 5
= zo(5,q)
*
+ ez,(E, q ) + e252(E,11) +  . .
Y = YO(5, 7) f ey1(5,7)
+ e2y2(E, 77) +
*
.
*
(6.3.49) (6.3.50)
The algebraic details of the solution are not presented here. The details for the firstorder solution are the same as in Section 6.2.6, with E = Tl and 7 = To. The reader is referred to Alfriend and Rand (1969) for details of the secondorder solution. Their results are in full agreement with those obtained in Section 3.1.5 using Whittaker's method. 6.3.4. LIMlTATlONS OF THIS TECHNIQUE The above examples demonstrate that by choosing the two variables appropriately the results of the twovariable expansion procedure agree with those of the derivativeexpansion method. In some cases more than two variables are needed to obtain uniformly valid expansions such as in the case
276 THE
METHOD OF MULTIPLE SCALES
of satellite motion around the smaller primary in the restricted problem of three bodies (Eckstein, Shi, and Kevorkian, 1966a) and the motion of an artificial satellite having a period commensurable with the rotational period of its primary (Shi and Eckstein, 1968). In the case of hyperbolic equations, this technique as well as the derivativeexpansion method applies only to dispersive wave problems and it does not provide solutions for nondispersive wave problems such as that discussed in Section 6.2.10. 6.4
Generalized Method
6.4.1. A SECONDORDER EQUATION WITH VARIABLE COEFFICIENTS Let us consider the following special secondorder problem (Nayfeh, 1964, 1965b) d2Y dy (6.4.1) lz(22 I1)2y = 0 dx2 dx
+
+
Y(0) = YO) = B (6.4.2) discussed in Sections 4.1.3 and 4.2.2 using the methods of matched and composite asymptotic expansions. As discussed in Section 4.1.3, the straightforward expansion possesses a nonuniformity at x = 0. The size of the region of nonuniformity is x = O(F). To treat this problem using the method of matched asymptotic expansions, an inner expansion valid when x = O(E) was introduced using the inner variable r] = x/e. This inner expansion was matched to the outer expansion and then a composite expansion was formed to give a uniformly valid expansion. To obtain a uniformly valid expansion using the generalized version of the method of multiple scales, we introduce the scales E = X
(6.4.3)
where g, is determined from the analysis. We require that go(0) = g,(O) = 0 so that go(z)+ z as x + 0, hence 17 approaches the inner variable X/Q. Then the derivatives with respect to x are transformed according to (6.4.5)
6.4.
GENERALIZED METHOD
277
These variables transform (6.4.1) into
.)
a2y
+
+ g; + €g; + . .
€(&
:(
+2E  + g g ; + € g ; + . .
4 (2E
+ I)[ (& + gi + E
+ 2y = 0
(6.4.7)
where primes denote differentiation with respect to the argument. Note that we expressed the z variables appearing in (6.4.1) in terms of 5; namely, we expressed 22 1 as 2E 1. Moreover, we expressed gn and its derivatives in terms of 5. Now we assume that there exists a uniformly valid asymptotic representation of the solution of (6.4.7) in the form
+
+
(6.4.8)
where
ynY7l1
0 in [0, I]. The case in which ‘ ( 2 ) vanishes in the interior of [0, 13 is called a turning point problem. Turning point problems are discussed briefly in Section 6.4.4 and in detail in Sections 7.3.1 through 7.3.9. If c = 0, this example will be the same as that treated in Section 4.1.3 using the method of matched asymptotic expansions. Since a(.) > 0, the nonuniformity is at 1: = 0. In Section 4.1.3, we introduced an inner variable X / E to determine an expansion valid in the region x = O(E), which was matched to an outer expansion. To determine a uniformly valid first approximation using the method of multiple scales, we assume that there exists an asymptotic representation for y of the form
Y = Yo(& rl)
where
E = 5,
+ EYd6, r ) + 
1 with 17 = d. E
g(x)+ x
*
.
as s + O
(6.4.36) (6.4.37)
6.4.
281 Substituting (6.4.36) and (6.4.37) into (6.4.34) and equating the coefficients of co and to zero, we obtain GENERALIZED METHOD
(6.4.38)
(6.4.39)
where we have expressed a(z), b(z), c(z), and g(z) in terms of E . Since g $ 0, the general solution of (6.4.38) is where
yo = A(5) y=
+ B(S)e+l)V
45)
(6.4.40) (6.4.41)
g’
As discussed in the previous section, y must be a constant; otherwise the derivatives with respect to 6 would produce terms proportional to y ‘ q in (6.4.39), hence make ylIyo unbounded as q + 00. For a uniformly valid expansion, we require that y = 1 without loss of generality. Hence g =[u(t)dt
since g(z)+ z as 20
(6.4.42)
Substituting for yo into (6.4.39) gives (a72
+ 2 gf2=  [ a ~ ’ + b~  C] + [g’B’ + (g”  l ~ ) B ] e  (6.4.43) ~
In order that yl/y, be bounded for all q , we require that aA’
g’B’
+ bA = c
+ (g”  b)B = 0
(6.4.44) (6.4.45)
The solutions of these equations are
(6.4.47)
where a, and 6, are constants of integration.
282
THE METHOD OF MULTIPLE SCALES
To first approximation, y is given by
bo
e S ~ [ b ( t ) / a ( t ) l d t e  ~  l S o Z a ( t ) d+ t
4%)
O(r)
(6.4.48)
The limits of the integrals in (6.4.46)and (6.4.47)were chosen so that a, and b, could be expressed in a simple manner in terms of the boundary conditions (6.4.35).Thus a, = 8 and Expansion (6.4.48)is a composite expansion which agrees with the inner and outer expansions obtained in Section 4.1.3 in the inner and outer regions, respectively. If we specialize (6.4.48)for the case a(.) = 1
+ 22,
b(x) = 2,
c(.)
=0
(6.4.50)
discussed in the previous section, we obtain
(6.4.51)
where we have used a, = 8 and (6.4.49).This expansion is in full agreement with the first term in the expansion obtained in the previous section.
6.4.3. A LINEAR OSCILLATOR WITH A SLOWLY
VARYING RESTORING FORCE The two cases discussed above can be treated by using either the method of multiple scales or the method of matched asymptotic expansions. Let us consider next an example that cannot be treated by the latter method; namely y” + b(€s)y = 0 (6.4.52) where b ( a ) # 0 and E is a small parameter. To obtain an expansion uniformly valid for large x, we assume that there exists an asymptotic representation for y of the form Y = >[(OY q) EYl(6, q) * * * (6.4.53) where ,$=Ex, q =  +g ( t ) ... (6.4.54)
+
+
E
This form of q was chosen in order that the frequency of oscillation w = dq/dx = g‘(6) = O(1). Substituting (6.4.53) and (6.4.54) into (6.4.52)and
6.4. GENERALIZED METHOD 283
equating the coefficients of
eo
and e to zero, we obtain
The general solution of (6.4.55) is (6.4.57)
where
(6.4.58) As argued in the previous two sections, we set y = 1 to obtain an expansion in which yl/yois bounded for all 7. Hence g
=J6m dt
Substituting for yo into (6.4.56), and remembering that y
a2y1 + y1 g" (av' )
= i(g"A
(6.4.59) =
1 , we obtain
+ 2g'A')e'" + i(g"B + 2g'B')e'q
(6.4.60)
In order that yl/yo be bounded for all 17, we require the vanishing of the coefficients of exp (&ill) on the righthand side of (6.4.60); that is
+ 2g'A' = 0 g"B + 2g'B' = 0 g"A
(6.4.61) (6.4.62)
The solutions of these equations are A = = 6, 0 B= 6 0 J g' J g' where E0 and 6, are constants of integration. If b(Ez) > 0, y is given by
where a, and b, are constants. If b(ez) < 0, y is given by
(6.4.63)
(6.4.64)
(6.4.65)
284 THE METHOD OF MULTIPLE SCALES
Expansions (6.4.64) and (6.4.65) are called the WKB approximation to the solution of (6.4.52) (see Section 7.1.3). These expansions are clearly not valid near a point where b(cx) vanishes. In fact, they tend to infinity as x approaches a zero of b(~x).The zeros of ~ ( E z ) are called turning points and are discussed in detail in Sections 7.3.1 through 7.3.9. An example of a turning point problem is treated in the next section using the method of multiple scales. If we change variables in (6.4.52) from x to 6, we find that (6.4.66)
which is a problem containing a large parameter 1.Thus the aboveobtained approximation is applicable to this problem as well. 6.4.4. AN EXAMPLE WITH A TURNING POINT Let us consider the problem y"
+ P ( 1  x)j(x)y = 0
(6.4.67)
where 1 is a large positive number andf(x) is regular and positive. The WKB approximation tends to infinity as x + 1 as can be seen from (6.4.64) and (6.4.65) if we let b ( d ) = (1  x ) f ( x ) and E = P. To determine an expansion valid everywhere using the method of multiple scales, we first determine the size of the nonuniformity. Thus we let 5 = (1  x)A" with v > 0 in (6.4.67) and obtain (6.4.68)
As A + co, the following different limits exist depending on the value of
y=o
d2Y + f ( l ) t y dt2
if v < g }
= 0 if
Y
(6.4.69)
v =Q
The last limit is the appropriate one because its solution has an exponential behavior for 5 < 0 (i.e., x > 1) and an oscillatory behavior for 5 > 0 (i.e., x < 1). Thus it can be used to connect (6.4.64) and (6.4.65) as the turning point is crossed. Therefore we assume that there exists an asymptotic representation o f the
6.4.
GENERALIZED METHOD
285
solution of (6.4.67) of the form (Cochran, 1962; Nayfeh, 1964, 1965b; Fowkes, 1968, Part I)
Y
where
= yo((,
f = 2,
with
7)
+ A  2 / 3 ~ 1 ( E , 7) + . . . + ...
(6.4.7 1)
h(5) > 0
(6.4.72)
7 =Pg(5)
g(5) = (1  2 ) / ? ( 5 ) ,
(6.4.70)
The functions of the independent variable z that appear in (6.4.67) are expressed in terms of 6, except 1  5 is replaced by ~ i l  ~ / ~ / hbecause (z) it reflects the nonuniformity. Therefore (6.4.67) becomes
Substituting (6.4.70) into (6.4.73) and equating the coefficients of 1213 to zero, we obtain
A41a
and
(6.4.74)
The general solution of (6.4.74) is
+
yo = A ( E ) ~ ~ / ~ ~ ~ / ~ [ Y (~E()~~) ~~ /1~i 2I ~ J  1 / 3 [ ~ ( s )(6.4.76) ~3~2~
where Jill3 are Bessel’s functions of order f 1/3 and
(6.4.77)
In order that yl/yo be bounded for all q , y = 1 as discussed in Sections 6.4.1 and 6.4.2. Therefore g’(5)WE) =
3rfcnl”2
(6.4.78)
where the negative sign in (6.4.77) was taken so that h ( s ) > 0. Multiplying both sides of (6.4.78) by (1  E ) l I 2 , we obtain Since g(1) = 0
g”2g’ = $[(1
 E )f ( E ) ] 1 / 2
g3l2= S,’[(I
 t ) f ( t ) ] 1 / 2d t
(6.4.79)
286
THE METHOD OF MULTIPLE SCALES
With yo known and y = 1, (6.4.75) becomes
a [(2g‘A’ = a7
+ g”A)Tf2J1/3(lj13/2)+ (2g’B‘ + g”B)$/2J1/3(~3’2)](6.4.80)
In order that yl/yo be bounded for all 7,the righthand side of (6.4.80) must vanish; that is
I
+ g“A = 0 2g’B’ + g”B = 0
2g‘A‘ Hence
(6.4.81)
(6.4.82)
where a and b are constants of integration. Thus, to first approximation
where a, and b, are constants. As x t 1
where Z, and 6, are constants. Since J,(t) = tV
+ o(tv)
as t + 0
we conclude that (6.4.83b) is bounded as x t 1. 6.4.5. THE DUFFING EQUATION WITH SLOWLY VARYING COEFFICIENTS Let us next consider the equation d2u dt2

where
+ a(E)u + B(E)u3 = 0 6 = Ef,
E
2/9, or if a(5) < 0 and < 0, (6.4.89) has no periodic solutions. (2) a(5) > 0, B(5) > 0. In this case p > 0 and (6.4.94) shows that y < 0. Hence the curve that determines y lies in the fourth quadrant. The solution for y exists for 0 < p < 03. (3) a(6) < 0, B(E) > 0. In this case p < 0 and (6.4.94) shows that y < 0. Hence the curve that determines y lies in the third quadrant. The solution for y exists if a < p < 419. Since the elliptic functions and integrals are usually tabulated for real Y such that 0 < Y < 1, alternative asymptotic solutions in terms of cn(q, Y ) and dn(ljr, Y ) are preferred in cases (2) and (3).
@(E)
6.4.6. REENTRY DYNAMICS The motion of a reentry rolling body with variable spin under the influence of nonlinear aerodynamic forces and slight center of gravity offset and aerodynamic asymmetries is governed by (Nayfeh and Saric, 1972a)
+ ic2x1E + ixz 151' 6
+=P p = e2v0
+ E Y ~ G+
E ~ Y ~ P ,
(6.4.105) (6.4.106)
G = Imaginary {Eea+}
(6.4.107)
+
where 6 = fl icc, [ E l is the sine of the total angle of attack, p is the roll rate, EKis the amplitude of the excitation due to aerodynamic asymmetry, and E is a small but finite quantity of the order of the sine of the initial total angle of attack. Here wo, K, y , pi,xi, and vi are slowly varying functions of time, and I and I, are constants. In the absence of damping and nonlinear terms (i.e., y = pi = xi = 0), the solution of (6.4.105) for constant p, K, and wo is
5 =~
~+ Azeimort ~
+
i
(01
~
EK
 P)(P
t

e'(H+90)
(6.4.108)
02)
where A, and A, are complex constants and w1.2
=pzz f
21
J(+F
+ 002
(6.4.109)
The frequencies w1and w, are called the nutation and precession frequencies.
292
THE METHOD OF MULTIPLE SCALES
For statically stable bodies (ie., coo2> 0) and positivep, w1 is positive while negative. Two cases arise depending on whetherp is near w1 or not. The first case is called roll resonance, and the forced response tends to infinity as p + wl.Before p approaches wl, the damping as well as the nonlinear aerodynamic forces significantly modify the response. The roll resonance case is discussed in this section for the case K = C2k,and we refer the reader to Nayfeh and Saric (1972a) for the nonresonant case. To determine an approximate solution to (6.4.105) through (6.4.107) when p M o1using the generalized version of the method of multiple scales, we make use of the fact that actual fight test data and sixdegreeoffreedom numerical calculations show that there are at least four time scales: a slow time scale T2 = cat characterizing K, coo, y , vi, xi, and pi;and three fast scales characterizing the nutation, precession, and forced components of the angle of attack. Thus we assume expansions of the form o2is
where wl(Tz) is the nutation frequency and w2(T2) is the precession frequency. In terms of these variables, the time derivatives are transformed according to da a a o1+w2+++2
dt
%I
aqz
a+
a aT2
(6.4.113)
(6.4.1 14)
Substituting (6.4.1 10) through (6.4.1 14) into (6.4.105) and (6.4.107) and
6.4. equating coefficients of like powers of
6,
GENERALIZED METHOD
293
we have
LO,) = 0
(6.4.1 15)
+ i(x1 f xz 1S1I2)t1
(6.4.116)
+ v2po+ v1 Imaginary (tlePi') (6.4.1 17)
where
+ w:
a
 i(
(6.4.118)
I The solution of (6.4.115) is
t1= A1(T2)e"1 + A2(~2)eiq2
(6.4.1 19)
Then (6.4.1 17) becomes
+ d a l sin (ql  4 + 8,) + a2 sin (qz  4 + 8,)l
(6.4.120)
where A,, = a, exp (ie,,) with real a,, and 8,. Since po w wl, v1  4 is a slowly varying function of time, and we consider it a function of T2.Now the solution of (6.4.120) contains terms that tend to infinity as ql, q2, or 4 + co (i.e., t + co), thereby invalidating our expansion for long times unless dP0
 = yo d T2
+ v2p0+ vial sin (ql  4 + 0,)
(6.4.121)
Then pz becomes P2
a2v = ___ cos(q2 Po  2
+ 6,  4)
(6.4.122)
294 THE METHOD OF
MULTIPLE SCALES
With El known (6.4.1 16) becomes
+ Qzeivz + (iq,uZ + ixz + y)A~&ea(2''q~a) + (iwZp2+ ix2 + y ) ~ l A ~ e i ( 2 v 2   4 1 )
L(E3)= Qleivl where
(6.4.123)
Q1 =  i ( q  w z ) dAi   iw;A, dT2
+
+ xl)+ [2iy + 2x2 + (wl + dpZIal2 + xz + %Pz)a,2)A2
i((W2pl
+ (iY
(6.4.125)
Secular terms in (6.4.123) will be eliminated if Q , = Qz= 0. Letting A , = a , exp (ion) with real a, and 0 , in (6.4.124) and (6.4.125) with Ql = Q , = 0 and separating real and imaginary parts, we obtain
(6.4.127)
(6.4.129) where
r = +ql
08,
+
(6.4.130)
(6.4.131)
(6.4.132)
6.4.
GENERALIZED METHOD
295
Combining (6.4.128) and (6.4.130) and introducing the detuning parameter a defined by po = w1
we have
+
6%
6.4.7. THE EARTHMOONSPACESHIP PROBLEM The next example is the onedimensional earthmoonspaceship problem discussed in Sections 2.4.2, 3.2.2, and 4.1.7 and given by
, t(0) = 0 (6.4.134) 1x The straightforward expansion for r , for small p , in terms of x is singular at x = 1, and the region of nonuniformity is 1 x = O ( p ) . Thus to determine an expansion valid for all z using the method of multiple scales, we introduce the two variables (Nayfeh, 1964, 1965a)

7"
E=x,
 (
With these variables (6.4.134) becomes J2( at
+IG)
at
at
1x P
5p
+
T2 7
(6.4.135)
where all functions of x are expressed in terms of E except the term 1  x, the source of the nonuniformity, which is expressed in terms of q as pq. Now we assume that t possesses the following uniformly valid expansion t .= t n ( f , 11)
+ ~ f i (11)t +, P ' G ( ~ ,7) + .
* *
(6.4.136)
Substituting (6.4.136) into (6.4.135) and equating coefficients of like powers of p , we obtain (6.4.137) (6.4.138) (6.4.139)
The general solution of (6.4.137) is (6.4.140)
296
THE METHOD OF MULTIPLE SCALES
where A is determined from the condition that f l / t obe bounded for all 7. The solution of (6.4.138) becomes
J2
+J
tl = A'(t)q
j  PI2sinh'&
m
As q+ co, (6.4.141) becomes fitl
=? /,[
 A'(6)Iq
+ E ( t ) (6.4.141)
i
+ it,/?  t3"In 2  + E ( t ) + O(q') (6.4.142)
Thus t , contains two terms which make t , singular as q +co; a term proportional to q and a term proportional to In (11). The first term can be eliminated if ~ ' ( 8=) & or A = $5312 a (6.4.143)
+
where Q is an arbitrary constant. As for the second term, In (9) is slowly varying with z and p although q is fast varying. Thus it should be expressed in terms of E ; that is
Then tl is singular as 8

1, and it will be bounded as 5
B ( t ) = $E3'/" In (1  t )
+ C(6)

1 if (6.4.144)
The function C(5) is determined by requiring that t,/t, be bounded as q + 00. Substituting the above solutions into (6.4.139), we obtain ,
As q
+
I
co, (6.4.145) becomes
+l3I2 215
C'(t) + O(q')
as 3f
00
(6.4.146)
6.4. GENERALIZED METHOD 297
Here again In
J.1 is expressed in terms of 5 . Consequently, (6.4.146) becomes as q  + w
In order that t J t , be bounded as 1;1+
00
or
where c is a constant of integration. Expressing to and t , in terms of x and using the initial condition t(x = 0) = 0, we obtain a = c = 0. Hence
$1
+ $x3/' + JZ  4 In k + O(pz) 1  JZ
(6.4.148)
We next discuss an alternative method (Nayfeh, 1965a) of determining A(E) and B(6). Since
is assumed to be uniformly valid for all x, it must reduce to the straightforward expansion (see Exercise 2.12)
JZ t = gx3'/"+ p
3x3I2
+ JZ
")
 4 In  + O(p2) (6.4.150) 1  Jx
away from x = 1 . We can use this condition to determine the functions A(E) and B(6) rather than use the condition that te/t,l be bounded for all 5 and
298
THE METHOD OF MULTIPLE SCALES
q. Expressing (6.4.149) in terms of x , and expanding for small p , we obtain
1+
+4
~ In ~4(1’ ~  B(x)
O(p2) (6.4.151)
11% In order that the first terms in (6.4.150) and (6.4.151) be the same A ( z ) = 8x3‘2
(6.4.1 52)
Then the second terms are the same if
Substituting these expressions for A and B into (6.4.149) and expressing the result in terms of x , we obtain (6.4.148) exactly. 6.4.8. A MODEL FOR DISPERSIVE WAVES We again consider Bretherton’s (1964) model equation
4t*+
L
2
+
$22
+ 4 = 43
(6.4.154)
The linearized equation admits the uniform wave train solution
4 = a cos 8
8 = kx
 wt,
w2 = k4  k2
(6.4.155)
+1
To determine a wave train slowly varying with position and time, we follow Nayfeh and Hassan (1971) by assuming that
4 = +d8, Xl, TI) + .41(8,
where
8 = e’((X1, Tl),
Xl, Tl)
X , = EX,
+
* * *
(6.4.156)
Tl = ~f (6.4.157)
In terms of the new variables 8, X I , and T,, the time and space derivatives become a 2  0 2  a2  2EW
at2
ad2
a2 aw a+ E 
aeaT,
aT,ae
€2
a2
aT:
ak a  + 2za 2 aeax, + ax,ae ax, a4  = k4  + 4€k3___ + 6 e k 2 ak a3 + . . . a84 a83 ax, ax, a83 a2 a2  k2+2ck
ax2
ao2
a4
a4
ax4
a2
E 
6.4.
GENERALlZED METHOD
299
Substituting (6.4.156) into (6.4.154) and equating coefficients of like powers of c, we have
~ ( 4 =~ (w' ) 3
+ k2)
*'+ ae2
ae4
(Eg)$!?a34 ax, ae3 a2+0
a2+0
L(+l)=+O +2a2k4k3ae aTl
+
+ +' = 0
k4 ?h0
ae ax,
(6.4.158) a93 ax,
6k2oak
(6.4.159)
The solution of (6.4.158) is taken to be
LO'
Substituting for where aA Q =2 i~ a TI
+ A(X,, T,)e" = k4  k' + 1
(6.4.160)
+o = A(X,, Q e "
where
into (6.4.159), we have
L(+,)= Q ( X , , T,)e"
+ A3e3" + CC
(6.4.161)
aA ak + 2ik(2k2  1)+ i am A + i(6k2  1) A + 3A26
ax,
aq
8x1
(6.4.162) The condition that must be satisfied for there to be no secular terms is Q = 0. To simplify this condition, we note that LOW'
= 2k3  k
(6.4.163)
where w' = dw/dk, the group velocity. Differentiating (6.4.163) with respect to XI, we have (6.4.164)
If 5 is twice continuously differentiable, w and k satisfy the compatabllity relationship am ak +=O or
a ~ , ax,
(6.4.165)
Hence
300
THE METHOD OF MULTIPLE SCALES
so that (6.4.164) can be rewritten as (6.4.166)
With (6.4.163) and (6.4.166) the condition Q = 0 can be simplified to aA
2
aT1
i2w'
aA
8x1
+ w " akA
3x1
3i
= A2A 0
(6.4.167)
Letting A = (112)~exp (ip) in (6.4.167) and separating real and imaginary parts, we have
a (w'a') + aTl ax,
=0
(6.4.168) (6.4.169)
The solution obtained in this section using the method of multiple scales is a different representation of the solution obtained in Section 5.8.1 by averaging the Lagrangian. In fact, the equations governing the amplitude and the wave number have exactly the same form. However, in Section 5.8.1 there is no phase, but the dispersion relationship (5.8.9) is amplitudedependent; in this section the dispersion relationship is independent of amplitude but the solution describes the phase variation. To show the equivalence of these representations, we expand 8 of Section 5.8.1 in the form
e = eo  E~
so that
0
= wo
+ E at
(6.4.170)
Substituting (6.4.170) into (5.8.9) and equating coefficients of like powers of E, we obtain W : = k t  kO2 1
a,!7
+
+w,=at
ax
The last equation is the same as (6.4.169)
3a2 8 ~ 0
6.4. GENERALIZED METHOD 301 6.4.9. THE NONLINEAR KLEINGORDON EQUATION The last example considered in this chapter is the equation
+
utf  u,, V’(u) = 0 (6.4.171) which was treated in Section 5.8.3 using Whitham’s method of averaging the Lagrangian. Our analysis follows that of Luke (1966). We assume that u possesses a uniformly valid expansion of the form
u(z, t ) = uo(0, Xl, Tl)
+ .u,(e, x,,Tl) i.
(6.5.172)
+ V’(u,)
(6.4.173)
* *
where 0, XI,and T , are defined in (6.4.157). Substituting (6.4.172) into (6.4.171), using the expressions for the derivatives from the previous section, and equating coefficients of like powers of E , we obtain aZu
(w2 
ae2
aZu + V ” ( U ~ ) U , ae
=0
(wZ k2)
0
yy’  y = 0
(22 + 1)y’ + y2 = 0 (d) EY” (e> ry” 7 y’ yn = 0 ; n is a positive integer
+
subject to the boundary conditions
Y(1)
Y(0) = a,
=B
6.11. Use the MMS to determine a firstorder uniform expansion for the problem ry“
+ a(2)y’ = 1
d o ) = a,
Y(1) = B
if a(z) has a simple zero at p in [O, I]. 6.12. Use the MMS to determine a firstorder uniform expansion for y”
+ P(1  s)y(z)y
=0
where n is a positive integer,f(z) > 0 and 1 >> 1. 6.13. Determine firstorder uniform expansions for (6.4.84) of the form
(a)
u = A ( S ) c n [ q , 4 8 1 and
(b) u
= A(E)dntrl, 4~31.
6.14. Consider the problem
ii where
4
+ oo2(Et)u = €us + Kcos 4
= ~ ( r f )Determine .
firstorder uniform expansions for the cases
(a) K = O(1) and w away from (b) K = O(1) and o m 30, (c) K = O(1) ahd w w 4 3 (d) K = O(e) and o w w,
oo,3w,,
and w0/3
306
THE METHOD OF MULTIPLE SCALES
6.15. Consider the problem defined by (6.4.105) through (6.4.107) with variable coefficients. Determine a firstorder uniform expansion when K = 0(1) and p is away from wl. Show that this expansion is not valid whenp M 0 or 2 9  w,, and determine uniform expansions for these cases (Nayfeh and Saric, 1972a). 6.16. Solve Exercise 5.14 using the MMS. 6.17. Determine a firstorder uniform expansion for small amplitudes for
+ 2x + 3x2 + 2y2 = 0 y + x + 26y + 4xy = 0
x y when 6 w 1. 6.18. Consider the problem
Determine a firstorder uniform expansion when
6.19. Consider the equation
Use the MMS to determine firstorder expansions for traveling waves if (a) the amplitude and phase vary slowly with position and time and (b) the wave number, frequency, amplitude, and phase vary slowly with position and time. 6.20. Nonlinear transverse oscillations in a homogeneous freefree beam with a nonlinear momentcurvature relationship are given by
where c and E are constants. Determine firstorder uniform expansions for small E using the MMS for the cases enumerated in Exercise 6.19. 6.21. Use the MMS to determine secondorder uniform traveling wave solutions for the problem defined in Exercise 5.15 and the cases of Exercise 6.19. 6.22. Consider again Bretherton's model equation
but with a general nonlinear function$ Use the MMS to determine firstorder uniformly valid expansions for small c for the case of nthharmonic resonance. Specialize your results for (a) secondharmonic resonance and (b) thirdharmonic resonance.
Perturbation Methods ALI HASAN NAYFEH Copyright Q 2004 WILEYVCH Valag GmbH & Co. KGaA
CHAPTER 7
Asymptotic Solutions of Linear Equations
In this chapter we describe new techniques and use some of the techniques described in the previous chapters to obtain asymptotic solutions of linear ordinary and partial differential equations. Interest here is in differential equations with variable coefficients. The approach is to utilize a small or large parameter and develop parameter perturbations or to utilize a small or large coordinate and develop coordinate perturbations. If the point at infinity is an ordinary point or a regular singular point of an ordinary system of differential equations, convergent solutions can be obtained in inverse powers of the coordinate, except in some special cases of the regular singular problem when one of the solutions may involve the logarithm of the coordinate. Our interest in this chapter is in the irregular singular case, when the solution must be represented by an asymptotic expansion. In the parameter perturbation case, the parameter can be small or large; the first case includes the case of slowly varying coefficients. The expansions in these cases are obtained using the LiouvilleGreen (WKB) transformation and its generalizations. The resulting expansions are valid everywhere except at certain points called turning or transition points. Expansions valid everywhere including the turning points are obtained using Langer’s transformation and its generalizations. The discussion of partial differential equations is limited to the case of the reduced wave equation with variable index of refraction. The expansion is first developed for the case in which the index of refraction deviates slightly from a constant, using the BornNeumann procedure, and the solution is then represented by Feynman diagrams. The resulting expansion is valid only for short distances, and its region of uniformity is extended using the renormalization technique. Then the geometrical optics approach and the method of smoothing are described. Since these problems are linear, a vast amount of literature exists on their asymptotic solutions as well as their mathematical justification. We describe
308
7.1.
SECONDORDER DIFFERENTIAL EQUATIONS
309
in this chapter the techniques of developing formal asymptotic expansions to the solution of the equations without any mathematical justification. Moreover, a selective number of articles is cited. For further references and mathematical rigor, the reader is referred to ErdClyi (1956), Jeffreys (1962), Cesari (1971), Bellman (1964), Wilcox (1964), Wasow (1965), Feshchenko, Shkil', and Nikolenko (1967), Wasow (1968), and Frisch (1968). Secondorder ordinary differential equations are treated first in Section 7.1, while systems of firstorder ordinary differential equations are discussed in Section 7.2. Turning point problems are then taken up in Section 7.3, while the reduced wave equation is treated in Section 7.4. 7.1. SecondOrder Differential Equations In this section we are concerned with the asymptotic development of the solutions of the equation (7.1.1) where E is a parameter which can be either small or large. We assume that p and q do not simultaneously vanish in the interval of interest. We discuss first the asymptotic solutions of this equation near an irregular singular point. Then we describe techniques for determining the asymptotic solutions of (7.1.1) when it contains a large parameter. Next we consider singular perturbation problems in which a small parameter multiplies the highest derivative. Finally, we describe techniques for obtaining asymptotic expansions when p , q , and r are slowly varying functions of x. 7.1.1. EXPANSIONS NEAR AN IRREGULAR SINGULARITY Let us investigate the asymptotic development of the solutions of (7.1.2) as x + co, if the point at infinity is an irregular singular point. Before doing so let us define a regular singular and an irregular singular point. Assume thatp(z) and q(s) can be developed in ascending powers of (2  so),zo < co, as P(X) = Po("  xJa[l P ~ ( Z 4 . . .I, Po # 0 (7.1.3) q ( z ) = qo(x  zo)'[1 qi(5  2 0 ) . . .I, qo # 0 The point Z o is called an ordinarypoint if a 2 0 and fi 2 0; otherwise it is called a singular point. A singular point is called a regular singular point if a 2  1 and ,6 2 2; otherwise it is called an irregular singular point.
