WAVE PHENOMENA: MODERN THEORY AND APPLICATIONS
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WAVE PHENOMENA: MODERN THEORY AND APPLICATIONS
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NORTH-HOLLAND MATHEMATKS STUDIES
97
Wave Phenomena: ModernTheory and Applications Edited by
C. ROGERS University of Waterloo Canada and
T. BRYANT MOODIE University ofAlberta Canada
1984 NORTH-HOLLAND -AMSTERDAM
NEW YORK
OXFORD
@ElsevierSciencePublishers B.V., 1984 All rights reserved. N o part of rhispublication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocoping, recording or otherwise, without the prior permission of the copyrighr o wner.-
ISBN: 0444875867
Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 IOOOBZ AMSTERDAM T H E NETHERLANDS Soledisiibutors f o r the U . S . A .and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A.
Library of Congress Calaloging in Publicalion Data
Main entry under title: Wave phenomena.
(North-Holland mathematics studies ; 97) Papers presented at an international meeting held at the University of Toronto, June 20-24, 1983 under the auspices of the Canadian Applied Mathematics Society. Includes bibliographical references. 1. Wave-motion, Theory of--Addresses, essays, lectures. I. Rogers, C. 11. Moodie, T. Bryant (Thomas Bryant), 111. Canadian Applied Mathematics Society. 1944IV. Series. 84-14142 QA927.W375 1984 531’ .1133 ISBN 0-444-87586-7
.
PRINTED IN T H E NETHERLANDS
PREFACE
T h i s volume c o n t a i n s t h i r t y f i v e c o n t r i b u t i o n s t o t h e i n t e r n a t i o n a l m e e t i n g Wave Phenomena: Modern Theory and A p p l i c a t i o n s h e l d a t t h e U n i v e r s i t y o f T o r o n t o June 20 - June 24, 1983 under t h e auspices of t h e Canadian A p p l i e d Mathematics S o c i e t y and s u p p o r t e d by t h e N a t u r a l Sciences and E n g i n e e r i n g Research Council of Canada. The B r i t i s h Council k i n d l y s u p p l i e d a d d i t i o n a l t r a v e l s u p p o r t . The purpose o f t h e m e e t i n g was t o b r i n g t o g e t h e r r e s e a r c h e r s o f i n t e r n a t i o n a l s t a t u r e i n v a r i o u s aspects o f wave t h e o r y , b o t h l i n e a r and n o n l i n e a r . The c o n t e n t s o f t h i s volume bear w i t n e s s t o t h e d i v e r s i t y o f p h y s i c a l phenomena embraced by t h e subject. The e d i t o r s w i s h t o express t h e i r deep g r a t i t u d e t o a l l c o n t r i b u t o r s whose a r t i c l e s made p o s s i b l e t h e p r e s e n t proceedings. E s p e c i a l thanks a r e due t o R.A. Ross f o r o r g a n i s a t i o n a l s u p p o r t and t o Helen Warren f o r h e r e x c e l l e n t e d i t o r i a l assistance.
vi
LIST OF CONTENTS
V
Papers
1
EQUATIONS OF EVOLUTION AND WAVES A1 an Jeffrey RATIONAL FUNCTION FREQUENCY EXTRAPOLATION I N ULTRASONIC TOMOGRAPHY F. S t e n g e r , M.J.
B e r g g r e n , S.A.
J o h n s o n and C.H.
NONLINEAR T R A V E L L I N G WAVES I N F L U I D S P.L.
19
Wilcox 35
Sachdev
SECOND ORDER WAVE INTERACTION W I T H LARGE STRUCTURES
49
M. Rahman ACCELERATION WAVES I N I S O T R O P I C CONSTRAINED THERMOELASTIC MATERIALS B.D.
71
Reddy
A LAGRANGIAN VIEW OF WAVE-MEAN FLOW INTERACTION
83
R. G r i m s h a w A THEORY OF DYNAMICAL FERROELECTRICITY
99
P e t e r J. Chen SOLITARY WAVE SOLUTIONS OF GENERALISED KORTEWEG-DE V R I E S EQUATIONS F.A.
GEOMETRICAL CRYSTAL ACOUSTICS R.S.D.
113
Thomas and H. C o h e n
RIEMANN I N V A R I A N T S A.M.
105
Howes
123
Grundland
VARIATIONAL FORMULATION OF THE SINGULAR SURFACE PROPAGATION I N NON-
193
S I M P L E E L A S T I C MATERIALS
Jacek T u r s k i FORCED INTEGRABLE SYSTEMS
163
D a v i d J. K a u p INHOMOGENEOUS PLANE WAVES I N INCOMPRESSIBLE E L A S T I C MATERIALS M i c h a e l Hayes
175
List of Contents
CAN ACOUSTIC FOCUSING GENERATE TURBULENT SPOTS?
vii
7 93
G r e g o r y A. K r i e g s m a n n a n d E d w a r d L. R e i s s NONLINEAR WAVES I N THE PELLET FUSION PROCESS
V.J. E r v i n , W.F.
199
Ames a n d E . Adams
ENTROPY PRINCIPLE, SYMMETRIC HYPERBOLIC SYSTEMS AND SHOCK WAVES
21 1
Tommaso R u g g e r i DIFFERENTIAL AND DISCRETE SPECTRAL PROBLEMS AND THEIR INVERSES P.J.
221
Caudrey
WAVEFRONT FIELDS I N THE SCATTERING OF TRANSIENT ELASTIC WAVES BY A
233
SURFACE-BREAKING CRACK J u l i u s M i k l o w i t z and Constantine Chazapis WIENER-HOPF ANALYSIS:
AN APPRAISAL AND A NON-STANDARD APPLICATION
253
H a r o l d L e v i ne STRESS NAVE PROPAGATION I N DISCRETE RANDOM SOLIDS
267
Martin Ostoja-Starzewski DO L I O U V I L L E ' S SOLUTIONS POSSESS TOPOLOGICAL CHARGE?
279
George L e i b b r a n d t DISLOCATIONS I N THE DYNAMIC ANALYSIS OF CRACKS
293
L o u i s M. B r o c k A BACKLUND TRANSFORMATION FOR A NONLINEAR TELEGRAPH EQUATION
299
B r i a n S e y m o u r a n d E r i c Varley ON LINEAR DISPERSIVE WAVES WITH D I S S I P A T I O N
307
Francesco Mainardi L I M I T I N G GASDYNAMIC THEORY AND I T S APPLICATIONS W.H.
SCATTERING OF ACOUSTIC WAVES T.S.
3 19
H u i a n d H.J. Van R o e s s e l
BY IMPEDANCE SURFACES
329
A n g e l 1 and R.E. K l e i n m a n
INTERACTING NONLINEAR WAVES I N A NEURAL CONTINUUM MODEL:
337
ASSOCIATIVE MEMORY AND PATTERN RECOGNITION M a r c u s S. Cohen SOLITON DYNAMIC STATES I N THE SINE-GORDON SYSTEM
peter Leth C h r i s t i a n s e n
349
viii
List of Contents
AN ASYMPTOTIC A N A L Y S I S O F ACOUSTIC SCATTERING B Y NEARLY R I G I D
361
OR SOFT TARGETS G r e g o r y A. K r i e g s m a n n , A n d r e w N o r r i s and E d w a r d L. R e i s s SYMMETRY REDUCTION FOR NONLINEAR WAVE EQUATIONS I N RIEMANNIAN AND PSEUDO-
373
RIEMANNIAN SPACES A.M.
G r u n d l a n d , J . H a r n a d and P. W i n t e r n i t z
INTEGRAL REPRESENTATIONS FOR SOLUTIONS TO SOME D I F F E R E N T I A L EQUATIONS
391
THAT A R I S E I N WAVE THEORY R o b e r t F. M i l l a r NONLINEAR WAVE PHENOMENA I N THE B R A I N
409
R o b e r t M. M i u r a SOLITON SOLUTIONS TO ZAKHAROV-SHABAT SYSTEMS BY THE REDUCTION METHOD
423
J . H a r n a d , Y . S a i n t - A u b i n and S. S h n i d e r M U L T I P L E SCATTERING BY SH WAVES BY RANDOMLY D I S T R I B U T E D D I S S I M I L A R
433
SCATTERE RS M a t t h e w F. M c C a r t h y and M i c h a e l M. C a r r o l l AN INTEGRAL EQUATION A R I S I N G I N D I F F R A C T I O N THEORY R.A.
Ross
453
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
EQUATIONS OF EVOLUTION AND WAVES
Alan J e f f r e y Department o f E n g i n e e r i n g Mathematics U n i v e r s i t y o f Newcastle upon Tyne U.K.
INTRODUCTION The purpose o f t h i s paper i s t o d i s c u s s some o f t h e many d i f f e r e n t e v o l u t i o n e q u a t i o n s which a r i s e i n c o n n e c t i o n w i t h wave m o t i o n , some d i r e c t l y and o t h e r s as a s y m p t o t i c a p p r o x i m a t i o n s . Wave m o t i o n i t s e l f may be regarded as t h e m a n i f e s t a t i o n o f some f o r m o f i n h e r e n t i n s t a b i l i t y i n a process, so perhaps i t i s t o be expected t h a t t h e e q u a t i o n s which govern i t s many d i f f e r e n t aspects b e l o n g t o no c l e a r l y d e f i n e d c l a s s . The e v o l u t i o n e q u a t i o n s d i s c u s s e d h e r e by way o f example emphasise t h i s v a r i e t y o f t y p e , w h i l e a t t h e same t i m e showing something o f t h e d i v e r s i t y of b e h a v i o u r w h i c h can a r i s e i n c o n n e c t i o n w i t h wave p r o p a g a t i o n . We s h a l l , f o r example, see a way i n which e q u a t i o n s o f d i f f e r e n t c o m p l e x i t y can d e s c r i b e t h e b e h a v i o u r o f t h e same i n i t i a l v a l u e problem, and how t h i s can cause a s o l u t i o n t o t h e s i m p l e r f o r m o f e q u a t i o n (degenerate s o l u t i o n ) t o " f o r g e t " some o f t h e i n i t i a l d a t a t h a t gave r i s e t o i t . Then, i n t h e h y p e r b o l i c case, t h e b e h a v i o u r o f an a c c e l e r a t i o n t y p e o f wave w i l l be seen t o i n v o l v e an o r d i n a r y d i f f e r e n t i a l e q u a t i o n o f t h e B e r n o u l l i t y p e . Depending on t h e c o e f f i c i e n t s o f t h i s e q u a t i o n , so t h e i n t e n s i t y o f t h e a c c e l e r a t i o n wave may, o r may n o t , become i n f i n i t e c a u s i n g t h e c o r r e s p o n d i n g s o l u t i o n o f t h e h y p e r b o l i c system t o produce a shock ( d i s c o n t i n u o u s ) s o l u t i o n . More extreme s t i l l i s t h e b e h a v i o u r o f a s i m p l e q u a s i l i n e a r h y p e r b o l i c system i n which i t w i l l be seen t h e s o l u t i o n i t s e l f becomes unbounded a f t e r a f i n i t e t i m e (blow-up t i m e ) d e s p i t e t h e i n i t i a l d a t a b e i n g bounded and C1. F o r e q u a t i o n s of t h i s t y p e no g e n e r a l i s a t i o n o f t h e s o l u t i o n i s p o s s i b l e which w i l l e n a b l e i t t o be extended beyond t h e blow-up t i m e . I m p l i c i t i n some o f t h e s e examples, and i n o t h e r s t h a t w i l l a l s o be discussed, i s t h e i d e a t h a t t h e s t r u c t u r e o f t h e s o l u t i o n t o an i n i t i a l v a l u e problem o f t e n s i m p l i f i e s a f t e r a s u i t a b l y l o n g t i m e . By o n l y s t u d y i n g t h e s o l u t i o n a f t e r a s u f f i c i e n t l a p s e o f t i m e we a r e , i n e f f e c t , c o n s i d e r i n g i t s f o r m a t l o c a t i o n s f a r removed f r o m t h e i n i t i a l l i n e . A c c o r d i n g l y , i f t h e s o l u t i o n i s c o n s i d e r e d t o d e f i n e a f i e l d , t h i s t y p e of s o l u t i o n may be termed a f a r f i e l d f o r t h e problem. Far f i e l d s i n v o l v e the asymptotic behaviour o f t h e s o l u t i o n t o the o r i g i n a l problem i n some a p p r o p r i a t e sense. As an example o f t h i s l a s t process we c o n s i d e r l o n g waves i n s h a l l o w w a t e r i n t h e cases when t h e f l u i d has a c o n s t a n t d e n s i t y , and when i t i s s t a b l y and c o n t i n u o u s l y s t r a t i f i e d . The a n a l y s i s f o r t h e s e d i f f e r e n t s i t u a t i o n s e s t a b l i s h e s t h a t t h e Korteweg-de V r i e s (KdV) e q u a t i o n governs t h e wave m o t i o n i n each case, though w i t h d i f f e r e n t c o n s t a n t c o e f f i c i e n t s . Thus t h e s o l i t o n s o l u t i o n s which a r e so f a m i l i a r i n t h e c o n s t a n t d e n s i t y case o c c u r a l s o i n a c o n t i n u o u s l y
1
A. Jeffrey
2 stratified fluid.
Rather than d i s c u s s t h e w e l l documented case o f KdV s o l i t o n s , we d e s c r i b e i n s t e a d a c o m p a r a t i v e l y new f o r m o f s o l i t o n c a l l e d t h e l o o p s o l i t o n which a r i s e s i n t h e case o f a v e r y f l e x i b l e rod. These l o o p s o l i t o n s have t h e t r u e s o l i t o n p r o p e r t y t h a t despite being s o l u t i o n s t o a nonlinear equation they preserve t h e i r i d e n t i t y e x a c t l y a f t e r i n t e r a c t i o n w i t h one a n o t h e r . The l o o p s o l i t o n s a l s o p r o v i d e a simple p h y s i c a l e x p l a n a t i o n o f t h e phase change t h a t o c c u r s as a r e s u l t o f such interaction. F i n a l l y , an example i s g i v e n o f a s t o c h a s t i c o r d i n a r y d i f f e r e n t i a l e q u a t i o n w h i c h c o u l d a r i s e i n wave p r o p a g a t i o n t h r o u g h a medium c o n t a i n i n g randomly d i s t r i b u t e d inhomogeneities. Although t h e random c o e f f i c i e n t i n t h e e q u a t i o n i s d e s c r i b e d by a g e n e r a l i s e d random t e l e g r a p h process, which i s a s t a t i o n a r y random process, t h e moments o f t h e s o l u t i o n depend on t h e t i m e , showing t h e s o l u t i o n i t s e l f does n o t e v o l v e as a s t o c h a s t i c process which i s s t a t i o n a r y i n t i m e . DEGENERATE SOLUTIONS AND FAR FIELDS The degeneracy o f s o l u t i o n s t h a t a r i s e s when c o n s i d e r i n g f a r f i e l d s i n wave p r o p a g a t i o n problems i s w e l l i l l u s t r a t e d by a s i m p l e l i n e a r example i n v o l v i n g t h e telegraph equation
i n which a , b and c a r e c o n s t a n t s . Here, when a , b > 0, t h e l a s t two terms r e p r e s e n t d i s s i p a t i v e e f f e c t s i n t h e system. I n terms of t h e wavenumber k and frequency
the function
= RefA e x p ( i ( k x
u(x.t)
w i l l be a s o l u t i o n o f (2.1) when
W,
-
wt))}
(2.2)
and k s a t i s f y t h e d i s p e r s i o n r e l a t i o n
w
T h i s shows t h e s o l u t i o n u s u a l l y e x h i b i t s t h e e f f e c t s o f b o t h d i s s i p a t i o n and d i s p e r s i o n , f o r (2.2) t h e n becomes
iI
i
u ( x , t ) = Re A e x p ( - a t / 2 ) exp i k x 2
& (4c2 k2
t
46
-
a2)?’))>.(2.4)
L e t us t a k e as t h e i n i t i a l c o n d i t i o n s f o r (2.1)
Then, u s i n g (2.41, a f t e r some c a l c u l a t i o n i t f o l l o w s by use o f t h e F o u r i e r i n t e g r a l t h a t t h e s o l u t i o n o f (2.1) s u b j e c t t o (2.5) i s u(x,t) = t
1
1 c exp(- a t / 2 )
Q(X - c t ) i
rtct &
exp(- at/E)[o(x + c t )
t
~ ( s )
x-ct
I([ a2-4b F] % ( c 2 t 2
-
(~-s)~)’]ds
3
Equations of Evolution and Waves
+ 1 c exp(- a t / 2 ) x
(c2t2 - (x
-
s)~)']~s
,
where I denotes t h e m o d i f i e d Bessel f u n c t i o n o f t h e f i r s t k i n d . The e n t i r e s o l u t i o n ( 2 . 6 ) i s seen t o decay l i k e e x p ( - a t / 2 ) as shown i n (2.4). However, t h e f i r s t group o f terms r e p r e s e n t s a damped s o l u t i o n o f t h e o r d i n a r y wave e q u a t i o n , w h i l e t h e o t h e r two terms r e p r e s e n t a decaying r e s i d u a l s o l u t i o n determined by t h e domain o f dependence [ x - c t , x + c t l on t h e i n i t i a l l i n e of t h e p o i n t ( x , t ) i n question. I f t h e i n i t i a l d a t a has compact s u p p o r t e x p r e s s i o n (2.6) s i m p l i f i e s , as may b e seen by c o n s i d e r i n g t h e s i t u a t i o n i n which o ( x ) i s a r e c t a n g u l a r p u l s e and ~ ( x =) c o n s t . o v e r t h e p u l s e b u t i s z e r o e l s e w h e w . The r e s u l t i s i l l u s t r a t e d i n F i g . 1 which shows t h e r e s o l u t i o n o f t h e i n i t i a l p u l s e i n t o two damped wedge-shaped s o l u t i o n s o f t h e o r d i n a r y wave e q u a t i o n moving t o t h e r i g h t and l e f t , l i n k e d by t h e decaying r e s i d u a l s o l u t i o n .
The p o s i t i o n and shape o f these p u l s e s s u i t a b l y l o n g a f t e r t h e s t a r t o f t h e d i s t u r b a n c e r e p r e s e n t s t h e s a l i e n t f e a t u r e o f t h e s o l u t i o n a t such a t i m e . We now show h e u r i s t i c a l l y how t h i s i n f o r m a t i o n may be determined d i r e c t l y f r o m (2.1) w i t h o u t t h e complication o f the r e s i d u a l s o l u t i o n being involved. I f t h e s o l u t i o n t o (2.1) i s considered t o d e f i n e a f i e l d , then i t i s appropriate t o c a l l t h e s o l u t i o n s describing these pulses t h e f a r - f i e l d s o f (2.1).
As t h e damped wedge-shaped s o l u t i o n s i n (2.6) p r e s e r v e t h e i r shape as t h e y propagate t h e y must be f r e e f r o m d i s p e r s i o n . From (2.4) t h i s can be seen t o be t h e case i f a2 = 4b when, i n c i d e n t a l l y , t h e t a i l i n ( 2 . 6 ) vanishes. S e t t i n g a2 = 4b i n ( 2 . 1 ) t h e n enables i t t o be w r i t t e n i n e i t h e r o f t h e two forms ( ' t c a x + -aa
]
( g - c a au x + l1a u ] = ~ ,
(2.7)
or
L e t t h e functions
u")
s a t i s f y t h e f o l l o w i n g two f i r s t o r d e r e q u a t i o n s
and (2.10) Then u = u(') a r e a u t o m a t i c a l l y s o l u t i o n s o f (2.1) i n which a' = 4b , though t h e y a r e degenerate s o l u t i o n s s i n c e ( 2 . 1 ) i s a second o r d e r e q u a t i o n whereas t h e u(*) a r e s o l u t i o n s o f f i r s t o r d e r e q u a t i o n s . The i n t e r p r e t a t i o n o f u ( * ) p r e s e n t s no problem, because (2.9) and (2.10) show t h e u(') a r e propagated along t h e C(*) c h a r a c t e r i s t i c s o f ( Z . l ) , respectively, given by (2.11) The
u(+)
s o l u t i o n o f (2.1) propagates t o t h e r i g h t and t h e
u(-)
solution t o
4
A. Jeffrey
4u
Figure 1 the l e f t . If or
v epresents e i t h e r u(+) or u ( - ) and t denotes the time along a C (+I C(-! characteristic, i t follows t h a t (2.9) and (2.10) are each o f the form
5
Equations of Evolution and Waves
*+ dt
1 a v = 0 2
-
along
.
characteristics
C")
(2.12)
S o l v i n g (2.12) a l o n g a c h a r a c t e r i s t i c g i v e s v ( t ) = v(0) exp(- at/2)
,
(2.13)
where v(0) i s t h e v a l u e o f v a t t h e p o i n t a t which t h e c h a r a c t e r i s t i c c u t s t h e i n i t i a l l i n e . T h i s i s t h e r e q u i r e d s o l u t i o n , f o r i t shows t h a t t h e ' n i t i a l shape o f v = u(') i s p r e s e r v e d as i t propagates a l o n g t h e r e s p e c t i v e C!)' The f u n c t i o n s cha a c t e r i s t i c s , though t h e s o l u t i o n s decays l i k e e x p ( - a t / 2 ) a r e a s y m p t o t i c s o l u t i o n s t o ( 2 . 1 ) , and t h e r e d u c t i o n from two t o one o f t h e u(' o r d e r o f the equation governing t h e f u l l s o l u t i o n t o t h e order o f the equation g o v e r n i n g t h e a s y m p t o t i c s o l u t i o n i s t y p i c a l o f such s o l u t i o n s . I n a sense we may say t h e f a r f i e l d s have " f o r g o t t e n " some o f t h e i n i t i a l c o n d i t i o n s which gave r i s e t o them.
.
7
I n passing, we remark t h a t a s p e c i a l case o f ( 2 . 1 ) a r i s e s when a = KO. and K i s l a r g e . For s e t t i n g E = 1/K , e q u a t i o n ( 2 . 1 ) becomes
,b
= Kg
(2.14) ax T h i s may be analysed r i g o r o u s l y i n t h e f o r m o f a s i n g u l a r l y p e r t u r b e d i n i t i a l v a l u e problem f o r which e r r o r e s t i m a t e s and an e x i s t e n c e theorem can be e s t a b l i s h e d . By c o n s t r u c t i n g a formal a s y m p t o t i c expansion i n terms o f E and u s i n g a c o n t r a c t i o n mapping p r i n c i p l e f o r o p e r a t o r s i n a Banach space a s o l u t i o n can be shown t o e x i s t c l o s e t o t h e a s y m p t o t i c expansion f o r s u i t a b l y s m a l l E . T h i s e q u a t i o n a r i s e s i n overdamped v i b r a t i o n problems [ll,i n t h e p r o p a g a t i o n of r a d i a t i o n t h r o u g h a h i g h l y a b s o r b i n g medium and a v a r i e t y o f s i t u a t i o n s i n continuum mechanics [2,31.
BREAKDOWN OF S O L U T I O N S U n l i k e t h e s o l u t i o n t o t h e t e l e g r a p h e q u a t i o n s which e x i s t s f o r a l l t > 0 , t h e s o l u t i o n t o n o n l i n e a r e v o l u t i o n e q u a t i o n s need n o t n e c e s s a r i l y e x i s t f o r a r b i t r a r i l y large t The s t a n d a r d example b y which t h i s may be i l l u s t r a t e d [2,41 i s t h e C1 s o l u t i o n o f
.
au % + f ( u ) jy= 0 , at
(3.1)
subject t o the i n i t i a l condition u(x,O) = @ ( X )
, --
1
and n e g a t i v e f o r
p*
i m p l y i n g t h a t (5.24) i s s a t i s f i e d f o r t h e e n t i r e range of
p*
if a >
1 7 . P*
This i n t u r n requires t h a t
(5.25) T h i s i s t h e necessary and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e of p e r i o d i c waves i n an i s o t h e r m a l s e l f - g r a v i t a t i n g medium. W i t h t h e d e f i n i t i o n (5.19) o f a , (5.25) d e l i n e a t e s t h e r e g i o n i n t h e A1 - A 2 p l a n e f o r which such s o l u t i o n s , we have from ( 5 . 2 5 ) t h a t t h e s o l u t i o n e x i s t . Furthermore s i n c e a = i n the region o f existence s a t i s f i e s t h e c o n d i t i o n
(F)' - U< 1
- A 2 < 1 ,
aO t h a t i s , t h e p r o p a g a t i o n speed o f t h e n o n l i n e a r p e r i o d i c wave i n a s e l f g r a v i t a t i n g medium i s always subsonic, and t h e l a r g e r A2 , t h e s m a l l e r t h e Yueh [151 has shown t h a t t h e same r e s u l t s h o l d f o r t h e maximum v a l u e o f U/ao p o l y t r o p i c case.
.
L o c a t i n g t h e o r i g i n a t t h e peak where wave, a c c o r d i n g t o ( 5 . 1 5 ) as
p+ =
1 + A1 , we have t h e p r o f i l e o f t h e
= -1+
5
2
X,
5
1
.
(5.26)
The second p a r t o f ( 5 . 2 6 ) f o l l o w s f r o m t h e assumption t h a t t h e non-dimensional wave l e n g t h o f t h e p r o f i l e i s 1 , and t h e l e n g t h o f t h e p r o f i l e f r o m t h e peak t o the valley i s The n o n l i n e a r d i s p e r s i o n r e l a t i o n f o r t h e p e r i o d i c s o l u t i o n 1 i s o b t a i n e d by p u t t i n g X, = when p * = 1 - A2 i n t h e f i r s t p a r t o f ( 5 . 2 6 ) :
k..
+F= 1
r+Al 1-A2
1-
f ( l - A2 ; a ) -
f(S ;a)
ds
(5.27)
47
Nonlinear Travelling Waves in Fluids
o r by t h e d e f i n i t i o n o f
B
, (5.28)
Here
xJ
=
a.
i s t h e Jean's length.
The appearance o f A2 , t h e a m p l i t u d e parameter, i n t h e d i s p e r s i o n r e l a t i o n (5.29) c l e a r l y b r i n g s o u t t h e n o n l i n e a r c h a r a c t e r o f t h e d i s p e r s i o n r e l a t i o n . Thus f o r each p a i r (A2 , A T ) o f t h e a d m i s s i b l e domain i n t h e (A1 , A2) p l a n e , we g e t a0 2 ~1 f r o m (5.19) and t h e r e f o r e t h e wave speed f r o m = a ; t h e wavelength i s Instead o f g i v e n by (5.29) w h i l e t h e wave p r o f i l e i s d e s c r i b e d by (5.26). t a k i n g (A1 , A2) as t h e two independent parameters o f t h e f a m i l y o f s o l u t i o n s , we may p r o f i t a b l y choose A2 and t o b r i n g o u t t h e p h y s i c s o f t h e problem
(r)
hJ
more c l e a r l y . F i g u r e 6 o f Yueh shows t h a t t h e r e a r e t h r e e d i s t i n c t r e g i o n s i n t h e a m p l i t u d e wavelength p l a n e : ( 1 ) a r e g i o n i n which a shock wave i s expected t o form, ( 2 ) t h e r e g i o n i n which t h e n e u t r a l ( p e r i o d i c ) waves e x i s t , and ( 3 ) a r e g i o n where t h e r e i s g r a v i t a t i o n a l i n s t a b i l i t y . I n t h e t h i r d case, i t i s e x p e c t e d t h a t t h e u n s t a b l e waves w i l l e v e n t u a l l y e v o l v e t o a n e u t r a l wave w i t h f i n i t e a m p l i t u d e . Yueh has f u r t h e r shown t h a t no s o l i t a r y wave can be found i n a homogeneous s e l f - g r a v i t a t i n g medium. A s i m i l a r s t u d y i s c a r r i e d o u t f o r a homogeneous plasma f o r which t h e g o v e r n i n g system o f PDEs i s s i m i l a r b u t t h e r e s u l t s come o u t t o be q u i t e d i f f e r e n t . 6.
CONCLUSIONS
I n t h i s r e v i e w we have a t t e m p t e d t o demonstrate, w i t h t h e h e l p o f s p e c i f i c examples, t h e e x i s t e n c e of e x a c t p r o g r e s s i v e o r t r a v e l l i n g wave s o l u t i o n s o f systems o f q u a s i l i n e a r inhomogeneous PDEs. The s o l u t i o n s may have d i s c o n t i n u i t i e s such as shocks, t h e y may r e p r e s e n t waves w i t h i n f i x e d o r f r e e b o u n d a r i e s o r may be p e r i o d i c p r o p a g a t i n g i n an i n f i n i t e domain. These examples would show t h a t s i m i l a r s o l u t i o n s may b e found f o r o t h e r systems, p a r t i c u l a r l y i n s e v e r a l ( s p a t i a l ) dimensions i n areas such as plasma p h y s i c s o r r o t a t i n g f l u i d s . The e x i s t e n c e o f i n t e r m e d i a t e i n t e g r a l s i s a fortunate circumstance w h i c h e n a b l e s t h e d i s c u s s i o n t o be reduced t o t h e f a m i l i a r phase p l a n e . F o r o t h e r problems i n more t h a n one dimension, which a w a i t s o l u t i o n , such i n t e r mediate i n t e g r a l s may n o t e x i s t and t h e d i s c u s s i o n would have t o be c a r r i e d o u t i n phase space o f t h r e e o r more dimensions. T h i s would pose mathematical c o m p l e x i t i e s s i n c e no s t a n d a r d d i s c u s s i o n o f t h e phase space seems a v a i l a b l e . The o t h e r aspect t h a t remains t o be t r e a t e d i s t h e a s y m p t o t i c s t a b i l i t y o f such s o l u t i o n s . We g i v e some f u r t h e r s o l u t i o n s o f t h e t y p e c o n s i d e r e d h e r e f o r w a t e r wave problems i n C161 and C171.
REFERENCES
111 B a r e n b l a t t , G.I. and Z e l ' d o v i c h , Ya.B.,
S e l f - s i m i l a r s o l u t i o n s as i n t e r -
48
P. L. Sachdev
mediate asymptotes, Ann. Rev. F l u i d Mech. 4 (1972) 285-312. 121 P e l e t i e r , L.A., A s y m p t o t i c b e h a v i o r o f s o l u t i o n s o f t h e porous media e q u a t i o n , S I A M J . Appl. Math. 21 (1971) 542-551. [ 3 1 Courant, R. and F r i e d r i c h s , K . O . , Supersonic Flow and Shock Waves, I n t e r s c i e n c e , New York (1948) 424-429. C41 Earnshaw, S., On t h e mathematical t h e o r y o f sound, Trans. Royal. SOC. London 150 (1860) 133-148. [ 5 ] S c h i n d l e r , G.M., Simple waves i n m u l t i d i m e n s i o n a l gas f l o w , S I A M J . Appl. Math. 19 (1970) 390-407. 161 Mackie, A.G., Two-dimensional q u a s i - s t a t i o n a r y f l o w s i n gas dynamics, Proc. Camb. P h i l . SOC. 64 (1968) 1009-1108. C71 Levine, L.E., Two-dimensional unsteady s e l f - s i m i l a r f l o w s i n gas dynamics, ZAMM 52 (1972) 441-460. 181 Freeman, N.C., Simple waves on shear f l o w : S i m i l a r i t y s o l u t i o n s , J . F l u i d Flech. 56 (1972) 257-263. 191 Sachdev, P.L., Exact, s e l f - s i m i l a r , time-dependent f r e e s u r f a c e f l o w s under g r a v i t y , J . F l u i d Mech. 96 (1980) 797-802. C l O l Longuet-Higgins, M.S., A c l a s s o f e x a c t , time-dependent, f r e e s u r f a c e f l o w s , J . F l u i d Mech. 55 (1972) 529-543. 1111 Longuet-Higgins, M.S., S e l f - s i m i l a r , time-dependent f l o w s w i t h a f r e e surface, J . F l u i d Mech. 73 (1976) 603-620. C121 B l y t h e , P.A., Kazakia, Y. and V a r l e y , E., The i n t e r a c t i o n o f l a r g e a m p l i t u d e s h a l l o w - w a t e r waves w i t h an ambient shear f l o w : N o n - c r i t i c a l f l o w s , J . F l u i d Mech. 56 (1972) 241-255. C131 Seshadri, V . S . and Sachdev, P.L., Q u a s i - s i m p l e wave s o l u t i o n s f o r a c o u s t i c g r a v i t y waves, Phys. o f F l u i d s 20 (1977) 888-894. C141 D r a z i n , P.G., N o n l i n e a r i n t e r n a l g r a v i t y waves i n a s l i g h t l y s t r a t i f i e d atmosphere, J . F l u i d Mech. 36 (1969) 433-446. C151 Yueh, T.Y., N o n l i n e a r waves i n a s e l f - g r a v i t a t i n g medium, S t u d i e s i n Appl . Math. 65 (1981) 1-36. E l 6 1 Sachdev, P.L. and Singh, M.C., E x a c t p e r i o d i c s o l u t i o n s o f l o n g wave e q u a t i o n s f o r a r o t a t i n g f l u i d , J . Appl. Mech., t o appear. C171 Sachdev, P.L. and Singh, M.C., Exact time-dependent f l o w s f o r nons h a l l o w w a t e r e q u a t i o n s (1983), t o be p u b l i s h e d . Appendix
The phase
y = 2 .
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
49
SECOND ORDER WAVE INTERACTION WITH LARGE STRUCTURES M. Rahman Department of Applied Mathematics Technical University of Nova Scotia Halifax, Nova Scotia B3J 2 x 4 Canada
The present study is mainly concerned with the investigation of second order wave forces on offshore structures. This paper extends Lighthill’s method for deep water waves to the waves in intermediate depth and shallow water cases and clearly demonstrates that all nonlinear boundary conditions including the radiation condition are satisfied by the scattered velocity potential. An exact expression for linear velocity potential applicable for shallow water waves has been used in these calculations. These results have been verified with those obtained by direct perturbation technique reported recently by Rahman and Heaps (1983) and with the experimental data. These comparisons show excellent agreement. INTRODUCTION The estimation of hydrodynamic wave interaction with large offshore structures has received considerable mathematical and engineering interest due to many practical applications to ocean engineering. Recently, the United States Commission on Marine Science and Engineering Resources have predicted that underwater structures would be built in connection with oil and gas recovery and production. The dynamic characteristics of these strueures could be determined provided an accurate prediction of the wave-induced forces on these structures is available. The mathematical form of the Morison equation (1950) has been applied to predict these forces for the last thirty years. However, certain difficulties in using it in the design and construction of offshore structures have been reported in the literature. Based upon the classical work on linear diffraction theory for water waves by Havelock (1940), MacCamy and Fuchs (1954) developed a linear diffraction theory for wave forces on large cylindrical piles immersed in the ocean. This study was also found to have limited applications because the nature of the water waves is inherently nonlinear. In a recent paper by Chakrabarti (1975), the nonlinear theory has been attempted for the case of a vertical cylinder and difficulties were found in satisfyinq the free surface boundarv condition and the radiation condition. The to this objection method of Raman and Venkatanarasaiah (1976), is not subject but is extremely complicated. The recent solution of Rahman (1981)has satisfied the free surface boundary conditions only at the surface of the cylinder. Hogben et a1 (1977) and Lighthill (1979) have published comprehensive review articles on the recent developments in the subject with a critical evaluation of both the methods and the results concerning the waves and hydrodynamic loading on offshore structures.
M. Rahman
50
Hunt and W i l l i a m s (1982) have r e c e n t l y d e m o n s t r a t e d a second o r d e r t h e o r y t o e v a l u a t e t h e wave l o a d i n g s on a c y l i n d e r . Another second o r d e r t h e o r y a l m o s t i d e n t i c a l t o t h a t o f Hunt and W i l l i a m s w a s r e p o r t e d by Rahman and Heaps (1983) by using formal p e r t u r b a t i o n technique. Both t h e s e t h e o r i e s seem t o produce identical results. However, d i f f i c u l t i e s were found i n e v a l u a t i n g t h e F o u r i e r B e s s e l i n t e g r a l a r i s i n g i n t h e q u a d r a t i c f o r c e s because of t h e h i g h l y complicated s i n g u l a r behaviour of t h e i n t e g r a n d . Motivated by t h e above d i s c u s s i o n , Rahman and C h a k r a v a r t t y (1981) have r e c e n t l y r e p o r t e d a t h e o r e t i c a l i n v e s t i g a t i o n o f t h e n o n l i n e a r wave l o a d i n g on o f f s h o r e s t r u c t u r e s extending t h e L i g h t h i l l ' s deep w a t e r t h e o r y t o t h e shallow w a t e r waves. These c a l c u l a t i o n s w e r e o n l y performed f o r t h e s m a l l kb v a l u e s where k is t h e i n c i d e n t wave number and b i s t h e c y l i n d e r r a d i u s . The p r e s e n t i n v e s t i g a t i o n i s mainly concerned w i t h t h e e x a c t second o r d e r c a l c u l a t i o n u s i n g t h e e x a c t e x p r e s s i o n f o r t h e l i n e a r v e l o c i t y p o t e n t i a l of d i f f r a c t i o n t h e o r y . These r e s u l t s have been v e r i f i e d w i t h t h o s e o b t a i n e d by I t is interestRahman and Heaps (1983) and compared w i t h t h e e x p e r i m e n t a l d a t a . i n g t o n o t e t h a t b o t h t e c h n i q u e s l e a d t o i d e n t i c a l r e s u l t s a n d comparison shows e x c e l l e n t aqreement. The a d v a n t a g e o f t h i s new t h e o r y i s t h a t t h e second o r d e r e f f e c t s of t h e Morison e q u a t i o n can be i n c o r p o r a t e d from t h e knowledge of t h e lirr?ar v e l o c i t y p o t e n t i a l a l o n e w i t h o u t c a l c u l a t i n g t h e q u a d r a t i c wave p o t e n t i a l as i t i s necessary i n t h e formal p e r t u r b a t i o n technique.
MATHEMATICAL FORMULATION We c o n s i d e r a l a r g e r i g i d v e r t i c a l c y l i n d e r o f r a d i u s b which is a c t e d upon by a t r a i n o f t w o d i m e n s i o n a l p e r i o d i c , p r o g r e s s i v e waves p r o p a g a t i n g on t h e I t i s assumed t h a t t h e f l u i d i s s u r f a c e o f f l u i d of a r b i t r a r y u n i f o r m d e p t h h . i n c o m p r e s s i b l e , i n v i s c i d , and t h e motion i s i r r o t a t i o n a l i n t h e r e g i o n bounded The by t h e f r e e s u r f a c e , r i g i d bottom boundary and t h e s u r f a c e o f t h e c y l i n d e r . governing d i f f e r e n t i a l e q u a t i o n s and t h e r e l e v a n t boundary c o n d i t i o n s may be found e l s e w h e r e i n c l u d i n g Rahman and Heaps ( 1 9 8 3 ) . A s d e m o n s t r a t e d by Rahman and C h a k r a v a r t t y ( 1 9 8 1 ) , t h e second o r d e r c o r r e c t i o n i n t h e l i n e a r f r e e s u r f a c e c o n d i t i o n may b e o b t a i n e d a s
where
Q, i s t h e v e l o c i t y p o t e n t i a l z is the v e r t i c a l axis
t is the t i m e
g i s t h e ' a c c e l e r a t i o n due t o g r a v i t y , and
( V @ i 2 is t h e f l u i d speed squared. Using t h e p e r t u r b a t i o n series (see L i g h t h i l l
(1979)), the velocity potential,
4 , and t h e f r e e s u r f a c e e l e v a t i o n , n , may b e e x p r e s s e d as
51
Second Order Wave Interaction with Large Structures
where qQ a r e t h e ordinary s o l u t i o n s s a t i s f y i n g t h e l i n e a r i z e d f r e e surface c o n d i t i o n s and I$ 4 , nq r e p r e s e n t t h e second o r d e r c o r r e c t i o n o f t h e o r d e r of t h e s q u a r e of t h e d i s t u r b a n c e s (and h i g h e r o r d e r c o r r e c t i o n s a r e n e g l e c t e d ) . Now u s i n q t h e T a y l o r s e r i e s e x p a n s i o n of @ a b o u t z = 0 , $ can b e e x p r e s s e d as
+ higher o r d e r terms The d i f f e r e n t q u a n t i t i e s i n t h e f r e e s u r f a c e c o n d i t i o n (1) may s i m i l a r l y b e o b t a i n e d and s u b s t i t u t i n g t h e s e v a l u e s i n t o (l), we o b t a i n t h e e q u a t i o n s f o r t h e l i n e a r p o t e n t i a l , $9., and t h e q u a d r a t i c p o t e n t i a l , + q , i n t h e form
a2+,
adl,
y + g - - = o
at
(5)
az
where
E q u a t i o n (6), a l t h o u g h d e r i v e d a t t h e e x a c t f r e e s u r f a c e , c a n b e a p p l i e d a t t h e u n d i s t u r b e d f r e e s u r f a c e z = 0 b e c a u s e b o t h s i d e s a r e q u a d r a t i c so t h a t e r r o r i n r e l o c a t i n g them a t t h e f r e e s u r f a c e i s o f s t i l l h i g h e r o r d e r .
PHYSICAL SOLUTION OF L I N E A R DIFFRACTION THEORY The l i n e a r d i f f r a c t i o n v e l o c i t y p o t e n t i a l , @ & , f o r a n incoming wave t r a i n i n t h e p r e s e n c e of a v e r t i c a l c y l i n d e r f o r d e e p w a t e r waves w a s g i v e n by Havelock ( 1 9 4 0 ) and h i s t h e o r y h a s been e x t e n d e d by MacCamy and Fuchs ( 1 9 5 4 ) f o r s h a l l o w w a t e r waves and t h e v e l o c i t y p o t e n t i a l i n t h i s c a s e may b e e x p r e s s e d i n t h e complex form
where Re stands f o r real p a r t ,
a
0
= 1
,
a
=
2(-i)n
,
i =
fi ( n
> 01,
and
i n which r i s t h e r a d i a l d i s t a n c e and a i s t h e a m p l i t u d e of t h e wave. g i v e n by t h e d i s p e r s i o n r e l a t i o n
Here w i s
52
M. Rahman
o2
gk t a n h kh
=
(11)
and H F ) i s t h e n t h o r d e r Hankel f u n c t i o n o f second k i n d , d e f i n e d by Hij2)(kr) = J ( k r )
-
i y (kr)
(12)
i n which J n ( k r ) and y n ( k r ) a r e B e s s e l f u n c t i o n s of f i r s t and second k i n d , respectively. Here a prime d e n o t e s t h e d i f f e r e n t i a t i o n .
($,
The s o l u t i o n $ f i n ( 8 ) i s i n complex form, a r e a l s o l u t i o n of which i s g i v e n by + $); where 6; i s t h e complex c o n j u g a t e o f $, may b e e x p r e s s e d i n the form io t
P.
where w
=
a w cosh k ( z + h ) a n An(kr) e s i n h kh n=-m
c o s nO
2k
i s d e f i n e d as
w
o
if n
-o
if n
= \
1
The f u n c t i o n an and An are d e f i n e d by ( 9 ) and 1101, r e s p e c t i v e l y f o r p o s i t i v e v a l u e s o f n , and by t h e r e l a t i o n s
a
-n
= n
*
* A-n
= An
*
*
f o r n e g a t i v e v a l u e s of n where a n and An are t h e complex c o n j u g a t e s of an and An, respectively. The summation i n ( 1 3 ) i n c l u d e s b o t h n = 0, and n = 0- i n o r d e r t o ( k r ) and e-iwt A. ( k r ) i n c l u d e t e r m s l i k e elwt A.
.
E q u a t i o n ( 9 ) may b e r e d e f i n e d as
a
n
= 6
n
e
-nni/2
where
60 = 1
n = O
6
n # O
= 2
WAVE FORCES FORMULATIONS
I n t h i s s e c t i o n w e w i l l b e mainly concerned w i t h t h e f o r m u l a t i o n o f wave f o r c e s on o f f s h o r e s t r u c t u r e s . Making r e f e r e n c e t o t h e work o f Rahman and C h a k r a v a r t t y (19811, t h e t o t a l f o r c e s may b e o b t a i n e d i n t h e form
where
Second Order Wave Interaction with Large Structures
53
FL i s t h e l i n e a r f o r c e
F
d
i s t h e second o r d e r dynamic f o r c e
F,
i s t h e second o r d e r w a t e r l i n e f o r c e , and
F
is the quadratic force
q
The a c c u r a c y o f t h e wave f o r c e s c a n b e g r e a t l y i n c r e a s e d by a d d i n g t o t h e l i n e a r f o r c e , F Q , t h e second o r d e r c o n t r i b u t i o n g i v e n b y t h e term (Fa + F, + Fq) which i s t h e key t o t h e u n d e r s t a n d i n g o f t h e i r r o t a t i o n a l flow t e r m i n t h e Morison equation.
In f o r m u l a t i n q t h e t o t a l f o r c e s w e have n e g l e c t e d t h e c o n t r i b u t i o n due t o v i s c o u s d r a g on t h e a s s u m p t i o n t h a t t h i s f o r c e may b e n e g l e c t e d when t h e r e l a t i v e dimension of t h e s t r u c t u r e i s l a r q e . Denoting t h e l i n e a r f o r c e by FQ which i s e q u a l t o t h e r e s u l t a n t of t h e t r a n s i e n t p r e s s u r e s of - P -
a%
at
w e can w r i t e
where s i s t h e body s u r f a c e and nx i s t h e d i r e c t i o n c o s i n e between t h e outward normal and t h e d i r e c t i o n of t h e f o r c e FQ. Here p i s t h e d e n s i t y o f t h e f l u i d . The second o r d e r f o r c e a r i s e s from t h e p r e s s u r e d i s t r i b u t i o n s d e r i v e d from t h e l i n e a r p o t e n t i a l , 61. These p r e s s u r e d i s t r i b u t i o n s have second o r d e r terms which e x e r t a l o a d i n g on a s t r u c t u r a l component. A c t u a l l y two f o r c e s c o n s t i t u t e One c o r r e s p o n d s t o t h e dynamic f o r c e , Fd, which t h i s second o r d e r c o n t r i b u t i o n . is t h e r e s u l t a n t o f t h e dynamic p r e s s u r e ,
o v e r t h e s u r f a c e o f t h e body. representation
Thus, t h e f o r c e , Fd, h a s t h e f o l l o w i n g i n t e g r a l
I t s h o u l d b e p o i n t e d o u t t h a t t h e p o t e n t i a l i n c l u d e s t h e r e s p o n s e s o f t h e body t o b o t h f l u c t u a t i n g m o t i o n s and t h e e x t e n s i o n a l m o t i o n s due t o waves.
Another f o r c e p r o v i d e s a second o r d e r c o n t r i b u t i o n t o t h e wave l o a d i n g on I t is s t r u c t u r e s whenever t h e body p e n e t r a t e s t h e f r e e s u r f a c e of t h e l i q u i d . d i r e c t l y a s s o c i a t e d w i t h t h e t r a n s i e n t and h y d r o - s t a t i c p r e s s u r e s of t h e i r r o t a t i o n a l f l o w s . According t o L i q h t h i l l (1979),t h i s a d d i t i o n a l second o r d e r h o r i z o n t a l x-component o f t h e f o r c e , F, a c t i n g a t t h e w a t e r l i n e w (where t h e s t r u c t u r a l component i n t e r s e c t s t h e f r e e s u r f a c e o f t h e f l u i d ) i s g i v e n by
54
M. Rahman
where t h e i n t e g r a n d i s t h e h o r i z o n t a l l y r e s o l v e d f o r c e p e r u n i t l e n g t h a c t i n g a t I t is t h e w a t e r l i n e and t h e i n t e q r a l g i v e s i t s r e s u l t a n t i n t h e x - d i r e c t i o n . i n t e r e s t i n g t o note t h a t t h i s equation is equivalent t o
a s p r e s e n t e d i n E q u a t i o n ( 3 0 ) o f Rahman and Heaps ( 1 9 8 3 ) .
SIMPLE APPROACH TO CALCULATING F 4 The s i m p l e approach t o c a l c u l a t i n g Fq c o u l d b e summarized a s f o l l o w s : W e w i l l assume t h e f o l l o w i n g form o f t h e l i n e a r p o t e n t i a l , $, and t h e q u a d r a t i c : potential,
%
$J
$
9" = R e [ @a. e
4
= R e [ @ e 4
iwt
I
2iwt
(22)
1
(23)
I n s e r t i n g t h e s e v a l u e s i n E q u a t i o n ( 6 ) , w e o b t a i n a f t e r some r e d u c t i o n
where 2 K = % =
9
4k t a n h kh
.
(25)
With t h e p o t e n t i a l due t o a u n i t t r a n s l a t i o n o s c i l l a t i o n o f t h e body i n t h e form Re t $ e2iwtl, it f o l l o w s t h a t on t h e s u r f a c e s of t h e body,
(26-ab)
where n
i s t h e x-component of a u n i t inward normal t o t h e s u r f a c e s .
w h i l e $, l i k e aq, s a t i s f i e s t h e r a d i a t i o n c o n d i t i o n at i n f i n i t y . us t o apply Green's e q u a t i o n
JJ [O V 2 $
v
q
-
$0 2 @
1 dV
9
=
J
suz=0
[Oq
2-
Furthermore,
This then l e a d s
a@
$
anq1 d s
where t h e s u r f a c e i n t e g r a l i s t a k e n o v e r t h e b o u n d a r i e s of t h e f l u i d ( s and z ' = 0 ) and n is t h e normal outward from t h e f l u i d . S i n c e b o t h 0 and Q s a t i s f y L a p l a c e ' s e q u a t i o n , t h e l e f t hand s i d e of v a n i s h e s so t h a t 7281 g i v e s
(28)
Second Order Wave Interaction with Large Structures
55
A f t e r p e r f o r m i n g t h e i n d i c a t e d i n t e g r a t i o n i n ( 2 9 ) and a p p l y i n g t h e c o n d i t i o n s
( 2 6 ) and ( 2 7 ) , i t f o l l o w s from ( 2 9 ) t h a t
ao
I
I@
$ [ + - K @ l d x d y = 4
z=O
q
s
n
ds
.
(30)
x
Thus, t h e q u a d r a t i c f o r c e , F c a n be w r i t t e n a s q
= Re
I Oq
[-2piw e 2 i w t
nx d s l
.
s
I n v o k i n g ( 3 0 ) and ( 2 4 ) , i t f o l l o w s t h a t
= Re
F
[(-2iwp) e
2iwt
CJ
ao
a$
1 z=O
z=o
az -
K 3 '
I dx dy
(33)
(34)
= Re
[-
I
PW
(VOQ)
2
1
+ 2
2
(tanh
2 kh-1) ( k $ L )
dx dyl
(35)
z=O where W = (-)a$
az
i s t h e v e r t i c a l v e l o c i t y on t h e f r e e s u r f a c e 2=0
a s s o c i a t e d w i t h t h e u n i t o s c i l l a t i o n o f t h e body. I n o r d e r t o determine t h e v e r t i c a l v e l o c i t y , W , it is necessary t o f i n d t h e s o l u t i o n f o r t h e p o t e n t i a l , $. A s s t a t e d e a r l i e r t h e f u n c t i o n , $, s a t i s f i e s t h e L a p l a c e ' s e q u a t i o n , t h e c o n d i t i o n (26ab) on t h e c y l i n d e r r = b , t h e f r e e s u r f a c e The s o l u t i o n , $, c a n c o n d i t i o n on z = 0 and t h e r a d i a t i o n c o n d i t i o n a t i n f i n i t y . be e x p r e s s e d i n t h e form
M. Rahman
56
where (P) denotes the Cauchy principal value of the integral and K1 is the modified Bessel function of order one. It is noted that the integral in (36) represents the standing wave part of the motion and the nonintegral part is the radiation component of the motion. The r derivative of the integral on r = b is a simple Fourier integral which can easily be evaluated as 1 - e2kz, and the r-derivative of the radiation component on r = b is simply 2ekz so that the potential I$ satisfies the boundary condition
Some simple algebraic calculation gives the real integral form of W
From the asymptotic nature of the Bessel function for small Kr and Kb, it turnsout that K (Kru) 1 b2 lim [UK, Kr+O Kb4
b2
Thus the integral in (37) is asymptotically equal to K - , and the nonintegral term is approximately equal to - 2 K b 2 / r when both Kr andrKb tend to zero. This
leads to the final asymptotic value of W as
.
Kb2
w----Ose
In order to estimate the effective range b we can define B by the equation B Re
J
2
Wdrl
= -
Kb
b
(39) i
r < B with which (39) holds,
B
en (r) ” cos 0
(40)
on the other hand, it follows from (37) that m
J b
K (Kbu)
m
2 Wdr = [ J
{-
0 2
+: I
2H(2) du
u K Ki(Kbu) l+u
0
(Kb)
1 cos e
K H ~ ’~( ~) b )
.
In evaluating the r-integral we have used the Bessel relation that
We next use the asymptotic result for small K b as
(41)
Second Order Wave Interaction with Large Structures
57
t o o b t a i n t h e a s y m p t o t i c r e p r e s e n t a t i o n f o r ( 4 1 ) i n t h e form
where
y is t h e E u l e r ' s constant. This r e s u l t a g r e e s w i t h ( 4 0 ) provided t h a t B = 1 (2e-Y)
K =
1 e-' 2
-
C k t a n h kh
(k t a n h k h )
-1
(45)
where C =
2
e-'
= 0.2807
a numerical c o n s t a n t
.
T h i s asymptc _ i c b e h a v i o u r i s of p r a c t i c a l i m p o r t a n c e t o second o r d e r i r r c a t i o n a l f l o w l o a d i n g s f o r v e r y s m a l l kb. I n t h e f o l l o w i n g however, w e are g o i n g t o p r e s e n t a n e x a c t second o r d e r s o l u t i o n f o r a r b i t r a r y kb.
EXACT CALCULATIONS FOR SECOND ORDER FORCES For a d i f f r a c t e d wave whose l i n e a r p o t e n t i a l i s of t h e form (13), t h e t o t a l h o r i z o n t a l f o r c e , F e , c a n b e o b t a i n e d from e q u a t i o n (18). T h e r e f o r e ,
2n FE =
a+,
0
1
1
8=0
z=-h
(-p
at
)
dzl
r=b
( - c o s 8) bd 8
.
A f t e r i n s e r t i n g t h e e x p r e s s i o n f o r $p, from (13), and p e r f o r m i n g t h e i n d i c a t e d i n t e g r a t i o n , F i t u r n s o u t t o b e t h e e x p r e s s i o n which h a s been o b t a i n e d by many r e s e a r c h e r s i n c l u d i n g MacCamy and Fuchs 1 1 9 5 4 ) . To c a l c u l a t e t h e second o r d e r c o n t r i b u t i o n s , F d , Fg, and Fq, t h e c a l c u l a t i o n s are summarized a s f o l l o w s . To e v a l u a t e t h e dynamic f o r c e , Fd, t h e f o r m u l a ( 1 9 ) may be r e w r i t t e n as
I n v o k i n g t h e e x p r e s s i o n f o r $ k from ( 1 3 ) and c a r r y i n g o u t t h e z - i n t e g r a t i o n of t h e e x p r e s s i o n (V$ )2, w e o b t a i n P,
M. Rahman
58
+
cos
I
(m-n)
2
+ D
m
-
B 11
[ c o s (m+n)O
n
+
cos ( m - n ) ~ ]
(48)
where t h e Wronskian p r o p e r t y of B e s s e l f u n c t i o n s g i v e s
- 2i
-iD,
Am(kb) =
= R
(nkb)
ffm (2
)'
(49)
e
(kb)
i n which
I n view of t h e s u b s e q u e n t 8 - i n t e g r a t i o n d e s c r i b e d i n e q u a t i o n ( 4 7 ) , w e r e q u i r e o n l y t h e c o e f f i c i e n t of cos 8 i n t h e d o u b l e summation i n (481 and t h i s y i e l d s t o the following
2
-
4a 9
c
T2k2b2 9,=0
[(l
-
2 kh s i n h 2kh
Q x [ E ~ (-1)
+
R ( R + 1) k 2 b2 ('
[c9,cos
2wt -
where
.iy'R+1 -
EL = (J
"i+l yi)'Ta.
sQ
+
2 kh s i n h 2kh )
sin 2wtIl
(52)
Second Order Wave Interaction with Large Structures
The B e s s e l f u n c t i o n s arguments b e i n g K b .
59
Then i n t e g r a t i n g w i t h r e s p e c t t o 8 ,
t h e t o t a l dynamic f o r c e , F d , may b e e x p r e s s e d as
k
.
- (-1) Cc, c o s 2 w t - s2 s i n 2wt)I
x [E,
(57)
The r e s u l t ( 5 7 ) i s t h e sum of t h e s t e a d y - s t a t e and o s c i l l a t o r y p a r t s which c a n be w r i t t e n as
Fss =
2
m
21)4a 1 nbk2
2
Fos = - %!% d
nbk2 x
[(1 -
8=0 "
c
, 2kh s i n h 2kh )
('
2kh , s i n h 2kh
(-1)9" { (1 -
k=O
Ic,
a. (9"+1) +
+
+
2kh s i n h 2kh
R(R+l) k2,,2
('
+
(58)
)I
2 kh s i n h 2kh
c o s 2 w t - S~ s i n 2 w t J
(59)
.
(60)
such t h a t ss
Fd = Fd
+
0s
Fd
T o c a l c u l a t e t h e w a t e r l i n e f o r c e , Fu, t h e f o r m u l a ( 2 0 ) may b e r e w r i t t e n a s
z=b Inserting t h e expression f o r
4,
i n t o (61), t h e t e r m i n (-1a$ at
written as
==o
may b e
r=b
+
Bm
-
O n ) 1 x [cos (m+n)O + cos (m-n) 8 1
.
(62)
I n view of t h e s u b s e q u e n t 8 - i n t e g r a t i o n g i v e n i n (61), we r e q u i r e o n l y t h e c o e f f i c i e n t of cos 8 term i n t h e d o u b l e summation i n ( 6 2 ) and t h i s y i e l d s t o t h e following
60
M. Rahman
where E ,,
CL, and SL a r e as p r e v i o u s l y d e f i n e d .
Then i n t e g r a t i n g w i t h r e s p e c t t o 0 , t h e t o t a l w a t e r l i n e f o r c e , F ,,
may b e
o b t a i n e d as 2
-
c
F w = - - 4pga nbk2
[EL
Q
+ (-1) ice
cos w t -
sQ s i n
.
