{"http:\/\/dx.doi.org\/10.14288\/1.0080439":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"McGorman, Robert Ernest","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2011-04-11T16:10:47Z","type":"literal","lang":"en"},{"value":"1972","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Science - MSc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"In this thesis we consider the propagation of internal waves\r\nin a rotating stratified unbounded ocean with randomly varying\r\nBrunt-V\u00e4is\u00e4l\u00e4 frequency, N.  Keller's method is used to obtain the\r\ndispersion relation for the mean wave field correct to second\r\n\r\norder in \u0404 when N is of the form N\u00b2 = N\u0585\u00b2 (1+\u0404\u03bc) where N\u0585 = constant,\r\n0<\u0404\u00b2<< 1 and \u03bc is a centered stationary random function of either depth or time separately. From the dispersion relation there are derived general formulas for the change in phase speed and growth or attenuation rates due to the random fluctuations \u03bc . These formulas are dependent on the statistics of \u03bc only through the autocovariance function.\r\nThe phase speed change and growth rate formulas for depth dependent \u03bc , which constitutes a model of the temperature and salinity fine-structure in the ocean, are presented for various special cases including the limiting cases of correlation lengths of \u03bc  that are long or short with respect to the wavelength. Observations at station P (50\u00b0N, 145\u00b0 W) indicate that, to a good approximation, the \u03bc are \"white noise\" and a close examination is made of the theoretical results for this case. With the aid of the station P data it is estimated that, although the phase speed changes are generally small, the amplitude of a wave increases (decreases) significantly in propagating upward (downward) through a depth of a few kilometers. In addition it is found that the mean\r\neffect of the depth dependent fluctuations \u03bc is to increase the \r\neffective Brunt-V\u00e4is\u00e4l\u00e4 frequency, or \"stiffen\" the fluid. This may explain why some recently observed frequency spectra of internal waves do not exhibit a sharp cut-off at N\u0585, the deterministic\r\ntheoretical upper bound for the wave frequency. Finally an attempt is made to asses the range of validity of Keller's method in the context of the present problem.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/33482?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"INTERNAL WAVES IN A RANDOMLY STRATIFIED OCEAN by ROBERT ERNEST McGORMAN B.Sc. (Honours M a t h e m a t i c s ) , M c G i l l U n i v e r s i t y , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f M a t h e m a t i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ITlcit&iswastiCsas T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada D a t e IAaaXZ \/+\/ 72 ABSTRACT i i I n t h i s t h e s i s we c o n s i d e r t h e p r o p a g a t i o n o f i n t e r n a l waves i n a r o t a t i n g s t r a t i f i e d unbounded ocean w i t h randomly v a r y i n g B r u n t - V a i s a l a frequency,N . K e l l e r ' s method i s u s e d t o o b t a i n t h e d i s p e r s i o n r e l a t i o n f o r t h e mean wave f i e l d c o r r e c t t o second 2 2 o r d e r i n \u00a3 when N i s o f t h e form where - c o n s t a n t , 0 < \" \u00a3 ^ < 1 and yx i s a c e n t e r e d s t a t i o n a r y random f u n c t i o n o f . e i t h e r d e pth o r t i m e s e p a r a t e l y . From t h e d i s p e r s i o n r e l a t i o n t h e r e a r e d e r i v e d g e n e r a l f o r m u l a s f o r the change i n phase speed and g r o w t h o r a t t e n u a t i o n r a t e s due t o the random f l u c t u a t i o n s yU . These f o r m u l a s a r e dependent on t h e s t a t i s t i c s of \/u o n l y t h r o u g h the a u t o c o v a r i a n c e f u n c t i o n . The phase speed change and growth r a t e f o r m u l a s f o r depth dependent yu , w h i c h c o n s t i t u t e s a model o f the t e m p e r a t u r e and s a l i n i t y f i n e - s t r u c t u r e i n t h e ocean, a r e p r e s e n t e d f o r v a r i o u s s p e c i a l c a s e s i n c l u d i n g t h e l i m i t i n g c a s e s o f c o r r r 1 a t i o n l e n g t h s o f JJ. t h a t a r e l o n g o r s h o r t w i t h r e s p e c t t o t h e w a v e l e n g t h . O b s e r v a t i o n s a t s t a t i o n P (50\u00b0N, 145\u00b0 W) i n d i c a t e t h a t , t o a good a p p r o x i m a t i o n , t h e y\\x a r e \" w h i t e n o i s e \" and a c l o s e e x a m i n a t i o n i s made o f t h e t h e o r e t i c a l r e s u l t s f o r t h i s c a s e . W i t h t h e a i d of the s t a t i o n P d a t a i t i s e s t i m a t e d t h a t , a l t h o u g h t h e phase speed changes a r e g e n e r a l l y s m a l l , t h ^ a m p l i t u d e o f a wave i n c r e a s e s ( d e c r e a s e s ) s i g n i f i c a n t l y i n p r o p a g a t i n g upward (downward) t h r o u g h a d e p t h o f a few k i l o m e t e r s . I n a d d i t i o n i t i s f o u n d t h a t t h e mean e f f e c t o f t h e depth dependent f l u c t u a t i o n s \/U i s t o i n c r e a s e t h e e f f e c t i v e B r u n t - V a i s a l a . f r e q u e n c y , or \" s t i f f e n \" t h e f l u i d . T h i s may e x p l a i n why some r e c e n t l y o b s e r v e d f r e q u e n c y s p e c t r a o f i n t e r n a l waves do n o t e x h i b i t a sharp c u t - o f f a t A\/0, t h e d e t e r -m i n i s t i c t h e o r e t i c a l upper bound f o r t h e wave f r e q u e n c y . F i n a l l y an a t t e m p t i s made t o a s s e s t h e range o f v a l i d i t y o f K e l l e r ' s method i n t h e c o n t e x t o f t h e p r e s e n t problem. TABLE OF CONTENTS i v Chapter 0: I n t r o d u c t i o n - 1 Chapter I : The Depth Dependent Case 3 1.1 F o r m a l D i s p e r s i o n R e l a t i o n 3 1.2 S i m p l i f y i n g t h e D i s p e r s i o n R e l a t i o n 5 1.3 P o l a r R e p r e s e n t a t i o n o f t h e D i s p e r s i o n R e l a t i o n 9 1.4 S p e c i a l Cases 12 Chapter I I : The Time Dependent Case 19 11.1 F o r m a l D i s p e r s i o n R e l a t i o n 19 11.2 S i m p l i f y i n g t h e D i s p e r s i o n R e l a t i o n .20 Chapter I I I F u r t h e r E x a m i n a t i o n o f t h e Case o f Depth Dependent \"White N o i s e \" F l u c t u a t i o n s 26 111.1 I n t r o d u c t i o n 26 111.2 D i s c u s s i o n o f t h e \"White N o i s e \" Growth Curves 27 111.3 D i s c u s s i o n o f t h e \"White N o i s e \" Phase Speed Curves 32 B i b l i o g r a p h y 37 A p p e n d i c e s : 39 A- The 0 t h Order D i s p e r s i o n R e l a t i o n 39 B- D e r i v a t i o n o f t h e Stream F u n c t i o n E q u a t i o n 40 C- K e l l e r ' s Method 44 D- The L i m i t i n g Case L << ~XQ 49 E- The L i m i t i n g Case ^a <'< L . \u2022 51 LIST OF TABLES v T a b l e I Sample Computed A u t o c o v a r i a n c e F u n c t i o n 13 LIST OF FIGURES F i g u r e 1 Growth and A t t e n u a t i o n R a t e s F i g u r e 2 R e l a t i v e Phase Speed Change f o r Passband I : -f2< Q~2< F i g u r e 3 R e l a t i v e Phase Speed Change f o r Passband l l : A o < < T * ' ACKNOWLEDGMENT v i i I am p l e a s e d t o thank my s u p e r v i s o r , p r o f . L.A.Mysak, p r e s e n t l y a S e n i o r V i s i t o r i n t h e Department o f A p p l i e d Mathe-m a t i c s and T h e o r e t i c a l P h y s i c s a t Cambridge U n i v e r s i t y , f