+ +
+ +
3 10
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
The above definitions indicate that the nature of a finite point such as zo can be decided almost at a glance. The nature of the point at infinity can be determined by transforming it first into the origin. Thus we let x = zl in (7.1.2) and obtain (7.1.4) The point at infinity is an ordinary point of the original equation if the origin is an ordinary point of the transformed equation; that is
which correspond to p(2) = 22'
q(x) = o
+O ( d )
as
(~~)
x+03
(7.1.5)
in the original equation. In order that infinity be a regular singular point of (7.1.2), the origin must be a regular singular point of the transformed equation; that is as
z+O
which correspond to p ( z ) = 0(z1) q ( 2 ) = o(z2)
as
x+w
(7.1.6)
Thus if p ( x ) and q(x) can be developed in descending power series of x as
+
+
= p02' . . . , q(z) = qoxs . . . , po and qo # 0 then the point at infinity is an ordinary point if /? 4, and either u =  1 withpo = 2 or a 2. In this case (7.1.2) has two convergent solutions in  1 and /? 2, powers of xl. If these conditions are not satisfied, and a the point at infinity is a regular singular point. In this case (7.1.2) has two of the form (Frobenius, 1875) convergent solutions in powers of z1
 1, x = m is an irregular singular point of the system. The behavior of the solution near an irregular singularity depends on whether all the eigenvalues of A. are distinct or not. We discuss in this section the case of distinct eigenvalues. If (7.2.1) is a scalar equation, it can be solved explicitly and its solution has the form y ( x ) = U(x)z“e*(z’ (7.2.3) where G is generally a complex constant, Q(z) is a polynomial of z having the form (0 if q =  1 Q ( x ) = Q+I (7.2.4) 12 m=l Qmxm and m
U(X) = 2 umxm m=O
(7.2.5)
In the case of systems of equations, the asymptotic solution still has the form (7.2.3) but G, Q,, and Urnare constant matrices. This expansion was called a “normal” solution by Thomi (1 883). To calculate the asymptotic expansions for the solutions of (7.2.1) and (7.2.2), we seek formal solutions of the form where
G
y = u(x)x‘eA(z)
is a constant,
(7.2.6) (7.2.7)
with I  , = 0 for m
2 0, and U(X)
=
m
2U,X~
m=O
as x + co
(7.2.8)
Here A is a scalar quantity, while y and u are column vectors. Substituting (7.2.6) through (7.2.8) into (7.2.1) and (7.2.2) and equating the coefficients
7.2.
SYSTEMS OF FIRSTORDER ORDINARY EQUATIONS
327
of equal powers of x, we obtain equations to determine successively A,, a, and u,. The first equation gives (A,  Aa+lZ)uo= 0,
I: identity matrix
(7.2.9)
For a nontrivial solution the determinant in (7.2.9) must vanish. This condition gives the following nthorder algebraic equation 140
 &+I[l = 0
(7.2.10)
If the eigenvalues of A , are distinct, (7.2.10) gives n distinct values for A,, which correspond to n linear independent solutions of the form (7.2.6). 7.2.2. ASYMPTOTIC PARTITIONING OF SYSTEMS OF EQUATIONS Sibuya (1958) developed the following scheme for simplifying the system of equations (7.2.1) by reducing them to some special differential equations whose solutions can be found more readily than those of the original system. To accomplish this we let (7.2.1 1) Y = W)V(X) where P is an n x n nonsingular matrix to be determined and v is a column vector. Hence (7.2.1) is transformed into

dv = zQB(z)v dx
where
A(x)P(z) 
(7.2.12) 59

or
dx
(7.2.13) The essence of the technique is to choose the matrix P(x) such that the matrix B(s) has a canonical Jordan form. To d o this we let m
B = z B m x  m as x+
a,
m=O m
(7.2.14)
P = z P m x  m as x + a , m=O
where Bm represents a Jordan canonical matrix. Substituting (7.2.14) into (7.2.13) and equating coefficients of like powers of 5 , we obtain (7.2.15) AoPo  P,B, = 0 m1
328
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
for rn 2 1, with Pmal= 0 if m then Po can be chosen such that
 q  1 < 0. If A, has distinct eigenvalues, (7.2.17)
B, = P,'AoP,,
is diagonal. We multiply (7.2.16) from the left with P;' to obtain
 W,Bo
BOW,
where
= Bm
+F,
(7.2.18)
w, = K I P r n
F , = P,'A,Po
+ PF1 2 (P,Bm, m1 S=l
(7.2.19)
 A,,PJ
 (m  4  l)Wmai (7.2.20)
If we denote the components of Fmby FC, we choose B, such that B: = F:,
B g = 0 for i # j
(7.2.21)
Since B, has distinct eigenvalues, (7.2.18) with (7.2.21) can be solved to determine W,, hence P,, from (7.2.19). In the case of multiple eigenvalues, we can partition the system of equations into simpler systems using the same scheme. We assume that A . has multiple eigenvalues and that there exists a matrix Po such that B, = P,'A,P,
(
= 'B
0
BF
)
(7.2.22)
where B:' has the eigenvalues iZi ( i = 1,2, . . . ,r) while BF has the eigenvalues 1, ( j = r 1, r 2, . . . , n) such that iZi Aj. Let us partition W,,, and F, according to
+
+
+
w: w:/
(7.2.23)
F, =
where Wil and Fkl are r x r matrices while W;' and Fk2 are (n  r) x (n  r) matrices. We choose
wt = W E = 0
and B:
=
FE,
B g = pg
(7.2.24)
Then (7.2.18) becomes
(7.2.25) These equations can be solved uniquely for Wi' and WE because BF and Bia do not have common eigenvalues.
7.2.
SYSTEMS OF FIRSTORDER ORDINARY EQUATIONS
329
As an example, we consider Bessel's equation 2 ' 
Letting
+
d2Y + xdY dx2 dx
2
(5
Y = u1,
 n 2)y
=o
(7.2.26)
dY = u2 dx
(7.2.27)
we transform (7.2.26) into
du
 = A(s)u dx
Hence q = 0, A,,, = 0 for m
> 2 , and
Since the eigenvalues of A, are f i ,
Po=
(
1 i
7
i
(I 3 1
and P 2 =
2
so that
Bo = PilAoPo = Hence from (7.2.20)
(7.2.30)
1
(
0 i
)
(7.2.31)
(7.2.32) Therefore from (7.2.21) and (7.2.25) we obtain
B1= With W,and B, known
( ), 1 1 0 2 0 1
w&(
1
0
)
1 0
(7.2.33)
(7.2.34)
330
ASYMPTOTIC SOLUTlONS OF LINEAR EQUATIONS
from (7.2.20). Hence
Substituting the expressions for B,, B,, and B2into (7.2.12),we obtain
Hence
v2
b
=Jexp
[ix
 i
1
(7.2.36)
+ 4n2
where a and b are constants. Since P = Po + 1PI X
+ 0(x2)= Po(z +:  w, ) +O(X3
(7.2.37) Therefore 1
+
%(
+ i
i)

exp [ix
 i]
82
+ O ( K ~ ” ) (7.2.38)
7.2.
SYSTEMS OF FIRSTORDER ORDINARY EQUATIONS
33 1
Note that we did not use P, in (7.2.37) and (7.2.38) because the error in v is O ( Z  ~ / ~To ) . compare these results with those obtained in Section 7.1.2, we expand exp [ f i ( l 4n2)/8s] in powers of xl. Since
+
ixi
+824n2) + 0(~5’2)
in agreement with (7.1.14) when n = 0. Higher approximations can be obtained by calculating the higher values of B, and W , in a straightforward, though tedious, way. The technique employed in Section 7.1.2 is very much easier to implement than the technique described in this section. 7.2.3. SUBNORMAL SOLUTIONS If A, of (7.2.1) has multiple eigenvalues, we cannot decouple all these equations by choosing B to be a diagonal matrix. Instead we partition this system of equations to obtain simpler systems of the form m
dv’ = z ~ B ~ ( z ) ~ ’ , B’ = 2 Bmiz  ~ dX
(7.2.40)
m=O
where the eigenvalues li of B l are different from l j of B ; for i # j . Thus corresponding to each single eigenvalue A,, B” is a scalar, and (7.2.40) can be solved. If q = 0 (7.2.41) where a is a constant and c, are known in terms of Brmfor r 2 2. Hence a normal solution exists corresponding to this eigenvalue of the original system (7.2.1). For a multiple eigenvalue 1, of multiplicity m,,B* has a rank of m,.It turns out that this reduced m, system of equations, hence the original system, may not have a normal solution corresponding to this eigenvalue. However, it may have what is called a subnormal solution of the form (7.2.3) through (7.2.5), but Q and U are expanded in powers of xIir with integer r. As an example, the equation dx
dx
(7.2.42)
332
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
has the general solution = aed/z(xa/4  x514) + bed;(23/4 + ~  5 1 4 )
(7.2.43)
which consists of two subnormal solutions. Equation (7.2.42) is equivalent to the system
dx
(7.2.44)
In this case
has the eigenvalue , I= 0 with a multiplicity of two, which precludes the existence of normal solutions. 7.2.4. SYSTEMS CONTAINING A PARAMETER Let us consider the system of n linear equations = A(x, e)y (7.2.45) dx where E is a small positive number, h is an integer, and A(%, E) is an n x n matrix which possesses the asymptotic expansion eh*
m
A(s, e) = 2 emA,(x) as m=O
0
E +
(7.2.46)
If h is zero or negative, y possesses asymptotic expansions of the form m
If h > 0 the asymptotic expansions of the solutions of (7.2.45) depend on whether the eigenvalues of Ao(x) are distinct in the whole interval of interest or not. A point at which A,(x) has multiple eigenvalues is called a turning or transition point; turning point problems are discussed in Section 7.3. If the eigenvalues of A,(%) are distinct, the asymptotic representations of n linearly independent solutions of (7.2.45) have the form Y = u(x, E) exp
[S
1
A(x, E) ds
(7.2.48)
7.2. SYSTEMS
OF FIRSTORDER ORDINARY EQUATIONS
where
I:€'A&) h
A(%, u(x,
=
(7.2.49)
.>
€)
333
71
m
= 2 E'rd(5)
(7.2.50)
7EO
Substituting (7.2.48) through (7.2.50) into (7.2.45) and (7.2.46) and equating coefficients of like powers of .s, we obtain equations that determine successively il,and u,. There are n linear independent solutions of the form (7.2.48) through (7.2.50) corresponding to the n eigenvalues of A o ( z ) ; that is, the solutions of (7.2.51) P O ( 4  AO(4Zl = 0 7.2.5. HOMOGENEOUS SYSTEMS WITH SLOWLY VARYING COEFFICIENTS In this section we consider the asymptotic solutions of
where
dY = A(5, .)y, dx m
A(6, c) = 2 E"A,(E) m=O
5 =EZ as
E
+
(7.2.52)
0
(7.2.53)
In this problem x is a fast variable while 6 is a slow variable. As in Section 7.2.2, we assume the existence of a nonsingular matrix Po(5)such that
where BY has the eigenvalues Ai (i = I , 2, . . . ,r ) and Bizhas the eigenvalues A j ( j = r + 1 , r + 2, . . . , n) such that Ai # Aj. In this case we can reduce the original coupled system of equations (7.2.52) into two decoupled systems of orders r and n  r . To do this we let (7.2.55)
where
dx
= B(&
E)V
(7.2.56)
334
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
or (7.2.57) We seek asymptotic representations of P and B of the form
where B , represents a block diagonal matrix. Substituting (7.2.58) into (7.2.57) and equating coefficients of like powers of E, we obtain
m1
F,
=
2 (P,B,,
81
 A,,P,)
 AmPo +dP,l
d5
(7.2.60)
As in Section 7.2.2, we choose B, and Po in accordance with (7.2.54), multiply the second of equations (7.2.59) from the left by Pi', and use (7.2.19) to obtain BOW,  W,B, = B, F, (7.2.61) where W, = PG'P,, F , = PG1Frn (7.2.62)
(zzi),
+
To solve (7.2.61) we partition F, and W , according to F, =
(
w, = 'w w') w: w2,2
(7.2.63)
where F: and WE are r x r matrices. If we choose W 11, = W E = 0 and B:=
''
Fm,
B:2=  F Z
(7.2.64)
(7.2.61) becomes BA'WE  WzB:' = F',2 BFWZ  WZB: = F:
(7.2.65)
(7.2.66) These equations can be solved uniquely for W;' and Wkl since B: and BF do not have common eigenvalues. If A , has distinct eigenvalues, one can use the above scheme to reduce the original system to an uncoupled system of n equations having the form (7.2.56) with B a diagonal matrix. The details are the same as in Section 7.2.2. An easier technique can be used to determine the asymptotic solutions of
7.3.
TURNING POINT PROBLEMS
335
the system (7.2.52) if A , has distinct eigenvalues. The asymptotic representation has the form (7.2.67) (7.2.68)
df, = A(E)
(7.2.69)
dx
where A ( f ) is an eigenvalue of A,( p and by (7.3.3) for z < p . These expansions are called outer expansions and break down near z = p. To determine the region of nonuniformity of these expansions, we let 5 = (z  p)Av with positive v in (7.3.1) and obtain
5% dF2
+ {1""'6fflu
+ 51']+ L  Z ' q , ~ + 6AVI)y
=0
(7.3.5)
As 1 + 00 the third term in (7.3.5) tends to zero for all positive v ; however, the resulting equation depends on the value of v. As 1 + co (7.3.5) tends to y=O d2Y
=0 dF2
if v < j if v > %
(7.3.6)
It is obvious that the first two limits are not acceptable because their solutions do not match the outer expansions (7.3.2) and (7.3.3). Therefore the acceptable limit is the distinguished limit with Y = 2/3 yielding the third equation in (7.3.6). If we let =
N%)
(7.3.7)
the firstorder inner solution is governed by (7.3.8) Its general solution is
y = u,Ai(z)
+ b,Bi(z)
(7.3.9)
where Ai(z) and W ( z ) are the Airy functions of the first and second kind, respectively. Let us now digress to give some properties of the Airy functions which we will need in the ensuing discussion. These functions have the following
7.3.
TURNING POINT PROBLEMS
337
integral representations see (e.g., ErdClyi, 1956, Section 4.6) Ai(z) = T
Bi(z) = T
S" S" O
O
+
cos (it3 z t ) d t [exp (it3
(7.3.10)
+ z t ) + sin ( i t 3 + zt)] d t
(7.3.11)
These functions can also be related to Bessel functions of order 1/3 according to Ai(z) = iJz[Ii/dc)
 11/dc)l =
(7.3.12)
77
(7.3.13) (7.3.14) (7.3.15)
where 5 = (2/3)z3l2. For large positive asymptotic expansions
z
these functions have the following (7.3.16)
1 Ai(z) = z  ~ sin / ~
,I;
1
Bi(z) = T2II4 cos
Jn
i5 + 3
(7.3.1 7)

(7.3.18)
3
5+
(7.3.19)
To match the inner solution (7.3.9) with the outer solution (7.3.2), we express the latter in terms of 6 = (z  p ) P and determine its limit as A 03. In this case z > ,u and
+ O(A2'3) Hence
+...
(7.3.20)
338
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
Expressing (7.3.9) in terms of
Y = a3Ai(
Its expansion for large
E , we obtain
6qf(p)) + b3Bi( E qf (p))
6,obtained
by using (7.3.17) and (7.3.19), is
+ b, cos (845) + ')] + .4 E3I2
*
(7.3.2 1)
Since the matching principle demands that (7.3.20) and (7.3.21) be equal, we obtain al = _[a,sinA116f116
J"
"I
"4 + b3cos4
7.3.2) becomes
(7.3.23)
where
To match (7.3.9) with the outer solution (7.3.3), we note that
Hence (7.3.3) becomes
(7.3.24) Since 5 is negative in this case, z = is positive, and the asymptotic behavior of the inner solution (7.3.9) for large z, obtained from (7.3.16) and (7.3.18), is
Evm
+ b3 exp (gJfe)
pa3 4 n
(7.3.66)
dT
in agreement with (7.3.60). Problems with two turning points have also been analyzed by Olver (1959), and Moriguchi (1959). Several turning point problems were treated by Evgrafov and Fedoryuk (1966), Hsieh and Sibuya (1966), Sibuya (1967), and Lynn and Keller (1970) among others. 7.3.4. HIGHERORDER T U R N I N G POINT PROBLEMS In this section we let q , = (z
. p)"f(x),
f(z)> 0 and
a is a positive real number (7.3.67)
To determine a single uniformly valid expansion, we let [ ' z = z' so that has the same number of zeros as q l . Hence
['2
(7.3.68) where the branches of the square root are chosen so that the regions where ql(x) > 0 and ql(x) < 0 correspond to z' > 0 and z' < 0, respectively. This transformation leads to the related equation (Langer, 1931) (7.3.69) The solution of (7.3.69) is
where v = (2
+ x)
Hence, to first approximation
346
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
McKelvey ( 1 955) expressed the asymptotic solutions of a secondorder turning point problem (i.e., c/. = 2 ) in terms of Whittaker's functions. A secondorder turning point problem arises in the diffraction by elliptic cylinders whose eccentricities are almost unity (Goodrich and Kazarinoff, 1963), and in the solution of the Schrodinger equation (Voss, 1933). The first analysis of a secondorder turning point problem was given by Goldstein (1931) using the method of matched asymptotic expansions as described in Section 7.3.1. 7.3.5. H 1G H ER APPROXI MAT 10N S So far in our presentation, only the first term in the asymptotic expansion has been obtained. There are four different approaches for the determination of the higherorder terms.
Lunger's Approach. The gist of this approach is always to relate the solution of the equation to be solved to that of some simpler but structurally similar problem that can be solved explicitly in terms of transcendental functions (Langer, 1949). The drawback to this approach is that it is unsuitable for numerical calculations because the coefficients of the asymptotic expansions are functions of the independent variable as well as the perturbation parameter. Moreover, the expansions are established using several transformations. Equivalent expansions can be obtained in an easier way by using Olver's approach as indicated later. CI1erry's Approach. In 1949 and 1950, Cherry developed a technique for obtaining the higherorder terms of a simple turning point problem, which has been transformed using the Langer transformation (Section 7.3.2) into d2V
cIz2
where
+ [A% + Ag(2, A ) ] u = 0
(7.3.72)
(7.3.73)
In Cherry's analysis all g , with even n are missing. We assume a formal expansion of the form u
=
A(z;
A)5,[P3+(2;
A)],
i
=
I, 2
where 5, and 5, are the Airy functions of the first and second kind, respectively. Since
TURNING POINT PROBLEMS
7.3.
347
and
(7.3.72) becomes
Equating the coefficients of
5, and dtild4 to zero, we obtain (7.3.76)
A” + 1g + =0
(7.3.77)
_ 1
(7.3.78)
 2)
A“$l$”
A
From (7.3.76) A= hence (7.3.77) becomes
 2)
+ A+“
=0
2A‘4‘
J
+I
+Ig +
(7.3.79)
To solve this equation we let (b = 2
+ A P $ , ( Z ) + I”&)
+.
*
.
(7.3.80)
and equate coefficients of like powers of 1 to obtain equations for the successive determination of 4,. The first two equations are
24;
+ + go = 0 244 + = g, $1
$2
whose solutions are
(7.3.81)
 z+;z  241+;
The lower limit in these expressions was chosen so that at z = 0. Therefore v is given by the formal expansion
6,
and
$2
are regular
348
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
This formal expansion is a uniform approximation to the solution of the original problem except in small neighborhoods of the zeros of v , where Cherry used the following modified formula u =
Ai(A2/32)(l
+ A2al + A%, + . . .) + @(A2/3z)(A2b, + A4b2 + . . .) dz (7.3.84)
If
w, 4= z: m
n=O
A2ng2n(Z)
an and b, are determined from (7.3.83) by expanding = z and using the relation d2Aildz2= A2zAi.
+
4‘ and Ai(Az134)about
Oluer’s Approach. Olver (1954) proposed to determine a complete asymptotic expansion by assuming that
+
u = A ( Z ;A ) < ~ ( A ~ ’ ~ z )B ( Z ;A)
151
1
..
Go(rz;rl) dr, dr, . . dr,
+. .
(7.4.18)

The averaged quantities (x(rl)x(r2) * x(r,)) depend on the configuration of the points rl, r,, . . . ,rm because for most random media there exists a correlation length I (ie., the values of x at points of separation larger than 1 are uncorrelated). To express the dependence on the correlation length, we expand these averaged quantities in the following cluster (cumulant) expansions (x(r,)x(rz)> = Wl, r2) (x(r,)x(rz)x(rs)>= Nr1, r2. r3) (x(rl)x(rz)x(r3)x(r4>)= W1,rz)R(r3,r4) k R(rl, r4)R(r2?
(X(rl)x(r2) . *
'
X(rm)>
=
2
kl+...+k,=m
+ W1, r3)R(rz,r4) r3)
R(gl, .
x R(ql, . . .
* *
7
+ R(rl, 9
(7.4.19)
r2, r3r r4)
gkl)R( = Go(r; ro)e'"z'Go
(7.4.55)
Gz = k4 Go(r; rl)Go(rl; r2)Go(rz;ro)R(rl; rz) drl dr,
(7.4.56)
s
This solution is valid for homogeneous as well as inhomogeneous random media. It is the same solution that would be obtained if we had recast (G) = Go 8 G Zinto an exponential. Using diagrammatic techniques, Tatarski (1964) and Frisch (1968) obtained the following BetheSalpeter equation for a centered Gaussian and general refractive index
+
=
+
where the intensity operator
M
I>a
; that is
'L/
(7.4.58) This equation was first introduced by Salpeter and Bethe (1951) for relativistic boundstate problems.
7.4.
WAVE EQUATIONS
373
7.4.3. RYTOV'S METHOD To obtain an approximate solution of
+
+
V2u k2[1 q(r)]u = 0 (7.4.59) we follow Rytov (1937) by assuming that (see also Tatarski, 1961, pp. 121128; Chernov 1960, pp. 5867) u = ev
(7.4.60)
thereby transforming (7.4.59) into Now we assume that y possesses the following asymptotic expansion m
(7.4.62) Substituting (7.4.62) into (7.4.61) and equating coefficients of like powers of E, we obtain (7.4.63) V2vo Vyo . Vyo = k2 V2y1
V2y, k
+ + 2 V y 1 . VyO = k2X
2Vy;
m=O
(7.4.64)
n 22
VynPm= 0,
(7.4.65)
These equations can be solved successively. The resulting expansion m
corresponds exactly to the Born expansion u =
2 P u n of (7.4.59). In fact,
n0
the Rytov expansion can be obtained from the Born series by recasting the latter into an exponential. Setting \
(7.4.66) expanding the exponential for small z, and equating coefficients of like powers of E , we obtain for the first terms u, = evO,
u1 = eWOyl, u3
= eYy3
u2 = evo(yz
+ YlY2 + 3Y13)
+ ty?) (7.4.67)
Thus the Rytov expansion is a renormalized Born expansion, hence it should be more uniformly valid than the Born expansion. However, this conclusion is controversial (Hufnagel and Stanley, 1964; deWolf 1965, 1967; Brown, 1966, 1967; Fried 1967; Heidbreder, 1967; Taylor, 1967; Strohbehn, 1968; Sancer and Varvatsis, 1969, 1970).
374
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
THE GEOMETRICAL OPTICS APPROXIMATION The object is to obtain an asymptotic solution for
7.4.4.
+
(7.4.68) V2u k2n2(r)u = O for large wave number k (ix., small wavelength 1 = 2a/k). For large k, (7.4.68) has an asymptotic expansion of the form (Keller, 1958) =pS(r)
m
2 (Wrn4nW
(7.4.69)
m=O
Substituting (7.4.69) into (7.4.68) and equating coefficients of like powers of k, we obtain VS * VS = n2(r) (eiconal equation) (7.4.70) 2 vs * vu, 2 VS Vu,
is
+ uov2s = 0 + umV% = V2u,1
(transport equations) for m 2 1
(7.4.71) (7.4.72)
Equation (7.4.70) can be solved using the method of characteristics; that
* do
=21vs
(7.4.73)
where o is a parameter and 1 is a proportionality function. Elimination of VS from the first and third equations gives
>
dr da 21da (
=2 ~ n ~ n
(7.4.74)
Then the solution of the second equation in (7.4.73) is (7.4.75)
where r(o) is the solution of (7.4.74) subject to the initial conditions r(aO)= r,, dr(ao)/da= io. Choosing 21 = nl and u to be the arc length along the rays, we rewrite (7.4.74) and (7.4.75) as
[n(r(o)) d do
1'
= Vn
da
s = so + r n [ r ( ~ ) ]d7 Jso
(7.4.76) (7.4.77)
7.4. Equation (7.4.71) becomes
WAVE EQUATIONS
375
along the rays. Its solution is (7.4.78) Similarly, the solution of (7.4.72) is
where c is a constant to be determined from the initial conditions. The expansion obtained in this section is not valid at a caustic (i.e., an envelope of rays), shadow boundaries, foci of the rays, and source points. In such regions neighboring rays intersect and the crosssectional area of a tube of rays becomes zero. Since the energy is conserved in a tube of rays, the amplitude of the field must be infinite in these regions. The unboundedness of the field at a caustic is shown below, and an expansion valid everywhere including a caustic is obtained in the next section. To show the breakdown of the expansion of this section at or near a caustic, we specialize it to the case n(r) = 1. In this case the rays are straight lines according to (7.4.76), and S = So  oo o according to (7.4.77). Now we express the solution of (7.4.71) and (7.4.72) in terms of a coordinate system with respect to the caustic which we assume to be smooth and convex. Figure 71 shows a point P outside the caustic and two rays passing through it. If we assign a direction to the caustic, this induces a direction for each ray which must be tangent to it at some point. Thus each point P outside the caustic lies on two raysone has left the caustic and the other is approaching the caustic.
+
Figure 71
376
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
In two dimensions we let s measure the arc length along the caustic and u measure the length from the point of tangency to the point P. Thus the position of P is determined by s1 and ul or s2 and 02.In terms of these coordinates, the position vector r of P can be expressed as
r = ro(s)
+ ue,
(7.4.80)
where r = ro(s) is the equation of the caustic, and el = dro/ds is a unit vector tangent to the caustic. Differentiating (7.4.80), we have
+ el d o + o e2 ds
dr = el ds because de,/ds = ple, where the variables from u and s to
P
is the curvature of the caustic. Changing
pI
5 = S
(7.4.81)
q=s+u
we obtain
716 dr =
Hence
dv +
e2 d t
(7.4.82)
P
af + P af
Vf = el all 11Since from (7.4.81)
17 
tate2
(7.4.83)
"1
+v  t a ta v[ t a E
ta7
(7.4.84)
we rewrite (7.4.83) and (7.4.84) as
v f =afe l au
+
(asaf  ;e2 
+ oo,then S(s, a) = s + a
If we take the special case S(s, uo) = s
(7.4.86) which is doublevalued corresponding to either the ray that has left or the one that is approaching the caustic. Using (7.4.85) and (7.4.86) in (7.4.78)
7.4.
WAVE EQUATIONS
377
and (7.4.79), we obtain (7.4.87)

which are unbounded as G 0 (i.e., a caustic). A modified expansion valid everywhere including the neighborhood of the caustic is obtained in the next section. 7.4.5. A UNIFORM EXPANSION AT A CAUSTlC To determine an expansion valid at the caustic, we first must determine the size of the region of nonuniformity and the form of the solution in this region. To do this we let u = y(r, k)eiks"' (7.4.89) in (7.4.68) with n(r) = 1 and obtain k Z y [l  (VS)']
+ ik(2VS . V y + y V 2 S ) + V 2 y = 0
(7.4.90)
Assuming that the k 2 term is the leading term, we obtain
VS . V S = 1 ik(2VS. V y
(eiconal equation)
+ yV2S) + V 2 y = 0
We take the solution of (7.4.91) as S = s
+
0.Hence
(7.4.91) (7.4.92)
(7.4.92) becomes
To analyze (7.4.93) near the caustic, we introduce the stretching transformation T = k"a, 1> 0 and obtain
The parameter 1, is determined by requiring that the highest power of k within the braces (terms neglected in the leading term of the straightforward expansion of the previbus section) be equal to the power of k in the first
378
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
term (is., the leading term in the ray expansion). That is, we let 1+A=41 Then as k
f
or
A=&
(7.4.95)
co, (7.4.94) tends to
(7.4.96) Letting 3
2
p = efr / 3 p V(Z), we obtain
2 7
z =P
(7.4.97)
FG
(7.4.98) whose solutions are the Airy functions A i (  z ) and Bi(z) (see Section 7.3.1). Had we analyzed the behavior of the solution near a shadow boundary, we would have found that it must be represented by Weber or parabolic cylinder functions. Now we can either match this inner expansion with the outer expansion obtained in the previous section and with another outer expansion inside the caustic (Buchal and Keller, 1960), or form a single uniformly valid expansion imitating the expansions for turning point problems (see Section 7.3). Following Kravtsov (1964a, b), Ludwig (1966), and Zauderer (1970b), we assume an asymptotic expansion of the form u = e"fl(r'(g(r, k)V[k2/3+(r)]
+ rk1/3 1 h(r, k)v'[k2'3+(r)]]
(7.4.99)
where 8 and 4 are determined from the analysis and V(z)is given by (7.4.98). Substituting (7.4.99) into (7.4.68), using the fact that V" zV' V = 0, and equating the coefficients of V and V' to zero, we obtain
+
+
+ +(V+)2  13  2k2+hV8 . V+ + ik[2V8. Vg + gV28 + 2 4 V + . Vh + +hV2+ + h(V+)2] + v2g = o
k2g [(V8)2
k2h
[(vq2+ + ( v + y
+ ik[2V+
*
Vg
 11  kZgve.
v+
+ gVz+ + 2V8 . Vh + hV28] + V2h = 0
(7.4.100) (7.4.101)
The coefficients of k2 in (7.4.100) and (7.4.101) vanish if (7.4.102)
7.4. WAVE EQUATIONS 379
To solve the resulting equations, we let
g(r, k ) = go@) h(r, k ) = h,(r)
+ k’gl(r)
+..
+ k’h1(r) + . . .
(7.4.103)
and equate coefficients of like powers of kl. The firstorder equations are
2V8 . Vg,
+ goV28 + 2 4 V 4 2 V 4 . Vg,
*
Vh,
+ go”’+
+ +h,V2+ + II,(V$)~ = 0 + 2Vf3. Vh, + h,V28 = 0
(7.4.104)
Equations (7.4.102) are equivalent to the eiconal equation (7.4.70), while (7.4.104)are equivalent to the transport equation (7.4.71).Similar expansions were obtained by Fowkes (1968, Part 11) by using the method of multiple scales. The system of equations (7.4.102)is a nonlinear system of equations which is elliptic where 4 < 0 (shadow region), hyperbolic where 4 > 0 (illuminated region), and parabolic where 4 = 0 (caustic curve or surface). For a convex analytic caustic, the systems (7.4.102) and (7.4.104) can be solved by expanding 8 and 4 in power series in terms of a coordinate system with respect to the caustic. To see the connection between the system (7.4.102) and the eiconal equation (7.4.70), we multiply the second equation in (7.4.102) by f2& (illuminated region where 4 > 0 is considered) and add the result to the first of these equations to obtain
(ve f J$V+)2
or
=
1
(VS)2 = 1
where
(7.4.105)
s* = 6 f &$3/2
(7.4.106)
Similarly, we multiply the second equation in (7.4.104) by &dT, add the result to the first of these equations, and use VO * V 4 = 0 to obtain where
2Ts+ vy* ’
+ [V2S* 7 $+”2(v+)”y* Y* = go f
43 ho
=0
(7.4.107) (7.4.108)
Equation (7.4.107) differs from the transport equation (7.4.71) only by the term &(1/2)41/2(V4)2 which makes the coefficient of y* bounded near the caustic. Replacing V and V’ by their asymptotic expansions for large argument, we recover the geometrical optics expansion of the previous section (4 > 0, illuminated region). Replacing V and V’ by their asymptotic expansions for
380
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
large argument for the case q4 < 0, we obtain an expansion in the shadow region which can be interpreted in terms of complex rays, phases, and transport coefficients. Uniform asymptotic expansions for wave propagation and diffraction problems were reviewed by Ludwig (1970b) and Babich (1970, 1971). The role of coordinate systems in rendering the expansions uniformly valid was investigated by Zauderer (1970a). A single uniformly valid expansion for a point source problem was obtained by Babich (1965), while three matched expansions for this problem were presented by Avila and Keller (1963). Problems of diffraction by convex objectswere treated by Zauderer (1964b), Buslaev (1964), Grimshaw (1966), Ludwig (1967), and Lewis, Bleistein and Ludwig (1967) among others. Diffraction by a transparent object was analyzed by Rulf (1968). Uniform expansions for the problem of diffraction near the concave side of an object (whispering galley modes) were obtained by Kravtsov (1964b), Matkowsky (1966), and Ludwig (1970a) among others. Characteristic transition regions in which two caustics are near each other are analogous to secondorder turning points. Their uniform expansions involve Weber or parabolic cylinder functions. Some of these problems were treated by Kravtsov (1965), Babich and Kravtsova (1967), Weinstein (1969), and Zauderer (1970a, b). Diffraction by a thin screen (Fresnel diffraction) was investigated by Wolfe (1967), Kersten (1967), and Ahluwalia, Lewis, and Boersma (1968). Multiple transition regions arise from the tangential intersection of two or more caustics and shadow boundaries, such as near terminated caustics (Levey and Felsen, 1967), cusps on caustics (Ludwig, 1966), and points of diffraction for smooth objects (Ludwig, 1967). Problems involving transition regions were treated by Zauderer (1964a, 1970a), Fock (1965), Rulf (1967), and Bleistein (1967) among others. 7.4.6. THE METHOD OF SMOOTHING
To apply this technique to (7.4.3) with n2(r) a centered random function,
we first convert it to the integral equation
u = M g  Ek2MXU where the operator M is defined by
(7.4.109)
(7.4.110) with Gothe freespace Green’s function. Frisch (1968) pointed out that many other linear equations of mathematical physics, such as the Liouville equation
7.4.