2wt11
(64)
&=O
T h i s r e s u l t may b e e x p r e s s e d a s t h e s t e a d y - s t a t e and o s c i l l a t o r y p a r t s a s follows : ss
0s
Fw = Fo
Fw
+
where
2
-
F,”” =
%
-
nbk
il=O
II
C
(-1)
{C,
cos 2ot
.
- SQ s i n 2wt)
(67)
Now combining t h e s t e a d y - s t a t e and o s c i l l a t o r y p a r t s o f Fd and Fo, we o b t a i n
2
Fos = -
m Z
2pga
x
!2 (-1)
[(3
-
sinh
2kh 2kh )
L=O
Tbk2
[cL c o s
+
k(L+l)
____ k2b2
2wt
- sR s i n
(’
+
2kh s i n h 2kh
.
2wtl
11
(69)
I t i s i n t e r e s t i n g t o n o t e t h a t t h e s e two r e s u l t s p e r f e c t l y a g r e e w i t h t h o s e
o b t a i n e d by d i r e c t p e r t u r b a t i o n t e c h n i q u e , To obtain t h e quadratic f o r c e , F
g
2n
m
F
q
1
= Re {
I
(-rdr)
b
pW
[(V@L)
2
1 +7
r e p o r t e d by Rahman and Heaps ( 1 9 8 3 ) .
t h e formula ( 3 5 ) i s r e w r i t t e n as f o l l o w s :
(tanh’ kh
2
- 1) ( k @ L )1
where t h e r e a l p a r t of W i s g i v e n by
K = - 4wL g
d0 z=o
0
= 4k t a n h k h , a s d e f i n e d i n ( 2 5 )
.
(70)
Second Order Wave interaction with Large Structures Invoking t h e expression f o r $%, ( C $ y ) 2 2
2
( v + y ) z = O=
2
a+L
a@l. + (--) +
2
m
a 9k c o t h kh
(-1
lz=O
aZ
m
I: I: 6 m=O n=O
2
a 9k c o t h kh 4
2 a 9k t a n h kh 4
c a n be c a l c u l a t e d and i s o b t a i n e d a s
2
[(x’ rao a@9.
61
m
,
~Q; ~O; [ c o s ((m+n)
m
z
6m6n Q,
~1,
m=O n=O
m
mn
22
+ B~ + O~
[ c o s ((m+n) P 2 +
B~ + B~
k r
m
z
I: m=O n=O
6 6 9 o [ c o s ((m+n) m n -m -n
+
am +
bn - 2 o t )
where
i n which Qm(kr) Qm(kr)=
(74) iJ:(kb)
P,(kr)
= J
m
+ y:(kb)
(kr) y'(kb)
m
as p r e v i o u s l y d e f i n e d i n ( 5 1 ) .
- J;(kb)
y (kr)
m
(75)
Here a prime d e n o t e s t h e d i f f e r e n t i a t i o n .
S i m i l a r l y t h e e x p r e s s i o n f o r ( k + e ) 2 i s q i v e n by
M. Rahman
62
I n view of t h e c o s 0 t e r m p r e s e n t i n t h e W e x p r e s s i o n , i n t h e 8 - i n t e g r a t i o n i n ( 7 ) we o n l y need t o c o l l e c t t h e c o e f f i c i e n t of c o s 8 term i n t h e d o u b l e summation of ( 7 2 ) and ( 7 6 ) . Thus, w e o b t a i n , a f t e r some r e d u c t i o n ,
may b e Then i n t e g r a t i n g w i t h r e s p e c t t o 8 , t h e t o t a l q u a d r a t i c f o r c e , F g’ e x p r e s s e d i n i n t e g r a l form
- -
m
F
q
= - 2pgka
+
2
-
f,
rdr
1 [3 t a n h kh OR QQ+l R=O
2 L (P+1) ( r - 1) OR QR+,]l k r
[ J m0
.9
[El - (-1)
K (Kur) du 1 { - uK'(Kub) I,+n lfu 1
J1(Kr)
. c o t h kh 1 2 Q i Qi+l
f
{CL cos 2ot
J;(Kb)
2
Ji
f
- Sl s i n 2 w t ) J
y l ( K r ) y;(Kb)
(Kb) + y ;
2
(78) (Kb)
Here t h e t e r m s a s s o c i a t e d w i t h EQ i s t h e s t e a d y - s t a t e p a r t and t h e t e r m s a s s o c i a t e d w i t h c o s 2 w t and s i n 2 w t c a n be r e c o g n i z e d t o b e t h e o s c i l l a t o r y p a r t of t h e F q' F o r t h e c o n v e n i e n c e of t h e p r e s e n t a t i o n a n d comparison w i t h t h e e x p e r i m e n t a l d a t a , t h e f o r c e e x p r e s s i o n s a r e made d i m e n s i o n l e s s and t h e i r d i m e n s i o n l e s s forms a r e g i v e n below
- Fd pgD3
2 kh ~ T ~ ~ ( D k/ L0 ) ~
R x [ E -~ (-1) {c,
a. (9"+1) k2b2
(1
c o s 2 w t - sa. s i n 2 w t 3 1
+
s i n2kh h 2kh )
I
Second Order Wave Interaction with Large Structures
63
Hence, t h e d i m e n s i o n l e s s form o f t h e t o t a l f o r c e may b e e x p r e s s e d a s
DISCUSSION OF RESULTS An e x a c t second o r d e r d i f f r a c t i o n t h e o r y h a s been d e v e l o p e d i n t h i s paper. L i g h t h i l l ‘ s t h e o r y f o r d e e p w a t e r waves h a s been e x t e n d e d t o t h e s h a l l o w water waves u s i n q an e x a c t e x p r e s s i o n € o r t h e v e l o c i t y p o t e n t i a l i n t h e c a s e of a s u r f a c e piercing circular cylinder. The t h e o r y h a s b e e n t e s t e d by comparison w i t h t h e r e s u l t s o f t h e a v a i l a b l e e x p e r i m e n t a l d a t a . The maximum h o r i z o n t a l f o r c e was e x p r e s s e d a s t h e a v e r a g e of t h e a b s o l u t e v a l u e s of t h e maximum p o s i t i v e f o r c e and t h e maximum n e g a t i v e f o r c e . The p r e s e n t t h e o r y d i f f e r s from t h e p r e v i o u s one ( s e e Rahman and Heaps (1982)) i n t h a t t h e q u a d r a t i c f o r c e , Fq i n t h e p r e s e n t p a p e r h a s been o b t a i n e d u s i n g G r e e n ’ s f u n c t i o n f o r m u l a t i o n w i t h t h e p r e v i o u s knowledge of t h e l i n e a r v e l o c i t y p o t e n t i a l , alone; thus avoiding t h e t ed i o u s algebr ai c c a l c u l a t i o n of t h e q u a d r a t i c v e l o c i t y p o t e n t i a l , which i s a b s o l u t e l y n e c e s s a r y i n t h e f o r m a l p e r t u r b a t i o n t e c h n i q u e u s e d i n o u r p r e v i o u s work. I t h a s been found t h r o u g h t h i s a n a l y s i s t h a t both t h e t h e o r i e s l e a d t o t h e i d e n t i c a l r e s u l t s except t h a t t h e i n t e g r a l r e p r e s e n t a t i o n o f t h e q u a d r a t i c f o r c e w a s t u r n e d o u t t o be i n s l i g h t l y d i f f e r e n t form. T h i s d i f f e r e n c e , however, h a s b e e n a t t r i b u t e d t o two d i f f e r e n t t h e o r e t i c a l developments. I n any c a s e , t h e f i n a l r e s u l t s w e r e found t o b e a l m o s t I n t h i s p a p e r t h e d r i f t f o r c e h a s b e e n d e f i n e d a s t h e sum o f s t e a d y identical. s t a t e p a r t o f t h e dynamic FZS, w a t e r l i n e F F and q u a d r a t i c f o r c e F F b e c a u s e of t h e absence o f t h e s t e a d y state p a r t i n t h e q u a d r a t i c f o r c e e x p r e s s i o n i n t h e p r e v i o u s t h e o r y (1983), t h e d r i f t f o r c e i n t h a t c a s e w a s o b t a i n e d a s t h e s t e a d y s t a t e p a r t of t h e dynamic and w a t e r l i n e f o r c e s and i s g i v e n by t h e e q u a t i o n ( 6 8 ) . F i g u r e s 1-3 show a comparison of t h e p r e d i c t e d r e s u l t s o b t a i n e d by t h e p r e s e n t t h e o r y and the p r e v i o u s t h e o r y w i t h t h e e x p e r i m e n t a l d a t a c o l l e c t e d by I t may b e n o t e d t h a t t h e p r e s e n t t h e N a t i o n a l Research C o u n c i l o f Canada, O t t a w a . theory l e a d s t o t h e b e t t e r c o r r e l a t i o n with t h e experimental d a t a . I n Figure 4 , comparison w a s made between t h e r e s u l t s o f t h e s e two t h e o r i e s w i t h e x p e r i m e n t a l d a t a o f C h a k r a b a r t i ( 1 9 7 5 ) . Again t h e p r e s e n t t h e o r y a p p e a r s t o c o r r e l a t e w e l l with t h e experimental d a t a .
M. Rahman
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Experiment *** L i n e a r Theory Second-Order Theory of Rahman and Heaps (1983) ----P r e s e n t Second Order Theory 0 0 0
m R tr a \
F i g u r e 1 Comparison of L i n e a r and Second Order Wave F o r c e s w i t h E x p e r i m e n t a l Data of Moyridge and Jamieson ( 1 9 7 6 )
65
Second Order Wave interaction with Large Structures Experiment * * * Linear Theory __ Second-Order Theory of Rahman and Heaps ( 1 9 8 3 ) P r e s e n t Second Order Theory 000
Figure 2
-----
Comparison of Linear and Second Order Wave Forces with Experimental Data o f Mogridge and Jamieson (1976)
66
M. Rahrnan Experiment * * * L i n e a r Theory Second-Order Theory o f Rahman and Heaps (1983) ----P r e s e n t Second Order Theory 000 ~
0
0.02
0.04
0.06
n/L
Figure 3
Comparison of L i n e a r and Second Order Wave F o r c e s
with E x p e r i m e n t a l D a t a of Mogridge and Jamieson (1976)
Second Order Wave Interaction with Large Structures
67 n u
3 Y
in d
0 rl
in
0
0
Fig. 4
Comparison of Linear and Second-Order Wave Forces with Experimental Data of Chakrabarti (1975)
M. Rahman
68
we have u s e d S i m p s o n ' s i n t e g r a t i o n scheme i n e v a l u a t i n g t h e d o u b l e i n t e g r a l i n t h e q u a d r a t i c f o r c e g i v e n by ( 8 2 ) . T h i s i n t e g r a t i o n i s performed i n c o n j u n c t i o n w i t h IMSL l i b r a r y t o e v a l u a t e t h e B e s s e l f u n c t i o n s of d i f f e r e n t o r d e r s arising in the integrand. The u - i n t e g r a l is v e r y s i m p l e b e c a u s e t h e i n t e g r a n d is a f u n c t i o n o f m o d i f i e d B e s s e l f u n c t i o n o f f i r s t o r d e r and of s e c o n d k i n d . Due t o i t s e x p o n e n t i a l l y decaying n a t u r e , t h e v a l u e o f t h e i n t e g r a l q u i c k l y converges t o a finite limit. However, t h e r - i n t e g r a l i s e x t r e m e l y d i f f i c u l t b e c a u s e o f t h e c o m p l i c a t e d c o m b i n a t i o n of d i f f e r e n t o r d e r s o f B e s s e l f u n c t i o n s o f f i r s t and second k i n d . I t h a s been found t h r o u g h o u r n u m e r i c a l a n a l y s i s t h a t t h e r - i n t e g r a n d i s h i g h l y o s c i l l a t o r y . The i n t e g r a t i o n h a s been c a r r i e d o u t from t h e s u r f a c e of t h e c y l i n d e r t o a l a r g e d i s t a n c e B. Extensive numerical tests i n d i c a t e t h a t beyond t h e l i m i t i n g B t h e c o n t r i b u t i o n o f t h e i n t e g r a l i s insignificant. Fortunately, t h e s i n g u l a r i t y a r i s i n g i n t h e u-integrand i n t h i s p a p e r h a s been removed,while i n t h e p r e v i o u s r e p o r t t h e Cauchy p r i n c i p a l v a l u e a t the singular point of t h e integrand w a s considered. The s o l u t i o n method f o r t h e d o u b l e i n t e g r a l s was programmed i n F o r t r a n V f o r t h e CDC Cyber 170 d i g i t a l computer a t t h e T e c h n i c a l U n i v e r s i t y o f Nova S c o t i a . A Simpson's i n t e g r a t i o n scheme h a s b e e n f o r m u l a t e d and was u s e d t o i n t e g r a t e w i t h very f i n e grid s i z e .
ACKNOWLEDGEMENTS The a u t h o r i s g r a t e f u l t o I m p e r i a l O i l o f Canada L i m i t e d and t o t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada f o r t h e i r f i n a n c i a l s u p p o r t l e a d i n g t o t h i s p a p e r . Thanks are also due t o M r . P a u l C o l b o u r n e , a summer research s t u d e n t , f o r h i s s k i l l e d c o m p u t a t i o n .
REFERENCES Morison, J. R . , O ' B r i e n , M . R . , J o h n s o n , J. W. and S c h a a f , S . A., "The F o r c e s E x e r t e d by S u r f a c e Waves on P i l e s " , P e t r o l e u m T r a n s a c t i o n s , A I M E , 1 8 9 , 1950, 149-154. Havelock, T. H . , " P r e s s u r e o f Water Waves on a F i x e d O b s t a c l e " , P r o c . Roy. SOC., A175, 1940, 409-421. MacCamy, R . C . and Fuchs, R . A., "Wave F o r c e s o n P i l e s , A D i f f r a c t i o n Theory", Beach E r o s i o n Board T e c h n i c a l Memo N o . 6 9 , 1954. C h a k r a b a r t i , S . K . , "Second-Order Wave F o r c e on Large V e r t i c a l C y l i n d e r " , J o u r n a l o f t h e Waterways, Harbours and C o a s t a l E n g i n e e r i n g D i v i s i o n , A X E , V O l . 101, No. WW3, 1975, 311-317. Raman, H . and V e n k a t a n a r a s a i a h , P., " F o r c e s d u e t o N o n l i n e a r Waves on V e r t i c a l C y l i n d e r s " , J o u r n a l o f t h e Waterways, Harbours and Coastal E n g i n e e r i n g D i v i s i o n , A X E , V o l . 1 0 2 , NO. WW3, 1976, 301-316. Rahman, M . , " N o n l i n e a r Wave Loads o n L a r g e C i r c u l a r C y l i n d e r s : A P e r t u r b a t i o n T e c h n i q u e " , Ad. Water R e s o u r c e s , 4 , 1981, 9-19. Hogben, N . , M i l l e r , B . L . , S e a r l e , J. W . and Ward, G . , " E s t i m a t i o n o f F l u i d Loading on O f f s h o r e S t r u c t u r e s " , N a t i o n a l M a r i t i m e I n s t i t u t e Report NO. 11, 1977, 1-60. L i g h t h i l l M. J . , "Waves and Hydrodynamic Loading", P r o c . Second I n t e r n a t i o n a l C o n f e r e n c e on t h e Behaviour o f O f f s h o r e S t r u c t u r e s , 1, 1 9 7 9 , 1-40.
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[91
Hunt, J. N. and Williams, A. N., "Nonlinear Diffraction of Stokes’ Water Waves by a Circular Cylinder for Arbitrary Uniform Depth", J. de Mechanique Theo. et Appl. 1(3), 1982, 429-449
[lo]
Rahman, M. and Heaps, H. s . , "Wave Forces on Offshore Structures: Nonlinear Wave Diffraction by Large Cylinders", J. Physical Oceanography, 1983 (in press).
[ll]
Rahman, M. and Chakravartty, I. C., "Hydrodynamic Loading Calculations for Offshore Structures", SIAM J. of Applied Mathematics, 41, 1981, 445-458.
[12]
Mogridge, G. R. and Jamieson, W. W., "Wave Loads on Circular Cylinders: A Design Method", Hydraulics Laboratory, National Research Council, Canada, MH-111, December, 1976, 34 pp.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.E. Moodie (eds.) 0 Elsevier Science PublishersB.V. (North-Holland), 1984
71
ACCELERATION WAVES I N ISOTROPIC CONSTRAINED THERMOELASTIC MATERIALS B.D. Reddy Department o f A p p l i e d Mathematics and Department o f C i v i l E n g i n e e r i n g U n i v e r s i t y of Cape Town Rondebosch 7700 South A f r i c a 1.
INTRODUCTION
I n a r e c e n t paper ( 1 ) p r o p a g a t i o n c o n d i t i o n s have been d e r i v e d f o r a c c e l e r a t i o n waves i n constrained h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s . A t t e n t i o n has been c o n f i n e d t o t h e cases o f homothermal and homentropic waves; f o r b o t h t h e p r o p a g a t i o n c o n d i t i o n s a r e analogues of t h e c l a s s i c a l F r e s n e l -Hadamard c o n d i t i o n . Growth e q u a t i o n s have a l s o been d e r i v e d f o r p l a n e waves p r o p a g a t i n g i n t o homogeneously deformed m a t e r i a l which i s a t r e s t ahead o f t h e w a v e f r o n t . I n t h e above-mentioned work, no assumption has been made r e g a r d i n g t h e symmetry o f t h e m a t e r i a l o r t h e d i r e c t i o n o f t h e wave normal, a l t h o u g h t h e l a t t e r may be r e s t r i c t e d when c e r t a i n c l a s s e s o f c o n s t r a i n t s a r e p r e s e n t ( ( I ) , e q u a t i o n (6.4)). I t would be i n t e r e s t i n g , though, t o examine t h e e f f e c t s o f m a t e r i a l symmetry on t h e r e s u l t s f o r t h e g e n e r a l case. I n t h e p r e s e n t c o n t r i b u t i o n we s p e c i a l i s e t h e r e s u l t s i n ( 1 ) t o t h e case i n which t h e m a t e r i a l i s isotropic, and re-examine t h e c o n d i t i o n s f o r p r o p a g a t i o n of a c c e l e r a t i o n waves i n t h e l i g h t o f t h e i s o t r o p y o f t h e m a t e r i a l . As i n t h e p r e v i o u s work, t h e c o n s t r a i n t s a r e a r b i t r a r y . I n S e c t i o n 2 b a s i c r e s u l t s f r o m ( 1 ) a r e summarised and t h e n s p e c i a l i s e d f o r i s o t r o p i c m a t e r i a l s . We t a k e f u l l advantage o f t h e i n h e r e n t s i m p l i c i t y o f r e s u l t s when t h e orthonormal bases used a r e t h e t r i a d s o f p r o p e r v e c t o r s o f t h e r i g h t - and l e f t Cauchy d e f o r m a t i o n t e n s o r s . R e s t r i c t i o n s imposed by t h e c o n s t r a i n t e q u a t i o n s on t h e p r o p a g a t i o n o f a c c e l e r a t i o n waves a r e examined i n S e c t i o n 3; here we p a r t i c u l a r i s e o u r s t u d y t o t h r e e s p e c i a l cases f o r which t h e c o n s t r a i n t e q u a t i o n s t a k e s i m p l e forms, and f o r which t h e a c c e l e r a t i o n wave i s a principaZ wave. We show f o r t h e s e t h r e e cases t h a t t h e c o n s t r a i n t s a r e ZinearZy dependent ( i n a sense t o be d e f i n e d ) , and f u r t h e r m o r e , t h a t a c c e l e r a t i o n waves a r e aZways trrmsuerse ( a n d a l s o homothema2 i f a d d i t i o n a l c o n d i t i o n s a r e s a t i s f i e d ) . These o b s e r v a t i o n s a l s o h o l d f o r a c c e l e r a t i o n waves propagating i n material which i s i n a s t a t e o f uniform d i l a t a t i o n . I n S e c t i o n 4 t h e p r o p a g a t i o n c o n d i t i o n s o b t a i n e d i n ( 1 ) a r e summarised, and we subsequently r e c o n s i d e r t h e s e c o n d i t i o n s i n t h e c o n t e x t o f i s o t r o p i c m a t e r i a l s , f o c u s s i n g a t t e n t i o n on t h e t h r e e s p e c i a l cases mentioned above. Analogues a r e established o f t h e r e s u l t s f o r unconstrained e l a s t i c m a t e r i a l s given e a r l i e r by Chen ( 2 ) f o r h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s and b y T r u e s d e l l ( 3 ) f o r t h e p u r e l y mechanical t h e o r y . N o t a t i o n . I n g e n e r a l , s c a l a r s a r e denoted by l o w e r - c a s e Greek l e t t e r s , v e c t o r s by lower-case Roman l e t t e r s and t e n s o r s o f o r d e r t w o and h i g h e r b y upper-case Roman l e t t e r s . E x c e p t i o n s a r e X f o r p o s i t i o n v e c t o r o f a p a r t i c l e , U f o r wave speed, and, ai f o r p r i n c i p a l s t r e t c h e s . The v e c t o r space o f t e n s o r s o f
72
B.D. Reddy
o r d e r p i s denoted b y EP ; we w r i t e E1 E E and Eo 5 R . Very o f t e n we s h a l l come across e x p r e s s i o n s o f t h e f o r m C a i q i B p i , and so we m o d i f y t h e r e p e a t e d i n d e x appears on one side usual summation c o n v e n t i o n as f o l l o w s : i f only of an e q u a t i o n , o r i f i t appears i n a group o f q u a n t i t i e s which do n o t form p a r t o f an e q u a t i o n , t h e n s m u t i o n is impZied o v e r t h e r e p e a t e d i n d e x . However, i f a r e p e a t e d i n d e x appears on b o t h s i d e s o f an e q u a t i o n , no s m u t i o n is i m p l i e d ; f o r example, t h e e q u a t i o n S i = g i b j ( c j d j ) r e p r e s e n t s Si = higi ,;bj ( cj dJ . ) , i = 1 ,... 3 .
a
2.
CONSTRAINED ISOTROPIC THERMOELASTIC MATERIALS
We t a k e as s t a r t i n g p o i n t t h e c o n s t i t u t i v e e q u a t i o n s f o r an u n c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l i n t h e a l t e r n a t i v e forms = paQ0/aF
,Q
=
- a$o/ae
(2.1)
= E ~ ( F , Q ), S = paEo/aF
,e
=
aEo/ae
(2.3)
,S
$ = $O(F,e)
or E
Here F = Grad x ( X , t ) i s t h e d e f o r m a t i o n g r a d i e n t , J, i s t h e s p e c i f i c f r e e t h e s p e c i f i c entropy, e the energy, S E E 2 ( V ) t h e P i o l a - K i r c h h o f f s t r e s s , temperature, E t h e s p e c i f i c i n t e r n a l energy and p t h e d e n s i t y i n t h e r e f e r e n c e c o n f i g u r a t i o n . We denote by V t h e subset o f R3 o c c u p i e d by t h e body i n t h e r e f e r e n c e c o n f i g u r a t i o n . The r e f e r e n t i a l h e a t f l u x v e c t o r l i s denoted by q , and i n ( 2 . 2 ) and (2.4) g E E(V) i s t h e t e m p e r a t u r e g r a d i e n t ae/aX , X being t h e p o s i t i o n vector o f p a r t i c l e s i n t h e reference configuration. For i s o t r o p i c t h e r m o e l a s t i c m a t e r i a l s t h e f u n c t i o n $0 i s a s c a l a r - v a l u e d i s t h e l i s t of p r i n c i p a l f u n c t i o n o f T A and e , where TA(A = 1,2,3) i n v a r i a n t s o f t h e r i g h t Cauchy-Green t e n s o r C = FTF , w h i l e qo i s a f u n c t i o n o f TA,e and t h e i n v a r i a n t s l g l , lFgl and lCgl . S i m i l a r c o n s i d e r a t i o n s a p p l y t o t h e f u n c t i o n EO i n ( 2 . 3 ) , t h e r o l e s o f temperature and e n t r o p y b e i n g interchanged. We t a k e t h e a l t e r n a t i v e b u t e q u i v a l e n t view t h a t t h e l i s t T A may be r e p l a c e d by t h e t h r e e principuZ s t r e t c h e s a i E R(V) as independent v a r i a b l e s , so t h a t t h e c o n s t i t u t i v e e q u a t i o n s now become
or E
= Eo(ai,n)
,S
= P(aEo/aai)qi
q = qo(ai
n
191
B pi
IFgl
,e
Icsl)
= aEo/an
.
, (2.6)
I n ( 2 . 5 ) and ( 2 . 6 ) $0 and EO a r e symetYic f u n c t i o n s o f ai , w h i l e p i and qi are, r e s p e c t i v e l y , t h e orthonormal t r i a d s o f p r o p e r v e c t o r s o f C and o f
73
Acceleration Waves in Therrnoelastic Materials
T
B=FF
.
Constraints a r e d e f i n e d v i a t h e f u n c t i o n s $' : E2 and t h e c o r r e s p o n d i n g c o n s t r a i n t e q u a t i o n s a r e = 0
$,(F,e) zB(F,e).g
,
CY
1 ,...,N
=
,6
= 0
x
= 1
R
+
R
and
,
z6 : E2 x R + E , (2.7)
,...,M .
(2.8)
On t h e assumption t h a t t h e c o n t r i b u t i o n o f t h e c o n s t r a i n t s t o S and ~1 do n o t c o n t r i b u t e t o t h e e n t r o p y p r o d u c t i o n , i t can be shown ( 1 ) t h a t t h e c o n s t i t u t i v e equations f o r thermoelastic m a t e r i a l s subject t o t h e constraints (2.7 - 8) t a k e t h e f o r m jl =
T(F,e,xCY) = jlo(F,e)
s
paji/aF ,
q
=
= q(F,e,g,rB)
,
-$aa
q =
XCY$"(F,e) ,
+
4' = aJl/axa
= qo(F,@,s)
+
(2.9)
,
u6zB(F.e) ,
(2.10) (2.11)
where jlO(F,e) i s d e f i n e d i n ( 2 . 1 ) 1 and A,, yB E R ( V ) are a r b i t r a r y f i e l d s o f s c a l a r m u l t i p l i e r s . Here, and h e n c e f o r t h , summation i s imp i e d o v e r r e p e a t e d Greek in d i ces . The c o n j u g a t e r e p r e s e n t a t i o n o f ( 2 . 9 - 11) i s a c h i e v e d b y d e f n i n g t h e augmented i n t e r n a l energy f u n c t i o n E
= ;(~,q.h~)=
ji
t
ne
, e , a;/an
,
$'
=
(2.12)
;
t h e n we f i n d t h a t S = pai/aF
q = q(F,v,g,yB,ACY)
.
a;/aA,
,
(2.13) (2.14)
I n g e n e r a l , c o n s t r a i n t s have some degree o f d i r e c t i o n a l i t y s p e c i f i e d b y one o r more v e c t o r f i e l d s . F o r example, w h i l e t h e c o n s t r a i n t $(R,B) = det F - f ( a ) which corresponds t o temperature-dependent c o m p r e s s i b i l i t y , e x h i b i t s no such directionality, the constraint
$(F,@) = IFel2 - g(a) c o r r e s p o n d i n g t o temperature-dependent e x t e n s i b i l i t y possesses t h e d i r e c t i o n a l dependence s p e c i f i e d by t h e u n i t v e c t o r f i e l d e ( X ) . Such v e c t o r s a r e n o t u s u a l l y s p e c i f i e d e x p l i c i t l y i n t h e l i s t o f arguments o f c o n s t r a i n t s , presumably because as a r u l e t h e y a r e independent o f t h e d e f o r m a t i o n . However, i f we a r e t o c o n s i d e r t h e consequences o f material isotropy, i t i s e s s e n t i a l t h a t we make c l e a r t h e dependence o f 4 on v e c t o r f i e l d s er by w r i t i n g $"(F,e,er)
..., .
= 0
,
(2.15)
where r = 1, re O f course, n o t a l l o f t h e er need be p r e s e n t i n each c o n s t r a i n t . The c o n s t r a i n t s (2.8) a f f e c t t h e c o n d i t i o n s f o r growth o f a c c e l e r a t i o n waves b u t t h e p r o p a g a t i o n c o n d i t i o n s a r e u n a f f e c t e d by them. S i n c e we a r e concerned c h i e f l y w i t h p r o p a g a t i o n c o n d i t i o n s , we do n o t c o n s i d e r (2.8) f u r t h e r .
74
B.D. Reddy
F o r i s o t r o p i c m a t e r i a l s $' i n (2.15) a r e i s o t r o p i c f u n c t i o n s o f F , e and e, ; a c c o r d i n g l y , i f we i n v o k e a r e p r e s e n t a t i o n theorem ((5), ( 6 ) ) f o r s c a l a r - v a l u e d i s o t r o p i c f u n c t i o n s we f i n d t h a t $
where
u
-a
= $
I
.
A = 1 ,. .3, r,A = 1 ,...,re
= 0
,
(2.16)
; here
, h,
f, = Fe,
and
, )h,
8 , f,
= Ce,
,, ,h,
f,
,
= h
= f .f
(2.17)
r . ha
(2.18)
.
a r e symmetric m a t r i c e s whose elements a r e t h e i n v a r i a n t s i n v o l v i n g er We assume t h a t l e r l = 1 and t h a t t h e s c a l a r s ( e e ) a r e independent o f t h e d e f o r m a t i o n , so t h a t t h e y do n o t appear i n t h e o f arguments o f i n (2.16). As i n t h e case o f (2.5) we r e p l a c e dependence on TA i n (2.16) by symmetric dependence on a i , so t h a t (2.16) becomes
7is.t
"U
$ (ai
, e , ,f
,h),
= 0
.
(2.19)
Using t h e i d e n t i t i e s aai/aF ah
k
where
rA / a F
r
=
= Fh
af,,/aF
+ k,
er e,
= k,
Bf,
,
er pi
= qi
= f,
+ f,
b er
er eA + f, 8
e,
8
hA + f, 8 h,
, (2.20)
,
r '
t o g e t h e r w i t h e q u a t i o n s (2.19 - 14), (2.5 - 6 ) and (2.16), t h e c o n s t i t u t i v e e q u a t i o n s f o r i s o t r o p i c c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l s become
-
JI = $(ai P -1
s
-
, e , xu
= (a$/aai)qi
,)h,,
f, 8
pi + k
+
~r
-JIO(ai
=
,e)
(aT/af,,)(f,
ere
e,
8
, e , f,, ,)h,
+ hccTu(ai + fA
Q
,
e r ) + (aT/ah,,)(k,
(2.21) 8
eA)
+f,sh,+f,sh,),
(2.22)
- aT/ae
(2.23)
17 =
or E
p
-1
s
= JI +
ne ,
E
€(ai , n . ~ ~ , f ,,~,'A)
=
= (a€/aai)qi
B pi + (a;/af,,)(f,
(ac/ahr,)(k,
B e,
e
+ k,
,
= a2an
+ fr
er e, =
a;/axu
'
(2.24)
8
e,
+ f,
8
h,
+ f, B h,)
.
8
e,)
+
(2.25)
, (2.26)
I n e q u a t i o n s (2.21) and (2.24) t h e f u n c t i o n s and 'E a r e assumed symmetrised w i t h r e s p e c t t o f r A and h r a ; t h a t i s , t h e i r dependence on f,A and ,h, is Representations f o r the heat f l u x vector v i a ( f r A + f A r ) and ( h r 4 + h A r ) have been o m i t t e d s i n c e t h i s v e c t o r f i e l d i s n o t r e l e v a n t t o o u r subsequent discussion.
.
3.
CONSEQUENCES OF THE CONSTRAINT EQUATIONS
We examine here necessary c o n d i t i o n s f o r t h e e x i s t e n c e o f a c c e l e r a t i o n waves, based on t h e c o n s t r a i n t e q u a t i o n s (2.7 - 8 ) . The s t a n d a r d r e s u l t s
75
Acceleration Waves in Thermoelastic Materials
I F 1 = 0 , 1il
=
0
,
1
[FI = U- s B n , [Sin = - PUS (3.1) w i l l be needed l a t e r ; here, s = 1x1 i s t h e a m p l i t u d e and n and U a r e t h e d i r e c t i o n and speed of p r o p a g a t i o n o f t h e wave r e l a t i v e t o t h e reference c o n f i g u r a t i o n . A s s o c i a t e d r e s u l t s i n v o l v i n g jumps i n t h e temperature, t h e e n t r o p y and i n t h e i r f i r s t d e r i v a t i v e s a r e
o
lei = I g I = on ,
[il= -
UO
, 1ni
,
=
o
[On1 = Hn ,
I:[
=
-
UH
,
(3.2)
where Q and H a r e , r e s p e c t i v e l y , t h e thermal and e n t r o p i c a m p l i t u d e s . a c c e l e r a t i o n wave i s homothema2 i f o = 0 and homentropic i f H = 0 .
An
When no assumption i s made a b o u t t h e degree o f m a t e r i a l symmetry, t h e c o n d i t i o n [ G a l = 0 implies t h a t (1) -a 2-a c . s = - u w e
(3.3) . For homothema2 waves ( 3 . 3 ) i m p l i e s where Fa = ( a r / a F ) n and I;"= t h a t a t most two c o n s t r a i n t s o f t y p e (2.7) may b e p e r m i t t e d i n o r d e r f o r a r e ZinearZy a c c e l e r a t i o n waves t o propagate, assuming t h a t t h e v e c t o r s independent. F o r c o n s t r a i n t s w h i c h a r e such t h a t a r / a F = V a B wa , t h a t i s when aza/aF i s a s i m p l e t e n s o r , t h e c o n d i t i o n wa.n = 0 i s a l s o s u f f i c i e n t f o r (3.3) t o h o l d w i t h o = 0, . Then any number o f c o n s t r a i n t s of t h i s t y p e a r e permitted provided t h a t Wa E I n } l .
aT/ae
ca
When c o n s t r a i n t s o f t h e f o r m (2.8) a r e p r e s e n t , b y e v a l u a t i n g t h e jump i n t h i s e q u a t i o n we f i n d t h a t o z B. n = O ,
(3.4)
which i s s a t i s f i e d i d e n t i c a l l y f o r homothermal waves, F n r non-hornothernlal \.caves ( 3 . 4 ) imposes a l i m i t a t i o n on t h e p e r m i s s i b l e wave normals, w h i l e ( 3 . 3 ) , when r e w r i t t e n i n t h e form
-N c .s
=
- u2w-N o , m'.s ;Nmu
=
=o N ;
o , - Nu;;
(5
= 1 ,...N-1
, (3.5)
I
shows t h a t a t w s t t h r e e l i n e a r l y independent c o n s t r a i n t s a r e p e r m i t t e d . However, when mu = (v' 8 w')n f o r some v' , w' E E ( V ) any number of such c o n s t r a i n t s a r e p e r m i t t e d i f wa E I n } '
.
An a l l - e m b r a c i n g s e t o f necessary c o n d i t i o n s f o r t h e e x i s t e n c e o f a c c e l e r a t i o n waves, which a p p l i e s a l s o t o t h e cases when dim{? , a = 1, ...,N } < N i s dim{?}
2
f o r homothema2 waves
3
otherwise
2
F o r isotropic m a t e r i a l s t h e c o n d i t i o n (a7/aai)qi.s)(pi.n)
+ 2(ap/ah,,)
.
[Gal = 0
t o g e t h e r w i t h (2.20)
2(a~/afrA)(eA.n)(fr.s) 2-0 C(eA.n)(kr.s) + (h,.n)(f,.s)l = -U w CI
imply t h a t
+
.
.
(3.7)
and ,h, As i t stands, e q u a t i o n Here we have made use of t h e symmetry o f A,f (3.7) i s t o o g e n e r a l i n n a t u r e f o r any u s e f u l r e s u l t s t o be deduced f r o m i t .
76
B.D. Reddy
However, t h e w a r e a number o f n o n - t r i v i a l s p e c i a l cases which m e r i t a t t e n t i o n , and f o r which ( 3 . 7 ) s i m p l i f i e s c o n s i d e r a b l y . We s h a l l a c c o r d i n g l y devote a t t e n t i o n t o t h e s e s p e c i a l cases, w h i c h concern f i r s t l y principal! waves p r o p a g a t i n g i n a c o n s t r a i n e d m a t e r i a l f o r which ( i ) = p ( a i , 0 ) : we r e f e r t o these as i s o t r o p i c c o n s t r a i n t s ; ( i i ) t h e v e c t o r s e r a r e c o i n c i d e n t w i t h p r o p e r v e c t o r s p i ; and ( i i i ) t h e v e c t o r s er l i e i n t h e subspace (n-1' A f o u r t h s p e c i a l case concerns waves p r o p a g a t i n g i n a m a t e r i a l which i s i n a s t a t e o f pure d i l a t i o n . I n what f o l l o w s , we say t h a t a s e t o f c o n s t r a i n t s i s ZinearZy independent ( r e s p . dependent) if the s e t is l i n e a r l y independent (resp. dependent).
.
{c"}
P r i n c i p a l waves, i s o t r o p i c c o n s t r a i n t s
Case ( i ) :
To be d e f i n i t e , suppose t h a t t h e wave normal = ( a p / a a l ) q 1 and ( 3 . 7 ) becomes
n
c"
(aT/aal)ql.s
=
-
U'P~
coincides w i t h
.
p1 ; t h e n
(3.8)
The c o n s t r a i n t s a r e l i n e a r l y dependent ( i n f a c t , c o l l i n e a r ) , and t h e r e i s consequently no r e s t r i c t i o n on t h e number w h i c h may be p e r m i t t e d i n o r d e r f o r a c c e l e r a t i o n waves t o propagate. A number o f o b s e r v a t i o n s a r e summarised i n t h e f o l 1owi ng THEOREM 1. -a
(ai,e)
I n an i s o t r o p i c t h e m a e l a s t i c material w i t h i s o t r o p i c constraints
= O
I
( a ) a l l homothemal waves are transverse; (hl if w"" = 0 ( i . e . none of th8 constraint functions 4" depends on temperature), then a12 waves are transtrerse; (ci if W" = 0 for onZy a , then an acceZeration waue is homothermal! and transverse; ( d ) i f W" z 0 f o r a l l a and (iBaa"/aa, - ;"aTB/aal) f 0 .for a t l e a s t one p k r ($",@j of c o n s t r a i n t s , t h & z m ' a c c e l e r a t i o ~wave is-both transverse and homothermal.
some
i s t r i v i a l . To p r o v e ( e l , suppose t h a t . Then (3.8) g i v e s ( a T a / a a l ) q l . s = 0 and so q1 .s= 0 and t h e wave i s t r a n s v e r s e . Thus ( a a B / a a l ) q l . s = 0 = -U21;'o , i n d i c a t i n g t h a t t h e wave i s homothermal. To prove I d ) , we have o n l y t o observe -5 -a t h a t ( W a $ / a a l - o"aaT5/aa1!~1.5 = 0 ; i f t h e t e r m i n b r a c k e t s i s non-zero f o r a t least-one p a i r o f c o n s t r a i n t s , t h e n t h e wave i s t r a n s v e r s e . B u t t h i s i m p l i e s = 0 f o r a l l ~1 . Hence t h e wave, i f i t e x i s t s , i s homothermal. t h a t U2,"o PROOF.
F
The p r o o f o f l a ) and
f'bi
O and o"@ z 0 f o r some a ,
5
We a l s o r e c o r d h e r e f o r completeness some r e s u l t s f o r generalised transverse waves, these b e i n g d e f i n e d as waves which a r e b o t h homothermal and homentropic. P r o p e r t i e s o f these waves have been d i s c u s s e d i n some d e t a i l by Chadwick and C u r r i e ( 7 ) , (8) i n t h e c o n t e x t o f u n c o n s t r a i n e d t h e r m o e l a s t i c media. When c o n s t r a i n t s are p r e s e n t t h e i d e n t i t y = -ajl/ae and e q u a t i o n s (3.1-2) y i e l d -U 2H = s.(a';/Fae)n
+ U'o
-
p-'o""[ial
(3.9)
. For and so a g e n e r a l i s e d t r a n s v e r s e wave s a t i s f i e s ps.(a2;/aFae)n = G"[i,l t h e s p e c i a l case b e i n g d i s c u s s e d n = p1 and a*q/aFae = (a2$/aaiae)qi Q) p i , so t h a t general i s e d t r a n s v e r s e waves i n u n c o n s t r a i n e d m a t e r i a l s a r e t r a n s v e r s e .
77
Acceleration Waves in Thermoelastic Materials
Furthermore, i f a22 l" = 0 , an a c c e l e r a t i o n wave i s a g a i n n e c e s s a r i l y t r a n s v e r s e i n b o t h t h e o r d i n a r y and g e n e r a l i s e d senses. Case ( i i ) :
P r i n c i p a l waves,
e,
= p,
We assume h e r e o f c o u r s e t h a t t h e r e a r e a t most t h r e e v e c t o r s ep . An e l e m e n t a r y a p p l i c a t i o n o f t h e p o l a r decomposition theorem l e a d s t o t h e r e p r e s e n t a t i o n s f r = aSqr and hi- = afpr The c o n s t r a i n t e q u a t i o n (2.7) c l e a r l y reduces t o a f u n c t i o n of t h e p r i n c i p a l s t r e t c h e s and t e m p e r a t u r e o n l y , w i t h t h e r e s u l t t h a t a l l o f t h e c o n c l u s i o n s reached i n case ( i ) h o l d here.
.
Case ( i i i ) :
P r i n c i p a l waves,
en
E
.
In}'
F o r t h i s case i t i s easy t o show t h a t hi- E I n } ' ; hence ( 3 . 7 ) reduces t o ( 3 . 8 ) and once again t h e r e s u l t s d e s c r i b e d i n Theorem 1 h o l d . Case ( i v ) :
State o f pure d i l a t a t i o n
Here a1 = a2 = a3 = a say, F = a1 and e l e m e n t a r y c a l c u l a t i o n s show t h a t f r A = a2erA , ,h, = a4erA where erA = gA.e, , and so $a reduces t o a f u n c t i o n o f a and e o n l y , w i t h -a (3.10) c = (ar/aa)n . Thus t h e c o n s t r a i n t s a r e c o l l i n e a r . O f course any orthonormal t r i a d p i i n t h e r e f e r e n c e c o n f i g u r a t i o n i s an e i g e n b a s i s , and f u r t h e r m o r e p i = q i . Suppose we choose p i s o t h a t p1 = n ; t h e n ( 3 . 3 ) becomes ( a p / a a ) q l . s =-U21ao which i s q u a l i t a t i v e l y t h e same as (3.8). C l e a r l y , then, Theorem 1 h o l d s h e r e as well. 4.
PROPAGATION CONDITIONS FOR I S O T R O P I C MATERIALS
We d e r i v e i n t h i s s e c t i o n p r o p a g a t i o n c o n d i t i o n s f o r homothermal and homoentropic waves i n i s o t r o p i c c o n s t r a i n e d m a t e r i a l s . We have shown i n ( 1 ) t h a t t h e concepts o f a d e f i n i t e c o n d u c t o r and a non-conductor a r e e a s i l y extended t o c o n s t r a i n e d m a t e r i a l ; s i n c e t h e thermal c o n d u c t i v i t y t e n s o r K = a i / a g = aqo/ag i s d e f i n e d t o be one f o r which = 0 , we have, as i n t h e case o f u n c o n s t r a i n e d m a t e r i a l s (1 1
6
u.(sym K ) u > 0
-q = O -
Vu
E r, o = 0
E
,
H = O .
(4.1
1
T h a t i s , a c c e l e r a t i o n waves i n a d e f i n i t e c o n d u c t o r (non c o n d u c t o r ) a r e homothermal ( h o m o e n t r o p i c ) . The f i r s t - o r d e r moduli
1,1 E
4 E (V)
-A = p a 2 j l / a F a F
f o r constrained m a t e r i a l s are defined by
,A
and t h e c o r r e s p o n d i n g a c o u s t i c t e n s o r s
f o r a l l vectors
u
HomokhemaZ waves. condition i s (1)
tju = , A{(.)}
iiru
8
A
2"
= pa daFaF
0,6
2
E
E (V)
n l n , Qu = A{u
d e n o t i n g t h e map
E2
8
(4.2) a r e d e f i n e d by
nln
(4.3) E2
.
When t h e c o n s t r a i n t s a r e ZinearZy independent t h e p r o p a g a t i o n p u 2 s = PUS
(4.4)
B.D.
78
-
where- P i s t h e projection operator and da defined by
Reddy
c'
I -
, c" being defined e a r l i e r
B
C".';j B = 6" B , 2, E span cC"} . (4.5) t h e s e t of spurious proper vectors [81 ; We c a l l E, E Is : 6s E span {?I} f o r s E E, , U = 0 . For N c o n s t r a i n t s there e x i s t a t o t a l of (3-N) proper vectors which have associated w i i h them non-zero proper numbers. These proper vectors form a basis of {span C C " } } ~ ; furthermore, s i n c e 0 i s real a n d symmetric the proper numbers a r e real and non-zero. We have seen in Section 3 t h a t t h e c o n s t r a i n t s a r e c o l l i n e a r f o r a number of special cases; on the o t h e r hand, ( 4 . 3 ) was derived i n ( 1 ) on the assumption t h a t t h e constraints are l i n e a r l y independent. We therefore derive here t h e propagation condition f o r materials with c o n s t r a i n t s which a r e c o l l i n e a r . To be d e f i n i t e , suppose t h a t T' = GaE where 7' a r e known s c a l a r s and E a mit vector. The preliminary form of t h e propagation condition, obtained from ( 3 . 1 ) 4 , is ( P U2 I - 0 ) s = -E"[iOll = -UvaCi'lz . (4.6) Y
c . s 7 0 ; fur-thermore, from equation UvaCXal = ?.Qs . The propagation
Since we are dealing with homothermal waves ( 4 . 5 ) we have 3 = c , and so we find t h a t condition thus becomes
iJu2 s
c
(I B c,ts (4.7) which i s just t h e propagation condition f o r a material with a s i n g l e c o n s t r a i n t . The difference here, though, i s t h a t whereas t h e quantity .Cil i s a b l e t o b e found f o r a singly constrained m a t e r i a l , t h e q u a n t i t i e s [ A 1 are indeterminate when there are a number o f c o l l i n e a r c o n s t r a i n t s (see also 'the remarks by Whi tworth ( 9 ) ) . =
I n t h e case of isotropic materials t h e tensor
The term (a$aa.)qi Ipi by Chadwick and bgden (10) t o t h e bases pi and qi term. The contribution of we find eventually t h a t p-'R
=
q i.
Ip
k
+. ,
A
i s found from
i s comiaZ with and so t h e techniques employed f o r t h e derivation of non-zero components r e l a t i v e f o r tensors l i k e E may be applied d i r e c t l y t o t h i s t h e remaining terms t o a r e readily evaluated and
[Taia;/aai-akaT/aak a 2i. - a 2k
(
I
+ 2 q i B e r B ($af,,qi
+ 2ajr/ah,,f, + 2aT/ahrAqi +
B e,
+ a;/ah,,
B pi B
L
f,
(aif,
B
B pi + a i q i
pi + aiqi 2 B eA + q i B h,)] B
e,]
w h, B q i B e, + 2f, B e, BX(aT/af,,)
2 ( k r B e,
+
f,
Ih a )
&(a;/ah,,)
(4.9)
79
Acceleration Waves in Thermoelastic Materials
where
&: R & =
-f
E2
qi
i s t h e tensor valued d i f f e r e n t i a l operator. pi a/aai + 2{f,
I
I
eA
a/af,,
+ (kn
I
e, + f,
I
h,)a/ahaA1.(4.10)
I t i s now a straightforward matter t o obtain t h e acoustic tensor corresponding
t o i s o t r o p i c m a t e r i a l s ; from (4.3)1 and (4.9) we have
N
Y
Here, 7 E f.n f o r any vector f , $,! 5 a$/aai and I v j ) E 4(u. I v . + u . I v . ) 1 J J J represents the symmetric p a r t of t h e simple tensor u i 8 v j . Once again we focus a t t e n t i o n on t h e special cases outlined in Section 3 , r e s t r i c t i n g our discussions t o homothermal waves in t h e f i r s t instance. For Cases ( i ) , ( i i ) and ( i i i ) t h e acoustic t e n s o r reduces t o -14 = o i q i @ q i P Qi = $ , 1 1 6 i l + ( 1 - S i l ) ( a i G , i - a l T , l ) ( / ( a i - a:)) N
c
and t h e vector
(4.12)
in ( 4 . 7 ) i s given by
-c =
.
(aT/aa,)q,
(4.13)
From ( 4 . 7 ) we therefore have THEOREM 2 .
In m i s o t r o p i c heat-conducting e l a s t i c body i n which t h e nomaZ t o
an acceleration wave coincides with t h e proper vector pi , E d for which t h e a -a c o n s t r a i n t s are such t h a t f i ) @ = $ ( a i , e ) ; o r f i i ) $a = $ a ( a j , e , e r ) w i t h e r = pr ; or (iii) ga = p ( a i , e , e r ) w i t h e r E { n P , there e x i s t two homothermal a c e e l e m t i o n waves. Both waves arc transverse, w i t h amplitudes q 2 , 43 and corresponding wave speeds Q$ , Q3k , provided t h a t 0 i s positive definite.
The above applies t o t h e case when the s e t of principal d i r e c t i o n s i s d i s t i n c t , t h a t i s , a1 a2 a3 a1 , when n = p1 and a1 f a2 z a3 any vector s E I n ) L i s an amplitude, while f o r a s t a t e of d i l a t i o n any vector orthogonal t o n i s again an amplitude. For these l a t t e r two cases t h e components of t h e a c o u s t i c tensor a r e found by proceeding t o the l i m i t i n (4.12) (see equivalent calculations in (10))and are e a s i l y shown t o be f
f
-
-1Qi = $ , l , 6 i l
P
+
for
(1
al
and
-1-
4,
-
= $,,16il
+
-
6il)(ai$.i f
a2 = a3
4(1 - 6 i l ) a -
1
-
2 alG,,)/(ai
-
2 all
(4.14)
’
(T,i
Y
+
a$,ii
-
(4.15)
80 for
B. D. Reddy
al = a2 = a3
=
a
or
al = ai
, i = 2
3 .
or
Hornentropic waves When d e a l i n g w i t h homentropic waves i t i s p r e f e r a b l e t o develop t h e t h e o r y u s i n g t h e form (2.24) - (2.26) o f t h e c o n s t i t u t i v e equations. The c o n d i t i o n e q u i v a l e n t t o (3.3) i s t h e n
&/axU)*l where
c"
= (a;/aF)n
.
*
o
=
CU.s
u ( a 2 g / a A a a A B ci,1 .
=
(4.16)
B u t we have shown ( 1 ) t h a t = w*a^, w /U ,.
a 2-E / a x U a x
(4.17)
B
where
( R e l a t i o n s h i p s between e n t r o p i c q u a n t i t i e s l i k e U' and t h e i r thermal c o u n t e r p a r t s ;;" are g i v e n i n d e t a i l i n ( 1 ) ) . The c o n s t r a i n t c o n d i t i o n t h e r e f o r e becomes, f o r t h e case o f N c o n s t r a i n t s , GN.s =
m" =
^-1
!J
^aAB
u w w Ci,I
,2.5
iNc" - w"cN , u
=
= 0
, (4.19)
1, ..., N - 1
.
We see from Hence, and h e n c e f o r t h , i n d i c e s a range o v e r 1 t o (N - 1 ) ( 4 . 1 9 ) ~ t h a t a maximum o f t h r e e c o n s t r a i n t s may be p e r m i t t e d i n o r d e r t h a t homentropic a c c e l e r a t i o n waves may propagate, assuming t h a t t h e v e c t o r s mu a r e l i n e a r l y independent. The p r o p a g a t i o n c o n d i t i o n f o r homentropic waves i s d e r i v e d u s i n g arguments s i m i l a r t o t h o s e used i n t h e d e r i v a t i o n o f ( 4 . 4 ) f o r homothermal waves, and i s 2
The t e n s o r
a*
pU
s = PQ*s
.
(4.20)
i s related t o the acoustic tensor
Q (eqn. (4.3)) by (4.21)
and and
i s t h e p r o j e c t i o n o p e r a t o r I - m B iu where i n t h e sense t h a t
mu
i s d e f i n e d i n (4.19)
ia i s r e c i p r o c a l t o 6"
mu.;,
= A',
, iT
c span{m* U 1
.
(4.22)
.
I n t h e e v e n t o f t h e r e b e i n g o n l y one c o n s t r a i n t , f' reduces t o I For a t o t a l o f N independent c o n s t r a i n t s t h e symmetry o f $* i m p l i e s t h e e x i s t e n c e o f (44) r e a l non-zero p r o p e r number pU2 w i t h c o r r e s p o n d i n g m u t u a l l y o r t h o g o n a l p r o p e r v e c t o r s f o r m i n g a b a s i s o f (span{m'})'
.
We t u r n now t o isotropic m a t e r i a l s , and t o t h e c o n s t i t u t i v e e q u a t i o n s i n t h e f o r m ( 2 . 2 4 ) - ( 2 . 2 6 ) . We know f r o m ( 3 . 8 ) t h a t f o r t h e s p e c i a l cases c o n s i d e r e d i n S e c t i o n 3, t h e c o n s t r a i n t s a r e c o l l i n e a r whether o r n o t t h e wave i s homent r o p i c . I t i s t h u s once again necessary t o r e d e r i v e t h e p r o p a g a t i o n c o n d i t i o n on t h e assumption t h a t t h e c o n s t r a i n t s a r e c o l l i n e a r . Suppose t h e n t h a t C' = vaC w i t h = 1 ; t h e p r e l i m i n a r y form o f the propagation condition i s e a s i l y shown t o be
81
Acceleration Waves in Thermoelastic Materials
2
(pU
- Q)
I
^ ^
-
s =
(4.23)
,
a = N
On t h e o t h e r hand (4.16) y i e l d s , w i t h
.
Ucv"Cia1
,
w b c i 1 = U-lj;N.S/,N
B
(4.24)
so t h a t *CIA *N-o ^N 2 . s = v c.s = v w (c.s)/w A
or
(LN2 where
u = 1,
^N"
... , N -
1
^u^N
w
)(c.s) = 0 ,
i,
4.25
~
.
^oAN
Thus w v' = w w o r c . s = 0 ; i f t h e f o r m e r c o n d i t i o n h o l d s , we o b t a i n f r o m (4.23) and (4.24) t h e c o n d i t i o n pU
2
s
*
(4.26
= Q*s
^N2 where Q* = Q - ( u / ( w ) ) c B c . Then, as i n t h e case o f homothermal waves t h e p r o p a g a t i o n c o n d i t i o n reduces t o t h a t f o r a s i n g l y - c o n s t r a i n e d m a t e r i a l . c.s = 0 , on t h e o t h e r hand, we f i n d f r o m (4.23) t h a t A
~
(pU2I where
b
:
E
-f
Ic}'
- 64)s
0
=
If
(4.27)
I- c
i s the projection operator
8
c
.
By f o l l o w i n g t h e s t e p s t a k e n i n e q u a t i o n s ( 4 . 9 ) t h r o u g h (4.12) for i s o t r o p i c and ?, , t h i s t i m e m a t e r i a l s we may deduce s i m i l a r r e s u l t s f o r t h e t e n s o r i n v o l v i n g t h e augmented i n t e r n a l energy f u n c t i o n . Then f o r t h e s p e c i a l cases t r e a t e d e a r l i e r i n t h e c o n t e x t o f homothermal waves, s i m i l a r r e s u l t s may a g a i n be deduced f a r homentropic waves. I t is i n s t r u c t i v e , though, t o proceed b y a s l i g h t l y d i f f e r e n t r o u t e and t o use t h e r e s u l t d e r i v e d i n C l l , t h a t =
- U-lf;i
c =
C" - ;-IN
Fi
Q
(4.28)
and
where
-u = pa2?/ae2
and
8
N
i=
(4.29)
pa2T/aFae
.
(4.30)
The c o r r e s p o n d i n g a c o u s t i c t e n s o r s a r e r e l a t e d by Q =
Q
- U-lMn
8
Fin
.
(4.31)
I n t h e case o f a p r i n c i p a l wave p r o p a g a t i n g i n an i s o t r o p i c m a t e r i a l w i t h c e r t a i n o r w i t h e,.n = 0 , we f i n d f r o m ( 4 . 1 2 ) , i s o t r o p i c c o n s t r a i n t s , w i t h e r = p, (4.29) and (4.30) t h a t
Q = Qi qi
8
qi
,
=
G;" q1
(4.32)
where
Qi and
=
iji - --1 ( a 2 $/aalae) 2 6il
= aTa/aal
- i-'Za(a2T/aalae)
(4.33)
.