o r h i s g r a c i o u s encouragement and a s s i s t a n c e i n t h e w r i t i n g o f t h i s t h e s i s . I n a d d i t i o n I s h o u l d l i k e t o thank P r o f . P.H.LeBlond f o r e x a m i n i n g t h e t h e s i s , and Dr.. R.E.Thomson f o r t h e use o f t h e d a t a . T h i s r e s e a r c h was a c c o m p l i s h e d w i t h f i n a n c i a l a s s i s t s - - e i n the form o f an H.R.MacMillan F a m i l y F e l l o w s h i p . CHAPTER 0: INTRODUCTION 1 I t has l o n g been known t h a t i n t e r n a l g r a v i t y wave m o t i o n jjf. e x i s t s i n t h e ocean (see C h a p t e r 5 o f P h i l l i p s ) . I n o r d e r t o p r o p a g a t e , t h e s e waves r e q u i r e a s t r a t i f i e d medium such as t h e ocean where l e s s dense s u r f a c e l a y e r s r e s t on more dense deeper l a y e r s . A measure o f t h e s t r e n g t h o f t h e s t r a t i f i c a t i o n i s t h e B r u n t - V a i s a l a f r e q u e n c y , N , d e f i n e d by t h e e q u a t i o n where o i s the a c c e l e r a t i o n due t o g r a v i t y and paii) i s t h e r e s t d e n s i t y o f t h e w a t e r as a f u n c t i o n o f depth H, w h i c h i s p o s i t i v e upwards. O b s e r v a t i o n s i n d i c a t e t h a t i n t e r n a l waves f r e q u e n t l y do n o t p r o p a g a t e w i t h c o n s t a n t a m p l i t u d e b u t grow t o t h e p o i n t o f b r e a k -i n g o r become damped o u t . P o s s i b l e mechanisms have been advanced t o a c c o u n t f o r t h i s b e h a v i o u r i n c l u d i n g d i f f u s i v e processes'\" 2 and i n t e r a c t i o n w i t h c u r r e n t s ' f In t h i s t h e s i s we s h a l l examii.. the e f f e c t on t h e waves o f random v a r i a t i o n s i n t h e q u a n t i t y N* by s t u d y i n g t h e d i s p e r s i o n r e l a t i o n . Many s t u d i e s o f i n t e r n a l waves have assumed a s t r a t i f i c a t i o n such t h a t l\\j d) \u2014 N0 , a c o n s t a n t . T h i s i s done f o r two r e a s o n s : f i r s t , i t i m p l i e s a smoothly v a r y i n g dependence o f d e n s i t y on depth, w h i c h may be a r e a s o n a b l e a p p r o x i m a t i o n t o t h e a c t u a l depen-dence a t l e a s t s e c t i o n a l l y ; second, i t r e d u c e s a \"E-dependent c o e f f i c i e n t i n the i n t e r n a l wave e q u a t i o n t o a c o n s t a n t , i . e . , a p u r e l y m a t h e m a t i c a l c o n v e n i e n c e . The p r o p e r t i e s o f f r e e wave 2 s o l u t i o n s f o r t h i s case a r e w e l l known and a r e summarized i n Appendix A. Here we s h a l l assume N1 i s s u b j e c t t o s m a l l random f l u c t u a t i o n s about a c o n s t a n t mean Nl. i n C h a p t e r I t h e random f l u c t u a t i o n s w i l l be depth dependent and i n C h a p t e r I I t i m e dependent f l u c t u a t i o n s w i l l be c o n s i d e r e d . I n C h a p t e r I I I we d i s -c uss g r a p h s o f t h e r e l a t i v e phase speed change and growth r a t e p l o t t e d u s i n g r e a l o c e a n i c d a t a i n t h e c o r r e s p o n d i n g f o r m u l a s f o r \" w h i t e n o i s e \" f l u c t u a t i o n s d e r i v e d i n C h a p t e r I . CHAPTER I : THE DEPTH DEPENDENT CASE 3 1.1 Formal D i s p e r s i o n R e l a t i o n I n A ppendix B we b e g i n w i t h t h e e q u a t i o n s o f mass and momentum c o n s e r v a t i o n o f an i n c o m p r e s s i b l e , i n v i s c i d f l u i d i n a r i g h t - h a n d e d system o f C a r t e s i a n c o o r d i n a t e s u n i f o r m l y r o t a t i n g w i t h a n g u l a r f r e q u e n c y \"\u00a3? about t h e 2 - a x i s , w h i c h i s v e r t i c a l and p o s i t i v e