WAVE EQUATIONS
381
for an ensemble of classic interacting particles, the Hopf equation for turbulence, and the FokkerPlanck equation, can be put in the above integral equation form. In general, we are not interested in the function u but in its projection Pu on a subspace of the original space over which u is defined. For example, in the problem of wave propagation in a random medium, Pu = ( u ) , while in the case of N interacting particles, Pu is the Nparticle velocity distribution function obtained by integrating u over all position coordinates. Let Pu = u,,
and
u, = ( I
 P)u
(7.4.1 11)
For wave propagation in a random media, u, is the coherent part of the field (mean part) while u, is the incoherent part (fluctuating part). For a deterministic g and a centered random x Pg = g ,
PM = M P ,
PXP = 0
(7.4.1 12)
Applying the operator P from the left on (7.4.109), we obtain U,
Applying the operator I U,
= ek2M(I
=
M g  ck2MPX(Pu  ck2MPXUz
= Mg
+ u,)
(7.4.1 13)
 P from the left on (7.4.109) also, we obtain
 P ) ~ ( u+ , u,) = d?M(Z
 P)xuC  ~ k ' M ( 1 P ) p ,
(7.4.114)
We solve (7.4.1 14) by formal iteration for u, in terms of u, to obtain tli =
m
2 [k'M(I
n1
Substituting for
U,
 P)x]''u,
(7.4.115)
into (7.4.1 13) gives
Since u, = Pucr this equation can be written as where
u, = Mg
+ MQu,
Q =  z € k 2 P X [  c k 2 M ( Z  P)x]"P n= 1
(7.4.1 16) (7.4.1 17)
For random operators (7.4.116) is just the Dyson equation (7.4.49) with Q the mass operator. This technique was called the method of smoothing by Frisch (1968) because it is the counterpart of the method of averaging (Chapter 5) in which the dependent variables have short and longperiod parts. This
382
ASYMPTOTIC SOLUTIONS OF LINEAR EQUATIONS
technique was first introduced by Primas (1961), Ernst and Primas (1963), and Tatarski and Gertsenshtein (1963) for random equations, and by Zwanzig (1964) for the Liouville equation. The firstorder smoothing is given by (7.4.116), with n = 1 in (7.4.117); that is Q = E W P ~ M (I P)xP = c2k4PxMxP Hence u, = Mg e2k4MP~MxPu, (7.4.118)
+
which coincides with the Bourett equation (7.4.54) for linear random media, and with the Landau equation for the Liouville problem (see, for example, PrigoginC 1962). For linear random media this first approximation was also obtained by Keller (1962) and Kubo (1963). Exercises 7.1. Determine asymptotic expansions for large x for the solutions of
(c) x2y” + xy’ + 2.(
 n2)y
(d) Y“
1 + (I +; y):+
( e ) y”
+ xy‘ + y = 0
=0
=o
7.2. Determine asymptotic expansions for large x for the solutions of
+
(b) y” f (gx 1)y = 0 (c) y“ xy’ yx3y = 0 (d) y” + ~‘/~y‘ + X~Y= 0
+
+
7.3. Consider the equation
y”  Ps2y = 0
(a) Show that the exact solution is
y = azm1 + b P z where m1,2 = 4 f %‘A2 + $. (b) Determine the first WKB approximation. (c) Compare this approximation with the exact solution (Jeffreys, 1962).
EXERCISES
383
7.4. Determine a first approximation to the eigenvalue problem
+ t Y ( z ) u = 0,
u”
f(z)
u(0) = u(1)
for large 1.
>0
=0
where Determine an equation for 4 and then determine the 7.6. Determine a uniform asymptotic expansion for ii
+
W2(Ct)U
+n
(Brull and Soler, 1966).
=f(ct)
where E is a small parameter and f is a bounded nonperiodic function of t. 7.7. Consider the problem ii
4
+ w 2 ( c t ) U = k cos 4
where = t ( d ) .Determine uniform asymptotic expansions for the cases (a) I is not near w , and (b) 1 = w for some t = to > 0 (Kevorkian, 1971). 7.8. Consider the problem ii
+ wz(cr)Jri w,(ct)wz(ct)u
 i[W1(Et)
=f ( E t )
where overdots denote differentiation with respect to t and c is a small parameter. Determine uniform asymptotic approximations to the solution of this equation if (a)f(d) = 0, (b)f(rt) is a bounded nonperiodic function oft, (c)f(ct) = k exp (ie) with 4 = A ( c t ) # w , , and (d) f ( r t ) has the same form as in (c) but t = w1 at some I = to.
7.9. Determine a uniform asymptotic expansion for the general solution of 2y”
+ 4 2 2 + 1)y’ + 2xy
= 0,
0
1 andf(r) > 0. (a) Determine an expansion valid away from r = 0 and r = 1 ; (b) determine expansions valid near r = 0 and r = 1;(c) match these three expansions, hence determine A, and form a uniformly valid composite expansion (Sellers, Tribus, and Klein, 1956). 7.15. Consider Exercise 7.14 again. (a) Determine an expansion valid away from r = 0 using Langer's transformation; (b) determine an expansion valid away from r = 1 using Olver's transformation; (c) match these two expansions to determine 2, (d) form a composite uniformly valid expansion; and (e) compare these results with those of Exercise 7.14. 7.16. Consider the problem y"  12(z  1)(2  z)y = 0 Determine an approximate solution to y if 1 >> 1, and y 7.17. Consider the problem u"
+ d2(1  z)"(x
 p)rnf(z)u
u(1) = u(p)
=0
< co for all x.
=0
EXERCISES
385
where 1 >> I , f ( . r ) > 0, n and rn are positive integers, and < 1 . Determine expansions valid away from z = and z = 1 using Olver’s transformation, match them to determine 1, and then form a composite expansion, 7.18. Consider the problem ru”
+ pu’ + W(l  r)mf(r)u
= 0,
u(1) = 0 and u(0)
> 1, f ( r ) > 0, 11 is a real number, and n and rn are positive integers. Determine expansions valid away from r = 0 and 1 using Olver’s transformation, match them to determine 1, and then form a composite expansion (Nayfeh, 1967a). 7.19. Given that f(z) > 0 and 1>> 1 , determine first approximations to the eigenvalue problems (a) s y ”
+ y’ + Pz(1  z)f(z>y = 0 + +
(b) r y ” y’ A2zn(l z)*f(z) = 0, rn and n are positive integers subject to the condition that y < rn for z 2 0. 7.20. How would you go about determining a particular solution for U”
+ ,4223~= n2g(z)
when 1 >> 1 if (a) g(0) # 0, (b) g(0) = 0 but g‘(0) # 0, (c) g(0) = g’(0)= 0 but g”(0) # 0, and (d) g(0) = g’(0) = g”(0) = O?
5. A molecular study on the depthdependent oxidation and condensation gradients of aged dammar and mastic varnish films
Abstract
The depthdependent oxidation and condensation gradients of extremely aged, ~ 55 µm thick dammar and mastic films are studied on the molecular level. Direct Temperatureresolved MS (DTMS), MatrixAssisted Laser Desorption/Ionisation TimeOfFlight MS (MALDITOFMS) and High Performance Size Exclusion Chromatography (HPSEC) were employed to analyse the films after 248 nm laserinduced removal of 3.5, 7, 11.5, 15, 20, 25 µm and 3, 6, 10, 12, 16, 20, 25 µm from the aged dammar and mastic films respectively. Electron ionisation (16 eV) DTMS total ion currents indicated that polarity and condensation decrease depthwise, since lower temperature is required to volatilise the incorporated polar compounds and to induce pyrolysis of the high MW condensed fraction as the thickness of both films reduces. The relative abundance of oxidised dammarane, oleanane and ursane type triterpenoids gradually decreases with depth. Multivariant Factor Discriminant Analysis (DA) quantified the oxidative gradient and showed that a depth of 15 µm from surface of the aged films is the threshold between highly and poorly deteriorated
Chapter 5
A molecular study on the depthdependent gradients
material. MALDITOFMS showed that UVinduced oxidation resulting in Aring openings at position C2 of the oleanane / ursane type molecules stops at the 15 µm below the surface. The upper layers completely absorb radiation with λ < 350 nm. HPSEC determined that the high molecular weight fraction becomes less prominent as a function of depth. All the data presented establish the depthdependent compositional gradients and determine that deep layers in the bulk of aged dammar and mastic films remain unaffected from autoxidative degradation processes. The fact that nonUVinduced oxidation and unaffected material were detected in the bulk of the laser ablated varnish films indicates that excimer laser ablation at 248 nm is a nonoxidative process.
5.1 Introduction Ageing of triterpenoid resins, such as dammar and mastic, under light, is in principle an oxidative process, leading, as a first step, to the conversion of triterpenoid hydrocarbon type carbon skeletons (Figure 2.2.1) to carbon skeletons with oxygen containing functional groups (Figure 2.2.2) (De la Rie 1988a, Van der Doelen 1999). Today, it is understood that these changes are only the beginning of an oxidative process leading to radical polymerisation (crosslinking or condensation), oxidative modifications in the side chain or in the functional groups on the tetra or the pentacyclic ring structures, shortening of the side chain, especially in triterpenoid compounds with the dammarane skeleton (Van der Doelen, et al. 1998a, Van der Doelen, et al. 1998b), and probably to eventual defunctionalisation, bond breaking and disintegration of the triterpenoid carbon skeleton (Boon and van der Doelen 1999).
250
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As already described above (Section 2.2), the extent of oxidative degradation is significantly affected by the presence or absence of UV wavelengths in the incident radiation. The main difference is that UV wavelengths lead to an opening of the Aring at position C2 of dammarane and oleanane/ursane type molecules (Figure 2.2.2c) (Van der Doelen, et al. 2000), while in the absence of UV wavelengths most of the oxidative products are associated with oleanane and ursane type triterpenoids oxidised at positions C11 and C28 (Figure 2.2.2b) (Van der Doelen, et al. 1998a). UV wavelengths have been employed for various studies on the ageing of the triterpenoid varnishes (De la Rie 1988b, Zumbühl, et al. 1998, Van der Doelen, et al. 2000, Scalarone, et al. 2003). It has been determined, herein, that for the study of extremely aged varnishes, UV wavelengths are essential during the ageing process (Section 2.3). Even freshly harvested triterpenoid resins, having been irradiated by sunlight during exudation from the bark of the resin trees (Koller, et al. 1997), contain dammarenolic acid and 20,24epoxy25hydroxydammaran3one (Mills and Werner 1955, Poehland, et al. 1987, De la Rie 1988a, Van der Doelen, et al. 1998a, Van der Doelen, et al. 1998b, Scalarone, et al. 2003), both having oxidised Arings. The presence of these highly degraded molecules in ‘fresh’ resins is explicable by the free radical chain reactions which were initiated in the resin teardrops during the harvesting period (Dietemann 2003). A DTMS study of 55 µm thick dammar and mastic films, aged under intense, accelerated deteriorative conditions1, showed that both films contain a mixture of a few intact triterpenoid molecules, some oxidised triterpenoids with Aring openings, evidently produced upon UV irradiation, and 1
500 h xenon arc exposure (λ > 295 nm) at 60°C, 45 days near an open window and over a month storage in the dark
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some oxidised oleanane/ursane type molecules generated upon nonUV irradiation (Section 2.3). It is should be stressed that UVinduced and nonUVinduced oxidised triterpenoids were found to be almost in equal relative amounts in the investigated films (Table 2.3.5). In Section 4.3.2 it was demonstrated that the distribution of carbonyl groups is not random in the thickness of the films but influenced by the optical absorption lengths at the various wavelengths contained in the incident light. In particular, upon ageing dammar and mastic films absorb completely wavelengths with λ < 350 nm at the uppermost 1015 µm surface layers, wavelengths with λ > 350 nm penetrate deeper and wavelengths λ > 400 nm fully penetrate the 55 µm films tested (Figure 4.3.2.6). A theoretical model based on earlier GC/MS findings on aged dammar and mastic indicated that, owing to the strong light intensity and the abundant presence of oxygen in the surface, most of the oxidative products are generated in the surface, while the reduced amounts of oxygen in the bulk result in termination reactions of free radicals producing nonoxidative crosslinks (Boon and van der Doelen 1999). In other words, it is suggested that the oxidative deterioration of aged dammar and mastic films reduces with depth. The most apparent clues towards this conclusion are that: –
aged dammar films thicker than 10 µm were found to be less deteriorated than thinner dammar films aged under the same conditions (De la Rie 1988b).
–
solubility tests of laser ablated, naturally aged varnishes on paintings using a KrF excimer laser and a mixture of cleaning solvents showed that polarity decreases as a function of depth (Theodorakopoulos and Zafiropulos 2003, Section 1.4).
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– the amount of free radicals in fresh mastic resin teardrops, harvested in the same period, increases as the teardrops decrease in size, indicating that mastic blocks the propagation of light in the bulk (Dietemann 2003). These findings are supported by theoretical models on the ageing of various organic materials. Thomson suggested the presence of unreacted substance across depth of light aged organic films and based on Beer’s Law postulated that the abundance of this unreacted material should increase with increasing distance from the surface (Thomson 1965, 1979). Other models have shown that the consumption of oxygen across depth is dependent upon a reducing rate of diffusion, which leads to ‘oxygen starvation’ in the bulk (Thomson 1978, Cunliffe and Davis 1982, Fukushima 1983). The integration of data resulting from reducing light propagation into films and from the reduction of oxygen availability with depth produced models showing that the overall degradation of organic substrates is reduced as a function of depth (Schoolenberg and Vink 1991). A comprehensive review about depthdependent degradation in various materials due to light propagation and oxygen diffusion has been already provided (Feller 1994a). Heatbodied oil coatings, such as a thick copal oil varnish film tested, are the exception of this rule, because of the prepolymerisation process (Mantell, et al. 1949, Mills and White 1994, Carlyle 2001), which leads to high degrees of saturation both in terms of absorption (Sections 2.4, 2.5 and 4.3.2) and in terms of bonding of the incorporated molecular structures (Chapter 6). The depthdependent gradients in oxidation and crosslinking across the thickness of aged dammar and mastic films have been determined experimentally by the changes 253
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Chapter 5
A molecular study on the depthdependent gradients
in the interaction of these films at consecutive depths with a KrF excimer laser (Zafiropulos,
et
al.
2000,
Theodorakopoulos
and
Zafiropulos
2003,
Theodorakopoulos, et al. 2005, Chapters 1 and 3). In particular, it was shown that (a) the ablation rate per pulse and the ablation yield per incident laser photon (Section 3.3.2), (b) the carbon dimers emitted in the ablation plume monitored by LIBS (Section 3.3.4), (c) the ablation step obtained using a common laser cleaning method (Section 3.3.5) and (d) the laser light transmitted towards the underlying substrate (Section 4.3.1.2), gradually change across the reducing thickness of the ablated films. UV/VIS and ATRFTIR measurements established the decreasing gradients in absorption, oxidation, and crosslinking as a function of depth (Section 4.3.2), and the results were in a good agreement with earlier findings (Zafiropulos, et al. 2000). Moreover, evidence was provided about the nonoxidative contribution of the 248 nm pulses of the KrF excimer laser to the remaining films, as has been also determined elsewhere (Castillejo, et al. 2002). The latter finding shows that the determined gradients are not a result of the interaction of the resin films with the UV laser photons but a genuine characteristic of aged dammar and mastic films. This chapter aims at the establishment of the compositional gradients of aged dammar and mastic films on the molecular level. The evaluation of the dammar and mastic films prior to and after ageing (Section 2.3) enables a comparison with the degradation state of the films across their depth profiles. The following work is based on
Direct
Temperatureresolved
MS
(DTMS),
MatrixAssisted
Laser
Desorption/Ionisation TimeOfFlight MS (MALDITOF MS) and High Performance – Size Exclusion Chromatography (HPSEC). Analytical data from earlier studies in
254
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Chapter 5
A molecular study on the depthdependent gradients
particular of aged and unaged natural resins, based on HPLCMS, GC/MS and DTMS, are employed for an efficient interpretation of the complex DTMS summation mass spectra (De la Rie 1988a, Papageorgiou, et al. 1997, Van der Doelen, et al. 1998a, Scalarone, et al. 2003). Quantification of the DTMS data is enabled with Multivariant Factor Discriminant Analysis. MALDITOFMS is employed to provide complementary molecular information on the ablated surfaces and interpretation of the MS is enabled by comparison with related analytical data (Zumbühl, et al. 1998, Dietemann 2003). Finally, SEC is employed to determine the molecular weight modifications across depth to support findings based on excimer laser ablation (Chapter 3) and ATRFTIR (Section 4.3.2). Interpretation of the SEC traces was based on previous molecular weight studies of aged dammar films (De la Rie 1988b, Van der Doelen and Boon 2000).
5.2 Experimental 5.2.1 Direct temperatureresolved mass spectrometry The laserablated zones were examined with DTMS in order to investigate the potential molecular variation across the depth profile. From each zone a thin film of approximately 4 mm2 was mechanically subtracted, then homogenized and brought in suspension with a few drops of ethanol. A volume of 23 µl of the mixture was applied to a Pt/Rh (9:1) filament (100µm diameter) of a direct insertion probe, and dried in vacuo (using the Purevap) by evaporation of the ethanol. After insertion of the probe in the ionisation chamber a gradual temperature increase of the filament was set at a rate of 1A/min to a final temperature of approximately 800ºC, while the MS was monitoring the evolved compounds in electron ionisation (EI) or ammonia
255
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A molecular study on the depthdependent gradients
chemical ionisation (NH3/CI) modes. The compounds were ionised at 16 eV energy in EI mode and 250 eV in NH3/CI mode, and analysed in a JEOL SX102 double focusing mass spectrometer (B/E) over a mass range from 201000 Dalton at a cycle time of 1 s. The acceleration voltage was 8 kV. The total ion currents and summation mass spectra were examined. 5.2.2 Factor discriminant analysis (DA) EIDTMS summation mass spectra were numerically analysed by factor discriminant analysis coupled with the FOMpyroMAP multivariate analysis program, that is a modified version of the ARHTUR package from Infometrix Inc. (Seattle, USA; 1978 release) and with FOM developed Matlab® (The Mathworks Inc., Natick, MA, USA) toolbox ChemomeTricks. The discrimination was based on a double stage component analysis (PCA) that is described elsewhere (Hoogerbrugge, et al. 1983). Mass spectra of triplicate (at least) measurements for each sample were inserted in the aforementioned software to enable discrimination. To minimise variance in the data due to variance in the mass spectrometer readings, the DT mass spectra of the same films were measured on the same day prior to DA analysis. 5.2.3 Matrixassisted laser desorption/ionisation TimeOfFlight mass spectrometry (MALDITOFMS) Sampling was enabled with ethanol wetted minute TLC plates coated with cellulose that were simply brought in contact with the surface of the aged, unaged and laser ablated films. The samples were coated with a thin layer of 2,5dihydroxybenzoic acid (DHB, Aldrich, Steinheim, Germany). A sample of neat DHB coating was also prepared for reference. The TLC plates were attached on the stainless steel MALDI
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Chapter 5
A molecular study on the depthdependent gradients
probe, which was then placed on the xyz translator of the TOFMS system (BruckerFranzen Analytik, Bremen, Germany) for the appropriate orientation into the ionisation chamber (107 mbar) of the system. The ion source consists of positively or negatively charged metal electrode that is the MALDI probe and a grounded accelerating grid at a ~ 2 cm distance. The accelerating potential was 19.9 kV in the reflex mode. As soon as the vacuum lock was fastened the probe was in position to face the laser beam at an angle of 60º with respect to the surface normal. The nitrogen discharge (N2) laser (337 nm) used had a 4 ns pulse duration and a repetition rate of 12 Hz (Photon Technology PL2300). Manipulation of the sample was enabled by monitoring the ion source via a CCD camera. Processing was carried out with Bruker software (DataAnalysis for TOF 1.6.g). 5.2.4 High Performance Size Exclusion Chromatograph (HPSEC) The study of the molecular weight changes, across the depth profiles of the light aged, laser ablated resin films was carried out with High Performance Size Exclusion Chromatography (HPSEC). Unaged, aged and laser ablated samples were dissolved in THF (~10mg/µl) and centrifuged. An amount of 20 µl was analysed on a Shimadzu HPSEC system, consisting of a SCL10AD vp control panel, an LP10AD vp pump, a DGU14A degasser, a SIL10AD vp autoinjector, a CTO10AS column oven and a FRC10A fraction collector (Shimadzu Benelux, ‘sHertogenbosch, The Netherlands). Two different detectors connected in series were used for the signal detection. These were a Shimadzu SPD10A vp UV/VIS detector operated at 240 nm and a Shimadzu RID10A refractive index detector in combination with a Shimadzu Class vp 5.03 software. The temperature in the system was 40ºC and the flow rate was 1ml/min.
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Finally, calibration was carried out with polystyrene standards (Polymer Laboratories) with an average mass ranging from 580 to 370,000 Daltons.
5.3 Results and Discussion 5.3.1 DTMS of the laserablated depthprofiles of the aged dammar and mastic films Tables 5.3.1.1 and 5.3.1.2 present the lists of identified compounds in the aged and laser ablated dammar and mastic films with DTMS. It should be noted that because of the sampling and the subsequent dissolution of the aged films in ethanol for the technical requirements of sampling for DTMS, the final mass spectra provide average readings with respect to the abundance of these compounds across depth. Both the aged ~ 55 µm dammar and mastic films (Appendix 2.7) are basically a mixture of triterpenoids with oxidised Arings at position C2 (5, 7, 152, 1720, 22 and 23), oxidised oleanane/ursane type triterpenoids at positions C11 and C28 (2428) and a series of unreacted triterpenoids (116). According to the optical absorption lengths at various wavelengths (Figure 4.3.2.6), this mixture indicates that the aged dammar and mastic films can be divided in three different parts in their thicknesses; namely a surface section that absorbed the UV wavelengths of the incident radiation, a deeper part that absorbed only the visible radiation and an underlying part in the bulk at which the transmitted visible light was of low intensity, which combined with the absence of oxygen did not lead to the destruction of the triterpenoid molecules.
2
Compounds 5, 7 and 15 are typical oxidised dammarane type molecules with an Aring opening at C2 (Mills and Werner 1955, Poehland, et al. 1987, De la Rie 1988a) and are present in triterpenoid resin films even prior to ageing (Van der Doelen, et al, 1998a, Dietemann 2003).
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Table 5.3.1.1 List of Compounds found in aged and laser ablated dammar resin films
Label Compound name
Mw
EI mass/charges*
NH3/CI mass/charges
1
Norαamyrone (3oxo28norurs12ene)
410
204, 410
428
2
Norβamyrone (3oxo28norolean12ene)
410
204, 410
428
3
Dammaradienone (3oxodammara20(21),24diene)
424
109, 205, 424
442
4
Dammaradienol (3βhydroxydammara20,24diene)
426
109, 189,207,408, 426
444
5
Dammarenolic acid (20hydroxy3,4seco4(28),24dammaradien3oic acid)
458
109, 440
476,458
6
Hydroxydammarenone (20hydroxy24dammaren3one)
442
109, 315, 355, 424
442
7
20,24epoxy25hydroxydammaran3one
458
143, 399
143
8
Oleanonic acid (3oxoolean12en28oic acid)
454
189, 203,248,409, 454
472
9
Ursonic acid (3oxo12ursen28oic acid)
454
189, 203,248,409, 454
472
11
Oleanonic aldehyde
438
203, 232, 409, 438
456
12
Ursonic aldehyde
438
203, 232, 409, 438
456
15
20,24epoxy25hydroxy3,4seco4(28)dammaren3oic acid
474
143
143
17
3,4seco28norurs12en3oic acid
428
204, 428
446
18
3,4Seco28norolean12en3oic acid
428
204, 428
446
19
3,4seco28norurs12en3,28dioic acid
472
203, 248 472
490
20
3,4seco28norolean12en3,28dioic acid
472
203, 248 472
490
21
Dihydrodammarenolic acid (20hydroxy3,4seco24dammaren3oic acid)
460
387, 456
478
3oxo25,26,27trisnordammarano24,20lactone
414
95, 99, 205, 315, 414
432
440
234, 275, 422, 440
458
468
217, 257, 276, 317, 482
486
440
234, 275, 422, 440
458
468
257, 276, 317, 482
486
24 25 26 27 28
17hydroxy11oxonorβamyrone (3,11dioxo17hydroxy28norolean12ene) 11oxooleanonic acid (3,11dioxoolean12en28oic acid) 17hydroxy11oxonorαamyrone (3,11dioxo17hydroxy28norurs12ene) 11oxoursonic acid (3,11dioxours12en28oic acid)
*
m/z in bold correspond to the highest relative intensity of the series according to previous findings (Van der Doelen, et al. 1998a)
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Chapter 5
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Table 5.3.1.2 List of Compounds found in aged and laser ablated mastic resin
Label Compound name
Mw
EI mass/charges
NH3/CI mass/charges
2
Norβamyrone (3oxo28norolean12ene)
410
204, 410
428
3
Dammaradienone (3oxodammara20(21),24diene)
424
109, 205, 424
442
5
Dammarenolic acid (20hydroxy3,4seco4(28),24dammaradien3oic acid)
458
109, 440
476,458
6
Hydroxydammarenone (20hydroxy24dammaren3one)
442
109, 315, 355, 424
442
7
20,24epoxy25hydroxydammaran3one
458
143, 399
143
8
Oleanonic acid (3oxoolean12en28oic acid)
454
189, 203,248,409, 454
472
10
Moronic acid (3oxoolean18en28oic acid)
454
189, 203,248,409, 454
472
11
Oleanonic aldehyde
438
203, 232, 409, 438
456
13
(Iso)masticadienonic acid (3oxo13α,14β,17βH,20αHlanosta8,24dien26oic acid or 3oxo13α,14β,17βH,20αHlanosta7,24dien26oic acid)
454
439, 454
472
14
3OAcetyl3epi(iso)masticadienolic acid (3αΑcetoxy13α,14β,17βH,20αHlanosta8,24dien26oic acid or 3αAcetoxy13α,14β,17βH,20αHlanosta7,24dien26oic acid)
498
438, 498
516
16
28norolean18en3one
410
163, 191, 410
428
18
3,4seco28norolean12en3oic acid
428
204, 428
446
20
3,4seco28norolean12en3,28dioic acid
472
203, 248 472
490
22
3,4Seco28norolean18en3oic acid
428
204, 428
446
23
3,4seco28norolean18en3,28dioic acid
472
203, 248 472
490
24
3oxo25,26,27trisnordammarano24,20lactone
414
95, 99, 205, 315, 414
432
440
234, 275, 422, 440
458
468
217, 257, 276, 317, 482
486
25 26
17hydroxy11oxonorβamyrone (3,11dioxo17hydroxy28norolean12ene) 11oxooleanonic acid (3,11dioxoolean12en28oic acid)
*
m/z in bold correspond to the highest relative intensity of the series according to previous findings (Van der Doelen, et al. 1998a)
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Chapter 5
A molecular study on the depthdependent gradients
100 Aged Dammar: surface 50
100 Aged Mastic: 50 Surface
0
0 100
 3 µm
50
% Relative Intensity
% Relative Intensity
100  3.5 µm 50 0 100  15 µm 50 0 100  20 µm 50
0 100
0 100
 25 µm
50
0
0
100
100
50
 16 µm
50
50
Unaged Dammar
0
Unaged Mastic
0 20
40
60
80
20
Scan No.
40
60
80
Scan No.
Figure 5.3.1.1 EIDTMS TIC’s of laserablated depth steps of (a) aged dammar and (b) aged mastic
Figure 5.3.1.1 presents the electron ionisation (16 eV) total ion currents (EIDTMSTIC’s) of the aged films prior to ablation and that of the laser ablated depthsteps after removal of 3.5, 15 and 20 µm from the aged dammar film and 3, 16 and 25 µm from the aged mastic film. The EIDTMSTIC’s of the unaged films are also presented for comparison reasons. The major shifts towards lower scan numbers across the successive depthsteps and especially of the volatile fraction that is desorbed via evaporation (Boon 1992) demonstrate that lower temperatures are required for sufficient desorption with depth. Consequently, the more material is removed from the surface of both aged varnish films the more volatile the remaining films become. The apparent shifts are so intense that after removal of 20 µm from the aged dammar and 25 µm from the aged mastic films the corresponding volatile fractions are evaporated with ~ 80% less heating compared to the temperatures that induced evaporation in the unaged and aged films prior to ablation. As discussed in Section 2.3, such shifts are 261
C. Theodorakopoulos
Chapter 5
A molecular study on the depthdependent gradients
related to changes in the polarity (Figure 2.3.1) (Van der Doelen 1999, Scalarone, et al. 2003). Moreover, the lower peak that emerges at high scan numbers represents the high molecular weight, crosslinked and condensed fraction of the analysed samples that desorbs in the ionisation chamber via pyrolysis at high degrees of temperature (i.e. T > 450°C) (Boon 1992). In both cases, pyrolysis occurs at lower temperatures, while the corresponding peaks become weaker with depth indicating that the high MW fraction becomes less polar and is reduced as a function of depth. It should be noted that the pyrolysis peak gives only an estimation of the high MW fraction, because at high temperature windows (> ~500°C) thermally stable material is formed corresponding to breakage of more or stronger bonds (Boon 1992) (see also Section 2.3). Despite this complexity the pyrolysis event is used often as a good indicator of the high MW fraction of aged dammar and mastic resins (Van der Doelen 1999, Scalarone, et al. 2003). According to the EIDTMS TIC’s then, light ageing of dammar and mastic films leads to lower degrees of oxidation, polarity and condensation or crosslinking as a function of depth. With regards to polarity this finding supports preliminary results across the laser ablated depth profile of hundredyearold natural resin varnishes, according to which solubility increases as a function of depth (Section 1.4, Theodorakopoulos and Zafiropulos 2003). In this particular example, the polarity of the remaining varnish across the laser ablated depthsteps was reducing almost logarithmically in the deeper parts of the tested varnishes (Figure 1.4.1). A similar phenomenon is observed here. Figure 5.3.1.2 shows the % volatilisation shifts against the logarithm of the laser262
C. Theodorakopoulos
Chapter 5
A molecular study on the depthdependent gradients
ablated depthsteps (lower xx’ axis) and the remaining film thickness (upper xx’ axis) (Table 5.3.1.3). The scan number of the EIDTMSTIC volatilisation peaks of the films prior to laser ablation was set as the maximum (100%). The corresponding plots quantify the shifts in the temperature windows that are responsible for the evaporation of the volatile fraction. An estimation of the degree of crosslinking versus depth is also provided based on the apex and broadness of the pyrolysis peak of the EIDTMSTIC’s. However, the corresponding SEC traces are more accurate with respect to the molecular weight modifications across depth (Section 5.3.4). Given that both intensity and broadness of the pyrolysis peaks enclose significant information about molecular weight, the areas of the pyrolysis peaks were measured. The plots of the pyrolysed high MW condensed fractions as a function of depth are presented in Figure 5.3.1.3. It is observed that the crosslinking gradient performs two apparent trends with depth in both films tested. The first trend is terminated at 15 µm from surface and from that point inwards new trends commence, which tend to reach the degree of pyrolysis determined prior to ageing. The outlined jump in the plots after removal of 15 µm from surface coincides with the change of the ablation step using a KrF excimer laser Table 5.3.1.3 Data extracted from EIDTMS TIC’s Laser ablated depth% Volatilization % Pyrolysis yield steps. Measured (Normalized with scan (Normalized with %area depth steps from surface (µm) no of aged surface) of aged surface)
Surface 1 2 3 4 5 6 7 Unaged
dammar 0 3.5 7 11.5 15 20 25 – 0
mastic 0 3 6 10 12 16 20 25 0
dammar 100 80 77 73 71 66 62 – 54
263
Mastic 100 98 95 94 92 89 88 84 77
dammar 100 86.44 81.76 80.42 78.96 50.68 46.79 – 41.80
mastic 100 85.13 82.59 79.64 78.79 78.35 59.61 58.52 53.24
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Chapter 5
A molecular study on the depthdependent gradients
Residual Resin Films (µm) 55
45
50
52
35
25
aged surface both films
100
% Volatilization (with reference to EIDTMS TIC's)
53
95 90 85 80
unaged mastic
75 70 65
Dammar Mastic
60
unaged dammar
55
0 0.01
2
3
4
5
6 7 8 9 10
20
30
Depthprofile (µm)
Figure 5.3.1.2 Gradient in volatilisation across depth. 55
45
40
35
30
aged surface (both films)
100
% Pyrolysis yield (with reference to EIDTMS TIC's)
50
25
20
Dammar Mastic
90
80
70
60
unaged mastic 50
unaged dammar 40 0
5
10
15
20
25
30
35
Depthprofile (µm)
Figure 5.3.1.3 Gradient in pyrolysis yield relative intensity across depth.