(4.34)
B.D. Reddy
82
^ - l ^ u ^ B = --1-a-B From t h e i d e n t i t y - p w w p w w (C11, eqn. ( 6 . 3 0 ) ) we a l s o deduce t h a t has t h e same s p e c t r a l r e s p r e s e n t a t i o n as Q , w i t h
a*
67 = Gi
+
6il
- -N u(W
-2
{aTN/aal
- --1-N u w a 2-a/aalae}
2
(4.35)
We thus have, u s i n g e q u a t i o n ( 4 . 2 5 ) onwards, 3. In an i s o t r o p i c heat-conducting e l a s t i c body in which t h e normal t o an acceleration wave coincides with ?he proper vector -pl ,-and f o r which t h e = $(ai,e) ; or ( i i ) = $‘(ai,e,er) with constraints are such t h a t (i) er = Pr ; or ( i i i l $ 01 = -a $ (aj,e,er) with e r E I n } * , there e x i s t three hornentropic acceleration waves, two of which are transverse w i t h F p Z i t u d e s 42 , q3 and corresponding wave speeds $ ; , Q? , provided that Q i s p o s i t i v e . d e f i n i t e . The t h i r d acceZeration wave i s longitudinal, w i t h wave speed
THEOREM
The d i s c u s s i o n f o l l o w i n g Theorem 2 has an o b v i o u s analogue here, t h e d e t a i l s o f which we o m i t . We a l s o observe, from (3.7) s u i t a b l y s p e c i a l i s e d o r i n d e e d f r o m Theorem 1, t h a t t h e t r a n s v e r s e waves d e s c r i b e d i n Theorem 3 a r e b o t h homentropic and homothermal. ACKNOWLEDGEMENT T h i s work was f i n a n c i a l l y s u p p o r t e d b y t h e Council f o r S c i e n t i f i c and I n d u s t r i a l Research o f South A f r i c a . REFERENCES
Reddy, B.D, The p r o p a g a t i o n and growth o f a c c e l e r a t i o n waves i n c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l s , J. E l a s t . ( t o appear). 2. Chen, P.J., Thermodynamic i n f l u e n c e s on t h e p r o p a g a t i o n and growth o f a c c e l e r a t i o n waves i n e l a s t i c m a t e r i a l s , Arch. Rat, Flech. Anal. 31 (1968) 228-254. 3. T r u e s d e l l , C., General and e x a c t t h e o r y o f waves i n f i n i t e e l a s t i c s t r a i n , Arch. Rat. Mech. Anal. 8 (1961) 263-296. R e p r i n t e d i n Problems o f N o n l i n e a r E l a s t i c i t y . Gordon and Breach (New York) 1965. 4. Chadwick, P. and Seet, L . T . C . , Second o r d e r t h e r m o e l a s t i c i t y f o r i s o t r o p i c and t r a n s v e r s e l y i s o t r o p i c m a t e r i a l s . I n Trends i n E l a s t i c i t y and T h e r m o e l a s t i c i t y (Nowacki A n n i v e r s a r y Vol . ) W o l t e r s - N o o r d h o f f (Groningen) (1971) 29-57. 5. Wang, C.-C., A new r e p r e s e n t a t i o n theorem f o r i s o t r o p i c f u n c t i o n s , Arch. Rat. Flech. Anal. 36 (1970) 166-19Z. 6. Chadwick, P. and C u r r i e , P.K., The p r o p a g a t i o n and growth o f a c c e l e r a t i o n waves i n h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s , Arch. Rat. Mech. A n a l . 49 (1972) 137-158. 7. Chadwick, P. and C u r r i e , P.K., On t h e p r o p a g a t i o n o f g e n e r a l i s e d t r a n s v e r s e waves i n h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s , J. E l a s t . 4 (1974) 301-315. 8. S c o t t , N.H., A c c e l e r a t i o n waves i n c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l s , Arch. Rat. Mech. A n a l . 58 (1975) 57-75. 9. Whitworth, A.M., Simple waves i n c o n s t r a i n e d e l a s t i c m a t e r i a l s , Q u a r t . J. Mech. Appl. Math. 35 (1982) 461-484. 10. Chadwick, P. and Ogden, R.W., A theorem o f t e n s o r c a l c u l u s and i t s a p p l i c a t i c n t o i s o t r o p i c e l a s t i c i t y , Arch. Rat. Mech. Anal. 44 (1971) 54-68. 1.
’See Chadwick and Seet ( 4 ) f o r a d e f i n i t i o n .
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
83
A LAGRANGIAN VIEW OF WAVE-MEAN FLOW INTERACTION R. Grimshaw
Department of Mathematics U n i v e r s i t y of Melbourne P a r k v i l l e , V i c t o r i a 3052 Australia
It i s shown how t h e wave a c t i o n e q u a t i o n i s r e a d i l y o b t a i n e d from a Lagrangian Formulation of t h e e q u a t i o n s of motion. The c o u n t e r p a r t t o t h e wave a c t i o n e q u a t i o n i s t h e mean-flow e q u a t i o n which c a n a l s o be simply d e r i v e d i n a Lagrangian Eormulation; i n t h i s e q u a t i o n t h e wave f o r c i n g c a n b e expressed e i t h e r by a r a d i a t i o n s t r e s s t e n s o r o r by t h e wave pseudomomentum. A f t e r d e v e l o p i n g t h e t h e o r y i n a g e n e r a l framework t h e r e s u l t s a r e a p p l i e d t o a compressLble f l u i d . In p a r t i c u l a r i t is shown t h a t , u s i n g s p h e r i c a l p o l a r coo r d i n a t e s , t h e wave a c t i o n e q u a t i o n and t h e mean-flow e q u a t i o n t a k e Eorms u s e f u l f o r a p p l i c a t i o n s f n s t r a t o s p h e r i c meteorology. F i n a l l y t h e t h e o r y i s a p p l i e d t o a s e t of e q u a t i o n s which d e s c r i b e t h e i n t e r a c t i o n of b a r o t r o p i c t o p o g r a p h i c , @-plane Rossby waves w i t h a mean Elow.
91.
INTRODUCTION
The i n t e r a c t i o n between waves and mean flows is a t o p i c w i t h r e l e v a n c e t o many d i E f e r e n t b r a n c h e s of p h y s i c s . I n r e c e n t y e a r s c o n s i d e r a b l e i n t e r e s t in t h i s s u b j e c t h a s developed w i t h i n t h e g e o p h y s i c a l f l u i d dynamics community. Here, as e l s e w h e r e , t h e b e n e f i t s of d e s c r i b i n g t h e L n t e r a c t i o n w i t h i n a L a g r a n g i a n framework a r e now g e t t i n g i n c r e a s i n g r e c o g n i t i o n . Thus, on t h e one hand, t h e r e s p o n s e of t h e waves t o t h e mean flow i s b e s t d e s c r i b e d by t h e wave a c t i o n e q u a t i o n , which i s a n out-growth from t h e a c t i o n p r i n c i p l e s of c l a s s i c a l mechanics. In t h e f l u i d dynamics c o n t e x t t h e c e n t r a l r o l e of t h e wave a c t i o n e q u a t i o n w a s f i r s t recognized i n t h e p i o n e e r i n g work of Whitham (1965, 1970) and B r e t h e r t o n and G a r r e t t (1968). On t h e o t h e r hand, t h e r e s p o n s e of t h e mean f l o w t o t h e waves i s d e s c r i b e d w i t h i n t h e c o n t e x t of a g e n e r a l i z e d Lagrangian-mean Eormulation, i n which t h e a c c e l e r a t i o n of t h e Lagrangian-mean f l o w due t o t h e waves i s d e s c r i b e d e i t h e r by t h e r a d i a t i o n s t r e s s t e n s o r , O K by t h e pseudomomentum. The p r e l i m i n a r y i d e a s were developed by D e w a r (1970) and B r e t h e r t o n (1971) and r e c e i v e d t h e i r f u l f i l m e n t i n t h e g e n e r a l i z e d Lagrangian-mean t h e o r y of A n d r e w s and McIntyre (1978a,b). A g e n e r a l review of b o t h wave a c t i o n and wavemean Elow i n t e r a c t i o n i s c o n t a i n e d in a r e c e n t review by Grimshaw (1984).
In t h i s a r t i c l e w e s h a l l f i r s t g i v e a b r i e f a c c o u n t of t h e g e n e r a l t h e o r y based o n a L a g r a n g i a n f o r m u l a t i o n of t h e e q u a t i o n s of motion in o r d e r t o s e t t h e s c e n e f o r t h e developments of subsequent s e c t i o n s . A more comprehensive a c c o u n t of t h e g e n e r a l t h e o r y i s d e s c r i b e d by Grimshaw (1984). Then i n 92 we s h a l l d e s c r l b e t h e i m p l e m e n t a t i o n of t h e g e n e r a l t h e o r y f o r a c o m p r e s s i b l e f l u i d u s i n g t h e g e n e r a l i z e d Lagrangian-mean t h e o r y of Andrews and P k I n t y r e (1978a,b). In 93 t h i s t h e o i y i s r e c a s t i n s p h e r i c a l p o l a r c o - o r d i n a t e s and we d e m o n s t r a t e how t h e wave a c t i o n e q u a t i o n and t h e Lagrangian-mean flow e q u a t i o n s a r e r e l a t e d t o t h e Eliassen-Palm r e l a t i o n s and t h e Charney-Drazin theorems of s t r a t o s p h e r i c meteorology. Then i n 94 w e d e s c r i b e how t h e g e n e r a l t h e o r y c a n b e used t o s t u d y b a r o t r o p i c t o p o g r a p h i c Rossby waves on a p-plane.
84
R. Grimshaw
To d e s c r i b e t h e g e n e r a l t h e o r y we s h a l l suppose t h a t t h e p h y s i c a l system is s p e c i f i e d by t h e vector-valued f i e l d Q ( t , x ) where t i s t h e time and xi ( i = 1, 2 , 3) a r e s p a t i a l c o - o r d i n a t e s . i I n t h e a b s e n c e of d i s s i p a t i o n t h e p h y s i c a l system obeys a v a r i a t t o n a l p r i n c i p l e w i t h Lagrangian d e n s i t y Then t h e e q u a t i o n s of motion a r e L(c$~, Qxi, Q; t , x i ) .
where t h e g e n e r a l i z e d f o r c e Q r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . Our d e r i v a t i o n of t h e wave a c t i o n e q u a t i o n w i l l b e based o n t h e i d e a s of Whitham (1965, 1970) b u t w i l l f o l l o w t h e development by Hayes (1970) more c l o s e l y . Thus, w e suppose t h a t @ ( t , x i ; 8 ) depends o n t h e ensemble parameter 8 such t h a t
e+
Q ( t , xi;
2n)
Q ( t , xi;
=
el,
(1.2)
and t h e n d e f i n e t h e a v e r a g i n g o p e r a t o r
For s i m p l i c i t y w e s h a l l d e n o t e t h e mean f i e l d by 3 . The a v e r a g i n g o p e r a t o r commutes w i t h d i f f e r e n t i a t i o n and h a s o t h e r s i m p l e and obvious p r o p e r t i e s (Andrews e n e x t d e f i n e t h e wave p e r t u r b a t i o n o r d i s t u r b a n c e f i e l d andAMcIntyre, 1978a,b). W by Q where
(1.4)
C$=?+C$
We s h a l l assume t h a t Q i s O(a) where a i s a measure of wave a m p l i t u d e , a l t h o u g h t h e r e i s a t p r e s e n t no r e s t r i c t i o n on t h e magnitude of a vis-a-vis t h a t oE I$,nor on t h e i r r e l a t i v e s c a l e s w i t h r e s p e c t t o t and xi. I f (1.1) is m u l t i p l i e d by $ e ( i . e . a$/ae) and t h e r e s u l t a v e r a g e d , i t may b e shown t h a t
(1.5a)
(1.5b)
where
=
@i
aL < Q e -->, wx
(1.5~)
i
a =< ieQ >.
(1.5d)
ai
Equation (1.5a) is t h e wave a c t i o n e q u a t i o n , * i s t h e wave a c t i o n d e n s i t y , is t h e wave a c t i o n f l u x I n d 8 is a t e r m r e p r e s e n t i n g t h e e f f e c t s of d i s s i p a t i o n . B o . t h k and 6, are O(a ) wave q u a n t i t i e s , and a r e t h e a p p r o p r i a t e g e n e r a l measures of wave a c t i v i t y . A p p l i c a t i o n s of ( 1 . 5 a ) depend on d e l i n e a t i o n of t h e f a m i l y C$(t,x i ; 0 ) . One c h o i c e which fs commonly used i s t o i d e n t i f y 8 a s a phase s h i f t , s o t h a t t h e a v e r a g i n g o p e r a t o r c a n b e i n t e r p r e t e d as a phase a v e r a g e . Thus w e p u t ,
Q a nd
.
A
=
$ ( t , xi;
s ( t , xi)
w = -s
- el,
(1.6a) (1.6b)
t'
Ki
= sxi'
85
A Lagrangian View of Wave-Mean Flow Interaction Here s ( t , x i ) is t h e phase, 4 i s p e r i o d i c i n s , w i s t h e wave frequency and i s t h e wavenumber. It f o l l o w s t h a t
Ki
a
(1.7)
i
and t h e wave a c t i o n e q u a t i o n (1.5a) t a k e s t h e form o b t a i n e d by Whitham (1970) i n t h e absence o f d i s s i p a t i o n . Another c h o i c e i s t o i d e n t i f y 9 w i t h a c o - o r d i n a t e so t h a t t h e a v e r a g i n g o p e r a t o r i s a c o - o r d i n a t e a v e r a g e . It i s remarkable t h a t t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) is f o r m a l l y e x a c t , v a l i d w i t h o u t any a p p a r e n t r e s t r i c t i o n o n wave a m p l i t u d e o r w i t h o u t any assumption t h a t t h e mean f i e l d i s s l o w l y v a r y i n g w i t h r e s p e c t t o t h e waves. There i s , of c o u r s e , a n a s y m p t o t i c e r r o r i n c u r r e d in t h e i d e n t i f i c a t i o n of t h e f a m i l y ( 1 . 2 ) (Hayes, 1970). Turning next t o t h e mean flow e q u a t i o n w e f i r s t d e f i n e t h e " u n d i s t u r b e d " Lagr a n g i a n by
Note t h g t a l t h o u g h Lo does n o t depend e x p l i c i t l y o n t h e wave f i e l d $, t h e mean f i e l d s 4 g e n e r a l l y c o n t a i n O(a2) wave-induced components. Next w e p u t
L~ i s a n o ( a 2 ) wave p r o p e r t y , and i t s e x p l i c i t dependence o n t , x i l i n c l u d e s i t s dependence on t h e mean f i e l d s 0. I f we now m u l t i p l y (1.1) by $. and a v e r a g e J t h e r e s u l t , we c a n show t h a t
aT. aT . -&! +&
=
-
+ ,
i
j
(1.10a)
j
where
(1.10b)
>
T.. = J1
- n,>bji'
(1.1Oc)
is t h e pseudomomentum, and -T.i t h e c o r r e s p o n d i n g f l u x . The s i g n s are H e r e -T a g r e e w i t h h i s t o r i c a l conventl!on (Andrews and M c l n t y r e , 1978b). A chosen similar e q u a t i o n can b e o b t a i n e d f o r t h e pseudoenergy by m u l t i p l y i n g ( 1 . 1 ) by 0, and a v e r a g i n g t h e r e s u l t (Grimshaw, 1984). Note t h a t when 9 is i d e n t i f i e d w i t h t h e wave a c t i o n d e n s i t y A (1.5b) r e d u c e s t o -Tjo, and t h e t h e c o - o r d i n a t e -x j’ wave a c t i o n f l u x 6 ( 1 . 5 ~ ) r e d u c e s t o -Tji; a l s o e q u a t i o n (1.10a) t h e n r e d u c e s t o t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) .
2:
To make f u r t h e r p r o g r e s s w e now assume t h a t - t h e mean f i e l d s v e l o c i t y ii and a vector-valued mean f i e l d h , where dh - + A dt
a; ij
i
h--=O,
ax
j
7
c o n s i s t oE a mean
(1.11)
86
R. Grimshaw
x
where u r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . In a p p l i c a t i o n s w i l l incorporate t h e e f f e c t s of mean d e n s i t y , mean f l u i d d e p t h e t c . (compare Garrett (1968) and Note t h a t i n t h e absence of d i s s i p a t i o n A i s a mean q u a n t i t y Dewar-(1970)). ( A = 1); however i t i s u s e f u l i n a p p l i c a t i o n s t o l e t h have a f l u c t u a t i n g , d i s s i p a t i v e component. We n e x t d e f i n e
which i s thg: time d e r i v a t i v e f o l l o w i n g t h e meanlmotion, and assume t h a t L1 only t h r o u g h i t s dependence on d4ddt; f u r t h e r a p a r t from t h e depends on dependence L1 o! Ci through d $ / d t any o t h e r e x p l i c i t denpendence o f L1 o n These h y p o t h e s e s ii i s b i l i n e a r i n ui and t h e d i s t u r b a n c e v a r i a b l e s Q and $x
+
OF
i
.
a r i s e from t h e f a c t t h a t t h e f u l l Lagrangtan i s u s u a l l y a t most q u a d r a t i c i n t h e v e l o c i t y f i e l d . The mean flow e q u a t i o n c a n now b e o b t a i n e d by a v e r a g i n g (1.1). The d e t a i l s of t h e d e r i v a t i o n a r e d e s c r i b e d by Grimshaw (1984), and t h e r e s u l t i s
a aLO aLoE ( Y ) +a F- b_ aui j j a;,
+ R
where
ij
=
~
a R i jax j
-T
ij
e + -
ah
iJ
(1.13b)
?A
Here R i j is t h e r a d i a t i o n stress t e n s o r , a l t h o u g h n o t e t h a t t h e terminology i s not u n i v e r s a l . A l s o e d e n o t e s t h e e x p l i c i t d e r i v a t i v e of L w i t h r e s p e c t t o xi when t h e d i s t u r b a n c e f i e l d s Q and t h e mean v a r i a b l e s Ci, A a r e a l l held c o n s t a n t . Equation (1.13a) c a n b e recognized as t h e mean momentum e q u a t on and d e m o n s t r a t e s t h e manner in which t h e r a d i a t i o n s t r e s s t e n s o r , a n O(a ) wave q u a n t i t y , i s r e s p o n s i b l e f o r t h e a c c e l e r a t i o n of t h e mean flow. An a l t e r n a t i v e t o (1.13a) i n v o l v i n g t h e pseudomomentum i n s t e a d of t h e r a d i a t i o n stress t e n s o r c a n b e The r e s u l t is o b t a i n e d from ( 1 . 1 3 a , b ) by u s i n g (1.lOa).
!
a (-=aLO
at
i
)
+ aa x( -u j j
aLo ) hi
>e
+
+ - < -aA ax. J
iJ
$x Q>.
+ax a no> i
(1.14)
i
T h i s i s t h e form p r e f e r r e d by Andrews and McIntyre (1978a). s e c t i o n by n o t i n g t h e a l t e r n a t i v e e x p r e s s i o n f o r Rij,
We c o n c l u d e t h i s
(1.15)
a7
A Lagrangian View of Wave-Mean Flow Interaction
Here (i3Ll/a$x2)d d e n o t e s t h e d e r i v a t i v e w i t h r e s p e c t t o constant.
J
S i m l l a r l y I t c a n b e shown t h a t
8,
ix
4 J
when d $ / d t i s h e l d
=
(1.16a)
Uik+4+
(1.16b)
where
92.
GENERALIZED LAGRANGIAN-MEAN
FORMULATION
The equati-ons of motion f o r a c o m p r e s s i b l e f l u i d a r e (2.la) (2.lb)
(2.lc) (2.ld)
where
Here x: i s t h e Eul r i a n c o - o r d i n a t e such t h a t a f l u i d p r t i c l e a t x i h a s The n o t a t i o n i s s t a n d a r d ; t h u s Qi i s t h e c o n s t a n t a n g u l a r v e l o c i t y v e l o c i t y ui. of t h e frame of r e f e r e n c e , @ ( x i ) i s t h e p o t e n t i a l f o r g r a v i t a t i o n a l and c e n t r i f u g a l f o r c e s , p(p, S ) i s t h e thermodynamic p r e s s u r e , p i s t h e d e n s i t y and S i s t h e e n t r o p y . The terms Xi and h r e p r e s e n t t h e e f f e c t s of d i s s i p a t i v e and d i a b a t ic e f f e c t s r e s p e c t i v e l y
.
The a p p r o p r i a t e L a g r a n g i a n f o r m u l a t i o n of t h e s e e q u a t i o n s i s t h e g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n of Andrews and McIntyre (1978a), which c o n t a i n s a comprehensive a c c o u n t of t h i s t h e o r y . Here we s h a l l g i v e only a b r i e f o u t l i n e ( s e e a l s o McIntyre (1977, 1980), Grimshaw (1979) o r Dunkerton (1980)). L e t xi b e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s and l e t c i ( t , xo) b e t h e p a r t i c l e displacements defined s o t h a t
Then, f o r any g i v e n ui t h e r e i s a u n i q u e " r e f e r e n c e " v e l o c i t y i l ( t , x . ) s u c h J t h e p o i n t x i moves w i t h t h a t when t h e p o i n t xi moves w i t h v e l o c i t y The material time d e r i v a t i v e ( $ . l a ) i s t h e n g i v e n by v e l o c i t y ui.
The g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n i s now o b t a i n e d by l e t t i n g mean v e l o c i t y , and r e q u i r i n g t h a t
= 0.
ui
be t h e
(2.4)
Thus (2.3) a g r e e s w i t h ( 1 . 1 2 ) . The r e a d e r should n o t e t h a t o u r n o t a t i o n d i f f e r s i n one s i g n i f i c a n t r e s p e c t from Andrews and McIntyre (1978a,b); h e r e ii d e n o t e s
R. Grimshaw
88
zt
t h e Lagrangian-mean v e l o c i t y , r a t h e r t h a n used by Andrews and McIntyre (1978a,b), who u s e t h e s i n g l e o v e r b a r t o d e n o t e E u l e r i a n means. No c o n f u s i o n should arise as E u l e r i a n means w i l l n o t b e d i s c u s s e d i n t h i s a r t i c l e .
-
Next w e d e f i n e a mean d e n s i t y p s o t h a t
au g+p - = i ax
0.
(2.5)
i
It i s a n immediate consequence of ( 2 . l b ) and ( 2 . 5 ) t h a t
PJ where
(2.6b)
= P,
J = det[
axi ~
ax.
1.
(2.6b)
J
We l e t K i j b e t h e i , j
-
c o - f a c t o r of J , and s o
Note t h a t U i j is t h e d e r i v a t i v e of 3 w i t h r e s p e c t t o B x i / a x . and aU /axj = 0 . J ij The momentum e q u a t i o n ( 2 . l a ) now becomes
-
dui p-+ dt
-
2 p ~ Q u ijk j k
ui = ui
(2.8a)
.) = pXI,
iJ
j
-
where
.
@ a + p- -a-axi ;+-(pK ax d5,
+dt
(2.8b)
The g e n e r a l i z e d L a g r a n g i a n m e a n e q u a t i o n s a r e t h u s ( 2 . 8 a ) , (2.5) mean f i e l d s a r e Gi, p and S and t h e wave p e r t u r b a t i o n f i e l d i s Lagrangian is
and ( 2 . 1 ~ ) . The A suitable
ci.
- 1 p{.j uiui
+
E
B
X’U
ijk i j k
-
@(xi)
-
E(p, S)}.
(2.9)
-
Here w e r e c a l l t h a t p i s d e f i n e d in terms of p by ( 2 . b a ) , and E ( p , S) is t h e i n t e r n a l energy p e r u n i t mass, so t h a t
(2.10)
where T is t h e temperature. V a r i a t i o n s i n x; ( o r e q u i v a l e n t l y i n ti) g i v e (2.8a) w i t h Q i = pXi. I n o r d e r t o k e e p the-correspondence w i t h t h e g e n e r a l t h e o r y o f $1 a s c l o s e as p o s s i b l e w e d e f i n e Qs = pT s o t h a t v a r i a t i o n s w i t h r e s p e c t t o S-can b e d e f i n e d . The v e c t o r A of $1 i s i d e n t i f i e d as t h e 2-vector w i t h components p and S , and w e n o t e t h a t ( 2 . 1 ~ ) and (2.5) h a v e t h e r e q u i r e d form (1.11).
89
A Lagrangian View of WaveMean Flow interaction The wave a c t i o n e u a t i o n c a n now b e o b t a i n e d from t h e g e n e r a l t h e o r y of 51, u s i n g ( 1 . 5 a , b , c , d ) and q l . l 6 a , b ) .
where
,. ac.,
.4= < p -
ae (2.11c)
8 =< -. p ( ac.,
ae Xi
as T>. +-
(2.11d)
ae
ThLs a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978b) a l t h o u g h t h e d i s s i p a t i v e term& h a s been w r i t t e n i n a d i f f e r e n t form h e r e . F o r l i n e a r i z e d wave motion, i t i s u s e f u l t o i n t r o d u c e t h e E u l e r i a n p r e s s u r e p e r t u r b a t i o n
%+
=
2 O(a ) .
(2.12)
It may t h e n b e shown t h a t
(2.13)
Note t h a t t h e second term h e r e i s i d e n t i c a l l y non-divergent and c a n b e o m i t t e d from (2.11a). To d e r i v e t h e mean flow e q u a t i o n s we f i r s t i d e n t i f y LO from (1.8).
Thus
where L i s g i v e n by ( 2 . 9 ) . The mean f l o w e q u a t i o n i s t h e n o b t a i n e d from ( 1 . 1 3 a , b ) , o r more d i r e c t l y by a v e r a g i n g ( 2 . 8 a ) . The r e s u l t i s
(2.15a) where
and
-
-
P = P(P,
S),
(2.15b) (2.15~)
It c a n b e v e r i f i e d t h a t Rij i s t h e same r a d i a t i o n stress t e n s o r d e f i n e d by An a l t e r n a t i v e form of (2.15a) i n v o l v i n g t h e pseudomomentum -Ti0 i n s t e a d (1.14). of t h e r a d i a t i o n stress t e n s o r c a n b e o b t a i n e d u s i n g t h e p r o c e d u r e d e s c r i b e d i n $1 and a v e r a g i n g ( s e e ( s e e (1.14), o r more d i r e c t l y by m u l t i p l y i n g ( 2 . l a ) by ax;/ax j Andrews and McIntyre (1978a)). The r e s u l t i s
90
R. Grimshaw
;a = + aTio - a t + - aa ~ (. U- j T i o ) Tjo 2 ax, J +
where
lt =
a + E> +m- < P
-
P
+ , ax,
(2.16a)
d5
EjklQ$,)>'
and
(2.16b)
(2.16~)
T h i s a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978a) a l t h o u g h t h e d i s s i p a t t v e terms on t h e right-hand s i d e of (2.16a) have been w r i t t e n i n a d i f f e r e n t form h e r e .
s3.
GENERALIZED ELIASSEN-PALM AND CHARNEY-DRAZIN THEOREMS
One of t h e most promising a r e a s f o r a p p l i c a t i o n of t h e g e n e r a l r e s u l t s of $2 i s s t r a t o s p h e r i c meteorology. With t h e a v e r a g i n g o p e r a t o r i n t e r p r e t e d as a z o n a l a v e r a g e , t h e c o u n t e r p a r t s of t h e wave a c t i o n e q u a t i o n (2.11a) and t h e mean f l o w e q u a t i o n (2.15a) ( o r (2.16a)) are u s u a l l y c a l l e d Eliassen-Palm and Charney-Drazin theorems r e s p e c t i v e l y , a f t e r t h e p i o n e e r i n g work of E l i a s s e n and Palm (1961) and Charney and D r a z i n (1981) ( f o r r e c e n t reviews see Andrews and & I n t y r e (1978c), McIntyre (1980), Dunkerton (1980) and Uryu (1980)). To o b t a i n t h e s e theorems i n t h e i r most g e n e r a l form w e must f i r s t e x p r e s s t h e e q u a t i o n s of motion i n s p h e r i c a l A s i m p l e and c o n v e n i e n t way of d o i n g t h i s i s t o i n t r o d u c e polar co-ordinates. s p h e r i c a l p o l a r c o - o r d i n a t e s i n t o t h e Lagrangian (2.9); t h i s i s t h e approach used by R r e t h e r t o n (1982) f o r small-amplitude waves. The c o r r e s p o n d i n g r e s u l t s i n c y l i n d r i c a l p o l a r c o - o r d i n a t e s are d e s c r i b e d by Grimshaw (1984). Thus w e l e t t h e c o - o r d i n a t e s x i of 51 b e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , h a n d v, where r i s t h e r a d i u s , h t h e c o - l a t i t u d e and v t h e l o n g i t u d e . These a r e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s , whose E u l e r i a n c o u n t e r p a r t s x a r e I. r ' , A' and v ' . The p a r t i c l e d i s p l a c e m e n t s are t h e n d e f i n e d by
The v e l o c i t y components Ln t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s a r e u
where
= _d _r ' dt
,
d h' dv' u2 = r' , u3 = r' sin h' dt ' dt
(3.2a)
(3.2b)
Here Ui are t h e mean v e l o c i t y components in t h e L a g r a n g i a n c o - o r d i n a t e d i r e c t i o n s , which must b e c a r e f u l l y d i s t i n g u i s h e d from t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s . The Lagrangian i s a g a i n g i v e n by ( Z a g ) , where, f o r s i m p l i c i t y , we s h a l l suppose t h a t t h e a x i s of r o t a t i o n i s i n t h e N-S d i r e c t i o n . Thus t h e L a g r a n g i a n i s
91
A Lagrangian View of Wave-Mean Flow Interaction
- 1
r2 s i n A PI- u u + 9 r ' s i n 2 i i
r f 2 s i n A ' PJ
where
=
u3 - ~(x;) - ~ ( p , s)},
i.
r2 s i n A
(3.3a) (3.3b)
and J i s a g a i n d e f i n e d by (2.6b) b u t now xi and xi a r e t h e s p h e r i c a l p o l a r coo r d i n a t e s . V a r i a t i o n s i n Si now g i v e t h e e q u a t i o n s o € motion.
-p { dul ~
dt
L
L
u2 - I u 3 - 2Q s i n h' u3 - -T K r
+ aQ ar
}
(3.4a)
(3.4b)
(3.4c) The c o u n t e r p a r t of ( 2 . 5 ) i s
w h i l e S a g a i n s a t i s f i e s (2.1.2). The r e s u l t i s
The wave a c t i o n e q u a t i o n c a n now b e o b t a i n e d from ( 1 . 5 a , b , c , d ) .
-a + - ( -a
at
where
=
-
art r2 s i n ~ p < u
+ r2
ae
1
sin h
;=
ui
axi
hi
+
rf
*i )
=
$ u2 + r'
l p ~ .
(3.6d)
ae
h ' = 1; h = r, h ' = r ' ; h = r s i n A, h ' = r ' s i n 1 2 2 3 3
A’.
(3.6e)
Here w e remind t h e r e a d e r t h a t xi a r e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , A and v. For l i n e a r i z e d wave motion w e i n t r o d u c e t h e E u l e r i a n p r e s s u r e - p e r t u r b a t i o n p' by ( 2 . 1 2 ) . It may t h e n b e shown t h a t
The second term i s i d e n t i c a l l y t h e b a s i c f l o w is z o n a l ( i - e . i n t e r p r e t e d as a z o n a l a v e r a g e e q u a t i o n (3.6a) r e d u c e s t o t h e Andrews and McIntyre ( 1 9 7 8 ~ ) .
non-divergent and c a n be o m i t t e d from (3.6a). When and z2 a r e O(a2)) and t h e a v e r a g i n g o p e r a t o r is ( i . e . 8 i s i d e n t i f i e d w i t h -v) t h e wave a c t i o n g e n e r a l i z e d Eliassen-Palm r e l a t i o n d e r i v e d by Also -(r2 s i n ?-,)-!& i s t h e a n g u l a r pseudomomentum.
The mean flow e q u a t i o n s c a n now b e o b t a i n e d from (1.13a) ( o r ( 1 . 1 4 ) ) ( a f t e r a l l o w i n g f o r t h e p r e s e n c e of t h e g e o m e t r i c a l f a c t o r s h i i n L ) , o r more d i r e c t l y by a v e r a g i n g ( 3 . 4 a , b , c ) . The e s s e n t i a l s t e p in t h i s l a t t e r approach is
(3.8a)
where
R
ij
=
6
..
(3.9b)
Here M is t h e z o n a l mean s p e c i f i c a n g u l a r momentum; € o r t h e r e l a t i o n s h i p between M and t h e z o n a l mean f l o w , see Dunkerton (1980). When t h e a v e r a g i n g o p e r a t o r is i n t e r p r e t e d as a z o n a l a v e r a g e ( i . e . 0 i s i d e n t i f i e d w i t h -v) t h e o f f - d i a g o n a l components of R v j are i d e n t i c a l t o Bq; simply compare (3.8b) w i t h ( 3 . 6 ~ ) . Note t h a t t h e d i a g o n a l components of R v j do not now a p p e a r in ( 3 . 9 ~ ) . With t h e s e i n t e r p r e t a t i o n s , and u s i n g ( 3 . 6 a ) , e q u a t i o n ( 3 . 9 ~ )becomes
A Lagrangian View of Wave-Mean Flow Interaction
dM dt
--
dt
(
r2L -?*--+ pr
.
93
(3.10)
pr2 s i n A
sin A
With t h e f u r t h e r r e s t r i c t i o n t o l i n e a r i z e d waves on a z o n a l b a s i c flow (3.10) r e d u c e s t o a g e n e r a l i z e d Charney-Drazin theorem ( s e e t h e s i m i l a r r e s u l t s o b t a i n e d by Andrews and McIntyre ( 1 9 7 8 ~ )and B r e t h e r t o n ( 1 9 8 2 ) ) . It f o l l o w s t h a t M w i l l change o n l y i n r e s p o n s e t o wave t r a n s i e n c e r e p r e s e n t e d by t h e term i n v o l v i n g k i n (3.10), o r due t o d i s s i p a t i v e e f f e c t s r e p r e s e n t e d by t h e terms i n v o l v i n g 3 and X3.
S4.
TOPOGRAPHIC, @-PLANE ROSSBY WAVES
The r e s u l t s of $2 and 53 have been o b t a i n e d w i t h o u t any r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e s c a l e of t h e waves, o r w i t h o u t making any of t h e a p p r o x i m a t l o n s such a s t h e h y d r o s t a t i c a p p r o x i m a t i o n , o r t h e q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n , o f t e n used i n g e o p h y s i c a l f l u i d dynamics. These v a r i o u s a p p r o x i m a t i o n s c a n b e used i n p o s t e r i o r i i n t h e e v a l u a t t o n of t h e wave a c t i o n d e n s i t y Jk ((2.11b) o r ( 3 . 6 b ) ) , t h e wave a c t i o n f l u x B 2 ( ( 2 . 1 1 ~ ) o r ( 3 . 6 c ) ) , e t c . However, i t i s i n s t r u c t i v e and sometimes more u s e f u l t o d e r i v e a wave a c t i o n e q u a t i o n and a mean flow e q u a t i o n d i r e c t l y from a n approximate s e t of e q u a t i o n s . In t h i s r e s p e c t t h e g e n e r a l t h e o r y of § l p l a y s a c o - o r d i n a t i n g r o l e . To i l l u s t r a t e t h i s a s p e c t w e s h a l l d e r i v e t h e wave a c t i o n e q u a t i o n and t h e mean f l o w e q u a t i o n s € o r a homogeneous f l u i d u s i n g t h e h y d r o s t a t i c and @-plane a p p r o x i m a t i o n s . In thLs c o n t e x t t h e wave p e r t u r b a t i o n s a r e g e n e r i c a l l y c a l l e d Rossby waves, and d e r i v e t h e i r wave-like c h a r a c t e r from a combination of t h e t o p o g r a p h i c s l o p e , t h e @ - e f f e c t and t h e s h e a r of t h e mean flow. Many a u t h o r s have s t u d i e d t h e i n t e r a c t i o n of Rossby waves w i t h mean flows ( s e e Dickinson (1978) f o r a review). From t h e p o i n t of view adopted i n t h l s a r t i c l e t h e p a p e r s by Grimshaw ( 1 9 7 7 ) , Rhines and Holland (1979), Young and Rhines (1980) and Huthnance (1981) a r e p a r t i c u l a r l y r e l e v a n t , a l t h o u g h n e a r l y a l l t h e r e s u l t s o b t a i n e d by t h e s e a u t h o r s a r e r e s t r i c t e d t o s m a l l a m p l i t u d e waves, o r s u b j e c t t o some o t h e r r e s t r i c t i o n c o n c e r n i n g t h e s t r u c t u r e of t h e mean flow. The r e s u l t s t o b e o b t a i n e d below c o n t a i n no r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e scale of t h e waves v i s a - v i s t h e mean f l o w , and hence c a n b e r e g a r d e d as a g e n e r a l i z a t i o n of t h e c o r r e s p o n d i n g r e s u l t s of t h e s e a u t h o r s . The E u l e r i a n e q u a t i o n s of motion f o r a homogeneous f l u i d i n t h e h y d r o s t a t i c a p p r o x i m a t i o n are
(4. l a )
(4.lb)
(4.1~) where
H = h(x', y ' )
f = f and
0
+
@1x '
-d= = + au , , + v a Ya ' dt
+
+
5,
(4.ld)
p2y1,
(4.le)
a
(4.1f)
94
R. Grimshaw
Here ( x ' , y ' ) a r e E u l e r i a n h o r i z o n t a l c o - o r d i n a t e s such t h a t a f l u i d p a r t i c l e a t (x’ y ' ) h a s h o r i z o n t a l v e l o c i t y components (u, v ) , w h i l e is t h e f r e e s u r € a c e d i s p l a c e m e n t , and h ( x ' , y ' ) i s t h e e q u i l i b r i u m d e p t h of t h e f l u i d . The terms F and G r e p r e s e n t t h e combined e f f e c t s of wind s t r e s s f o r c i n g and d i s s i p a t i o n . One immediate and useful consequence of ( 4 . 1 a , b and c ) i s t h e p o t e n t i a l v o r t l c i t y equation
IBS 5
asI 5
and f o r m u l a ( 1 7 ) h o l d s i f and o n l y i f sgn N3 # sgn RE/aE
- -
or
sgn N3 = sgn 13ElaE and I N 3 \ < IRE/aE\
- -
- -
.
Otherwise, t h e r e i s no domain s w i t c h i n g . I t i s a s i m p l e m a t t e r t o i n f e r t h e i m p l i c a t i o n s o f these conditions concerning the r a t e law (14). The c o n s t i t u t i v e r e l a t i o n s ( 8 ) and ( 9 ) a r e a l s o g e n e r a l i z a t i o n s o f thos; which I f we appear i n t h e open l i t e r a t u r e c o n c e r n i n g p o l e d f e r r o e l e c t r i c m a t e r i a l s . suppress t h e e x p l i c i t dependences on N3 and assume t h a t t h e t i m e s c a l e i s such
A Theory of Dynamical Ferroelectricity
103
t h a t a l l t h e t r a n s i e n t processes have reached e q u i l i b r i u m , t h e n ( 8 ) and ( 9 ) may be condensed so t h a t t h e y a r e s i m i l a r t o t h o s e i n t h e open l i t e r a t u r e e x c e p t f o r t h e These terms account f o r t h e a d d i t i o n a l s t r e s s and e l e c t r i c terms h3ijN3 and ki3N3. displacement due t o domain s w i t c h i n g . ACKNOWLEDGMENT T h i s work was supported by t h e U.S. Department o f Energy under c o n t r a c t No. OE-AC04-76-DP00789 t o Sandia N a t i o n a l L a b o r a t o r i e s , a Department o f Energy f a c i l i t y FOOTNOTES 1. 2. 3.
4.
The r e s u l t s g i v e n h e r e a r e due t o Chen [l]. See, e.g., J a f f e , Cook and J a f f e 121, p. 180. Thus f a r we have n o t seen e x p e r i m e n t a l evidence o f e x t e n s i v e domain s w i t c h i n g due t o expansive s t r a i n . See, e.g., T i e r s t e n [3].
REFERENCES
[ l ] Chen, P . J., Three dimensional dynamic e l e c t r o m e c h a n i c a l c o n s t i t u t i v e r e l a t i o n s f o r f e r r o e l e c t r i c m a t e r i a l s , I n t . J. S o l i d s S t r u c t . 16, (1980) 1059-1067. [2] J a f f e , B., Cook, W . R. J r . , and J a f f e , H., P i e z o e l e c t r i c ceramics (Academic p r e s s , London-New York 1971 ) [3] T i e r s t e n , H.F., L i n e a r p i e z o e l e c t r i c p l a t e v i b r a t i o n s (Plenum Press, New York 1969).
.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
105
SOLITARY WAVE SOLUTIONS OF GENERALIZED KORTEWEG-DE VKIES EQUATIONS F. A. Howes
Department of Mathematics U n i v e r s i t y of C a l i f o r n i a a t Davis Davis, CA 95616 U.S.A.
We b e g i n w i t h a d i s c u s s i o n of t h e e x i s t e n c e of s o l i t a r y wave s o l u t i o n s of t h e s c a l a r Korteweg-deVries (KdV) and modified Korteweg-deVries e q u a t i o n s , i n c l u d i n g t h e d e r i v a t i o n of formulas f o r wave amplitude and speed when t h e waves move a g a i n s t a nonzero "background". This i s followed by a c o n s i d e r a t i o n of r e l a t e d q u e s t i o n s f o r system a n a l o g s of (KdV). Our approach i n v o l v e s t h e c o n s i s t e n t u s e of t h e asymptotic t h e o r y of second-order s i n g u l a r l y p e r t u r b e d boundary v a l u e problems, and so i t o f f e r s a convenient a l t e r n a t i v e t o more t r a d i t i o n a l a n a l y t i c a l and numerical methods.
SOLITARY WAVE SOLUTIONS OF THE KDV AND MODlFIED KDV EQUATIONS e q u a t i o n i n t h e form
Let us c o n s i d e r f i r s t t h e Korteweg-deVries
- + -2uux + E u = xxx Ut
0,
e
i s a s m a l l p o s i t i v e parameter and s u b s c r i p t s d e n o t e p a r t i a l d i f f e r e n -
tiation.
I n o r d e r t o s t u d y s o l u t i o n s of (KdV) which r e p r e s e n t permanent waves
where
moving t o t h e r i g h t w i t h speed c
-
u = u ( ~ , e )= u ( x , t , g ) .
2
0, we i n t r o d u c e t h e v a r i a b l e s 5
The new v a r i a b l e s a l l o w us t o w r i t e (KdV)
2
x
-
c t and
as t h e ordinary
d i f f e r e n t i a l equation 2
e u/'/ = ( c u ) '
-
2 (u / 2 ) f ,
' =
d/d5,
and a n i n t e g r a t i o n reduces t h i s t h i r d - o r d e r e q u a t i o n t o t h e second-order one
Here Urn i s t h e c o n s t a n t l i m i t i n g s t a t e of l i m u ( ~ ) ( ~ , E= ) 0, f o r k
IF1 (KdV) -4"
equation, N
variables
5
2
1.
u6,e)
l i m u(f,e)
t h a t is,
=
Um and
151 - b r n
(Note t h a t i f w e had s t a r t e d w i t h t h e unscaled
xt + uux + uxxx = 0,
z g(x-ct),
u,
w e would a l s o o b t a i n (1.1) by d e f i n i n g t h e
= T(x,t,g),
and proceeding a s above.)
Thus t h e s t u d y of permanent wave s o l u t i o n s of t h e Korteweg-deVries duces t o t h e s t u d y of t h e asymptotic b e h a v i o r ( a s s i n g u l a r l y p e r t u r b e d boundary v a l u e problem
g
-+
+
equation re-
0 ) of s o l u t i o n s of t h e
106
F.A. Howes
where
'p
I f cpu(Um) > 0 t h e n Urn is ob-
i s a smooth f u n c t i o n s a t i s f y i n g cp(U m ) = 0.
-
v i o u s l y a maximum p o i n t of t h e p o t e n t i a l energy f u n c t i o n a l @ ( u ) = mathematical t h e o r y of (1.2)
J:: based on phase-plane
(O'Malley (1976)),
cp(s)ds.
t e l l s us t h a t i f t h e r e i s a f i n i t e U-value u* such t h a t @(u+c)= ip(U ) m ’p(u*)
# 0, t h e n t h e problem (1.2) h a s a s o l u t i o n u
= u ( ~ , E as )
E -t
The
m
O+
except t h a t
analysis, =
0 and
satisfying
(1.3) lim
u(O,E)
+
=
u*.
E - t o
This s o l u t i o n c l e a r l y r e p r e s e n t s a s o l i t a r y wave of amplitude )u;k-U,I; wave of e l e v a t i o n ( d e p r e s s i o n ) i f u*
> Urn (u" < U,).
i t is a
We n o t e i n p a s s i n g t h a t i n
addition t o the solution satisfying (1.3), there are solutions u =
of (1.2)
w i t h "spikes" a t r e g u l a r l y spaced p o i n t s i n any f i n i t e i n t e r v a l (-L,L)
C (-m,m),
that is, lim c
-+ o+
G(5,s)
=
u,,
5
for
i n (-L,L),
except t h a t l i m +;(Tk,c) s + o
,
n=2,3,...
k=l,Z,...,n-l
= u",
f o r [k
(O'Malley (1976)).
=
L(Zk/n-l),
These f u n c t i o n s correspond t o p e r i -
odic c n o i d a l wave s o l u t i o n s o f e v o l u t i o n e q u a t i o n s l i k e (KdV),
a s discussed, f o r
example, by Karpman (1975). 2 Returning t o t h e s p e c i f i c problem (1.1) we s e e t h a t f o r ~ ( u )= c(u-Urn)- i ( u 2 -Urn),
- u;
cp(Um) = 0 and y U ( u ) = c
c o n s e q u e n t l y , 'pu(U,)
>
0 only i f Urn
U r n .
It
i s i n s t r u c t i v e t o r e w r i t e t h i s amplitude-speed r e l a t i o n a s c = Urn thus, t h e l a r g e r t h e amplitude a d d i t i o n , (1.4) since
c
(1.4)
of t h e wave, t h e g r e a t e r i t s phase speed.
In
i m p l i e s t h a t t h e s o l i t a r y wave of amplitude -3Um i s s t a t i o n a r y ,
i s t h e n zero.
s t a t e (KdV)
A
+ A13 ;
C l e a r l y t h i s s o l i t a r y wave i s a s o l u t i o n of t h e s t e a d y -
equation u u x
+ E 2uxxx
= 0, which e x i s t s provided Urn
0.
h2 ; then
.
f h
h
2
r
will be denoted
jm
t # 0
,
Corresponding
ern
, where jm
a unit vector.
E
m
To find a decay equation for the jump magnitudes jm , we differentiate equation (3.1) partially with respect to relations (2.6)2,
(C
dvm d-r
dv Dv m m C-30~ + 3 w v m I)jm 1 m m Dt ~
djmErn 2 +(3epv I - v 0 - er) _ _ m m d-r - ( r - pv 2I)u(m) = o , m m 3 where we have let 0
=
take jumps, and apply compatibility
(2.8), and (2.10), obtaining
v A--@-v
m m
,
t
(t n + n t )Aab
a b
a b
DjmEm + ( r - ~ P 2mVI) D t
ern (3.5)
117
Geometrical Crystal Acoustics
and
u(~) 3
u3
denote the value of
associated with
the product derivatives, multiplying through by
j E Expanding some of m m , which satisfies
E
m de /d-r = E~ DE /Dt = 0 , eliminates several terms of (3.5) but not that in m m m m djm/d-r , which makes (3.5) a partial differential equation. This term is
E~
eliminated by choosing e 2em pvm
e m
=
= ET
m
o
to satisfy Em
.
(3.7)
In 1111, it is shown that the integral curve of the velocity vector
a
so
determined is a bicharacteristic curve 1131. A s well as this elimination, this value of
e
the term in
allows one simplification of the coefficient of E
T
m
jm E~
and also of
0 dE /dT as follows. m
If (3.7) is differentiated in the direction
T
,
then the symmetry of
allows
@
the result to be written dv 2-pv dem
m
d-r
+2e
*v +2e m dT m m
The Frenet formulas (2.2) and (3.6)
1
T
d Em
P ~ = Z E O - + E
md-r
dT
(3.8)
md-rm
allow dO/d.r to be found, and (3.6)2 allows
simplification to
Progress
2cm ( r - A)
+
2nln2 Z(iill d - A 22) + (n22 - nl) 2 %(n12 d
dr
=
so
far allows (3.5) to be written
+A,,)
(3.9)
+
T d vm j mE m In 1n2 dT (All
Addition and subtraction of
- AZ2)
-
1 2 2 7 (n2 - nl)
n1n2d(A12+A21)/dv
d
(Al2
+ AZ1)
-C
~
.E
~
to the matrix factor of (3.9)
allows it to be resolved into (3.10)
However, the Frenet formulas ( 2 . 2 ) applied to the orthogonal trajectories Km
of
Cm
allow
dI /dv
-: :_- n n If the identity symmetry of
r
a b
to be found, and (3.6)1 __ dAab
dv
+
k 0 m
allows simplification to
.
= 0 is differentiated in the direction ET(r-pv2 m m allows the result to be written
(3.11)
v , the
of which the first term vanishes; hence (3.12)
118
R.S. D. Thomas and H. Cohen
by (3.11) and (3.7).
Dividing (3.8) by
and using (3.10) and (3.12)
pvij,
allows (3.9) to be written n
1
D En jLpv’
dv de m = c v - - m + 2(kmem ---) m m m dr dv
m Dt
The skew-symmetry of
-
A12
Application of (2.1) t o dv/dT
causes the last term of (3.13) to vanish. and comparison with (2.3)
T =
(3.13)
.
2
shows that
- kv (3.14) from the second term of (3.13) using (2.1) allows
Using (3.14) to eliminate k, (3.13) to be reduced to
which integrates to our first main result,
(3.15) To be any use, an equation like (3.15) needs calculable quantities.
The
curvature c is a feature of the particular wave-curve geometry, and we shall m return to it in Section 5, but the other quantities are calculable quite generally. By finding the unit eigenvector
E
m
explicitly and direct
substitution into (3.7), one finds that
,
2epv
=
.r’(All+A 22) v
f
=
[vT(All - A22)~IC~T(A11 - A22)v1
f f/h
(3.16)
where
+ I V ~ ( A ~ ~ + A ~ ~ T) (A12+A21)v1 VI~T
(3.17)
,
and the sign chosen on the right side corresponds to that chosen in (3.4) for the v
on the left side.
dem/dr in terms of
c m
The Frenet formulas (2.2) allow the calculation of and the derivatives of the
’ab.
In the remaining sections, wave modes will be treated entirely separately, and so the subscript m
will be dropped.
The sign
t
will continue to be used as
in (3.16). 4.
EXAMPLES
A s a first example, we consider an initial wave curve that is straight proceeding
through a half plane
Po
in which
the wave curve remains parallel;
or
p
c =
k
Aab
= 0.
or both vary in such a way that Let v = (-l,O)T
,r
=
(O,l)T, for a
119
Geometrical Crystal Acoustics
f r o n t b e g i n n i n g on t h e i s such t h a t
e # 0
x2
-
I f w e assume t h a t t h e medium
t = 0.
axis a t t i m e
i n i t i a l l y a l l along t h e f r o n t , then u n t i l
e
=
0
along
e a c h r a y we can u s e t h e i d e n t i t y de/d-r = e d(llne)/d.c
=
D(!Lne)/Dt - v d(llne)/dv
t o r e d u c e e q u a t i o n (3.15) t o ?
I
;
D lln jLepvJ = Dt which i s i n t e g r a b l e because
d Rn e
dxl Dx / D t = v
When e q u a t i o n ( 4 . 1 ) i s
along each ray.
1
integrated,
e # 0
v a l u e of
d e t e r m i n e s t h e o b l i q u e d i r e c t i o n i n which t h e wave moves, i t h a s no
e
c a n c e l s because i t a p p e a r s on b o t h s i d e s .
So, w h i l e t h e
e x p l i c i t e f f e c t o n decay;
1/2 v 2 j = v2(0) j ( 0 ) ~ p ( ~ ) v ( o ) / P v ~ (0)
where
,
(4.2)
i n d i c a t e s i n i t i a l v a l u e s , a c o n v e n t i o n we u s e throughout.
The second
example i s t h e p a s s a g e of a wave curve t h r o u g h an i s o t r o p i c r e g i o n example of one i n which
as a n Po I n t h i s c a s e ( 3 . 1 3 ) i s b e t t e r s i m p l i f i e d by
e :0 .
eliminating the last t e r m a s before, applying ( 2 . 1 ) , involving
e
and e l i m i n a t i n g a l l terms
t o obtain
which i n t e g r a t e s t o
1/2 2 2 v j = v (0) j ( 0 ) {p(O)v(O)/pv) exp
1
c/2dw
,
(4.3)
where t h e i n t e g r a l i s w i t h r e s p e c t t o a r c l e n g t h a l o n g t h e o r t h o g o n a l t r a j e c t o r i e s K
of t h e wave c u r v e s , which a r e t h e r a y s when
e
=
0.
Homogeneity, t h e t h i r d example, w i l l have a s e c t i o n t o i t s e l f .
5.
ANISOTROPIC HOMOGENEITY
W e c o n s i d e r t h e c o n s t a n c y of
speeds
v, the discriminant
and
p
,
h2
hab
of (3.1) i n
Po.
The d i s t i n c t wave
and t h e t a n g e n t i a l component
e
of t h e r a y
v e l o c i t y a r e s t i l l g i v e n r e s p e c t i v e l y by ( 3 . 4 ) , ( 3 . 3 ) , and ( 3 . 7 ) , b u t are a l l now f u n c t i o n s of
v
alone.
S e v e r a l d e r i v a t i v e s w i l l b e r e q u i r e d , a l l of which
c a n be o b t a i n e d by d i r e c t i o n a l d i f f e r e n t i a t i o n and a p p l i c a t i o n of t h e F r e n e t formulas.
From ( 3 . 3 ) u s i n g t h e symmetry of dh/dV = 2 k € / h
From (3.4) u s i n g t h e symmetry of
All
+
=
AZ2
- AZ2
- 2cf/h
and
.
h12
+
AZ1
come
(5.1)
and (5.1) come
dv/d-c = - c e df/d.r = c ( h 2 - k), where
dv/dv = k e S i m i l a r l y from (3.17)
,
, dh/dT
All
.
(5.2)
120
R.S.D. Thomas and H. Cahen
T + C V ~ ( A ~ ~ + A ~ ~ ) V I (.Al2 C T +A21)r1
+
2[rT(A12
+hZl)vI
.
2
Lastly, from ( 3 . 1 6 ) , (5.212, (5.1)2, and ( 3 . 4 ) comes de/d-r = cg ,
15.4)
where 2pvg
=
2
+
(2epv) /2pv2 T
-
T
2
2pv
(5.5)
3 k 2f /h + E/h 2
(All+A22)r
.
The initial consequence that we draw from the above is that the rays are straight lines; substitution of (5.2)
into (2.3)
from which it follows that
so
2 and
r,v,e
which is accordingly straight.
a
1 shows that Dv/bt
=
,
0
are all constant along a ray,
The second consequence presumes non-zero
curvature of the wave curve. LEMMA.
The radius of curvature of a smooth wave curve is a linear function of
time along a ray. Proof:
The proof proceeds by calculation of
Dc/Dt.
Application of the Frenet
formulas (2.2) to themselves gives 2
_d _v_ _ _ _ drdv
2
dk (-++)T
-
dvdr
d?
.
dc dv
Equation ( 2 . 4 ) on the other hand gives the value dc/dv
=
k2
+
c2
(k2+c2)r.
Accordingly,
- dk/dT
(5.6)
Comparison of (3.14) with (5.2)2 shows that
.
ec = vk We note that the existence of a radius of curvature requires c # 0 , since v # 0 , the conditions When
e # 0 , from ( 2 . 1 ) , Dc/Dt
The form of
e
function of nl
, given
=
&/at 2
c (v-g)
=
0 and
=
0 , (5.6), and (5.7),
In the latter case, e :0 =
,
(5.7) that,
0 are equivalent.
=
.
(5.8)
v , is a simple enough
is substituted for n2
has isolated zeros along a curve with straight so that k
k
by (3.16) as a function of
2 +d(l-nl)
if
e
so
as appropriate that it
c # 0 , unless it is identically zero.
the rays are all orthogonal trajectories but still
dk1d-c = 0 and s o from (5.6),
c2 = dc/dv = v-l Dc/Dt In consequence, the radius of curvature l/c holds because g = 0 by ( 5 . 4 ) .
. is linear in
t;
(5.9) in fact (5.8)
121
Geometrical Crystal Acoustics
It remains to consider a ray along which
along which
e
=
0 and which is surrounded by rays
e # 0 , corresponding to an isolated zero of
e
as a function of
Since the curvature of the wave curves along that ray are continuous with
nl. the curvatures along neighbouring rays, they must be governed by the same
linear law (5.8) though with
g
0 as above.
=
Equation (5.8) integrates to give
-- 1 -- 1 c(s,t) c(s,O)
+
(5.10)
’
(g-v)t
and the lemma is proved. This result corresponds to the expected shrinkage in the radius of curvature of a circular wave curve in the simple case with
e
=
g
=
0.
The second main result is easily deduced from the lemma and equations ( 3 . 1 5 ) , (5.4), and
h/Dt
,
= 0
v 2. 3 where
r
=
each ray.
l/c(s,O)
=
and
v 2 (o)j(0)r1 * v
=
v(O),
{r g
In the case of isotropy, g
the expected one cl21.
= g(0) =
t
to
0
.
[ g ( o ) -v(o)lt~-~ ~
(5.11)
since they are constant along
, and the expression (5.11) reduces
co
The negative exponent indicates a limited range of
validity in one time direction whenever The approach of
+
r/(v-g)
caustic, an envelope of rays.
g # v ;
t
must not reach r/(v-g).
indicates that the ray is approaching a The consequent concentration o f rays makes
measurements undependable. The attention of solid-state physicists, who seem to be the persons most likely to be concerned with such measurements, especially in materials of cubic symmetry
(Cll =
c22 ;
C16
=
cZ6
=
0) , has been called
to this formation of caustics elsewhere by the present authors C141.
In the case of a straight wave curve, neither the curvature nor the magnitude of the wave varies with time. or is not zero, c constant by (3.15).
From
c
=
0 , (5.7) gives k
=
0. Then as
is constant by (5.9) or (5.8) respectively, and vj
e
is is
R.S.D. Thomas and H. Cohen
122 REFERENCES
111 Musgrave, M.J.P., Crystal acoustics (Holden-Day, San Francisco, 1970). 121 Friedlander, F.G., Sound pulses (Cambridge University Press, Cambridge 1958).
131 Every, A.G., Ballistic phonons and the shape of the ray surface in cubic crystals, Phys. Rev. B 24 (1981) 3456-3467. C41
Thomas, T.Y., Plastic flow and fracture in solids (Academic Press, New York and London, 1961).
[51
Karal, F.C. and Keller, J.B., Elastic wave propagation in homogeneous and inhomogeneous media, J. Acoustical SOC. Amer. 31 (1959) 694-705.
C61 Keller, H . B . ,
Propagation of stress discontinuities in inhomogeneous
elastic media, SIAM Rev. 6 (1964) 356-382. C71
Wright, T.W., Acceleration waves in simple elastic materials, Arch, Rational Mech. Anal. 50 (1973) 237-277.
C81
Bowen, R.M. and Wang, C.-C.,
Acceleration waves in orthotropic elastic
materials. Arch. Rational Mech. Anal. 47 (1972) 149-170. 191 Truesdell, C. and Toupin, R.A., The classical field theories, in Fliigge, S. (ed.), Handbuch der Physik, V o l . III/l.
(Springer, Berlin-Gattingen-
Heidelberg-New York, 1965). ClOl
Cohen, H. and Wang, C.-C., Compatibility conditions for singular surfaces, Arch. Rational Mech. Anal. 80 (1982) 205-261.
1111 Cohen, H. and Thomas, R.S.D., Plane wave propagation and evolution for quasilinear hyperbolic systems, Utilitas Math. 24 (1983), to appear. El21
Cohen, H. and Suh, S.L., Wave propagation in elastic surfaces, Journ. Math. and Mech. 19 (1970) 1117-1129.