upward. A b a s i c s t a t e o f h y d r o s t a t i c e q u i l i b r i u m i s assumed and upon t h i s s m a l l p e r t u r b a t i o n s i n t h e f l u i d v e l o c i t y , p r e s s u r e and d e n s i t y a r e imposed. I f t h e r e s u l t i n g e q u a t i o n s a r e l i n e a r i z e d and a t t e n t i o n i s r e s t r i c t e d t o two d i m e n s i o n a l m o t i o n then t h e stream f u n c t i o n i s shown t o s a t i s f y t h e e q u a t i o n C\u00ab + + = 0 (i-1) where x i s the h o r i z o n t a l c o o r d i n a t e and \u00b1 i s t h e t i m e . We assume depth dependent random f l u c t u a t i o n s i n N2 o f t h e form where yu(2) i s a zero-mean, w i d e - s e n s e s t a t i o n a r y random p r o c e s s \" and \u00a3 i s a s i z e parameter such t h a t 0< \u20ac.*<< 1 . R e s t r i c t i n g a t t e n t i o n t o the case o f h a r m o n i c time dependence we s e t T h i s g i v e s e q u a t i o n 1-1 t h e form where a d e t e r m i n i s t i c o p e r a t o r , and a \" s m a l l \" random o p e r a t o r . E q u a t i o n 1-2 i s i n a form s u i t a b l e f o r t h e a p p l i c a t i o n o f K e l l e r ' s method' ' ' o f w h i c h we g i v e an a c c o u n t i n Appendix C. There we show t h a t , c o r r e c t t o second o r d e r i n \u00a3\" , t h e d i s p e r s i o n r e l a t i o n o f the mean wave < > (where < > i n d i c a t e s ensemble average) i s where JA.'1 i s t h e i n t e g r a l o p e r a t o r d e f i n e d by CO -oo w i t h Gr , t h e Green's f u n c t i o n , b e i n g the s o l u t i o n o f JAG- =\u2022 Scx-x'\/fa-z') S i n c e ^\/A, i s t r a n s l a t i o n a l l y i n v a r i a n t i n x and Z, t h e Green's f u n c t i o n i s a d i s p l a c e m e n t k e r n e l i n cM.1, t h a t i s , GrCx.K', 2,Z') = Crcx-x', z - z.'J so i t i s n e c e s s a r y t o c o n s i d e r o n l y E q u a t i o n 1-4 can be s o l v e d by t a k i n g a d o u b l e F o u r i e r t r a n s f o r m o f b o t h s i d e s , i . e . m u l t i p l y b o t h s i d e s by Q~c(kx- ^^v) and i n t e g r a t e from -oo t o -hco s u c c e s s i v e l y w i t h r e s p e c t t o * and 2, . The i n v e r s i o n i n JL can be r e a d i l y p e r f o r m e d t o y i e l d the F o u r i e r t r a n s f o r m i n o f Q , w h i c h we denote by G(k,z) , so t h a t U(K,IL) - ... -where H i s the H e a v i s i d e f u n c t i o n and we have s e t d = \u00bbvg and f o r c o n v e n i e n c e . I n t h e d e t e r m i n i s t i c t h e o r y t h e r e a r e two passbands o f cJ 3 d e f i n e d by i n e q u a l i t i e s : r e p r e s e n t s passband I waves and r e p r e s e n t s passband I I waves. I n e i t h e r case CZ > O. 1-2 S i m p l i f y i n g t h e D i s p e r s i o n R e l a t i o n I t i s now p o s s i b l e t o s u b s t i t u t e At, J\/i'1 Jf i n t o 1-3 t o f i n d t h e d i s p e r s i o n r e l a t i o n e x p l i c i t l y . S e t 7 = e - \u00ab Y * \u00ab * ' \u00bb ^ e < \u00ab x , \/ e ; and Then and <9U(*)L j\u00b1f*d*. -<43*lJJGrtx-x.', 2 - 2 7 \u2014 CO S i m p l i f y i n g -co T h i s can be w r i t t e n CO \u2014 OC3 Make t h e change o f v a r i a b l e s OL\" - x- x' Then t h i s g i v e s Now t a k i n g i n t o a c c o u n t t h a t t h e i n t e g r a l i s a f u n c t i o n o f Z, the d i f f e r e n t i a t i o n can be pe r f o r m e d t o g i v e \u2014 CO CO \u2014CO U s i n g t h e s t a t i o n a r i t y o f jua) we d e f i n e t h e a u t o c o v a r i a n c e f u n c t i o n , Tc?\";, by Vet\") = )yjLC I-?.