(Figure 3.3.5.3), as well as with the minimisation of the carbon dimer emission as recorded with LIBS (Figure 3.3.4.2). From a photomechanical point of view, excimer laser ablation of organic materials is dependent on the degree of polymerisation and crosslinking (Srinivasan and Braren 1989, Zafiropulos 2002). Thus, all the results
264
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Chapter 5
A molecular study on the depthdependent gradients
presented in the present work demonstrate that below the 15 µm surface layers neither of the aged dammar and mastic films is heavily crosslinked. Figures 5.3.1.4 and 5.3.1.5 present the ammonia chemical ionisation (NH3/CI) DTMS summation mass spectra of the aged dammar and mastic films prior to laser ablation (i), after removal of ~12 and 25 µm from surface (ii, iii) and prior to ageing (iv). The electron ionisation (EI) DTMS summation mass spectra of the aged films prior to ablation, after removal of increasingly thicker surface layers ranging from 3 to 25 µm, and prior to ageing are presented in Figures 5.3.1.6 (dammar) and 5.3.1.7 (mastic). In NH3/CI mode (250 eV) ammonia adduct molecular ions [M+NH4]+, [M+H]+ and [M+NH4H2O]+ cations are obtained resulting in molecular weight information, while in EI mode (16 eV) minor fragmentation is obtained resulting in structural and more detailed information. Interpretation of the mass spectra was based on earlier DTMS studies (Van der Doelen, et al. 1998a, Van der Doelen 1999, Van der Doelen and Boon 2000, Scalarone, et al. 2003), as well as on GC/MS and HPLCMS (Van der Doelen, et al. 1998a, Van der Doelen, et al. 1998b). The most characteristic molecular ions and ion fragments of the identified compounds are presented in Tables 5.3.1.1 and 5.3.1.2. Details of the incorporated compounds (Figure 2.2.1) and the ion fragments thereof are given in Section 2.3. It is demonstrated that the thicker the layer removed from the highly deteriorated surfaces, the more the molecular configuration of the remaining films tends to resemble the composition of the resins prior to ageing. The depthdependent shift towards a less oxidative state is evident even after removal of only 3.5 and 3 µm from the aged dammar and mastic films respectively, although the molecular changes are 265
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A molecular study on the depthdependent gradients
376
334
20
(i)
460 476
224 254
428
(Aged Surface) 40
446
490
(11.5 microns) 224 254
334
376
460 476
(ii)
490 0
(25 microns) 224 254
376
472
334
428 460 426 476
(iii)
458
50
442
rel. abundance %
50
426 428 442
0
486
0
224
334
254
376
458
50
460 442 428 426 472
(Unaged)
(iv) 476
0 200
300
400
500
mass/charge
Figure 5.3.1.4 NH3/CI DTMS summation mass spectra of the aged dammar film (total thickness: ~ 55 µm) prior to ablation (i), after removal of 11.5 µm (ii) and 25 µm (iii) from surface, and the NH3/CI DTMS of the same film prior to ageing (iv) (i)
428
(Aged Surface)
50
408
458
100
442
472 476 486 490
0
(ii)
442
50
408
458 472
428
(12 microns)
476 486 490
0
100
428
(25 microns)
50
408
(iii) 458 472
rel. abundance (%)
100
444
486
0
428
(Unaged)
408
(iv)
458 472
50
444
486
0 200
300
400
500
mass/charges
Figure 5.3.1.5 NH3/CI DTMS summation mass spectra of the accelerated aged mastic (total thickness: ~ 55 µm) prior to ablation (i), after removal of 12 µm (ii) and 25 µm (iii) from surface, and the NH3/CI DTMS of the same film prior to ageing (iv)
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more obvious with increasing depth. The unreacted compounds observed are the dammarane type triterpenoids (3, 4, 5, 6 and 7) with EI marker peaks at m/z 109, 143, 424 and NH3/CI marker peaks at m/z 442, 446, 478 and the oleanane/ ursane triterpenoids (1, 2, 8, 9, 10, 11 and 12) with EI marker peaks at m/z 189, 203, 204, 248, 409, 410 and NH3/CI marker peaks at m/z 428, 456, 472. These marker peaks are evidently increasing with depth. Although both types of triterpenoids are contained in both dammar and mastic, molecules with the dammarane skeleton are more abundant than those with the oleanane / ursane skeleton in dammar resin, while the opposite occurs in mastic resin. The recovery of the ion fragments corresponding to these marker compounds is more impressive in dammar, because of the abrupt reduction of the triterpenoid compounds that is observed in dammar resin upon ageing (Section 2.3.1, Van der Doelen, et al. 1998a, Van der Doelen 1999, Scalarone, et al. 2003). Oxidised compounds such as the oleanane / ursane type molecules with the Aring opening at position C2 (17/18, 19/20 for dammar and 18/22, 20/23 for mastic) with EI marker peaks at m/z 428, 472 and NH3/CI marker peaks at m/z 446, 490, which are in trace amounts in the aged films prior to ablation seem to disappear after removal of the uppermost 15 µm in both cases. Since these particular compounds are only formed upon UV irradiation during accelerated ageing (Van der Doelen, et al. 2000), it is indicated that the UV wavelengths of the incident radiation do not penetrate deeper than a 15 µm surface zone of the films. This finding is in a very good agreement with the optical absorption lengths, lα, determined for the aged dammar and mastic films tested (Figure 4.3.2.6), showing that both films absorb completely the incident
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A molecular study on the depthdependent gradients
radiation with λ < 350 nm at depths shorter than 15 µm from surface. This observation is confirmed with MALDITOFMS (Section 5.3.3). 143
80
399
428
454 456 470 472
408
359 355 371
482
300
350
400
450
248
424 426
95
234
257 275
315
359 355 371
414
408 410
189
40
399
109
387
60
500
( 3.5 microns) (ii)
204
80
20
250
440 454 456 470 472
200
428
150
143
99
482
0 100
150
200
350
400
450
500
( 7 microns) (iii) 248
109
189
99
95
235
257 275
315
359 371 355
426 424 440 408 454 456 470 472
204
60
414
80
20
300
143
100
40
250
399
0 100
150
200
100 80
20
300
350
400
450
500
( 11.5 microns) (iv) 204 248
60 40
250
143
109
424 426
189
95
235
257
315
275
355
359 371
408 399
440 454 456 470 472
% rel. abundance
315
0
100
% rel. abundance
234 257 275
414
95
424 426 440
387
109
40 20
248
410
204
60
100
% rel. abundance
(Aged Surface) (i)
99
% rel. abundance
100
0 100
150
200
250
300
350
400
450
500
mass/charge
Figure 5.3.1.6 EIDTMS summation mass spectra of the accelerated aged dammar film (total thickness: ~ 55 µm) prior to ablation (i), and after removal of 3.5 µm (ii), 7 µm (iii) and 11.5 µm (iv) from surface.
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143
% rel. abundance
100 80
424 426
60 40
408
189 257
234
20
315
275
410 440
355359 371 399
454
150
200
300
350
400
450
60
424 248
109
189
20
235 257 275
315
355
440
408 359 371 399
454
0 150
200
250
300
350
143
100
400
470
450
500
( 25 microns) (vii) 204
80 60
500
( 20 microns) (vi) 204
80
40
250
143
100
424 248
109
40
189
20
235
257 275
315
359 355 371
410 408
440 454 470
0 100
150
200
250
300
350
400
450
500
424
100
% rel. abundance
470
0
100
% rel. abundance
248
109
100
% rel. abundance
( 15 microns) (v) 204
(Unaged) (viii)
204
80 60 40
109
143
189
248 315
20
355
408
440 454
0 100
150
200
250
300
350
400
450
500
mass/charge
Figure 5.3.1.6 (continued) EIDTMS summation mass spectra of the accelerated aged dammar film after removal of 15 µm (v), 20 µm (vi) and 25µm (vii) from surface and the corresponding EIDTMS of the unaged film (viii) (total thickness: ~ 55 µm).
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95 109
20
399 395
250
300
350
468 470 472 484 498
400
450 500 ( 3 microns) (ii) 408
143
426
95 109
424
248
218 235
395 275 315 327
20
440 454
422
191203
399
163
60 40
200
468
355
484498
0 150
40
250
143
300
350
400 408
248
80 60
200
163
218 235
395 275
20
500
( 6 microns) (iii) 424
191 203
95 109
450
440 454
399
100
468
315 327 355
484 498
0 150
200
250
100 80
143 95
60
163
109
203 191
300
350
400
450 500 408 ( 10 microns) (iv)
248
424 440
218 235
40
315 327
273
20
395 355
399
100
454 468 484
498
0 100
150
200
250
350
400 408
100
143
80 163
203 191
95 109
218 235
20
450 ( 12 microns)
500 (v)
424
248
60 40
300
395
315 327
275
399
% rel. abundance
150
80
100
% rel. abundance
315 327 355
0
100
% rel. abundance
218
275
100
% rel. abundance
410 424 426 440 454 422
40
203 205 191
428
163
408
234
% rel. abundance
80 60
(Aged Surface) (i)
248
143
100
440 454 468
355
484 498
0 100
150
200
250
300
350
400
450
500
mass/charge Figure 5.3.1.7 EIDTMS summation mass spectra of the accelerated aged mastic film (total thickness: ~ 55 µm) prior to ablation (i), and after removal of 3 µm (ii), 6 µm (iii), 10 µm (iv) and 12 µm (v) from surface.
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408 ( 16 microns) (vi) 143
60
163
424
191 203
95 109
440454
218 235
315 327 355
275
20
399
80
40
248
484 498
0 100
150
200
250
300
350
400 408
60 40
163
191
203
95109
218
235
20
315
440 454 468
327
355
395
484 498
0 100
150
200
250
100
% rel. abundance
450 500 ( 20 microns) (vii)
424
248
399
143
80
400
450
408
191
440
163
95 109
500
( 25 microns) (viii)
424
203
143
60
20
350
248
80
40
300
218 235
399
% rel. abundance
100
275
315327
355
454 468
395
484
0 100
150
200
250
203
100
300
350
400
60
20
191
143
40 109
163
450
248
424440 454 218 235 315
150
200
(ix)
408
327
355
395
468 484 498
0 100
498 500
(unaged)
80
95
% rel. abundance
468
395
399
% rel. abundance
100
250
300
350
400
450
500
mass/charge Figure 5.3.1.7 (continued) EIDTMS summation mass spectra of the accelerated aged mastic film after removal of 16 µm (vi), 20 µm (vii), 25 µm (viii) from surface and the corresponding EIDTMS of the unaged film (ix) (total thickness: ~ 55 µm).
The DTMS summation mass spectra also detect traces of some oleanane / ursane type molecules oxidised at positions C11 and C28, which are listed in Tables 5.3.1.1 and 5.3.1.2 as compounds 2528 for dammar and 25 and 26 for mastic. The dammarane type compound 24, which is present in both films, has the same type of side chain
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oxidation (Figure 2.2.2b, Van der Doelen, et al. 1998a). Marker peaks of these particular oxidation products (2428) are monitored at m/z 234, 317, 414, 482 in the EI mode and at m/z 432, 458, 486 in the NH3/CI mode. These particular oxidation products have been detected by Van der Doelen and coworkers (1998a, 1999) in aged varnishes sampled from a large number of paintings kept in several galleries and museums. These products are reproducible in dammar and mastic resins upon light ageing excluding UV wavelengths or using an elaborate ageing process involving dissolution of the resins in a mixture of solvents with photosensitisers followed by irradiation in a fluorescent tube device (Van der Doelen and Boon 2000). Judged from the relative intensity of the corresponding m/z marker peaks, the abundance of these compounds is found to be equivalent to that of the UVinduced oxidation products with the oxidised Arings in both the aged dammar and mastic 55 µm films prior to ablation (Tables 2.3.5 and 2.3.6). The marker peaks of the nonUV induced oxidation products 2428 are in particular notable in the summation MS of the remaining films after removal of the 15 µm highly deteriorated surface layers. Some (minor) traces are also observed in the films even prior to ageing, indicating that the films prior to ageing were somewhat oxidised, as shown in Section 2.3. According to the presented data, the molecular changes are apparent by monitoring the gradual recovery of the abundances of the ion fragments corresponding to the unreacted, nonoxidised dammarane (3, 4, 5, 6 and 7) and oleanane / ursane type triterpenoids (1, 2, 8, 9, 10, 11 and 12) with depth. The molecular changes are less pronounced in the uppermost 15 µm, which absorb highly the UV wavelengths of the incident radiation. Strong molecular changes between surface and deeper parts of the
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films are observed after the removal of the highly deteriorated surface layers, especially in the case of dammar, where after removal of 20 µm from surface the characteristic ion fragments of the unreacted compounds produce relative intensities comparable to those of the same film prior to ageing. After removal of 25 µm from the surface of the aged mastic film the ion peak m/z 143, which is the characteristic ion fragment of the ocotillone type molecules, such as compound 7, decreases, indicating that the dammarane type molecules and their oxidation products influence the composition of mastic less than that of dammar. This is a genuine characteristic of mastic resin since the oleanane / ursane molecules are more abundant than the dammarane ones (Van der Doelen, et al. 1998a, Van der Doelen 1999, Scalarone, et al. 2003). The demonstrated molecular changes across the depthprofiles of the aged films determine that with increasing depth the composition of the films is gradually shifted towards a less deteriorated state similar to that prior to ageing. These findings are in a very good agreement with the results of the excimer laser ablation (Chapter 3), the laser induced transmission, the UV/VIS and the ATRFTIR measurements (Chapter 4). 5.3.2 Multivariant Factor Discriminant Analysis (DA) of the DTMS data across the depthprofiles of the aged dammar and mastic films. The observation of the depthdependent compositional gradients in the evidently complex DTMS summation mass spectra of the aged dammar and mastic films basically allows a qualitative demonstration of the molecular changes across depth. Multivariant factor discriminant analysis (DA) (Hoogerbrugge, et al. 1983, Windig, et al. 1983) was employed to quantify the modifications observed via the multivariate
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mass data. Figure 5.3.2.1 shows the first discriminant function (DF1) mass spectrum for the EIDTMS summation MS series of the remaining films after laser ablation at 159 194
248 232
189
(+)
354 375 385 387 413 431 438 456 472
dammar
315 342
224
399
440
()
355 205 109
143
410
203
(a) 426
204 424
(+) 472
387 413 428 431
mastic 354
0.2 0.0
109 143
0.4
82
468
438
315 355 235
399
()
189 218 191 440
0.2
454
0.6 0.8
(b)
439 203 424
1.0
248 408
Figure 5.3.2.1 Projections of the EIDTMS data classified with MFDA for the aged, laser ablated and unaged films of dammar (a) and mastic (b).
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successive depthsteps of both dammar (a) and mastic (b). The DTMS mass spectra of the unaged films are also included, so that the two opposite compositional cases, i.e. agedunaged, would mark the discrimination boundaries of the inserted analytical data. In short, judging from the molecular ions of the identified compounds that are listed in Tables 5.3.1.1 and 5.3.1.2, DA analysis shows that across the thickness of dammar, oleanane / ursane molecules 1 and 2, i.e. [M]+· = 410, and the dammarane molecules 3, 4 and 6, i.e. [M]+· = 424, 426, 442 respectively, peak intensely at the negative side of DF1. The characteristic and commonly most abundant ion fragments of the oleanane and ursane molecules 8, 9, 11 and 12, such as m/z 189, 203 and 409, peak at the negative DF1 side. The molecular ions of the latter compounds, i.e. [M]+· = 454 for 8, 9 and [M]+· = 438 for 11, 12 and the characteristic ion fragment at m/z 248, peak at the positive side of DF1, possibly due to overlapping with ion fragments of oxidation products of these molecules. The molecular ions of the UVinduced oxygenated products with the characteristic Aring openings, i.e., 17/18, ([M]+· = 428), 19/20 ([M]+· = 472) and 21 ([M]+· = 460), peak weakly at the positive DF1 side. Similarly, m/z 234, 317, 414 and 482 representing the nonUVinduced oxidation products (24, 25, 26, 27 and 28) peak in the negative side of DF1. The molecular ions of oxygenated compounds 5, 7 ([M]+· = 458) and 15 ([M]+· = 474) with oxidised Arings at C2, which are also detected in the composition of unaged dammar (Mills and Werner 1955, Poehland, et al. 1987, De la Rie 1988a), peak weakly at the positive DF1 side. The ionic fragments of their dammarane skeletons, i.e. m/z 109 and 440 (5),
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A molecular study on the depthdependent gradients
m/z 399 (7) and the ionic fragment m/z 143 derived from the ocotillone side chain (7, 15) peak at the negative DF1 side. The first discriminant function mass spectrum of the corresponding EIDTMS mass spectra of mastic is somewhat different compared to the DA results of dammar. In agreement with the DA analysis of dammar, oleanane molecule 1, i.e. [M]+· = 410, and the dammarane type molecules 3 and 6, i.e. [M]+· = 424, 442 respectively, peak intensely at the negative side of DF1. In contrast to dammar, the molecular ions of the oleanane type molecules 8, 10 ([M]+· = 454) and 11 ([M]+· = 438), peak at the negative DF1 side, confirming that these molecules are abundant throughout the depth of mastic. The same molecular ion with that of 8 and 10 corresponds to (iso)masticadienonic acid (13), the abundant presence of which is monitored at the intense marker peak m/z 439 on the negative DF1 side. Compound 14, which has an ionic fragment at m/z 438 that is also the molecular ion of 11, does not seem to play any significant role in the final discriminant analysis, since its molecular ion at m/z 498 coincides with the DF1 axis. The same occurs with m/z 163 that is characteristic of compound 16, although m/z 191, also corresponding to 16, peaks clearly at the negative DF1 side. In addition, the molecular ion of compounds 5 and 7 ([M]+· = 458) peaks at the positive DF1 side as in the case of dammar. The ionic fragments of these compounds, characteristic of their dammarane skeletons, i.e. m/z 109 and 440 for 5 and m/z 143, 399 for 7, peak at the negative side, although their relative intensity is lower than that in dammar. Finally, the molecular ions of the UVinduced oxidised triterpenoids 18/22, 20/23 (m/z 428, 472) and the marker peaks of the nonUVinduced oxidation
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A molecular study on the depthdependent gradients
0 .6 0.6
*1 *1
0.5 0 .5
5
*1 *1
5
Discrimination Function 2
0 .4 0.4 0 .3 0.3
4
4
0 .2 0.2
*9 *9
0 .1 0.1 *9 *9
00
77 88 4 4
77 77
8 6 4 2 0
2
4
6
8
0 . 6
Discrimination Function 2
0.3
(a)
33
0 10
0 . 4
0 . 2
YAxisTitle
0.6 0.4 0.2
0.4
33
33
66 66
10
0 . 4 0.4
*1 *1
22
88
0 . 2 0.2
0.5
44
2 2
55 6688
0 . 1 0.1
0 . 3 0.3
22
*9 55 *9
0
0.2
0.4
0.6
0.8
1
1 .2
0 0.2 0.4 0.6 0.8 1 Discrimination Function 1
1 .4
1.2
1.4
7 7 XAxisTitle
7
0.2
5
0.1
*10 *10 9
0 0.1 *10 *10
0.2
9 9
7 8 8 8 8
6 5 5 6 6 6
2 24 4 3 4 43 2 2 5 33 1 1 1 1
0.3
(b)
9
0.4 0.2
0.1
0 0.1 0.2 0.3 0.4 0.5 Discrimination Function 1
0.6
0.7
Figure 5.3.2.2 Score maps of DF2 versus DF1 of dammar (a) and mastic (b) depth steps. The ascending numbers represent the compositions of the successive laser ablated depth steps. Data points number (1) correspond to the aged surfaces, while numbers (9) for dammar and (10) for mastic to the unaged materials. The consecutive shifts across the DF1 from the very deteriorated surface layers towards the less deteriorated bulk, which resembles the composition prior to ageing, are evident.
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A molecular study on the depthdependent gradients
products 24, 25, 26 (m/z 414, 234, 276 and 482) peak weakly at the positive DF1 side, since these compounds are found in traces across the depth of the aged mastic film. Figure 5.3.2.2 presents the plots of the MS data points as coordinates in the score map of the first discriminant function (DF1) versus the second discriminant function (DF2). The geometric distance between the data points is an accepted measure of the demonstrated compositional differences (Hoogerbrugge, et al. 1983, Windig, et al. 1983). Therefore this distance is utilised here to illustrate the chemical changes across the depthprofiles of the tested films. The gradual shifts of the data points across DF1 represent the compositional gradient that is observed with the EIDTMS mass spectra. The data points from the lower to the higher numbers correspond to the consecutive depthsteps from the surface towards the bulk, while the highest numbers represent the MS data of the resins prior to ageing. Figure 5.3.2.3 demonstrates the data points along DF1 as a function of depth for the ~ 55 µm, aged dammar and mastic films. Both cases reveal a remarkable compositional gradient, which decreases from the surface. These results quantify the molecular modifications observed with DTMS and verify previous findings on the excimer laser ablation (Theodorakopoulos and Zafiropulos 2003, Theodorakopoulos, et al. 2005, Chapter 3) and the determined gradients in the optical properties characteristics of both films across depth (Zafiropulos, et al. 2000, Section 4.3.2). Although the delineated trends of the aged dammar and mastic films are not similar, there is a very significant common characteristic. As shown in Figure 5.3.2.2, DA analysis placed the origin of DF1 coordinate at a depth corresponding to 15 µm from the original aged surfaces. This is not irrelevant with the change of the ablative interaction upon 248 278
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Chapter 5
A molecular study on the depthdependent gradients
DTMS discrimination function 1
1.2
Depth steps
1.0 0.8 Thresholds of highly aged and less deteriorated material
0.6
(a)
0.4 0.2 0.0 fresh dammar: independent signal from film thickness
0.2 0.4 0
5
10
15
20
25
30
35
40
45
Depth from surface (µm)
0.2
Thresholds of highly aged and less deteriorated material
DTMS discrimination function 1
0.1 0.0 0.1 0.2
(b)
0.3 0.4 fresh mastic: independent signal from film thickness
0.5
Depth steps 0.6 0
5
10
15
20
25
30
35
40
45
Depth from surface (µm) Figure 5.3.2.3 MS projections of DF 1 as a function of depth in the ~ 55 µm thick, aged dammar (a) and mastic (b) films.
nm pulses (Sections 3.3.2, 3.3.3, 3.3.5). At this particular depth the ablation step changes upon KrF excimer laser pulses in both films (Figure 3.3.5.3). Since photochemical ablation is basically affected by the degree of oxidation and
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A molecular study on the depthdependent gradients
crosslinking of the laser irradiated material (Srinivasan 1994), these findings indicate that the 15 µm depth from surface of both the aged dammar and mastic films is the threshold between highly and poorly deteriorated material. This finding supports earlier findings by De la Rie (1988b), who determined that 10 µm dammar films deteriorate more than thicker films aged under the same conditions. It should be also noted that the data points corresponding to the deepest laserablated depthsteps (2025 µm from surface) emerge near the data points of the unaged films. Given that the total film thicknesses were of the order of 55 µm, it should be expected that below a depth of 25 µm from the original surface the composition of the films would eventually become identical to that prior to ageing. 5.3.3 Matrix assisted laser desorption/ionisation (MALDI) timeofflight (TOF) MS Some quantification of the molecular changes – resulting in the compositional depthdependent gradients – of the aged dammar and mastic films was obtained with MFDA based on the DTMS data. However, the changes monitored provided mean compositional information because the samples required for the DTMS analysis contained the remaining films. MALDITOFMS, using 2,5dihydroxybenzoic acid (DHB) as the matrix, was carried out to shed light on the composition of the ablated surfaces of the successive depthsteps across the thicknesses of the aged dammar and mastic films (Figure 5.3.3.1). Unfortunately, the low signaltonoise ratio of the mass spectra obtained did not permit analysis of mastic and all of the depth steps of
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A molecular study on the depthdependent gradients
(i)
dammar. The problem of low signaltonoise ratios of the MALDI/MS was also
f
M
M
d
M
ab
M
c
M
encountered in earlier studies of aged
h
?
(ii)
e? g M
?
M? M
triterpenoid resin films and therefore similar measurements were carried out
M
i
ab
MM
j M
c ? ? M
M
f g
?
(iii) M
with graphite instead of the matrix
M
(GALDI/MS), which results in spectra f M
a M 460
M
M
470
480
c
? M 490
? M
(iv)
g
M
500
mass/charge
Figure 5.3.3.1 MALDITOFMS of the DHB matrix (i), the aged dammar surface (ii), the surface of the film after laser ablation of the uppermost 15 µm (iii) and the surface of the dammar film before ageing (iv). Despite the relatively low peaktonoise ratio some molecular changes between the aged surface and the ablated surface are discriminated. For the assignments refer to the text.
without chemical noise (from DHB) (Zumbühl, et al. 1998, Dietemann 2003). However, the novel TOF technique with the TLC plates (see Section 5.2.3), employed for the present analysis, is different MALDI/MS
from
the
method.
established It
has
been
observed that with the present MALDITOFMS technique the interference of DHB is minimal, while all the m/z peaks generated correspond to the material on the TLC plates. Hence, the low signaltonoise ratio is most likely ascribed to the minimal subtraction of resin from the TLC plate. This phenomenon has been observed to be typical of the highly aged resin films (AMOLF, internal report). In the present work crucial conclusions were drawn from the mass spectra at the surface of the aged dammar film prior to ablation, the surface of the remaining film after removal of 15 µm and the surface of the unaged dammar resin. The identification of the mass peaks is tentative and proposed by comparison with data from the
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literature (Zumbühl, et al. 1998, Dietemann 2003). A representative mass / charge range observed in the MALDITOF mass spectra is within m/z 450 to 510, at which the sodiated ions [M+Na]+ of unreacted and oxidised triterpenoids are discriminated from the corresponding ion peaks of DHB (Zumbühl, et al. 1998). However, even within this range there are several m/z peaks attributed to DHB (marked with M in Figure 5.3.3.1), which can potentially interfere with the sodiated molecular ions of characteristic triterpenoid compounds (Zumbühl, et al. 1998). A relatively intense peak of DHB is at m/z 451 (not shown), where the sodiated molecular ions of UVinduced oxidation products 17/18 (428 Da, Tables 5.3.1.1, 5.3.1.2) with oxidised Arings are detected (Dietemann 2003). Other overlapping DHB mass / charges are at m/z 461 prohibiting detection of oleanonic and ursonic aldehydes 11/12 (438 Da), and m/z 477 that coincided with [M+Na]+ of oleanonic and ursonic acids, 8/9 (454 Da) (Dietemann 2003). The highest DHB peak within the examined range was at m/z 480 near a marker peak of dammarenolic acid, 5 (458 Da) at m/z 481 due to the [M+Na]+ (Dietemann 2003). In the unaged dammar film surface (Figure 5.3.3.1iv) peaks a, c, f and g at m/z 465, 481, 497 and 497.7 are discriminated respectively. Peak a at m/z 465 is assigned to the sodiated molecular ion of hydroxydammarenone (6) (442 + 23 = 465), c at m/z 481 to [M+Na]+ of dammarenolic acid, 5 (458 Da), and f at m/z 497 to [M+Na]+ of compound 15 (474 Da) (Dietemann 2003). Peak g at m/z 497.7 is attributed to the molecular ion of an oxidised dammarenolic acid 5, [M+Na+O]+, i.e 458+23+16 = 497 (Zumbühl, et al. 1998). Some changes compared to the MALDI MS of the unaged dammar film are monitored in the mass spectrum of the aged film surface (Figure 5.3.3.1ii). It is shown that the
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sodiated molecular ion peak of hydroxydammarenone, 6 (m/z 465, a) is less intense than in the unaged film. This is in good correspondence with the DTMS data showing that there is an abundance reduction of dammarane type triterpenoids upon ageing (Van der Doelen, et al. 1998a). Dammarenolic acid, 5, is recognised by marker peaks, m/z 481, c, which is attributed to [M+Na]+, and m/z 497.7, g, which is attributed to [M+Na+O]+, with the latter peak being more intense compared to the unaged dammar film. The high intensity at m/z 497 (f) indicates that the dammarane type degradation compound 15, which is generated during the biosynthesis of triterpenoid resins upon sunlight irradiation during exudation from the tree (Koller, et al. 1997, Dietemann 2003), is also abundant in the aged film. In the surface of the aged dammar film there are some peaks, such as m/z 467 (b), 483 (d), 495 (e) and 499 (h), which are not detected prior to ageing. These peaks correspond to the sodiated molecular ions of dammarenediol with a molecular weight of 444 Da (Mills and Werner 1955, Mills 1956, Poehland, et al. 1987, De la Rie 1988a), which was not reported in the present DTMS study, (b: 444 + 23 = 467), dihydrodammarenolic acid 21, (460 Da, d: 460 + 23 = 483), the UVinduced oleanane / ursane secoproducts with oxidised Arings 19/20, (472 Da, e: 472 + 23 = 495) and 20,24epoxy25hydroxy3,4secodammaran3oic acid that is a UVinduced dammarane type secoproduct with a molecular weight of 476 Da (Van der Doelen, et al. 2000, Scalarone, et al. 2003). The sodiated molecular ion of this particular compound peaks at m/z 499 (476 + 23), h. The MALDI MS of the film after the removal of the 15 µm surface layers is shown in Figure 5.3.3.1iii. It is observed that the marker peaks of the UVinduced secoproducts, at m/z 483, 495, 499 (d, e, h) are not present at this particular depth. Instead,
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peaks at m/z 463 and m/z 491 (i and j respectively) emerge, which are not detected in the mass spectrum of the aged or on the surface of the unaged dammar. These peaks represent the sodiated molecular ions of the oxidation compounds 25/27 (440 Da) and 26/28 (468 Da) respectively (Dietemann 2003). These are typical compounds formed upon oxidation of oleanane/ursane triterpenoids in the absence of UV wavelengths and have been reported in aged dammar varnishes on paintings (Van der Doelen, et al. 1998a, Van der Doelen 1999). This finding is in line with the decreasing number of carbonyl groups according to the ATRFTIR measurements (Section 4.3.3), as well as with the decreasing optical densities determined with the UV/VIS measurements (Section 4.3.2). Given that the optical absorption lengths, lα , of both the aged dammar and mastic films at λ > 350 nm are longer than 15 µm from the surface (Figure 4.3.2.6), it is suggested herein that Aring oxidation occurs under irradiation with λ < 350 nm. This suggestion agrees with findings by Van der Doelen and coworkers (1999, 2000), who determined the Aring oxidation upon irradiation with λ > 315 nm and the nonUVinduced oxidation upon irradiation with λ > 380 nm. Regardless of the wavelengths employed, it is demonstrated here that the bulk of aged dammar and mastic films is protected from UV irradiation anyway. 5.3.4 A study on the molecular weight across depth of aged dammar and mastic films with High Performance  Size Exclusion Chromatography (HPSEC) Although a decreasing gradient in the condensed, high molecular weight fraction of the aged dammar and mastic films was indicated with the weakening of the EIDTMS TIC’s pyrolysis peak with depth (Figure 5.3.1.3), size exclusion chromatography was carried out to determine depthwise molecular weight changes. Interpretation was
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carried out using earlier SEC studies of triterpenoid varnishes (De la Rie 1988b, Van der Doelen 1999). De la Rie (1988b) used Gel Permeation Chromatography (GPC), which is a synonymous method to SEC, and determined an average increase in the molecular weight of accelerated aged dammar resins up to 1500 Da. Zumbühl and coworkers (1998) also reached the same conclusion measuring the mass/charges of light aged dammar films on the basis of GALDI/MS. Van der Doelen and coworkers (1999, 2000), used SEC with UV (240 nm) and VIS (400 nm) detectors to detect compatibility of naturally aged varnishes and accelerated aged resins. In the present work, SEC traces at 240 and 400 nm of the unaged, aged and laser ablated dammar and mastic films were studied to detect MW changes across depth. The same quantity of material was used from all the films tested and dissolved in THF (~10mg/µl), while about 20 µl of the prepared solutions were injected in the column. It was therefore expected that the final chromatograms would provide information about the molecular weight distribution across depth as a function of the absolute absorbance at 240 and 400 nm. However, this was not possible because of the depthdependent gradients of the optical properties of the aged films (Figures 4.3.2.1b, 4.3.2.2b), which affected the intensities of the resulting SEC traces accordingly. Therefore, all the SEC traces were normalised at a % relative absorption. Figure 5.3.4.1 presents the SEC traces at 240 and 400 nm of the dammar film prior to and after accelerated ageing. The molecular weight increase upon ageing is evident with both detectors. Absorption at both wavelengths is separated in a small fraction of sesquiterpenoids (~ 200 Da), the group of triterpenoid molecules (400/500 Da), which being the most abundant compounds absorb strongly, a small fraction of dimerised
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and oxidised triterpenoid molecules (900/1000 Da) and traces of the condensed high MW fraction at 10 kDa, that are mainly formed upon the radical polymerisation of polycadinene (Van Aarssen, et al. 1990). The highest absorption at 240 nm of the aged dammar film is obtained at 900/1000 Da, while the triterpenoid molecules (400/500 Da) absorb slightly less than their dimers, forming a double peak within the range 400 to 1000 Da. This double peak broadens towards the higher MW regime, up to 15 kDa. Absorption of the ~ 200 Da group is critically reduced upon ageing indicating that both low MW components and triterpenoid molecules participate in the formation of longer molecular chains during degradation or that the low MW fraction escaped the film (Boon and van der Doelen 1999). The SEC trace at 400 nm of the aged film shows that the group of molecules absorbing intensely in the blue have a molecular weight in the order of 400 Da. This MW points to triterpenoid molecules, while the second intense peak at 800/900 Da shows that oxidised and dimerised triterpenoids absorb strongly in the blue region. SEC traces of aged varnishes from paintings at 400 nm detected a MW increase absorbing strongly at molecular weights greater than 1000 Da, while artificially aged dammar resins in solution absorbed intensely at ~ 2000 Da with shoulders at higher molecular weights (Van der Doelen 1999, Van der Doelen and Boon 2000). This mass increase was associated with yellowing. In the present work, the strongest absorption in the blue was obtained at 400 Da.