1131 Courant, R. and Hilbert, D., Methods of mathematical physics (Interscience, New York, 1953).
1141 Thomas, R.S.D. and Cohen, H., Caustics in geometrical crystal acoustics, Phys. Rev. B 28 (1983), to appear.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
RIEMA"
123
INVARIANTS
A . M . Grundland
McGill U n i v e r s i t y Department o f Mathematics and S t a t i s t i c s M o n t r e a l , Quebec Canada
T h i s p a p e r p r e s e n t s t h e g e n e r a l i z e d Riemann i n v a r i a n t s method of c o n s t r u c t i n g s o l u t i o n s f o r n o n l i n e a r and n o n e l l i p t i c s y s t e m of P . D . E . ' s . The approach adopted h e r e i s b a s e d on t h e method o f c h a r a c t e r i s t i c s . The main f e a t u r e o f t h i s method i s i n t r o d u c t i o n o f new dependent v a r i a b l e s ( c a l l e d Riemann i n v a r i a n t s ) which remain c o n s t a n t a l o n g a p p r o p r i a t e c h a r a c t e r i s t i c c u r v e s o f t h e b a s i c system o f e q u a t i o n s . T h i s method e n a b l e s us t o r e d u c e t h e number o f dependent v a r i a b l e s f o r some problems, so t h e s o l v i n g o f t h e b a s i c system o f e q u a t i o n s i s s i m p l i f i e d . We describe a procedurefor finding the existence conditions f o r s o l u t i o n s which have a p o s t u l a t e d form and p r e s e n t t h e i r p h y s i c a l interpretation. INTRODUCTION This paper has a methodological character;
i t concerns ways of c o n s t r u c t i n g
c e r t a i n classes of solutions of p a r t i a l d i f f e r e n t i a l equations.
I t c o n t a i n s an
a t t e m p t a t t h e d e s c r i p t i o n and a n a l y s i s [from t h e formal p o i n t o f view) o f t h e phenomena o f p r o p a g a t i o n and n o n l i n e a r s u p e r p o s i t i o n o f waves i n media.
We
assume t h a t t h e problem can be d e s c r i b e d by a n o n e l l i p t i c system of q u a s i l i n e a r
P.D.E.’s o f t h e f i r s t o r d e r as follows
where
s = 1 , ..., m 1-1 = 1,
. . . ,n
j = 1,. . . , L 1
x =
i s t h e number o f e q u a t i o n s , i s t h e number o f independent v a r i a b l e s , i s t h e number o f unknown f u n c t i o n s ( L E E; u = u ( x ) = ( U 1(x) ,...,u a. ( x ) )
,..., xn )
(X
The E u c l i d e a n s p a c e
m),
H.
E clRn - t h e s p a c e o f independent v a r i a b l e s - w i l l b e c a l l e d
a p h y s i c a l s p a c e , and t h e s p a c e hodograph s p a c e .
5 E
H
c R L - t h e s p a c e o f dependent v a r i a b l e s - a
Let us assume t h a t t h e c o e f f i c i e n t s
a ';
and
bS
o f t h e system
(1.1) are smooth f u n c t i o n s of t h e v a r i a b l e u d e f i n e d on an open s e t 0 moreover t h e i n i t i a l c o n d i t i o n s f o r t h e system ( 1 . 1 ) b e smooth, i . e .
c
H .
Let
A.M. Grundland
124
1 2 u(xo,x ,..,xn) = u (x) 0
E
1 n-1 C (R 1
We a r e l o o k i n g f o r s o l u t i o n s d e s c r i b i n g t h e p r o p a g a t i o n and n o n l i n e a r s u p e r p o s i t i o n The approach adopted i s based
of waves t h a t can b e r e a l i s e d i n t h e above s y s t e m s .
on a g e n e r a l i z e d method of c h a r a c t e r i s t i c s which o r i g i n a t e s i n t h e works of B. Riemann ( 1 8 6 9 ) .
H e was t h e f i r s t t o f o r m u l a t e t h e problem o f p r o p a g a t i o n and
s u p e r p o s i t i o n of s i m p l e waves ( l a t e r c a l l e d Riemann w a v e s ) .
Riemann c o n s i d e r e d a
h y p e r b o l i c s y s t e m o f homogeneous e q u a t i o n s w i t h two v a r i a b l e s
( d e p e n d e n t and i n -
dependent) and found t h e s o l u t i o n s c o r r e s p o n d i n g t o t h e s u p e r p o s i t i o n of s i m p l e waves i n t h e c a s e o f p a r a l l e l d i r e c t i o n o f p r o p a g a t i o n .
S i n c e then t h e problemof
Riemann waves and t h e i r s u p e r p o s i t i o n h a s been s t u d i e d by many a u t h o r s e . g .
R.
Mises ( 1 9 5 8 ) , J . L i g h t h i l l ( 1 9 6 8 ) , G . Whitham ( 1 9 7 4 ) , L e i b o w i t z and S e e b a s s ( 1 9 7 4 ) . I n t h e c u r r e n t l i t e r a t u r e one c a n f i n d many a t t e m p t s a t g e n e r a l i z i n g t h e methodof Riemann i n v a r i a n t s and i t s v a r i o u s a p p l i c a t i o n s ; a r e v i e w o f t h e s e problems can b e found i n t h e p a p e r s of T. T a n i u t i and A. J e f f r e y (19641, A . J e f f r e y ( 1 9 7 6 ) , Rozdestvenski and N .
yanienko ( 1 9 6 8 ) .
I n t h i s f i e l d t h e r e e x i s t s a Polish, theo-
r e t i c a l approach which e x t e n d s t h e method o f Riemann i n v a r i a n t s . w i t h t h e works of J . Bonder (19561, f o l l o w e d by M.
It o r i g i n a t e d
B u r n a t (1969-1974) and w a s
r e c e n t l y developed by 2 . P e r a d z y n s k i (1970-1981) and W. Zajaczkowski (1974-1980)
.
The r e s u l t s o b t a i n e d f o r the homogeneous s y s t e m were s o p r o m i s i n g t h a t i t seemed w o r t h w h i l e t o t r y t o e x t e n d t h e method and check i t s e f f e c t i v e n e s s f o r t h e c a s e of nonhomogeneous P.D.E. s. 1974-1983).
T h i s was done i n t h e a u t h o r ’ s p a p e r s (A.M.
Grundland
The a i m o f t h e p r e s e n t p a p e r i s t o g i v e a b r i e f o u t l i n e o f r e s u l t s
f o r q u a s i l i n e a r system o f P . D . E . ’ s .
The s t a r t i n g p o i n t f o r t h i s r e s e a r c h was an a l g e b r a i z a t i o n ( f o l l o w i n g t h e p a p e r s o f M. Burnat (1966),P e r a d z y n s k i ( 1 9 7 0 ) , A. Grundland ( 1 9 7 4 ) ) o f t h e system of e q u a t i o n s (1.11, which c a n b e w r i t t e n i n t h e form:
,uj E axP
DEFINITION.
A matrix
L3
u
{LJ: a . ” L 1 P l u
=
bs}
.
(1.2)
, s a t i s f y i n g t h e above c o n d i t i o n s a t a p o i n t uo
E
H,
w e s h a l l c a l l an i n t e g r a l e l e m e n t of t h e s y s t e m (1.1).
The m a t r i x
L =
l $l
i s a m a t r i x of t h e t a n g e n t mapping
by t h e f o r m u l a :
E 3 (6~’) The mapping
du(x)
-
( 6 ~ 1 ) E TUH
, where
du(x):
I auj 6u- = -
ax’
d e t e r m i n e s an e l e m e n t of t h e l i n e a r s p a c e
-
E
fixP
+ T
.
H
given
(1.3)
L ( E , T ~ ~ ,~ )which
125
Riernann Invariants
may be i d e n t i f i e d w i t h t h e t e n s o r p r o d u c t
E*
Tuff@E* (where
i s t h e dual t o
E)
Each element o f t h i s t e n s o r p r o d u c t i s a f i n i t e sum o f t h e s i m p l e t e n s o r s , i . e . L = y@X
ff
to
where
h
E*
E
a t the point
DEFINITION.
i s a c o v a r i a n t v e c t o r and
y
E
T
H
i s a vector tangent
u.
I n t e g r a l element
i s c a l l e d a simple element, i f
Lj
u
rankllL~11 = 1 ,
i . e . when t h e c o r r e s p o n d i n g t e n s o r i s t h e s i m p l e one.
L
I n o r d e r t o d e t e r m i n e a s i m p l e i n t e g r a l element
x
E
E*
(
s
=
we s h o u l d seek y
E
T
H
and
satisfying
1,..., m ,
,...,n ,
0
f o r homogeneous s y s t e m ,
j = 1,..., 1
p = 1
The n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r e x i s t e n c e o f a non-zero s o l u t i o n
y
Of
t h e equations (1.4) i s rankl/a?’A
1
< I.,
111
(if m
=
I. t h e n det(a?’X
)=O)
I
f o r homogeneous systems 11.5)
I ’ ,bS/I = r a n k ~ ~ a ~ p X( ipf~ m~ ,= 9. rank//aspA
then
yj = (a;’Au)-lbs)
f o r nonhomogeneous s y s t e m s . The r e l a t i o n s ( 1 . 4 ) and ( 1 . 5 ) are c a l l e d t h e wave r e l a t i o n and t h e d i s p e r s i o n r e l a t i o n ( F i s z d o n and Peradzynski 1976) r e s p e c t i v e l y . If t h e c o v e c t o r
X satisfies the
dispersion r e l a t i o n (1.5) then t h e r e e x i s t s a p o l a r i z a t i o n v e c t o r s a t i s f y i n g t h e wave r e l a t i o n ( 1 . 4 ) . y
and
A.
Hence
exists a vector a c t e r i s t i c vector
H
x
*
E
3
(u,X)
-+
y E TUH Thus t h e r e e x i s t s a r e l a t i o n between d i r e c t i o n s y ( u ,A)
E
T H.
A vector
y
f o r which t h e r e
such t h a t homogeneous system ( 1 . 4 ) h o l d s w i l l b e c a l l e d c h a r -
A
.
Simple e l e m e n t s of t h e s y s t e m ( 1 . 4 ) w i l l b e d e n o t e d by
f o r t h e homogeneous system ( t h a t i s when b ( u ) homogeneous s y s t e m (when
0 ) and by
b ( u ) f 0) r e s p e c t i v e l y .
yo
@
Xo
y@A
f o r t h e non-
Now w e w i l l show t h a t t h e s i m p l e
i n t e g r a l e l e m e n t s can b e used i n c o n s t r u c t i o n o f t h e s o l u t i o n o f t h e system ( 1 . 1 ) .
SIMPLb RIEMiWN WAVES. Let t h e mapping
u: U
+
ff
(U
c
E)
b e a s o l u t i o n o f t h e system ( 1 . 1 ) .
126
A.M. Grundland
DEFINITION.
o f t h e system ( 1 . 1 ) i s c a l l e d a simple wave ( f o r a
u
The s o l u t i o n
homogeneous system) or a s i m p l e s t a t e ( i n t h e c a s e o f a nonhomogeneous system) i f t h e t a n g e n t mapping
du(x)
i s a s i m p l e element a t each p o i n t
x0
E
D.
I n t h a t c a s e it f o l l o w s from t h e rank mapping theorem (Dieudonn6, 1960)
u(D)
that: R
-t-
f(R)
E
c
H
H
r.
i s a one-dimensional submanifold, i . e . some curve
r.
be a p a r a m e t r i z a t i o n o f
Then:
u(x)
=
f(R(x)),
Let
where
a J dfJ aR i s a s c a l a r f u n c t i o n C a l l e d a R i e m a n n i n v a r i a n t . Since -= - a xu dR a x u ’
R(x)
then for
t h e homogeneous system ( 1 . 1 ) t h e t a n g e n t v e c t o r :
i s a characteristic vector. y(R).
Since
lated t o
u(x)
satisfied.
be a c h a r a c t e r i s t i c c o v e c t o r r e l a t e d t o covectors proportional t o
dR(x)
Let u s assume t h a t t h e f o l l o w i n g e q u a l i t y h o l d s :
y(R).
= C(x)X(R(x)).
X(R)
Let
s a t i s f i e s eq. (l.l),
I f ranklla~’yj11
This is a typical
=
n
-
( i . e . g e n e r i c ) c a s e o f t h e system when we have more (m > n).
Now we show t h a t t h e r e e x i s t s a f u n c t i o n
ip
following equation holds:
R(x)
Indeed,
=
t h e above assumption i s a u t o m a t i c a l l y
1
e q u a t i o n s t h a n t h e independent v a r i a b l e s
o f a s i m p l e wave.
dR(x)
are re-
=
ip(O(x)),
of one v a r i a b l e such t h a t t h e
where
dO(x) = (1 + SAli
o(x)
x’)Avdxv,
=
Xu(R(x))xu thus
dO
A
i s t h e phase
so
dR = 0 ,
K R(x) = cp(O(x))
i n t h e neighbourhood o f t h e r e g u l a r p o i n t s where
dO(x)
doesn‘t
vanish. Let u s c o n s i d e r now a smooth c u r v e r : u = f ( R ) i n t h e hodograph s p a c e H w i t h df . S i n c e from (1.4) it f o l l o w s t h a t ranklla~’yj11 = q y(R) =
the tangent vector
k = 9. - q
then t h e r e e x i s t
ponding t o one v e c t o r (1.4) w i t h t h e same THEOREM 1.
y
y).
Let t h e c u r v e
given by O . D . E .
l i n e a r l y independent c o v e c t o r s
(i.e.
@Ai,
i = 1,
...,k
. . . , Xk
corres-
Then t h e f o l l o w i n g theorem h o l d s .
r
of the class
C1
be such t h a t t h e v e c t o r
(2.1) i s a characteristic vector.
independent c o v e c t o r s
A1,
a r e s o l u t i o n s of t h e equations
A A (R) := ( A l (R)
,. . . , A k ( R ) )
y(R)
Let a l s o a system o f l i n e a r l y be related t o the vector
y(R)
by t h e wave r e l a t i o n
(where q u a n t i t i e s
Er
a r e r e g a r d e d as v a r i a b l e s dependent on
x)
and l e t
cp(zA)
127
Riernann invariants
b e an a r b i t r a r y smooth f u n c t i o n of
can b e r e s o l v e d w i t h r e s p e c t t o v a r i a b l e s then the function
u(x)
I f the system
variables.
k
R (i.e. u = u ( x ) ,
and
u
R = R(x))
i s a s o l u t i o n o f t h e homogeneous system ( 1 . 1 ) .
This solution i s called, f o r
k = 1,
a p l a n a r s i m p l e Riemann wave and f o r
a n o n p l a n a r s i m p l e Riemann wave ( Z . Peradzynski 1 9 7 1 ) .
The s c a l a r f u n c t i o n
k z 1 R(x)
i s c a l l e d t h e Riemann i n v a r i a n t . Proof:
we o b t a i n
Differentiating (2.3)2
k
k dfj ulxu(x) = R,
dR
where
k 1 cp, s=1 ns
u(x)
~i =
yj.
s=l
k 1 - E s=l
dXsvxV cp,
* -
ns
’
dR
dXsvxV
dR #
o f t h e system ( 2 . 3 ) . the point
x
1 which i s j u s t t h e c o n d i t i o n o f l o c a l r e s o l v a b i l i t y
Then
U ~ , ~ U ( X )
i s a s i m p l e element o f t h e system ( 2 . 4 ) a t
o f t h e form k
du(x) = y
@
( Z
s=1
,A
su
)dxU
E
TUH
@
E*,
(2.5)
that is
where t h e q u a n t i t i e s
5’
$ 0
are c o n s i d e r e d as v a r i a b l e s depending on
b y v i r t u e o f t h e wave r e l a t i o n ( 2 . 2 ) , a f u n c t i o n o f t h e form
uJ
=
x.
fJ (R(s))
Thus is a
s o l u t i o n of t h e homogeneous e q u a t i o n s ( l . l ) , and t h e P f a f f i a n system ( 2 . 5 ) i s a c o m p l e t e l y i n t e g r a b l e one (Dieudonng, 1 9 6 0 ) .
Q.E.D.
THEOREM 2 . Let S b e a k = {A}-dimensional s u r f a c e i n a p h y s i c a l s p a c e E and 1 1 I, E C ( S , R ) b e a given f u n c t i o n . If t h e t r a n s v e r s a l i t y c o n d i t i o n i s s a t i s f i e d , i . e .
128
A.M. Grundland
//
XA(o ( x ) )
XES
I
a r e l i n e a r l y independent,
(2.6)
TXS
t h e n i n some neighbourhood
U
f o r which
S
3
u
E
1 C (U,H)
e x i s t s , a nonplanar
simple wave ( 2 . 3 ) s a t i s f i e s t h e boundary c o n d i t i o n :
= f(p)
Proof:
In some neighbourhood
xo
E
where
S,
R
(2.7)
i n t h e a p p r o p r i a t e map, t h e r e i s
A S = {xk = 0 , x - a r b i t r a r y }
Iu)
where
If t h e c o n d i t i o n ( 2 . 6 ) i s s a t i s f i e d , t h e n t h e m a t r i x Thus, we can change t h e dependence o f t h e f u n c t i o n
:= {AluIk}
IIAt(R)II AA
on
R
i s not degenerate
.
by l i n e a r t r a n s -
formation i n t h e f o l l o w i n g way A XB(R)
=
A AB
.
Then t h e c o n d i t i o n ( 2 . 7 ) h a s t h e form
P(X because R(.)
xk = 0 .
A
1
1 A k A A ipCAA(R)X ,...,A A(R)x 1 = cp(x 1 ,
=
Hence, i n such a chosen map we have
rp = p ,
so t h e f u n c t i o n
i s given i n t h e i m p l i c i t form A k R(x) = R(x , x )
In a p a r t i c u l a r c a s e when
= p(X
A
Q.E.D.
u (R(x))x’).
k = {A} = 1 t h e boundary c o n d i t i o n ( 2 . 7 ) can be g i v e n
on c e r t a i n c u r v e s . The s u r f a c e s o f constant R (i,e. t h e l e v e l o f t h e f u n c t i o n
REMARK.
a r e assumed t o be k-dimensional h y p e r p l a n e s .
The f u n c t i o n
hyperplane o f dimension (n-k) which i s a t t a c h e d a t t h e p o i n t forms xo
E
AA(p(xo)). S,
If t h e point
then t h e function
R
x
p(x,) xo
R(.))
determines t h e and d e f i n e d by
l i e s on s u c h a h y p e r p l a n e , p a s s i n g t h r o u g h
must be given b y
The t r a n s v e r s a l i t y c o n d i t i o n means t h a t t h e f a m i l y of k-dimensional h y p e r p l a n e s covers a c e r t a i n neighbourhood o f s u r f a c e s solution
R(.)
S.
Hence, i n t h i s neighbourhood, t h e
i s u n i q u e l y determined by t h e boundary c o n d i t i o n
RIS = p .
129
Riemann Invariants
Let us s t u d y now a p l a n a r simple Riemann wave (when k = 1 i n t h e e q u a t i o n ( 2 . 3 ) ) .
fT
Changing t h e dependent v a r i a b l e s
xu ( E ) : =
Au(R),
i n t h e e q u a t i o n ( 2 . 3 ) and d e n o t i n g
= (p-loR
we o b t a i n R(x)
=
XU(R(x))xu = x v ( k ( x ) ) x u .
So, by v i r t u e o f t h e e q u a t i o n ( 2 . 3 ) , we have
and
Upon performing t r a n s f o r m a t i o n s
the relation ( 2 . 3 ) for
k = 1 t a k e s t h e form u(x) = r ( k ( x ) ) ,
dI‘
- = v(k),
where
(2.9)
dR
Each s o l u t i o n ( 2 . 3 ) o f t h e system (1.1) can b e p r e s e n t e d i n t h e e q u i v a l e n t form (2.9).
The form ( 2 . 9 ) a p p e a r s t o be s i m p l e r from t h e p o i n t o f view o f c a l c u l a t i o n s .
There a r e two ways o f u s i n g Theorem 1. Method 1.
One can f i n d a v e c t o r f i e l d
ranklla?’yy311 < n .
Then one f i n d s
y(u)
u = f(R)
E
T,H
which i s c h a r a c t e r i s t i c , i . e
as a n i n t e g r a l curve o f t h e f i e l d
y
by means o f i n t e g r a t i n g t h e system o f O . D . E . ’ s df’ = dR Also one p u t s
y(R):= y(f(R))
t h e vector
y(R).
Method 11.
One d e t e r m i n e s
y’(f(R)). ~
and f i n d s a n a r b i t r a r y c o v e c t o r
u = f(R)
related t o
as an i n t e g r a l curve o f t h e system o f e q u a t i o n s
dl
ranklla?’((f(R)) df’ 7
X(R)
< n.
A.M. Grundland
130
I t i s i n g e n e r a l an o v e r d e t e r m i n e d s y s t e m , f o r example, f o r
m = n
i t may b e
reduced t o one p a r t i a l d i f f e r e n t i a l e q u a t i o n d e t (a?’ I
for
R
unknown f u n c t i o n s .
y ( R ) := d f and f i n d s
determines
Next one
h(R)
as
dR
was mentioned above (Method I ) . Let
M
g:= ( . 1 .) : MxM
(1,m).
=
m + 1 dimensional
+
R1
be m n 0
By
(x
(n 2 2 )
symmetric s c a l a r p r o d u c t w i t h s i g n a t u r e
be a degenerate
,. . . , xm )
Minkowski s p a c e , and l e t
we d e n o t e
Cartesian coordinates i n
M
so t h a t t h e
s c a l a r p r o d u c t i s d e f i n e d by (xly)
=
xvy
u
=
xOyO -
m Z x1. y 1. i=l
W e use a l s o a standard notation
Now we prove t h e f o l l o w i n g s t a t e m e n t . LEMMA 1.
If the function
o f Minkowski s p a c e
R(x)
d e f i n e d on some neighbourhood o f t h e p o i n t
satisfies
M
mZ h=O
then f o r
5
E
R’
x0
x
R R
x
(2.10)
=0,
and
m z. = 1
R.
R. . n . - Sni, j = 1 11 J C
n . :=
where
2 ,
(2.11)
Ro
t h e following i d e n t i t y
m
Z
(R. . - z . n . - z . n . 1 i , j = i 11 1 3 J
= 45
2
m
-45
Z R..n.n. i , j = l 11 1 3
+
m + 2
Z
i , j ,k,R=l
m - 2 holds.
R . R. n n . n n . + ik J Q k 1 R 1
m
R . .R. n . n + C R. .R.. i , j , k = l 11 I k 3 k i , j = i 11 11 Z
(2.12)
131
Riemann Invariants
From ( 2 . 1 0 ) and ( 2 . 1 1 ) we o b t a i n t h e f o l l o w i n g formulae 2 Z Ri 2 i = l n, = ~- 1, 1
Proof:
m z.n. =
C
k=l
R. n n . ik k 1
-
(2.13)
0
of t h e metric I.1.).
If a function
THEOREM 5 .
point
xo
where
q(.,..,.)
E
0
R: 0 c M
i s a real f u n c t i o n o f
b a s i s o f an i s o t r o p i c s u b s p a c e each
i,j
E
can b e d e f i n e d i n a neighbourhood of t h e
R1
-+
i n t h e i m p l i c i t form by t h e e q u a t i o n ( 2 . 3 )
{l,..,k}
T(R)
c
k M
(Ai(R)IAj(R)) = 0 ) ,
2:
v a r i a b l e s and o f dimension
k
(Al(R),..,Xk(R)) 5
then t h e function
min(p,n-p) R(x)
is a solution
o f t h e system
v 2R where
V2
(P ,n-p).
= 0,
is a
(i.e. for
(VR~VR = )o
d e n o t e s t h e Laplace-Beltrami o p e r a t o r w i t h r e s p e c t t o s i g n a t u r e
135
Riemann invariants
T h i s theorem may be e a s i l y proved by d i f f e r e n t i a t i o n o f t h e e q u a t i o n s ( 2 . 3 ) 2 . P h y s i c a l l y , t h e nonplanar s i m p l e Riemann wave [ 2 . 3 ) which a d m i t s a F o u r i e r t r a n s form ( e . g . i f
i s L1 i n each o f t h e v a r i a b l ’ e s ) can be c o n s i d e r e d as a s u p e r -
q
p o s i t i o n o f an i n f i n i t e number of p l a n e waves r e l a t e d t o t h e same c h a r a c t e r i s t i c curve given by O . D . E .
(2.1).
So we have
k [P(nl,.
.
f
’Tik)
=
___ $(yl,...,yk) ( 2 T )l l / k i
exp(-i
C
Y,q,ldYl...dYkj
c1=1
and
where
qa:=
A
c1U
xu.
NONLINEAR SUPERPOSITION OF RIEMANN WAVES I N A HOMOGENEOUS SYSTEM.
The a l g e b r a i z a t i o n o f t h e system o f homogeneous e q u a t i o n s (1.1) allows u s t o c o n s t r u c t more g e n e r a l c l a s s e s o f s o l u t i o n s , r e p r e s e n t i n g n o n l i n e a r s u p e r p o s i t i o n o f simple Riemann waves i . e . double waves and i n g e n e r a l k-waves. Let 1 ,. . . ,y @Ap be s i m p l e i n t e g r a l elements w i t h l i n e a r l y independent A 1, . ,A k P and yl, . . . , y k r e s p e c t i v e l y . We propose now a form o f t h e s o l u t i o n u f o r which
. .
ylaX
t h e t a n g e n t mapping
where
St f 0
du(x)
i s t h e sum o f simple i n t e g r a l e l e m e n t s .
i s c o n s i d e r e d as a f u n c t i o n of
x.
The s i m p l e i n t e g r a l elements
I t was proved (M.
Burnat 1969, 2. ys@Xs r e p r e s e n t a simple Riemann wave. P e r a d z y i n s k i , 1970) t h a t t h e s o l u t i o n s o f t h e p o s t u l a t e d form (3.1) e x i s t and can be w r i t t e n i n Riemann i n v a r i a n t s .
I n t h e l a t t e r case i t means t h a t w e make t h e
a d d i t i o n a l assumption t h a t t h e commutator o f t h e v e c t o r f i e l d s
ys,yt
is a linear
combination o f t h e s e f i e l d s , i . e .
If t h e s e c o n d i t i o n s a r e s a t i s f i e d we may change t h e l e n g t h of t h e v e c t o r s
y,
so
that
TY,.Y~I and t h e v e c t o r s
yl,...,yk
= 0,
c o n s t i t u t e a holonomic system and t h e r e e x i s t s a
(3.2)
136
A.M. Grundland
parametrization of t h e surface tangent t o t h e f i e l d s u(x)
=
f(R(x))
where
R(x):=
.
yl,.
,yk
1 k k (R ( x ) , . . . , R (x)) E R
,
(3.3)
such t h a t
The system (3.4) i s an over-determined solutions).
Consequently,
du(x) =
C
system of P.D.E.s
(which does not alwayshave
af
aRS
7 dRS, where dRS = 1 dx’, 11 ’=I ax
s=l aR
which t o g e t h e r w i t h t h e assumption t h a t
yl, ...,yk
a r e l i n e a r l y independent,
l e a d s t o a system o f P f a f f i a n forms dRs(x)
=
CsAs(R(x))
The f i e l d s of c o v e c t o r s AS =
As(R)dx’ II !J=l
E
E*).
As
where
cs(x) f 0,
s
E
{l,
. . . ,k l .
become f u n c t i o n s o f t h e p a r a m e t e r
(3.5)
R (i.e.
I n v o l u t i v i t y c o n d i t i o n s f o r t h e system (3.5) have been
a l r e a d y i n v e s t i g a t e d ( Z . Peradzynski 1970).
Namely, t h e system ( 3 . 5 ) h a s s o l u t i o n s
( i s i n t e g r a b l e ) i f t h e following conditions a r e s a t i s f i e d .
A
sftEI1, where
as
t
and
From e q s . (3.6)
f3:
axs = atA s s s t + BtA , t
(3.6)
..., k ) a R
a r e given f u n c t i o n s o f
by formulae,
R
t h e c o m p a t i b i l i t y c o n d i t i o n s ( t h a t i s , t h e symmetry o f t h e second
d e r i v a t i v e s ) are o f t h e form
If l i n e a r l y independent v e c t o r s
one v e c t o r
y
S’
As,.
t h e n t h e above problem f o r
( Z . Peradzynski 1971).
. . ,A?
1 (s)
k
’
s = 1,
...,k
are r e l a t e d t o
waves admits a g e n e r a l i z a t i o n
The system of P f a f f i a n forms (3.5) may b e r e p l a c e d by the
137
Riemann Invariants
following
where
E:[,)
$ 0,
R = (R
,... , R k ) ,
1
s
E
{l, ...,k 1
.
( 3 . 6 ) , we have t h e f o l l o w i n g c o m p a t i b i l i t y c o n d i t i o n s
In t h i s c a s e , i n s t e a d o f
= 0,
R‘
where j ( r ) = 1 ,..., i [ r ) , I t h a s been proved (H.
k ( s ) = 1 ,...,i(s).
Btjttger 1978, Z . Peradzynski 1978) t h a t i n t h e c a s e
d e s c r i b e d by Riemann i n v a r i a n t s t h e s o l u t i o n
u
decays i n an
e x a c t way i n t o
s i m p l e Riemann waves b e i n g o f t h e same type’)
a s a t t h e i n i t i a l moment.
Thismeans
t h a t t h e r e s u l t o f n o n l i n e a r s u p e r p o s i t i o n o f t h e simple Riemann waves c o n s i s t s o f
k
waves o f t h e same k i n d b u t w i t h d i f f e r e n t amplitudes and, i n g e n e r a l , with
d i f f e r e n t d i r e c t i o n s of propagation.
So w e have
and t h e t y p e of t h e s i m p l e Riemann waves. s u p e r p o s i t i o n o f t h e s i m p l e Riemann waves.
a
c o n s e r v a t i o n lawfor t h e number
I n t h i s case we can speak about e l a s t i c The Cauchy problem f o r t h e system ( 3 . 5 )
h a s been f o r m u l a t e d and s o l v e d u s i n g t h e Riemann f u n c t i o n s .
(A.
Grundland and
R . Zelazny 1983, Z . Peradzynski 1979).
We w i l l p r o v e t h a t f o r an a r b i t r a r y
k 2 1
a l l the solutions of t h e
system (3.5) can b e o b t a i n e d [ t a k i n g i n t o account ( 3 . 6 ) and ( 3 . 7 ) ) by r e s o l u t i o n 1 Rk o f t h e system with respect t o t h e variable R
,...,
AS(R)*xu = Ys(R),
u
where
Vs
R = (R
are t h e a r b i t r a r y f u n c t i o n s o f c l a s s
1
,...,Rk ) , C1
of
k
(3.8)
variables.
Indeed,
d i f f e r e n t i a t i o n o f t h e e q u a t i o n s (3.8) g i v e s
If t h e system (3.5) i s s a t i s f i e d , t h e n t h e l e f t hand s i d e i s p r o p o r t i o n a l t o
and t h e r i g h t hand s i d e i s a l i n e a r combination o f
f$
0,
then
Xr(R)
for
r # s.
As(R).
Thus i f
138
A.M. Grundland
(3.10) From (3.9) and from t h e c o m p a t i b i l i t y c o n d i t i o n s (3.6) w e have s s
u A (R)x’ ri.l
s r
+ B h (R)x’
=
r u
ws
(R) ___ aRr
,
t h e r e f o r e , by v i r t u e o f t h e e q u a t i o n (3.8) (3.11) Then from t h e e q u a t i o n (3.9) aAS(R)xU
(--a$’
-
a RS
’a RS
)dRS = A s ,
axs (R) xu where
(-
-
a RS
aRS
i s t h e resolvability condition.
I f 0
Thus w e have
equation ( 3 . 5 ) . Let us i n v e s t i g a t e now t h e c o m p a t i b i l i t y c o n d i t i o n s f o r t h e system ( 3 . 1 1 ) . I t i s a system of
for
s
# r #
t.
k(k-1)
e q u a t i o n s f o r k unknown f u n c t i o n s .
We have
Symmetry o f t h e second d e r i v a t i v e s g i v e s
Hence t h e s e c o n d i t i o n s a r e i d e n t i c a l l y s a t i s f i e d by v i r t u e o f ( 3 . 7 ) .
Let t h e f u n c t i o n s
be s o l u t i o n s o f t h e e q u a t i o n s
p s = ps(R)
aps= -u s a Rr
then t h e functions
for
r
+ s,
ns:= exp(pS) s a t i s f y
allS + a Rr Introducing the notation
$”:=
qsJISr
= 0.
from (3.11) we obtain
(3.12)
139
Riemann invariants
ajlS ~
s-r
for
B,$
=
# s;
r
(3.13)
aRr and t h e e q u a t i o n s ( 3 . 7 ) t a k e t h e form (3.14)
Let us n o t i c e , t h a t i n formulae (3.13) and (3.14) t h e v a r i a b l e
x
does n o t a p p e a r
explicitly.
5
I f f o r example
1
i n t h e equation (3.5), then
E 0
R
1
= const.
Y
From t h e
-
c o m p a t i b i l i t y c o n d i t i o n s (3.6) and ( 3 . 7 ) f o r t h e c o v e c t o r s Xs(R) := As(Rb,K), 2 k s E 12, k } where R = (R ,..., R ) we o b t a i n t h e f o l l o w i n g e q u a t i o n s ,
...,
where
-s
-S
- -- -Bst -Btr "r
- -s-r
a?
Functions
$
2
Brat
,..,qk
+
-s
aa
aa
a?
a 2
L E - - --
-5-5
Brat,
for
0
t
# s # r
E
12,
s a t i s f y t h e c o n d i t i o n s ( 3 . 8 ) and (3.11) f o r r 1 2 k R = R1 and by t h e f u n c t i o n R = (R ,..., R )
s o l u t i o n i s given by
...,k } . 2
0
2.
Thus t h e
o b t a i n e d by
r e s o l u t i o n o f t h e system
In p a r t i c u l a r , f o r
k
=
1 we o b t a i n t h e formula (3.8) f o r a s i m p l e Riemann
wave ( 2 . 9 ) .
REMARK 2 .
As we a l r e a d y know, s o l u t i o n o f t h e e q u a t i o n ( 3 . 5 ) can be o b t a i n e d by
i n t e g r a t i o n o f t h e system ( 3 . 1 1 ) . degenerate
Conversely, we w i l l show now, t h a t e v e r y non-
s o l u t i o n o f t h e system (3.5) can be o b t a i n e d u s i n g t h i s method.
Indeed, t h e e q u a t i o n (3.5) i m p l i e s t h a t f o r
k dqS = Xs(R) + If
gr f 0 ,
q S ( x ) : = AE(R(x))x'
aAs(R)x'
'-
C
CPXr(R),
R:= (R
1
w e have
,. . , R k )
then k dqs =
C
r= 1
pSdRr,
for
s
=
l,..,k,
(3.15)
140 where
A.M. Grundland
vs
a r e f u n c t i o n s d e f i n e d on
1
E.
Additionally functions R ,..,Rk are 1 k R k + l, . . ,Rn). f u n c t i o n a l l y i n d e p e n d e n t , t h u s we can add them t o t h e map (R , . . , R ,
Then (3.15) means, t h a t we have t h e f o l l o w i n g c o n d i t i o n s i n t h i s map.
So t h e f u n c t i o n s REMARK 3.
ns
can be e x p r e s s e d as f u n c t i o n s of R ,...
0
I f (on a c e r t a i n domain At(R)x’
= 0,
R:=
E)
c
(R
1
axs
.
t h e equation
,..,R k ) ,
can be r e s o l v e d with r e s p e c t t o t h e v a r i a b l e a s o l u t i o n of equations (3.5).
RK
s
E
{i ,..,k l ,
R( i . e .
(3.16)
R = R(x)),
t h e n we o b t a i n
Indeed, i f t h e e q u a t i o n s (3.16) are s a t i s f i e d , t h e n
k
(R) xu
dRS +
Xs(R) +
a RS
ahz(R)x’ dR’
C
=
0,
r+s=l aR’
b u t from t h e c o m p a t i b i l i t y c o n d i t i o n s (3.6) and t h e e q u a t i o n s (3.16) we have
axsu (R)xU
= 0.
a Rr Hence, i f t h e c o n d i t i o n o f r e s o l v a b i l i t y
axsu ( ~ ) x ’ l a RS i s s a t i s f i e d , then covector RS = Rs(x)
i s proportional t o
Xs(R)
s a t i s f y t h e equations ( 3 . 5 ) .
a f a m i l y (parametrized by p o i n t s
# 0,
a
E
E)
i . e . , functions
E we obtain
o f s o l u t i o n s o f t h e system ( 3 . 5 ) ,
These s o l u t i o n s have a s i n g u l a r i t y a t t h e p o i n t EXAMPLE.
dRS,
Through a t r a n s l a t i o n i n
x = a.
Let us c o n s i d e r now a system of P f a f f i a n forms dR1(x) = S1(x)X1(R’(x),R2(x)), (3.17) dR2(x) = ~ 2 ( x ) X 2 ( R 1 ( x ) , R 2 ( x ) ) ,
where :
141
Riemann Invariants
(a,B = a r e a r b i t r a r y f u n c t i o n s
of t h e i r arguments), f o r which t h e c o m p a t i b i l i t y
c o n d i t i o n s (3.6) a r e a u t o m a t i c a l l y s a t i s f i e d
ah2
-
1
1
1
B - a (cos R , s i n R , 0) + 0 1( - s i n ~ l , c o sR
aR2
1
,
0) =
(%-a) _ -- a
x1 +
- aA . 2
6-a
%-a
The g e n e r a l s o l u t i o n o f t h e system (3.17) has t h e form
where
1
1
2
2
$ (R ) , $ (R )
a r e a r b i t r a r y d i f f e r e n t i a b l e f u n c t i o n s o f t h e i r arguments.
Thus from e q u a t i o n (3.18) we o b t a i n 1
-atJ2 _
- 0,
aR
Hence t h e c o n d i t i o n s (3.11) are s a t i s f i e d .
To simply t h e formulae we use t h e
following n o t a t i o n
The s o l u t i o n s can be w r i t t e n i n t h e e q u i v a l e n t form $'(R1)
= a(R 1)Xu(R 1 1 1) x 1 ,
A. M. Grundland
142 that is
1
-1 1
( R ) = x cos R1 + x'sin
I$
2
(R
2
- 91(R1I
1 + x3i(R 1) ,
2 1 3 R~ + x cos R + ( B ( R ~ ) - ~ ( R ' ). ) ~
= -xlsin
1 cr(R ) = aR1
For t h e s a k e of s i m p l i c i t y we t a k e equations (3.17).
R
2 8(R ) = bR2
and
i n the
In t h i s c a s e e q u a t i o n ( 3 . 1 6 ) has t h e form 1 x cos R'
+
x 2 s i n R'
+ ax3 = 0,
1 . 2 2 1 - x s i n R1 + x cos R1 + (bR -aR )
=
0.
Hence we have
From t h e above e q u a t i o n s ,
and ax2 + (bR2 - aR1)xl - r x l + x 2 t g R1 =
+ (bR2 - aR2)x2 - x1 + r x
ax'
where
a r : = bR2 - aR
1
.
2’
So w e have
r =
1
((x112
+
(x
2 2
2
- a (x
ax
3 2 %
I I .
F i n a l l y w e o b t a i n a s y s t e m of e q u a t i o n s 1 + 2 1 R (x) = a r c t g " 1 2 ' x + rx R 2 (x) =
1
((x1)2+(x2)2-a2(x3)2)' +
bx
The l e v e l s of f u n c t i o n s
(3.19)
R1
and
o r i g i n of t h e c o o r d i n a t e system.
R2
1 2 arc tg rx + x 1 2 x + rx
.
are the straight lines intersecting a t the
143
Riemann Invariants
SIMPLE STATES I n t h e p a p e r (A. Grundland 1974) i t h a s been proved t h a t t h e r e e x i s t s o l u t i o n s o f t h e nonhomogeneous system (1.1) f o r which d e r i v a t i v e s du(x) mapping
u
of t h e
would be nonhomogeneous s i m p l e elements i f
du(x) = yo@ho where
a?’y;A: 7
=
bS,
yo
E
TuH, h
0
=
o
A dx
p
u
E
E
*
.
(4.1)
The c o n d i t i o n s o f c o m p a t i b i l i t y ( t h a t i s t h e symmetry o f t h e second d e r i v a t i v e s Schwarz lemma) a r e o f t h e form d(du(x))
dy0
=
A
A o + yo@dho = 0
modulo e q u a t i o n s ( 4 . 1 ) ,
where
:=E
Weuse t h e n o t a t i o n X ,
ax yi . -
Namely t h e system ( 4 . 1 ) h a s a s o l u t i o n ( i s
i = i aul
integrable) i f
hi)
A
ha = ( ) . T h i s means t h a t t h e d i r e c t i o n o f
i s constant
‘h
YO along t h e f i e l d where
a = a(u)
yo,
# 0,
and t h e n , by t h e n o r m a l i z a t i o n , one
gets
Ao
ho
-f
aho
c o n s t a n t along the f i e l d
Thus t h e image o f t h e s o l u t i o n i s a curve
u = f(R)
1 and y,, + a y,,, yo, i . e .
tangent t o t h e f i e l d
y
0;
moreover, one can choose i t s p a r a m e t r i z a t i o n s o t h a t
AS
0
A (f(R))
=
*fl A = c o n s t , and
i s a s o l u t i o n o f a nonhomogeneous e q u a t i o n (1.1) , t h e n
Hence, a s a Riemann i n v a r i a n t w e can choose l i n e a r f u n c t i o n
R(x)
=
-0 v A>,x
.
Equatior
A.M. Grundland
144 (4.4) determines a s o l u t i o n
u
=
f(R)
of the field
u = u(x)
c o r r e s p o n d i n g t o t h e i n t e g r a l curve
Such s o l u t i o n s w i l l be c a l l e d s i m p l e s t a t e s .
yo,
The n o t i o n o f t h e “ s i m p l e wave” we r e s e r v e e x c l u s i v e l y f o r s o l u t i o n s o f One o f t h e arguments f o r s u c h an approach i s t h a t t h e
homogeneous systems ( 1 . 1 ) .
a c c e p t e d d e f i - n i t i o n o f a s i m p l e s t a t e can be used f o r e l l i p t i c e q u a t i o n s a s w e l l . Such e q u a t i o n s have no s o l u t i o n s which can be i n t e r p r e t e d p h y s i c a l l y a s waves.
On
t h e o t h e r hand i t s h o u l d be n o t i c e d , t h a t some phenomena which a r e d e f i n e d a s waves i n t h e t r a d i t i o n a l ( p h y s i c a l ) terminology can b e d e s c r i b e d i n terms o f t h e notion of a simple s t a t e .
For example, a l o c a l i z e d d i s t u r b a n c e p r o p a g a t i n g i n a
nonhomogeneous system and c a l l e d a s o l i t a r y wave ( o r a s o l i t o n i n some c a s e s ) corresponds t o a s i m p l e s t a t e .
Physically, a simple s t a t e describes a disturbance
which p o s s e s s e s , u n l i k e a s i m p l e wave, a w e l l d e f i n e d p r o f i l e as w e l l as c o n s t a n t v e l o c i t y and d i r e c t i o n o f p r o p a g a t i o n .
Moreover a s i m p l e s t a t e h a s no auJ of the solution are
gradient catastrophy ( i . e . a l l p a r t i a l derivations
I----)
axu bounded i n t h e e n t i r e domain).
NONLINEAR SUPERPOSITION OF R I E M A ” WAVES I N A NONHOMOGENEOUS SYSTEM.
I t i s p o s s i b l e t o g e n e r a l i z e t h e proposed method o f c o n s t r u c t i n g s o l u t i o n s t o t h e c a s e o f many waves i n nonhomogeneous systems. I t r e q u i r e s t h e s o l u t i o n o f t h e system o f P f a f f i a n forms to be a sum o f homogeneous and nonhomogeneous s i m p l e i n t e g r a l elements i n t h e form:
gr f 0
where q u a n t i t i e s t h a t vectors simple element elements
yoah
x.
We assume
F i e l d s o f nonhomogeneous
( r e p r e s e n t i n g a s i m p l e s t a t d and o f homogeneous simple
The e x i s t e n c e c o n d i t i o n s f o r s o l u t i o n s o f t h e P f a f f i a n forms (5 . l )
with a r b i t r a r y v a r i a b l e s R. Zelazny 1982
1.
a r e l i n e a r l y independent.
( r e p r e s e n t i n g Riemann waves) s a t i s f y t h e e q u a t i o n s :
y,@A’
respectively.
are c o n s i d e r e d a s v a r i a b l e s dependent on
yo,yl, ...,yk
.
5‘
were c h t a i n e d b y A . Grundland 1974, A . Grundland and
I t i s n e c e s s a r y t o c o n s i d e r t w o cases:
The c a s e o f two independent v a r i a b l e s :
145
Riemann Invariants
where c o v e c t o r 2.
E
T*H
The c a s e o f many
has t h e p ro p ert y
(u > 2)
2)
0
=
<w,y,>
independent v a r i a b l e s .
ol,B
= 0,
E
{O,l,..,kl
r
E
(5.4)
tl,..,kI,
0
(5.5)
6a '
where t h e r e i s no summation w i t h r e s p e c t t o
a;.
These r e l a t i o n s a r e t h e n e c e s s a r y
and s u f f i c i e n t c o n d i t i o n s f o r e x i s t e n c e o f t h e s o l u t i o n s o f t h e system ( 5 . 1 ) .
They
g u a r a n t e e t h a t t h e s e t o f s o l u t i o n s o f t h e system (5.1) depends on k a r b i t r a r y f u n c t i o n s o f one v a r i a b l e a c c o r d i n g t o t h e C a r t a n t h e o r y of systems i n i n v o l u t i o n . A p h y s i c a l i n t e r p r e t a t i o n o f s o l u t i o n s ( 5 . 2 ) and (5.3) c o v e r s a l s o g e n e r a t i o n of
waves a s a r e s u l t o f t h e i r s u p e r p o s i t i o n on a s i m p l e s t a t e .
Generation can t a k e
p l a c e , f o r example, when t h e commutators (5.2) o f t h e v e c t o r f i e l d s
y,
and
y
B
are l i n e a r combinations (with c o e f f i c i e n t s n o t n e c e s s a r i l y c o n s t a n t ) o f t h e s e fields
y,
and
yB
and a l s o o f o t h e r f i e l d s
connected with t h e s e f i e l d s
y E , ...,yo.
I t means t h a t waves
yp,..,yu t a k e p a r t i n t h e s u p e r p o s i t i o n .
Thus, as
an e f f e c t o f t h e s u p e r p o s i t i o n o f two s i m p l e waves we o b t a i n new waves ( o f a t y p e o t h e r t h a n t h o s e a t t h e i n i t i a l moment).
I f t h e s e new waves do n o t v a n i s h asymptot-
n
i c a l l y f o r l a r g e time ( t : = x " f m ) ,
then t h e e f f e c t
o f g e n e r a t i o n i s permanent ( i n
accordance with t h e a c c e p t e d terminology i t i s "a n o n e l a s t i c s u p e r p o s i t i o n " ) .
In
t h e c a s e d e s c r i b e d by Riemann i n v a r i a n t s ( 5 . 4 ) and (5.5) t h e c o n s e r v a t i o n law f o r t h e k i n d and q u a n t i t y o f waves h o l d s ( s o c a l l e d e l a s t i c s u p e r p o s i t i o n ) ( A . Grundland and R . Zelazny 1982).
I t h a s been a l s o proved (A.
Grundland a R . Zelazny 1983)
t h a t such s o l u t i o n s , r e s u l t i n g from t h e s u p e r p o s i t i o n o f many simple waves p r o p a g a t i n g on t h e s i m p l e s t a t e , decay i n an e x a c t way i n t o simple waves on t h e s t a t e , of t h e same k i n d as t h o s e e n t e r i n g t h e i n t e r a c t i o n . Consider now, as a n i l l u s t r a t i o n , a p r o c e s s o f p r o p a g a t i o n o f t h e Riemann wave in the
nonhomogeneous system ( 1 . 1 ) by means o f t h e p r o c e d u r e s d e s c r i b e d above.
W e propose a form o f t h e s o l u t i o n
u
f o r which t h e t a n g e n t mapping
sum o f a homogeneous and a nonhomogeneous simple elements
du(x)
is a
A.M. Grundland
146
where yo
51 $ 0
and
y1
i s c o n s i d e r e d as a f u n c t i o n o f
c o n s t i t u t e a holonomic system, i . e . t h a t t h e p a r a m e t r i z a t i o n
=
0 1 f(R , R )
of t h e surface, tangent t o t h e f i e l d s
yo, y l ,
u
Consequently,
du(x) =
af a dRo + af dR1 1 aR
y o , y1
(5.7)
is such t h a t
which t o g e t h e r w i t h t h e assumption t h a t
aR
a r e l i n e a r l y i n d e p e n d e n t , l e a d s t o a system o f P f a f f i a n forms dR dR
0
0
0
As
=
1
= A (R , R ) ,
1
=
h:(f(R))dx’
where
A
0
A
A
1
# 0,
c1(x) f 0
(5.9)
5 1A 1(R 0, R 1) .
The f i e l d s o f t h e c o v e c t o r s (i.e.
We s h a l l assume t h a t t h e v e c t o r s
x.
E
AS
E*).
0 1 R : = (R , R )
become f u n c t i o n s o f t h e p a r a m e t e r
The c o m p a t i b i l i t y c o n d i t i o n s (5.5) f o r t h e system
(5.9) t a k e t h e form (5.10)
where
ao, a l , B0,
B1
are g i v e n f u n c t i o n s o f
consequence o f e q u a t i o n s (5.10)1 of t h e e q u a t i o n
% aR
= ao, t h e n
R.
and ( 5 . 1 0 ) 2 . ho(R)
If
a1 # 0 ,
then (5.10)3
Let the f u n c t i o n
= A(R1)exp q ( R ) .
v
is a
be a solution
Thus we c o n s i d e r t h e
system (5.9) i n t h e form dR 0 = A(Rl)exp(cp(R)),
where
X(R 1)
A
dA(R1) +
o,
dR1
(5.11)
f o r which t h e c o m p a t i b i l i t y c o n d i t i o n s (5.10) a r e a u t o m a t i c a l l y s a t i s f i e d . THEOREM 6 .
A l l s o l u t i o n s o f t h e system (5.11) can b e o b t a i n e d by r e s o l u t i o n , w i t h 1
respect t o the variables
Ro
and
R
, o f t h e system
Riemann Invariants
Xu(R1)xu = JI(R1)
+
1;
147
1 exp(-cp(r,R ) ) d r , (5.12)
dh (R1)xu
RU
dR1
0
u
where Proof:
1
1 exp(-cp(r,R ) ) d r ,
aR1
$(.) is a n a r b i t r a r y d i f f e r e n t i a b l e f u n c t i o n o f
R1
I n d e e d , i f (5.11) i s s a t i s f i e d , t h e n dh (R1)xu
11-1 dCX (R ) x 3 = dXu(R1)xu dR1 + u dR1
-
h(R1) = exp(-cp)dR 0
+
(
) dR1
)J
dR1
acp 0 7exp(-cp)dR
d2Xu(R1)xu +
aR
.
'
+ -)dR
(
d(R1)2 dXu (R1)xu
Therefore both functions
A (R1)xu
and
can b e e x p r e s s e d as f u n c t i o n s
u of
0 R (x)
and
dR1
1
R (x). (5.13)
D i f f e r e n t i a t i n g t h e e q u a t i o n s ( 5 . 1 2 ) we o b t a i n
ax+. O 1
X
aR1 (5.14)
ax
0
dX -
2
dR1
where t h e f o l l o w i n g n o t a t i o n
v
h a s been u s e d .
0
:=
1
X [R ) x u , u
1
dXu(R 1) Xu
S u b s t i t u t i n g (5.14)
dR1
d2Xu (R1) xu
2
, v
v :=
:=
d(R1I2
'
i n t o t h e system o f e q u a t i o n s (5.11) we o b t a i n
A.M. Grundland
148
(5.15) Note, t h a t ( 5 . 1 5 ) 3 (5.15)2.
i s t h e c o m p a t i b i l i t y c o n d i t i o n f o r t h e system ( 5 . 1 5 ) 1
The g e n e r a l s o l u t i o n o f e q u a t i o n s ( 5 . 1 5 ) 3 h a s t h e form
x 1(R)
= -
f 0
1
a'(r;R aR
1 d$ (R1) e x p ( - q ( r , R ) ) d r + -, dR1 0
i s a c o n s t a n t of i n t e g r a t i o n w i t h r e s p e c t t o R , depending on 1 Having t h e f u n c t i o n x (R) we can s o l v e t h e system (5.15)1 and (5.15)2.
where
and
Ji
Thus by v i r t u e of (5.131,
(5.16) R
1
.
(5.16) and ( 5 . 1 7 ) t h e g e n e r a l s o l u t i o n of t h e system
(5.11) h a s t h e form ( 5 . 1 2 ) .
Q.E.D.
I t h a s been proved ( A . Grundland and R. Zelazny 1983) t h a t t h e above s o l u t i o n s
can be g e n e r a l i z e d t o t h e c a s e o f many Riemann waves i n t h e nonhomogeneous system ( 1 . 1 ) and t h e Cauchy problem f o r t h e s y s t e m (5.1) h a s been f o r m u l a t e d and s o l v e d u s i n g t h e Riemann f u n c t i o n s .
Examples o f such s o l u t i o n s f o r t h e e q u a t i o n s of
gasdynamics and magneto-hydrodynamics can be found ( A . Grundland 1974 a , b , 1978, 1983, A . Grundland and R. Zelazny 1982, 1983). ACKNOWLEDGMENTS.
I am g r e a t l y i n d e b t e d t o P r o f e s s o r s B . Lawruk and P . W i n t e r n i t z
f o r very i n s p i r i n g d i s c u s s i o n s d u r i n g t h e study of this s u b j e c t .
149
Riemann Invariants
FOOTNOTES. 1)
A s i m p l e wave (2.9)
of matrix
2)
SU
a. X 7 v
i s of t h e i - t h t y p e i f
y =
i s the i - t h eigenvector
.
The p a r e n t h e s i s d e n o t e s t h e c o n t r a c t i o n o f t h e covector E
TtH
with t h e vector
y
a
t
T H. u
A.M. Grundland
150
REFERENCES Bonder, J . , 1 9 5 6 - 1 9 5 7 , A p p l i c a t i o n d e s ondes s i m p l e s 2 l a r e c h e r c h e s des 6 c o u k m e n t c o m p r e s s i b l e i s e n t r o p i q u e s non s t a t i o n a i r e .
Actes du 9-Congress I n t e r .
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B u r n a t , M., 1966, Theory of s i m p l e waves f o r n o n l i n e a r systems o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s and a p p l i c a t i o n s t o gasdynamics, Arch. Flech. s t o s . , Burnat, M . ,
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B u r n a t , M., 1971, The method o f Riemann i n v a r i a n t s and i t s a p p l i c a t i o n s t o t h e t h e o r y o f p l a s t i c i t y , P t . 1 , Arch. Mech. s t o s . , 23. C a r t a n , E . , 1953, Sur l a s t r u c t u r e des groupes i n f i n i s de t r a n s f o r m a t i o n s . C h a p i t r e I : Les systZmes d i f f g r e n t i e l l e s e n i n v o l u t i o n , G a u t h i e r - V i l l a r s , Paris. C i e c i u r a , G . Grundland, A.M.,1982 Geometrical a s p e c t s of t h e s y s t e m 1VvI2 = a ( v ) , 2 V v = B(v) and a p p l i c a t i o n s t o t h e n o n l i n e a r wave e q u a t i o n . P r e p r i n t I n s t i t u t e f o r T h e o r e t i c a l P h y s i c s , I . F . T . 9 , 1982 Warsaw U n i v e r s i t y , pp.1-14. Courant, R . , H i l b e r t , D . ,
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l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . C o n d i t i o n s of i n v o l u t i o n , B u l l . Acad. P o l . S c . S 6 r . t e c h . 4 , pp. 177-185.
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order p a r t i a l q u a s i - l i n e a r d i f f e r e n t i a l e q u a t i o n s - a l g e b r a i c a s p e c t s . Examples from gasdynamics, Arch. o f Mech. 26, pp. 271-296. Grundland, A . M . , 1975, A l g e b r a i c p r o p e r t i e s o f nonhomogeneous e q u a t i o n s o f magnetohydrodynamics i n t h e p r e s e n c e of g r a v i t a t i o n a l and C o r i o l i s f o r c e s . of s o l u t i o n s
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t h e "Cauchy problem f o r d i s s i p a t i v e media", Ed. W . K o s i n s k i . Wroclaw, P u b l i s h e d i n Ossolineum, i n P o l i s h , pp. 195-209. 1981. N o n l i n e a r s u p e r p o s i t i o n o f s i m p l e waves i n nonhomogeneous
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22. e q u a t i o n s , B u l l . Acad. P o l . S c . S d r . t e c h . , Peradzynski, Z . ,
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a p p l i c a t i o n i n gasdynamics ( i n R u s s i a n ) , Publ. M i r . Moscow. Whitham, G . , 1974. L i n e a r and N o n l i n e a r h a v e s , John Wiley and Sons, New York. Zajaczkowski, W . ,
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153
VARIATIONAL FORMULATION OF THE SINGULAR SURFACE PROPAGATION I N NONSIMPLE ELASTIC MATERIALS Jacek T u r s k i Department o f Mechanical E n g i n e e r i n g * McGi 1 1 U n i v e r s i t y L l o n t r e a l , P . Q . H3A 2K6
The g e n e r a l i z e d framework o f t h e v a r i a t i o n a l formalism o f continuum mechanics i s a p p l i e d t o t h e f o r m u l a t i o n o f dynamical c o n d i t i o n s f o r t h e p r o p a g a t i o n o f a s i n g u l a r s u r f a c e i n nonsimple e l a s t i c m a t e r i a l s . C o n s i d e r i n g t h e f i r s t v a r i a t i o n o f t h e f u n c t i o n a l i n t h e doma i n where t h e s i n g u l a r s u r f a c e i s p r o p a g a t i n g , t h e W e i e r s t r a s s Erdmann c o n d i t i o n s a r e o b t a i n e d . These c o n d i t i o n s a r e e q u a t i o n s f o r t h e jump o f d e n s i t i e s o f momentum and o f energy across t h e s i n g u l a r s u r f a c e . F o r t h e case o f a m a t e r i a l o f grade 2 t h e s e r e l a t i o n s a r e o b t a i n e d h e r e f o r t h e f i r s t t i m e . Using t h e proposed v a r i a t i o n a l f o r m u l a t i o n an e x t e n s i o n o f r e s u l t s t o a m a t e r i a l o f any grade N can be o b t a i n e d . 1.