\")> Now 8 and -$UCZj\/4z*S2--z<';> ^ ~ P2\u00bb \u00a3u< zJyU ( * -Thus \u2014 Q3 - 0 0 and i n each case the i n t e g r a t i o n o v e r produces GcCk.z\") i n t h e above i n t e g r a l s . So, c o r r e c t t o second o r d e r i n \u2022\u20ac , t h e d i s p e r s i o n r e l a t i o n r e d u c e s t o -oo U s i n g i n t e g r a t i o n by p a r t s we can w r i t e -co as the i n t e g r a t e d term v a n i s h e s a t 0 and CO . C o l l e c t i n g terms we o b t a i n -e\\t[0.-<:**]fed% o^JT^^zjrczje^Jz^O (1-5) ~ 03 as the d i s p e r s i o n r e l a t i o n f o r t h e mean f i e l d <'V'), c o r r e c t t o Oce1). We n o t e t h a t t h e Oct*) terms, w h i c h a r e t h e c o r r e c t i o n s t o the d i s p e r s i o n r e l a t i o n f o r i n t e r n a l waves i n t h e non random f l u i d ( i . e . t h e above w i t h \u00a3 - O ), a r e dependent on the passband o f &? 1-3 P o l a r R e p r e s e n t a t i o n o f t h e D i s p e r s i o n R e l a t i o n I n d i s c u s s i n g t h e Oth o r d e r d i s p e r s i o n r e l a t i o n i n Appendix A we employ a p o l a r r e p r e s e n t a t i o n o f t h e wavenumbers, i . e . k = Wo co<u9 and where \u00a3g0 i s r e a l . We want t o e x t e n d t h i s r e p r e s e n t a t i o n t o t h e OcelJ d i s p e r s i o n r e l a t i o n , i . e . s e t k^^C^G (1-6) < 10 and I - Xs+me -cd\/z (1-7) where 6 i s a r e a l a n g l e and B\u20ac may be complex. S u b s t i t u t i n g 1-6 and 1-7 i n t o T - 5 and s i m p l i f y i n g we o b t a i n r ^ L ^w* -<llfrr: ^ ( ^ ^ % * ) e ****** -co CI-8) Now i f we f i x an a n g l e o f p r o p a g a t i o n , 0 , and s p e c i f y t h e q u a n t i t i e s ffjf, A\/0) (\/ and f c ? ; t h e n e q u a t i o n 1-8 i s an e q u a t i o n f o r ?6 w h i c h can be s o l v e d by i t e r a t i o n (see K e l l e r and Veronis\") , S e t t i n g \u20ac = 0 g i v e s the 0 t h o r d e r s o l u t i o n o f Appendix A d?o - \u2022 ^ , \u2022 CT-Q) S u b s t i t u t i n g ^ 0 f o r i n t h e 0 C\u00a3l) terms g i v e s t h e f i r s t i t e r a t i o n s o l u t i o n (which i s v a l i d i f the Oc(?) terms a r e s m a l l , see d i s c u s s i o n Appendix C) S i n c e swtfe**^- and CoxL,(a z j a r e even f u n c t i o n s o f oc , t h e a b s o l u t e v a l u e s i g n s can be dropped. F o r c o n v e n i e n c e s e t o< = Mo s*m e and T a k i n g t h e square r o o t o f b o t h s i d e s o f 1-10 and u s i n g the b i n o m i a l e x p a n s i o n on the R.H.S. (assuming the Oct1) terms a r e s m a l l r e l a t i v e t o 1 ) , we o b t a i n o \u201400 \u00b0^ - c a Hence i t r e a d i l y f o l l o w s t h a t <>\u2022 -Too . - e^i^Vd) %Xn l^idi (I-11) -co and 2 6 0 - ffcfrt*)(l-Co<L2*2)dti -<=Mfrti)(l + Co\u00ab.2^)di (1-12) -\u00b0\u00b0 -co 1-4 S p e c i a l Cases F i r s t o f a l l we s h a l l assume t h a t t h e f l u c t u a t i o n s i n jud.) a r e \" w h i t e n o i s e \" so t h a t O b s e r v a t i o n s t a k e n on F e b r u a r y 23, 1971 and J u l y 29, 1970 a t weather s h i p s t a t i o n P (50\u00b0N, 145\u00b0W) and a n a l y z e d by R i c h a r d Thomson o f Environment Canada i n d i c a t e t h a t t h e a s s u m p t i o n i s w e l l j u s t i f i e d o v e r c e r t a i n depth r a n g e s . The q u a n t i t y N2(D was computed a t 10 meter i n t e r v a l s from t h e ocean s u r f a c e t o a depth o f 1500 m e t e r s . The r e s u l t s i n d i c a t e t h a t \/\\J2d) e x h i b i t s a s t r o n g t r e n d w i t h d e pth i n t h e upper 700 m e t e r s . However, between the d e p t h s o f 700 m and 1500 m \/\\\/ 2<Z) a p p e a r s t o have l i t t l e t r e n d w i t h mean s*e~Z and s t a t i o n a r y f l u c t u a t i o n s , Cju.it), about M> w i t h \u20ac^^0(^L) . F u r t h e r m o r e , t h e a u t ^ - j o v a r i a n c e f u n c t i o n s f o r b o t h t h e summer and w i n t e r d a t a computed u s i n g t h e mean, \/V0 , over t h i s range i n d i c a t e t h e \u20ac\/ua)'s a r e \" w h i t e n o i s e \" t o a good a p p r o x i m a t i o n . We i n c l u d e f o r i l l u s t r a t i o n i n t a b l e I the a u t o c o v a r i a n c e f u n c t i o n computed from t h e w i n t e r d a t a . 