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100
% Relative absorption
80
aged at 240 nm unaged at 240 nm aged at 400 nm unaged at 400 nm
60
40
20
0 10000
1000
100
Molecular Weight (Da) Figure 5.3.4.1 SEC traces of the aged and unaged dammar films at 240 nm and 400 nm.
Figure 5.3.4.2 demonstrates the SEC traces at 240 nm of consecutive depthsteps down to 15 µm from the surface of the aged dammar film and the SEC trace of the same dammar film prior to ageing. It is observed that the molecular weight across the highly deteriorated surface layers of the aged dammar film does not change significantly, which is in agreement with the intense carbon dimer emission upon 248 nm laser ablation (Figure 3.3.4.2) as well as with the constant ablation step in the 15 µm surface layers (Figure 3.3.5.3). The results agree also with the DTMS data of the 15 µm surface layers of the aged dammar film, showing that there is little change in the abundance of the mass/charges corresponding to the triterpenoid molecules, (400/500 Da). A small shoulder peak, increasing at about 200 Da on the SEC traces at 240 nm (Figure 5.3.4.2) of the ablated depthsteps, indicates the presence of some low MW products with depth, which eventually escape in the atmosphere (Boon and van der Doelen 1999). Figure 5.3.4.3 shows the SEC traces of the same depthsteps of the
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aged dammar film at 400 nm and the SEC trace of the film prior to ageing. Two facts are apparent in this chromatogram. First, ageing results in a slight increase of the MW 100
36
% relative intensity
80
1 60
40
2 20
0 100000
10000
1000
100
Molecular Weight (Da) Figure 5.3.4.2 HPSEC plots of dammar at 240 nm before ageing (1), after ageing (2) and those of the laser ablated depth steps between 3.5 and 15 µm (36). 100
2 80
% relative intensity
36 60
1 40
20
0 100000
10000
1000
Molecular Weight (Da) Figure 5.3.4.3 HPSEC plots of dammar at 400 nm. For comparison the traces of the unaged (1), aged (2) and the ablated depths from 3.5 to 15 µm (36) are demonstrated.
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of the triterpenoid compounds (400/500 Da), which is related to oxidation. Second, there is an abrupt decrease in the absorption from the crosslinked fraction of the dimerised triterpenoids (900 Da). These products participate in the formation of yellow compounds absorbing strongly in the blue and are responsible for the colouration effect in the aged dammar (Van der Doelen and Boon 2000). The decrease in the absorption at 400 nm between the aged surface and the layers below, especially at 900 Da, indicates a decreasing degree of yellowing with depth, which has been determined in laser ablated aged varnishes on paintings (Theodorakopoulos and Zafiropulos 2003, Section 1.4). 100
% Relative absorption
80
aged at 240 nm unaged at 240 nm aged at 400 nm unaged at 400 nm
60
40
20
0 100000
10000
1000
Molecular Weight (Da) Figure 5.3.4.4 HPSEC traces of the aged and unaged mastic film at 240 and 400 nm
Figure 5.3.4.4 presents the SEC traces of mastic prior to and after ageing at 240 nm and 400 nm. At 240 nm the unaged film absorbs strongly at 400 Da owing to the triterpenoid molecules. There is a shoulder peak at ~ 1000 Da caused by oxidised triterpenoid dimers and an apparent hump at ~ 10 kDa that represents a condensed fraction of mastic. At 400 nm the unaged film absorbs strongly at ~ 300 Da owing to
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unsaturated triterpenoid ketones and quinones (Formo 1979) that are responsible for yellowing (Figure 2.2.4). The fraction of 1000 Da absorbs less intensely at 400 nm than at 240 nm, while there is also a small fraction at 10 kDa that absorbs the radiation equally at both wavelengths. Upon ageing absorption due to triterpenoids molecules (~400 Da) is strong in both wavelengths, but a shift towards higher MW indicates oxidation. The increased absorption at 1000 Da (240 nm) and ~ 900 Da (400 nm), indicates the increasing concentration in oxidised triterpenoid dimers (De la Rie 1988b). The broadening of the peaks suggests the existence of triterpenoid trimers, tetramers and longer crosslinked chains that appear as shoulders at 10 kDa. Absorption at 400 nm is relatively strong from about 3000 Da to almost 80 kDa compared to the SEC trace at 240 nm. According to Van der Doelen and Boon (2000) these long molecules absorb strongly in the blue and are responsible for the yellowing effect of the triterpenoid varnishes. Indeed, unlike the aged dammar film, the aged mastic film obtained a stronger yellow hue after several weeks in the dark following an extreme light ageing procedure, which initially bleached the yellow chromophores (Chapter 2). Therefore, the depthdependent molecular weight decrease of the aged mastic film is better illustrated with the SEC chromatograms at 400 nm (Figure 5.3.4.5). The SEC traces of the unaged mastic film and the aged film prior to ablation are shown for reasons of comparison. It is observed that there is an evident gradient in the absorption from the high MW fraction across depth. In particular, the abundances of the 900/1000 Da components and the condensed fraction in the high MW regime up to 80 kDa are gradually decreasing with depth. As in the case of DTMS, sampling for SEC included the remaining varnish film (total thickness ~ 55 µm) resulting in average molecular weight determination of the remaining films. Therefore, the 290
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absolute molecular weight as a function of depth will most likely display a sharper gradient than that demonstrated in Figure 5.3.4.5.
Relative absorption at 400 nm (%)
100
aged surface  16 µm
80
 3 µm 60
40
 27 µm 20
unaged 0 100000
10000
1000
100
Molecular Weight (Da) Figure 5.3.4.5 HPSEC traces at 400 nm of the aged and unaged and laser ablated mastic film at successive etchings at 3, 6, 9, 12, 16, 20, 25 and 27 µm from the surface. The apparent MW reduction across the depth profile is evident.
5.4 Rationalisation of triterpenoid varnish ageing Recently an ageing model of triterpenoid varnishes has been proposed (Boon and van der Doelen 1999). The model was based on experimental observations of molecular studies carried out with GCMS and LCMS analysis of fresh and aged triterpenoid varnishes and orientational studies on sterols, which have comparable carbon skeletal ring structures as the triterpenoid molecules. In view of the current mass spectrometric and chromatographic analytical data based on DTMS, MALDIMS, HPSEC, UV/VIS and ATRFTIR (Chapter 4) on successive depths of the excimer laser ablated triterpenoid films, an updated schematic model on the degradation of these varnishes is proposed. The current findings are integrated into the previous model, aiming at a better understanding of the nature of aged triterpenoid varnishes. In line with the 291
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previous model (Boon and van der Doelen 1999), we assume the existence of a considerable degree of orientation of the triterpenoid molecules forming flat sheetlike structures on a molecular level before the degradation process begins. A first phase in triterpenoid varnish degradation corresponds to photoinitiation and the early oxidation stages. As soon as the varnish is exposed to light, the triterpenoid molecules lying on the surface varnish layers absorb the short wavelengths of the incident ambient radiation and hence the longer wavelengths (λ > 350 nm) are transmitted to the triterpenoids in the bulk. In line with Beer’s Law, absorption of light from the resin compounds reduces logarithmically across the depth of the film. The depthdependent qualitative and quantitative reduction of light leads to decreasing degrees of excitation. Thus, the free radicals formed across depth are quenching as the light propagates deeper into the bulk. Given the findings in the present work it is understood that the free radicals in the surface layers are more abundant than in the bulk because more intense bondbreakage occurs in the surface. Certainly, the presence of oxygen and the mobility of the radicals into the triterpenoid films play a significant role in the degradation process and the delineated quenching with depth. The excited molecules and the free radicals generate new covalent bonds between each other forming so new high molecular weight compounds. Because of the gradient in light propagation (Chapter 4), the extent of this condensation process decreases accordingly with depth. At the same time, the oxygen diffusion from the airsurface interface into the bulk is also decreasing (Thomson 1965, 1979), thereby leading to high degrees of functionalisation on the surface layers and oxygen depletion in the bulk. Because of 292
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the absorption of the short wavelengths (λ < 350 nm) from the surface layers only, different oxidation products are formed in the surface and the deeper layers, which is supported by the existence of Aring oxidation at C2 in the surface layers and oleanane / ursane type molecules oxidised at C11 and C12 in the bulk (Section 5.3.3). The degree of oxygen consumption across depth is important for the propagation reactions (Section 2.2), which originally are initiated in an decreasing rate across the depth profile of the film (top to bottom) (Schoolenberg and Vink 1991). Hence, depending on the degree and the type of oxidation, water molecules are attracted by the oxidised triterpenoids across the film thickness. This attraction leads to hydrogen bonding between the polar groups and the water molecules creating weak hydrogen bonded ‘crosslinks’ (Boon and van der Doelen 1999), the abundance of which decreases with depth. The sheetlike orientations of the triterpenoid compounds are disturbed, especially at the surface, as soon as these structural modifications take place from the first stages of degradation. Both types of crosslinking reduce the mobility of the triterpenoid compounds especially on the surface, thereby leading to surface stiffness, the degree of which is accordingly reducing with depth. At the same time an equivalent gradient in polarity across thickness is also formed. At a second phase, the varnish proceeds to further degradation that is led by further oxygen consumption, bond breaking and polymerisation processes. The already modified triterpenoid molecules at the uppermost layers of the film are further degraded forming smaller molecular species, which no more have the characteristic triterpenoid carbon ring skeleton, as determined by analytical data based on GC and HPLC (Van der Doelen, et al. 1998b, Boon and van der Doelen 1999). The high
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degree of functionalisation of the triterpenoids at the surface results in even stronger absorption of the short wavelengths by the outer layers, which prevents penetration of UV wavelengths into the bulk. The overall propagation of light into the film is shortened further and Beer’s Law is no longer valid for all the film thickness because of the abrupt optical changes across depth. This is now well understood, because of the decreasing trends of the absorption coefficients progressing into the depths of aged mastic and dammar films (Section 4.3.2) (Zafiropulos, et al. 2000). According to the previous theoretical model on triterpenoid ageing, small degradation products, such as functionalised carbon compounds (aldehydes, ketones and acids) yielded from bond breaking and ring disintegration, escape to the atmosphere (Boon and van der Doelen 1999). At the same time bondbreakage, excitations and oxygen consumption is continued gradually in the underlying layers although all trends would be decreasing with time (Thomson 1965, 1979). Hence, further attraction of water molecules is consistent with the degree and the depth of oxygen consumption across the depthprofile. Crosslinked fractions have reduced molecular weight with depth. Crosslinks at the surface may well contain more polar that apolar groups, changing gradually the plasticity of the varnish at the exposed surface. A final phase corresponds to further degradation of highly deteriorated triterpenoid varnish. Elimination reactions in the high molecular weight condensed fractions may lead to formation of aromatic centres, further influencing the light absorbing properties of the surface layers. On the surface, residual polar triterpenoid and polar parts of high molecular weight species are linked with water molecules. The high concentration of the three dimensional covalentbonded polymer network may lead to
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a certain degree of insolubility of the surface layers. Further exposure to radiation leads to depolymerisation of the surface by continuous bondbreakage resulting from excitation of the absorbing functional groups attached to the condensed surface layers. On the other hand, the bulk is preserved in a somewhat ‘unaged’ state, since all the deteriorative actions are carried out in the uppermost layers. Both the extent of polarity and crosslinking in the deep layers are comparable to that of the starting film, protecting the substrate from the deteriorative consequences of the ambient conditions.
5.5 Conclusions In conclusion, deterioration of dammar and mastic varnish films formed upon ageing is critically dependent on their thicknesses. Based on the results shown above it is determined that thick dammar and mastic films protect photosensitive substrates from deteriorative UV light wavelengths and from exposure to oxygen, preserving them in the best possible condition for a considerable lifetime. The transition from the highly degraded surface to the less reacted or even unreacted bulk layers, is a matter of a depthdependent gradient both in terms of oxidation, as shown herein with DTMS, and in terms of the photochemically induced high MW condensed material, detected by HPSEC. The presented results support all the findings demonstrated above on the change of the interaction of the aged resin films with the UV laser photons (Figures 3.3.2.2, 3.3.2.4, 3.3.3.1), the decreasing carbon dimer emission upon consecutive laser pulses (Figures 3.3.4.2a, b), the reduction of the ablation step (Figure 3.3.5.3), the change of the laserinduced transmission rate (Figures 4.3.1.2.2a, b, 4.3.1.2.4), the decreasing optical densities (Figures 4.3.2.1b, 4.3.2.2b, 4.3.2.4b) and the decreasing
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number of carbonyls (Figures 4.3.3.2i, ii, 4.3.3.3a, b, 4.3.3.4) as a function of depth, on the molecular level. Oxidation not only reduces gradually as a function of depth, but the transition from highly oxidative and polar states in the surface towards nonoxidative states in the bulk is, in addition, qualitative. Based on the MALDITOFMS results, it is determined herein that UVinduced oxygenated triterpenoid compounds, such as oleanane / ursane type molecules with oxidised Arings (Figure 2.2.2c), are formed only in the 15 µm surface layers despite the fact that both the 55 µm films tested were irradiated with light of λ > 295 nm for about 500 hours (Section 2.7.2.2). At less than 15 µm below surface only oleanane / ursane type molecules with side chain oxidation at positions C11 and C28 (Figure 2.2.2b); these are detected as typical compounds of oxidised varnishes found in paintings (Van der Doelen, et al 1998a). This finding is in line with the optical absorption lengths of light with λ < 350 nm that is completely absorbed in the 15 µm surface layers (Figure 4.3.2.6), which indicates that wavelengths shorter than 350 nm are responsible for the Aring oxidation. Hence, wavelengths longer than 350 nm penetrate in the deeper layers of the film, as also shown in Figure 4.3.2.6. Similar results have been reported by (Feller 1994b), who noted that the penetration depth of 350 nm radiation is ten times that of 320 nm into aromatic polyester films. Although oxidation is considered as the main problem of natural resin varnishes (Section 2.2), the decreasing gradient in the high MW condensed fraction that is formed during photochemical degradation should not be underestimated. Provided that high MW crosslinks are insoluble (Stolow 1985, De la Rie 1988b), the suggestion that upon chemical treatments only the volatile compounds are removed and the condensed material remains on the treated surface (Boon and van der Doelen 1999) is fully supported here. As discussed in Section 296
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2.2.2, this potential influences the appearance of the substrate, especially if this is paint, because of the overall rise of the MW and the high viscosity (De la Rie 1987, Berns and De la Rie 2002, 2003) formed in the new varnish, which eventually will be integrated with the high MW condensed fraction of the older varnish that remains on the surface. This possibility and the fact that no oxidative contribution of the 248 nm laser pulses to the remaining films was detected is a good argument for the excimer laser ablation of aged varnish films.
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5.6 References Berns, R. S. and De la Rie, E. R. 'The relative importance of surface roughness and refractive index in the effects of varnishes on the appearance of painitngs'. In Preprints of the 13th triennial meeting of the ICOM Committee for Conservation, Ed. V. R., Vol. I, James & James (Science Publishers) Ltd, Rio de Janeiro, (2002), 211216. Berns, R. S. and De la Rie, E. R., 'The effect of the refractive index of a varnish on the appearance of oil paintings', Studies in Conservation 48 (2003) 251262. Boon, J. J., 'Analytical pyrolysis mass spectrometry: new vistas opened by temperatureresolved insource PYMS', International Journal of Mass Spectrometry and Ion Processes 118/119 (1992) 755787. Boon, J. J. and van der Doelen, G. A. 'Advances in the current understanding of aged dammar and mastic triterpenoid varnishes on the molecular level'. In Firnis: Material  Aesthetik  Geschichte, International Kolloquium, Braunschweig, 1517 Juni 1998, Ed. A. Harmssen, HertogAntonUlrichMuseum, Braunschweig, (1999), 92104. Carlyle, L. A. The artist's assistant: Oil painting Instruction manuals and handbooks in Britain 18001900: with reference to selected eighteenth century sources, Archetype Publications, London, 2001. Castillejo, M., Martin, M., Oujja, M., Silva, D., Torres, R., Manousaki, A., Zafiropulos, V., van den Brink, O. F., Heeren, R. M. A., Teule, R., Silva, A., and Gouveia, H., 'Analytical study of the chemcial and physical changes induced by KrF laser cleaning of tempera paints', Analytical Chemistry 74 (2002) 46624671. Cunliffe, A. V. and Davis, A., 'Photooxidation of thick polymer samples. Part II: The influence of oxygen diffusion on the natural and artificial weathering of polyolefins', Polymer Degradation and Stability Vol. 4 (1982) 17  37. De la Rie, E. R., 'The influence of varnish on the appearance of paintings', Studies in Conservation 32 (1987) 113. De la Rie, E. R., 'Stable Varnishes for Old Master Paintings', PhD Thesis University of Amsterdam, (1988a). De la Rie, E. R., 'Photochemical and thermal degradation of films of dammar resin', Studies in Conservation 33 (1988b) 5370.
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Dietemann, P., 'Towards more stable natural resin varnihses for painintgs', PhD Thesis Swiss Federal Institute of Technology, Zurich, (2003). Feller, R. L. 'Depth of Penetration of Light into Coatings; and Influence of Sample Thickness and Oxygen Diffusion'. In Accelerated Aging: Photochemical and Thermal Aspects, Ed. D. Berland, The Getty Conservation Institute, USA, (1994a), 5661 and 135137. Feller, R. L. 'Depth of Penetration of Light into Coatings'. In Accelerated Aging: Photochemical and Thermal Aspects, Ed. D. Berland, The Getty Conservation Institute, USA, (1994b), 56  61. Formo, M. W. 'Paints, varnishes and related products: Discolouration'. In Baile's Industrial Oil and Fat Products, Ed. D. Swern, Vol. 1, John Wiley & Sons, New York, (1979), 722724. Fukushima, T., 'Deterioration Processes of Polymeric Materials and their Dependence on Depth from Surfaces', Durability of Building Materials Vol. 1 (1983) 327  343. Hoogerbrugge, R., Willig, S. J., and Kistemaker, P. G., 'Discriminant analysis by double stage principal component analysis', Analytical Chemistry 55 (1983) 17101712. Koller, J., Baumer, U., Grosser, D., and Schmid, E. 'Mastic'. In Baroque and Rococo Lascquers, Ed. J. Koller, Vol. 81, Arbeitshefte des Bayerischen Landesamtes fuer Denkmalpflege, Karl M. Lipp Verlag, Muenchen, (1997), 347358. Mantell, C. L., Kopf, C. W., Curtis, J. L., and Rogers, E. M. 'Oil Varnishes'. In The technology of natural resins, John Wiley & Sons, Inc., (1949), 265319. Mills, J. S. and Werner, A. E. A., 'The chemistry of dammar resin', Journal of Chemical Society (1955) 31323140. Mills, J. S., 'The constitution of the natural, tetracyclic triterpenes of dammar resin', Journal of Chemical Society (1956) 21962202. Mills, J. S. and White, R. The Organic Chemistry of Museum Objects, 2nd edition, ButterworthHeinemann, Oxford, 1994. Papageorgiou, V. P., BakolaChristianopoulou, M. N., Apazidou, K. K., and Psarros, E. E., 'Gas chromatographicmass spectrometric analysis of the acidic triterpenic fraction of mastic gum', Journal of Chromatography A 769 (1997) 263273.
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N., and T. Taniuti (1970). Reductive perturbation method for nonlinear wave propagation m inhomogeneous media Il. J. Phys. Soc. Japun, 29,209214.[78] Ashky, H. (1967). Multipie pcaliag in flight vehicle dynamic analysis  a preliminary look. AIAA Paper No. 67560.(2331 Avila, G. S. S., and J. B. Keller, (1963). The high frequency asymptotic field of a point source in an inhomogeneous medium. Cbmm Aue AppL Ahth.. 16,363381. [ 3801 Babich, V. M. (1965). A point source in an inhomogeneow medium. 2. V y d . Mar. f Mut. Fyz., 5,949951;EpglishtrcmsL,usSRCbmput~rkloth.Mth.Ph~8., 5,247251.[380] Babich, V. M. (1970). MarhemtiuaI ptoblcms in Wave Ropugation Thawy, Part I. Plenum, New York. [ 361,3801 Babich, V. M. (1971).hfathemaffalh b k m s in Wave hpugation Theury. Part II. Pknum, New York. [361.380] Babich, V. M., and T. S. Kravtsova, (1967). Propagation of wave film type oscillations of quantizedthfclmcrs.P W . 31,204210. [380] Balcscu,R. (1963). Smt&tidMech&sof CkgdAvrider Why, New York. [367.369) BaU, R. H. (1964). Nonlinea~theory of waves in a cold homogeneous plasma without magnetic &ld. Stauford Microwave Laboratory Rept. No. 1200. [234] Bankoff, S. G. See Chang, Akius, and Bankoff. Barakat, R., and A. Houston, (1968). Nonlinear periodic capiliarymvity waves on a fluid of fdte depth.J. Ceophyg Rer., 73 65456554.[771 Barcilon. V. (1970). Some inertial modifications of the h e a r viecous theory of steady rotating fluid flowaPh~aFhddr. 13,537544.[235] Barnu, R. B. (1970). Converlpnce of the von Zeipel procedure. CehtiulMe&, 2,494504. (1921 Barua, S . N. (1954). sbcondary flow in a rotating stm&#t pipe. Roc. Roy. Soc. (London), A227,133139.[79] Bauer, H. F. (1968). Nonlineat response of elastic plates to pulse excitations. J. AppL Mech., 35,4752.[S8] Bellman, R. (1955). Remuch problems. Buu Am Math Soc., 61,192.[22] Bellman, R. (1964). PaMkrtfon Techniques in Mathemrrtia, Physics, and Engineering. Holt, New York. [3091 Benney, D. J. (1965). The flow induced by a disk oscillating about a state of steady rotation. QUd. J. M.APPL M t h . , 18,333345.[235] Benney, D.J. (1%6). Long waves on liquid h . J . Moth. amfPhyx 45,150155.[41] Benney, D. J.(1%7). The asymptotic bellavior of nodinear dispersive waves. J. Marh. und P&&, 46,115132.[234] Benney, D. J. See Ablowitz and €bnney. Bcnney, D. J., and A. C. Newcll, (1967). Sequential time closures for interacting random waves. J. b¶iath. and my&, 4,363393.(2341 Bcnney, D. J., and A. C. Nenen, (1969). Random wave closures. Stud. Appl. Moth.,48, 2953.[234] m y , D. I., and G. J. R o b s , (1969). Wave instabilities. Sfud. AppL Muth.,48, 377385. [2351 Benney, D. J. and P. G. Wman, (1%6). Nonlinear mteractions of random waves in a dhpcrsb medium.Awz Roy. Soc. (London). 9,301320. [234] Bethe, H. A. See Salpeter and &the.
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389
Bisshopp, F.(1969). A modified stationary principle for nonlinear waves. J. Diff. Equations, 5,592405. [216] Bleistein, N. (1967). Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singulatities. J. Math. Mech., 17,533359.[380] Bleistein, N. See Lewis, Bleistein, and Ludwig. Boersma, J. See Ahluwalia, Lewis, and Boersma. Bogoliubov, N. N. See Krylov and Bogoliubov. Bogoliubov, N. N., and Y. A. Mitropolski, (1961). Asymptotic Methods in the Theory of Nonlineor Osci&@on& Gordon and Breach, New York [ 168,174J Bohlin, K. P. (1889). Ube~eine neue Annaherungsmethode der Storungstheorie, Akad Handl. BUmng.. 14,(Afdl, Stokholm). [56] Born, M. (1926).Quantum mechanics of impact processes. 2.Phyr, 38,803827.(3621 Bourret, R. C. (1962a). Propagation of randomly perturbed Filds. &n. J. Phys., 40, 782790.[367,369,372] Bourret, R. C. (1962b) Stochastically perturbed fwlds, with applications to wave propagation in random media. Nuovo Cimenro, 26, 131.[ J 367,369,3721 Brau, C. A. (1967). On the stochastic theory of dissociation and recombination of diatomic molecules in inert dilutens. J. Chem Phyr, 47,11531163.[235] Breakwell, 1. V., and L. M. Perko, (1966). Matched asymptotic expansions, patched conics, and the computation of interplanetary trajectories. In &ogress in Asfromutics and A m m u t i a , Vol. 17, Method in AJhrodymmics and Celestid Mechanics (R. L. Duncombe and V. G. Szebehely, Eds.), Academic, New York, pp. 159182.[ 1381 Breakwell, J. V., and R. Pringle, Jr.(1966). Resonances affecting motion near the earthmoon equilateral libration points. In Rogress in Astronmrticsand Aeronautics, Vol. 17,Methods in A d y ~ m i c s a n CelesrrblMechanics d (R. L. Duncombe and V. G. Szebehely, Eds.), Academic, New York,pp. 5574.[ 1991 Bretherton, F. P. (1962). Slow viscous motion round a cylinder in a simple shear. J. a i d Mech., 12,591613.[110] Bretherton, F. P. (1964). Resonant interactions between waves. The case of discrete oscillations. J. FluidMech., 20,457479. [217,227,267,298] Bretherton, F. P. (1968). Propagation in slowly varying waveguides hoe. Roy. Soc. (London), A 302,555576.1216) Bretherton, F.P., and C. J. R. Garrett (1968).Wavetrains in inhomogeneous moving media. Roc. Roy. Soc fLondon).A302,529554. [216J BriUouin, L. (1926).Remarques sur la mecanique ondulatoire. J. Phyr Rudium, 7,353368. [ 114,315,3391 Brofman, S. See Ting and Brofman. Brofman, W. (1967). Approximate analytical solution for satellite orbits subjected to small thrust or drag.AIAA J., 5,11211128.[233] Bromberg, E. (1956). Nonlinear bending of a circular plate under normal pressure. Comm Pure A p p l Mkrh.. 9,633659.[ 144,1461 Brown, W. P., Jr. (1966). Validity of the Rytov approximation in optical propagation calculations.J. Opt. Soc. A n , 56,10451052.[373] Brown, W.P., Jr.(1967). Validity of the Rytov approximation. J. Opt. Soc. A m 57,15391543. [ 373J
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B d M. A., and A. I. S o h , (1966). A new perturbation techniquefor differential equations with ~ p p t p m c t e r s m . t.Appf. M t h . , 24,143151. [383] B u W R. N., and J. B. Keller, (1960). Boundary layer problems in diffraction t h m u . Comm Rue A& Math., 13,85144. I3781 Budiansky, B. See Amazigo, Budianslry, and C d e r . Bumside, R. R. (1970). A note on Lighthill’smethod of strained coordinates. SIAMJ. Appf. Math. 18,318321. [79,107] Buslaev, N. S. (1964). On formulas of short wave asymptotics in the problem of diffraction by convex bodies. Ak d. N d S S R M y Mat. Inst. Steklov, 73,14117. [ 3801 Butler, D. S., and R. J. Gribben (1968). Relativistic formulation for nonlinear waves in a nowniform plasma. J PhslM P h p . 2,257281.1234) Caldirola, P., 0. De Barbier&and C. Maroli, (1966). Electromagnetic wave propagation in a d y ionized plasma. I. Nuovo cimento, B42,266289.[2361 Campbell, J. A., and W. H. Jefferys, (1970). Equivalence of the perturbation theories of Hori and Deprit. Cele#&l Mech. 2,467473. [202] Carlini, F. (1817). Ricerche Julia convergenza della serie che serve alla soluzione del problem di Keplero. Memoria di Francesco Carlini, Milano; also kcobi’s Ges Werke, 30,197254. I3151 7,189245;alSo A m Na&i&tm, Carrier, G. F. (1953). Boundary layer problems in applied mechanics. Advan Appl. Mech., 3,119. [119] Carrier, G. F. (1954). Boundary layer problems in applied mathematics. C o r n R u e Appl. Math., 7,ll17 (1141 Carrier, G . F. (1966). Gravity waves on water of variable depth. J. Fluid Mech., 24, 641659. [234] Carrier, G. F. (1970). Singular pertubation theory and geophysics. SUMRev.. 12,175193. r1101 Carrier, G. F. See Amazigo, Budiansky, and Carrier. Cashwell, E. D. (1951). The asymptotic solutions of an ordinsuy differential equation in which the coefficient of the parameter is sbguk.Pa&J. Moth.. 1,337353. [3581 Caughey, T. K. See Tso and Caughey. Caughey, T.K., and H. J. Payne, (1967). On the response of a class of selfexcited oscillators to stochastic excitation. Int J. NonLhear Mech.. 2,125151.1235 J Cesari, L. (1971). Avmptotic Be?wvior and Stabiliv Roblems in Ordinmy Differential E q w t h s SpringerVmlag,New York. [309] Chang, K. S., R. G. Akins, and S. G. Bankoff, (1966). Free convection of a liquid metal from a uniformly heated vertical plate. Ind. Ens. Chem Fundurn,5,2637.1791 Chen, C. S. (1971). Parametric excitation in an inhomogeneous plasma. J. Pbsma Phys., 5, 107113. [235] Chen, C. S., and G. J. Lewak, (1970). Parametric excitation of a plasma. J. pbsntrr Phys, 4, 357369. [235] Cheng, H.and T. T. Wu (1970). An aging spring. Stud. Appf.Math., 49,183185. [232] Cheng, H. K., J. W. Kirsch, and R. S. Lee, (1971). On the reattachment of a shock layer produced by an instantaneous energy release. J. Fluid Me&. 48, 241263, [235] cheng, s. 1. See Goldburg and cheng. Chernov, L. A. (1960). Wave Proprgcrtion in a Random Medium Dover, New York. [361, 3731
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39 1
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393
Dyson, F. I. (1949). The radiation theories of Tomonaga, Schwinger, and Feynman. P h y ~ Rev., 75,486502. (3711 Eckhaus, W. (1965). Studies in Nonlinem Stability Theory. SphgerVuhg, New York. 11621 Eckstein, M. C. See Shi and Eckstein. Eckstein, M. C. and Y. Y. Shi (1967). Lowthrust elliptic sphl najecta*tsof a arrtenite of wriable ~ S S AIAA . J., 5,14911494.12331 Eckstein, M. C., Y. Y. Shi, and I. Kcvmkian, (1966a). A uniformly valid asymptotic representation of satellite motion around the smaller primary in the lestricted threebody problem. In Rmgress in Asbonmrricr a d Aavnautics. Vol. 17, Methods in Astraiynamfcs and Celcsthl Meelvrnicr (R.L. Lhmcombe and V. G. Szabehely, Eds.), Academic, New York, pp. 183198. (233.276) Eckstein, M. C., Y. Y. Shi, and J. Kevorkian, (1966b). Satellite motion for arbitrary eccentricity and inclination around the smaller primary in the restricted threebody problem. Amon J.. 71,248263. [233] Eckstein, M. C.,Y. Y. Shi, and J. Kevorkian (1966~).Use of the energy integral to evaluate higherorder terms in the time history of sateIUte motion. Ashon. J., 71, 301305. 12331 Einaudi, F. (1969). Singular perturbation analysis of acoustiqpavity waves. mys. 8, 12,752756. [78] Einaudi, F. (1970). Shock formation in acoustic gravity waves. J. Geophyr Re&, 75, 193200. [78] Emanuel, G. (1966). Unsteady, diffusing, reacting tubular flow with application to the flow in a glow discharge tube. Aerospace Corporation Rept. No. TR669 (62402019. (991 Emery, V. J. (1970). Eikonal approximation for nodinear equations. J. Math. Phyr, 11, 18931900. [234] Erddyi, A. (1956).Asymptotic Expansion& D o m , New York (18,309,3371 Erddyi, A. (1960). Asymptotic solutions of differential equations with transition points or singularities. J. W t h . M y s , 1,1626.13581 Erdclyi, A. (1961). An expansion procedure for singular perturbations. Atti A d ScL Torim, Cl. Sci Fk Mat. Nat., 95,651672. (113,1211 Emst, R. R., and H. Primas (1963). Nuclear magnetic resonance with stochastic highfrequency fmlds.Helv. Phyr Ackr., 36,583600. ( 382) Espedal, M. S. (1971). The effects of ionion collision on an ionacoustic plasma pulse. J. P h s m Phyr, 5,343355. ( 781 Euler, L. (1754). Novi CornmenWi'iAmd. Scf. Pe?Popolitanae, 5, 205237; Opcrcr Om?&, Ser. I. 14,585617. I l l ] Evensen, D. A. (1968). Nonlinarr vibrations of beams with various boundary conditions. AL4A J., 6,370372. ( 1061 Evgzafov, M. A., and M. B. Fedoryuk, (1966). Asymptotic behavior of solutions of the equation w"(z)  p(z'. A)w(z) = 0 as A + in the complex aplane. U@.&t.Nauk, 21,350. [ 3451 Faa De Bruno, F. (1857). Note sur une nouvek formule de c a l d differentiel. Quart. J. Pure Appl. Math., 1,359360. [ 1921

394 REFERJ3NCES AND AUTHOR INDEX Fedory~k,M. B. See Evgrafov and Fedoryuk. Felsen, L.B. See Levey and Felsen. Feshcheniro, S. F., N. I. S W , and L. D. N h h k o , (1967). Asymptoric Methods in the Tireoty of Linear Dtffacnthr Equationx. American Elseer, New York. [309, 319) Feynman, R. P. (1948). Spacetime approach to nomlativistic qUantilm mechanics. Rev. bhdmplr)rr, 20,367387. [%I Fife, P. Sea Pierson and Fife. F&, V. A. (1965). Electronwgneetic Diffnrctn and Ropagution Roblems. P e w o n , New York. [3801 Fowkes, N. D. (1968). A singulpr perturbation method. Parts I and 11. Quart. AppL Math, 26,5749,71435. [232,234,379,285] Fowler, R. H., and C. N. H. Lock, (1921). Approximate solutions of linw differential equations. Roc. London mth. Soc., 20,127147. [321,322] Fowler, R. H., E. C.Callop, C. N. H.Lock,and H. W. Richmond (1920). The aerodynamics Roy. Soc. London, 221,295387. [ 321,3221 of a 8pinning ShelL tlha. Fox, P. A. (1953). On the use of coordinate perturbations in the solution of physical problems. Project DIC 6915 Technical Rept. No. 1, Massachusetts Institute of T ~ hn o l o g yCambridge. , [98] Fox, P. A. (1955). Perturbation theory of wave propagation based on the method of characteristics.J. Math. andPhyr, 34,133151. [57,87,89] Fraenkel, L. E. (1969). On the method of matched asymptotic expansions, Parts 1111. Roc Chrnbr&&e€?hilSot, 65,209231,233251,263284. [ 1191 Freeman, N. C., and R. S. Johnson, (1970). Shallow water waves on shear flows. J. Fluid Me&, 42,401409. [2341 Fried, D. L. (1967). Test of the Rytov approximation. J. Opt. Soc. Am., 57, 268269. [3731 Friedrichs, K. 0. (1942). Theory of viscous fluids. Fluid Dynamics. Brown University Press. Providence, R. I., Chapter 4. [1191 Friedrichs, K. 0. (1955). Asymptotic phenomena in mathematical physics. BulL A m Math SOC.61,485504. [18] Frieman, E. A. (1963). On a new method in the theory of irreversible processes. J. Math. Phys., 4,410418. [230] Frieman, E. A., and P. Rutherford, (1964). Kinetic theory of a weakly unstable plssma. A m my&, 28,134177. (2351 Frisch, U. (1965). Wuue Promation in Random Medm. Institut d'Astrophysique, Paris. r367.3691 Frisch, U. (1968). Wave propagation in random media. h b a b i l i v Methods in Applied Mathem& Vol. 1 (A. T. BharuchaReid, Ed.), Academic, New York, pp. 75198. (309,361,364,365,367,372,380,381]
Frobenius, G. (1875). Veber die mguGren Integrale der linearen Differentialgleichungen.J. Reine Angew. Math., 80,317333. [310] Furutsu, K. (1963). On the statistical theory of electromagaetic waves in a fluctuating medium. J. Re&NatL Bur. S t a M s , Sec. D,67,303323. [ 367,3691 Gallop, E. G. See Fowler, Qhp, Lock, and Richmond. Galloway, J. J., and F. W. Crawford, (1970). Lagrangian derivation of wavewave coupling coefficients. Proc. 4th European Conference on Controlled Fusion and Plasma Physics, Rome (CHEN), 161. [217]
’
REFERENCES AND AUTHOR INDEX
395
Galloway, J. I. and H. Kim, (1971). Lagrangian approach to nonlinear wave interactions in a warm plasma. J. Phsma Phys., 6 5372.[217] Gans, R. (1915). Propagation of light through an inhomogeneous media. Ann. Phys., 47,
709736.[ 114,3391
Garrett, C . J. R. (1968). On the interaction between internal Bavity waves and a shear flow. J. FluidMech. 34, 711720.I2161 Garrett, C. J. R. See Bretherton and Garrett. Germain, P. (1967). Recent evolution in problems and methods in aerodynamics. J. Roy. Aeron. Soc., 71,673491.[110,235] Gertsenshtein, M. E. See Tatarski and Gertsenshtein. Giacaglia, G. E. 0.(1964).Notes on von Zeipel’s method. NASA X54764161.[ 1911 Golay, M. J. E. (1964). Normalized equations of the regenerative oscillatornoise, phaselocking, and pulling. ROC. IEEE, 52,13111330.[258] Goldberg, P., and G. Sandri, (1967). Power series of kinetic theory. Parts I and 11. Phys. Rev., L154,188209.[236] Goldburg, A., and S. I. Cheng, (1961). The anomaly in the application of PoincakLighthillKuo and parabolic coordinates to the trailing edge boundary layer. J. Math.