INTRQDUCTION
I n t h i s paper t h e t r e a t m e n t i s e x c l u s i v e l y i n E u c l i d e a n spaces u s i n g s t a n d a r d t u c l idean c o o r d i n a t e s . Consider a f u n c t i o n a l
where D if a f i n i t e domain i n R 3 , $!$(X,t) a dynamical v a r i a b l e $ E a $ / a t , $,,-_a$/ax . . and L t h e Lagrangian d e n s i t y . I n continuum mechanics L=T-W where
.
T=1/2p0(X)$’ i s t h e k i n e t i c energy p e r u n i t volume i n R 3 and W t h e deformation energy p e r u n i t volume i n R 3 . F o l l o w i n g N o l 1 , f o r s i m p l e e l a s t i c m a t e r i a l s
W=W(X,$,,) whereas f o r nonsimple e l a s t i c m a t e r i a l s W=W(X,$,,, Q’LK.. . , G * L ~ L ’ . . .iN). I f t i i s t h e o r d e r of t h e h i g h e s t g r a d i e n t a c t u a l l y p r e s e n t as an argument o f W, t h e c o r r e s p o n d i n g m a t e r i a l i s c a l l e d t h e m a t e r i a l o f grade N. The t h e o r y of e l a s t i c m a t e r i a l s o f grade 2 has been c o n s i d e r e d by Toupin [l]. L e t a r e g u l a r s u r f a c e C ( t ) moving i n t h e domain D a t t i m e t d i v i d e D i n t o t h e subdomains D + ( t ) and D - ( t ) . The s i d e o f t h e s u r f a c e which i s i n t o u c h w i t h t h e r e g i o n D + ( t ) i s l a b e l e d w i t h a s u p e r s c r i p t (+] w h i l e t h e o t h e r s i d e i s l a b e l e d w i t h ( - ) . F o r a g i v e n f u n c t i o n $(X,t) XED l e t @ and $- denote t h e f i n i t e values o f 4 a t any p o i n t o f t h e s u r f a c e C ( t ) which i s approached from D + ( t ) and D - ( t ) respect i v e l y . I t i s assumed i n g e n e r a l , t h a t t h e dynarnical v a r i a b l e @ of (1.1) i s cont i n u o u s i n D, d i f f e r e n t i a b l e i n D + ( t ) , D - ( t ) and $+#$-, ($,L)+#($,L)-. . . e t c , i . e . C ( t ) i s t h e s i n g u l a r s u r f a c e s r e l a t i v e t o t h e f i e l d $. I n o r d e r t o o b t a i n t h e v a r i a t i o n o f f u n c t i o n a l (1.1) f o r such a c l a s s o f f u n c t i o n $ one has t o express variational quantities
154
J. Turski
by surface operators: surface d i f f e r e n t i a l s , normal variation of s u r f a c e , . . . e t c . From these expressions one can get the variational conditions of compatibility. These conditions a r e complementary t o those known in t h e theory of s i n g u l a r surfaces, i .e. geometrical and kinematical conditions of compatibility. Note t h a t the concepts of t h e theory of s i n g u l a r surfaces and i t s applications t o continuum mechanics are a v a i l a b l e in numerous references, f o r example in [21,[31,[41.
The generalization presented here of t h e c l a s s i c a l Weierstrass-Erdrnann conditions 151 of the variational problem a r e t h e dynamical compatibility conditions i .e. conditions f o r jump of momentum a n d energy across s i n g u l a r surfaces C ( t ) . The general f e a t u r e of t h i s formulation of the dynamical conditions f o r t h e propagat i o n of s i n g u l a r surfaces (shock waves and acceleration waves) in nonsimple e l a s t i c materials a r e i l l u s t r a t e d s u f f i c i e n t l y well by materials o f grade 2 and t h e analysis of higher grade materials i s only t h a t much more complicated in d e t a i l . 2.
VARIATIONAL CONDITIONS O F COMPATIBILITY
Let RcR3 be a n open connected s e t and UcR2 an open s e t . The points o f R and U a r e denoted by ( X L ) , L=1,2,3 and ( u a ) a = 1 , 2 , respectively. Assume t h a t the mapping 3 x : U x T + R xL = xL( ua t )
(2.1
1
9
i s the one-parameter family of regular surfaces which a r e o r i e n t a b l e and f o r every t of the interval T divide R i n t o two non-empty subsets d ( t ) and Q - ( t )i . e . R=R*(t)UR-(t)UX(t). The p o s i t i v e unit normal NL t o E f t ) i s directed toward t h e region n*(t). For s i m p l i c i t y t h e s e t of p o i n t s of C ( t ) will be a l s o c a l l e d a surface. Let a mapping
€1
(-E,
3
s
&--
x(u,t,s)
E
R
x ( u . t . 0 ) :x ( u , t ) be a deformation of the surface ( 2 . 1 ) . The variation of the surface C i s defined as the d i f f e r e n t i a l of ( 2 . 2 ) a t s=O, i . e .
The normal variation of t h e surface (2.1) i s defined by L
Ix
=
L K N NKdx
For a given surface, tensor-valued functions ation defined by d @ = 9’s
i.e.
1
s=o
@(x L( ua , t , s ) , t , s ) = @ ( au , t , s )
have vari-
Singular Surface Propagation in Nonsimple Elastic Materials
155
The normal v a r i a t i o n o f t h e s u r f a c e t e n s o r @ i s d e f i n e d by
Note t h a t q u a n t i t i e s (2.4) and (2.5) a r e independent o f t h e c h o i c e o f t h e s u r f a c e parametrization. I n g e n e r a l , if t h e surface p a r a m e t e r i z a t i o n i s n o t t h e normal one, t h e n i t f o l l o w s t h a t
L L i a 6x ( 6 s ) = 6X (6s) + x adu 8@(6s) = 6 @ ( S s ) t @;a6ua
where @;a = @,L
x,,L
.
L t Consider a t e n s o r - v a l u e d f u n c t i o n @(X , t ) , XEQ which i s d i f f e r e n t i a b l e i n R ( t ) , R - ( t ) , a n d has t h e f i n i t e values $t and @- a t any p o i n t X k C ( t ) which i s approached from Q t ( t ) and 0 - ( t ) , r e s p e c t i v e l y . I n general @ has a jump across I ( t ) u s u a l l y denoted by
so [@I i s a f u n c t i o n o f p o s i t i o n on t h e s u r f a c e C ( t ) and t i m e t. When [I@n#O o r some d e r i v a t i v e s o f @ a r e d i s c o n t i n u o u s across C ( t ) , t h e n r e l a t i v e t o t h e f i e l d @ t h e s u r f a c e i s c a l l e d s i n g u l a r . Using t h e n o t a t i o n
K K K where D L @ = ( ~ L - N L N )@,L i s t h e surface g r a d i e n t o f @ and D@=N$ , K t h e normal grad i e n t of @, t h e n by Hadamard's Lemma: (DL@)*=DL@*, t h e f o l l o w i n g c o n d i t i o n s a r e obtained
K K where BL=-DLN i s t h e second fundamental form o f t h e surface C. From t h e d e f i n i t i o n o f t h e normal v a r i a t i o n o f t h e s u r f a c e t e n s o r s ( 2 . 5 ) t h e f o l l o w i n g c o n d i t i o n s a r e obtained
156
J. Turski
( 6 $ 1 , ~ ) ' = DL(b@')
-
DL{(D$)'8E1 + NLB(D@)'
JK
J K i
+ N ~ S D ~ ( ~ c ) D -~ $NL(@,JKN + N where 8C=NL6x
a
(2.9)
L
On t h e b a s i s o f e q u a t i o n s ( 2 . 8 ) and ( 2 . 9 ) t h e c o n d i t i o n s which a r e c a l l e d t h e v a r i a t i o n a l c o n d i t i o n s o f c o m p a t i b i l i t y a r e f o r m u l a t e d as f o l l o w s
(2.11) Note t h a t i f t h e parameter o f t h e d e f o r m a t i o n i s t i m e , t h e n r e l a t i o n s (2.10) and (2.11) become t h e well-known k i n e m a t i c a l c o n d i t i o n s o f c o m p a t i b i l i t y o f o r d e r one and two, r e s p e c t i v e l y . 3.
VARIATION
OF FUNCTIONALS I N A DOMAIN WITH THE PROPAGATING SINGULAR SURFACE
Assume t h a t t h e dynamical v a r i a b l e con t inuous f u n c t i o n R
3
3 Q 3X
L
i
-x
= $I
i
L ( X ,t)
9 o f the f u n c t i o n a l (1.1) i s a vector-valued,
E
E
3
where E 3 i s E u c l i d e a n space w i t h s t a n d a r d c o o r d i n a t e s . Consider a system (D,C,$) where C i s t h e s i n g u l a r s u r f a c e r e l a t i v e t o t h e v a r i a b l e $ i n t h e f i n i t e domain D c R . For t h e v a r i a t i o n o f t h e f u n c t i o n a l (1.1) t h a t i s g i v e n by 6L =
w ds j s=o
one can w r i t e
where dA i s a E u c l i d e a n area element o f C i n R 3 . By assuming f o r convenience i n t h e p r e s e n t a n a l y s i s , t h a t C ( t ) , t r T a r e c l o s e d s u r f a c e s such t h a t C ( t ) c D and by u s i n g r e l a t i o n s (2.8) and ( 2 . 9 ) as w e l l as t h e i n t e g r a l i d e n t i t y :
i DLfdA
c
=
- 1
z
K NLfBkdA
which h o l d s under above assumptions, t h e f o l l o w i n g r e s u l t s a r e o b t a i n e d :
Singular Surface Propagation in Nonsimple Elastic Materials
(A)
The e q u a t i o n ( 3 . 1 ) w i t h t h e Lagrangian o f t h e f o r m L(X,$,
becomes
dV 1 t2
J’
f
I)J,~)
157
D
It1
i 8 aL where E ( L ) = -(T)
a t oqi
aL - (-----) ’+i,L
,
,L
a
~
.
a r e t h e E u l e r e q u a t i o n s , U ( N ) = N L x x i s t h e normal v e l o c i t y o f C i n R 3 and t h e f o l l o w i n g n o t a t i o n s have been used 3.3)
3.4)
i s a q u a n t i t y independent o f t h e c h o i c e o f t h e s u r f a c e p a r a m e t r i z a (N) t i o n , and bri, and 81 a r e m u t u a l l y independent v a r i a t i o n s . A c c o r d i n g t o Du BoisReymond’s Lemma t h e v a r i a t i o n o f L g i v e n by ( 3 . 2 ) vanishes f o r a l l v a r i a t i o n s 6$ i such t h a t 6 i i I =0, =O if and o n l y i f
Note t h a t U
I
The e q u a t i o r s b ) and c ) of ( 3 . 5 ) a r e t h e g e n e r a l i z e d Weierstrass-Erdmann c o n d i t i o n s .
(B)
The e q u a t i o n (3.1) w i t h t h e Lagrangian o f t h e form L(X,$,$,L,$,LK)
6~ =
I 1 T D
E i ( L ) ~ $ i dVdt -
Ii T aD
(Tic‘@i
f
HiD6Gi)dAdt
becomes
J. Turski
158
where
i a aL E ( L ) =-(-) a t a+i
2L - [-aqi,L
1 1
aL
( p
aqi,LK ,K ,L
a r e E u l e r e q u a t i o n s and i T
iL
iLKN
T NL - DL(H
=
- Hi BK K
I n t h e above e x p r e s s i o n s t h e f o l l o w i n g n o t a t i o n s have been i n t r o d u c e d (3.9) (3.10)
(3.11)
According t o Du Bois-Reymond's.Lemrna t h e v a r i a t i o n o f L q i v e n by e q u a t i o n ( 3 . 6 ) vanishes f o r a l l v a r i a t i o n s 6JI' such t h a t 0 1 ) ~ = 0, D?J$~ = 0, 6$i) = 641 =
I,,
i f and o n l y i f
t2
tl
o
a)
E ~ ( L =)
b)
[Ti
c)
[(Ti+PiU(NjD$il
d)
[Hill
+ PiU(N)rI = 0
=
o
Hi+ = Hi-
+ ULI + fHi$i,LKNLNK]
if = 0
uoq if
= 0
(3.12)
o
=
iWill #
0
The c o n d i t i o n s b ) c ) and d) o f (3.12) a r e t h e g e n e r a l i z e d Weierstrass-Erdmann c o n d i t i o n s f o r t h e c o r r e s p o n d i n g v a r i a t i o n a l problem. 4.
DYNAMICAL CONDITIONS OF COMPATIBILITY I N MATERIALS OF GRADE 2
The m o t i o n o f QcR3i n t h e p h y s i c a l space E 3 ( E u c l i d e a n space w i t h s t a n d a r d c o o r d i n a t e s ) i s a one-parameter f a m i l y o f homeomorphisms
R
3X-
x = $,(X)
= @(X, t )
E
E
3
(4.1)
o
159
Singular Surface Propagation in Nonsirnple Elastic Materials
where R i s c a l l e d t h e space o f m a t e r i a l p o i n t s . Consider t h e system ( Q , X , $ ) where R i s t h e m a t e r i a l o f grade 2 and C i s t h e s i n g u l a r s u r f a c e r e l a t i v e t o t h e m o t i o n I+ o f s1. The dynarnical p r o p e r t i e s o f such syst i m e a r e p r e s c r i b e d by t h e f u n c t i o n a l
By t h e r e s u l t (B) o f t h e paragraph 3 t h e Weierstrass-Erdrnann c o n d i t i o n s i n (3.12)
f o r t h e f u n c t i o n a l ( 4 . 2 ) become
i i Here T and H a r e g i v e n by e x p r e s s i o n s ( 3 . 7 ) and ( 3 . 8 ) , r e s p e c t i v e l y , i n which t h e t e n s o r ( 3 . 9 ) g i v e n by
i s t h e g e n e r a l i z e d K i r c h h o f f s t r s s s t e n s o r and t h e t e n s o r (3.10) g i v e n by HiLK -
aw
(4.7)
a@i ,LK
i s the hyperstress tensor. By a p p l y i n g i n e q u a t i o n s ( 4 . 3 ) - ( 4 . 5 ) t h e formula f o r energy
and t h e f o l l o w i n g k i n e m a t i c a l c o n d i t i o n s o f c o m p a t i b i l i t y
one can c o n s i d e r two cases:
(i)
f o r t h e s i n g u l a r surface such t h a t [$,n#O, w a v e f r o n t one has
i . e . i n t h e case o f a shock
160
J. Turski
( i i ) f o r t h e s i n g u l a r s u r f a c e such t h a t IJ$iJ=O, 7.e. i n t h e case o f an a c c e l e r a t i o n wavefront one has [Tin
=
0
,
aHil
= 0
(4.11)
(4.12) The e q u a t i o n s ( 4 . 8 ) - ( 4 . 1 2 ) a r e t h e dynarnical c o n d i t i o n s o f c o m p a t a b i l i t y f o r t h e c o r r e s p o n d i n g cases. The e q u a t i o n s (4.8) - (4.12) can be w r i t t e n i n t h e s p a t i a l f o r m by u s i n g d e s c r i p t i o n i n t h e p h y s i c a l space E 3 . hus, l e t as i s usual (4.13) h j i k = J-l HiLK j
k
P y hK
(4.14)
be t h e g e n e r a l i z e d Cauchy s t r e s s t e n s o r and t h e h y p e r s t r e s s t e n s o r i n t h e space i Here J=det(q,,-) i s a J a c o b i a n . U s i n g r e l a t i o n s ( 4 . 1 3 ) and i (4.14), t h e s p a t i a l f o r m s of Ti and H t h a t a r e g i v e n by ( 3 . 7 ) and ( 3 . 8 ) , respect i v e l y , a r e o b t a i n e d by E3, respectively.
(4.15) hji = h j i k nk
(4.16)
where ti
= T
i dA
(4.17)
i s t h e t r a c t i o n p e r u n i t area i n E 3 and (4.18) i s t h e s u r f a c e q u a n t i t y whose mechanical meaning w i l l be discussed a t t h e end o f t h i s s e c t i o n . I n t h e above n . i s t h e u n i t normal t o t h e s i n g u l a r s u r f a c e u and da i s t h e E u c l i d e a n area e l e m e n t ’ o f u where u i s t h e image of t h e surface C g i v e n by m o t i o n $. The t e n s o r b i j which appears i n (4.15) i s t h e second.. fundamental f o r m of t h e surface ci and d.hJi denotes t h e s u r f a c e d i v e r g e n c e o f hJ’. Note t h e d i f J f e r e n c e between d e f i n i t i o n s (4.17) and (4.78) g i v e n above and t h e d e f i n i t i o n s (10.17) i n [l]. Using e q u a t i o n s (4.15)
-
(4.18) t h e dynarnical c o n d i t i o n s o f c o m p a t i b i l i t y f o r t h e
Singular Surface Propagation in Nonsirnple Elastic Materials
161
c o r r e s p o n d i n g cases ( i ) and ( i i ) have s p a t i a l forms as f o l l o w s (iv)
on a shock w a v e f r o n t
nup$!
=
- Itin
(4.19)
nuel = -
utiq
(4.20)
(hJin.)+ = J
ihJi
o
nj)- =
(4.21)
( i i v ) on an a c c e l e r a t i o n w a v e f r o n t ntin
=
unen =
o ,
-
[hJin.n J
.. hJ'l$.
I n t h e above p = J - ' p o
=
0
(4.22)
.I
(4.23)
1 ,J
i s t h e mass p e r u n i t volume i n t h e space E 3 and
i s t h e energy p e r u n i t valume i n E 3 and
u=
yn)-
.i
$ ni
i s t h e normal component o f t h e v e l o i s t h e l o c a l speed o f p r o p a g a t i o n where u (n) c i t y o f t h e s u r f a c e a. If i t i s assumed t h a t t h e d e f o r m a t i o n energy W(X,$,L, that
$,)
has t h e f o r m such
(4.24)
where parentheses e n c l o s i n g i n d i c e s i and j i n d i c a t e s y m m e. .t r i z a t i o n w i t h r e s 7 e c t t o t h e s e i n d i c e s , t h e n i t f o l l o w s t h a t i n (4.23) one has hJ1=-hiJ e q u a t i o n becomes
ilcre
W.
.=-w..
1J
J1
and t h i s
i s t h e v o r t i c i t y t e n s o r t h a t i s g i v e n by
1 ' w.. = -($ ij
2 i , j - 'j,i)
.. and. hJ' i s i n t e r p r e t e d as a c o u p l e - t r a c t i o n p e r u n i t a r e a o f t h e s u r f a c e a. (See e q u a t i o n ( 4 . 1 8 ) ) . The r i g h t hand s i d e o f e q u a t i o n ( 4 . 2 5 ) i s t h e n e g a t i v e o f t h e jump o f r a t e o f w o r k i n g o f t h e c o u p l e - t r a c t i o n . Note t h a t t h e general form o f t h e
162
J. Turski
d e f o r m a t i o n energy W(X,+,,,
@,,,I such
t h a t ( i ) c o n d i t i o n (4.24) and ( i i ) r o t a -
t i o n a l i n v a r i a n c e o f W h o l d , was d i s c u s s e d i n [l]. I n g e n e r a l , e q u a t i o n (4.23) can be w r i t t e n as
.. where d . . = l / Z ( $
.+$. . ) i s t h e d e f o r m a t i o n r a t e t e n s o r and mJi=hCJ1’,
i , J .J,1 symmetric p a r t o f hJ1, i s t h e c o u p l e - t r a c t i o n . 1J
*
i.e. theanti-
T h i s i n v e s t i g a t i o n was c a r r i e d o u t w h i l e a u t h o r was r e s i d e n t i n t h e Department o f Mathematics a t M c G i l l U n i v e r s i t y .
REFERENCES Toupin, R.A. T h e o r i e s o f E l a s t i c i t y w i t h Couple-stress. Mech. Anal. 17 (1964) 85-112.
Arch. R a t i o n a l
The C l a s s i c a l F i e l d T h e o r i e s . Handbuch d e r T r u e s d e l l , C. and Toupin, R.A. P h y s i k , V o l . III/l ( S p r i n g e r - V e r l a g , B e r l i n and New York 1960). Thomas, T.Y. Concepts f r o m Tensor A n a l y s i s and D i f f e r e n t i a l Geometry. (Academic Press, New York, London, 1961). Eringen, A.C. and Suhubi, E.S. New York, London 1974) Gelfand, 1963).
I.M. and Fomin, S.V.
Elastodynamics, Vol. 1 (Academic Press, C a l c u l u s of V a r i a t i o n s ( P r e n t i c e - H a l l , I n c .
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
163
FORCED INTEGRABLE SYSTEMS David J. Kaup Clarkson Colleqe of Technology Potsdam, New York 13676 U.S.A.
A discussion is given of boundary value problems for nonlinear integrable systems. The forced nonlinear Schroedinger equation and the forced Toda lattice are used as examples. It is demonstrated that one can quite well solve such systems by using reasonable approximations. As an example, the birthrate for solitons in the forced Toda lattice is solved approximately and it is shown that the result compares quite favorably with the actual birthrate. ITJTRODUCTION It has now been some tine (Gardner et al. (1967)), (Gardner et al. (1974)) since we have been able to solve one-dimensional evolution equations. And it also has been some time, albeit shorter, since we have been able to solve integrable multidimensional problems (Ablowitz an$ Haberman (1975)), (Kaup (1981)) , (Fokas and Ablowitz (1984)). So where do we go from here? One way is to study integrable systems when they are being forced by localized sources. And that is what I wish to report on here. In a sense, this i s not really new. There are already examples of systems similar to forced systems being solved, although the forced nature of the system was not necessarily obvious earlier on. What I shall attempt to do first is to discuss three of these examples and to explain why they are analogous to forced integrable systems. For example, the phenomenon known as "self-induced transparency" ( S I T ) (PlcCall and Hahn (1969)), (Lamb (1973)),(Kaup (1977)) can be considered. to be a forced system. To make this clearer, consider Figure 1 which illustrates the geometry of the SIT system. Along the horizontal axis lies the material which is composed of simple two-level atoms. From the left comes an electromagnetic wave whose frequency can resonate with the two-level atoms. The standard manner for solving this problem is to solve an eigenvalue problem in time for the scattering data, using the envelope of the incident electromagnetic field as a potential. Thus one maps the electromagnetic field envelope into a scattering space, which is analogous to taking a Fourier transform. Now one can propagate this scattering data forward in space in the standard manner, and how this scattering data evolve in space is not a simple harmonic motion as in the case of the KdV equation or the nonlinear Schroedinger equation. Rather its exact evolution depends on the initial state at t = 0 of these two-level atoms. Even in the case where all the atoms are considered to be initially in the ground state, the time-dependence of this scattering data is not simple. As it evolves in x, the continuous spectrum will
164
D.J. Kaup
t {Vacuum
}
Figure 1 The geometry of the SIT system. A coherent electromagnetic pulse is directed from the left onto the two-level atoms located in the shaded half-space x > 0. The final state of the atoms will be determined by the envelope, b(x,t-x/v ) , of the EM pulse and the initial state of 9 the atoms. v is the group velocity of the EM pulse. g
be selectively absorbed. Thus one may consider the initial state of the atoms to be "forcing" the evolution of the scattering data. In other words, the evolution of the scattering data is being determined by a localized (here, at t = 0) quantity. The reader is warned not to become confused by the interchange of space and time in the treatment of the SIT system. Normally, as in solving the nonlinear Schroedinger equation or the KdV equation, one would solve an eigenvalue problem in space, and evolve it forward in time. But in solving the S I T system, we solve an eigenvalue problem in time and evolve it forward in space. This method is used simply because we do not know how to solve the much more complicated spatial eigenvalue problem of SIT. But let me point out that when one considers general forced systems, one must be prepared to tackle such ’complicated’ eigenvalue problems. More shall be said on this later. For now, let me give a couple more examples. Another example is "simple harmonic motion" (Kaup (1978)). This system consists of a fundamental wave and its harmonic interacting in a nonlinear medium. When the group velocities of the fundamental and the harmonic are significantly different, then the problem becomes a two-dimensional problem where the natural coordinates are the characteristic coordinates. This case is illustrated in Figure 2 where the fundamental wave propagates along the horizontal axis (x) and the harmonic propagates along the tilted axis ( T ) . This system is solvable by using an inverse scattering transform (IST) in the X-direction, where one would solve for the scattering data from the X-eigenvalue
Forced Integrable Systems
165
Figure 2 The geometry of the simple harmonic generation system. The envelope fundamental, ql, propagates along the X-axis while the envelope of the harmonic, q 2 , propagates along the tilted -r-axis. The harmonic is generated by the nonlinear interaction of the fundamental with itself. In this figure, the wavy lines indicate the general direction of flow of the two types. In the above example, no harmonic (q2) is present initially (at T = 0 ) . problem, and then propagate it forward in T. In this case, the initial envelope of the harmonic, if any, is the initial data. The initial envelope of the fundamental (its value along x = 0 ) then acts as a forcing agent on the scattering data, forcing it to change as it evolves in T . Again we have the evolution of the scattering data being determined by another or second envelope. Thus this second envelope can be said to be forcing the system. This forcing term is localized along the -r-axis. And this is what I mean by forcing. The forcing term is to be localized. Better examples will follow. But I wished to bring out this example because it is similar to SIT and the next example which is stimulated Raman scattering (SRS) (Kaup (1983)), (Steudel (1983)) . The SRS system and the similar Two-Photon Propagation system have been solved more recently. This system has a very high degree of symmetry between the time and the space coordinates. In fact, one can exchange the two and the equations remain form-invariant. As in SIT, we have an incident electromagnetic wave and a material composed of two-level atoms. But now the interaction is second-order, and we also have a second electromagnetic wave called the Stoke s wave. As in the SIT system we can consider the incident electromagnetic wave to be incident from the left and then to be forcing the SRS system to respond. But most significantly, the method of solution of the SRS system requires one to solve two eigenvalue problems. Thus, there are two IST s involved instead of only one. Now this system is one step up from the previous two examples. In those examples, the evolution of the scattering data could be determined by a simple
166
D.J. Kaup
quadrature. But in the SRS system, one uses the first IST to determine the initial scattering data. Then a second IST is required for determining how these initial scattering data are to be evolved. Note that the time evolution of this system is not of the conventional type. The time-dependence of the initial scattering data is determined by a second independent set of scattering data. And in this respect, the SRS system resembles and really is a forced integrable system. However, it is also a very special forced system. In fact, it is so special that one could even say that it was a degenerate forced system, and it is exactly this degeneracy which allows this system to be solved so easily. This degeneracy arises because in order to specify the time-dependence of the scattering data, one needs to specify only quantity (the polarization state of the atoms) and no more. This comes from the fact that the SRS equations are only first-order partial differential equations. On the other hand, if they were of a higher order, then more than one quantity would have to be specified in order to determine the time-dependence of the scattering data. But as we shall see later, these profiles would then be independent and thus one would be immediately into a nonlinear consistency problem, and it is this consistency problem which has always stifled other previous attempts to treat forced integrable systems .
one
AN EXAMPLE
-
THE FORCED NONLINEAR SCHROEDINGER EQUATION
To illustrate the last comment, let us look at an example. take the nonlinear Schroedinger equation (NLS)
Let us
how since the nonlinearity is not essential for this example, we shall first omit it and review the linear solution. Let us take q to be driven by some source at x = 0 where for t > O , q(O,t) = Q(t)r
2
and let the initial value for x
(2)
0 be
q(x,Of = S(X).
(3)
As is known, the specification of these two functions uniquely determingthe solution of the linear equation. An outline of the linear solution will be of value for understanding what is required for the nonlinear solution. So we start with expressing q(x,t) for x 2 0 as a Fourier transform m
l l
q(x,t) = 2Tr
-m
A(k,t)eikxdk,
where we take A to be given by m
A(k,t) =
1
q(x,t)e 0
From iqt = -qxx, it follows that
- ikxdx
(4)
167
Forced Integrable Systems
2 iA - k A = P(t) t where Q(t) is given by Eq.(2) and
+
ikQ(t),
(7)
P(t) = qx(O,t).
Now note that, according to Eq.(6), it seems as though we may need to independently specify P as well as Q and S in order to determine how A evolves in time. But such is not the case. Instead, one can show (for the linear case at least) that P(t) is indeed dependent on Q(t) and S(x). To demonstrate this, the solution of Eq. (6) is
where by Eqs.(3) and ( 5 1 , m
A(k,O) =
1
S(x)e-ikxdx. 0
Now from Eqs.(4) and (7) P(t) =
&
m
ikA(k,t)dt. -m
which from Eq.(8) and due to the symmetry in the k-integral, reduces to m
+
&
m
lmk2dk ltdsQ ( s )exp [ik2 (s-t)1
,
LII
-m
,
’0
which gives P(t) wholly in terms of S and Q. So, although Eq.(6) may require that one knows what P(t) is, one can eventually evaluate P(t) in terms of S and Q. Now, what changes occur when we consider the nonlinear case? Well, similar to Eq.(9), we may map the initial data, S(x), into the scattering data. This is accomplished by solving for the t = 0 eigenfunctions of the Zakharov-Shabat (ZS) equations vlx
+
icvl = S(x)v2,
(lla)
between x = 0 and x = +a. From the values of these eigenfunctions at x = 0 and as x + m , one can then construct the scattering data (Ablowitz et al. (1974)). The scattering data are the nonlinear generalizations (Ablowitz et al. (1974)) of the Fourier amplitude, A(k), defined by Eq.(5). To determine the time-dependence of the scattering data, we must solve the nonlinear generalizations of Eq.(6), which are the "ABC" equations. These equations describe how the ZS eigenfunctions evolve in time, and for the NLS they are
168
D.J. Kaup
These e q u a t i o n s a l s o d e t e r m i n e t h e e v o l u t i o n o f t h e s c a t t e r i n g d a t a s i n c e t h e s c a t t e r i n g d a t a a r e d e f i n e d i n t e r m s of t h e s e eigenfunctions. For t h e t i m e e v o l u t i o n o f t h e s c a t t e r i n g d a t a , w e o n l y r e q u i r e E q . ( 1 2 ) e v a l u a t e d a t x = 0 and f o r x + +a. S i n c e w e demand the f i e l d s to vanish a s x +m, this part is trivial. On t h e o t h e r hand, a t x = 0 , Eq. ( 1 2 ) becomes -f
2
5
=
iv
= +(2i5Q* + P*)vl
2 t
(25
Q*Q)v,
+
ivlt
-
(2icQ
-
-
P)v2
(2c2 + Q*Q)v2
(13b)
w i t h P ( t ) and Q ( t d ) e f i n e d a s b e f o r e i n Eqs. ( 2 ) a n d ( 7 ) . T h i s i s c e r t a i n l y n o t s o t r i v i a l t o s o l v e a s was E q . ( l O ) . One s h o u l d n o t e t h e a n a l o g y between Eqs. ( 6 ) a n d ( 1 3 ) . Each d e p e n d s on P and Q. But as w e have a l r e a d y d e m o n s t r a t e d , P i s n o t i n d e p e n d e n t of Q and S i n t h e l i n e a r case. Presumably t h e s a m e i s t r u e i n t h e n o n l i n e a r case, n.nd o n e now h a s t h e problem o f how c a n o n e d e t e r m i n e P i n t h e nonl i n e a r c a s e , g i v e n Q and S. Although a f i n a l answer i s s t i l l t o b e g i v e n , p r e l i m i n a r y c a l c u l a t i o n s have a l r e a d y v e r i f i e d t h a t d i r e c t l y from t h e a n a l y t i c p r o p e r t i e s o f t h e ZS e i g e n f u n c t i o n s one may d e t e r mine a l l t h e i n i t i a l d e r i v a t i v e s o f P ( t ) , namely P t ( 0 ) , P t t ( 0 ) , .
. ..
So i t i s e x p e c t e d t h a t it w i l l be o n l y a matter o f t i m e b e f o r e w e w i l l b e a b l e t o r e c o v e r P ( t ) i t s e l f d i r e c t l y from S and Q .
THE FORCED TODA LATTICE AND THE BIRTHRATE OF ITS SOLITONS Although w e e x p e c t e v e n t u a l l y t o b e a b l e t o o b t a i n P ( t ) d i r e c t l y f r o m Q ( t ) a n d S ( x ) , t h e r e q u i r e d p r o c e d u r e m i g h t be s u f f i c i e n t l y complex so a s t o be a l s o u s e l e s s from a n u m e r i c a l o r a p p l i e d p o i n t of view. I n t h i s c a s e , w e would want t o a s k w h e t h e r o r n o t t h e r e i s some o t h e r method whereby w e c o u l d b y p a s s h a v i n g t o d e t e r m i n e P ( t ) . I t t u r n s o u t t h a t one c a n d o t h i s t o a c e r t a i n d e g r e e and t h a t i n many cases t h i s d e g r e e might b e q u i t e a d e q u a t e . The b a s i c i d e a i s t o r e a l i z e t h a t t h e o n l y t h i n g p r e v e n t i n g u s from o b t a i n i n g t h e time-dependence of t h e s c a t t e r i n g d a t a i s knowing what P ( t ) i s . So i f w e c o u l d make a r e a s o n a b l e g u e s s a s t o what P ( t ) w a s , t h e n w e c o u l d a p p r o x i m a t e P ( t ) by u s i n g t h i s g u e s s , and t h e n d i r e c t l y s o l v e f o r t h e t i m e dependence of t h e s c a t t e r i n g d a t a . Of c o u r s e , t h i s would o n l y b e a n a p p r o x i m a t e s o l u t i o n , b u t it c o u l d w e l l be a d e q u a t e and c o u l d a l s o p e r h a p s g i v e some i n s i g h t i n t o t h e a c t u a l s o l u t i o n . A t t h e moment w e h a v e no r e s u l t s f o r t h e NLS;
however, w e do h a v e s u c h n u m e r i c a l r e s u l t s f o r t h e f o r c e d Toda l a t t i c e (Kaup ( 1 9 8 4 ) ) , (Kaup and Neuberger ( 1 9 8 4 ) ) . So I s h a l l now d i s c u s s t h o s e r e s u l t s and u s e them t o i l l u s t r a t e s u c h a p p r o x i m a t e methods.
By t h e f o r c e d Toda l a t t i c e , I mean t h e s e m i - i n f i n i t e ( n = 0 , 1 , 2 , . . . ) Toda l a t t i c e (Toda ( 1 9 7 0 ) ) where t h e z e r o t h p a r t i c l e i s c o n s t r a i n e d These e q u a t i o n s a r e t o move i n some p r e s c r i b e d manner.
Qn = exp(C?nn-l-Qn) where n = 1,2,3,
...,
-
exp(Qn-Qn+l)
Q o ( t ) i s t o b e g i v e n and a s n
-+
+m,
Qn
-+
0.
169
Forced Integrable Systems
B (t)is the forcing term. T h i s l a t t i c e h a s b e e n s t u d i e d a s a model 0 f o r " m o l e c u l a r d y n a m i c s " ( H o l i a n and S t r a u b ( 1 9 7 8 ) ) . I n t h i s c a s e , Then a t t = 0 , one g i v e s t h i s o n e l e t s Qo ( t ) b e z e r o f o r t < 0 . z e r o t h p a r t i c l e a l a r g e f o r w a r d and c o n s t a n t v e l o c i t y . This causes t h e z e r o t h p a r t i c l e t o ram i n t o t h e o t h e r p a r t i c l e s , t h e r e b y c r e a t i n g a s h o c k wave. The s i z e o f t h i s s h o c k wave d e p e n d s on t h e v i o l e n c e w i t h which t h e z e r o t h p a r t i c l e i s rammed i n t o t h e l a t t i c e . T h i s s h o c k wave h a s a s i z e which a l s o i n c r e a s e s l i n e a r l y i n t i m e . The s h o c k i s s u p e r s o n i c and c o n s i s t s m o s t l y o f s o l i t o n s ( H o l i a n e t a l . ( 1 9 8 1 ) ) . And a s i t e v o l v e s , more a n d more s o l i t o n s a r e formed i n t h e s h o c k . - A t y p i c a l example o f a wide s h o c k i s shown i n F i g u r e 3 where b (= -?jQn-l) i s shown v s n . The q u a n t i t y b l ( = -+ao) i s j u s t t h e n n e g a t i v e o f o n e - h a l f o f t h e v e l o c i t y w i t h which t h e z e r o t h p a r t i c l e i s b e i n g rammed i n t o t h e l a t t i c e . The s h o c k i s t h e r e g i o n from What one would l i k e t o d o i s t o be a b l e t o n = 75 up t o n = 1 4 0 . p r e d i c t t h i s s t r u c t u r e . A s a s t a r t , w e s h a l l a d d r e s s simply t r y i n g t o p r e d i c t t h e b i r t h r a t e of t h e s o l i t o n s i n t h i s s h o c k wave. (I s h o u l d p o i n t o u t t h a t from l o o k i n g a t F i g u r e 3 i t i s not a t a l l o b v i o u s t h a t t h e r e a r e s o l i t o n s p r e s e n t i n t h e shock. That i s t r u e b e c a u s e t h e y h a v e b e e n rammed t o g e t h e r i n t o a n a p p a r e n t i n c o h e r e n t lump. However, t h e y a r e t h e r e ( H o l i a n a n d S t r a u b ( 1 9 7 8 ) ) , and i n t h i s one shock, t h e r e a r e a b o u t 60 s o l i t o n s p r e s e n t . ) ~
~
_.
0 1
t = 64.0
b,= -1.95
t '
I
Figure 3 P l o t of bn v s t i n t h e f o r c e d Toda l a t t i c e f o r bl = -1.95 where -2bn+l
i s t h e v e l o c i t y of t h e nth p a r t i c l e .
envelope s t r u c t u r e t o t h e r i g h t .
a t t
=
64.0
Note t h e r e g u l a r
D.J. Kaup
170
The r e q u i r e d m a t h e m a t i c s f o r t h i s a n a l y s i s is t o be p u b l i s h e d i n J . Math. Phys. (Kaup (1984)) , (Kaup and Neuberger (1984)), s o h e r e I s h a l l o n l y d e s c r i b e t h e s e c a l c u l a t i o n s and g i v e t h e m a j o r r e s u l t s . I n o r d e r t o d e t e r m i n e t h e time-dependence o f t h e Toda l a t t i c e scatt e r i n g d a t a , one must s o l v e t h e e q u a t i o n
where
x
='i(z
+
1
-).
(16)
I n E q . ( 1 6 ) , z i s t h e e i g e n v a l u e f o r t h e Toda l a t t i c e s c a t t e r i n g d a t a . 1. The s o l i t o n s p e c t r u m o c c u r s a l o n g t h e r e a l z - a x i s where I z I (The c o n t i n u o u s s p e c t r u m o c c u r s on t h e u n i t c i r c l e i n t h e complex z - p l a n e where I z I = 1 . ) Equation required t i o n can (15) w e
(15) i s e x a c t l y a n a l o g o u s t o E q . ( 6 ) and a s i n E q . ( 6 ) , w e are t o know o n e a d d i t i o n a l p i e c e o f i n f o r m a t i o n b e f o r e a s o l u be o b t a i n e d . I n Eq.(6) w e h a d t o know P(t). Here i n Eq. a r e r e q u i r e d t o know (Qo- Q1) ( t ) i n s t e a d . W e d o know w h a t
Q o ( ti)s b e c a u s e t h e z e r o t h p a r t i c l e i s f o r c i n g t h e s y s t e m . B u t w e do n o t know b e f o r e h a n d what Q , ( t )i s . However, w e may a p p r o x i m a t e it. And t o see how b e s t t o a p p r o x i m a t e i t , l e t u s l o o k a t F i g u r e s 4 and 5 2 where w e p l o t 4 a 2 = e x p ( Q - Q ) a s a f u n c t i o n of t f o r d i f f e r e n t
0
1
v a l u e s of bl.
-
2.a-
1
-
bd
=
-0.5
1.5;
-
4a;
1 1.0-
-
-
a.5-
-
8.0
~
E
i n i t i a l ringing.
l
l
I0
l
20
~
l
30
l
l
40
l
[
50
l
l
60
l
~
~
171
Forced Integrable Systems
12.5-
18.0.
ti
.s
4a;
S O
2.5
0.0
A p l o t of
2
4a2 vs t when bl
A s i s c l e a r f o r bl
= -0.5
=
Figure 5 -2.0 showing t h e a s y m p t o t i c o s c i l l a t i o n s
i n F i g u r e 4 , t h e q u a n t i t y e x p ( Q o - Q,)
r a p i d l y a p p r o a c h e s a c o n s t a n t v a l u e of 2 . 2 5 ,
2 2 in
( = 4a )
w h e r e a s f o r bl = -2.0
F i g u r e 5 w e see t h a t i t d o e s n o t a p p r o a c h a c o n s t a n t v a l u e b u t r a t h e r So t h i s s u g g e s t s o s c i l l a t e s a b o u t an a v e r a g e v a l u e of a r o u n d 9 . 0 . t h a t w e c o u l d a p p r o x i m a t e e x p ( Q - Q,) by c o n s t a n t v a l u e s u s i n g 2 . 2 5 0 When w e do t h i s , w e may t h e n f o r b l = - 0 . 5 and 9 . 0 f o r bl = - 2 . 0 . s o l v e E q . ( 1 5 ) , from w h i c h o n e may p r e d i c t t h e s o l i t o n s p e c t r u m a s a f u n c t i o n o f t i m e (Kaup and N e u b e r g e r ( 1 9 8 4 ) 1 c o m p a r i n g i t t o t h e a c t u a l s p e c t r u m . When w e d o t h i s f o r t h e s e t w o v a l u e s of bl, w e o b t a i n t h e r e s u l t s shown i n F i g u r e s 6 and 7 . I n F i g u r e 6, w e see t h a t f o r bl = - 0 . 5 , t h e a g r e e m e n t between t h e a p p r o x i m a t e s p e c t r u m ( s o l i d l i n e ) and t h e a c t u a l s p e c t r u m ( d a s h e d l i n e ) i s i n d e e d q u i t e good. T h i s is n o t t o o s u r p r i s i n g s i n c e o u r a p p r o x i m a t i o n f o r e x p ( Q o - Q1) w a s i n d e e d q u i t e e x c e l l e n t ( s e e F i g u r e 4 ) t o s t a r t w i t h . F o r bl = - 2 . 0 ,
F i g u r e 7 shows t h a t t h e a g r e e m e n t is now n o t q u i t e s o
g o d , a l t h o u g h t h e p r e d i c t e d spectrum d o e s c l o s e l y f o l l o w t h e a c t u a l spectrum. Presumably had w e i n c l u d e d t h e o s c i l l a t i o n s t h a t a r e p r e s e n t i n exp(Qo- Ql) (see F i g u r e 51, t h e n t h e a g r e e m e n t would h a v e b e e n much b e t t e r .
S t i l l , f r o m t a k i n g e x p ( Q o - Q,)
t o be a c o n s t a n t
172
D.J. Kaup
one d o e s o b t a i n a r e a s o n a b l e p r e d i c t i o n o f t h e a c t u a l
f o r bl = - 2 . 0 ,
s o l i t o n spectrum.
---Actual -Predicted
(dashed) (solid)
Actual (dashed)
//
_____--
---.
_____---
w -10
Z
I
-08
-06 -04 - 0 2
Figure 6 The s o l i t o n b i r t h r a t e when bl = - 0 . 5 a s p r e d i c t e d ( s o l i d
Figure 7 The s o l i t o n b i r t h r a t e when bl = - 2 . 0 a s p r e d i c t e d ( s o l i d
l i n e ) and a s it actually is (dashed l i n e ) .
l i n e ) and a s it a c t u a l l g i s (dashed l i n e ) .
SUMMARY I n c o n c l u s i o n , I hope t h a t I h a v e d e m o n s t r a t e d t o you t h a t t h e r e a r e s t i l l some i n t e r e s t i n g a s p e c t s o f i n t e g r a b l e s y s t e m s t o b e s t u d i e d . Here I have d i s c u s s e d f o r c e d i n t e g r a b l e s y s t e m s which a r e a form o f boundary v a l u e problems f o r n o n l i n e a r i n t e g r a b l e s y s t e m s . A s a c l o s i n g t h o u g h t , I should l i k e t o pose t h e q u e s t i o n of whether or n o t s u c h s y s t e m s a r e i n t e g r a b l e o r c o u l d t h e y d e m o n s t r a t e any of t h e p r o p e r t i e s of n o n i n t e g r a b l e s y s t e m s ? A t t h e moment I d o n ' t t h i n k t h a t For i n s t a n c e , a s demonstrated f o r t h e r e i s a c l e a r answer t o t h i s . t h e sine-Gordon s y s t e m , i f one f o r c e s i t w i t h i n a f i n i t e r e g i o n (Eilbeck e t a l . ( 1 9 8 1 ) ) chaos develops. But, a s I have i n d i c a t e d h e r e , g i v e n l o c a l i z e d f o r c i n g , t h e s y s t e m j u s t m i g h t w e l l be f u l l y integrable. C l e a r l y , more work n e e d s t o b e d o n e , a n d u n d o u b t e d l y i t w i l l be. ACKNOWLEDGMENTS The a u t h o r w i s h e s t o t h a n k D r . on t h e f o r c e d NLS e q u a t i o n .
P a u l Hansen f o r v a l u a b l e d i s c u s s i o n s
T h i s r e s e a r c h w a s s p o n s o r e d by t h e A i r F o r c e O f f i c e of S c i e n t i f i c R e s e a r c h , A i r F o r c e Systems Command, USAF, u n d e r G r a n t o r C o o p e r a t i v e Agreement Number AFOSR-82-0154. The U n i t e d S t a t e s Government i s
Forced Integrable Systems
173
authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints for governmental purposes. This material is based upon work supported by the National Science Foundation under Grant No. MCS-8202117. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation. REFERENCES Ablowitz, M.J. and Haberman, R., Resonantly coupled nonlinear evolution equations, J. Math. Phys. 16 (1975) 2301-2305. Ablowitz, M.J., Kaup, D.J., Newell, A.C. and Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249-315. Eilbeck, J.C., Lomdahl, P.S., and Newell, A.C., Chaos in the inhomogeneously driven sine-Gordon equation, Phys. Lett. 87A (1981) 1-4. Fokas, A.S. and Ablowitz, M.J., On the inverse scattering of the time-dependent Schroedinger Equation and the associated Kadomtsev-Petviashvili (I) Equation, INS Preprint # 2 2 , Clarkson College (to appear in Stud. Appl. Math., 1984). Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M., Plethods for solving the Korteweg-deVries equation, Phys. Rev. Lett. 19 (1967) 1095-1097. Gardner, C.S., Greene, J.M., Kruskal, M.D., and Miura, R.M., Kortewep-deVries equations and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974) 97-133. Holian, B.L. and Straub, G.K., Molecular dynamics of shock waves in one-dimensional chains, Phys. Rev. B18 (1978) 15931608. Holian, B.L., Flaschka, H. and McLaughlin, D.W., Shock waves in the Toda lattice: Analysis, Phys. Rev. A24 (1981) 25952623. Kaup, D.J., Coherent pulse propagation: A comparison of the complete solution with the McCall-Hahn theory and others, Phys. Rev. A16 (1977) 704-719. Kaup, D.J., Simple harmonic generation: An exact method of solution, Stud. Appl. Math. 59 (1978) 24-35. Kaup, D.J., The solution of the general initial value problem for the full three-dimensional three-wave resonant interaction, Physica 3D (1981) 374-395. Kaup, D.J., The method of solution for stimulated Raman Scattering and two-photon propagation, Physica 6D (1983) 143-154. Kaup, D.J., The forced Toda lattice: An example of an almost integrable system, INS Preprint #32, Clarkson College (to appear in J. Math. Phys., Jan. 1984).
174
D.J. Kaup
[14] Kaup, D.J. and Neuberger, D.H., The soliton birthrate in the forced Toda lattice, INS Preprint #33, Clarkson College (to appear in J. Math. Phys., Jan. 1984). [15] Lamb, G.L., Jr., Phase variation in coherent-optical-pulse propagation, Phys. Rev. Lett. 31 (1973) 196-199. [16] McCall, S.L. and Hahn, E.L., Self-induced transparency, Phys. Rev. 183 (1969) 457-485. [17] Steudel, H., Solutions in stimulated Raman Scattering and resonant two-photon propagation, Physica 6D (1983) 155-178. 1181
Toda, M., Waves in nonlinear lattice, Prog. Theor. Phys. Suppl. No. 45 (1970) 174-200.
Wave Phenomena: Modern Theory and Applications
C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Hollaml), 1984
175
INHOMOGENEOUS PLANE WAVES I N INCOMPRESSIBLE ELASTIC MATERIALS Michael Hayes Department of Mathematical Physics University College Dub1 i n
Gibbs b i v e c t o r s [l] a r e used t o g i v e a d e s c r i p t i o n of inhomogeneous plane waves i n a n i s o t r o p i c homogeneous I t i s shown
incompressible l i n e a r e l a s t i c m a t e r i a l s .
t h a t i f t h e slowness b i v e c t o r i s n o t i s o t r o p i c then t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y polari s e d wave propagates, and conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e corresponding wave i s c i r c u l a r l y polarised. INTRODUCTION
1.
This i s a sequel t o a previous paper
121 where t h e corresponding problem
of inhomogeneous plane waves i n compressible a n i s o t r o p i c e l a s t i c and compressi b l e i s o t r o p i c v i s c o e l a s t i c m a t e r i a l s was considered. t h a t instead of specifying a d i r e c t i o n
a s i n d e a l i n g with homogeneous
p l a n e waves, f o r inhomogeneous plane waves a p a i r
n
where
i s a u n i t v e c t o r and
m
The e s s e n t i a l i d e a i s
@, E)
is perpendicular t o it.
is specified,
Associated with
t h i s p a i r i s an e l l i p s e , t h e d i r e c t i o n a l e l l i p s e o f t h e b i v e c t o r
1. + IIJ. Thus i n s t e a d o f s p e c i f y i n g a d i r e c t i o n 3, a d i r e c t i o n a l e l l i p s e i s s p e c i f i e d . The corresponding slowness b i v e c t o r and
4
I t i s seen t h a t
are real.
is written Te"
S = Te [m
+
11.)where T
i s determined from t h e s e c u l a r
equation and t h e corresponding amplitude b i v e c t o r i s determined as an eigenb i v e c t o r of t h e a c o u s t i c a l t e n s o r . I t i s seen, i n t h e u s u a l way t h a t t h e r e a r e i n general j u s t two waves which may propagate f o r a given choice of t h e d i r e c t i o n a l e l l i p s e of t h e slowness bivector.
The corresponding amplitude b i v e c t o r s
arthogonal:
A.B =
0.
&,
(say) a r e mutually
Hence f o r a given d i r e c t i o n a l e l l i p s e of t h e slowness
b i v e c t o r t h e p a r t i c l e displacements corresponding t o t h e two waves l i e on p l a n e s which may not be orthogonal.
Also t h e e l l i p s e s corresponding t o
e i t h e r displacement when p r o j e c t e d onto t h e p l a n e of t h e d i r e c t i o n a l e l l i p s e
M. Hayes
of t h e slowness b i v e c t o r a r e s i m i l a r and s i m i l a r l y s i t u a t e d with major a x i s perpendicular t o t h e major a x i s of t h e e l l i p s e of t h e slowness b i v e c t o r . On t h e assumption t h a t t h e slowness b i v e c t o r i s not i s o t r o p i c , it i s shown
t h a t t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y p o l a r i s e d wave propagates.
Conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e
corresponding wave i s c i r c u l a r l y p o l a r i s e d .
Thus a s t h e r o o t s coalesce t h e
amplitude b i v e c t o r s merge and t h e o r t h o g o n a l i t y condition A.A- = -
0, so t h a t t h e wave i s c i r c u l a r l y p o l a r i s e d .
A.B
= 0
becomes
The condition t h a t t h e
r o o t s be double leads t o a q u a r t i c with complex c o e f f i c i e n t s .
If t h i s q u a r t i c
has a r e a l r o o t then a c i r c u l a r l y p o l a r i s e d inhomogeneous plane wave may propagate with a p p r o p r i a t e slowness.
The superposition of t h e displacements corresponding t o t h e p a i r of waves with given d i r e c t i o n a l e l l i p s e i s a l s o considered.
For motion p a r a l l e l t o t h e
plane o f t h e amplitude b i v e c t o r of e i t h e r wave it i s seen t h a t f o r e q u i d i s t a n t p o i n t s on c e r t a i n l i n e s t h e t o t a l displacement e l l i p s e s a r e similar and s i m i l a r l y s i t u a t e d with r e s p e c t t o each o t h e r and with r e s p e c t t o t h e p r o j e c t i o n of t h e slowness b i v e c t o r onto t h e plane of t h e amplitude b i v e c t o r .
52. -
EQUATIONS OF MOTION The c o n s t i t u t i v e equations a r e taken t o b e t.. = 11 =
u . .
' d i j k e Uk,,t
-p 6 . ij
(2.11
'
0 ,
1,1
where dijk9. =
djik,t
Here t h e s t r e s s e s a r e denoted by displacements by
dk,tij
=
=
dij,tk
*
t i j , t h e e l a s t i c c o n s t a n t s by
and t h e comma denotes d i f f e r e n t i a t i o n : u
u.
The summation convention i s used throughout.
Also
p
dijka.
,
the
~ E, auk/axk ~
.
is a scalar, the
h y d r o s t a t i c p r e s s u r e which i s t o be determined from t h e equations of motion and t h e boundary conditions.
Equation ( 2 . 2 ) expresses t h e f a c t t h a t a l l deformations
i n t h e body a r e i s o c h o r i c .
The equations of motion a r e given by 2 a u
at.. *
=
I
P
y
i
at
i n t h e absence o f body f o r c e s , where
(2.41
>
p
is t h e material density.
Inserting
177
lnhomogeneous Plane Waves
(2.1) i n t o ( 2 . 4 ) g i v e s dijki
13. -
k , a j - P , ~=
a 2u i / a t
P
2
.
ISHOMOGEiiEOUS I'LAiiE WAVES
Now i t i s assumed t h a t t h e displacements
a r i s e due t o t h e propagation ui of an i n f i n i t e t r a i n of inhomogeneous p l a n e waves i n t h e body. Thus
Here
A i s a bivector:
&
&+ +
=
The planes of constant phase a r e amplitude a r e = constant.
z-.~
I&-
2,
and so i s
s+.x=
t h e slowness b i v e c t o r .
c o n s t a n t , and t h e planes of constant
The period of t h e wave is
Equation
27r/w.
(3.1) d e s c r i b e s an i n f i n i t e t r a i n of e l l i p t i c a l l y p o l a r i s e d inhomogeneous
plane waves.
For f i x e d
x(= x*
say) t h e displacement v e c t o r
’1
e l l i p s e which i s s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of
A+ exp(- w
t h e e l l i p s e whose conjugate semi-diameters a r e
A-expc-
uZ-.f).
l i e s on an
&,
z-.~*) and
namely
A t any given time t h e displacement v e c t o r i s along one semi-
diameter of t h e e l l i p s e and t h e p a r t i c l e v e l o c i t y i s p a r a l l e l t o i t s conjugate semi-diameter.
t
As
t h e e l l i p s e i s from
i n c r e a s e s t h e sense i n which t h e p a r t i c l e moves along
&+ t o
A-.
The p l a n e of p o l a r i s a t i o n i s determined by
t h e plane of t h e e l l i p s e of t h e amplitude b i v e c t o r
Circularly polarised i f Linearly p o l a r i s e d i f
Te"
S = where
5
The slowness b i v e c t o r
T
vector.
g
angles t o problem.
and
$
(5+,):I
are real.
and t a k i n g
m.:
.
The wave i s
-A.A _
=
0 ;
(3.2)
&A&
=
0.
(3.33
When equation ( 3 . 2 ) is s a t i s f i e d t h e b i v e c t o r "nullt'.
A -
A -
i s s a i d t o be " i s o t r o p i c "
or
may be w r i t t e n
__
= 0, n.n = 1
,
(3.4)
By f i x i n g on a p a r t i c u l a r choice of t h e u n i t
5 t o be o f a r b i t r a r y [chosen) magnitude and a t r i g h t
1 , i t will be seen t h a t T and 4 a r e determined by an eigenvalue 5 and may be regarded a s t h e p r i n c i p a l axes of an e l l i p s e .
I 78
M. Hayes
Then
S/T
(z+,g-) i s
and
13i s
el$@ +
=
a p a i r of conjugate semi-diameters of t h i s e l l i p s e
t h u s a p a i r of conjugate semi-diameters of a similar and s i m i l a r l y
s i t u a t e d e l l i p s e whose major and minor axes a r e
(m .+):1
axes of t h e e l l i p s e of
times t h e major and minor
Also
S+
= T(cos$m
-
sin$n-),
S-
= T(sin@
+
cos@),
The angle
T
2 3 ; = T(cos2$m2 + s i n $)
12-1 =
,
z f .
T(sin2$m2 + cos $)
(3.5)
between t h e planes o f constant phase and t h e planes of
8
constant amplitude i s given by tan 8
=
[Z]
m
(3.6)
(m 2 - l ) c o s $ s i n $
The condition (2.2) gives
A.
2
=
0.
(3.7)
This means i n general t h a t t h e e l l i p s e of t h e amplitude b i v e c t o r
2
e l l i p s e o f t h e slowness b i v e c t o r l a r t o each o t h e r .
S
A
and t h e
may not l i e on p l a n e s which a r e perpendicu-
Also, t h e p r o j e c t i o n of e i t h e r
(5 say)
upon t h e plane o f
i s an e l l i p s e whose aspect r a t i o ( r a t i o of major t o minor a x i s ) i s equal t o
t h e aspect r a t i o of t h e e l l i p s e of t h e minor a x i s o f t h e e l l i p s e of
5
and whose major a x i s i s perpendicular t o
2.
A A. A =
Exceptionally it may happen that
i s a s c a l a r and a l s o t h a t polarised.
84. -
I t i s seen i n
2. 55
=
0
i s p a r a l l e l t o 2: A = as, where so t h a t t h e wave i s c i r c u l a r l y
a
that t h i s i s a p o s s i b i l i t y f o r i s o t r o p i c bodies.
THE PROPAGATION CONDITION
Here it i s assumed t h a t t h e d i r e c t i o n a l e l l i p s e of eigenvalue equation f o r t h e determination of that t h e r e a r e j u s t two non-zero eigenvalues.