13 TABLE I, SAMPLE COMPUTED AUTOCOVARIANCE FUNCTION DEPTH LAG, Z \/m \u20ac2 T< 2 ) 0 0.19580 10 .... 0.09219 20 0.01881 30 0.06659 40 0.02786 50 0.00000 60 0.02475 70 0.0659 3 80 0.02659 90 0.03095 100 0.06968 110 0.04162 120 0.01218 130 0.0597\" 140 0.0750b 150 0.01380 S u b s t i t u t i n g f o r i n f o r m u l a s 1-11 and 1-12 y i e l d s and U s i n g e q u a t i o n 1-9 and r e c a l l i n g t h e v a l u e s o f Fl and oi. we can. w r i t e and \/\/m i 4?ol = \u2014 \u00a3 \/ 1 , i . , 2 \\l C*C<rz*Q-U\u00bb\\*Q (1-1+) 14 We denote t h e d i m e n s i o n l e s s q u a n t i t y dV by cT where I f 5 0 i s the phase speed o f t h e Oth o r d e r wave and S i s t h e phase speed o f t h e Oct2) wave t h e n and when i s n e g a t i v e t h e wave grows, t h u s we can r e w r i t e 1-13 and 1-14 as - ^ _ ^ a - i s ) and . .Equations 1-15 and 1-16 a r e d i s c u s s e d a t l e n g t h i n C h a p t e r I I I where t h e y appear as e q u a t i o n s I I I - l and I I I - 2 . A t t h i s p o i n t we j u s t m e n t i o n t h a t from 1-16 i t i s c l e a r t h a t , i n d e p e n d e n t o f the (j2 passband, waves p r o p a g a t i n g w i t h an upward component o f d i r e c t i o n grow and waves p r o p a g a t i n g w i t h a downward component decay w i t h t h e magnitude o f t h e growth or decay dependent on t h e passband o f <T o n l y t h r o u g h t h e d i m e n s i o n l e s s q u a n t i t y \u00a3 d . Next we c o n s i d e r the s p e c i a l case o f 1-11 and 1-12 f o r w h i c h 15 w h i c h i s t h e a u t o c o v a r i a n c e f u n c t i o n o f a U h l e n b e c k - O r n s t e i n p r o c e s s . The r e s u l t i n g i n t e g r a l s a r e e a s i l y e v a l u a t e d as t h e y have the form o f L a p l a c e t r a n s f o r m s . E q u a t i o n s 1-11 and 1-12 become T? f t y , 1-14- j\u00a3T\u00a3JLr*\u00a3. . Co*d - ^ 7 2 Kef 4t*\\ - I + ^ ^ ^ ^ L<r*-fl 4 . C c * c o * M ) - S ^ , * 6 ) Z j + J7^ + * ' U l W ' \u00ab c * a \u00ab e a ' 1 7 ) and I f we s e t 1 'ZL and then as L\u2014>0 we have Mjym P ^ - r Su) L e t t i n g t h e d i m e n s i o n l e s s q u a n t i t i e s dLjCt.L\u2014^0 we have t h e l i m i t i n g case o f 1-17 and 1-18 16 and w h i c h i s a r e c a p t u r i n g o f e q u a t i o n s 1-15 and 1-16 o f t h e \" w h i t e n o i s e c a s e . F i n a l l y , we d e f i n e a q u a n t i t y , L_, c a l l e d t h e c o r r e l a t i o n l e n g t h o f yuciL} ; i t i s t h e s m a l l e s t l e n g t h such t h a t i z i > L ==> 17*) ^ o I t I s now . p o s s i b l e t o deduce the form o f t h e d i s p e r s i o n r e l a t i o n i n t h e two l i m i t i n g c a ses o f s h o r t and l o n g c o r r e l a t i o n l e n g t h w i t h r e s p e c t t o t h e w a v e l e n g t h . F i r s t we examine t h e case o f s h o r t c o r r e l a t i o n l e n g t h , i . e . ^~\/~^0 ^  1 where Wo = 27!\/7\\0- N o n - d i m e n s i o n a l i z i n g t h e depth c o o r d i n a t e Z w i t h L_, i . e . s e t t i n g Z = L ^ * i t i s now p o s s i b l e t o r e w r i t e 1-11 and 1-12 as -.\u00a3*<xLfrCLi.*) (2<xL**)di* ('I-19) and -\u00b0\u00b0 -co -\u00a3 zc<Ljr(L?.