Mech., 10,529535.[79]
Goldstein H. (1965).C&ssiml Mechunics AddisonWesley, Reading, Mass. [ 1811 Goldstein. S. (1929).The steady flow of viscous fluid past a f j e d spherical obstacle at small [ 1411 Reynolds numbers. Roc. Roy. Soc. (London). A123,225235. Goldstein, S. (1931). A note on certain approximate solutions of linear differential equations of the second order. Roc. London Math. SOC.,33,246252.(3461 Goldstein, S. (1969). Applications of singular perturbations and boundarylayer theory to simple ordinary differential equations. In Conmktions to Mechunics (D. Abir, Ed.), Pergamon, Oxford, pp. 4167. [317] Good, R. H.See Miller and Good. Goodrich, R. F., and N. D. Kazarinoff, (1963). Diffraction by thin elliptic cylinders. Michigan Mark. J., 10,105127.[346] GoreWr, G., and A. Witt, (1933). Swing of an elastic pendulum as an example of two parametically bound linear vibration systems. J. Tech. P k p (USSR,) 3, 244307. ~351 Green, G. (1837). On the motion of waves in a variable canal of small depth and width. Trens, Cambridge Phil. Soc., 6,457462. (3141 Green, G. S., and A. K. Weaver, (1961). The estimation of the threedimensional gyrations of a ballistic missile descending through the atmosphere. Royal Airuaft Establishment Tech. Nore G . W. 596 (London). I3211 Gretler, W. (1968). Em indirekte Methode zur Berechnung der ebenen Unterschallstromung. 1.Miamque, 7,8396.[ 89) Gribben, R. J. See Butler and Gribben. Grimshaw, R. (1966). Highfrequency scattering by finte convex regions. Comm Rue A p p l Math., 19, 167198.(3801 Grlmshaw, R. (1970). The solitary wave in water of variable depth. J. Fluid Mech. 42,
639656.(2161
Guiraud, J. P. (1965). Acoustique geomktrique, bruit balistique des avions supersoniqueset focalisation. J. Mebanique, 4,215267.(891
396
REFERENCES AND AUTHOR INDEX
Gundersen, R. M. (1967). Selfsustained thermally driven nonlinear oscations in magnctohydrodynamic flow. Znt. J. Eng. Sci,5,205211.[89] Gly&n, H. (1893). N o u v e k rscerches SUT les &s empby& dans les thiories des p l a n h . Actrr Mirth. 9,l168.(561 b, L. S. (1965). On the free vibration of a beam on a nonlinear elastic foundation. J. AppL Mech., 32,445447. (58,1051 Hanks, T . C. (1971). Model relating heatflow value near, and vertical velocities of mass transport beneath, ocean rises. J. Ceophys Reg. 76,537544.[ 1561 Hasegawa, H. S. See Savage and Hasegawa. Hassan, S. D. See Nayfeh and Hassan. Heidbredcr, G. R. (1967). Multiple scattering and the method of Rytov. J. Opt. Soc. A n , 57,14771479.13731 Henrard, J. (1970). On a perturbation theory using Lie transforms. Cekstk?l Mech., 3, 107120.[202] Heynatz, J. T.See Zierep and Heynatz. Hirschfelder, J. 0.(1969). Formal RayleighSchrikIhger perturbation theory for both desnerate and nondegenerate energy states. Znt. J. Quantum C?&?m.3,731748. t711 Holstein, V. H. (1950). Uber die aussere and h e r e Reibungsschicht bei Storungen hminarer Stromungen. ZAMM, 30,2549. [3531 Holt, M. (1967). The collapse of an imploding spherical cavity. Rev. Roumaine, S d Tech. Ser. Mec AppL. 12,407415. [78] Holt, M.,and N. J. Schwartz, (1963).Cavitation bubble collapse in water with finite density behind the interface. Phya Fluidr. 6,521525.[78] Hoogstraten, H. W. (1967). Uniformly valid approximations in twodimensional subsonic thia aitfoil theory. L Eng. Muth., 1,5145. [99] Hoogstraten, H. W. (1968). Dispersion of nonlinear shallow water waves. J. Eng. Muth., 2, 249273.(2341 Hori, G. I. (1966). Theory of general perturbations with unspecified canonical variables. PUbL Astron. Soc Japan. 18,287296,[199,201,202,212] Hori, G. I. (1967). Nonlinear coupling of two harmonic oscillations. PubL Astron. Soc. Japn, 19,229241.[199,201,212] Hori, G. I. (1970). Comparison of two perturbation theories based on canonical transformations. PubL Astron. Soc. Japan, 22,191198.[ 2021 Horn, J. (1899). Ueber eiw h e a m DifYerentialgleichung zweiter ordnung mit einezn wiuhulichen Pa?ameter.Mzth. Ann. 52,271292.(3161 Horn, J. (1903). Untersuchung der Intergrale einer linearen Differentialgleichung in der Umgebung ciner Unbestimmtheitsstelle vexmittelst succeSSiver Annahemngen. Arch. Mirth. Phy~, 4,213230.[312] Hoult, D. P. (1968). EulerLagrangerelationship for random dispersive waves. Phys.Fluid%, 11,20822086.[ 2343 Houston, A. See Barakat and Houston. Howe, M. S. (1967). Nonlinear theory of openchannel steady flow past a solid surface of f~tewavegroupshape. J. FkktMech. 30,497512.[216] Howland, L. See Van Hove, Hugenholtz, and Howland.
REFERENCES AND AUTHOR INDEX
397
Hsieh, P. F., and Y. Sibuya, (1966). On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients. J. Mkth. Anal AppL, 16,84103. (3451 Hufnagel, R. E., and N.R. Stanley (1964). Modulation transfer function associated with image transmission through turbulent media. J. Opt. Soc. A m . 54,5261. [373] Hugenholtz, N. See Van Hove, Hugenholtz, and Howland. Imai, I. (1948). On a refmement of the W. K. B. method.Phys. Rev.. 74, 113. [ 324, 3501 Imai, I. (1950). Asymptotic solutions of ordinary linear differential equations of the second order.Phys Rev.. 80, 1112. [350] Ince, E. L. (1926). Ordimry DifferentialEquations Longmans, Green, London. [ 5 , 3141 Jacobi, C. G. J. (1849). Versuch einer Berechnung der grossen Ungleichheit des Saturns nach einer strengen Entwickelung. Astron. Nu&. 28,6594. [ 312) Jacobs, S. J. (1967). An asymptotic solution of the tidal equations. J. Fluid Mech., 30, 417438. [234] Jahsman, W. E. (1968). Collapse of a gasfiUed spherical cavity. J. Appl. Mech., 95,579587. [781 Jakob, M. (1949). H a t Tnma;er, Vol. 1. Wiley, New York. I3421 Jefferys, W. H. See Campbell and Jefferys. Jeffreys, H. (1924a). On certain approximate solutions of linear differential equations of the second order. Roc. London Muth. Soc., 23,428436. [ 114,3391 Jeffreys, H. (1924b). On certain solutions of Mathieu’s equation. Roc. LondonMufh. Soc. 23,437476. [114,339] Jeffreys, H. (1962). Asymptotic Approximations. Oxford University Press, Oxford. [ 309, 342,3821 Johansen, K. F., and T. R. h e , (1969). A simple description of the motion of a spherical pendulum. J. Appl. Mech., 36,40841 1. (2241 Johnson, R. S. See Freeman and Johnson. Kain, M.E. See Kane and Kahn. Kamel, A. A., (1969a). Expansion formulae in canonical transformations depending on a small parameter. Celesziul Me&, 1,190199. [199,202,206,212] Kamel, A. A. (1969b). Perturbation theory based on Lie transforms and its application to the stability of motion near sunperturbed earthmoon triangular libration points. SUDAAR Rept. No. 391, Stanford University. [ 199,202,2061 Kamel, A. A. (1970). Perturbation method in the theory of nonlinear oscillations. Celestial Mech., 3,90106. [ 199,201,202,206) Kamel, A. A. (1971). Lie transforms and the Hamiltonization of nonHamiltonian systems. CelestialMech. ,4,397405. [ 2021 Kamel, A. A. See Nayfeh and Kamel. h e , T.R.See Johansen and Kane. Kane, T. R., and M. E. Kahn, (1968). On a class of twodegreeoffreedom oscillations. J. Appl. Mech., 35,547552. [ 1851 Kaplun, S. (1954). The role of coordinate systems in boundarylayer theory. Z. Angew. Math. phys, 5, 111135; also Chapter 1 of Kaplun (1967). [SO, 1141 Kaplun, S . (1957). Low Reynolds number flow past a circular cylinder. J. Mkth. Mech., 6 , 595603; also Chapter 3 of Kaplun (1967). [ 1141
398
REFERENCES AND AUTHOR INDEX
Kaplun, S. (1967). FIuid Medronics and Singuhr Perturbations (a collection of papers by S. Kaplun, edited by P. A. Lagerstrom, L. N. Howard, and C. S. Liu), Academic, New York. [114,1191 Kaplun, S., and P. A. Lagerstrom, (1957). Asymptotic expansions of NavierStokes solutions for small Reynolds numbem. J. Murh. Me& 6, 585593; also Chapter 2 of Kaplun
(1967). [114] Karal, F. C., and J. B. Keller, (1964). Elastic, electromagnetic, and other waves in a random medium. J. mrh.Rhys. 5,537547. [369] Karpman,V. I. See Al'tshul' and Karpman. Karpman, V. I., and E. M. Km&kal' (1969). Modulated waves in nonlinear dispersive media. soviet Phys. 28,277281. [216] Kawakami, I. (1970). Perturbation approach to nonlinear vlasov equation. J. Phys. SOC. Japn. 28,505514. [216] Kawakami, I., and T.Yagishita, (1971). Pertubation approach to nonlinear Vlasov equation. 11. Nonlinear plasma oscillation of fmite amplitude. J. P?iys Soc. Japan. 30,244253. [2161 Kazarinoff, N. D., (1958). Asymptotic theory of second order differential equations with two simple turning points.Arch. Rat. Me&. Anal. 2,129150. [ 3441 Kazarinoff, N. D. See Goodrichand off. Kazarinoff, N. D., and R. W. Mc Kelvey, (1956). Asymptotic solution of differential equations in a domain containing a regular singular point. Con J. Math., 8, 97104. [3581 Keller, J. B. (1958). A geometrical theory of diffraction, calculus of variations and its applications. Aoc. Symp. AppL Mkth.. 8,2752. [ 3741 Keller, J. B. (1962). Wave propagation in random media. Proc. Symp. Appl. Marh., 13, 227246. [ 369,3821 Keller, J. B. (1968). Perturbation Theory. Lecture notes, Mathematics Department, Michigan State University.[58,150] Keller, J. B. See Avila and Keller, Buchal and Keller; Karal and Keller; Lynn and Keller; Millman and Keller. Keller, J. B., and S. Kogelman, (1970). Asymptotic solutions of initial value problems for nonlinea partial differential equations. SIAMJ. AppL Marh., 18,748758. [234] Keller, J. B., and M.H.MiUmau, (1969). Perturbation theory of nonlinear electromagnetic wave propagation.P?iys Rev.. 181,17301747.1771 Keller, J. B., and L. Ting, (1966). Periodic vibrations of systems governed by nonlinear partial differential equations. Comm Rue. AppL Murh. 19,371420. [77] Kelly, R. E. (1965). Stability of a panel in incompressible, unsteady flow. AZAA J., 3, 11131118. [233] Kelly, R. E. (1967). On the stability of an inviscid shear layer which is periodic in space and time. J. Fhrid Mech., 27,657689. [2351 Kelly, R. E. See Maslowe and Kelly. Kemble, E. C. (1935). A contribution to the theory of the B.W.K. method. Phys, Rev.. 48, 549561. [339] Kersten, P. H. M. (1967). Diffraction of an electromagnetic wave by a plane screen. Ph. D. Thesis. Technbche Hochschull, Eindhoven. [380]
rnN%.ENCES AND AUTHOR INDEX
399
Kevorkian, J. (1966a). The two variable.expansion procedure for the approximate solution of certain nonlineax differential equations. Spurn Marhenrrrt&x Part 3. (J.B. Rosser, Ed.) American Mathematical Society, Providence, R.I.,pp. 206275. (231,232,233, 273) Kevorkih, J. (1966b). von Zeipel method and the twwariable expansion procedure. Adtron J., 71, 878885. [231] Kevorkian, J. (1971). Passage through resonance for a omxbmmional oscillstox with alowly varying frequency. SCQMJ.AppL hizth.. 20,364373. [233,383] Kevorkian, J. See Cole and Kevorkian; Eckstein, Shi, and Kevorkian; Lagcrstrom and Kevorkian. Kiang, R. L. (1969). Nonlinear theory of inviscid Taylor instability near the cutoff wave number. Phys. Fluids, 12,13331339. [235] Kim, H. See Galloway and Kim. Kirchhoff, G. (1877). Zur Theorie des Cbndenazttas, Berlin, Akad., Monatsber. pp. 144162. [114] Kirsch, J. W.See Cheng. Kirsch, and La. Klein, J. S. See Sellers, Tribus, and Klein. Klimas, A., R. V., Ramnath, and G.Sandri, (1970). On the compatibility problem for the uniformization of asymptotic expansions. 1. Math. AnaL Appl., 32,482504. 12321 Kogelman, S. See Keller and Kogelman. Kraichnan, R. H. (1961). Dynamics of nonlinear stochastic systems. J. Mzth. Phys, 2, 124148. [ 367) Kramers, €I A. . (1926). WeUenmechanik und ha&* Quantisierung. 2. Phys., 39, 828840.[114,315,339) Kravtsov, Y. A. (1964a). A modification of the geometrical optics method. Rudbjizika, 7, 664673 (in Russian). [378] Kravtsov, Y. A. (1964b). Asymptotic solutions of Maxwell's equations near a caustic. Rudiofuika, 7,10491056 (in Russian>. [378,380] Kravtsov, Y. A. (1965). Modification of the method of geometrical optics for a wave penetrating a caustic. &&&ah, 8,,659667. I3801 Kravtsova, T. S. See Babich and Kravtsova. Krushkal', E. M. See Karpman and Kndkal'. Kruskal, M. (1962). Asymptotic theory of Hamiltoninn and OW systems with all solutions nearly periodic. J. Math. My&, 3,806828. /I681 W l w , N., and N. N. Bogoliubov (1947). Introduction to Nonlinear Mechanic& Princeton University Press, Princeton, N. J. [165,174] Kubo, R. (1963). Stochastic Liouville equation. J. Mufh. Phy&,4,174183. [ 3821 Kuiken, H. K. (1970). Inviscid film flow over an inclined surface originated by strong fluid injection. J. Fluid Me&, 42,337347. [99] Kuo, Y. H. (1953). On the flow of an incompmssible viscous fluid past a flat plate at moderate Reynolds numbers. J. Math and My%,3 5 83101. [ 78) Kuo, Y. H. (1956). Viscous flow along a flat plate moving at high supersonic speeds. J. A m n S d , 23,125136. I781 Kuzmak, G. E. (1959). Asymptotic solutions of nonlimar ascond order differential equations with variable coefficients.J. AppL Math. Me&, 23, 730.744. [232,2871
400 REFERF,NCESAND AUTHOR INDEX Lcina, J. (19693. New CBllOnicBL perturbation method for complete set of integrals of
motion. Cu&. J. tlhyr. B19,130133. (1991 Lacine,J. (1969b). New COllOniCPl perturbation method for complete set of integrals of motion. Ann. Ahys, 51,381391. [ 1991 Lagerstrom, P. A. See Kaplun and Lagerstrom. Lqerstrom, P. A., and J. D. Cole, (1955). Examples illustrating expansion procedures for the Nav&&tokw equations. J. Rat. Meah. A n d , 4,817882. [ 1401 Lagerstrom, P. A., and J. Kevorkian, (1963a). Earthtomoon trqjectories in the restricted body problem.J. M&nt4ue, 2,189218. [138] approximation to the two Lagerstrom, P. A., and J. Kevorkian, (1963b). Mat&ed&c fixed forcecater problem.Altron. L68,8492. (44.1381 Landahl, M. T.See Rubbert and Laadahl. Langer, R. E. (1931). On the asymptotic solutions of differential equations, with an application to the Besael functions of large complex order. Zhzm A m Murh. Soc., 33,2344. [339,340,3451 Langer, R. E. (1934). The asymptotic solutions of certain finear ordinary differential equations of the second order. Zhm A m Mfh.Soc, 36,90406. [339,340] Langer, R. E. (1935). On the asymptotic solutions of ordinary differential equations with reference to the Stokes phenomenon about a singulpi point. Pam A m Mth. Soc., 37,397416. [ 3581 Langer, R. E. (1949). The asymptotic solutions of ordinary bear differentialequations of the second order, with special reference to a turning point. Zhzm A m Mark Soc., 67,461490. [ 346,3481 Langer, R. E. (1957). On the asymptotic solutions of a class of ordinary differential equations of the fourth order, with specint reference to an equation of hydrodynamics. Ih.urs.Am. Wh. Sot. 84,144491. [360] Langer, R. E. (1959a). Formal solutions and a related equation for a class of fourth order differential equations of a hydrodynamic type. Zhzm A m Murk Soc., 92,371410. [ 3601 Langer, R. E. (1959b). The asymptotic solutions of a linear differential equation of the second order with two turning points. Zhzm A m Mfh.Soc.. 90,113142. (3441 Laplace, P. S. (1805). On the figure of a large drop of mercury, and the depression of mercury in a glass tube of a great diameter. In CClrrHl Mechunics (transl. by Nathaniel Bowditch, Boston, 18391, Chelsea, New York,1966. [ 1141 Latta, G. E. (1951). Singular perturbation problems. Ph.D.Thesis, California Institute of Technology. [114,1451 Latta, G. E. (1964). Admced Ordimwy DiffwentizlEquutiora Lecture notes, Stanford University. I3171 LedovsEsja,L. B. See Zabreiko and LedovaLaja. Lee, D. H.,and L. hi. Sheppard, (1966). An approximate secondorder wing theory. AIAA L, 4,18281830. [78,89] Lee, J. H. S. See Akinsete and Lee. Lee, R. S. Sa Cheng, Kirsch,and Lee. Legns, J. (1951). Application dz la Athode de Lighthill ‘a un ikoulement plan supersonique. C o w .R e d , 233,10051008. [78,86]
REFERENCES AND AUTHOR INDEX 401 Legras, J. (1953). Nouvelles applications de la mithode de LighthiU APdtude des ondes de choc. O.N.E.R.A. PubL No.66. [78,86] Lesser, M. B. (1970). Uniformly valid perturbation series for wave propagation in an inhomogeneous medium. J. Aooust. Soc. A m . 47,12971302. [W] Le Verrier, U. J. J. (1856). Sur la determination des longitudes terrestres. Paris, C0m;pt. Rend., 43,249257. [ 168) Levey, H. C. (1959). The thickuess of cylindrical shocks and the PLK method. Qurut. AppL Math., 17,7793. [52,99, 1001 Levey, L., and L. B. Fefsen, (1967). On transition functions Occurring in the theory of diffraaion in inhomogeneous media. J. Inst. M t . AppL, 3,7697. [3801 Levinson, N. (1969). Asymptotic behavior of solutions of nonlinear differential equations. Shld. AppL Math.. 48,285297. [21,22] Levinson, N.See Coddington and Levinson. Lewak, G. J. (1969). More uniform perturbation theory of the vlapov equation. J. Phsma PhyS., 3,243253. [78] Lewak, G. J. (1971). Interaction of electrostatic waves in collisionless plasmas. J. Phsma Phys, 5,5143. I2351 Lewak, G. J. See Chen and Lewak;Zawadzki and Lewak. Lewis, R.M. See Ahluwalia, Lewis, and Boersma. Lewis, R. M.,N. Bleistein. and D. Ludwig, (1967). Uniform asymptotic theory of creeping waves. Comm €We Appl. Math., 20,295328. [380] Lick, W. (1969). Twovariable expansions and singular perhubation problems. SZAMJ. AppL Mth., 17,815825. [89] Lick, W. (1970). Nonlinear wave propagation in fluids. Annual Review of Fluid Medmnics, Vol. 2 (M.van Dyke, W. G. Vinciuti, and J. V. Wehausen, Eds.), Annual Reviews, Palo Alto, Calif., pp. 113136. [235] Liepins, A. A. See Sanders and Liepins. Lighthill, M. J. (1949a). A technique for rendering approximate solutions to physical problems uniformly valid. Phil Mag.,40,11791201. [42,57,77,80,87,108] Lighthill,M. J. (1949b). The shock strength in supersonic “conical fields.” I412Mag., 40, 12021223. [78] Lighthill, M. J. (1951). A new approach to thin airfoil theory. A e o n Quart., 3, 193210. [981 Lighthill, M. J. (1961). A technique for rendering approximate solutions to physical problems uniformly valid. Z. F7ugWisk. 9,267275. [57,77,991 Lighthill, M. J. (1965). Contributions to the theory of waves in nonlinear dispersive systems. J: Inst. Math. AppL. 1,269306. [216] Wthill, M. J. (1967). Some special cases treated by the Whitham theory. h c . Roy. Soc., (London). A m , 2853. [216] Lin, C. C. (1954). On a perturbation theory based on the method of characteristics.J. Math andPhys, 33,117134. [57,87,89] Lin, C. C. (1955). The Theory of Hydroctmamic StubW. Cambridge University Press, Cambridge. [ 3601 Lin, C. C., and A. L. Rabenstein (1960). On the asymptotic solutions of a claw of ordinary differential equations of the fourth order. Zhzm A m Mth. Soc., 94,2457. I3601
402
REFERENCES AND AUTHOR INDEX
Lin, C. C., and A. L. Rabenstein (1969). On the asymptotic theory of a class of ordinary differentid equations of the fourth order. 11. Existence of s o l u t i o ~which ~ ~ are approximated by the fcKmal solutions. snd.ApPr Mkth.,48,311340. [360] Lindstedt, A. (1882). Ueber die Integration eiuer fur die strorungs~rkwichtigen DifferenWgbkhung. An, Nuah., 103, h i . 211220. [56] Lindzen, R. S. (1971). Equatorial planetary waves in shear: Part 1. J. Atmos. S C L 28, 609622. [2341 Liouville, J. (1837). Second m6moixe sur le d&eloppment des fonctions en shies dont divers tenncs mt 'bsujettis a satisfaire 'a une &me kuationdiff&ntielle du second ordm contenant un paramktre variable. J. Mkth. h e A&, 2,1635.[314] Lock,C. N. H. See Fowler and Lock;Fowler, Gallop, Lock,and Richmond. Lowll, S. C.(1970). Wave propagation in monatomic lattices with anharmonic potential. hx.Roy. Soc (London). A318,93106. [217] Ludwig, D. (1966). Uniform asymptotic expausions at a caustic. &nun Rue AppL Muth., 19,215250. [378,380] Ludwig, D. (1967). Uniform asymptotic expansion of the fwld scattered by a convex object at high frequencies. C o r n Pure AppL W M . 20,103138. [380] Ludwig, D. (197Oa). Diffraction by a circular cavity. J. W t h . P h p . 11,16171630.[380) Ludwig, D. (1970b). Uniform asymptotic expansions for wave propagation and diffraction problems. SIAMRCV., 12,325331. [380] Ludwig, D. See Lewis, Bleistein and Ludwig. Luke, J. C. (1966). A perturbation method for nonlinear dispersive wave problems. Boc. Roy. Soc. (London). A292,403412. [234,301) Lynn,R. Y. S., and J. B. Keller (1970). Uniform asymptotic solutionsof second order linear ordinary differentid equations with tumiug points. Comm Rue AppL W f h . , 23, 379408.[345] Lyusternik, L. A. See V&k and Lyusternik. Mc Goldrick, L. F. (1970). On Wilton's ripples: A special case of resonant interactions. J. FhddMech., 42,193200. [2341 Mc Intyre, J. E. (1966). Neighboring.optimal terminal control with discontinuous forcing functions.ALQA J., 4,141148. [79] Mc Kelvey, R. W. (1955). The solution of second order linear ordinary differentialequations about a turning point of order two. Zhzm. A m Muth. Soc., 79,103123. (3461 Mc Kelvey, R. W. See Kazariuoff and Mc Kelvey. Mc Namara, B., and K. J. Whiteman (1967). Invariants of nearly periodic Hamiltonian systems. J. Mth. Phy~.,8,20292038. [ 1991 Mahony, J. J. (1962). An expansion method for singular perturbation problems. J. Aushvrlian Wf. Soc,2,440463. [232,303] Malleus, W. V. R.;and G. Veronis (1958). Finite amplitude cellular convection. J. Fluid Meek. 4,225260. [77] Maroli, C. (1966). Kinetic theory of highfrequency resonance gas discharge breakdown. Nuovo CYmmto, B41,20&224.[236] Maroli, C. See Calttirola. De Barbieri, and Maroli; De Barbjeri and Maroli. Maroli, C., and R. Pozzoli (1969). Penetration of highfrequency electromagnetic WVH into a slightly ionized plpsma Nuow Cirnento.B61,277289. [235]
REFERENCES AND AUTHOR INDEX 403 Maslowe, S. A., and R. E. Kelly (1970). Finite amplitude oscillations in a KelvinHelmholtz flow. Int. J. NonLineor Mech., 5,427435.(771 Matkowsky, B. J. (1966). Asymptotic solution of partial differential equations in thin domains. Ph.D.Thesis, New York University. (3801 Matkowsky, B. J., See Re& and Matkowsky. Maxwell, J. C. (1866). On the viscosity or internal friction of air and other gases. Phil. l h n s Roy. Soc. London, 156,249268.11141 Mei, C. C. See Chu and Mei. Meirovitch, L. (1970).Methods of Amlytiarl Dymmks MarawHill, New York. [ 1811 Melnik, R. E. (1965). Newtonian entropy layer in the vicinity of a conical symmetry plane. A M J., 3,520522.[79] Mendelson, K. S. (1970). Perturbation theory for damped nonlinear oscillations. J. Muth. Phy~.,11,34133415.(2241 Mersman, W. A. (1970). A new algorithm for the Lie transformation. CelestiuI Mech., 3, 8189.[202] Merman, W. A. (1971). Explicit recursive algoxithms for the construction of equivalent canonical transformations. Celestial Mech..,3,384389.[202] Messiter, A. F. See Spriggs, Messiter, and Anderson. Mettlex, E. (1959). StabilitZtsfragen bei freien SchwingUngen mechanischer Systeme. IngenieWArchiv., 28,213228.[ 168,1881 Meyer, J. W. (1971). Rayleigh scattering of a kser beam from a massive relativistic twolevel atom. Phys. Rev.,A3,14311443.(2351 Miller, S. C., and R. H. Good (1953). A WKBtype approximation to the SchrSdinger equation. Phys Rev., 91,174179.[ 3441 Millman, M. H. See Keller and Millman. Millman, M. H., and J. B. Keller (1969). Perturbation theory of nonlinear boundaryvalue problems. J. Math. phys, 10,342361.[77] Mitchell, C. E. (1971). Analysis of a combustion instability problem using the technique of multiple scales. AIAA J., 9,532533.[235] Mitropolski, Y. A. (1965). Problems of the A&yymptotic Theory of Nonstutionury Vibmtions. Daniel Davey, New York. [ 1741 Mitropolski,Y. A, See Bogoliubov and Mitropolski. Montgomery, D., and D. A. Tidman (1964). Secular and nonsecular behavior for the cold plasma equations. Phys Fluas, 7,242249.[ 1781 Moriguchi, H. (1959). An improvement of the WKB method in the presence of turning points and the asymptotic solutions of a class of Hill equations. J. Phys Soc. k p a n , 14,17711796.[342,345,350]