Te"
5
i s given.
i s obtained.
The
I t i s seen
I f t h e s e a r e n o t equal it i s
shown t h a t t h e corresponding eigenbivectors a r e orthogonal. has t h e form (3.1), t h e n from the equations o f motion (2.5)
Now i f follows t h a t
p
must have a s i m i l a r form.
Thus write
it
179
lnhomogeneous Plane Waves
p
=
P expiw
(5. x
.
-t)
(4.1)
I n s e r t i n g t h e expressions (3.1) and (4.1) i n t o t h e equations of motion and P
using c3.5) t o e l i m i n a t e
2. 2 SO,
and assuming
leads t o
Using (3.4) t h i s may be w r i t t e n
(4.3) where
Q.
lk
*
m,
1 ~ k R ks j
=
'
%i
=m+ip,m .n = O , n .n = l .
S Thus, i f
* *
= d..
(4.41
a r e given, equation (4.3) i s an eigenvalue problem f o r t h e
determination of
and t h e corresponding eigenbivector
T, $
A.
The propagation condition (4.3) l e a d s t o t h e s e c u l a r equation d e t (pT-' f o r the determination o f det(Q)=0
e-21'
-
6ik
[Te'$)
= 0,
f o r given
*
*
s .s - * *
since det
6.i k)
1
y, y'
,
a r e a r b i t r a r y and 2.2 = 1,
1.11=
r.fl= 0 .
I n both c a s e s t h e
planes o f constant phase a r e orthogonal t o t h e planes of constant amplitude.
5
c Cp/p) homogeneous t r a n s v e r s e wave i s recovered. As
m
-+
a,
then from ( 5 . 9 ) ,
For given " +
15,and m
-+
a, m, A
s a t i s f y i n g (S.?),
the waves are c i r c u l a r l y p o l a r i s e d .
-f
y
12 +
or
and t h e usual
y'f,
y'
may be chosen s o t h a t
Thus, from equation (5.8),
take
y
given
bY (5.10)
Then (5.11) corresponds t o a c i r c u l a r l y p o l a r i s e d wave. c i r c l e i n t h e p l a n e spanned by
m
2
> 1,
take
yt
given by
m
and ;{
The displacement v e c t o r lies on a
+- (-2-1) 1 4E} m
.
Similarly, f o r
(5.12)
Then (5.13)
183
Inhomogeneous Plane Waves
corresponds t o a c i r c u l a r l y p o l a r i s e d wave, t h e c i r c l e of p o l a r i s a t i o n l y i n g i n a plane spanned by
86.
m m-m
1 and
1 4 IS}
{ -i (1 - -2)
m
STRUCTURE OF THE ACOUSTICAL TENSOR.
.
CIRCULARLY POLARISED WAVES.
Here t h e s t r u c t u r e o f t h e a c o u s t i c a l t e n s o r i s considered more f u l l y .
In
i s not i s o t r o p i c , it i s shown t h a t i f t h e s e c u l a r
particular,assuming t h a t
equation has a double r o o t then a c i r c u l a r l y p o l a r i s e d wave may propagate i n t h e material.
Also, it i s shown, assuming
S
.S 9
0, t h a t i f a c i r c u l a r l y p o l a r i s e d
wave propagates then t h e s e c u l a r equation has a double r o o t .
i s n o t i s o t r o p i c i s made, f o r , i n g e n e r a l , i n
The assumption t h a t
d e r i v i n g t h e form of t h e a c o u s t i c a l t e n s o r expressions o f t h e form enter.
I t was seen i n
o b t a i n a s o l u t i o n with
[s@SJ/(s.s)
f o r an i s o t r o p i c m a t e r i a l t h a t i t i s p o s s i b l e t o
35
5.2 =
0.
However, i n g e n e r a l , i f t h e d e t a i l a d s t r u c t u r e
of t h e e l a s t i c c o e f f i c i e n t s i s n o t given then it is e s s e n t i a l t o assume t h a t
S
i s not i s o t r o p i c . Now s i n c e any b i v e c t o r
where
a
may be w r i t t e n [l]
5
=
+
ib), a.b =
0,
i s a s c a l a r , i n general complex, we may, without l o s s of g e n e r a l i t y ,
write
where
T,
0,
respectively.
m
a r e r e a l and Then, assuming,
defined by equation (4.4),
i, 2.S
a r e u n i t v e c t o r s along t h e x and y axes 2 0 so that m $: 1, it i s seen t h a t Q,
may b e w r i t t e n * *
and has components given by
184
M. Hayes
Thus t h e matrix
4
has t h e form
where
One eigenvalue of
X1, AZ,
(0)
191
i s zero s i n c e
and t h e o t h e r two, denoted by
a r e given by
A,
t h e corresponding e i g e n b i v e c t o r s being
where
= 0
given by
tl, t2 a r e given by
Now tl t2 =
and hence
m2
-
1
,
(6.9)
185
Inhomogeneous Plane Waves
=
A . B-
(6.10)
0.
A1so
A.2 -
B.5
0,
=
=
(6.11)
0.
In t h e s p e c i a l case when (6.12) it follows from equation (6.4) t h a t
6
Cii)
=
(:i ; Q3i)
and t h e non-zero eigenvalues a r e bivectors
CO,
0, y)
and
Q,,
(1, m, 0)
and
(6.13)
,
a + imB
where
w i t h corresponding eigen-
is arbitrary.
y
The f i r s t of
t h e s e corresponds t o a l i n e a r l y p o l a r i s e d wave, t h e second t o an e l l i p t i c a l l y p o l a r i s e d wave.
(Recall
m
2
9 1).
CIRCULARLY POLARISED WAVES
F i n a l l y , the p o s s i b i l i t y of equa
eigenva ies i s considered.
The eigenvalues a r e equal provided L
a
(6.14)
so that (6.15) Then A1
=
From equation (6.4),
A2
=
(a + imB + Q3,)/2
(Q) i s now given by
.
(6.16)
M. Hayes
186
where the upper and lower s i g n s correspond t o t h e upper and lower s i g n s i n (6.15).
C(say) i s given by
The corresponding eigenbivector
(6.18) It is clear that
5.C =
0,
and thus the wave corresponding t o t h e double r o o t
is c i r c u l a r l y polarised. I t may be noted t h a t t h e c i r c l e s of p o l a r i s a t i o n corresponding t o t h e d i f f e r e n t s i g n s i n equation (6.15) and t h e r e f o r e a l s o i n (6.18), a r e g r e a t c i r c l e s i n t h e u n i t sphere.
They a r e described i n opposite senses.
most e a s i l y seen i n t h e s p e c i a l c a s e
_C = _i r i-k
are unit circles i n the
-
xz
if it i s p o s s i b l e
-
when m = 0.
This i s Then
plane, described i n opposite senses.
Also, r e t u r n i n g t o equation (.6.14), which i s t h e condition f o r double r o o t s ,
this condition may be w r i t t e n , using equation (6.5),
CQ,,
+
2 1 m Q12
-
m
2
Q2,
-
-
(1
2
as 2
m 1 Q33}
(6.19) This is a q u a r t i c i n real root for
m
m
with complex c o e f f i c i e n t s .
I f t h i s equation has a
then t h e r e a r e two corresponding c i r c u l a r l y p o l a r i s e d waves.
In general it w i l l not possess r e a l r o o t s f o r
m.
Assuming t h a t 5 i s not i s o t r o p i c it has been shown t h a t i f t h e a c o u s t i c a l tensor Q) has double r o o t s then t h e corresponding eigenbivector i s i s o t r o p i c and accordingly a c i r c u l a r l y p o l a r i s e d inhomogeneous wave may propagate. shown, assuming t h a t possesses an
Now, it i s
S
i s not i s o t r o p i c , t h a t i f t h e a c o u s t i c a l t e n s o r (Q) i s o t r o p i c eigenbivector then 9 has a double eigenvalue. n
Now from (4.4) t h e a c o u s t i c a l t e n s o r
9
has t h e form (6.20)
cij
=
6ij
-
SiSj
‘mSm
(6.21)
187
Inhomogeneous Plane Waves
and (6.22)
0
Without l o s s of g e n e r a l i t y , l e t
be given by
(6.23)
i s t o be an i s o t r o p i c eigenbivector of
Now
@)
so t h a t it s a t i s f i e s (6.24)
C6.25)
(6.26)
without l o s s of g e n e r a l i t y .
_A.S_
5
= 0,
I t i s assumed that
2.5
0
and since
may be assumed t o have t h e form (6.27)
where
i s some s c a l a r .
6
2.2
Now
i’ 6 -1
6
2
=
-1
-6
and
=
(
and
&:)
-1
62+1
0
( 6 * + l ) b -la -16c
- t b +(6’
+tb)
,
(6.28)
-16
2 (6 -1)b -if -6g
(6 -1)a - t b -6c -:(a
62,
=
2
(6 - 1 ) ~- t g -6h
+ l ) f -16g
-6(b + t f )
-6(c
+tg)
(6.29)
Then from (6.24) it follows t h a t
,
2tb
=
f - a
61
=
6(a + tb) - cc + 18)
,
(6.301
188
M. Hayes
and (6
-1)a - t b -6c
(S2 + l ) b - i a -16c
62G =
[-&:a On expanding r o o t Csince
17. -
+
-1
a -6g
(6
2 6 (a+2tb)+Ca+ib)-t6g
= 0,
2
-1)c-Ig-dh
-tc+(6
-
-t6Ca + ib)
tb)
IQ - e r l 191
( ~ 5+ l~) b
2
6(c + 1g)
i t i s seen t h a t t h e s e c u l a r equation has a zero
X
= 0 ) , and a double r o o t
given by ( 6 . 3 0 ) 2 .
SUPERPOSITION OF WAVE TRAINS WITH COMMON DIRECTIONAL ELLIPSE
In t h i s s e c t i o n t h e s u p e r p o s i t i o n of two wave t r a i n s with common d i r e c t i o n a1 e l l i p s e i s considered.
S
For given f i x e d
p a r a l l e l t o t h e plane of t h e e l l i p s e of
*
t h e motion i n any plane
is examined.
The displacement o f
a p a r t i c l e a r i s e s a s a l i n e a r combination of t h e two b a s i c displacements and (The sum of two b i v e c t o r s i s
hence, i n general, w i l l a l s o be e l l i p t i c a l . also a bivector.)
The e l l i p s e s f o r d i f f e r e n t p a r t i c l e s w i l l g e n e r a l l y d i f f e r .
However, it i s seen t h a t t h e e l l i p s e s a t c e r t a i n p o i n t s on a c e r t a i n l i n e are i d e n t i c a l both i n o r i e n t a t i o n a n d i n t e r m s of t h e lengths of t h e i r p r i n c i p a l axes. I t i s assumed t h a t
where
a,,
B,
a, b
5, A,
a r e given by
I t i s e a s i l y checked that
a r e assumed t o be r e a l .
A.B=
A.S-1 = B.S-2 = 0. Also Tle 161 , T2e1" of t h e s e c u l a r equation ( 4 . 5 ) f o r given 2 = corresponding eigenbivectors a r e The t o t a l displacement
E* =
A
and
a r e assumed t o be t h e s o l u t i o n s (ol/B)
u_"Csay) in t h e p l a n e of
A expiw C Tle'"(crx
+ iBy]
+ {B - (B.n) - - n) expiu{T2e"2(ax
+
ii,
and t h e
respectively.
-
&
may be w r i t t e n
t 1 + ’By)
-
t)
,
(7.21
189
lnhomogeneous Plane Waves
A
2 is t h e u n i t normal t o t h e p l a n e of
where
given by
- s i n d i + acossk
z =
2 2 2 l ( s i n S+a cos 6)
(7.3)
’
Now
Bcos6
-
bfisind[ii
=
IacosSj + sin8k) ]
,
(7.41
(1+B c o s S ) and
-s* -
=
(s”.fl)” 5 6
l[lL
-
’c0s6 ~ a c o s +~ si i n & & )I . 2 2 (1+B cos 6)
Notice t h a t t h e e l l i p s e s of t h e p r o j e c t i o n s of
and
5
(7.5)
upon t h e plane of
are s i m i l a r and s i m i l a r l y s i t u a t e d . Now
u*
may be w r i t t e n
+ a-’b@sin6{iL
-
(acossi
BCOSG
+
sin6k) 1 exp
IWP]
(1+6 cos 6)
,
(7.61
where (7.71 I t i s c l e a r t h a t f o r given
x, y,
t h e displacement
%* l i e s on an e l l i p s e .
The axes of t h e e l l i p s e w i l l v a r y from p o i n t t o p o i n t f o r p o i n t t o point. the ellipse a t
However, if exp(iwb) M
has p r i n c i p a l axes along
t h e ellipses at the points axes of t h e e l l i p s e of
5
M
u v a r i e s from
i s purely r e a l a t a point
i
M(say)
and (ctcosdi + sin@).
then
Thus
have p r i n c i p a l axes p a r a l l e l t o t h e p r i n c i p a l
and a l s o of course t o t h e p r i n c i p a l axes of t h e
e l l i p s e of t h e p r o j e c t i o n of
2.
Of course t h e s e e l l i p s e s a t t h e p o i n t s
M
will not be i d e n t i c a l but they do have t h a t one f e a t u r e i n common, t h a t t h e i r p r i n c i p a l axes a r e p a r a l l e l .
The p o i n t s
M
a r e any p o i n t s o n t h e e q u i d i s t a n t
190
M. Hayes
d
parallel lines
2
:
where
k,
:
wk[a[cos$)x $
-
p(sin$)y]
=
qs,
q
0,
...
+1, f 2 ,
(7.8)
a r e given by T e 2
‘42
- Tle
Consider t h e p o i n t s o f i n t e r s e c t i o h
32 with t h e l i n e s
:
.
(7.9) M*(say)
(asin+)x + ( ~ c o s $ ] y = The term
exp(iwu)
p o i n t s and has t h e same value a t each.
of a l i n e
7
constant
(7.10)
i s p u r e l y r e a l a t each of t h e s e
Thus f o r t h e s e p o i n t s
M
*
the
displacement e l l i p s e s a r e a l l i d e n t i c a l and are s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of t h e p r o j e c t i o n of
ACKNOWLEDGMENT.
5
upon t h e p l a n e of
A.
This work was supported by t h e National Board f o r Science
6 Technology under Grant 19/79.
191
Inhomogeneous Plane Waves
REFERENCES [l]
Gibbs, J.W. Elements of Yector Analysis, 1881, 1884 ( p r i v a t e l y p r i n t e d ) E pp 17 - 90, Vol. 2 , p a r t 2 , S c i e n t i f i c Papers, Dover Publications, New York, 1961.
[Z]
Hayes, M. Inhomogeneous Plane Waves. appear)
[3]
Synge, J . L . The Petrov C l a s s i f i c a t i o n of G r a v i t a t i o n a l F i e l d s . Dublin I n s t . f o r Adv. S t u d i e s , No. 15, Dublin, 1964.
Arch. R a t ' l Mech. Anal. 1984 ( t o
A,
Corn.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Mwdie (eds.)
0 Elsevier Science Publishers B.V. (North-Holland). 1984
193
CAN ACOUSTIC FOCUSING GENERATE TURBULENT SPOTS?
Gregory A. Kriegsmann and Edward L . R e i s s Department of E n g i n e e r i n g S c i e n c e s and Applied Mathematics Northwestern U n i v e r s i t y Evanston, I l l i n o i s 60201 USA
Experimental r e s u l t s , see e . g . [1-3] and r e f e r e n c e s g i v e n t h e r e , s t r o n g l y s u g g e s t t h a t c o h e r e n t f l u i d s t r u c t u r e s , which o r i g i n a t e i n t h e v i s c o u s s u b l a y e r s of w a l l t u r b u l e n t boundary l a y e r s , b u r s t and v i o l e n t l y e j e c t f l u i d i n t o t h e t u r b u l e n t regime of t h e boundary l a y e r . F u r t h e r m o r e , i t is b e l i e v e d t h a t t h i s b u r s t i n g produces most of t h e t u r b u l e n c e i n t h e boundary l a y e r . These e x p e r i m e n t s f u r t h e r s u g g e s t t h a t t h e c o h e r e n t s t r u c t u r e s , which a r e sometimes c a l l e d t u r b u l e n t s p o t s , are l o c a l i z e d r e g i o n s of i n t e n s e v o r t i c i t y t h a t p r o p a g a t e streamwise i n t h e sublayer. Because of t h e i r v o r t i c i t y , t h e s p o t s g r a d u a l l y l i f t as t h e y propa g a t e downstream where t h e y suddenly b u r s t and e j e c t t h e i r f l u i d upward. The p r e c i s e mechanism f o r t h e f o r m a t i o n of t h e s e s p o t s h a s n o t been c l e a r l y a s c e r t a i n e d , a l t h o u g h s e v e r a l c o n c e p t s have been proposed [l-31. I n t h i s p a p e r , w e d i s c u s s a p o s s i b l e new mechanism f o r t h e f o r m a t i o n of t h e s e s p o t s which w a s o r i g i n a l l y p r e s e n t e d i n Ref. [ 4 ] . S i n c e t h e b u r s t i n g p r o c e s s i s v i o l e n t , i t g e n e r a t e s a p r e s s u r e f i e l d w i t h a r e l a t i v e l y broad frequency spectrum, as e x p e r i m e n t a l r e s u l t s confirm. In [ 4 ] , we considered t h e b u r s t s a s localized s o u r c e s of energy which, among o t h e r i m p o r t a n t e f f e c t s , p r o p a g a t e a c o u s t i c waves i n t h e s u b l a y e r . The waves produced by t h e h i g h f r e q u e n c y components of t h e b u r s t s w e r e a n a l y z e d i n [ 4 ] u s i n g t h e methods of g e o m e t r i c a l a c o u s t i c s , see e . g . [ 5 ] . For s i m p l i c i t y , w e c o n s i d e r e d a two d i m e n s i o n a l a c o u s t i c p r o p a g a t i o n probl e m i n a s h e a r l a y e r which i s p a r a l l e l to t h e w a l l z = 0. The l a y e r a t t a i n e d a uniform f l o w p a r a l l e l t o t h e w a l l f o r z > D , where D i s t h e "depth" of t h e s u b l a y e r . The b u r s t w a s modeled a s a l i n e s o u r c e p e r p e n d i c u l a r t o t h e f l o w d i r e c t i o n , o r more g e n e r a l l y as a s o u r c e t h a t is l o c a l i z e d a l o n g t h i s l i n e , b u t i s spanwise uniform. Then w e l i n e a r i z e d t h e two d i m e n s i o n a l f l u i d dynamics e q u a t i o n s f o r i n v i s c i d , i s e n t r o p i c , c o m p r e s s i b l e f l o w a b o u t t h i s s h e a r flow t o o b t a i n t h e a c o u s t i c p r o p a g a t i o n e q u a t k o n s . The l i n e s o u r c e was assumed t o l i e i n s i d e t h e s h e a r l a y e r a t x = 0, z = z < D , as shown i n F i g u r e 1.
In o u r a n a l y s i s w e s t u d i e d t h e downstream a c o u s t i c f o c u s i n g o f t h e high-frequency components of t h e s o u r c e , by t h e flow g r a d i e n t of t h e s h e a r l a y e r . Thus, t h e s o l u t i o n o f t h e a c o u s t i c p r o p a g a t i o n problem w a s c o n s t r u c t e d i n t e r m s of t h e r a y s and w a v e f r o n t s of g e o m e t r i c a l a c o u s t i c s . The r a y s emanating from t h e s o u r c e , which are c a l l e d t h e primary r a y s , are shown i n F i g u r e 1 f o r a t y p i c a l set of f l u i d p a r a m e t e r s and s o u r c e l o c a t i o n . S i n c e t h e r a y s c a r r y energy ( n o i s e ) from t h e s o u r c e t o t h e f i e l d , i t i s s e e n t h a t some o f t h e energy i s e i t h e r r a d i a t e d d i r e c t l y t o t h e f a r f i e l d away from t h e w a l l , o r i t i s f i r s t p r o p a g a t e d upstream and t h e n " t u r n e d " by t h e f l o w t o t h e downstream f a r f i e l d . However, some of t h e energy i s t r a n s m i t t e d by t h e primary r a y s d i r e c t l y t o ' t h e w a l l and t h e n r e f l e c t e d a s shown i n F i g u r e 2 . Some of t h e r e f l e c t e d energy i s t h e n r a d i a t e d t o t h e f a r f i e l d . However, t h e r e are two one-parameter f a m i l i e s of r e f l e c t e d r a y s t h a t a r e t r a p p e d i n a c h a n n e l a d j a c e n t t o t h e w a l l which do n o t e s c a p e t o t h e f a r f i e l d a s i s i n d i c a t e d i n F i g u r e 3. Each r a y i n t h i s channel i s a l t e r n a t e l y r e f r a c t e d toward t h e w a l l by t h e s h e a r f l o w g r a d i e n t , and t h e n
194
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Kriegsmann and E. L. Reiss
reflected from the wall. These two families of rays then form an infinite sequence of caustics downstream of the source, as shown in Figure 3. The amplitude of the acoustic field is calculated by the method of geometrical acoustics along the rays and is given in 1 4 1 . However, as is well known, the geometrical acoustics approximation for the amplitude is unbounded on the caustics, s o that this approximation is invalid on and near the caustics. A boundary layer method, similar to the one developed in [ 6 ] for the Helmholtz equation, is then employed to determine the acoustic amplitude on and near the caustics. It is shown in [ 4 ] that the amplitude of the field on and near the caustics is O(k2l3) larger than the geometrical acoustics field away from the caustics for large k. Here k is the dimensionless wave number that is defined by k Z wD/c, where c is the ambient sound velocity in the shear layer, and w is the circular frequency of the time periodic component of the source. More significantly, the vorticity of the acoustic field on and near the caustics is O ( k 7 / 6 ) larger than the vorticity of the geometrical acoustics field away from the caustics. Thus, our analysis shows that a localized disturbance (the line source, which models a bursting spot) of unit amplitude can generate an array of narrow regions downstream of the source and surrounding each caustic in which the vorticity intensity is extremely large in the acoustic approximation. These local regions of intense vorticity can serve as nuclei for the "birth" of future turbulent spots. That is, the process of turbulent spot formation and hence of turbulence production in the boundary layer may be a self-induced phenomenon - old bursts give birth to new spots. To substantiate this conjecture it is necessary to consider the three-dimensional propagation from a point source in the shear flow, possibly including the low frequency spanwise vorticity in the sublayer flow. Since the amplitudes on the caustics are unbounded as k + m , this suggests that a nonlinear analysis is required to describe the further evolution of the local vortex nuclei of the acoustic theory. We are presently studying the possibility of eliminating or diminishing the caustic fields by appropriate wall motions which may be induced by driven or compliant walls. Furthermore the formation and bursting of the turbulent spots is closely related to the noise field emitted by turbulent boundary layer fluctuations of a moving vehicle submerged in a fluid. Thus, any mechanism for reducing the bursting of turbulent spots may substantially reduce the viscous drag of the vehicle and its concomitant radiated noise. Finally, if the sound source is interpreted as noise produced by components, such as propulsors or local vibrations of the vehicle structure, then the results presented in [ 4 ] give estimates of the near and farfield noise resulting from the vehicle& motion. The large self-noise corresponding to the caustic field, which is close to the wall, may mask weak signals received from distant sources. Then knowing the caustic field it can be filtered from the total received signal to determine the weak incident signal. ACKNOWLEDGEMENTS This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR 80-0016A, by the National Science Foundation under Grant No. MCS8300578 and the Office of Naval Research under Contract No. N00014-83-C-0518,
195
Can Acoustic Focusing Generate Turbulent Spots?
REFERENCES
1.
Offen, G . R . and K l i n e , S . J., A proposed model of t h e b u r s t i n g p r o c e s s i n t u r b u l e n t boundary l a y e r s , J. F l u i d Mech. 70 (1975) 209-228.
2.
B u s h n e l l , D . M . , Hefner, J . N . and Ash, R. L . , E f f e c t of compliant w a l l motion on t u r b u l e n t boundary l a y e r s , Phys. of F l u i d s 20 (1977) S 3 1 - S 4 8 .
3.
W i l l m a r t h , W. W. and Bogar, T . J . , Survey and new measurements of t u r b u l e n t s t r u c t u r e s n e a r t h e w a l l , Phys. of F l u i d s 20 (1977) S9-S21.
4.
Kriegsmann, G . A. and Reiss, E . L . , A c o u s t i c wave p r o p a g a t i o n i n a s h e a r f l o w , J. Acoust. SOC. of A m e r . , i n p r e s s .
5.
K e l l e r , J . B., L e w i s , R. M. and S e c k l o r , B. D., Asymptotic s o l u t i o n s of some d i f f r a c t i o n problems, Comm. on Pure and Appl. Mech. 9 (1956) 207-265.
6.
Buchal, R . N . and K e l l e r , J . B . , Boundary l a y e r problems i n d i f f r a c t i o n t h e o r y , Comm. on Pure and Appl. Math. 1 3 (1960) 85-114.
FIGURE CAPTIONS
F i g u r e 1:
The primary r a y s emanating from t h e s o u r c e p o i n t x = 0, z = z , f o r a t y p i c a l s u b s o n i c Mach number o f the s h e a r flow.
Figure 2:
Reflected rays.
Figure 3 :
R e f l e c t e d r a y s i n w a l l c h a n n e l t o show c a u s t i c f o r m a t i o n .
196
G.A. Kriegsmann and E. L. Reiss
Can Acoustic Focusing Generate Turbulent Spots?
197
WAVE PHENOMENA: MODERN THEORY AND APPLICATIONS
Wave Phenomena: Modern T h e o r y and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
199
NONLINEAR WAVES I N THE PELLET FUSION PROCESS
V.
J . E r v i n , W.
F.
and
Ames
E.
Adams
I n s t i t u t f c r Angewandte Mathematik Universitat Karlsruhe 75 Karlsruhe, F e d e r a l R e p u b l i c of Germany
School o f Mathematics G e o r g i a I n s t i t u t e o f Technology A t l a n t a , GA 3 0 3 3 2 U.S.A.
A g a s dynamic model o f t h e p e l l e t f u s i o n p r o c e s s
having a t i m e - i n v a r i a n t s o u r c e t e r m is s t u d i e d Some e x a c t s o l u t i o n s by means of g r o u p a n a l y s i s . of t h i s n o n l i n e a r s y s t e m a r e c o n s t r u c t e d f o r The s p e c i f i c ( p h y s i c a l ) v a l u e s of p a r a m e t e r s . d e v e l o p m e n t o f m u l t i p l e s h o c k waves i s demonstrated i n several cases analytically. Additional numerical r e s u l t s i l l u s t r a t e t h e e v o l u t i o n of s i n g u l a r i t i e s .
1.
INTRODUCTION
In t h e p e l l e t fusion process, a spherical p e l l e t is f i r e d i n t o a c o n t a i n m e n t chamber and t h e n bombarded w i t h p u l s e s of l a s e r energy.
When h i t by a l a s e r p u l s e , a s h o c k wave p r o p a g a t e s t h r o u g h
t h e p e l l e t a s t h e temperature rises.
[A t y p i c a l p e l l e t configura-
t i o n i s s k e t c h e d below.] 2 2 5 . 7 q @,
c7-0 67.2 q
\\
L PA
c 4 -,
’B,
;%, s
\
void
\\
II Q3*““8%
,I
\ \
/
\
%
I
I
I
/
\
I \
/ \ I
F i g . 1.
A Typical P e l l e t Configuration.
The o u t e r m o s t l a y e r of t h e p e l l e t , c o n s i s t i n g of l e a d , p r e v e n t s t h e o u t w a r d e x p a n s i o n of t h e i n n e r l a y e r s : h e n c e , a s t h e
Lithium-Lead
l a y e r h e a t s u p , it e x p a n d s inward f o r c i n g t h e DT f u e l t o w a r d t h e center. The s h o c k wave, a f t e r p a s s i n g t h r o u g h t o t h e i n n e r l a y e r , i s reflected.
O n s t r i k i n g t h e o u t e r s u r f a c e of t h e p e l l e t ,
another
p u l s e a r r i v e s and c o n s t r u c t i v e l y a d d s t o t h e s h o c k wave a l r e a d y i n
200
V.J. Ervin et a/.
the pellet. The p r o c e s s c o n t i n u e s u n t i l t h e c o n c e n t r a t i o n of f u e l and t h e t e m p e r a t u r e , a t t h e c e n t e r , i s h i g h enough f o r t h e f u s i o n r e a c t i o n
t o t a k e place. 2.
THE MODEL
I n t h i s a n a l y s i s , i n s t e a d of m o d e l i n g p u l s e s o f e n e r g y , w e assume t h a t t h e p e l l e t is s u b j e c t t o a t i m e - i n v a r i a n t e n e r g y s o u r c e n g i v e n by B0r
.
Using t h e p r i n c i p l e s o f f l u i d ( g a s ) d y n a m i c s , and u t i l i z i n g t h e s p h e r i c a l symmetry of t h e p r o b l e m , t h e e q u a t i o n s d e s c r i b i n g t h e process are:
aU +
au +
at
:E
-
ar
*+
RTp-'
ar
-1
aT
+ u ar + z ( y - 1 ) T u r
+
R aT =
ar
(y-l)T
o
2U at =
B 0 rn
where p ( r , t ) i s t h e d e n s i t y i n s i d e t h e p e l l e t , u ( r , t ) i s t h e v e l o c i t y of t h e s h o c k wave, T ( r , t ) is t h e temperature i n s i d e t h e p e l l e t ,
r
is the radial variable,
t
is time,
R
i s t h e g a s c o n s t a n t for l e a d , and
y, B0
a r e t h e r a t i o o f s p e c i f i c h e a t s and s o u r c e s t r e n g t h , respectively.
The b o u n d a r y / i n i t i a l p(r,O)
c o n d i t i o n s are: u(r,O) = 0 ,
= p0(r),
T(r,O)
= To(')
and u ( 0 , t ) = 0 3.
WAVE VELOCITIES
The wave v e l o c i t i e s f o r e q u a t i o n s ( 1 ) - ( 3 ) a r e
c1 = u,
C2
= u-(yRT) 1 / 2
,
c 3 = u+(yRT)1 / 2
.
Thus f o r y , R , T # 0 t h e s y s t e m i s TOTALLY HYPERBOLIC.
4.
APPLICATION O F THE DILATATION GROUP ( A m e s [l], Bluman and C o l e [21, O v s i a n n i k o v I311 Assume p =
asp,
a-
s-
E-
x-
u = a u, T = a T, r = a r, t = a t ,
201
Nonlinear Waves in the Pellet Fusion Process
with parameter a .
The invariants of the group are
E Z O ( 1 ) - ( 3 ) imply that
The invariance of the P.D.E.’s
E = -n+1 3
- -_2- (n+l) A
-
2-n 3
Following from the transformation described by ( 4 ) and subject to the conditions (5)- ( 7 ) , equations (1 - (3) become f’
h’
+
(2+@+E)fg E
E
f’ h g’ - A n(gg’ + R E f
+
Rh’
n(gh’ + (y-1)g’h)
+
( -6+
-
2(y-1) + E a (y-1))gh = O o ,
(10)
where f = f(q), g = g ( n ) , h = h(n). NOW, let us investigate the boundary/initial conditions under the dilatation transformation (i) p(r,O) = po(r) From ( 4 ) p (r,t) = r"IEf ( n ) , so that p(r,O) = ralEf ( 0 ) . a Hence assuming that p 0 (r) = Ar gives a/& = a and f ( 0 ) = A. (ii) u(r,O) = 0 Since u(r,t) = ralEg(q) it follows that u(r,O) = r’"g(O), whereupon g(0) = 0. (iii) T(r,O) = To(r) then T(r,O) = r2/3 (n+l)h (0) Hence Since T(r,t) = r"’h(q)
.
with To(r) = Br 2/3(n+1) it follows h(0) = B. (iv) u(0,t) = 0 Recall u(r,t) = rBiEg(q) = r R / E 9(=)t = r (n+l)/3g (tr(n-2)/3) r whereupon lim r(n+1)/39(t,r(n-2)/3) = r+O > 2. provided n -
202
V.J. Ervin et al.
CONSTRUCTION O F EXACT SOLUTIONS
5.
Now
C o n s i d e r t h e c a s e when n = 2 .
B = 1 n = 2 implies -
-
6 -
'
E
= 2
;
-
E q u a t i o n s ( 8 ) (10) t h e n become
+
f'
+
g'
(3
+
g2
+
h'
+
E)fg = 0
(2
+
E)Rh = 0
(3y-l)gh
Bo
=
w i t h i n i t i a l c o n d i t i o n s f(0) = A , g ( 0 ) = 0 and h ( 0 ) = B . From (11) it f o l l o w s t h a t r)
f ( r ) ) = A e x p [ ( 3 + ):
g(SIdS1
’J 0
From ( 1 2 ) h = - 4 ' (2
2
+
+
g E)R
whereupon
-
=
h’
g" (2
Equation
+
+
2gg' F)R
(12) gives
gl ( 0 ) = - ( 2 + ~ ) R B .
(17)
S u b s t i t u t i n g ( 1 5 ) and (16) i n t o ( 1 3 ) y i e l d s g"
+
( 3 Y + l ) g ' g + ( 3 y - l ) g 3 = -8,(2+:)R,
with i n i t i a l conditions
g(0) = 0
,
g ' ( 0 ) = -(2
(18)
.
+
(19)
The change o f v a r i a b l e s'
= ugs,
(20)
gives = lls(g'
s"
2 + IJg ) ,
(21)
and = IJs(g"
s'"
I n p a r t i c u l a r , f o r y = 5/3
v
= 2,
2 3
+ 3 u g ' g + LJ g 1 .
(22)
( c o r r e s p o n d i n g t o a monatomic gas) and
e q u a t i o n ( 1 8 ) becomes s"'
+ 280 ( 2 +
:)Rs
= 0,
w i t h i n i t i a l c o n d i t i o n s s (0) = 1, s ' ( 0 ) = 0 , s" ( 0 ) = - 2 ( 2 The s o l u t i o n of
( 2 3 ) and ( 2 4 ) i s
s(r)) =
where
cle -"
+
z r
e ( F ' 2 ) r ) ( c2c o s Fr) 2
+
+
(23 1 9)RB. (24)
c3 s i n Fr)), ( 2 5 ) 2
203
Nonlinear Waves in the Pellet Fusion Process (r
F = ( 2 f i O ( 2 + ,)R)
1/3
and c 1 , c 2 , c 3 a r e g i v e n by
From e q u a t i o n ( 2 0 ) ri
s(n) = e x p [ 2
.f
g(S)dSI
0
and
where c4
For y =
2 -, 3
=d-
w e choose 1~
and
Cp
= Tan -1 c2
1 i n equations
=
c3 (20)-(22).
(30) Equation
(18) t h e n becomes s"'
(n) +
a0(2
with i n i t i a l conditions s ( 0 )
=
+
(31)
Z)Rs(n) = 0,
1, s ' ( 0 )
=
0 , s" ( 0 ) = - ( 2 t
E ) R B .( 3 2 )
With G:?-,G#O,
(33)
t h e s o l u t i o n f o r s ( q ) becomes
where t h e c o n s t a n t s c 1 , c 2 , c 3 a r e g i v e n by
where
Jd-
4 = Tan -1 c2 and 2 c4 = C, I n t h e case y = 1 / 3 , e q u a t i o n ( 1 8 ) becomes
.
'
g " ( n ) + 2g' ( n ) g ( q ) = - B 0 ( 2 + F ) R . Integrating, we obtain t h a t f i r s t order d i f f e r e n t i a l equation g'(n) + (g(q))' = -B0(2
+ Fa )
R ~+ c .
(37) (38)
204
V.J. Ervin et a/.
Applying t h e i n i t i a l c o n d i t i o n s g ( 0 )
+
we o b t a i n t h a t c = - ( 2
:)RB.
0 and g ’ ( 0 ) = - ( 2
=
+
a)RB, E
t h e s e i n t o equation (38)
Substituting
there results g ’ (II)
+
(g(II)l2= -D
0
(2
+
E)R(n
+
B r)
(39)
0
Upon a p p l y i n g e q u a t i o n s ( 2 0 ) and (21) w i t h il = 1, e q u a t i o n ( 3 9 )
becomes a
+ B0(2 + r ) R ( r i
S”(II)
B
+ ~ ) s ( I I ) = 0,
(40)
0
w i t h i n i t i a l c o n d i t i o n s s ( 0 ) = 1, s ’ ( 0 ) = 0. Making t h e change of v a r i a b l e
5
+
= q
becomes
B
,
B , equation (40) 60
B s ’ (--) a0
= 0 (42) 60 E q u a t i o n ( 4 1 ) i s a B e s s e l e q u a t i o n of o r d e r 1/3, whose s o l u t i o n i s
s(-)
=
1
s ( C ) = C1’2[clJl/3(fi(C))
+ C~J-~,,~(O(E))I,
(43)
where
For t h e a p p l i c a t i o n of i n i t i a l c o n d i t i o n s , it i s c o n v e n i e n t t o s e t
no
=
=
(6 ( 2 + Ia) R ) 1/2 2 ( B ) 3 / 2
(46)
0 60 = 1 implies C ~ J ~ , , ~ + (CQ~~J )- ~ , ~= ( (-B-) Q 6 0 ~1 /)2
60
Then s ( - B) and
(47)
BO B
s ’ (--)
= 0 gives
(48)
BO [C1J1/3 (Qo)+C2J-1/3
( n o )I
+
~ ~ ~ O O [ C ~(RO)+C2J-1/3 J ; / ~
(no)I
= 0-
U s i n g e q u a t i o n ( 4 6 ) and ( 4 7 ) e q u a t i o n ( 4 8 ) r e d u c e s t o C1Ji/3(Qo)
+
C2J-1/3(Q
Hence c1 and c 2 a r e g i v e n by
0) =
-
1 60 1 / 2 3 R 0 (--)B
.
(49)
205
Since t h e functions J
(n) and
( Q ) a r e l i n e a r l y independent 1/3 s o l u t i o n s of ( 4 1 ) t h e n t h e Wronskian ( J1 / 3 ( Q ) r ~ - 1 / 3(a)) R=fi 0. 0 Hence c1 and c2 a r e u n i q u e l y d e t e r m i n e d by ( 5 0 ) . Using t h e r e l a t i o n s h i p
113
(5)
=
1
1
- Jr+,(z)
Jr-l(Z)
S’
J-
5 - 1 / 2 {C1[JlI3
(\))
+
(51)
= 2Jk(Z)
+ 3 RJ-
z
2/3
3 (0) - 7 RJ
4/3 (n) (52)
3 + c2[J(Q) + 1/3 2 and t h e r e f o r e g(u) =
1
q(n +
-e/3
-
(fi)
- 2 3
+ 3 RJ-2,3(R)
-1(c1[J1/3(h2) -)a 0
QJ
c1JlI3
(a) +
3 RJ (a) I } 2 2/3 f2J4/,(Q)l
c2J-1,3 ( Q ) (53)
where
6.
SINGULARITIES OF g ( q ) I n e a c h of t h e cases p r e s e n t e d a b o v e ,
f u n c t i o n g ( n ) w a s g i v e n by s (q)/ps(q). whenever s(n) = 0 .
(y = 5/3,
2/3,
1/31,
the
Thus g ( n ) i s s i n g u l a r
The s i n g u l a r i t i e s o f g ( q ) may b e i n t e r p r e t e d a s
t h e “ t i m e “ when t h e f u s i o n p r o c e s s o c c u r s .
The p r o c e d u r e would be
v i r t u a l l y u s e l e s s i f , a f t e r two atoms f u s e d t o g e t h e r , t h e p r o c e s s What w e would l i k e t o see happen i s a c o n t i n u a t i o n of
stopped.
f u s i o n a f t e r an i n i t i a l f u s i o n h a s t a k e n p l a c e . T h i s l e a d s u s t o i n v e s t i g a t e t h e z e r o s of s(n).
For y
=
5 3
F = (2a0(2
y = 2
+
s(n) = cle-Fn CY
+
;)R)
s(q) = cle -GQ G =
( a0 ( 2
a
+
+ -)R)
c 4 e ( F / 2 ) Q s i n (J5 T
~n + 4);
1/3 c4e(G/2)Qsin(q
+
4);
1/3 t
and f o r
Y
=
1
3
s(n) = ( n +
&)1’2[~lJl,,3(Q(n))
+
C2J-1/3(f2(Q))I;
206
V.J. Ervin et a/. =
Q(n)
(B0(2
+ *),'I2
$(q
E
+
a) B 3/2 . 0
I n t h e f i r s t two c a s e s i t i s o b v i o u s t h a t s ( n ) w i l l have i n f i n i t e l y A c t u a l l y a s q becomes l a r g e t h e s e z e r o s w i l l
many z e r o s f o r q > 0 .
o c c u r a l m o s t p e r i o d i c a l l y w i t h p e r i o d s 4 n / n F and 4 n / f i G tively.
respec-
F o r y = 1 / 3 l e t u s examine e q u a t i o n ( 4 0 )
U n l i k e t h e p r e v i o u s t w o cases, i n which t h e z e r o s o c c u r almost periodically, Theorem'' [ c f . 4 1 Theorem: U"
+
the zeros of
S(Q)
( f o r y = 1/3)
o c c u r "more
To see t h i s w e a p p l y t h e " I n t e r l a c i n g of Zeros
frequently."
.
I f f ( x ) and g ( x ) a r e l i n e a r l y i n d e p e n d e n t s o l u t i o n s o f
p(x)u'
+
g ( x ) u = 0,
t h e n f ( x ) must v a n i s h a t o n e p o i n t between
any two s u c c e s s i v e z e r o s o f g ( x ) .
I n o t h e r words, t h e z e r o s of f ( x )
and g ( x ) o c c u r a l t e r n a t i v e l y . Observe t h a t
i s a s o l u t i o n t o equation ( 4 1 ) . a r e s e p a r a t e d by a p p r o x i m a t e l y n.
i n c r e a s e s by T , d e c r e a s e s .
5 t h e z e r o s o f J 1 / 3 ( ') A s q i n c r e a s e s t h e amount by
For l a r g e
Hence t h e z e r o s of w ( q ) o c c u r "more
S i n c e t h e z e r o s of s ( n ) and w ( q ) a r e i t f o l l o w s t h a t t h e z e r o s of s ( q ) must a l s o o c c u r "more f r e q u e n t l y " a s rl i n c r e a s e s .
f r e q u e n t l y " as rl i n c r e a s e s . interlaced,
7.
NUMERICAL RESULTS
I n a d d i t i o n t o t h e a n a l y t i c r e s u l t s , p r e v i o u s l y d i s c u s s e d , some numerical s o l u t i o n s f o r t h e f i r s t o r d e r system ( e q u a t i o n s ( 8 ) - ( 1 0 ) ) have b e e n c a l c u l a t e d on t h e G e o r g i a Tech CYBER u s i n g t h e DGEAR package.
The g r a p h s w e r e made by a TERAK micro-computer.
F i g u r e 2 p r e s e n t s t h e s o l u t i o n s for f , g , and h w i t h i n i t i a l d a t a f ( 0 ) = 5.10, and p,
g ( 0 ) = 0 and h ( 0 ) = 0 . 0 1 ,
for n
=
2,
CX/E
= 2
= 3.47.
F i g u r e 3 p r e s e n t s t h e s o l u t i o n s f o r f , g , and h w i t h i n i t i a l d a t a f ( 0 ) = 8.16,
g ( 0 ) = 0 , and h ( 0 ) = 0 . 0 1 ,
for n = 2,
CX/E
= 5
and O o = 3 . 1 0 . The l a s t F i g u r e , 4 ,
r e s u l t s from t h e same i n i t i a l d a t a a s t h a t
f o r F i g u r e 2 b u t w i t h n = 7, a / ~= 2 a n d (3, = 5.95. The m u l t i p l e p e a k s o f t h i s f i g u r e a r e n o t o f p h y s i c a l o r i g i n . They w e r e i n t r o -
207
Nonlinear Waves in the Pellet Fusion Process
d u c e d by t h e TERAK c o m p u t e r a s i t a t t e m p t e d t o " s t e p " o v e r t h e singularity. REMARKS
The s u b s t i t u t i o n d e s c r i b e d i n e q u a t i o n ( 2 0 ) e n a b l e d u s t o
r e w r i t e t h e n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s ( 1 8 ) and ( 3 9 ) a s l i n e a r e q u a t i o n s o f one h i g h e r o r d e r . U n i q u e n e s s of t h e s o l u t i o n s f o r p r o b l e m s ( 2 3 ) - ( 2 4 )
,
( 3 1 )- ( 3 2 ) ,
a n d ( 4 1 ) - ( 4 2 ) i m p l i e s u n i q u e s o l u t i o n s o f e q u a t i o n s ( 1 8 )- ( 1 9 ) f o r t h e cases y = 5 / 3 , I n t h e case n = 2 , p ( r , t ) =r"/'f(t),
2/3,
and 1/3.
t which i m p l i e s t h e s e p a r a b l e s o l u t i o n 2 u ( r , t ) = r g ( t ) , and T ( r , t ) = r h ( t ) . T-
=
REFERENCES
1.
A m e s , W. F . , N o n l i n e a r P a r t i a l D i f f e r e n t i a l E q u a t i o n s i n E n g i n e e r i n g , Academic P r e s s , N e w York, V o l . I ( 1 9 6 5 ) , Vol. I1 (1972).
2.
Bluman, G. W . a n d C o l e , J . D . , S i m i l a r i t y Methods f o r D i f f e r e n t i a l E q u a t i o n s , S p r i n g e r , 1974.
3.
O v s i a n n i k o v , L . V . , Group A n a l y s i s of D i f f e r e n t i a l E q u a t i o n s , ( W . F. Ames, E d . ) Academic P r e s s , N e w York, 1982.
4.
B i r k h o f f , G. a n d R o t a , G . - C . , Ordinary D i f f e r e n t i a l Equations ( 3 r d E d . ) , W i l e y , N e w York, 1 9 7 8 ( p . 3 8 ) .
V.J. Ervin et al.
208
150
Figure 2.
Nonlinear Waves in the Pellet Fusion Process
Figure 3 .
209
210
V.J. Ervin et al.
F i g u r e 4.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
21 1
ENTROPY PRINCIPLE, SYMMETRIC HYPERBOLIC SYSTEMS AND SHOCK WAVES
Tommaso Ruggeri D i p a r t i m e n t o d i Matematica - U n i v e r s i t a d i Bologna V i a V a l l e s c u r a 2 , 40136 Bologna, I t a l y
INTRODUCTION The o b j e c t o f t h i s p a p e r i s t o g i v e r e l a t i o n s between two arguments a p p e a r i n g i n t h e l i t e r a t u r e w i t h d i f f e r e n t c o n c l u s i o n s and a p p a r e n t l y w i t h o u t common features. The former argument i s r e l a t e d t o a g e n e r a l q u e s t i o n of s y s t e m s o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f h y p e r b o l i c t y p e ; t h e l a t t e r c o n c e r n s t h e r o l e of t h e " e n t r o p y p r i n c i p l e " i n continuum t h e o r i e s . W e s t a r t with a b r i e f survey of these s u b j e c t s . W e c o n s i d e r a g e n e r i c q u a s i - l i n e a r f i r s t o r d e r s y s t e m of t h e t y p e :
AO(U)
au/at +
A'(%)
ayax,
=
gcg
+
-t
f o r t h e unknown N-column v e c t o r u ( t , x ) : u : ( u ~ , u ~ , . . . , u; ~x ) ~b e l o n g i n g t o are r e a l N X N m a t r i c e s which are f u n c t i o n s of u , t h e source ; Ao , A' term f ( 5 ) i s a l s o a v e c t o r of RN , T i n d i c a t e s t h e t r a n s p o s e and t h e E i n s t e i n c o n v e n t i o n on t h e i n d e x i s u s e d . By i d e n t i f y i n g t h e t i m e t w i t h xo , i t i s p o s s i b l e t o w r i t e t h e s y s t e m i n t h e a b b r e v i a t e d form: Rn
D e finition of hyperb o li cit y : The s y s t e m (1) i s h y p e r b o l i c i n t h e t - d i r e c t i o n , i)
d e t A"
#
0
-f
ii) Vv
E
Rn
:
if:
,
11 +v 11
= 1
,
t h e f o l l o w i n g e i g e n v a l u e problem
has only r e a l proper values h ( c h a r a c t e r i s t i c v e l o c i t i e s ) and N l i n e a r l y i n d e p e n d e n t e i g e n v e c t o r s r (Av = AiVi)
.
D e f i n i t i o n o f c o n s e r v a t i v e system: The s y s t e m (1) i s s a i d t o b e " c o n s e r v a t i v e " ( o r b e t t e r : a s y s t e m o f b a l a n c e l a w s ) , i f t h e r e e x i s t f o u r v e c t o r s Fa of RN , such t h a t t h e m a t r i c e s A" are gradients of w i t h r e s p e c t t o a s u i t a b l e f i e l d u : Aa = , i . e . e x h i b i t t h e form:
.ca
ac"/ag
T. Ruggeri
212
W e o b s e r v e t h a t a s y s t e m of t y p e ( 2 ) is v e r y common i n Mathematical P h y s i c s , b e c a u s e i t r e p r e s e n t s t h e l o c a l form (when d i f f e r e n t i a b i l i t y c o n d i t i o n s h o l d ) o f an i n t e g r a l b a l a n c e . O n t h e o t h e r hand i t i s w e l l known t h a t f o r t h i s k i n d o f system i t i s p o s s i b l e t o c o n s i d e r a more g e n e r a l c l a s s of s o l u t i o n s : " t h e weak Shock waves, t h a t a r e v e r y s o l u t i o n s " , f o r which c o n t i n u i t y i s n o t n e c e s s a r y . i m p o r t a n t i n some p h y s i c a l p r o b l e m s , b e l o n g t o t h i s c l a s s .
D e f i n i t i o n of symmetric h y p e r b o l i c s y s t e m s : A system o f t y p e (1) i s s a i d t o b e symmetric i f
is positive definite; i) A" = (Aa)T; ii) A" (from l i n e a r a l g e b r a any symmetric s y s t e m i s a u t o m a t i c a l l y h y p e r b o l i c ) The symmetric s y s t e m p l a y s an i m p o r t a n t r o l e c o n c e r n i n g t h e w e l l - p o s i n g o f t h e Cauchy problem; i n f a c t t h e r e e x i s t some theorems e n s u r i n g e x i s t e n c e , u n i q u e n e s s For i n s t a n c e w e quote t h e and c o n t i n u o u s dependence f o r t h e l o c a l Cauchy problem. r e s u l t due t o F i s c h e r and Marsden 111 i n 1 9 7 2 : "Any symmetric s y s t e m , t h e i n i t i a l d a t a of which a r e chosen i n a Sobolev s p a c e HS w i t h s 2 4 , h a s a u n i q u e s o l u t i o n 2 E Hs , i n t h e neighbourhood o f t h e i n i t i a l manifold". The r e a d e r who i s i n t e r e s t e d may a l s o see, i n a more g e n e r a l c o n t e x t , t h ? i m p o r t a n t r e s u l t by V o l p e r t and Hudiaev 12
1.
Supplementary c o n s e r v a t i o n law ( e n t r o p y p r i n c i p l e ) : I n 1 9 7 1 F r i e d r i c h s and Lax 131 f o r m u l a t e d t h e f o l l o w i n g q u e s t i o n : o f system ( 2 ) i s compatible w i t h t h e following assumptions?
which t y p e
i ) a l l t h e s o l u t i o n s o f ( 2 ) a l so s a t i s f y a n o t h e r s u p p l e m e n t a r y c o n s e r v a t i o n law: aaha(:) ii) choosing
N D F R
.
u -
=
Fo , ho -
= gcg
(3)
t
i s a s t r i c t l y convex f u n c t i o n of
u i n a convex domair) -
Under t h e s e h y p o t h e s e s t h e y p r o v e d t h a t t h e r e e x i s t s a p o s i t i v e d e f i n i t e NxN m a t r i x H ( u ) , s u c h t h a t t h e new s y s t e m H ( g ) . { a aFa(%) o b t a i n e d from ( 2 ) t i m e s H
,
-
f(g)}=
0
(4)
i s a symmetric h y p e r b o l i c one.
The i m p o r t a n t r e s u l t o f t h e s e a u t h o r s h a s , however, t h e f o l l o w i n g d i s a d v a n t a g e s :
1) The c o n s e r v a t i v e form of t h e new symmetric s y s t e m ( 4 ) i s l o s t ; 2 ) W e have n o t an e x p l i c i t e x p r e s s i o n f o r t h e f u n c t i o n a l dependence o f respect the f i e l d g
Fa
with
.
The former r e s t r i c t i o n d o e s n ' t a l l o w u s e o f t h e p e c u l i a r p r o p e r t i e s o f symmetric s y s t e m s c o n c e r n i n g weak s o l u t i o n s and,in p a r t i c u l a r , s h o c k s . I n f a c t , many s y s t e m s o f e v o l u t i o n i n t h e macroscopic t h e o r i e s of c o n t i n u a are w r i t t e n i n t h e form ( 2 ) , b u t t h e f u n c t i o n a l dependence o f w i t h r e s p e c t t o t h e f i e l d u i s completely a s s i g n e d o n l y when t h e c o n s t i t u t i v e r e l a t i o n s a r e known. One o f t h e m o s t i m p o r t a n t problems i s t o e s t a b l i s h t h e g e n e r a l c r i t e r i a f o r c o n s t i t u t i v e relations.
ra
I n t h e modern approach t o thermodynamics t h e e n t r o p y i n e q u a l i t y i s r e g a r d e d as a c o n s t r a i n t o n t h e c o n s t i t u t i v e r e l a t i o n s and n o t as an i d e n t i f i c a t i o n of a privileged t i m e orientation. F i r s t s u p p o r t e r s o f t h i s p o i n t o f view were
213
Entropy Principle and Shock Waves
l a t e r , a more g e n e r a l p r i n c i p l e w a s proposed by M i l l e r i n
Coleman and No11 141;
151. The s i t u a t i o n i s s u b s t a n t i a l l y t h e f o l l o w i n g : b o t h t h e e n t r o p y d e n s i t y and t h e e n t r o p y f l u x are t h o u g h t o f a s t h e c o n s t i t u t i v e r e l a t i o n s of t h e s e t o f f i e l d v a r i a b l e s and for a l l thermodynamical p r o c e s s e s i . e . , f o r a l l t h e s o l u t i o n s of t h e p a r t i a l d i f f e r e n t i a l s y s t e m , t h e f o l l o w i n g i n e q u a l i t y must b e i d e n t i c a l l y satisfied ~~
2
t
pS
+ a.
Qi
= s t 0
is t h e m a s s density).
(p
I n t h e l i m i t o f t h e c o n s t i t u t i v e f u n c t i o n s t h a t l o c a l l y depend on t h e f i e l d , t h i s problem i s c o m p l e t e l y t h e same a s t h e p r e v i o u s one, t h r o u g h t h e i d e n t i f i c a t i o n s ho = -pS , and h' = -0'
.
I n t h i s case i t i s no l o n g e r i m p o r t a n t t h a t t h e s y s t e m b e symmetric b u t on t h e c o n t r a r y i t i s i m p o r t a n t t o g i v e an e x p l i c i t c h a r a c t e r i z a t i o n o f f u n c t i o n a l and ha w i t h r e s p e c t t o t h e f i e l d 5 and t h e r e f o r e a complete dependence of c h a r a c t e r i z a t i o n of t h e compatible c o n s t i t u t i v e e q u a t i o n s .
c"
Now we q u o t e a theorem t h a t g i v e s a d i f f e r e n t p r o o f t o t h e , o n e b y F r i e d r i c h s and I t i s f r e e from t h e p r e v i o u s o b j e c t i o n s and a t t h e same t i m e , p e r m i t s a Lax. connection w i t h t h e entropy p r i n c i p l e . M A I N FIELD AND GENERATORS OF A SYMMETRIC SYSTEM
Theorem:
" A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a c o n s e r v a t i v e s y s t e m
t o be compatible w i t h a supplementary c o n s e r v a t i o n i n e q u a l i t y
i s t h a t t h e c o n s t i t u t i v e r e l a t i o n s are a l l o n e s f o r which f o u r s c a l a r s and a p r i v i l e g e d s e t o f f i e l d v a r i a b l e s x' , e x i s t s u c h t h a t :
F"
= ahi"/a
The components o f t h e "main f i e l d "
11’
h'"
are r e l a t e d t o
Moreover i f i t i s p o s s i b l e t o choose
u’
o
.
(5)
are t h e "Lagrange m u l t i p l i e r s " , s u c h t h a t :
u ' . d F-a and t h e ( p o t e n t i a l s
=
;
h'"
2 dh"
ha
,
by:
as f i e l d = aho/au
5 , we
h a v e from ( 6 )
,
(8)
ro
and if ho i s a convex f u n c t i o n o f 5 = ( o r l i k e w i s e , i f h'O i s a convex u’) , t h e n t h e o r i g i n a l s y s t e m i s a v e r y s p e c i a l , symmetric one f u n c t i o n of i n t h e f i e l d 5':
T h e r e f o r e t h e l o c a l Cauchy problem i s w e l l p o s e d " .
u’
which S u b s t a n t i a l l y , t h i s theorem g u a r a n t e e s t h a t there e x i s t s a f i e l d symmetrizes t h e o r i g i n a l system w i t h o u t l o s i n g t h e c o n s e r v a t i v e s t r u c t u r e and w i t h respect t o t h i s f i e l d , t h e s y s t e m assumes t h e s p e c i a l form ( 9 ) . W e o b s e r v e t h a t a l l matrices i n ( 9 ) are H e s s i a n o n e s and h q 0 i s t h e Legendre c o n j u g a t e
214
T. Ruggeri
function of
ho ( s e e ( 7 ) ,
(8) f o r
c1 =
0).
The systems of t y p e ( 9 ) have been considered f i r s t by Godunov 161 i n 1961, who observed t h a t t h e system f o r a p e r f e c t f l u i d and the Euler-Lagrange equations a r e s u s c e p t i b l e t o t h i s form. Subsequently t h e same author noted t h a t t h e systems of this form (9) always admit an equation ( 3 ) a s a consequence 17
I.
The i n t r o d u c t i o n of t h e f i e l d u’ through t h e eq. system i s due to B o i l l a t 181 in-1974.
( 8 ) f o r a g e n e r i c hyperbolic
5’ a r e introduced a s t h e Lagrange m u l t i p l i e r s and new p r o p e r t i e s appear i s due t o Ruggeri and Strumia 19 i n 1981 and a technique which holds a l s o f o r non-hyperbolic systems was presented q u i t e r e c e n t l y in )10).