*) con (2o<Lz*) Jz* (1-20) -CO By Ap p e n d i x D we have t h a t i n t h e l i m i t i n g case L < < & f % J = l - ^j^Hud* + 2*5Bfh*,d* +Ocp> (1-22; _oo ^ -co and = -2\u20ac*<xfr<\u00bbdf: +0(p? (1-22) -co where I f the t e r m i n 1-21 i n v o l v i n g t h e i n t e g r a l -co i s s m a l l (which seems p l a u s i b l e f o r s h o r t c o r r e l a t i o n l e n g t h s ) t h e n 1-21 and 1-22 a r e q u a l i t a t i v e l y the same as t h e \" w h i t e n o i s e \" f o r m u l a s 1-15 and 1-16 s i n c e the i n t e g r a l f R i ] ^ > 0 as i t i s one h a l f o f t h e power spectrum of ju(\u00b1) e v a l u a t e d a t t h e o r i g i n . Now i n t h e case o f l o n g c o r r e l a t i o n l e n g t h , i . e . L \/ ^ > > I 18 we can o b t a i n t h e a s y m p t o t i c e x p a n s i o n s f o r t h e i n t e g r a l s i n 1-11 and 1-12 i n v o l v i n g t h e s i n e and c o s i n e f u n c t i o n s by r e p e a t e d i n t e g r a t i o n by p a r t s . By Appendix E we have t h a t i n t h e l i m i t i n g case L>> , w i t h 3eo - ' d and = J.f Cayd* -oo -p sty 7 - i . \u00a3*r'\u00b0>. f-^- CaC*6 \u00a3lH\u00b0) and v a l i d f o r 6 n o t near 6= O or 9 - 77\". I n v i e w o f t h e e x p e r i m e n t a l d a t a mentioned e a r l i e r i n t h i s c h a p t e r and t h e remarks o f Appendix C we cannot e x p e c t t o t r e a t t h e case L > > i n t h e oceans. However, f o r m u l a s 1-23 and 1-24 may have some r e l e v a n c e t o i n t e r n a l wave m o t i o n i n s t e l l a r a tmospheres. CHAPTER I I : THE TIME DEPENDENT CASE 19 I I . 1 Formal D i s p e r s i o n R e l a t i o n As i n Cha p t e r I t h e e q u a t i o n s a t i s f i e d by t h e stream f u n c t i o n $(X,*,t) i s i \u00ab + i \u201e \u00ab + +f - % itt - q.'^ = o (u-i) T h i s time we assume HZ=-N0(l+^\/i^) where yuct) i s a g a i n a zero-mean, w i d e - s e n s e s t a t i o n a r y random p r o c e s s . T h i s g i v e s e q u a t i o n I I - l t h e form ' IJL + Jf) $ = 0~ (11-2) where and I n t h i s case t h e d i s p e r s i o n r e l a t i o n t o second o r d e r i n \u20ac becomes where eAC1 i s the i n t e g r a l o p e r a t o r d e f i n e d by CO \u2014 CO w i t h Gr, t h e Green's f u n c t i o n , b e i n g t h e s o l u t i o n o f As i n Ch a p t e r I , the Green's f u n c t i o n i s a d i s p l a c e m e n t k e r n e l Q(X.X',i , ^ ' , t , t ' ) - &<X.-X ,JT:-\u00b1',-t-t.') so i t i s n e c e s s a r y t o c o n s i d e r o n l y M G = JcxxfcvStt) (JT-+) We s o l v e by t a k i n g a t r i p l e F o u r i e r t r a n s f o r m o f b o t h s i d e s o f I I - 4 and d e f i n e Gr(kt*.a) \u2022= JJJ G<.x.*tt) e dxdidt -ca U s i n g t h e c a u s a l i t y p r o p e r t y , Cr = 0 f o r ~t < 0 , t h e i n v e r s i o n i n <r can be p e r f o r m e d t o y i e l d t h e F o u r i e r t r a n s f o r m i n x and ~z. o f G : \u201e , -H(t) Usn cut w i t h d= ^\/g and \/\/ A\/ok1- +W-+Uf\\L f o r c o n v e n i e n c e . I I . 2 S i m p l i f y i n g t h e D i s p e r s i o n R e l a t i o n I t i s now p o s s i b l e t o s u b s t i t u t e ,jA> , dC~J and c\/\/\" i n t c T l -t o f i n d t h e d i s p e r s i o n r e l a t i o n e x p l i c i t l y . S e t and Then and V _ -c(kxi- ?z -crt) CO ' e dx'dt'c\/t' S i m p l i f y i n g , CO -A3 T h i s can be w r i t t e n CO \u2014 C o d x ' c \/ i ' d l t ' Make t h e change o f v a r i a b l e s x\"-x-x' 2\" = Z - Z' Then t h i s g i v e s T2 = - \u00a3 \u00bb Y . \u00ab r - * v . y ^ - \u00ab ^ 7 e - \" \" \" \" - \" ^ , g . ^ ^ . i - C O Now t a k i n g a c c o u n t o f the i n t e g r a l b e i n g a f u n c t i o n o f t , t h e d i f f e r e n t i a t i o n can be per f o r m e d t o g i v e CO U s i n g t h e s t a t i o n a r i t y o f yU(\u00b1) we d e f i n e t he a u t o c o v a r i a n c e f u n c t i o n Fct\") by Then T2 =,-e*\/V*C-kz + i-jq -u^jy^a.ttyrtthe'^t* -oo 4 -* -co The d i s p e r s i o n r e l a t i o