Morino, L. (1969). A perturbation method for treating nonlinear panel flutter problems. MAA J., 7,405411. [234] Moms, W. D. (1965). Laminar convection in a heated vertical tube rotating about a parallel axis. J. Fhtid Mech., 21,453464.(791 Morrison, H. L. See Richmond and Morrison. Morrison, I. A. (1966a). Comparison of the modifKd method of averaging and the two variable expansion procedure. SIAMRev., 8,6645. 12311
404 REFERENCES AND AUTHOR INDEX Mordson, J. A. (1966b). Gcosralized method of averand the von zeipel method. In &ogres in Amornutics and A m m t i e r . VoL 17. Mcthodr in Asbrodynrunicr ond ccluthlM .(R. L. Duacombs and V. G. Szebehdy, Eda), Acadedc, New York, pp. 117138. [168,1911 M o w M. P. (1%8). Traveling load on a cylincticrl sbsll. J. Aaowt. Soc. Am. 44, 16641670. [233] Mortell, M. P. (1969). Waves on a spherical shell. J. Aoourt. Soc Am. 45,144149. [233] Morten, M. P. (1971). Resonant therm&acmstic osciDotionn Int. J. Eng. S d , 9,175192. I891 Mortell, M. P., and E. Varley (1971). Finite amplitude waves in boundedmedia: Noalinear free vibrations of an elastic panel. Roc. Roy. Soc ( L o d n ) . A318,1691%. I901 Morton, B. R. (1959). Laminar convection in uniformly heated horizontal pipes at low Rayleigh numbem Quort. L Me& Ap& k t h . 12,410426. [79] MulhoJland, R. J. (1971). Nonlinear oscillations of a thirdorder dafsrcntial equation. Int. J. NonLhw Me&, 6,279294. [ 1051 Murphy, C. H. (1963). Free flight motion of symmetric micsiles. JMlistic Reqarch Laboratorbs Rept. No. 1216, Abodeem hoviug Grounds, Md. (3211 Munay, J. D. (1968). On the effect of drahuge on free surface osdllptionr. A& Sci Re&. 19,234249. [2341 Muss, S. A. (1967). Integral constraints in wealdy nonlinear periodic system.SIAM J. AppL k t h . , IS, 13241327. I2321 Musen, P. (1965). On the high order effects in the methods of KrylovBogoliubov and Pom&. J. AItron S d , 12,129134. [168,192,199] Nair, S., and S. NematNmer (1971). On Wte amplitude waves in heterogeneous elastic soIids.Int. J. Eng. S d , 9,10871105. [94] Nayfeh, A d ~ n ,and S. NematNasser (1971). Thennoelastic waves in solids with thermal relaxation.Ado Mechunk, 12,5369. [45,136] Nayfeh, A. H. (1964). A gewralized method for treating shgular perturbation problems. Ph. D.Tbgis, Santfurd Udvadty. [232,276,280,285,295] Nayfeh, A. H. (1965a). A comparison of three perturbation methods for the earthmoonspaceship probbm. AL4A J.. 3,16821687. [82,103,233,295,297] Nayfeh, A. H. (1965b). An expansion method for treating sjngular perhubation problems.J. k t h . My&.6,19461951. [232,234,276,280,285,384] Nayfeh, A. H. (1965~).A paturbation method for treating nonlinear oscillation problems. J. kloth. ondMy&.44,368314. [230,232] Nayfeh, A. H. (19659. Nonlinear oscillations in a hot electron plasma. Phyo Fhjdp. 8, 18961898. (230,2341 Nayfeh, A. H. (1%6). Taboff from a circular orbit by a anall thrust. In &ogres in Arbolurcticr and Acnmcrutia. Vot 17, Metho& in Asmdynsmia ond &leatiul Mechanics (R. L. Duncornbe and V. G. !kebehely, Eds.), Academic, New York, pp. 139152 199.109,233,3051 Nayfeh, A H. (1967a). hplptotk solutions of an ejgenvalue problem 0 turning poisltsheet transfer in a tube. J. ddirtrh ond fiya, 16,349354. (3851 Nayfeh, A. H. (1967b). The van &r Pol 0aCill.tor with delayed amplitude limiting. ptoe. IhXE, 55,111112.[231.232,258]
REFERENCES AND AUTHOR INDEX
405
Nayfeh, A. H. (1968). Forced osdllations of the van der Pol oscillator with delayed amplitude limiting.IEEE Tranr GWdt Th, 15,192200. [230,232,258] Nayfeh, A. H. (1969a). A multiple time scaling aaalysis of rwntry vehicle roll dynamics. AIM J.. 7,21552157. (233,3201 Nayfeh, A. H. (1969b). On the nonliDear LambTaylor instability. J. Fhrfd Mech., 38, 619631. [235] Nayfeh, A. H. (197Oa). Characteristic exponents for the triangular points in the elliptic restricted problem of three bodies. AlAA J.. 8,19161917. [68] Nayfeh, A. H.(1970b). Finite amplitude surface waves in a liquid layer. J. Flicid Mech., 40, 671684. (2341 Nayfeh, A. H. (1970~).Nonlinear stability of a liquid jet. H y a &ids, 13, 841847. [99, 2351 Nayfeh, A. H. (197Od). Triple and quintupledimpled wave profrles in deep water. Phya WS 13,, 545550. [234] Nayfeh, 4. H.(1971a). Thirdharmonic resonance in the interaction of capillary and gravity waves. J. FhridMech.. 48,385395. [234] e ~ the equilateral libration points. AIAA Nayfeh, A. H. (1971b). Twotoone r e s o ~ n c near J., 9,2327. [233] Nayfeh, A. H.,and S. D. Hassan (1971). The methd of multiple scales and nonlinear dkpmive waves. J. Fluid Mech., 48,463475. [234,298] Nayfeh, A. H., and A. A. Kamel (1970a). Stability of the triangukr points in the elliptic restricted problem of three bodies. A U A J., 8,221223. [64,66] Nayfeh, A. H. and A. A. Kamel (1970b). Threetoone resonances near the equilateral libration points. AIAA J., 8,22452251. [233] Nayfeh, A. H., and W. S. S d c , (1971a). Nonlinear KehrinHelmholtz instability. 1. Fluid Mech., 46,209231. [235] Nayfeh, A. H., and W. S. Saric (1971b). Nonlinear resonances in the motion of rolling reentry bodies. AIM Paper No. 7147. [233,305] Nayfeh, A. H., and W. S. Saric (1972a). An analysis of asymmetric r o w bodies with nonlinear aerodynamics.ALAA J., 10,10041011. [233,291,292,306] Nayfeh, A. H.,and W. S. W c (1972b). Nonlinear waves in a KelvinHelmholtz flow. J. FhtidMech., 55,311327. [234] NematNasser, S. See Adnan Nayfeh and NematNasser; Nair and NematNasser. Neubert, J. A. (1970). Asymptotic solution of the stochastic Helmholtz equation for turbulent water. J. A m s t . Soc. Am, 48,12031211. [334] Newell, A. C. (1968). The closure problem in a system of random gravity waves. Rev. C e o p h y ~6,l31. , (2341 Newell, A. C. (1969). Rossby wave packet interacti0ns.J. Fluid Mech., 35,255271. (234) Newell, A. C. See Bemey and Newell. Newell, A. C., and J. A. Whitehead (1969). Finite bandwidth, fUrite amplitude convection. J. W M e c h . , 38,279303. (235) Nienhuis, G. (1970). On the microscopic theory of Brownian motion with a rotational degree of freedom. PhyJla. 49,2648. (2361 Nikolenko. L. D. See Feshchenko, Shkil', and Nikolenko. Nocrdlinger, P. D., and V. Petrosian (1971). The effect of cosmological expansion on selfgravitatingensembles of particles. Astrophya J.. 168,l9. [232]
406 REPERENCES AND AUTHOR INDEX Ockendon, J. R. (1966). The separation of Newtonian shack layers. J. Fhrid Me&., 26, 563572. [79] Olvsr, F. W. J. (1954). The asymptotic solution of linur differential equptions of the second order for large valuea of a ppramcter and t b m P b t i c a W o n of1 w w of lslgC order. piML IZMI. Roy. See. LondDn Sa. A, 247,307368. [341, 348,3581 Ohnr, F. W. J. (1959). Uniform asymptotic e~pansiOnafor Weber pprabolic cylinder functirms of b g a orders. J. Reg NutL Bw.stcmchrdr, 63B, 132169. [ 3451 for cssfpin nontineareecoadorder 0’M.lley. R. E., Jr. (19680). A boundary value differential equations with a mall parameter. Arch. Rat. Me& A d , 29, 6674. (2331 Omalley, R. E., Jr. (1968b). Topics in singular perturbations. Ad~tan.Marh., 2, 365470. [ 110,2331 O’Malley, R. E., Jr. (1971). Boundary layer methods for nonlincpr initial value problems. SUMRev., 13,425434. [1451 Omalley, R.E., Jr. See Ackerberg and O’Malley. OSCW, C. W. (1910). &x die S t o w F d and uber eiw verwandte Aufgabe in der Hydrrdynamk Ark mr. Artron. 0 s . 6, . No. 29. [ 1401 Oswatitsch, V. K. (1965). Ausbreitungsprobleme.ZAMM. 45,485498. [89] Pandey, B. C. (1968). Study of cylindrical piston problem in water using PLK method. ZAMP, 19,962963. [86] Parker, D. F. (1969). Nonlinearity, relaxation and diffusion in acoustics and ultrasonics.J. FhridMeali. 39,793815. (2341 Parker, D. F., and E. Vatley (1968). The interaction of fdte amplitude deileciion and stretchiug waves m drstic memlmmes and strings. Q w f . J. Mech.AppL Wth., 21, 329352. [go] Payne, H.J. See Caughey and Payne. Pearson, J. R. A. See Roudman and Pearson. Pcdtowsky, J.(1%7).Fluctua~winds and the ocean circulation. Z ‘ e h , 19,25@257. [77] Perko, L. M. (1969). Highsr order averaging and related methods for perturbed periodic and quasiperbdic systems. SIAMJ. Appl Math.. 17,698724. I2311 Perko, L.M. See Brsakwcll and Perko. Pertosian, V. See N0err and Petrosian. peyret, R. (1966). b e n t quasi unidimensionnel dans un ~ t e dewplasma ‘a ondes L M.m*, 5,471415. (2341 Peyret, R. (1970). Etude de 1’6coulement d’un fluide umducteur dans un canal par la mGthode dcs belles multiples.J. M k n i q u e , 9,6197. [2351 Pierson, W. J., and P. Fife (1961). Some nonliaeer properties of longaested periodic waves with lengthsnear 2.44 centimeters. J. Caopnva Reg. 66,163179. [771 pilrc, E. R. (1964). On the rektedupation method of asymptotic approximation.Quwt.J. Mech.AppL Math., 17,105124.369379. [242,344] Pohc& H. (1892). New McthodsofCckrtiplMCchmJa.VOLIIII (English transl.), NASA TTF450,1%7. [ 10,561 Potter, M. C. See Reynolds and Potter. Pozzoli, R. See Mudi and Pozzoli.
REFERENCES AND AUTHOR INDEX
407
Pmndtl, L. (1905). h e r Flhsigkeitsbewegung bei sehr klemer Reibung. Procedugs Third Internat. Math. Koagr., Haidelberg, pp. 484491. Motion of fluids with very little viscosity. Tech.Memo N.A.C.A. (English ttansl.), No. 452,1928. [ 113) Rasad, R. (1971). Effect of ion motion on parametric oscillations of a cold plasma in a magnetic fkld. J. Phmm Plryr, 5,291302. (2351 F5gogin6, I. (1962). NonequUilrrfurnStCrtMcal Meehonicg Wiley, New York. (367, 369, 3821 Primas, H. (1961). ijber quantenmeehdshe Systeme mit ehem stochastischen Hamiltonoperator. Helv. my&Ado. 34.3657. [ 3821 Primas, H.See Ernst and Primas. Pringle, R. Jr. See Breakwell and Pringle. Pritulo, M. F. (1962). On the determination of uniformly accurate solutions of differential equations by the method of perturbation of coordinates. J. AppL Mbth. Mech.. 26, 661667. [57,95,369] Proudman, I. (1960).An example of steady laminuflow at large Reynolds number. J. * i d Mech.,9,593602. [54,156] Proudman, I., and J. R. A. Pearson (1957). Expansions at Smpu Reynolds numbers for the flow past a sphere and a ciwular cylinder. J. FfuidMech., 2, 237262. 1114, 141, 1441 Puri, K. K. (1971). Effect of viscosity and membrane on the oscillations of superposed f l ~ & J. AppL Plry~.,42,9951000. (2351 Rabenstein, A. L. (1959). The determiuation of the inverse matrix for a basic reference equation for the theory of hydrodynamic stability. Arch. Rot. Me& Anal, 2, 355366. (3601 Rabensteiq, A. L. See Lin and Rabenstein. Rajappa, N. R. (1970). Nontineat theory of Taylor instability of superposed fluids. J. my& Soc. Jupon, 28,219224. (771 Ramamthan, G. V., and C . Sandri (1969). Model for the derivation of kinetic theory. J. Murk my&.10,17631773. (2361 Ramnath, R. V. (19700). A new analytical approximation for the ThomasFemi model in atomic physics. J. &rh. AML AppL, 31,285296. (2351 Ramnath, R. V. (1970b). Transition dynamics of VTOL aircraft. AIAA J., 8, 12141221. (2331 Ramauth, R. V. (1971). On a class of nonlinear differential equations of astrophysics. J. Moth. AWL AppL. 35,2747. [235] Ramnath, R. V.See Klimap, Ramnath, and Sandxi. Ramnath, R. V., and G. Sandri (1969). A g e m d .m d multiple scales approach to a class of linear differentialequations.J. &th. AML AppL. 28,339364. (2321 Rand, R.H.See Alfnend and Rand. Rand, R. H.. and S. F. Tmng (1969). On the stability of a differential equation with appli&tion to the vibrations of a particle m the p k . J. AppL Mech.,36,311313. 11041 Rao, P. S. (1956). Supersonic bangs. Aeron. Quorr., 7,135155. 1781 Rarity, B. S. H. (1969). A theory of the propagation of internal gravity waves of finite amplitude. J. W M e c h . , 39,497509. [216]
408
REFERENCES AND AUTHOR INDEX
Rasmussen, M. L. (1970). Uniformly valid approximations for nonlineiu oscillations with small damping. In?. J. NORLinemMe&.. 5,687696. [2321 Rayleigh, Lord (1912). On the propagation of waves through a stratified medium, with special reference to the question of reflection. BOCRoy. Soc. ( L o h n ) , A86, 208226.[114,339] Rayleigh, Lord, (1917). On the reflection of light from aregularly stratifkd medium. Roc. Roy. Soc ( L o d n ) , A93,565577. [95,367,368] Rehm, R. G. (1968). Radiative energy addition behind a shock wave. Phys. Fluids, 11, 18721883.1891 Reiss, E. L. (1971). On multivariable asymptotic expansions. SlAM Rev., 13, 189196. I2321 Reiss, E. L., and B. J. Matkowsky (1971). Nonlhear dynamic buckling of a compressed elnstic column. Qwrt. Appl Mth., 29,245260. [233] Reissner, E., and H. J. Weiuitschke (1963). Finite pure bending of circular cylindrical tubes. pumt. A@ Wtk,20,305319. (541 Reynolds, W. C., and M. C. Potter (1967). Finiteamplitude instability of parallel shear flows. J. FluidMech.. 27,465492.11621 Richmond, H. W.See Fowler, M o p , Lock, and Richmond. Richmond, O., and H.L. Morrison (1968). Application of a perturbation technique based on the method of characteristics to axisymmetric plasticity. 1 AppZ. Mech., 35, 117122. [90] Rogister, A. (1971). Parallel propagation of nonlinear lowfrequency waves in high8 plasma. M ~ sW . S 14,27332739. (2351 Rogister, A. See Dobrowolny and Rogister. Roskes, G. J. See Benuey and Roskes. Ross, L. W.(1970). Perturbation analysis of diffusioncoupled biochemical reaction kinetics. SLAMJ. Appl. Mth., 19,323329. (791 Rubbed, P. E., and M. T. Landahl(1967). Solution of the transonic airfoil problem through parametric differentiation. ALAA J., 5,470479.1235 J Rulf, B. (1967). Relation between creeping waves and lateral waves on a curved interface. L Math. M p . , 8,17851793. [380] Rulf, B. (1968). Uniform asymptotic theory of m ra c tion at an interface. Comm Pure AppL Mth., 21,6776. [ 3801 Rutherford, P. See Frieman and Rutherford. Rytov, S. M. (1937). Mr act i o n of light by ultrasonic wave. Zzv. Akad. Nuak SSSR Ser. Fiz. No. 2,223259 (in Russian). [ 373) Saffman, P. G. See Benney and Wman. Sakurai, A. (1965). Blast wave theory.InBasicDevebpments in Fluid Dymmiks, Vol. I (M. Holt, Ed.), Academic, New York, pp. 309375. [78] Sakurai, T. (1968). Effect of the plasma impedance on the time variation of the inverse pinch.J. B y & Soc. Japn. 25,16711679.1781 Salpeter, E. E., and H. A. Bethe (1951). A relativistic equation for boundstate problems. Phys. Rev., 84,12321242. (3721 Saner, M. I. See Varvatsis and Sancer.
REFERENCES AND AUTHOR INDEX
409
Saucer, M. I., and A. D. Varvatsis (1969). An investigation of the renormalization and Rytov methods as applied to propagation in a turbulent medium. Northrop Corporate Laboratories Rept. No. 69281. [ 3731 Sancer, M. I., and A. D. VarvataiS (1970). A comparison of the Born and Rytov methods. Roc. IEEE, 58,140141. [373] Sanders, J. L., Jr., and A. A. Liepius (1963). Toroidal membrane under internal pressure. AIAAJ., 1,21052110. [356,357] Sandri, G. (1965). A new method of expansion in mathematical physics. Nuovo amento, B36,6793. I2301 Sandri, G. (1967). Uniformization of asymptotic expansions. In Nonlinaar W & l Differentrbl Equations: A Symposium on Methods of Solutions gV. F. Ameq Ed.), Academic, New York, pp. 259277. [230] Sandri, G. See Coldberg and Saudri; Klimas, Ramnath, and Sandri; Ramanathan and Sandri; Ramnath and Sandri. Saric, W.S. See Nayfeh and Saric. Savage, J. C., and H. S. Hasegawa (1967). Evidence for a linear attenuation mechanism. Geophysics, 32,10031014. (781 Schechter, H. B. (1968). The effect of threedimensional nonlinear resonances on the motion of a particle near the earthmoon equilateral libration points. Semnd Compibtwn of Papers on Tkajectoty Am&sis and Guidcrnce Theory. NASA PM6721, 229344. [ 1991 SchrSdinger, E. (1926). QuantisierUng als Eigenwertproblem. Ann. Phys., 80,437490. [56, 711 Schwartz, N.J. See Holt and Schwartz. Schwertassek, V. R. (1969). Grenzen von Mitnahmeberkhen. ZAMW, 49,409421. [232] Scott, A. C. (1970). Propagation of magnetic flwc on a long Josephson tunnel junction. Nuow Cimento, B69,241261. [221] Scott, P. R. (1966). Equations of the oscillator with delayed amplitude limiting. Roc. IEEE, 54,898899. [ 2581 Searl, J. W. (1971). Expansions for singular perturbationsJ. Inst. Muth. Appl. 8,131138. [2331 Sellers, J. R., M. Tribus, and J. S. Klein (1956). Heat transfer to laminar flow in a round tube or flat conduitThe Graetz problem extended. Dun& A W E , 78, 441448. (3841 Sethna, P. R. (1963). TransKnts in certain autonomous multipledegreeoffreedom nonlinear vibrating systems. J. Appl Mech.,30,4450. [ 168) Sethna, P. R. (1965). Vibrations of dynamicat systems with quadratic nonlinearities. J. AppL Me&.., 32,576582. [ 188,2251 Shabbar, M. (1971). Sideband resonance mechanism in the atmosphere supporting Rossby waves. J. Atmoa Sci. 28,345349. (2341 Shen, C. N. (1959). Stability of forced oscillations with nonlinear secondorder terms. J. Appl Mech.,26,499502. [lo41 Sheppard, L. M. See Lee and Sheppard. Shi, Y. Y. See Eckstein and Shi;Eckstein, Shi,and Kevorkian.
410 RE
AND AUTHOR INDEX
S N Y. Y., and M. C. Eclrrtsin (1966). Ascent or dsllcsnt from satellite orbit by low thrust.
AL4AA 4,22032209. [233] Shi, Y. Y., and M. C. Eckstsjn (1968). Application of dDeulpr pertuxbath methods to ptobbma A m I;, 73,275289. [233,276] Shlcarofdqr, I. P. (1971). Moduisd Born back scattehg from turbulent p h m u ~ ~ : Attonuation kdhg to saturation and aosepddatiua 1Iplrdyo Sct, 6,819831. 13671 Shkil’, N. I. Ses Ferhchenlro, shidl’, and Nikohko. Shniad, H. (1970). The equtvllcPcc of von Zeipel mappings and Lis transforms. Celestial M&, 2,114120. (2021 Sbrestba, G. M.See T d and shrerth.. Sihya, Y. (1958). Sur r h c t i o n anslytisue d’un sy& d’&uations diff6rentielles ordinaims li&ires amtenant un paramhe. J. FBC scicnc~.Uni~.T o m , 7, 527540. [3271 Sibuya, Y. (1963~).Asymptotic solutions of a liaeu ordinruy ditkmtid equation of nth order about a 3impb turning In Z n t a m S~ y m p ~ dDdffcrmthr ~~ Equa&ns mi N o d k a r Mcahrrrricr (J. P. La SaUm mnd S. Lafsch~tz.Eds.), Academic, New York, pp. 485488. [360) Sibuya, Y. (1963b). Simplification of a liwar ordinary differentid equation of the nth order at a tumbg point. Ar& f i t . Me& A d , 13,206221. [3601 Sibuya, Y. (1967). Subdomhutsolulionsof the differenthl equation y”  A’ Oe a,) Or U ~ ) . . . ~  U ~ ) Y =AO~.k d h t l r119,235271.[345] . Sibuya, Y.See Hdch and Sbuya. Simmons, W. F. (1969). A variational method for weak maonant wave interactions. Roc Roy. Soc (I;onaon)),AJo9,551575. [216] Sirignano, W. A., pad L. Crocco (1964). A shack wave model of unstable racket combustors. AIAA 1.. 2,12851296. [78) Sirovich, L. See Chcmg and Sirovich. SiVaSU‘ A. See Tang and Sivasubramanian. Soh, A. I. See Brull and SOlQ. Spriggs. J. H. A. F. Mcssitcr, and W. J. Anderson (1969). Membrane flutter paradoxAn explnaation by bation methods. ALQA J., 7,17041709. I234J Stanley, N. R. !ke Hufnagd and Stanley. Stceh, C. R. (1965). On the asymptotic solution of nonhomogeneousordinary differential equations with a parameter. Qutvt.A& Mth., 23,193201. [353] Stetn, D. P. (197Qr). Dirtct tranrformationa 1. Wth. Aby~., 11, 27762781. [1WI Stern, D. P. (1970b). K ” s perturbation method. J. Mth f i y k . 11,27712775. [ 169) Stem, D. P. (1971a). A new formulption of CMoIljCpl perturbation theory. CelerthlMeJI., 3,241246. [199] St, D. P. (1971b). cla+dcp1 a d i a h t i ~theow. J. AWh. phyr, 12, 22312242. [169] Stem, D. P. (1971~).The canonization of nicc vari.bbk J. Math. my&, 12, 22262231. [911 Stewatson, K., and J. T. Stuart,(1971). A nonlineat instability theory for a wave system in plane Pohsuillc flow.1FMd Me&. 48,529545. (2351
mt.
REFERENCES AND AUTHOR INDEX
4 11
Stoker, J. J. (1957) Water W m s . Wiley, New York. 157,771 Stokes, G. G. (1851). On the effect of the internal friction of fluids on the motion of pendulums. %ns. GmibMge Phil. Soc,9,&106. [ 301 Stokes, G. G. (1857). On the discontinuity of arbitrary constants which appear in divergent developments. CIlrnbrWgePhiL lhms.. 10,106128; all.thpem,4,77109. 13121 Stone, P. H. (1969). The meridional structure of baroclinic waves. J.Atmos. Sci., 26, 376389. [ 234) Strohbehn, J. W. (1968). Comments on Rytov’s method. 1. Opt. Soc.A m , 58, 139140. (3731 Struble, R. A. (1962). Nonlineur Dflferent&zlEquutionr McGrawHill, New York. [171, 223) Stuart, J. T. (1958). On the nonlinear mechanics of hydrodynamic stability. J. FhridMech., 4,l21. (1621 Stuart, J. T. (19606). Nonlinear effects in hydrodynamic stability. Roc. Xth Int. Cong. Appl. Mech. Stresa, Italy. [ 1621 Stuart, J. T. (1960b). On the nonlinear mechanics of wave disturbances in stable’and unstable pardel flows. Part 1. The basic behaviour in plane Poiseuille flow. 1. h i d Mech., 9,353370. [ 1621 Stuart, J. T. (1961). On threedimensional nonlinear effects in the stability of parallel flows. Adwn. A w n . &A. 3,121142. Pergamon, Oxford. (1621 Stuart, J. T. See Stewartson and Stuart. Sturrock, P. A. (1957). Nonlinear effects in electron plasmas. Roc.Roy. Soc. (London), A242,277299. [230] Sturrock, P. A. (1958). A variational principle and an energy theorem for small amplitude disturbances of electron beams and of electronion plasmas. Ann. Phys.. 4,306324. [2161 Sturrock, P. A. (1962). PIoJm HydromgnetiC, Stanford University Press, Stanford, California. (2161 Sturrock, P. A. (1963). Nonlinear theory of electromagnetic waves in plasmas. Stanford University Microwave Laboratory Rept. No. 1004. [ 2301 Swanson, C. A. (1956). Differential equations with singular points. Tech. Rept. 16, Contract Nonr.220(11), hpartment of Mathematics, California Institute of Technology. [3581 Sweet, J. (1971). Impulse of a ring with nonlinear matsial behavior. AIAA J., 9,332334. [581 Tam, C. K. W. (1969). Amplitude dispersion and nonlinear instability of whistlers. phys. Fluids, 12,10281035. [234] Tam, C. K. W. (1970). Nonlinear dispersion of cold plasma waves. J. P h m . MYS.,4, 109125.12341 Tam, K. K. (1968). On the asymptotic solution of the OrrSommerfeld equation by the method of multiple scales. L FhridMech., 34,145158. I233.3601 Tang, T. W.,and A. Sivasubramanian, (1971). Nonlinear instability of modulated waves in a magnetoplasma. Phys. Fluids, 14,444446. I 2171 Taniuti, T. See Asano and Taniuti. Tatarski, V. 1. (1961). Wuve Ropagntion in 4 Zbbukwt Medium McGrawHill, New York. [361,369,373]
4 12 REFERENCES AND AUTHOR INDEX Tatarslci, V. I. (1964). Propagation of electromagnetic waves in a medium with strong mtant fluctuatiom Sovict ~ Y SJETP . @I@& 1, 19, 946953.
[367,369,371,3721
Tamski, V. I., and M. E. Gertrenshteiu, (1963). Propagation of waves m a medium with strong ti^^ Of the refr+ctive index. Sodt WS. Jm &&), 179
458469.[382]
Taussig, R. T. (1969). Macroscopic quasilinear theory of highfrequency radiation in a cold pkrmra P h y ~ Fhdds. . 12,914922.[234] Taylor, L. S. (1967). On Rytw’s method. Radio M,2,437441. I3731 Tempk, G. (1958). Lineariution and delhuization. Proceedings of the International Congrea~of Mathsmoticc, Edinburgh, pp. 233247.[941 Te& R. M., and G. M. shrsrth, (1965). Lamiuar flow throughacbannelwith uniformly p o r ~ u walls s of different permeability.AppL S d Re&,A15,440468. [54,156] Thorn, J. (1883). iiber Intsgrale zwaiter Gattung. 1. Reine A. Miah., 95, 241250.