A more general approach, i n which
1
D e f i n i t i o n of "Generators" : The systems of type (2) with t h e supplementary conservation l a w ( 3 ) a r e completely determined i f w e know t h e 2N+4 o b j e c t s : [E' , h " , f-} In f a c t from (5.1) we determine and from ( 7 ) and (5.2) h" and g Therefore w e c a l l t h e s e q u a n t i t i e s "generators".
c"
.
.
Now it i s necessary t o d i s t i n g u i s h two d i f f e r e n t kind of problems. Problem I I f one knows t h e supplementary conservation law and t h e c o n s t i t u t i v e r e l a t i o n s , i t i s very easy t o e v a l u a t e t h e generators from (8) and ( 7 ) and t h e r e f o r e t h e symmetric form of the system. I n t h i s s i t u a t i o n it is p o s s i b l e to use t h e s p e c i a l s t r u c t u r e of t h e system ( 9 ) and prove some important theorems on shock waves. Problem I1 I f we regard the supplementary law a s an "entropy p r i n c i p l e " t h e theorem becomes only q u a l i t a t i v e , because "a p r i o r i " 5’ and h ' " a r e not known. I n o t h e r words we know t h a t t h e system i s i n t h e form ( 9 ) , o r e q u i v a l e n t l y
b u t what i s not evident a r e t h e r e l a t i o n s between t h e q u a n t i t i e s t h e physical q u a n t i t i e s appearing i n t h e problem.
g’
and
hIa
and
I n t h e next s e c t i o n , w e p r e s e n t f i r s t some r e c e n t r e s u l t s on shock waves r e l a t e d t o problem I and then w e p r e s e n t some p o s s i b l e approaches t o t h e second problem. Most of t h i s l a s t p a r t i s completely new.
CONSEQUENCES FOR SHOCK WAVES Under t h e hypotheses o f t h e previous theorem, i t i s p o s s i b l e t o prove some i n t e r e s t i n g p r o p e r t i e s on shock waves:
i) t h e shock v e l o c i t i e s
s
a r e proved t o s a t i s f y t h e i n e q u a l i t i e s :
i n f min u6D k where
X
c s < sup Max
(k)
ucD k -
a r e the c h a r a c t e r i s t i c v e l o c i t i e s .
This theorem ( B o i l l a t and Ruqqeri 1111)
, ensures
t h a t t h e Rankine-Hugoniot
21 5
Entropy Principle and Shock Waves
equations :
([:
1
d e n o t e s t h e jump; +(xu)
=
0
i s t h e e q u a t i o n of t h e shock s u r f a c e
r
;
and n . = $i/lV$l , s = - $ t / / V $ / a r e r e s p e c t i v e l y t h e u n i t normal and t h e v e l o c i t y o f l t h e shock s u r f a c e ) , admit o n l y t h e t r i v i a l s o l u t i o n [%I = 0 T h e r e f o r e n o n - v a n i s h i n g s h o c k s take p l a c e o n l y i f t h e i r i f (11) i s n o t s a t i s f i e d . s p e e d i s g r e a t e r t h a n t h e s m a l l e s t c h a r a c t e r i s t i c s p e e d and smaller t h a n t h e g r e a t e s t one.
0,
= a$/axa
A c o v a r i a n t v e r s i o n o f t h i s theorem i s s t a t e d i n 191 and as a consequence i n a r e l a t i v i s t i c t h e o r y Is1 2 c , where c i s t h e l i g h t v e l o c i t y i n vacuum.
ii) F o r m a l l y t h e Rankine-Hugoniot e q s . are o b t a i n e d from t h e f i e l d e q s . ( 2 ) t h r o u g h t h e c o r r e s p o n d e n c e r u l e aa -f [ However t h i s r u l e d o e s n o t work when it i s a p p l i e d t o t h e s u p p l e m e n t a r y e q . ( 3 ) ; i n f a c t
.
d o e s n o t , i n g e n e r a l , v a n i s h . I t i s known t h a t t h e p o s i t i v e s i g n a t u r e on q f o r a f l u i d implies t h e growth o f thermodynamic e n t r o p y a c r o s s t h e s h o c k . T h i s i s t h e r e a s o n why t h e c o n d i t i o n I- Z 0 i s o f t e n c a l l e d i n t h e l i t e r a t u r e t h e " e n t r o p y growth c o n d i t i o n " . Here, t h i s c r i t e r i o n p i c k s up t h e p h y s i c a l s h o c k s among t h e s o l u t i o n s o f t h e Rankine-Hugoniot e q s . i n s t e a d of s e l e c t i n g t h e c o n s t i t u t i v e e q s . , a s i n t h e d i f f e r e n t i a b l e c a s e . U s u a l l y t h i s growth i s p r o v e d t h r o u g h a r t i f i c i a l v i s c o s i t y c o n s i d e r a t i o n s I12 ; a p r o o f which u s e s o n l y c o n v e x i t y arguments f o r k-shocks i s made i n 1131 , 191.
I
iii) The knowledge o f t h e jump o f t h e f i e l d Example:
o n l y t h e s c a l a r f u n c t i o n q on r i s enough t o d e t e r m i n e b e h a v e s a s a s o r t of " p o t e n t i a l " f o r t h e shock 1131. : q
u’
Perfect Fluid
A s a t e s t of t h e p r e v i o u s c o n s i d e r a t i o n s , w e p r e s e n t t h e s i m p l e and w e l l known
example o f t h e e q s . o f a p e r f e c t f l u i d . The e q s . i n c o n s e r v a t i v e form are:
+
atp
[at - + +
( p +v )
atE
+
+ div +
o
d i v (pv) =
( m a s s balance)
T = 0
(momentum eq. ) -f
d i v { ( ~ + p ) v=} 0
(energy balance)
2
-f
where T = pv ti v + p I ; E = p ( e + v /2) and p , v , e mass d e n s i t y , v e l o c i t y , i n t e r n a l energy and p r e s s u r e . I n t h i s c a s e t h e supplementary l a w i s t h e at(ps)
The s y s t e m (13) and t h e e q . following identifications:
+
,p
(13)
,
are r e s p e c t i v e l y t h e
entropy balance:
+
d i v (psv) =
o
(14) are e q u i v a l e n t t o ( 2 ) and ( 3 ) , t h r o u g h t h e
I n t h i s case, s i m p l e c a l c u l a t i o n s g i v e t h e f o l l o w i n g g e n e r a t o r s :
216
T. Ruggeri
where 0 and G a r e r e s p e c t i v e l y t h e absolute temperature and t h e f r e e enthalpy ( t h e chemical p o t e n t i a l : G = e - 8s + p/p) . The convexity of ho = -pS respect t o (p,B) , i . e . : c > 0 P
,
(aV/apg)
< 0
can be proved i f
(V
= l/p
,
c
P
G(p,0)
i s a concave function with
is the s p e c i f i c heat a t constantpressure).
These c o n d i t i o n s are t h e u s u a l thermodynamic s t a b i l i t y conditions a t equilibrium. and a l l previous Therefore t h e system (13) i s symmetric i n t h e main f i e l d u ' considerations hold (on t h i s s u b j e c t s e e 161, 1121, 1141, -191, 1151). Several o t h e r examples i n t h i s a r e a are considered i n t h e r e c e n t l i t e r a t u r e v i z : h y p e r e l a s t i c s o l i d s I16 ; r e l a t i v i s t i c f l u i d s and magnetofluids 19 , I17 ; nonl i n e a r electrodynamics 118 , 1 9 , I20 , s u p e r f l u i d s 121 , mixture of f l u i d s I 2 2
I
I I I
I
I
I
I
I.
The reader who i s i n t e r e s t e d i n more a n a l y t i c a l arguments on t h e s e s u b j e c t s about the entropy growth and t h e weak s o l u t i o n s , may read t h e important n o t e s by Dafermos 1231 . CONSTRUCTION OF SYSTEMS OF BALANCE LAWS WHIM ADMIT A SUPPLEMENTARY LAW: REPRESENTATION THEOREMS The p h y s i c a l l y s i g n i f i c a n t examples examined s o f a r were p e r t i n e n t t o t h e f i r s t problem, where t h e supplementary law was "a p r i o r i " known t o g e t h e r with t h e associated c o n s t i t u t i v e equations. For i n s t a n c e , i n t h e previous c a s e of a p e r f e c t f l u i d we have followed t h e c l a s s i c a l t h e o r y ; i n o t h e r words t h e Gibbs r e l a t i o n w a s supposed t o hold and t h e concept of absolute temperature, entropy and i n t e r n a l energy was "a p r i o r i " accepted. The a i m was t o o b t a i n a new symmetric hyperbolic conservative system i n o r d e r t o have q u a l i t a t i v e information on t h e s o l u t i o n (well-positing of t h e Cauchy problem, some p r o p e r t i e s o f shock waves and so on ) . A l t e r n a t i v e l y i f we look a t t h e problem from t h e p o i n t o f view of " r a t i o n a l thermodynamics", w e d o n ' t accept t h e s e concepts a s p r i m i t i v e ones and by considering t h e entropy i n e q u a l i t y a s a c o n s t r a i n t on t h e c o n s t i t u t i v e equations, t h e Gibbs r e l a t i o n emerges a u t o m a t i c a l l y a s an i n t e g r a b i l i t y c o n d i t i o n , and t h e absolute temperature a s an i n t e g r a t i n g f a c t o r . Clearly t h i s approach provides a q u a l i t a t i v e jump from t h e view p o i n t o f t h e axiomatic thermodynamics, even though it does not change t h e p r a c t i c a l r e s u l t s . Moreover, s o m e t i m e s t h e t r u e equations a r e not known "a p r i o r i " because they r e p r e s e n t new models of some p h y s i c a l phenomena. For i n s t a n c e , f o r a viscous, h e a t conducting f l u i d , t h e problem i s t o -f determine t h e c o n s t i t u t i v e r e l a t i o n s f o r t h e h e a t f l u x q and f o r t h e viscous tensor C 5 ( C , ,) .
...
13
I n t h e c l a s s i c a l approach t h e s e axe t h e Fourier and Navier-Stokes equations which, a s it i s w e l l known, d e s t r o y t h e h y p e r b o l i c i t y of t h e f i e l d system with t h e consequence t h a t an i n f i n i t e wave propagation speed i s obtained. In o r d e r t o e l i m i n a t e t h e paradox and hyperbolize t h e system, t h e non-equilibrium thermodynamics or, following M u l l e r ' s d e f i n i t i o n , t h e "extznded i r r e v e r s i b l e thermodynamics" 1241, aims a t determining e v o l u t i o n s f o r q and C , preserving t h e h y p e r b o l i c i t y of t h e e n t i r e system. The b a s i c i d e a of t h i s approach is: one c o n s i d e r s t h e non-equilibrium entropy d e n s i t y ( d i f f e r e n t from t h e equilibrium one) n o t only t o depend on t h e two thermodynamic v a r i a b l e s b u t a l s o on $ and C ,
217
Entropy Principle and Shock Waves
c o n s e q u e n t l y m o d i f y i n g t h e Gibbs r e l a t i o n . As t h i s i s not the r i g h t place t o p u r s u e s u c h a s u b j e c t , we r e f e r t h e i n t e r e s t e d r e a d e r t o t h e work 1101 where t h e a u t h o r u s e s a t e c h n i q u e connected w i t h t h e g e n e r a t o r s t o set up a n e v o l u t o r y symmetric h y p e r b o l i c s y s t e m . However, w e want t o show how t h e s t r u c t u r e theorem, mentioned above, can g i v e some i n f o r m a t i o n a b o u t new models (which are c o m p a t i b l e w i t h wave p r o p a g a t i o n ) d e s c r i b e d b y more g e n e r a l b a l a n c e e q u a t i o n s which admit a supplementary c o n s e r v a t i o n l a w ( n o t n e c e s s a r i l y t h e e n t r o p y o n e ) . The t e c h n i q u e i s b a s e d on some simple h" w i t h r e s p e c t t o t h e components r e p r e s e n t a t i o n theorems f o r t h e f o u r - v e c t o r of t h e main f i e l d 5’
.
L e t u s b r i e f l y i l l u s t r a t e t h e methodology w i t h an example, i n o r d e r t o make t h e method c l e a r e r .
F o r example, w e s e e k t o d e t e r m i n e t h e most g e n e r a l s t r u c t u r e of a d i f f e r e n t i a l s y s t e m o f b a l a n c e e q u a t i o n s where t h e components o f t h e f i e l d a r e a s c a l a r f a c t o r and a R3-vector; moreover the s y s t e m must b e c o m p a t i b l e w i t h a s u p p l e mentary l a w t o b e d e t e r m i n e d . A t f i r s t s i g h t t h e problem i s e x t r e m e l y complex. I n f a c t one must w r i t e a s y s t e m o f t h e t y p e ( 2 ) , formed b y a s c a l a r e q u a t i o n and a v e c t o r i a l o n e , and t h e n must c h a r a c t e r i z e t h e s u b c l a s s which i s c o m p a t i b l e w i t h a s u p p l e m e n t a r y l a w ( 3 ) . One c o u l d u s e theorems on r e p r e s e n t a t i o n , b u t t h e s e would b e e x t r e m e l y complex even i n t h i s s i m p l e c a s e , and what i s more, it would b e e x t r e m e l y d i f f i c u l t t o c h a r a c t e r i z e t h e i n t e g r a b i l i t y c o n d i t i o n s so t h a t e v e r y s o l u t i o n o f ( 2 ) s a t i s f i e s ( 3 ) . Now w e show how w e can make u s e o f the q u a l i t a t i v e theorem a c c o r d i n g t o which t h e m o s t g e n e r a l s y s t e m ( 2 1 , when i t i s compatible w i t h a supplementary l a w , h a s t h e s t r u c t u r e such t h a t is t h e g r a d i e n t o f h q a w i t h r e s p e c t t o t h e f i e l d '; F o r t h e c a s e u n d e r examination, w e know c e r t a i n l y t h a t u’ must have f o u r components formed by a s c a l a r f a c t o r and a R3-vector: '; :
ca
.
(F,l)T .
3
h t 0 i s a s c a l a r f a c t o r and h I i i s t h e g e n e r i c component o f a R - v e c t o r , the unique p o s s i b l e r e p r e s e n t a t i o n f o r them i s :
As
h'O = p ( S , z )
;
i 2 hQi= v(6,z)h ; z = A /2
.
I n t h i s way, t h e problem i s c o n s i d e r a b l y s i m p l i f i e d , as we have had t o r e p r e s e n t only the p o t e n t i a l s h'" and n o t t h e complete s e t o f t h e v e c t o r s Fa Now from (5.1), i t f o l l o w s t h a t t h e most g e n e r a l s y s t e m i s o f t h e form:
.
and e v e r y s o l u t i o n of (15) w i l l s a t i s f y , by ( 7 ) , ( 3 ) , ( 5 . 2 ) , t h e s u p p l e m e n t a r y law:
(where t h e s u b s c r i p t s d e n o t e t h e d e r i v a t i v e s w i t h r e s p e c t t o t h e a r g u m e n t s ) . I n t h i s way, even though t h e q u a n t i t i e s which are i n v o l v e d h a v e , a s o f y e t , n o p h y s i c a l meaning, w e h a v e e s t a b l i s h e d the s p e c i a l i z e d s t r u c t u r e ( i t i s i m p o r t a n t to n o t i c e t h a t t h e c o e f f i c i e n t o f t h e main p a r t s are r e l a t e d t o t h e d e r i v a t i v e s of o n l y two s c a l a r f u n c t i o n s p , u) f o r t h e s y s t e m (15) which a u t o m a t i c a l l y and "a p r i o r i " s a t i s f i e s t h e s u p p l e m e n t a r y l a w (16). F u r t h e r m o r e , t h i s makes u s u n d e r s t a n d t h a t , i n d e p e n d e n t of t h e p h y s i c s o f t h e d i f f e r e n t q u a n t i t i e s i n v o l v e d , s y s t e m s w i t h t h e same number o f c o n s e r v a t i v e e q s . and c o m p a t i b l e w i t h a
21 8
T. Ruggeri
supplementary law, must have t h e same s t r u c t u r e . Now, i f w e wish t o s p e c i a l i z e t h e s y s t e m (15) t o t h e s y s t e m f o r t h e i s e n t r o p i c p e r f e c t f l u i d , t h e n e c e s s a r y i d e n t i f i c a t i o n s are: P = p5
Set
E
=
,
pv
i
5lJ + 2zFi 5 E =
= v
5’A
-
l~
= pzA
i
, pv iv j
= vsij
+
+
~
z
niAj , a = B = O .
, and w e o b t a i n , a f t e r s t r a i g h t f o r w a r d c a l c u l a t i o n s ,
2 P ( e + v / 2 ) , w i t h e such t h a t :
d e = p dp/p
2
: p = w = p
;
+
2 l = v ; ~ = e + p / p - v / 2 .
I n o t h e r w o r d s , we f i n d t h a t t h e supplementary e q u a t i o n i s t h e e n e r g y conservat i o n one ( i n t h e m e c h a n i c a l case, t h e e n e r g y e q u a t i o n p l a y s t h e same r o l e as t h e e n t r o p y one i n thermodynamical case) and t h e c o n s t i t u t i v e c o n s t r a i n t a r i s e s a u t o m a t i c a l l y from t h e i d e n t i f i c a t i o n . Heat e q u a t i o n w i t h f i n i t e v e l o c i t i e s
Now, w e w i s h t o c o n s i d e r a d i f f e r e n t problem, d e s c r i b e d by t h e same number o f b a l a n c e e q u a t i o n s : t h i s i s , the case o f a r i g i d c o n d u c t o r w i t h h e a t p r o p a g a t i o n . I f w e a i m t o o b t a i n h y p e r b o l i c e q u a t i o n s , i n t h i s case, t h e staze o f t h e s y s t e m i s c h a r a c t e r i z e d by t h e t o t a l energy 6 and by t h e h e a t f l u x q
.
The f i r s t e q u a t i o n of (15) must b e i d e n t i f i e d w i t h t h e e n e r g y c o n s e r v a t i o n equation
+
a&/at +
div q
o ,
=
w h i l e t h e remaining o n e s of ( 1 5 ) are t h e e q u a t i o n s f o r I t i s s e e n t h a t ( 1 7 ) can a l s o b e w r i t t e n a s
+
aE/at
-+
div
(EW)
=
o
( n o t known "a p r i o r i " ) .
q
+ ;
(17)
+
+
w = q/E
.
(18)
T h i s is s i m i l a r t o t h e mass b a l a n c e e q u a t i o n , p r o v i d e d w e c o n s i d e r t h e f o l l o w i n g duality:
+
p + E ;
+
V + W .
From w h a t h a s been s a i d , ( 1 5 . 2 ) w i l l be s i m i l a r t o t h e e q u a t i o n o f momentum f o r t h e f l u i d , by making u s e o f t h e d u a l i t y mentioned above. In f a c t , equation ( 1 5 . 2 ) becomes:
a
gj
+ aiiqiqjlE
+
o(E)
t
2'1 =
gq j
.
(19)
The e q u a t i o n ( 1 6 ) w i l l assume t h e mzaning of t h e e n t r o p y i n e q u a l i t y . Then, i f one r e q u e s t s t h a t a t e q u i l i b r i u m ( q = 0) t h e e n t r o p y s a t i s f i e s t h e Gibbs r e l a t i o n , and s e t t i n g , as u s u a l , E = p c 0 , one f i n d s , a f t e r s t r a i g h t f o r w a r d o v is t h e following: c a l c u l a t i o n s , t h a t t h e unique c o n s t i t u t i v e e q u a t i o n f o r O ( E ) 0 = (p
C /5)
o v
109 8
and moreover
8 (po
T
= -p
c
/ X T ~
o v
i s t h e c o n s t a n t m a s s d e n s i t y , cv i s t h e s p e c i f i c h e a t a t c o n s t a n t volume, i s a c o n s t a n t parameter, and x i s t h e thermal c o n d u c t i v i t y ) .
I t f o l l o w s t h a t t h e e q u a t i o n ( 1 9 ) becomes:
Entropy Principle and Shock Waves
21 9
I t i s r e m a r k a b l e t o o b s e r v e t h a t when T -t 0 e q . ( 2 0 ) g o e s o v e r + i n t o t h e F o u r i e r e q u a t i o n , and i f w e n e g l e c t i n e q . ( 2 0 ) t h e q u a d r a t i c t e r m i n q it becomes t h e Maxwell-Cattaneo e q u a t i o n ( w i t h a n o n - c o n s t a n t r e l a x a t i o n t i m e ) :
+
T(e)a q
t
+ xve
-+
= -9 ; T = x T e / p o c v
.
The s u p p l e m e n t a r y law becomes:
.
The e n t r o p y i n e q u a l i t y i s a u t o m a t i c a l l y f u l f i l l e d u n d e r t h e (Seq = cv l o g 8) u s u a l c l a s s i c a l c o n d i t i o n : t h e t h e r m a l c o n d u c t i v i t y must b e n o w n e g a t i v e . It i s e a s y t o p r o v e t h a t t h e system (171, ( 2 0 ) i s a symmetric and h y p e r b o l i c o n e , when T > 0
.
T h e c o n s t r u c t i v e arguments which w e have u s e d h e r e are t h e s u b j e c t o f a work i n p r o g r e s s w h e r e i n t h e i n t e r e s t e d r e a d e r can f i n d d e t a i l e d c a l c u l a t i o n s which h e r e h a v e been o m i t t e d .
The a i m o f t h i s work w a s o n l y t o g i v e t h e b a s i s of t h e method and t o show how a p p r o a c h e s a r i s i n g i n wave t h e o r y can b e u s e d t o s e t up m a t h e m a t i c a l w e l l - p o s e d models. Acknowledgement T h i s work was s u p p o r t e d by t h e C . N . R . and by INFN - S e z . di Bologna.
(Gruppo N a z i o n a l e p e r l a F i s i c a Matematica)
T. Ruggeri
220 REFERENCES :
151
28, 1 (1972). Volpert, A.I. and Hudiaev, S.I., Math. USSR Sbornik 2, 571 (1972). Lax, P.D. and Friedrichs, K.O., Proc. Nat. Acad. Sc. USA 68, 1686 (1971). Coleman, B.D. and N o l l , W., Arch. Rat. Mech. Anal. 2, 1 (1963). Mcller, I . , Arch. Rat. Mech. Anal. 40, 319 (1971).
161
Godunov, S . K . ,
Sov. Math. 2n 947 (1961).
171
Godunov, S . K . ,
Russ. Math. Surv.
181
Boillat, G., C. R. Acad. Sc. Paris 278-A , 909 (1974).
191
Ruggeri, T. and Strumia, A., Ann. Inst. H. Poincar6
111 121 131
141
Fischer, A. and Marsden, D.P., Comm. Math. Phys.
1101 Ruggeri, T . , Acta Mech.
47,
17,145
(1962).
2, 65
(1981).
163 (1983).
111) Boillat, G. and Ruggeri, T., C.R. Acad. S c . Paris
z ,257 (1979).
1121 Lax, P.D., in: Contributions to non linear functional analysis; Zarantonello (ed.) (Academic Press, New York 1971). 1131 Boillat, G., C.R. Acad. Sc. Paris
E ,409
1141 Fusco, D . , Rend. Sem. Mat. Modena
2, 223
(1976).
(1979).
1151 Ruggeri. T . , Appunti di propagazione ondosa, ba Scuola Estiva di Fisica Matematica del C.N.R. - Dip. di Mat. - Univ. Bologna (Sett. 1981).
2, 271 (1980). Phys. 2, 1824 (1981).
1161 Boillat, G. and Ruggeri, T., Acta Mech. 1171 Ruggeri, T. and Strumia, A., J. Math. 1181 Boillat, G., C.R. Acad. Sc. Paris 1191 Strumia, A., Lett. Nuovo Cimento
E ,259
36, ( 1 7 ) ,
(1980).
569 (1983).
1201 Boillat, G. and Venturi, G., I1 Nuovo Cimento
3, (3),358
(1983).
1211 Boillat, G. and Muracchini, A., On the symmetric conservative form of Landau’s superfluid equations, (to appear in ZAM?, 1984). 1221 Virgopia, N. and Ferraioli, F., On the shock wave generating function in a simple mixture of gases (submitted in I1 Nuovo Cimento).
14,
1231 Dafermos, C . , Arch. Rat. Mech. Anal. 0, 167 (1979); J . Diff. Eqs. 202 (1973); Hyperbolic systems of conservation laws, Div. of Appl. Math. Brown Univ. Providence USA (March 1983). 1241 Liu, I-Shih and Miiller, I., Arch. Rat. Mech. Anal. 83, ( 4 ) , 285 (1983).
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland).1984
22 1
DIFFERENTIAL AND DISCRETE SPECTRAL PROBLEMS AND T H E I R INVERSES P.J. Caudrey Department o f Mathematics U.M.I.S.T. P.O. Box 88, Manchester M60 1QD U.K.
1Je c o n s i d e r t h e d i f f e r e n t i a l and d i s c r e t e s p e c t r a l problems
A
and g a r e m a t r i c e s which where u i s a column v e c t o r and d e p e n d o n t h e complex s p e c t r a l parameter 5. M a t r i x a l s o depends on e i t h e r t h e r e a l v a r i a b l e x o r t h e i n t e g e r v a r i a b l e n as shown. I n each case a s e t o f s p e c t r a l d a t a i s d e f i n e d and a method o f s o l v i n g t h e i n v e r s e problem, i . e . t h a t o f r e c o n s t r u c t i n g B from t h i s data, i s d e s c r i b e d . 1.
INTRODUCTION
The success o f t h e s p e c t r a l t r a n s f o r m o r " i n v e r s e s c a t t e r i n g method" i n s o l v i n g v a r i o u s n o n l i n e a r e v o l u t i o n e q u a t i o n s i s w e l l known E l - 3 1 . Many s p e c t r a l problems have been used f o r t h i s purpose 14-141. N e a r l y a l l o f them can be w r i t t e n i n one o f two forms depending on whether i t i s a d i f f e r e n t i a l o r a d i f f e r e n c e problem. These two forms a r e examined i n t h i s paper. I n 5 2 t h e d i r e c t s p e c t r a l problems a r e c o n s i d e r e d and t h e s p e c t r a l d a t a i s d e f i n e d i n 53. A method o f s o l v i n g t h e i n v e r s e s p e c t r a l problem i s e x p l a i n e d i n 54. The p r a c t i c a l use o f t h e s p e c t r a l t r a n s f o r m i s demonstrated i n 555 and 6. The Boussinesq e q u a t i o n i s s o l v e d i n 5 5 u s i n g a d i f f e r e n t i a l s p e c t r a l t r a n s f o r m and t h e Toda l a t t i c e e q u a t i o n s a r e s o l v e d i n 56 u s i n g a d i s c r e t e s p e c t r a l t r a n s f o r m . 2.
THE DIRECT SPECTRAL PROBLEMS
We s h a l l c o n s i d e r t h e s p e c t r a l problem 1 4 1
a
u(x) = ax -
Id( 1. 1
If w ( x ) i s real the locations o f these poles a r e very r e s t r i c t e d .
Either (5.19 (5.20
ck
where in both cases e i t h e r Ck'
and e i t h e r n / 3
=
sk
(5.21
= B[ 71
o r - 12’. < arg 5 The time evolution of the spectral data can be deduced from the f a c t t h a t
0 $ , on
t h e multivalued functions a(.)
and
satisfy Re f ; [ : ] G O
Br-y (and B r P ) . and hence, according t o ( 6 b ) , , will satisfy (4),for
j , for
v 4 m v pro-
Here and hereafter subscript 1 o n equation numbers indicates equations f r o m [ l , P a r t 11, e.g., (5)1.
J. Miklowitz and C. Chazapis
236
and
from (13)1, which leads t o the double -transformed displacements 3 s ,y , p ) and %(s ,y , p ) in (14)]for B . 0 .
I t is important t o point out now t h a t the branch functions a(.)
and @ ( s ) . defined in (7)1 and
(8)1, respectively, are analytic all along Brs, having Re[$;]]>O Re[$E]]O
B
and the
coefficients of these exponential functions vanish as y+=. Similarly, (14), for % and % a r e bounded as y +-, the critical terms being those involving exp[-a(s)y] and exp[-p(s)y] where
Re[i[Z]].cO and the coefficients of these exponential functions vanish as y-r-. I t is of further
-
that according t o the principle of reflection interest t o note from the results of (9)1 and on BrsO, so t h a t (14)1 forms a natural conjugate of ( l Z ) l . This leads to a real formal solution f o r the dispIacements over BrsL,as well as other conjugations in the s-plane. a ( s ) = - a ( s ) and @(s)=-@m
T h e Surjuce-Breaking Crack P r o b l e m . We now apply the basic techniques of [I], involving transient wavefront approximations t o elastic waves scattered by cracks, t o the problem of [3,4],i.e., the surface-breaking crack depicted in Fig. 1. Then, as pointed out earlier, the surface-breaking crack problem has a scattered Aeld t h a t decomposes into symmetric and antisymmetric fields with respect t o the plane of the crack, generating a pair of boundaryvalue problems f o r the quarter plane. For the surface-breaking crack we have the following symmetric and antisymmetric boundary-initial value problems for the quarter plane, (cf. Fig.
2). U,(O.Y
f)
= 0.
uz(O,y,t)= -uST(y,O,t)
and
osy ud (0 , y st
-y
cd),
Consider next t h e Rayleigh wave disturbance on t h e edge x = 0. Here as i n the half-plane problem one would expect a nondecaying in space ( y ) one-dimensional disturbance. Hence, it is assumed t h a t =R(OIY,P) = C ( p ) e x p ( - k R l ' ! / ) DRiR(osYIp)= D t P ) exP(-kRiY), I
from which one also has
(22)
Wavefront Fields in Elastic Wave Scattering
243
both in O s y < d , i.e., along the crack faces. The edge of the crack a t y = d is a source of diffracted dilatational and Rayleigh surface wavefronts that propagate from y=d t o y=O (cf Fig 2). and indeed keep diffracting from y=O and y=d ad inflnitum as time progresses Furthermore, the crack edge a t y=d generates the diffracted dilatational wavefronts that propagate in the domain d S y < m from y =d to -, ad infinitum being generated by the diffracted dilatational wavefronts on the crack faces when they reach the crack edge y = d as time progresses Since the boundary conditions in d c y <m are mixed [o,(O,y.t)=u(O,y t ) = 01, the Rayleigh surface waves are not present In d ~ y < = cf ,
"?, Fig
B 51
For these additional diffracted wavefronts, assuming their time behavlor is always the same as the basic wavefronts derived from the time transformed edge (x=O)unknowns (20)-(23), we can derive these additional wavefronts from forms similar to (20)-(23) First, we have for the domain d Iy = p i f r > r P 9 = q i f r > r 9 P and t h e multi-index
,
Zi(kr) = j n ( k r )
,
p< = p i f r < r P 9 = q i f r < r 9 P’
EOsll, m=2 f o r m>O
9
33 1
Scattering of Acoustic Waves by Impedance Surfaces
= n+m+j. The f u n c t i o n s i n t h e s e r i e s ( 2 . 6 ) could be r e o r d e r e d s o t h a t t h e summation i s c a r r i e d o u t over a s i n g l e i n d e x , s e e ( 7 1 f o r e x p l i c i t d e t a i l s . Furthermore we d e f i n e a modified G r e e n ' s f u n c t i o n as
is employed w i t h
where N may be +m and Ll i s again a multi-index a s i n ( 2 . 8 ) . S i n g l e and double l a y e r d i s t r i b u t i o n s w i t h a modified k e r n e l may be d e f i n e d which have t h e u s u a l p r o p e r t i e s ; t h a t is, l e t
an N
(2.11)
N
(D.LI)(P):= J
u(q)
a?.. 2
(p,q)dsq,
P
C
3
\{(I), ~
9
an
and
where j=O,l. W e w i l l d e a l a t each s t a g e w i t h m o d i f i c a t i o n s i n v o l v i n g o n l y T i n i t e l y many terms a l t h o u g h we w i l l comment l a t e r on t h e asymptotic b e h a v i o r o f t h e expansion c o e f f i c i e n t s ( s e e below). We w i l l t h e r e f o r e drop t h e s u p e r s c r i p t N from t h e s e o p e r a t o r s . We remark t h a t K. and S. a r e completely c o n t i n u o u s J J o p e r a t o r s on L ( a n ) p r o v i d e d of c o u r s e t h a t t h e s e r i e s o c c u r r i n g i n ( 2 . 9 )
([a])
2
converge if N-.
L 2 (D - \ B a )
The o p e r a t o r s S1 and D1 a r e c o n t i n u o u s o p e r a t o r s from L
2( a n ) t o
where Ba i s a b a l l of r a d i u s a>O, and t h e jump c o n d i t i o n s a r e
an? (2.14)
lim D . p = Tp + @ u , J
3
P+P?
pCaD ,
where K)f i s t h e L ( a n ) a d j o i n t of K. and - denotes complex c o n j u g a t e . J 2 J t h e s e r e l a t i o n s h o l d p o i n t w i s e almost everywhere on an.
Note t h a t
Using t h e s e jump r e l a t i o n s , one may proceed by s t a n d a r d arguments t o d e r i v e boundary i n t e g r a l e q u a t i o n s whose s o l u t i o n s may t h e n b e u s e d t o r e p r e s e n t a s o l u t i o n t o t h e given boundary v a l u e problem. If one employs t h e unmodified fundamental s i n g u l a r i t y ( 2 . 6 ) , t h e r e s u l t i n g i n t e g r a l e q u a t i o n p o s s e s s e s a m u l t i p l i c i t y of s o l u t i o n s a t a c e r t a i n d i s c r e t e set o f v a l u e s o f t h e number k. A s h a s been p o i n t e d out many t i m e s , t h i s l a c k of uniqueness i s a n a r t i f a c t of t h e i n t e g r a l e q u a t i o n method,for t h e boundary v a l u e problem i t s e l f e x h i b i t s unique s o l u t i o n s . This s i t u a t i o n i s i n d e e d t h e m o t i v a t i o n for s t u d y i n g m o d i f i c a t i o n s . For t h e boundary v a l u e problem ( 2 . 1 ) - ( 2 . 3 ) t h e r e s u l t i n g boundary i n t e g r a l equation i s
(2.15)
( I + K +oS.)v = f j
J
.
T: S. Angel1 and R. E. Kleinman
332
The r e s u l t s of [ 4 ] show t h a t , for j = O , one must i n t r o d u c e a second i n t e g r a l e q u a t i o n , i n v o l v i n g t h e normal d e r i v a t i v e of t h e o p e r a t o r Do, i n o r d e r t o g u a r a n t e e uniqueness of s o l u t i o n s for all v a l u e s of t h e wave number k . By u s i n g a modified k e r n e l , we may a v o i d t h e c o n s i d e r a t i o n of t h i s second e q u a t i o n . Indeed, w e proved t h e f o l l o w i n g theorem i n [9]. Theorem: k#k'
I f Ink20 and Imko>O. and i f la
e9.
+:I O ,and aee,=O,
t h e n (I+K +US )w=O if and only i f w=O. 1 1
We may t h e n use t h e unique s o l u t i o n v of t h e i n t e g r a l e q u a t i o n ( 2 . 1 5 ) t o r e p r e s e n t a s o l u t i o n o f t h e o r i g i n a l boundary v a l u e problem i n D i n t h e form o f a s i n g l e layer u = S v inD+ 1 and t h e r e b y o b t a i n t h e b a s i c e x i s t e n c e and uniqueness theorem. Theorem:
L e t aD be a c l o s e d Lyapunov s u r f a c e of index 1 and l e t I m k >-O , Re k>O.
Then for e v e r y fcL ( a n ) t h e r e e x i s t s a s o l u t i o n utC (D )nL ( a D ) of t h e Helmholtz 2 2 + 2 e q u a t i o n ( 2 . 1 ) and s a t i s f y i n g t h e Robin boundary c o n d i t i o n ( 2 . 2 ) i n t h e g e n e r a l i z e d L - s e n s e , and t h e r a d i a t i o n c o n d i t i o n (2.3), provided 1600.
-
2
3.
COMPLETE FAMILIES AND THE NULL FIELD EQUATIONS
Having e s t a b l i s h e d t h e e x i s t e n c e and uniqueness theorem, we may t u r n t o a
IRI=O d e f i n e d i n ( 2 . 7 ) .
d i s c u s s i o n of t h e f a m i l y of t h e r a d i a t i n g f u n c t i o n s {veIm 9,
It i s w e l l known ( s e e e.g. Vekua [lo]) t h a t t h e s e f u n c t i o n s form a complete family i n L2(aD). Moreover it f o l l o w s e a s i l y from t h e o r t h o g o n a l i t y of s p h e r i c a l harmonics on s p h e r e s t h a t any f i n i t e s u b s e t i s a l i n e a r l y independent s e t . t h e e x i s t e n c e and uniqueness theorem we can e s t a b l i s h t h e f o l l o w i n g r w h i c h g e n e r a l i z e s known r e s u l t s (see e . g . [111 and [12]). meorem:
me
family
1s
+
L
Using
cmzie I=o i s a complete and l i n e a r l y independent J
s u b s e t of L ( aD) p r o v i d e d I m k l O , otLm( aD), I&o_>O and Re k>O. 2 The s o - c a l l e d " n u l l f i e l d e q u a t i o n s " f o r t h e Robin problem, w r i t t e n i n t e r m s of t h e complete l i n e a r l y independent f a m i l y (2.7), can be d e r i v e d e a s i l y from t h e Helmholtz r e p r e s e n t a t i o n . Indeed, i f u i s t h e unique s o l u t i o n of t h e Robin problem ( 2 . 1 )- ( 2.3), t h e n
(3.1) 0 = l(u aD
ai,
and t h e r e f o r e
ai, This l a t t e r s e t o f e q u a t i o n s i s r e f e r r e d t o a s t h e null f i e l d e q u a t i o n s , t h e u s u a l t r e a t m e n t of which i n v o l v e s approximating u on aD w i t h a ( f i n i t e ) l i n e a r combination of elements of a complete family and s o l v i n g t h e r e s u l t i n g a l g e b r a i c ave i l e For s i m p l i c i t y , we w i l l use t h e f a m i l i e s {vi] and {- an + ma}.W e s h o u l d emphasize t h a t t h e s e f a m i l i e s do n o t , i n g e n e r a l , form a b a s i s for L2(aD).
system.
333
Scattering of Acoustic Waves by impedance Surfaces
A s f a r a s t h e a u t h o r s a r e aware, t h e r e are no g e n e r a l c o n d i t i o n s on aD which w i l l g u a r a n t e e such b a s i c i t y ; c e r t a i n l y such r e s u l t s w i l l i n v o l v e s p e c i f i c geometric c o n d i t i o n s on t h e boundary of t h e r e g i o n ( s e e e . g . [13]). N e v e r t h e l e s s , we c o n s i d e r , f o r e a c h N=0,1,2,..
., the
subspaces o f L2( a n ) ,
ave
generated, respectively,
and
choose f o r each N , f u n c t i o n s u r e s p e c t i v e l y i n L (an). Thus 2
N
e N L by {vLl,el=o and {-an
~ i band
fN,%,
%
and
N
+ ~ ev ~ } ~ and ~ Iwe= may ~ ,
which b e s t approximate u and f
and
The completeness o f t h e f a m i l i e s of expansion f u n c t i o n s i n s u r e t h a t , w i t h t h e s e choices f o r c o e f f i c i e n t s ,
and
We
now
extend
N t h e d e f i n i t i o n ( 3 . 3 ) of u t o a l l o f D+ and d e f i n e a f u n c t i o n
9N by
so that
N (3.8)
?&+ an
u $ = ~ fN ,
We may t h e n u s e G r e e n ' s f u n c t i o n s for t h e D i s i c h l e t and Robin problems, denoted by y D and yR r e s p e c t i v e l y , t o o b t a i n t h e r e p r e s e n t a t i o n s
and
s p h e r e of r a d i u s A chosen s u f f i c i e n t l y l a r g e I n t e g r a t i n g t h e s e r e l a t i o n s over t o c o m p l e t e l y c o n t a i n t h e r e g i o n D- and r e c a l l i n g t h e o r t h o g o n a l i t y p r o p e r t y of
334
TS. Angel1 and R. E. Kleinman
em t h e f u n c t i o n s {vQ)
lz0
for some c o n s t a n t
( a similar estimate holding f o r t h e d i f f e r e n c e I c Q - d i N i l ) ,
c1
on s p h e r e s , we a r e l e d t o t h e e s t i m a t e , u s i n g (2.81,
m
where u ( p ) =
(3.12)
c , v i ( p ) , r =A.
C
IQ]=O
dr)
l i m ciN) = lirn N-ro N-ro
We a r e t h e n l e d t o t h e c o n c l u s i o n t h a t
P
= cL
.
F i n a l l y , w e c o n s i d e r t h e c a s e when t h e d a t a f i s of t h e form
(3.13) f = -
au' ~
an
i
-
UU
where u1 r e p r e s e n t s a n i n c i d e n t f i e l d . This i s t h e form i n which t h e b o u n d a r y . d a t a o f t e n a p p e a r s i n s c a t t e r i n g problems. It i s u s u a l l y convenient t o r e p r e s e n t u1 i n t e r m s of incoming waves a,
where A i s t h e r a d i u s of a c i r c u m s c r i b i n g s p h e r e which c o n t a i n s no s o u r c e s . Then t h e n u l l f i e l d e q u a t i o n s ( 3 . 2 ) become
where, for n o t a t i o n a l convenience, we have r e p l a c e d t h e multi-index i n t r o d u c e t h e approximations
2 with
t.
Now
N
where, a g a i n , t h e a Q L ,a r e chosen s o t h a t t h e r i g h t hand s i d e r e p r e s e n t s t h e b e s t approximation i n
q, t o
t h e g i v e n l e f t hand s i d e .
=
N
c
The n u l l f i e l d e q u a t i o n s t h e n a r e
N
c
( R I = O IL’)=O
where we use t h e r e l a t i o n
aD
aD
N
aeIaEQ,
J
aD
ave ($
+ av:)veds
Scattering of Acoustic Waves by Impedance Surfaces
335
Using m a t r i x n o t a t i o n , (3.18) i s o f t h e form
(3.19) Q c
= Q n
N a '
This e q u a t i o n h a s t h e obvious s o l u t i o n
(3.20) c =
nN a
i n which form nN i s o f t e n r e f e r r e d t o as t h e T-matrix o r t r a n s i t i o n m a t r i x f o r t h e Robin problem s i n c e it t r a n s f o r m s t h e c o e f f i c i e n t s a = ( a , , ) of t h e i n c i d e n t wave (3.14) i n t o t h e c o e f f i c i e n t s c = ( c k y ' ) of t h e s c a t t e r e d f i e l d (3.16). W e n o t e an a d d i t i o n a l p r o p e r t y enjoyed by t h e elements aNQ R ,of t h e
T-matrix. I n [l] we show t h a t f o r f i x e d Q and R ' t h e sequence of approximations has a l i m i t i . e .
Furthermore t h e Green's f u n c t i o n f o r t h e Robin problem h a s t h e f o l l o w i n g r e p r e s e n t a t ion
when b o t h p and q l i e o u t s i d e any s p h e r e c o n t a i n i n g t h e boundary an. Thus we have e s t a b l i s h e d t h e c o n n e c t i o n between t h e T-matrix and t h e G r e e n ' s f u n c t i o n .
ACKNOWLEDGMENT T h i s work w a s s u p p o r t e d ur,der NSF Grant No. MCS-82-02-033.
336
T.S. Angell and R.E. Kleinman
REFERENCES A n g e l l , T.S. and Kleinman, R . E . ,
The Helmholtz Equation w i t h L
Boundary 2 Values, U n i v e r s i t y of Delaware Applied Mathematics I n s t i t u t e Technical Report 136A. Mikhailov, V.P., On t h e Boundary Values of t h e S o l u t i o n s of E l l i p t i c E q u a t i o n s , Appl Math Optim, 6 ( 1 9 8 0 ) 193-199. Mikhailov, V . P . , On D i r i c h l e t ' s Problem f o r E l l i p t i c Equations of t h e Second Order, D i f f E q u a t i o n s , 1 2 (1976) 1877-1891. A n g e l l , T . S . and Kleinman, R . E . , Boundary I n t e g r a l Equations f o r t h e Helmholtz Equation: The Third Boundary Value Problem, Math Meth i n t h e Appl s c i , 4 ( 1 9 8 2 ) 164-193. Miranda, C . , P a r t i a l D i f f e r e n t i a l Equations of E l l i p t i c Type, 2nd Rev. Ed. (Springer-Verlag, New York, H e i d e l b e r g , B e r l i n , 1970 ). Leis, R . ,h e r d a s Neumannsche Raudwert problem fer d i e Helmholtzsche Schwingungsgleichung, Arch R a t i o n a l Mech Anal, 2 (1958) 101-113. Kleinman, R.E., Roach, G.F., and StrEm, S., The Null F i e l d Method and Modified Green's F u n c t i o n s , t o a p p e a r . Mikhlin, S.G., Mathematical P h y s i c s : An Advanced Course (North Holland Pub. Co., Amsterdam 1970). A n g e l l , T.S. and Kleinman, R . E . , Modified G r e e n ' s F u n c t i o n s and t h e T h i r d Boundary Value Problem for t h e Helmholtz Equation, J. Math Anal and A p p l i c s , 97 (1) ( 1 9 8 3 ) 81-94. Vekua, I.N., On Completeness of a System of Metaharmonic F u n c t i o n s , Dokl. Akad Nauk. SSSR 90 (1953) 715-178. M i l l a r , R . F . , On t h e Completeness of S o l u t i o n s t o t h e Helmholtz Equation, M I A J . Appl. Math., 30 (1) (1983) 27-38. Colton, D . , Far F i e l d P a t t e r n s f o r t h e Impedance Boundary Value Problem i n Acoustic S c a t t e r i n g , Applic Anal, 16 ( 1 9 8 3 ) 131-139. K r i s t e n s s o n , G . , Ram, A . G . , and Strljm, S . , Convergence of t h e T-Matrix Approach i n S c a t t e r i n g Theory 11, J. Math. Phys, 24 (11) (1983) 2619-2631.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) Q Elsevier Science Publishers B.V. (North-Holland). 1984
337
INTERACTING NONLINEAR WAVES I N A NEURAL CONTINUUM MODEL: ASSOCIATIVE MEMORY AND PATTERN RECOGNITION Marcus S. Cohen Department o f Mathematics U n i v e r s i t y o f Kentucky L e x i n g t o n , Kentucky 40506 U.S.A.
Coupled wave e q u a t i o n s a r e d e r i v e d f o r t h e i n t e r a c t i o n o f graded p o t e n t i a l s i n a n e u r a l continuum. Waves d i v e r g i n g from two d i f f e r e n t sensory i n p u t p a t t e r n s i n t e r f e r e t h r o u g h a q u a d r a t i c memory n o n l i n e a r i t y , i m p r i n t i n g a parameter g r a t i n g . A r e p l i c a o f one o f t h e s e i n p u t waves i s Bragg-ref l e c t e d a t t h e g r a t i n g , r e c o n s t r u c t i n g t h e a s s o c i a t e d wave. Chains and c y c l e s of a s s o c i a t e d p a t t e r n s a r e r e c o n s t r u c t e d by m u l t i p l e Bragg r e f l e c t i o n , s u g g e s t i n g mechanisms f o r ass o c i a t i v e memory and p a t t e r n r e c o g n i t i o n . INTRODUCTION U n t i l t h e l a t e 1 9 6 0 ' s , most o f n e u r o p h y s i o l o g y c o n s i s t e d o f e x p e r i m e n t a l and theor e t i c a l i n v e s t i g a t i o n o f t h e " a l l o r none" p r o p a g a t i o n of a c t i o n p o t e n t i a l s i n axons.
There i s a g r o w i n g body o f o p i n i o n , however, t h a t these r e p r e s e n t t h e l o n g
d i s t a n c e " t e l e g r a p h y " o f sensory and motor i n f o r m a t i o n i n a d i g i t i z e d form, and t h a t t h e i n t e g r a t i v e f u n c t i o n s o f t h e b r a i n must i n v o l v e t h e spread and n o n l i n e a r i n t e r a c t i o n o f graded p o t e n t i a l s i n d e n d r i t i c t r e e s ( 1 , Z ) d r i t i c o r c o l a t e r a l axo-somatic synapses ( 3 ) .
and across dendro-den-
Employing a s i m p l e continuum model
f o r a n e u r a l t i s s u e c o n s i s t i n g o f c e l l s w i t h Hodgkin-Huxley t y p e membrane conductances connected b y r e c i p r o c a l l a t e r a l synapses (such as i s found i n t h e r e t i n a and sensory p r o j e c t i o n a r e a s ) we analyze t h e n o n l i n e a r i n t e r a c t i o n s o f s u b t h r e s h o l d graded p o t e n t i a l s , i.e. s m a l l a m p l i t u d e d e p a r t u r e s f r o m t h e r e s t i n g p o t e n t i a l Vr. Since Vr
i t s e l f changes w i t h i n t e g r a t e d a c t i v i t y on a s l o w t i m e s c a l e T , we d e f i n e
a q u i e s c e n t r e s t i n g p o t e n t i a l 1, t h e " b i a s p o i n t " i n t h e p r o l o n g e d absence o f exc i t a t i o n , which p l a y s t h e r o l e of a c o n t r o l parameter i n t h e model. f e a t u r e f o r o u r purposes i s t h a t , as v a l u e ,A,
The e s s e n t i a l
X i s r a i s e d past t h e threshold ( c r i t i c a l )
t h e homogeneous r e s t s t a t e becomes u n s t a b l e , and s m a l l a m p l i t u d e s t a b l e
waves b i f u r c a t e a t f r e q u e n c y wc and wavenumber kcgo. t i a l wayes."
We c a l l these "graded poten-
The s i t u a t i o n i s s i m i l a r t o a r e a c t i o n - d i f f u s i o n system whose homo-
geneous f o r m a d m i t s a Hopf b i f u r c a t i o n (4,5); however, t h e s y n a p t i c c o u p l i n g i s much more c o m p l i c a t e d t h a n d i f f u s i o n , p r o c e e d i n g as i t does t h r o u g h two i n t e r m e d i a t e v a r i a b l e s , t h e p r e - and p o s t s y n a p t i c t r a n s m i t t e r c o n c e n t r a t i o n s , w i t h d e l a y s .
338
M.S. Cohen
NONLINEAR SYNAPTIC TRANSFER F i r s t , we develop a s i m p l e model o f a chemical synapase which embodies i n a s k e l e t a l f o r m t h e known p h y s i o l o g y ( 3 ) : 1.
) Q t h a t depends on p r e Pre s y n a p t i c v o l t a g e and t h e s t a n d i n g pool o f t r a n s m i t t e r Q i s p r e s y n a p t i c P r e s y n a p t i c t r a n s m i t t e r Q i s r e l e a s e d a t r a t e K(V
vesicles. 2.
Q i s r e p l e n i s h e d a t a c o n s t a n t r a t e b.
3.
T r a n s m i t t e r r e l e a s e d f r o m c e l l s m-1 and m + l q u i c k l y d i f f u s e s a c r o s s t h e syna p t i c c l e f t s and i s bound by t h e l e f t and r i g h t p o s t s y n a p t i c membranes o f c e l l Bound p o s t s y n a p t i c t r a n s m i t t e r P o r R i s r e l e a s e d
m r e s p e c t i v e l y ( F i g u r e 1).
and degraded a t a r a t e p r o p o r t i o n a l t o i t s c o n c e n t r a t i o n .
Thus
1)
m- 1 m-1 m-1 m+l m+l m+l Qt = -K( V ) Q + b , Qt = -K( V ) Q + b
2)
m m-1 m-1 m Pt = K( V ) 4 - LP,
4.
P o s t s y n a p t i c conductance f o r some i o n i c s p e c i e s B ( n o m i n a l l y , C h l o r i d e ) w i t h
m Rt
=
m + l rn+l m -K( V ) Q - RR
r e v e r s a l p o t e n t i a l VB i s p r o p o r t i o n a l t o bound t r a n s m i t t e r . The p o s t s y n a p t i c g e n e r a t o r c u r r e n t s a r e t h u s
m 3)
m m
I = oP(V
m J =
- VB)
m m R(V - V B )
Note t h a t Ipost c o n t a i n s an " e s s e n t i a l " q u a d r a t i c n o n l i n e a r i t y : t h e p r o d u c t o f a f u n c t i o n o f v o l t a g e a t one p o i n t and t h e v o l t a g e a t a n o t h e r . L a t e r a l membrane r e s i s t a n c e i s i g n o r e d i n t h i s lumped model.
Departures f r o m t h e
m a
r e s t i n g l e v e l o f t h e n e t p o s t s y n a p t i c g e n e r a t o r c u r r e n t i n c e l l m, i + j m m ( I - I r ) + ( J - J r ) , a r e t h u s s i m p l y r e t u r n e d t h r o u g h t h e transmembrane conductance Z m m m c a u s i n g a s h i f t away f r o m t h e r e s t i n g p o t e n t i a l v :V-Vr expressed by t h e f u n c t i o n a l r e 1a t i o n
m t ~('){i(tl)
where Z
m
+j(tl),
m m i(t11) + j ( t ' l ) ,
m i(tg'1)
L i s t h e l i n e a r i z e d membrane conductance, 2'')
in
+j(t"l)}+
...
i s a quadratic nonlinearity
r e p r e s e n t i n g "delayed r e c t i f i c a t i o n " and Z ( 3 ) i s a c u b i c n o n l i n e a r i t y r e p r e s e n t i n g s a t u r a t i o n o f some i o n i c channel. t h e r e s t i n g p o t e n t i a l Vr.
The phenomenological conductances Z depend on
I n b o t h t h e Hodgkin-Huxley (6,7)
and Fitzhugh-Nagumo
models, Z ( L ) becomes u n s t a b l e v i a a growing o s c i l l a t i o n a t frequency urn (Hopf b i f u r c a t i o n ) as Vr i s r a i s e d beyond " t h r e s h o l d " Am;
t h e a m p l i t u d e i s l i m i t e d by t h e
interacting Waves in a Neural Continuum Model
339
Figure 1 A Line of Cells With Reciprocal Lateral Synapses cubic nonlinearity Z ( 3 ) ( 8 ) .
V r i t s e l f changes over a slow time scale T i f these
osci 1 1a t i ons pers i st: m m m vr = x + D ( * ) { ~ ( T )I , ~ ( T I I;TI ) 5)
m where 6 ( * ) i s a quadratic functional of v giving the e f f e c t of integrated a c t i v i t y in c e l l m upon i t s r e s t i n g potential and i s t h e quiescent value of the r e s t i n g p o t e n t i a l . This e f f e c t a r i s e s from the " r e c t i f i c a t i o n " properties of t h e quadratic synaptic and membrane n o n l i n e a r i t i e s ; o s c i l l a t o r y a c t i v i t y separates charge which changes the ionic balances across the membrane (Nernst p o t e n t i a l s ) and thus the r e s t i n g potential ( s e e ( 9 ) f o r a d e t a i l e d d i s c u s s i o n ) . As we s h a l l see below, i t i s t h i s "memory" nonlinearity b ( ' ) t h a t i s responsible f o r imprinting t h e shortterm memory t r a c e in our model. PROPAGATION EQUATIONS FOR HARMONIC WAVES W e give here the basic equations governing t h e spread of graded potential waves, an o u t l i n e of the perturbation method, and a statement of the main r e s u l t s ; the reader i s r e f e r r e d t o ( 9 , l O ) f o r a more d e t a i l e d account. The method follows t h a t of Hopf (11). The parameter deviation from c r i t i c a l i t y A-Xc and a l l t h e dynamical variables in each c e l l m a r e expanded in powers of a small parameter E , which s c a l e s the amplitude of b i f u r c a t i n g waves. Instead of the usual expansion of the
2
frequency W = W + E W ~ + E w 2 . - - with w1=0. however, we define a slow time s c a l e C d a z-,a and derive modulation T = E2t , so the time d e r i v a t i v e operator -becomes -+E dt at aT equations f o r the r a t e o f change o f the amplitude o f waves a t t h e fundamental f r e quency w with slow time T . Since we a r e looking f o r quasi-periodic s o l u t i o n s , we C f i r s t expand each v a r i a b l e in each c e l l as a Fourier s e r i e s
M.S. Cohen
340
and t h e n expand each F o u r i e r c o e f f i c i e n t as a power s e r i e s i n
E:
i
7)
m
v r- A
=
E
A-Ac
=
€1 + E 2 f 1 2
v 1+ E 2
m V2t."
x
We a l s o t r a n s f o r m t h e phenomenological r e l a t i o n 4) t o t h e frequency domain, obt a i n i n g f o r each harmonic n a t each o r d e r ( k ) ,
where we sum o v e r each i n d e x such t h a t mtr=s+u+w=n and a + b = c+d+e= k . Through t h e expansions 6 ) , 7 ) , and 8 ) , we may o b t a i n s o l u t i o n s t o 1 ) - 5 ) a t each o r d e r ( k ) f o r each harmonic n.
W e may t h e n reassemble t h e c o n t r i b u t i o n s a t t h e
f i r s t few o r d e r s i n a d j a c e n t c e l l s t o o b t a i n approximate coupled e q u a t i o n s f o r t h e p r o p a g a t i o n o f o s c i l l a t i o n s vneinWt
a t frequency n
from c e l l t o c e l l .
Finally,
we pass t o t h e continuum l i m i t t o o b t a i n an approximate p r o p a g a t i o n e q u a t i o n f o r m For example, we o b t a i n a t O ( E ) V1=O, A1=O, and v(x,t,T).
L
where
r
x(s)
= "(vy.-vB)c
J 7
and
g=aP
C
are evaluated a t
Vr=Ac
Recognizing
as t h e d i s c r e t e f o r m o f t h e L a p l a c i a n , t h i s becomes i n t h e continuum l i m i t
where
I
341
Interacting Waves in a Neural Continuum Model
and h i s t h e s p a c i n g between t h e c e l l s . O(E)
9)
C2vJi1('x,T) + k 2 ( X C , i i w c ) v
For N s p a t i a l dimensions, we o b t a i n a t
= 0
k1
A t 0 ( c L ) , we o b t a i n no c o n t r i b u t i o n t o t h e fundamental a m p l i t u d e ; however, we do o b t a i n D.C.
and second harmonic terms
11) vh2) = F ( i l o ) v i l ) v i i )
,
F(iw) = F*(iw)
= H(iw)vkl( l ) v +1 (l)
12) ~1.)
f r o m t h e m i x i n g o f terms a t + i w and a t - i w t h r o u g h t h e q u a d r a t i c n o n l i n e a r i t i e s . The importance o f t h e D.C.
t e r m i s t h a t i t d r i v e s charge s e p a r a t i o n across t h e
membrane, l e a d i n g t h r o u g h ( 5 ) t o a d r i f t slow t i m e T ( s e e ( 9 ) ) .
13)
D(2){~l:)(;,T’)~i:)(jt,T’);T}
i n Vr w i t h
2
Thus, t o O ( E ) ,
v ~ ( z , T ) - A :E 2 v 2 = E 2 ~ ( 2 ) {1 v ( 1 -) 1 v ( 1 ) ; ~.~
We a l s o f i n d h2 > O f o r t h e c l a s s o f models t h a t g i v e s m a l l a m p l i t u d e b i f u r c a t i n g waves, c o r r e s p o n d i n g t o a s u p e r c r i t i c a l b i f u r c a t i o n ; t h u s E2
X;l(X-Xc)
f o r small
X-Xc
.
3 A t O ( E ) we o b t a i n a new c o n t r i b u t i o n t o t h e fundamental t h r o u g h t h e c u b i c n o n l i n e a r i t y , gvil)vIi)vil),
where g i s complex, and A 3 = V 3 = 0 .
No new c o n t r i b u t i o n s t o
4
t h e fundamental appear a t O ( E ) . 4 Reassembling a l l t h e c o n t r i b u t i o n s t o v1 up t o O ( E ), we f i n a l l y o b t a i n a t r u n c a t e d m o d u l a t i o n e q u a t i o n f o r s l o w l y v a r y i n g waves vl(;.T)eiWt quency w 14)
a t t h e fundamental f r e -
=w : C
a
i p h aT v +
where v - v l , U = X 2 ( ak 2
v2 v + Ckc2 + v(U + F{v*v;TI + g v * v l v
v*-vel,
p=c2=Xi1(X-Ac),
= 0
h = ( a k2
and FIv*v;T) = ( a k 2/ a A ) c D ( 2 ) { ~ * ~ ; T 1 .
k 2 = k 2 (Xc,iwc), C
342
M.S. Cohen
f Irn
k2
Re k 2
A C
Figure 2 D i s p e r s i o n r e l a t i o n k2(A,iw). k: k 2 ( h c , i u c ) and (ak2/aw)c a r e b o t h r e a l . (Decaying waves broken, growing waves sol i d 1 i n e s . ) 2 k (i,i+w)
i s t h e l i n e a r d i s p e r s i o n r e l a t i o n 10) c h a r a c t e r i z i n g p l a n e waves
z-%=k 2 and
-
To a s c e r t a i n t h e s t a b i l i t y o f propaC -t ++ g a t i n g waves, we i n t r o d u c e a l o n g space s c a l e X = v z and reexpand v(x,X,T) =
ei(k'xtwt),
where
v(')(z,?,T)
+pv(')(z,;,T)
15)
+ k;v(O)
$v(o)
(x,y,z).
=;
We o b t a i n a t O ( 1 ) :
+O(p2).
= 0 ;
t h i s has planewave s o l u t i o n s v'')(;,?,T)
++ =A(?,T)eikax, where
llKll 2 = kc .
For X=X
C'
waves a t wC n e i t h e r grow n o r decay i n t h e i r d i r e c t i o n s o f propagation; t h u s , k c i s 2 r e a l . Since we r e q u i r e waves a t W Z W ~t o decay, (ak / a w ) c = 2 k c ( a k / a w ) c must be r e a l ; t h u s , t h e group v e l o c i t y V = - ( a o i / a k ) c i s r e a l . g o f growing waves about wc,kc (see F i g u r e 2 ) .
For X > X
C'
t h e r e i s a band
A t O(u) we o b t a i n an inhomogeneous e q u a t i o n f o r v(') in t h e s e l f - a d j o i n t ope:a$or 2 V2 + kc; t h e Fredholm a l t e r n a t i v e t e l l s us t h a t t h e terms on t h e R.H.S. i n eikSx 2 for k must vanish, g i v i n g a n e t m o d u l a t i o n e q u a t i o n C
16)
Ac :AT
V
-V
E - v - ~ A = -i 3 ( U g
x
+ F{A*A;T} + gA*A)A
2 kc
i s t h e r a y c o o r d i n a t e i n t h e frame moving a t V i n t h e where 5 E T - V-'^k.? 9 g t i o n . The "energy" e q u a t i o n
^k
direc-
t h e n t e l l s us t h a t a ) waves w i l l grow i n 5 f r o m since
I A l2 = O
if Im U > O ,
A2 > O ( c o n v e c t i v e i n s t a b i l i t y ) .
t h e n o n l i n e a r terms, i f
i.e.
2 i f ( a I m k /aA)c>O,
The a m p l i t u d e w i l l be l i m i t e d b y
343
Interacting Waves in a Neural Continuum Model
b ) lm g < 0 ( h a r d l i m i t t h r o u g h t h e c u b i c n o n l i n e a r i t y ) , and/or
2
c ) t h e d e l a y f u n c t i o n a l F{ / A \ ;T] accumulates a s u f f i c i e n t n e g a t i v e i m a g i n a r y 2 p a r t as \ A ] p e r s i s t s i n t i m e ( A u t o m a t i c Gain C o n t r o l , o r A.G.C.). a ) and b ) o r c ) a r e f u l f i l l e d by t h e c l a s s o f models o f i n t e r e s t t o us here; we conclude t h a t bifurcating waves are stabZe (see ( 9 ) f o r a s p e c i f i c example based on t h e Hodgkin-Huxley scheme). Now a l i n e a r i z e d model o f t h e charge d i s p l a c e m e n t c u r r e n t s across t h e membrane i n response t o t h e D.C.
'I
i n t e r f e r e n c e " term v o - p l v ( o ) 1 2
r e s t i n g p o t e n t i a l V,-x
where A,
a
(9) predicts a s h i f t i n the
w i t h i n t e g r a t e d a c t i v i t y g i v e n by
z ~ D ( ~ ) {
F o r C 1 ;
,
for 0 < x < 1
,
=
;
f o r x = 0,1
;
a n d t h e r e s u l t i n g s c a t t e r e d f i e l d s a r e o u t g o i n g f r o m t h e t a r g e t a s Ixl+m.
In
( 2 . 2 ) , t h e s u b s c r i p t s x d e n o t e d i f f e r e n t i a t i o n , a n d we h a v e u s e d t h e f o l l o w i n g notation,
where C
and C
respectively.
t
a r e t h e sound v e l o c i t i e s o f t h e s u r r o u n d i n g f l u i d and t a r g e t , The i n t e r f a c e c o n d i t i o n s ( 2 . 2 )
a r e c o n t i n u i t y c o n d i t i o n s on t h e
a c o u s t i c p r e s s u r e and v e l o c i t y .
A wave 0 ' .
which i s d e f i n e d b y ,
i s i n c i d e n t from
x=-m
on t h e t a r g e t .
The r e s u l t i n g s c a t t e r e d f i e l d c o n s i s t s o f
a r e f l e c t e d wave i n x l which
is o u t g o i n g a s x+=. i n Section 4.
The e x a c t s o l u t i o n o f t h i s s c a t t e r i n g p r o b l e m i s p r e s e n t e d
I t is e a s i l y o b t a i n e d f o r any v a l u e o f E .
However,
f o r exposi-
t o r y p u r p o s e s we assume t h a t i t i s " s i m p l e r " t o s o l v e ( 2 . 2 ) f o r t h e r i g i d t a r g e t . T h u s , f o r E=O t h e s o l u t i o n o f
( 2 . 2 ) t h a t i s i n r e s p o n s e t o t h e i n c i d e n t wave
( 2 . 4 ) a n d t h a t s a t i s f i e s t h e o u t g o i n g wave c o n d i t i o n s a s Ixl+m i s a wave r i g i d l y r e f l e c t e d a t x=O a n d p r o p a g a t i n g t o x=--m.
(2.5)
I t is g i v e n by,
I
o
,
x > l .
H e r e , t h e b a c k g r o u n d f i e l d O a is t h e sum of t h e i n c i d e n t a n d r l g i d l y r e f l e c t e d B -ikx ) f o r xO, b e c a u s e t h e t a r g e t i s r i g i d s o t h a t waves ( e t h e r e is no t r a n s m i t t e d f i e l d .
W e e x p r e s s t h e s o l u t i o n of t h e n e a r l y r i g i d s c a t t e r i n g p r o b l e m E # O
(2.2)
as
364
G.A. Kriegsrnann eta/.
(2.6)
Then,by inserting (2.6) into (2.2) and using (2.5), we deduce that the scattered potentials $a and
Ot
must satisfy the Helmholtz equations (2.2a) and (2.2b),
respectively, the boundary conditions:
a
+x
(2.7b)
=
t
t
+x ,
= E + ~
,
for x
1
=
and the appropriate radiation conditions on Qa as
;
Ixl+m.
We now reformulate this scattering problem for $a and t problem for $ . To do this we express the potential
and the outgoing radiation conditions as Ixl-,
- ikx Rse $a
i
and T
t @I
it is given by
,
for x 5 0 ,
,
for x
=
! T elkx
where R
as a boundary value in terms of values of
Since $a satisfies the Helmholtz equation (2.2a)
on the target boundaries x=@,1.
(2.8)
Ot
s
>
1
,
are the scattered reflection and transmission coefficients, to
be determined. The total reflection and transmission coefficients are related to R
and Ts by
(2.9)
R
1 + Rs
,
T
Z
T
By inserting (2.8) into the boundary conditions (2.7) and eliminating R
and T
from the resulting four equations, we obtain the required boundary value problem t for Q as
(2.10b)
When (2.10 is solved then R
and T
are given by
Acoustic Scattering b y nearly Rgid or Soft Targets
(2.11)
R
l t
s
k x (0) ,
=-$
T
=-
and hence the scattered potential ’$
3.1
365
is given by (2.8)
The Outer Expansion
To solve (2.10), we first seek an asymptotic expansion of $ t in the form
Then by inserting (3.1) into (2.10) and equating coefficients of the same powers t of E , we find, in the usual way, that satisfies
The solution of (3.2) is given by
(3.3)
where for u=1,2, ..., the functions JI ( X I
the constants k
are defined by
are given by V
(3.4b)
k,, = v n / n ,
and the parameters B
are defined by
The functions $ (x) and parameters k the "in vacuo" target problem
are the eiqenfunctions and eigenvalues of
G.A. Kriegsmann et a/.
366 (3.5.3)
2 2 Jixx+knJi
=
0 ,
O < x < l
The eigenfunctions are normalized by the condition ($u,@u)
=I, where we use the
inner product,
for any two functions f and g.
All the eiqenvalues of (3.5) are simple.
For
more complex scattering problems the Corresponding eiqenvalues of the target may be multiple, see e.g. [l].
Higher order terms in (3.1) can similarly be computed. In particular, t t satisfies (3.2a) and the boundary conditions Q ( O ) = i Q (O)/k and 2 1.x t t t Q2(1)=-iQ (l)/k. Thus, Q, is proportional to l/k and our outer expansion l,x becomes, to leading order
(3.6a)
Q t (x,k;E)
=
t
E $ ~ ( x , + ~ )O ( E
2 /k)
It is clear from (3.3) that the outer expansion (3.6.3) is invalid as k+k
u=1,2,3... because then
Qi+-
as k-k
.
v ' Furthermore, it i s invalid as k+k EO 0 2
because the second term in (3.6a) is O ( E /k).
The corresponding leading order outer expansions for R
and T
which follow from S'
(2.11), (3.4) and (3.6a), are
(3.6b)
These functions become singular as k+k , u=0,1,2, ...
3.2
.
The Inner Expansion
We now obtain asymptotrc representations of Qt which are valid for k near k fc,r each fixed u=1,2,...
.
Accordlngly, we define the "inner variable" a by
367
Acoustic Scattering by nearly Rigid or Soft Targets
k = k (1+Ea)
(3.7)
This Lmplies that a=O(1) when k-k = O ( E ) .
Then we seek an asymptotic expansion
for $t, valid for a=0(1),in the form
Inserting (3.7) and (3.8) into (2.10) and equating coefficients of the same powers of
E,
we find, for example, that Uo and U1 satisfy:
(3.9a)
U
(3.10a)
Ulxx
(3.10b)
U1(0)
Since k
+ k 2n2U
v
oxx
+
0
0
=
-2k2an2U
2 2 k n U v 1 2
=
+
,
=
' 1u k
Ox
is an eigenvalue of (3.51,
f o r O < x < l ,
,
(0)
it
f o r O < x < l ,
U](l)
=
- 1.
UOX(l)
.
V
follows from (3.9) that
is to be determined.
where the constant A
,
When (3.11) is inserted into (3.10)
an inhomogeneous eigenvalue problem must be solved to determine U (x). To 1 insure the existence of U (x) a solvability condition must be satisfied. Appli1 cation of this condition leads to a linear equation for A . Omitting all details we finally obtain
(3.12)
Thus, to leading order, and for each fixed v=1,2 ...
(3.13a)
$t
=
A
$J
v v
(x)
+
O(E)
, we have the inner expansion
.
The corresponding inner expansions, to leading order, for R (2.,11) with k=k
V’
(3.4a), (3.7), (3.12), and (3.13a).
and T
They are
follow from
G.A. Kriegsrnann et al.
368 (3.13b)
R~
=
i f i A"
(3.13~)
T
=
-ifi
+
O(E)
.
+ o(E)
Finally, we investigate the solution of (2.10) when k is near k
0
. We intro-
duce the new stretched variable a by
(3.14)
k
a~
=
and seek an asymptotic expansion for Q
t
valid for a = 0 ( 1 ) , of the form
InsertLng (3.14) and ( 3 . 1 5 ) into (2.10) and equating the coefficients of the same powers of
(3.16)
we find that U1 satisfies
E,
Ulxx = 0, O < x < l ,
U (O)+iaU (0)= 2ia lx 1
,
Ulx(l)-icuU (1) = 0 1
.
The solution of (3.16) is
and thus, to leading order, (3.15) gives (3.17b)
Qt
.
eQ0(x,a)+O ( E 3 )
=
The corresponding inner expansions, to leading order, for R
and T
valid near k=O, follow from (2.11), (3.14), ( 3 . 1 7 ) , and (3.18a).
(3.18)
3.3
= --
R S
2i~ + k+2i~
2
O ( E ) ,
T
=
~
2 1 ~ e-lk + k+2ie
which are They are
2
O(E )
.
The Matching Conditions
In the method of matched asymptotic expansions it is assumed that there exists an overlap interval in which the outer and inner expansions are valid asymptotic approximations o f the solution of the scattering problem.
This implies that in
this interval the expansion coefficients must satisfy matching conditions.
They
are obtained by first expressing the outer expansions in terms of the stretched variable rr by using (3.7) or (3.14) and then reexpanding the result as a new
Acoustic Scattering by nearly Rigid or Soft Targets
power serLes in
E
to give
at
(3.19)
369
m
=
1,
-t $j(XtU)E1
.
i=O
The series (3.19), which we call the residual expansion, is the outer expansion in terms of the stretched variable in the overlap interval.
The matching
conditions are then given by
1im
(3.20)
, j
1
=
0,1,..
Omitting all details, we can show that the conditions of (3.20) are identically satisfied.
3.4
and T
Similar calculations and results follow for R
.
The Uniform Expansion
The composite expansion of the method of matched asymptotic expansions is defined as the sum of the outer and inner expansions minus the residual expansion.
Thus,
by employing (3.3). (3.6)-(3.8), (3.11), and (3.12) we obtain to leading order
(3.21.3)
(3.21b)
These expressions are uniformly valid in an interval containing the simple eigenvalues k
or ko, respectively.
the k=k. ( j # v ) . 3
It is valid for any other k bounded away from
We obtain an asymptotic expansion that is uniformly valid for
all k by summing the last term in (3.21) over all values of v .
We find that
at
becomes
(3.22a)
$t
=
2~ [k(1-X)+IE k+21~
1 k2-k2
m=l
Similar considerations and calculations for R
m
(3.22b)
R
=
-2i~-+21 [k+iie
m=l
{
+
and T
- -k k2-k2
(k-k
m
21fiE ) (k-k
give the uniform results
ie (k-k ) (k-k +2ie) m m
il
370
G.A. Kriegsrnann et al.
(3.22~)
A uniform asymptotic expansion of the scattered potential now follows by
inserting (3.16)-(3.17), (3.22b) and ( 3 . 2 2 ~ )into (2.8).
4.
A Comparison of Exact and Asymptotic Results
The exact solution of the scattering problem (2.10) is
0,
(4.1)
for any value of
(4.2)
t =
~
a(l+an) eikn(x-l) A
+
(an-1)
A
-ikn(x-l)
e
20, where 2 2 2an cos(kn) - i(l+a n ) sin(kn)
A
.
By inserting these results into (2.11) we obtain the exact scattered reflection and transmission coefficients, respectively as
(4.3)
RE S
(4.4)
TS
E
=
=
-2an cos(kn)
+
2ia2n2 sin(kn)
A
2ane
-ik
/n
.
We now show the relationships between the exact results (4.1)-(4.3) and the uniform asymptotic results by taking various limits.
We first consider the outer limit in (4.3) where kfk m is fixed and a+O. gives
(4.5)
R~
=
This
.
-2ia cot(kn) + O(E 2)
S
In the same limit the uniform asymptotic approximation (3.22b) is reduced to the outer expansion (3.6b).
The equivalence between these two results is demon-
strated by inserting the Mittag-Leffler expansion for cot(kn) ,
(4.6)
into ( 4 . 5 ) .
cotikn)
=
2
['
+ 2
m=l
4 1 k -k
E
A similar agreement can be shown in this limit between TS and ( 3 . 6 ~ ) .
37 1
Acoustic Scattering b y nearly Rigid or Soft Targets
The outer limit of t r$E
(4.7)
r$g from
(4.1) i s
- 2 ~ sin kn(x-1) + sin (kn)
=
2
O(E )
.
To demonstrate the equivalence between (4.7) and the outer limit (3.6ai of (3.22a) we first rewrite (4.7) as
where g(x) is defined by,
sin kn(x-1) sin(kn)
-2(1-x) - 2
z
g(x)
(4.8b)
Since g(x) is a smooth function and q(l)=g(O)=O, it can be expanded in an eigenfunction series using the IJ (x). A straightforward calculation shows that the m 2 2 /(k -k ) . Inserting this result in (4.8) shows
expansion coefficients are
m
that (4.7) and (3.6) are identical.
We now consider the inner limit where a
(k-k )/(Ek ) and v > 0 are fixed, and
=
V
E + 0.
In this limit the exact scattered reflection coefficient (4.3) is
(4.9)
which agrees exactly with the inner limit of RS given by (3.13b).
Similar
agreement is obtained for the transmission coefficient.
I n the inner limit the exact potential given by (4.1) is
t ’ E
(4.10)
=
-2 sin VTX n[kua+21)
f
O(E)
.
This is identical to the inner limit given by (3.12) and (3.13) of the uniiorm asymptotic approximation (3.22~1). Similarly, we can show that the inner limit of the exact result and of the uniform asymptotic approximation are identical when a
=
k/E O ( 1 ) and
E
+
0. E
Finally, we have made graphs of RE and RS, and of TS and TS, to numerically evalS uate the accuracy of the uniform asymptotic approximations. For E < . 1 and k in the interval O 0 )
Pla was i n t r o d u c e d , b u t n o t i d e n t f i e d . a c y l i n d e r f u n c t i o n o f order Kp(p,a)
(14)
p;
It i s not d i f f i c u l t t o v e r i f y t h a t
K
more p r e c i s e l y ,
v
is
=
Y i s t h e Neumann f u n c t i o n , which can be r e p l a c e d by a Hankel f u n c t i o n i f P d e s i r e d . T h i s r e s u l t seems t o be new.
Here
The c r o s s p r o d u c t o f Bessel f u n c t i o n s t h a t appears i n (14) a r i s e s i n t h e e x p r e s s i o n f o r t h e Green's f u n c t i o n f o r a c i r c u l a r c y l i n d e r .
A t t h i s time i t i s
u n c l e a r whether (14) would be o f use i n summing t h i s , o r o t h e r , s e r i e s . i s such t h a t
J (ka) = 0 o r P
representations f o r
J (kp) &
If
Y ( k a ) = 0, t h e n (13) and (14) p r o v i d e i n t e g r a l
P
and
Y (kp), r e s p e c t i v e l y . P
Other Bessel f u n c t i o n r e l a t i o n s can be based on a d i f f e r e n t r e s u l t from
[lo]:
k
399
Integral Representations
i f t h e Cauchy d a t a f o r t h e c y l i n d r i c a l wave f u n c t i o n
the radial line
a r e g i v e n on
> 0, i t i s found t h a t
8 = 0, 0
Cp(kP)
(15)
Cp(kp)eipe
=
eie
-
C (ks)
iJp -i0
2 2 J o ( k A + p -2sp cos e)ds.
Pe
V a r i o u s r e s u l t s can be found by i n c r e a s i n g 1 e =7 ,one o b t a i n s 2 1 s i n ( 7 pn) (16)
CJkd
=
-7
c o n t i n u o u s l y from
0
0.
If
-
C (ikt) t J o ( k m ) d t ,
p
iJ -P
where t h e p a t h o f i n t e g r a t i o n i s i n d e n t e d below t h e p o s s i b l e s i n g u l a r i t y a t t = 0.
C
By s e t t i n g
Re p > 0), we f i n d
= J
C L P
,
I.r > 0 ( o r even
k = 1, and assuming t h a t
which corresponds t o a r e s u l t g i v e n by Watson “1, p. 374, ( 4 ) ; t h e omission o f 1 a factor of on t h e r i g h t - h a n d s i d e i n Watson’s r e s u l t has been noted by H e n r i c i [4] p. 292, and o t h e r s .
I f we s e t
C
P
= J
P
and
k = i i n ( 1 6 ) , we
get
which may be regarded as a r e l a t i o n i n v e r s e t o ( 1 7 ) .
s = 0
( 1 5 ) , t h e i n t e g r a t i o n p a t h becomes a l o o p around On l e t t i n g
C
P
= J
P
( p > 0)
n e g a t i v e r e a l a x i s from
0
from pe-in
in
0 =
TC
to
pein.
and c o l l a p s i n g t h i s l o o p o n t o t h e segment o f t h e to
-p
t h i s i s a s p e c i a l case o f f o r m u l a
(p
> 0 ) , one o b t a i n s t h e r e s u l t
11.3.40
As a f i n a l Bessel f u n c t i o n example, l e t Then
When one s e t s
i n [19]. C
P
= J
,
@
p > 0, 8 = 3n/4 i n ( 1 5 ) .
400
R. F. Millar
p (ber t+ibei t )
(20)
sin(3d4) = Im
JJP)
which expresses
Jo
,
) I Jo(J(p-t)(p+it))dt
t
i n terms o f t h e K e l v i n f u n c t i o n s g i v e n by
J (p)
P
b e r t + i b e i t := J (tei3r/4). P P P R e s u l t s f o r Mathieu f u n c t i o n s can a l s o be o b t a i n e d from ( 1 2 ) . r e l a t i o n s g i v e n by Volkmer [7]
I n particular,
and by Meixner e t a1 c20] a r e r e a d i l y found.
It
i s o n l y necessary t o p r e s c r i b e Cauchy d a t a f o r a separated s o l u t i o n t o t h e H e l m h o l t z e q u a t i o n on a c o o r d i n a t e a r c i n an e l l i p t i c c o o r d i n a t e system. rectangular coordinates by
f =
( c , ~ )o f
c cosh p cos t ~ , v = c s i n h
s i n 0, t h e n a separated s o l u t i o n i s a
product o f a s o l u t i o n t o Mathieu's equation i n m o d i f i e d Mathieu e q u a t i o n i n
If the
a p o i n t i n t h e p l a n e a r e g i v e n i n t h i s system
On s e t t i n g
p,.
0 and a s o l u t i o n t o t h e p = is
where
0
i s real, it i s
seen t h a t t h e p r o d u c t c o n s i s t s e n t i r e l y o f s o l u t i o n s t o t h e Mathieu e q u a t i o n
+
(b-
f o r t h e same p a i r o f values o f
kc
Y"(x)
(21)
1 2 2
7k c
COS
~ x ) Y ( x )= 0
,
and t h e s e p a r a t i o n parameter
b.
The f o r m
of t h e f i n a l r e s u l t does n o t depend on whether a h y p e r b o l i c a r c o r an e l l i p t i c a r c i s chosen f o r t h e choice o f
0
C. = 0
I n p a r t i c u l a r , i f v and w a r e any s o l u t i o n s t o ( 2 1 ) , C w i t h t h e r e p r e s e n t a t i o n (12) l e a d s e v e n t u a l l y t o
for
the relation
i n which
Z := k c d c o s 6-cos( t ~ + ~ ) ) ( c o e-cos( s +-a)).
By d i r e c t computation, i t may be v e r i f i e d t h a t any p a i r o f t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s t h a t s a t i s f y (22) a r e s o l u t i o n s t o (21) f o r t h e same parameter p a i r
kc, b.
I f one i n t e g r a t e s t h e f i r s t i n t e g r a l i n ( 2 2 ) by p a r t s , i t i s seen t h a t
40 1
Integral Representations
V(~)W(@ = ) v ( o ) ~ ( $ - ~+)
(23)
1
J
$+0
Cv'(O)w(e) + v(o)w'(e)lJo(Z)de
@-
0
p. 72, (10). On t h e o t h e r hand, i f t h e f i r s t and t h i r d i n t e g r a l s i n (22) a r e combined, t h e r e r e s u l t s
a r e l a t i o n g i v e n e a r l i e r i n [20],
a r e s u l t obtained previously i n
(71.
B o t h Volkmer and Meixner e t a1 c o n s i d e r t h e s p e c i a l case i n which yl(0)
= 0, y i ( 0 ) = 1.
By f i x i n g
where
Then
i n ( 2 5 ) , t h e y show t h a t two homogeneous i n t e g r a l e q u a t i o n s may b e
w.
o b t a i n e d f o r c e r t a i n p e r i o d i c Mathieu f u n c t i o n s i s known;
v = yl,
The form o f t h e s e e q u a t i o n s
what seems t o be new i s t h a t t h e e i g e n v a l u e parameters a r e g i v e n
e x p l i c i t l y i n terms o f values o f t h e f u n c t i o n
yl.
The p r e s e n t w r i t e r i s n o t aware o f any f u r t h e r s t u d i e s o f ( 2 3 ) o r ( 2 4 ) . Clearly,
w
i f a solution
any o t h e r s o l u t i o n
v
i s known, each g i v e s an e x p l i c i t r e p r e s e n t a t i o n f o r
i n terms o f
v(0)
r e l a t i o n s a r e o b t a i n e d by i n t e r c h a n g i n g
and v
and
v'(0).
w
Evidently the inverse
i n ( 2 3 ) or ( 2 4 ) .
i t i s not d i f f i c u l t t o obtain nonlinear equations f o r solutions.
by t a k i n g
i n which
v = w
and
Zo := Z ( 4 = o
=
.
Then
i n ( 2 3 ) , we f i n d
Moreover,
F o r example,
R. F. Miiiar
402
and, i f
y2
i s a s o l u t i o n t o (21) w i t h
y 2 ( 0 ) = 1, y;(O)
= 0,
It i s n o t known whether a s t u d y o f t h e s e n o n l i n e a r e q u a t i o n s has been made.
But
c e r t a i n l y i t i s t e m p t i n g t o s p e c u l a t e t h a t t h e work o f Volkmer [ 7 ] and o f Meixner e t a1 [20]
does n o t exhaust a l l t h e p o s s i b i l i t i e s f o r (23) o r ( 2 4 ) .
I n p r i n c i p l e , by s e p a r a t i n g a l l b u t two o f t h e v a r i a b l e s , t h e methods d e s c r i b e d above can be a p p l i e d t o e q u a t i o n s i n more t h a n two independent v a r i a b l e s .
More
s p e c i f i c a l l y , Volkmer [8] has r e c e n t l y a p p l i e d h i s t e c h n i q u e t o study c e r t a i n p r o p e r t i e s o f Lam;
f u n c t i o n s , which a r i s e when t h e H e l m h o l t z e q u a t i o n i s
separated i n e l l i p s o i d a l c o o r d i n a t e s . A n a l y t i c Continuation I t i s o f t e n u s e f u l t o know t h e e x t e n t t o which a n a l y t i c c o n t i n u a t i o n i s p o s s i b l e
f o r t h e s o l u t i o n t o an e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n .
For example, t h e
p o s i t i o n o f s i n g u l a r i t i e s determines t h e convergence p r o p e r t i e s o f some s e r i e s representations.
T h i s i s i n t i m a t e l y r e l a t e d t o t h e R a y l e i g h h y p o t h e s i s , which
has a c o n s i d e r a b l e l i t e r a t u r e C15-j.
Methods based on (12) have been used i n
t h i s connection, and a r e o u t l i n e d i n [21]. Another area i n which knowledge o f p o i n t s o r r e g i o n s o f n o n a n a l y t i c i t y i s o f i n t e r e s t i s t h a t o f i n v e r s e problems.
Here we r e f e r s p e c i f i c a l l y t o t h o s e t h a t
can be f o r m u l a t e d as problems f o r t h e H e l m h o l t z e q u a t i o n w i t h i n a c c u r a t e Cauchy data.
A problem o f t h i s s o r t w i l l be d i s c u s s e d l a t e r , b u t f i r s t we s h a l l o b t a i n
from ( 1 2 ) a few u s e f u l f o r m u l a s under t h e assumption t h a t a t l e a s t some o f t h e Cauchy d a t a a r e known p r e c i s e l y .
I n essence, these r e s u l t s w i l l p r o v i d e r u l e s
f o r r e f l e c t i n g a s o l u t i o n across an a n a l y t i c boundary c o n d i t i o n . With r e g a r d t o ( 1 2 ) , we make t h e f o l l o w i n g o b s e r v a t i o n s : ( i ) The v a r i a b l e l i m i t s o f i n t e g r a t i o n suggest V o l t e r r a i n t e g r a l s . ( i i ) S i n c e r = 0 a t each l i m i t o f i n t e g r a t i o n , i t may be p o s s i b l e t o choose a p a t h i n t h e s-plane on which
r
i s real.
403
Integral Representations
I n fact,
i n t h e s i t u a t i o n t h a t we c o n s i d e r , we s h a l l be a b l e t o choose s p e c i f i c
paths o f i n t e g r a t i o n t h a t l e a d t o r e a l i n t e g r a l e q u a t i o n s t h a t can be i n v e r t e d expl i c it l y
.
The s i m p l e s t case occurs when t h e a r c C s h a l l confine a t t e n t i o n t o t h i s .
i s p a r t o f a s t r a i g h t l i n e , and we
( A n o t h e r f a i r l y s i m p l e case a r i s e s when
C
is
a c i r c u l a r arc; b u t we have been unable t o o b t a i n e x p l i c i t l y i n v e r t i b l e e q u a t i o n s i n t h e s e circumstances.) x
and
= V(x,y)
+ J
t h e o r i g i n , and i f
y
u(x,y)
V,W
Here
0
V(x,a)
v(x,y)
:= $u(x+iy,o)
(31)
w(x,y)
:= $uI1(xtiy,o
Y
&
+ J
J0(k@?)da
0
W(x,o)Jo(kAq)da.
even i n y, t h a t a r e d e f i n e d b y
a r e harmonic f u n c t i o n s
(30)
i s an a r c o f t h e c - a x i s t h a t c o n t a i n s
a r e r e a l , t h e n ( 1 2 ) may be reduced t o
Y
(29)
C
If
u(x-iy,O)l
t
+
u (x-iy,O)] r)
,
V(x,O)
,
= u(x,O)
w(x,O)
,
= u (x,O) rl
.
Although we now have t o deal w i t h f a i r l y s i m p l e i n t e g r a l s , t h e r e i s a p r i c e t o pay: we have l o s t t h e f l e x i b i l i t y i n d e f o r m i n g paths o f i n t e g r a t i o n t h a t we had i n ( 1 2 ) . Consequently, ( 2 9 ) w i l l n o t g e n e r a l l y be s u i t a b l e f o r mapping o u t t h e detailed singularity structure.
I f Cauchy d a t a
are prescribed a n a l y t i c a l l y , then (29) u(x,O), u (x,O) r) determines u ( x , y ) , a t l e a s t i n a neighbourhood o f t h e z - a x i s . Suppose, however, y > 0
that
b u t s u f f i c i e n t l y s m a l l , and t h a t e i t h e r
prescribed analytically. either on or
0 V.
V(X,O)
5
0
5y
or
-
or
W(X,U) u(x,-y)
0 5 U’Y -, ( 2 9 ) can be
i s known f o r
y = 0.
is
( y > 0)
and, i f
More s p e c i f i c a l l y ,
if
Thus
u ( x , ~ ) i s known
s o l v e d f o r t h e unknown
W
t h e n can be found from
which f o l l o w s f r o m ( 2 9 ) , and which may be regarded as a r u l e f o r across
u (x,O) r)
even by measurement
The v a l u e o f
u(x,O)
(Impedance-type c o n d i t i o n s can a l s o be handled.)
W
i s known, t h e n
e f lection
Jr/k ) / J?
v ( t ) := V ( x ,
s a t i s f i e s t h e V o l t e r r a i n t e g r a l e q u a t i o n o f t h e second k i n d
R. F. Millar
404
(33) i n which
F
i s known and i n v o l v e s
u(x,a)
(0
5 0 5y)
and
W.
The s o l u t i o n
by Laplace t r a n s f o r m methods i s
(34) so
V
i s determined.
If
V
i s known, t h e n d i f f e r e n t i a t i o n o f ( 2 9 ) shows t h a t
w ( t ) := W(x,JF/k)/(PkJf-) involving
U ~ ( X , ~ ()0
s a t i s f i e s ( 3 3 ) w i t h given right-hand side
5
5 y)
and
V.
Thus
W
G(c)
may be determined.
When an a n a l y t i c D i r i c h l e t o r Neumann c o n d i t i o n i s p r e s c r i b e d on an a r c o f t h e c - a x i s t o g e t h e r w i t h a p p r o p r i a t e values o f
u
in
s t a b l e c o n t i n u a t i o n across t h i s a r c i s f e a s i b l e .
y > 0, i t i s c l e a r t h a t
Unfortunately, t h i s i s not t h e
s t a t e o f a f f a i r s f o r i n v e r s e problems w i t h i n a c c u r a t e Cauchy d a t a , t o which we now d i r e c t our a t t e n t i o n .
I n v e r s e Problems I n v e r s e problems f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s f r e q u e n t l y r e q u i r e t h e s o l u t i o n t o Cauchy problems w i t h i n a c c u r a t e d a t a .
As examples, we may mention
t h e c o n t i n u a t i o n o f an a n a l y t i c f u n c t i o n o u t o f some domain, i n v e r s e g r a v i t a t i o n a l problems, o r i n v e r s e c l a s s i c a l s c a t t e r i n g problems.
Further
c o n s i d e r a t i o n i s r e s t r i c t e d t o problems where d a t a a r e known a p p r o x i m a t e l y i n some l i m i t e d r e g i o n i n c l u d i n g o n l y p a r t o f i t s boundary.
The r e p r e s e n t a t i o n
( 1 2 ) appears t o be w e l l s u i t e d f o r f o r m u l a t i n g i n v e r s e problems o f t h i s k i n d i n t w o independent v a r i a b l e s .
We c o n s i d e r now an i n v e r s e s c a t t e r i n g problem i n
which Cauchy d a t a a r e known only a p p r o x i m a t e l y on a segment T h i s i s presumably t h e case i n
C
o f the c-axis.
many p r a c t i c a l s i t u a t i o n s where t h e c h o i c e o f
data surface i s a t t h e d i s c r e t i o n o f t h e i n v e s t i g a t o r . Suppose t h a t d i f f e r e n t media occupy n o t fill a l l o f medium i n
< 0
q < 0.)
q
> 0 and q < 0.
The wave number t h r o u g h o u t
(The l o w e r medium need
11 > 0
c o n s i s t s o f m a t e r i a l w i t h wave number
is
K(> 0),
k ( > 0).
The
i n which t h e r e
may be imbedded i n t e r n a l i n h o m o g e n e i t i e s o r v o i d s , cracks, o r o t h e r sources o f s c a t t e r i n g a t unknown l o c a t i o n s .
I t may have f i n i t e b u t unknown depth.
A
405
Integral Representations
> 0 and, from measurements on
s c a l a r f i e l d i s i n c i d e n t from
tl =
O+ (and,
> 0), we wish t o deduce i n f o r m a t i o n about t h e medium i n
perhaps, elsewhere i n
rl < 0. D e p a r t u r e s from homogeneity w i l l generate s i n g u l a r i t i e s i n t h e r e l e v a n t
s o l u t i o n t o t h e H e l m h o l t z e q u a t i o n w i t h wave number
K
in
T
< 0.
Our goal i s
t o l o c a t e these s i n g u l a r i t i e s . Let
u(')
be t h e t o t a l s o l u t i o n i n
w i t h wave number K.
In
> 0; i t s a t i s f i e s t h e Helmholtz e q u a t i o n
rl
< 0, t h e t o t a l s o l u t i o n i s u ( ~ ) ,w i t h wave number
It i s assumed t h a t t h e t r a n s i t i o n c o n d i t i o n s on
i n which into
k.
a
and
11 < 0, and
p
are constants.
u ( ~ )i n t o
hen
and
W(l)
can be c o n t i n u e d a n a l y t i c a l l y
i n terms o f even harmonic f u n c t i o n s
t h a t a r e d e f i n e d i n accordance w i t h ( 3 0 ) and ( 3 1 ) :
A representation f o r
u ( ~ ) i s o b t a i n e d from ( 3 6 )
s u p e r s c r i p t s ( 1 ) by ( 2 ) , and
k
by
K.
and
u ( ~ ) i s expressed i n terms o f
V(l)
(37)
by replacing
I n accordance w i t h ( 3 5 ) , t h e harmonic
f u n c t i o n s a r e r e l a t e d by
and
a r e o f t h e form
> 0.
A t t h i s stage, we use ( 2 9 ) t o express V(l)
u(l)
n = 0
and
W( 1 )
.
406
R. F. Millar
I t i s e v i d e n t t h a t t h e problem has been reduced t o t h a t o f f i n d i n g two harmonic
functions
V(l)
and
W(l)
t h a t a r e even i n
known a p p r o x i m a t e l y on a segment o f
and which, t h r o u g h ( 3 7 ) , a r e
11
f r o m measurements o f
= O+
u
and
u
1’
There a r e methods t h a t use a p r i o r i bounds t o s t a b i l i z e s o l u t i o n s t o such Cauchy problems f o r t h e L a p l a c e e q u a t i o n i n a s p e c i f i e d domain [ 2 2 ] ; here, however, t h e domains o f e x i s t e n c e o f Although
V(')
and
and
W(')
W(')
a r e unknown.
i n d i v i d u a l l y may be s i n g u l a r , t h e combination o f them
i n (36) i s n o t s i n g u l a r i n
y > 0.
( I t i s assumed t h a t t h e i n c i d e n t e x c i t a t i o n i s a n a l y t i c ; i f i t were s i n g u l a r anywhere i n 11 > 0, we c o u l d deal w i t h t h e nons i n g u l a r s c a t t e r e d component.)
We c o n j e c t u r e t h a t t h i s p r o p e r t y may be u s e f u l
f o r s t a b i l i z i n g t h e s o l u t i o n , t h e i d e a b e i n g t o use measurements on
to
11 = 0
g i v e a f i r s t a p p r o x i m a t i o n t o t h e Cauchy data, and t o employ ( 3 6 ) w i t h measured ( y > 0)
u(')(x,y)
t o improve t h e a p p r o x i m a t i o n .
To t h i s end, t h e f o l l o w i n g
procedure i s suggested: 1.
Represent
even i n
0
W ( 1 ) ( x , ~ ) as s e r i e s o f harmonic f u n c t i o n s t h a t a r e
V(')(X,O),
and t h a t form a s e t t h a t i s complete on
c o e f f i c i e n t s be { v }, {wn}, r e s p e c t i v e l y . n ( = u ( l ) ( x , O ) ) and W ( l ) ( x , O ) Measure V ( l ) ( x , O ) 2. a p p r o x i m a t i o n s ?,,, Gn t o t h e t r u e c o e f f i c i e n t s vn e s t i m a t e s o f t h e form 3.
Measure
u(')(x,y)
iivn
-
inii
< 6, liwn
-
(= u ( l ) ( x , O ) ) and
T
wn,
t o give
and e r r o r
< 6.
Gnii
on l i n e s o f c o n s t a n t
L e t t h e undetermined
C.
y
( y > 0)
t o give constraints
on t h e t r u e c o e f f i c i e n t s , t h r o u g h ( 3 6 ) .
4.
Attempt t o choose o p t i m a l values f o r t h e c o e f f i c i e n t s , f o r example by
l i n e a r programming o r l e a s t squares methods. 5.
Use t h e V(')
and
W(l)
so o b t a i n e d t o f i n d
~ ( ~ ) ( x , y )i n y < 0
by
means o f ( 3 9 ) , and t h u s determine t h e approximate l o c a t i o n o f s i n g u l a r i t i e s i n y < 0. I t has n o t y e t been a s c e r t a i n e d whether t h i s procedure w i l l l e a d t o a u s e f u l
r e s u l t , b u t i t i s planned i n v e s t i g a t e t h i s aspect o f t h e i n v e r s e problem soon. CONCLUDING REMARKS I n t h i s paper, we have g i v e n r e p r e s e n t a t i o n s f o r s o l u t i o n s t o some second- and f o u r t h - o r d e r d i f f e r e n t i a l equations.
For t h e Helmholtz equation, various
p o s s i b l e a p p l i c a t i o n s have been d e s c r i b e d i n areas l i k e s p e c i a l f u n c t i o n s , a n a l y t i c continuation,
and i n v e r s e s c a t t e r i n g t h e o r y .
It i s expected t h a t
s i m i l a r a p p l i c a t i o n s f o r o t h e r e q u a t i o n s can be t r e a t e d i n t h e same manner.
407
Integral Representations
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409
NONLINEAR WAVE PHENOMENA I N THE BRAIN
R o b e r t M.
Miura
Departments of Mathematics and Pharmacology and T h e r a p e u t i c s I n s t i t u t e of Applied Mathematics U n i v e r s i t y of B r i t i s h Columbia Vancouver, B.C. V6T 1Y4 Canada
U n d e r s t a n d i n g t h e m u l t i t u d e of phenomena a s s o c i a t e d w i t h t h e b r a i n remains one of man's g r e a t e s t s c i e n t i f i c and i n t e l l e c t u a l challenges. Among t h e s e phenomena are n o n l i n e a r waves s u c h as n e r v e i m p u l s e s which p r o p a g a t e r a p i d l y down an i n d i v i d u a l n e r v e c e l l axon and s p r e a d i n g c o r t i c a l d e p r e s s i o n which p r o p a g a t e s s l o w l y through a p o p u l a t i o n of n e r v e cells. Nerve impulse p r o p a g a t i o n h a s been s t u d i e d e x t e n s i v e l y w h i l e s p r e a d i n g d e p r e s s i o n h a s o n l y r e c e n t l y been modelled matheW e d i s c u s s v a r i o u s m a t h e m a t i c a l , n u m e r i c a l , and matically. I t i s our hope m o d e l l i n g a s p e c t s of t h e s e wave phenomena. t h a t t h i s knowledge c a n be p u t t o a u s e f u l p u r p o s e , e.9.. c o n t r o l l i n g or a n n i h i l a t i n g t h e s e waves. INTRODUCTION I
N o n l i n e a r "wave" phenomena o c c u r t h r o u g h o u t n a t u r e and, a s s c i e n t i s t s , our j o b i s t o d i s c o v e r t h e i r o c c u r r e n c e , t h e mechanisms by which t h e waves a r e g e n e r a t e d and p r o p a g a t e d , and, i n g e n e r a l , t h e i r s i g n i f i c a n c e and q u a l i t a t i v e and q u a n t i t a t i v e b e h a v i o r under v a r i o u s c o n d i t i o n s . The approach t o t h e s e problems by t h e m a t h e m a t i c a l m o d e l l e r is t o p r o p o s e a r e p r e s e n t a t i o n of t h e p h y s i c a l , c h e m i c a l , o r b i o l o g i c a l s y s t e m i n terms of some s u i t a b l e m a t h e m a t i c a l f o r m u l a t i o n . Ideall y , one t h e n s t u d i e s t h e model u s i n g a v a r i e t y of m a t h e m a t i c a l , a p p r o x i m a t i o n , and n u m e r i c a l t e c h n i q u e s , v a l i d a t e s t h e model e q u a t i o n s by q u a l i t a t i v e and quant i t a t i v e comparisons w i t h e x p e r i m e n t a l d a t a , and f i n a l l y a t t e m p t s t o p r e d i c t new b e h a v i o r p r e v i o u s l y unseen or u n n o t i c e d . A l l of t h e waves we see i n n a t u r e are t r a n s i e n t phenomena; however, t h e r e a r e many which c a n be viewed a p p r o x i m a t e l y a s s t e a d y p r o g r e s s i n g waves, i.e., as waves which move a t a c o n s t a n t speed w i t h unchanging waveform. Among t h e s t e a d y p r o g r e s s i n g waves are s o l i t a r y waves which a r e e s s e n t i a l l y s p a t i a l l y l o c a l i z e d p u l s e s moving on a c o n s t a n t background, w a v e f r o n t s which have a s p a t i a l l y l o c a l i z e d f r o n t and have a s y m p t o t i c a l l y d i f f e r e n t v a l u e s as x + f m , p e r i o d i c waves which have f i n i t e s p a t i a l and t e m p o r a l p e r i o d s , and m u l t i p l e waves which c a n be viewed as a g r o u p of s o l i t a r y - l i k e waves moving t o g e t h e r . ~
2
I n t h i s p a p e r , w e d e s c r i b e t w o t y p e s of n o n l i n e a r wave phenomena i n t h e b r a i n However, b e f o r e d o i n g t h i s , w e f i r s t make some g e n e r a l remarks on t h e b r a i n .
. A
f u n d a m e n t a l u n d e r s t a n d i n g of t h e b r a i n remains one of man's g r e a t e s t i n t e l l e c t u a l and s c i e n t i f i c c h a l l e n g e s . As t h e " c e n t e r " of t h e "mind, " " c o n s c i o u s n e s s , " and "memory," the b r a i n c o n t i n u e s t o confound p h i l o s o p h e r s and s c i e n t i s t s a l i k e . On t h i s t o p i c , t h e world famous n e u r o s u r g e o n , W i l d e r P e n f i e l d ( 1 9 7 5 ) w r o t e : "--consciousness and t h e r e l a t i o n s h i p of mind to b r a i n are probl e m s d i f f i c u l t to s t u d y i n a n i m a l s . C l i n i c a l physicians, on t h e o t h e r hand, i n t h e i r approach t o man, may hope w i t h r e a s o n t o push toward a n u n d e r s t a n d i n g of t h e p h y s i o l o g y of memory and t h e p h y s i o l o g i c a l b a s i s of t h e mind and c o n s c i o u s n e s s . "
410
R. M. Miura
Our immediate g o a l i s more modest b u t s t i l l e x t r e m e l y d i f f i c u l t - t o u n d e r s t a n d t h e mechanics of t h e b r a i n . To t h i s end, i t i s noted t h a t t h e e x p e r i m e n t a l lite r a t u r e c o n t a i n s a w e a l t h of i n f o r m a t i o n and d a t a which have n o t been examined by theoreticians. A s y n e r g y of e x p e r i m e n t a l f i n d i n g s and t h e o r e t i c a l s t u d i e s w i l l g r e a t l y enhance our b a s i c u n d e r s t a n d i n g of t h e b r a i n . The b a s i c u n i t i n t h e b r a i n i s t h e n e r v e c e l l , o r neuron as i t i s a l s o c a l l e d . T h e r e are a b o u t 1 0 l 1 n e u r o n s i n t h e human b r a i n ( H u b e l , 1 9 7 9 ) . I n a d d i t i o n , t h e s e neurons are s u r r o u n d e d by n e u r o g l i a l c e l l s of which t h e r e are e s t i m a t e d t o be a t l e a s t 10 t i m e s as many as neurons and which make up a b o u t h a l f of t h e volume of t h e b r a i n ( K u f f l e r and N i c h o l l s , 1 9 7 6 ) . However, t h e primary f u n c t i o n of t h e s e n e u r o g l i a l c e l l s i s n o t known, and hence t h e y are g e n e r a l l y i g n o r e d i n any d i s c u s s i o n of b r a i n f u n c t i o n - undoubtedly t h i s w i l l t u r n o u t t o be e r r o n e o u s as more d a t a on g l i a l c e l l s a r e accumulated. There are many d i f f e r e n t t y p e s of n e u r o n s , b u t i n g e n e r a l t h e y c o n s i s t of a c e l l body o r from which e x t e n d "processes" o r branches. Most of t h e s e b r a n c h e s are c a l l e d d e n d r i t e s and s e r v e t o r e c e i v e s i g n a l s from o t h e r n e r v e c e l l s and t o i n t e g r a t e t h e s e s i g n a l s a t t h e c e l l body b e f o r e p a s s i n g on s i g n a l s t o o t h e r neurons. One d i s t i n g u i s h e d branch i s called the a x ~ n and t h r o u g h i t s many b i f u r c a t i o n s s e r v e s a s t h e o u t p u t branch S i g n a l s are s e n t o u t a l o n g t h i s branch a t v e r y high s p e e d s f o r t h e nerve c e l l . t o o t h e r c e l l s , e.g., n e i g h b o r i n g neurons or muscle c e l l s . The geuron boundary i s d e f i n e d by a t h i n l i p i d b i l a y e r membrane ( a p p r o x i m a t e l y 50 A i n t h i c k n e s s ) embedded w i t h v a r i o u s p r o t e i n s and t h i s membrane i s c a p a b l e of p e r m i t t i n g t h e movement of i o n s a c r o s s it and a l s o of s e p a r a t i n g e l e c t r i c a l c h a r g e s . I n the normal r e s t i n g s t a t e , t h e r e i s a nonzero e l e c t r i c a l p o t e n t . i a 1 a c r o s s t h i s memb r a n e c a l l e d t h e membrane p o t e n t i a l w i t h a t y p i c a l v a l u e of a b o u t -70 mV ( m i l l i v o l t s ) , negative i n s i d e r e l a t i v e t o t h e outside. The r e g i o n o u t s i d e t h e neurons i s c a l l e d t h e e x t r a c e l l u l a r space and t h e i n s i d e r e g i o n s of neurons are c a l l e d the i n t r a c e l l u l a r space. These neurons a r e c o l l e c t e d t o g e t h e r i n the major c o m p o n e n t s of t h e b r a i n , t h e r i g h t and l e f t hemispheres of t h e cerebrum, which a l m o s t c o m p l e t e l y f i l l s t h e s k u l l , and t h e c e r e b e l l u m , which i s a much smaller p a r t of t h e b r a i n l o c a t e d under t h e back p a r t of t h e cerebrum. F i n a l l y there i s t h e b r a i n s t e m which e x t e n d s between t h e cerebrum and t h e s p i n a l c o r d . The cerebrum and t h e c e r e b e l l u m both have a c o r t e x , i.e., a t h i n l a y e r near t h e s u r f a c e ( l i k e t h e p e e l of an o r a n g e ) . T h i s c o r t e x i s c a l l e d g r e y matter and c o n s i s t s mainly of t h e nerve c e l l b o d i e s and d e n d r i t i c trees ( a n d , of c o u r s e , g l i a l cells). Beneath t h e c o r t e x i s t h e w h i t e matter c o n s i s t i n g of axons from t h e nerve cells. These axons are u s u a l l y covered w i t h a f a t t y s u b s t a n c e c a l l e d myelin from which t h e name w h i t e matter a r i s e s . The two t y p e s of n o n l i n e a r wave phenomena to be d e s c r i b e d h e r e a r e t h e a c t i o n The p o t e n t i a l o r n e r v e i m p u l s e and s p r e a d i n g c o r t i c a l d e p r e s s i o n (SD f o r s h o r t ) . a c t i o n p o t e n t i a l c o r r r s p w i s ko r\ membrane p o t e n t i a l c b r i ~ r ewhi 1, while
(4.10)
and, finally,
Randomly Distributed Dissimilar Scatterers
443
(4.11)
Obviously, equations (4.4) are extremely complicated when a, and az are arbitrary angles. However, when the preferred directions of the families of scatterers are in the direction of propagation or orthogonal to it, or when one is in the direction of propagation and the other is orthogonal to it, equations (4.4)uncouple, since in these cases = $B ( i j ) = 0 and we have
Since these equations have non-trivial solutions if and only if the determinants of their coefficients vanish, it appears that two waves, one corresponding to each of the values of K which follow from these equations, may propagate. However, on substitution from (4.1) into (3.12) we find that for the particular orientations under consideration, ( w ) depends only on Xe and Ye and so the root which follows from (4.11) determines the wave number of the coherent wave which may propagate in the composite. Next we turn our attention to the case when the scatterers of family one are aligned while those of the second family are randomly oriented. It now follows that equations (3.21) reduce to
(k2- K z )
(4.13)
where
.,d;jA
=
'
WCij)
R n j e m T L m . m,n,
% ( Vn m) =
Rn.E
I m
~ i ,dii) ,m , n .
(4.14)
Finally, if the preferred directions of both sets of scatterers are randomly oriented, equations (3.22) reduce to
(kZ-KZ)
[I]
=
.FJ(21)
"""] k:] d 2 2 )
(4.15)
(4.16)
Clearly equations (4.15) and (4.16) are uncoupled. An elementary calculation shows that the mean displacement in the composite is determined by Xe and Ye and is independent of Xo and Yo.Thus, the wave number of the coherent wave which may propagate in the composite is determined by the equation (4.15).
M. F. McCarthy and M. M. Carroll
444
5 . WAVE PROPAGATION I N T H E R A Y L E I G H LIMIT In the Rayleigh or low frequency limit, the size of the scatterers is assumed t o be small when compared t o the wavelength of the incident disturbance. In this limit, it suffices to take only the lower order terms in the expansions of the Bessel and Hankel functions. On adopting this procedure, i.e. retaining only the lowest order terms in kaj and Kai, i = 1,2, we find
where { = K / k . Next, in the Rayleigh limit, it follows from (2.8) and Appendix A that the only nonvanishing components of the T matrix of a scatterer whose boundary is described by the equation r = .fro), where a is the radius of the circular cylinder which just circumscribes the scatterer, are To,
i(d - 1 )
=-
2
k2a21,
TI1 = -T1,-1 =
i(l -m)k2a21 2(1 + m)(l - A 2 )
(5.2)
where (5.3)
and m = ~1 l p , d = p l p A may be regarded as a measure of the skewness of the scatterer while its area is f f ’ 212. In order t o obtain specific results for particular types of scatterers we note the following particular cases:
(i) Circular Cylinders: f ( 8 ) = 1,
I =r/2,
A =0,
(5.4)
(ii) Elliptic cylinders:
fro)= ( C o s 2 6 + a 2 / b z Sin6 \-l'z,
(1 - m)(b - a ) ab I =- , A = 2a ( 1 + m ) ( b+ a )
(5.5)
where a,b (b>a) are the semi-axes of the ellipse. (iii) Rectangular Cylinders:
I = 2b/a,
A=-
4(1 - m )
r(l + m )
tan:l
b-a b+a '
where 2a and 2 b ( 1
and r e p l a c e ( 5 )
(7)
-m
I f we i n t r o d u c e branch c u t s y = 0 , 1x1 t 1 i n t h e z-plane , then u s i n g p r o p e r t i e s o f t h e Macdonald f u n c t i o n g i v e n i n E r d g l y i e t a l . (1953), we o b t a i n the results If z
+
x i i0
f o r any r e a l
x
Ko(a(z-t))
-f
Ko(alx-tl) T i n H(t-x)Io(a(t-x))
Ko(a(t-z))
+
Ko(alx-t()
*ix
,
H(x-t)Io(alt-x))
, where ,x
H(x) = 1
>
0
= o , x < o and
Io(x)
Define
i s t h e m o d i f i e d Bessel f u n c t i o n .
F(z) =
r1
$ ( t ) Ko(a(z-t))dt +
-m
Then,writing
F,(X)
=
[
$ ( t ) Ko(a(t-z))dt
l i m F(z) , F-(x) = l i m F(z) z+x+i 0 z+x-i0 F,(x)
j-'
= F-(x) =
.
, we
(8)
have
$(t) Ko(a/x-tl)dt
, -1
< x < 1 ,(9)
1
-m
1
.-1
F ( x ) = -nE(x)
+
- in
'
Io(a(t-x))$(t)dt
,x
< -1
(10)
X
F - ( x ) = -nE(x)
Io(a(t-x))$(t)dt
t i a
,x
< -1
X
F+(x) = -ITE(X)
I,,(a(t-xj)$(t)dt
t in
,
x > 1 (11 1
F - ( x ) = -nE(x) -
i a
r"
Io(a(t-x))$(t)dt
,
x > 1
455
An Integral Equation Arising in Diffraction Theory
From ( 9 ) , ( l o ) , ( 1 1 ) we have
,
F + ( x ) + F - ( x ) = -2nE(x)
- F-(x) = 0
F,(x)
,
1x1 > 1
1x1 < 1
These c o n d i t i o n s a r e r e p l a c e d by F i ( X ) + F ' ( X ) = - 2 ~ E ' ( x ) , 1x1 > 1 F k ( x ) - F'(x)
= 0
, 1x1
< 1
Now f r o m ( 8 ) i t can be shown t h a t
F'(z)
= c f osh
(az)
fi
1
, z + =
(13)
and
The d e t e r m i n a t i o n o f p r o b l em. I f we w r i t e
F ' ( z ) = O(l0g (1-z))
,z
F ' ( z ) = O(1og ( l + z ) )
,z
F'(z)
F'(z) =
+
1
-f
-1
(14)
s a t i s f y i n g ( 1 2 ) , ( 1 3 ) and ( 1 4 ) i s a Riemann-Hilbert
e-az +
eaz , t h e n t h e f u n c t i o n s
Z
A(z) , B(z)
s a t i s f y t h e Riemann-Hi1 b e r t problem - n x ~ ' ( x ) e ~ ', 1x1 > 1
A,(x)
+ A-(x)
=
A+(')
- A-(x)
= 0
A(z) = O(&),
z
+
B