[311.326]
Tidman, D. A. See Montgomery and Tidman. Timoshenko, S., and S. WomowskyWger, (1959). Theory of Phres und She&, 2nd ed., McGrawHill,New York. [361 Tin& L. See Kdlcr and Ting. Ting, L., and S. Brofman, (1964). On takeoff from circular orbit by mail thrust. ZAhXf,
44,417428. [233] Tollmien, W. (1947). Asymptotische Integration der StonmgsdiBerentidgkichung ebener S t r 6 m v bei hohan Reynoldschen zahlea. ZAMM, 2S/27, 3350, 70.83. [3W] T n h q M. Ste !Mers, Tnbus, and Klein. Tsen& S. F. See Rand and Tseng. Tsien, H.S. (1956). The Pomcar&LighthillKuo method. Adwn. Appl. Me&, 4,281349. [78,80,991 Tso, W. K., and T. K. Caughey, (1965). Parametric excitation of a nonlinear system. J. AppL Me&, 32,899902.[2241 Tumptlrin, S. A. (1959). Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroedrl rbellr and Propdler blpdes.AppL A W L Me&. (H &%t.Me&, A W E mvrdj 23,15491565. [353,356] Usher, P. D. (1968). Coordinate stretching and interface location. 11. A new PL expansion. J. COnpntterPhy~.,32939.(951 Usher, P. D. (1971). Necegtary conditions for applicability of Pd~~&Lighthiuperhubtion theory. Qauwt, A& firh., 28,463471. [79] V a d b b w h , R. (1962). On the PLK method and the supersonic bluntbody problem. J. A m a S d . 29,185206. [99] Van der Corput, J. G. (1956). Asymptotic developments I. Fundamental theorems of asymptotic+ J. AmL bhth., 4,341418. [18] Van der Corput, J. G. (1962).Asymptotic Expmdons Lecture notes, Stanford University. I121 Van der Pol, B. (1922). On a type of osciuation hysteresisin a simple triode generator.Phil 43,177193. [3)
REFERENCES AND AUTHOR INDEX 4 13 Van der Pol, 8. (1926). On oscillation hysteresis in a simple triode generator. Phil Mug..43, 70@719.[164] Van der Pol, B. (1927). h e r Relaxations sch. 1 . b . h h t l Tekgr. Te&pk. 28, 178184.134) Van Dyke, M. D. (1952). A study of secondorder supersonic flow theory.N.A.C.A. Rept. No. 1081. [271 Van Dyke, M. (1964). &mubution Methods in F?a& Mechmica Academic, New York. ~ll0,114,119,130] Van Hove, L.(1955). Quantummechanicalperturbations giving rise to a statistical transport equation. Hysicn, 21,517540.[369] Van Hove, L. (1957). The approach to equilibhun m quantum statistics. P h y b , 23, 441480. [3691 Van Hove, L., N. Hugenholtz, and L. Howland, (196l).Quuntum Theory ofMunyporticle System Benjamin, New York. (3671 Van Wijngaarden, L.(1968). On the oscillations near and at resonance m open pipes. J. Eng. &rh., 2,225240.1891 Van Wijngaarden, L.See Verhagen and Van Wijngaarden. Varley. E.See Mortell and Varky; Parker and Varley. Vawatsis, A. D. See Sancer and Varvatsis. Varvatsis, A. D., and M. 1. Sancer, (1971). On the renormalization method in random wave propagation. Rudio Sci, 6,8797. [ 3723 Vasil'eva, A. B. (1959). On repeated differentiation with respect to the panmeter of solutions of systems of ordinary differential equations with a small parameter in the derivative. fit. Sb., 48,311334 (in Russian). [ 114,121J Vasil'eva, A. B. (1963). Asymptotic behavior of solutions of certain problems for ordinary nonlinear differential equations with a small parameter multiplying the highest dexivatives. Usp. Mar. Nuuk, 18, 1586 (in RueSian); Rurdon f i t h . SrpveYs. 18, (1963) 1381. [114] Verhagen, J. H. G.,and L. Van Wijngaarden, (1965). Non l i n a osciIlatioIlg of fluid in a container. 1.W M e c h . , 22,737751. 1891 Veronis, C . See Malleus and Veronis. Visik, M. I., and L. A. Lyusternilq (1957). Regular degeneration and boundary layer for linear differential equations with small paraiierer. Usp. Mt. Nu&, 12,3122 (in Russian); Am. Math. Soc. Transl., Serv.2,20,239364,1962. [ 114,1441 Volosov, V. M.(1961). Higher approximations in averaging. Sovfer Moth. Do& 2,221224. 11681 Volosov, V. M. (1962). Averaging in systems of ordinary differential equations. RugZlrn Murh. surveys, 7, 1126. [ 1681 Von Zeipel, H. (1916). Movements of minor phuets. Ark. Mzr. Aslron. Stockholm, 11, NO. 1,l58, NO. 7 , 1 4 2 . [189] Voss, W. (1933). Bsdingunsen F% das Aufbten dcs Ramsauereffektes. 2. Phyr, 83, 581618. [ 3461 Wasow, W. A. (1953). Asymptotic solution of the differential equation of hydrodynamic stability in a domain containing a transition poiut. A m Mth., 58,222252.13601 Wasow, W. A. (1955). On the convergence of an approximation method of M. J. Lighthill. J. Rat. Meek Anal. 4,751767. [79]
m&,
4 14 REFERENCES AND AUTHOR INDEX Wmw, W. A. (1965). Asymptotic Expansions for Ordintuy Differential Eqwtions Wiley, New York. [110,309,318,358] W w w , W. A. (1968). Connecti~nproblems for asymptotic series. BulL Am. Math. Soc.. 74, 831853. (3091 New York. Watson, G. N. (1944). A Trertise on the Theory of Besse2 Fwtctiollg Ma&, [3531 Watson, J. (1960). On the nonlinear mechanics of wave diptwbances in stable and unstable p d e l flows. Part 2. The development of a solution for plane PoiseuiUe flow and for plane Couette flow. J. FIuid Me&, 9,371389. ( 1621 Weaver, A. K. See Green and Weaver. Weinitchke, H.J. See Reispner and Weinitschke. Weinstein, L. A. (1969). Open Resonutors and Open Waveguides Golem Press, Boulder, Colorado. ( 3801 Wentzel, G. (1926). Eine Verallgemeinerung der Quantenbedingung fur die Zwecke der Wellenmedwnik. Z. my&,38,518529.[114,315,339] Weyl, H. (1942). On the differential equations of the simplest boundarylayer prohlems. Ann. Math., 43,381407. 11141 Whitehead, A. N. (1889). Second approximations to Viscous fluid motion. Quart. J. Math., 23,143152. 1311 Whitehead, J. A. See Newell and Whitehead. Whiteman, K. J. See Mc Namara and Whiteman. Whitham, G. B. (1952). The flow pattern of a supersonic projectile. C o r n Pure Appl. Math., 5,301348. [57,78,87] Whitham, G. B. (1953). The propagation of weak spherical shocks in stars. Comm Pure AppL Math., 6,397414. [57,78,87] Whitham, G. B. (1965a). A general approach to linear and nonlinear waves using a Lagrangian. 1. FluidMech.. 22,273283. (216,2211 Whitham. G. B. (1965b). Nonlinear dispersive waves. mu.Roy. Soc. (London), A283, 238261. [226] Whitham, G. B. (1967a). Nonlinear dispersion of water waves. J. mUid Mech., 27,399412. 12161 Whitham,G. B. (1967b). Variational methods and applications to water waves. Roc. Roy. Soc (London),A299,625. I2161 Whitham, G . B. (1970). Twetiming, variational principles and waves. J. a i d Mech.. 44, 373395. 12161 Whittaker, E . T. (1914). On the general solution of Mathieu’s equation. Edinburgh Math. SOC b., 32,7580.[62] Whittaker, E. T. (1916). On the adelphic integral of the differential equations of dynamics. Proc. ROY. SOC. Edinb~rgh.37,95116. 11991 Whittaker, E. T. (1937). Anulytial Dynamics of Particles and Rigid Bodies, 4th ed., Cambridge Univemity Press, Cambridge. ( 1991 Wilcox, C . H. (1964). Asymptotic Solutions of Dvfwentkzl Equations and Their A p p b t i o m Wiley, New York. 13091 Wilcox, C. H. (1966). Perhubrtion T h e m and its Appliaations in Qwnium Mechanics. Why, New York. [711
REFERENCES AND AUTHOR INDEX
41 5
Whmte, R. T.,and R. T. Davis, (1970). Perturbation solution of a hyperbolic equation governing longituainal wave propagation in cextain nonuniform bars. J. Acoust. Soc. Am., 47,13341337. I2341 Witt, A. See Gore& and Witt. WoinowskyKrieger,S. See Timoshenko and WoinowskyKrieger. Wolfe, P. (1967). A new approach to edge diffraction. SIAM J. Appl. Math., 15, 14341469. [ 3801 Wu, T. T. See Cheng and Wu. Wu, Y. T. (1956). Twodimensional sink flow of a Viscous, heatconducting, compressible fluid; cylindricalshock waves. Q w t . Appl. Math., 13,393418. [loo] Yagishita, T.See Kawakami and Yagishita. Ying, S.J. See Chu and Ying. Yuen, M. C. (1968). Nonlinear capillary instability of a liquid jet. J. Fluid Mech., 33, 151163. [99] Zabreiko, P. P., and L. B. Ledovskaja, (1966). Higher approximations of the BogolinbovKrylov averaging method. Dokl. A h d . Nuuk. SSSR,171,14531456. [ 1681 Zauderer, E. (1964a). Wave propagation around a convex cylinder. J. Muth. Mech., 13, 171186. [380) Zauderer, E. (1964b). Wave propagation around a smooth object. J. Murh. Mech., 13, 187199. [380] Zauderer, E. (1970a). Boundary layer and uniform asymptotic expansions for diffraction problems. S W J .Appl. hkth., 19,575600. I3801 Zauderer, E. (1970b). Uniform asymptotic solutions of the reduced wave equation. J. Moth. Anal Appl., 30,157171. [378,380] Zawadzki, E. M.,and G. J. Lewak, (1971). Penetration to second order of an electrostatic field into a warm plasma. J. P k s m Phys.. 5,7387. I781 Zierep, V. J., and J. T. Heynak, (1965). Ein analytisches Verfahren zur Berechnung der nichtlinearen Wellenausbreitung. Z B ,45,3746. (891 Zwaan, A. (1929). lntensitaten im CaFunknspektrum. Ph. D. Thesis, Utrecht. (3391 Zwanzig, R. (1964). On the intensity of three generalized master equations. Physicu, 30, 11091123. [382]
Perturbation Methods ALI HASAN NAYFEH Copyright Q 2004 WILEYVCH Valag GmbH & Co. KGaA
Subject Index
Acoustic, 77, 78 Aerodynamic, 110 Airfoil theory, 98,99,235, 303;see Supersonic airfoil theory Airy’s equation, 312,336 Airy’s functions, 49,336,378 Algebraic equation, 2,57,74,95,327 Algorithm, 168,171,199,209 for canonical systems, 212214 generalized, 202206 simplified, 206208 Anomaly, 79 Aperiodic motion, 189 Astrophysics, 235 Asymptotic expansion, 23,78 of Airy’s functions, 337 of Bessel’s function, 16 defiition of, 12 divergent, 16 elementary operation on, 1819 uniform, 1719 uniqueness of, 14 see UlSO Asymptotic series Asymptotic matching principle; see Matching Asymptotic partitioning, 327331 Asymptotic sequence, 12,14,16,18,19 factorial, 12 fractional powers in, 136, 137 logarithms in, 137,144 Asymptotic series, 1012 defiition of, 1 1 versus convergent series, 1516 see Asymptotic expansion Attenuation, 46,78 Averaging, method of, 159227 generalized method of, 168171,191,
211,223225,231
KrylovBogolinbov, method of, 165168
KrylovBogolinbovMopolski,method Of, 174179,183,194,211,212, 223,224,246,248 Struble’s method of, 171174,176,183, 194,223 using, canonical variables, 179189 Lagrangian, 216222 Lie series and transforms,200246 von Zeipel transformation, 189200 van der Pol’smethod of, 164165 see ulso Smoothing Beam, niration of, 105,106,155,226,306 Bearing, slider, 54,125128 Bellman, equation of, 22 Benard problem, 77 Bending of, shells and tubes, 54,353;see nlso Unsymmetricalbending of plates Benney’s technique, 3842 Bernoulli’s equation, 83 Besselfunctions, 1,5,6,15,312,315,383 asymptotic expansion of, 16,312314,
329331
integral representation of, 16,314 zeros of, 21 BetheSalpeter equation, 372 Blunt body problem, 99 Boltzmann’s equation, 236 Born approximation, 362 Born expansion, 308,361367,373 renormalization of, 367372 Boundary conditions, loss of, 31, 34,37,
38,54,111,114,122
transfer of, 27 Boundarylayer,18,23,79,111,112,147,
233
location of,114116,122 Prandtl, 34
417
418
SUBJECT INDEX
problem with two, s, 128133 stability of, 353 Branch point, 82 Bretherton's equation, exercises involving, 227,306 treated, by variational approach, 217221 by method of multiple scales, 266269, 298300 wavewave interaction for, 219221,266269 Brownian motion, 236 Buckling, 233 Canonical, averaging, variables, 179189 equations, 180,199 Jordan, form, 327 mixed, variables, 199 system, 190,201 transformation, 181,187,191 variables, 181.183,184,195,202,216, 224 Caustic, 234,375380 Change, 111,113 of characteristics, 89 see also Sharp change; Type change of Characteristic, parameters, 89,94 wave speeds,91 Characteristic exponent, 58,62,66 Characteristics, expansion in terms of, 57, 8694,303 method of, 374 Circuit, electronic, see van der Pol oscillator Cluster expansions, 363 Compatability relationship, 217,222 Composite expansion, 114,144,384,385 construction of, 121 for earthmoonspaceshipproblem, 139 for equation with variable coefficients, 125 for simple example, 121122 for slider bearing, 128 for thermoelastic waves, 136 for unsymmetrical bending of plates, 133 Composite expansions, applied to turning point problems, 348350 method of, 144154,317,318 Composite solution, 113 Conservation form ofequations, 226 Coordinate, optimal, 5051 parabolic, 79
perturbations, 1,47,21, 309314,379 role of, systems, 23,4952,54, 380 see olso Strained coordinates Correlation, 363 Cosmologicalexpansion, 233 Cycle, limit, 35 Cylinder, elliptic, 346 a solid, expanding, 8386 Cylindrical,functions, 358,378,380 jet, 99 Degenerate, 72, 74 Derivativeexpansion procedure, 302 applications of, 243269 description of, 230,236240 limitations of, 269270 see ulso Multiple scales, method of Detuning, 250 Diagram, 366,367 bare, 364 connected, 370,371,372 double, 365 dressed, 365 Diffraction, 346,380 Diffusion equation, 38 Dirichlet problem, 38 Discontinuity, see Singularity Dispersion relationship, 178,217,218, 219, 220,222,227,266,301 Dispersive waves, 78,234, 303 long nonlinear, 3842 see ulso Bretherton's equation; KleinGordon equation; Thermoehstic waves; Wavewave interaction Divisor, small, 196 Domain, effect of, on nonuniformity of expansions, 38,42 infinite, 2431 Duffing equation, 5 0 , s 1 exercises involving, 54,104,105,223, 224,304 with slowly vary coefficients, 286 straightforward expansion for, 2425 treated, by averaging using canonical variables, 182183 by derivativeexpansion procedure, 243245 by generalized method of multiple scales, 286291 by KrylovBogoliubov method, 167
SUBJECT INDEX
4 19
by KrylovBogoliubovMitropo~~ method, 175176 by LindstedtPoincard method, 5860 by renormalization, 9596 by Struble's method, 171174 by twovariable expansion procedure, 271273 by von Zeipel's procedure, 192194 Dyson equation, 371, 381
treated by method of matched asymp totic expansions, 139144 Flutter, 234 Foci, 253 FokkerPlanck equation, 235, 381 Fourier, 45,167,178 Frequency, 56,58,96, 165,252 Fresnel diffraction, 380 Frobenius, method of, 5,310
Earthmoonspaceship problem, 233,302, 303 exercises involving, 53, 107 illustrating limitations of method of strained coordinates, 102103 straightforward expansion for, 4 3 4 5 treated, by Lighthill's technique, 8283 by method of composite expansions, 153154 by method of matched asymptotic expansions, 137139 by method of multiple scales, 295298 Eccentricity, 64, 233, 346 Edge layer, 111 Eiconal equation, 374, 377, 379 Eigenvalue, 56 Eigenvalue problem, linear, 6871 quasilinear, 7176 Elastic, 46, 58,90, 353 waves, 8994 Elliptic equation, 37,42,98, 189,234, 303, 360 Energy level, 56, 58 Entropy layer, 79 EulerLagrangeequation, 216, 217, 218, 220,222
Gauge, function, 7,8 transformation, 232 Gaussian, 363, 367,369,372 Generalized expansion, see Composite expansion Generalized version of method of multiple scales, applications of, 276302 description of, 232,241243 limitations of, 302303 Generalized method of averaging, see Averaging, method of Generalized vector, 179 Generating function, 181,184, 189,190, 192,195,196,200,202,215 Generating vector, 201 Geometrical optics, 308,361, 374377, 379 Geophysics, 110 Graetz problem, 384 Green's function, 362, 364, 380 double, 364 Group velocity, 179,219,220,267,299
Faa de Bruno operators, 192 Feynman diagrams, 308,361372 night mechanics, 233 Floquet theory, 60,62,64,66 Flow, down an inclined plane, 3842,99 hypersonic, 79 jet, 99 past a body, 3334, 113 through a channel, 54,56,216 see also Flow past a sphere; Supersonic airfoil theory Flow past a sphere, exercise involving, 158 straightforward expansion for, 2831
HamiltonJacobi equation, 181, 182, 183, 186,190,199 Hamiltonian, 223,224,225 definition of, 180 for D u f f i equation, 182 for Mathieu equation, 184 for swingii spring,186 transformation of, using von Zeipel procedure, 189200 using Lie transforms, 202, 212216 Harmonic balance, 218 Harmonic resonance, 219221, 234, 262269 Harmonic wave, 76 Heat, 45,79,84,156,342,384 problem for, equation, 150152 Helmholtz equation, 234 Hill's equation, 60
420
SUBJECTINDEX
Hopf equation, 381 Hydraulicjump, 89 Hyperbolic equation, 37,42,57,99,379. See also Dispersivewaves; Elastic waves; and Supersonic airfoil theory
Jacobi elliptic functions, 189 Jerky oscillations, 34 Jordan, form, 327 matrix, 327 Josephson tunnel, 221
Inclined plane, see Flow, down an inclined Plane Indicial equation, 311 Induction, 13 Infiite domain, as a source,of nonuniformity, 2331,229 of uniformity, 38,42 Inhomogeneous, 361 problems with turningpoints, 352359 Initial, boundary value problem for heat equation, 150152 layer, 23 Inner and our expansions, see Matched asymptotic expansions, method of Inner expansion, 110,112,114,119,145,
Kamel's algorithm. 171 Kamel's method, 224
146,148
for bending of plates, 129132 d e f i t i o n , 117118 for earthmoonspaceship problem, 139 for equation with variable coefficients, 122124
for simple example, 117118 for slider beating, 127 for thermoelastic waves, 134135 for turning point problems, 336 Inner limit, 112,118,119,129,131,139 Inner region, 113,122,146,153 Inner solution, 112,113 Inner variables, 119,120,121,144,149, 153,154
choice of, 114116, 122124, 126127, 134135,137138
generalized, 145 Instability, 78,269. See also Model for nonlinear w b i l i t y ; Stability Integral, 10.11,12 differential equation, 18 equation, 370,380 of motion, 24,199 relations, 99 Intermediate limit, 119 Intermediate matching, see Overlapping Jacobian matrix, 202
KelvinHelmholtz,77,235
Kernel, 370 Kleincordon equation, 175,234 exercises involving, 105,226 treated, by averaging using Lagrmgh, 221222
by KrylovBogoliubovMitropohki method, 178179 by method of multiple scales, 301302 by method of strained parameters, 7677
KrylovBogoliubov technique, 165168,223 KrylovBgoliubovMitropolskitechnique, 174179,183,194,211,212,223, 224,246,248,303
Kruskal's technique, 168,191 Lagrange equations, 179,180 Lagrangian, 179
averaging using, 216222,301 for Bretherton's equations, 2 17 for KleinGordon equation, 221 for swinging spring, 186 Lamd coefficients of elasticity, 45,90
Laminar, 39,54
Landau, equation, 382 symbols, 8,9 Langer's transformation, 308,346,384 for firstorder turning point problems, 339341
for generalization of, 341342 successive, 350 Latta's technique, 144154 Layer, 23,79,111; see also Boundary layer Libration points, see Stability of elliptic triangular points Lie series and transforms, 171,199,200216,223,225,303
Lie triangle, 205,206 Lighthill's technique, 57,7795 exercises involving, 107,108 limitations of, 79,98100,107,109
SUBJECT INDEX Limit, 7 cycle, point, or solution, 35,99, 109 distinguished, 336 Oseen’s, 140 Stokes’, 140 see also Inner limit; Intermediate limit; and Outer limit Limitations of, method, of composite expansions, 153154 of matched asymptotic expansions, 144, 145,155,156,303,339 of multiple scales, 269270, 275276, 302303 of strained coordinates, 79,98103, 107,109,110,302,303 Struble’s method, 174 von Zeipel’s procedure, 199 LindstedtPoincard technique, 56,57,5860,95,96,185,200 Linear damped oscillator, treated by multiple scales, 228243 Linearization, method of, 57,9495 LiouviUe, equation, 236, 380, 382 problem, 314 LiouviUeGreen approximations, 49, 320, 335 higher, 315317 successive, 324325 LiouviUeGreen transformation, 308, 315, 340 Logarithms, 7,12,45,83,137,144,308, 311 Lommel functions, 353 Long period part, 169,191,209,214 Lunar motion, 60; see also Earthmoonspaceship problem Mach number, 26,83,85 Matched asymptotic expansions, method of, 37,48,51,78,110144,148,153, 154,235,317,346,378,384 applied to turning point problems, 336339 limitations of, 144,155,156,303 Matching, 51,110,114,115,116
asymptotic, principle, see van Dyke’s principle as guide to forms of expansions, 141 intermediate, 119 Kaplun’s, principle, 119
421
Randtl, procedure, 112,118 refined, 118119 Matching of inner and outer expansions, bending of plates, 130132 caustic problem, 378 earthmoonspaceship problem, 139 equation with variable coefficients, 124 now past a sphere, 141144 simple example, 12012 1 slider bearing, 127 term by term, 130 thermoelastic waves, 135136 turning point problems, 337339,343344 Mathien equation, 339 exercises involving, 53,104, 223, 303, 304 treated, by averaging using canonical variables, 183185 by LindstedtPoincard technique, 6 0 6 2 by method of multiple scales, 253257 by von Zeipei’s procedure, 194200 by Whittaker’s method, 6264 Maxwell’s heat conduction law, 45 Membrane, 356 Method, of strained coordinates, see Strained coordinates, method of of strained parameters, see Strained parameters, method of of multiple scales, see Multiple scales, method of Missile dynamics, 233 exercises involving, 304, 305, 306 linear, 320321 nonlinear,291295 Model for nonlinear instability, 50 exercises involving, 55 illustrating limitations of method of strained coordinates, 99100 straightforward expansion for, 2526 treated, by method of multiple scales, 264266 by method of renormalization, 9697 Momenta, 179,182,190,199,224 Moon,see Earthmoonspaceship problem; Lunar motion Multiple scales, method of, 51,97, 100, 153,221,228307,315, 317,339, 368,384 applied, to caustic, 379
422
SUBJECT INDEX
to equations with slowly varying coefficients, 318320 to OrrSommerfeld equation, 360 NavierStokesequations, 28,33, 39 Neumann, expansion, 361; see also Born expansion series, 364 Newtonian theory, 79 Nondispersive waves, 78,269270,303. See also Dispersive waves; Elastic; Shock waves; and Supersonic airfoil theory Nonuniformity, in airfoil theory, 28,50 in asymptotic expansions, 1618 bibending of plates, 37 in Born’s expansion, 367 dependence of, on coordinates, 4951 on size of domain, 2431,38,42 in Duffii’s equation, 2425,4950 in earthmoonspaceshipproblem, 45 in equation with constant coefficients, 32 in flow past a sphere, 31 in geometrical optics approximation, 376377
in interior of domain, 137 in jet instability, 99 in linear oscillator, 229 in model for nonlinear instability, 26,50 in relaxation oscillations, 35 in slider bearing, 126 in shift in singularity, 43,79 in thermoelastic waves, 4 7 4 8 in turning point problems, 49,284,315, 336
variable exhibiting, 57,285 see a h Boundary layer; Region of nonuniformity; Sources of nonuniformity; and Type change of Node, 252 Normal solution, 58,60,66,311,326,331, 332
Nodip condition, 33,34 Olver’s transformation, 341,384,385 Operator, 206 adjoint, 164 Faa de Bruno, 192 intensity, 372 mass, 371,381 selfadjoint. 163,164
Optics, see Geometrical optics Optimal, control, 79 coordinate, 50,51 Orbit, 64.99 Orbital mechanics, 233 Order symbols, 8,9 OrrSommerfeld equation, 233,360 Oscillations, 25,34,89,226,232,303 Oseen’s equation, 141 Oseen’s expansion, 140144 Oseen’s limit process, 140,141 Oseen’s variable, 143 Outerexpansion, 110,112.114,119,146 for bending of plates, 128129 deftnition of,117 for earthmoon spaceship problem, 45, 138
for equation with variable coefficients, 124
for simple example, 117 for slider bearing, 126 for thermoelastic waves, 4 7 4 8 Outer limit, 112,117,119,128,138,144 Outer region, 113,122 Outer solution, 111,113,115 Outer variable, 119, 121,144,145 Overlapping, 113,119 Parabolic coordinates, 79 Parabolic cylinder function, 344,378 Parabolic equation, 37,38,99,379 Paradox, 31 Parameter, 89 large, 315; see also Turning point problems perturbation, 14,16,308 see a h Small parameter multiplying highest derivative; Strained parameters, method of Parametric resonance, see Mathieu equation; Stability of elliptic ttiangularpoints Parameters, 201; see also Variation of parameters Parametrization, 88,92 Partitioning,see Asymptotic partitioning Pascal triangle, 205 Pendulum, 103,224; see Swinging spring Period, 169,191,192 Periodic, 58,60 motion, 189
SUBJECT INDEX orbit, 99 solutions, 34, 77 see Mathieu equation; Stability of elliptic triangular points Perturbations, coordinate, 47, 308314 parameter, 14, 16, 308 Phase, 178, 380 rapidly rotating, 168, 201, 234 speed, 76,71 Plasma, 58, 78,216,217,226,234, 235, 367 Plasticity, 90 Plates, see Unsymmetrical bending of plates PoincardLighthillKuomethod, see Lighthill‘s technique Poisson ratio, 36,47 Potential function, 26,83,89 Prandtl’s boundary layer, 113 Prandtl’s technique, 111, 114 Pritulo’s technique, see Renormalization, method of Quantum, 58, 371; see also Schrodinger equation Random, 361,362,363,365,367, 369, 372,380,381,382 RankineHugonoit relation, 84 Ratio test, 6, 7, 10, 11, 16 RayleighSchrodingermethod, 56, 71 applied to eigenvalue problem, 6871 RayleighTaylorinstability, 77, 235 Rayleigh wave speed, 47, 134 Reaction kinetics, 79 Reentry dynamics, see Missile dynamics Regionof nonuniformity, 17,18, 19,23, 79 for earthmoonspaceshipproblem, 137138 for equation with constant coefficients, 116 for flow past a sphere, 140 near a caustic, 377378 for thermoelastic waves, 134135 for turning point problems, 284, 336 Related equation, 341 Relaxation oscillations, 3435 Renormalization, method of, 57,9598, 308,361, 367372 exercises involving, 103,106,107,108
423
Resonance, 89, 233,248 external, 225 internal, 185, 225 in linear systems, 321324 near, 195 perfect, 189 parametric, 235 passage through, 233 see also Harmonic resonance Reynold’s equation, see Slider bearing Reynolds number, 40,360 high, 33,130 small, 28,29,139 Rigidity, flexural, 36 RitzGalerkin procedure, 58 Rossby wave, 77,234 Rytov’s method, 361,373 Saddle point, 252 Satellite, 233,276 Scales, 51, 110,230,231, 232,233, 242, 303; see also Multiple scales, method of Scattering, 57,95,235, 361, 364, 367, 368 Schrodinger equation, 56,71,160, 339, 342,344,346 Secular terms, 2326 elimination of, 6548,261, 288, 302 Selfsustained oscillations, 89; see also van der Pol oscillator Series, see Asymptotic series; Lie series and transforms Shadow boundary, 378 Sharp change, 103,110,303 Shell, 233 Shift, 42,77,79,106 in singularity, 103 exercises involving, 52,53, 106, 107 straightforward expansion for, 4243 treated, by Lighthill’s technique, 7982 by renormalization, 98 by Temple’s technique, 9495 Shock waves, 52,78,83, 84,85,99, 100, 235 Short period part, 169,191,214 Singular,17,57,78,81,231 Singular perturbation, 17,99,340 dependence on region size, 38,42 see also Nonuniformity Singular point, definition of, 309,310
424
SUBJECT INDEX
expansion near an irregular, 309312,326327 regular, 5,308 Singularity, 85,86 as a jump discontinuity, 32,35 branch point, 82 essential, 7 growing, 23,42,43,45,95 logarithmic, 45,83,102,137 tuning point with, 358359 worst, 81,98 see also Nonuniformity Skin layer, 111 Slider bearing, 54,125128 Small parameter multiplying highest derivative, 23,3137,99, 317318 in limitations of method of strained coordinates, 99,100102 Smoothing, method of, 308,361,380382 Solvability condition, 152,288,302; see Secular terms Sonic boom, 78 Sources of nonuniformity, 2355, 140 Spaceship, see Earthmoonspaceshipproblem Speed, 56,57,58,76,77,83,90,91 Sphere, see Flow past a sphere Spherical, 28,78,224 Spring, 232. See also D u f f i i equation; swinging spring Stability, 60,77,78,99,100,162164,235, 252,353,360. See also Mathieu equation; Model for nonlinear stability; Stability of elliptic triangular points Stability of elliptic triangular points, treated, by method of multiple scales, 259262,275 by method of strained parameters, 6466 by Whittaker’s technique, 6 6 6 8 Statistical mechanics, 236 Stochastic, 361,367 Stokes’ expansion, 140 Stokes’ limit process, 140 Stokes’ variable, 143 Stokes’ solution, 30 Strained coordinates, method of, 51.52, 56109,110.114.156,235,302, . . 303
Strained parameters, method of, 56,5877, 78,79,97,99,100,103,105,106 Strainitg, of characteristics, 87,89 dependent variable, 99,108 function, 57,78,79,81,87,99,101,102, 103,107 Stratification, 216 Stream function, 28,29,33,40,57,113, 140 Stretching transformation, for bending of plates, 128 for caustic, 377378 for dependent and independent variables, 134 for earthmoonspaceship problem, 137138 for equation with variable coefficients, 122,123 for heat equation, 151 for turning point problems, 284,336 Struble’s method, 171174,176,183,194, 223 StuartWatsonEckhaustechnique, 162164 Subnormal solution, 311,331332 Supersonic airfoil theory, 50,78,93 straightforward expansion for, 2628 treated, by Lighthill’s technique, 8689 by renormalization, 9798 swinging spring,exercises involving, 105, 225 treated, by averaging Hamiltonian, 185189 by Lie series and transforms, 214216 by method of multiple scales, 262264 Temple’s technique, 57,9495 Thermoelastic waves, straightforward expansion for, 4548 treated by method of matched asymptotic expansions, 133137 ThomasFermi model, 235 Three body problem, 199,233,276. See also Earthmoonspaceship problem; Stability of elliptic triangular points Triangle, 205,206 Triangular points, 233; see also Stability of elliptic triangularpoints Transfer of boundary condition, 27 Transform, see Lie series and transforms Transformation, canonical, 181,190,
SUBJECT INDEX 191,199 contracting, 140 Deprit, 202 Hori, 202 Lie, 201 LiouvfileGreen, 49,315,340 near identity, 51,57,168,190,201 stretching, 111,113,114,116,117,377 von Zeipel, 171,191,199,202 Transition, 380; see Turning point problems Transition curves, exercises involving, 104, 105,225,303,304 for libration points, 6468, 259262.275 for Mathieu’s equation, 6064, 183185, 194200,253257 Transport equation, 374, 379 Tunneling effect, 342 Turning point problems, 114,308,309, 335360 definition of, 49,122,284,335 exercises involving, 53, 305, 383, 384, 385 near caustic, 377380 treated by method of multiple scales, 232, 233,284286 Twobody problem, 201 Twovariable expansion procedure, 243, 302 applications of, 270275 description of, 231, 240241 limitations of, 275276 see also Multiple scales, method of Type change of, 23,3742
Uniformity, see Nonuniformity Uniformization, 94, 107, 232 Unsymmetrical bending of plates, exercises involving, 155 straightforward expansion for, 3537 treated by method of multiple scales, 128‘133 van der Pol, method of, 164165,166 van der Pol oscillator, exercises involving, 52,54,104,223,224,303,304 straightforward expansion for, 3 4 , 3435
425
treated by generalized method of averaging, 169171 by KrylovBogoliubov technique, 167168 by KrylovBogoliubovMitro~l~i technique, 176177 by Lie series and transforms, 209212 by method of multiple scales, 245253, 272275 van der Pol oscillator with delayed amplitude liiting, 104,257259,304 van Dyke’s matching principle, 114, 119 mechanics of, 120122 Variables, see Canonical variables; Inner variables; and Outer variables Variation of parameters, 52,159222 Variational, approach, 216222, 227 equations, 172,183, 184, 190 Vibrations, 114,232. See also Oscillations; Waves Vlasov’s equation, 78, 216 von Zeipel’s procedure, 189200, 202,231 Water waves, 56,57,58,77,216,217,234 Wave equations, 360382 Wave number, 58,77,99,360 Waves, 58,60,77, 78,89,90,94, 216, 217, 233,234,235. See also Bretherton’s equation; Dispersive waves; Elastic waves; Flow down an inclined plane; KleinGordon equation; Non dispersive waves; F’lasma; Shock waves; Supersonic airfoil theory; and Thermoelastic waves Weber function, 345, 378,380 Whitham’s method, 216222 Whittaker’s functions, 346 Whittaker’s method, 6244,6668, 104, 105,185,200 WKBJ approximation, 49,308, 315, 320, 335,339,382 successive, 324325 Wronskian, 160 Young’s modulus, 36