n , c o r r e c t t o second o r d e r i n \u00a3, i s t h u s CO A 0 0 S e t t i n g \u00a3 = 0 and r e a r r a n g i n g t h e terms o f e q u a t i o n I I - 5 g i v e s the Oth o r d e r d i s p e r s i o n r e l a t i o n as i n Appendix A, where we had Cf, k , 4% r e a l and JL - M -td\/z I t i s a g a i n p h y s i c a l l y m e a n i n g f u l t o s u b s t i t u t e t h e above form f o r Jl i n t o e q u a t i o n I I - 5 , t h e g e n e r a l d i s p e r s i o n r e l a t i o n t o OCe2) , and t h e n s o l v e the e q u a t i o n f o r (T . Making t h e s u b s t i t u t i o n we o b t a i n -era* - td\/Jd + ^^[f- -r ffj?* - c^j]\\ 24 T h i s r e d u c e s t o g-Z(k'+ Si + <t*\/4) - fli'l? - \/VAV <*%.) * \u00a32A\/\u201e*[kx- <%\u00abr'-r>-\u00b1 (r'-i'Jl*-E q u a t i o n I I - 6 i s an e q u a t i o n f o r & , and can be s o l v e d by-i t e r a t i o n . S e t t i n g \u20ac = O g i v e s t h e Oth o r d e r s o l u t i o n where we choose the p r i n c i p a l b r a n c h o f t h e square r o o t f u n c t i o n f o r d e f i n i t e n e s s . R e p l a c i n g <T by (T0 i n t h e Oct*) terms and r e a r r a n g i n g y i e l d s t h e a p p r o x i m a t e s o l u t i o n 25 ,00 D i v i d i n g b o t h s i d e s by (f0 and u s i n g the b i n o m i a l a p p r o x i m a t i o n f o r t h e square r o o t we can w r i t e o \u2014 \u2022 j \/ $jyyxO-0ir \u2022 -\u00a3<T0\u00a3K I (t + ir0d r ' c t ; + CM r'U, + <h r ? t i j etVr' W ( I T - 7 ) F o r a g i v e n c h o i c e o f a u t o c o v a r i a n c e f u n c t i o n f t t ) , \"Rel^-l] g i v e s t h e r e l a t i v e change i n the phase speed o f t h e mean wave, and ^\/m{%-B~^-} r e p r e s e n t s t h e growth or a t t e n u a t i o n f a c t o r . I f b o t h a r e p o s i t i v e t h e n t h e mean wave t r a v e l s f a s t e r and grows, i t i s a l s o p o s s i b l e from e q u a t i o n I I - 7 t o d e r i v e a s y m p t o t i c r e s u l t s f o r the l i m i t i n g c a s e s o f l o n g and s h o r t ( w i t h r e s p e c t t o t h e wave p e r i o d ) c o r r e l a t i o n t i m e s s i m i l a r t o e q u a t i o n s 1-21, 1-22, 1-2 3 and 1-24 o f Chapter I . CHAPTER I I I : FURTHER EXAMINATION OF THE CASE OF DEPTH 26 DEPENDENT \"WHITE NOISE\" FLUCTUATIONS I I I . l I n t r o d u c t i o n I n t h i s c h a p t e r we s h a l l be co n c e r n e d p r i m a r i l y w i t h t h e case o f \" w h i t e n o i s e \" y \" ( 2 l ) v a r i a t i o n s and t h e e q u a t i o n s and (nr.' 2.) w h i c h a r e f o r m u l a s 1-15 and 1-16 r e s p e c t i v e l y o f Ch a p t e r I . I n o r d e r t o o b t a i n a g r a p h i c a l r e p r e s e n t a t i o n c o r r e s p o n d i n g t o t h e s e f o r m u l a s we employed t h e d a t a from s t a t i o n P (50\u00b0N, 145\u00b0W) 2 -8 _ y d i s c u s s e d i n s e c t i o n 1.4. We p u t T -1.22 *I0 sec t h r o u g h o u t c o r r e s p o n d i n g t o 50\u00b0N l a t i t u d e , No - 3* io %c2 f o r t h e passband I ca s e , and i n the absence o f any d a t a we chose No = 10 $*c f o r t h e passband I I c a s e . R e p r e s e n t a t i v e v a l u e s o f cr1 were s e l e c t e d f o r b o t h passbands t o g i v e t h e v a l u e s o f asymptote a n g l e , 9# = to**1\\fc^ i n d i c a t e d on the c u r v e s . We t h e n l e t 9 r u n o v e r t h e range 0, o.ie^,... 0.98# as I I I - 1 ( I I I - 2 ) i s even (odd) i n 9 , and computed by machine t h e R.H.S. o f I I I - l and I I I - 2 f o r t h e s e r a d i a n a n g l e s o f p r o p a g a t i o n . The p r i n t - o u t was hand p l o t t e d on graph paper, smooth c u r v e s were drawn t h r o u g h t h e p o i n t s and t h e c u r v e s were i n k e d o n t o t r a c i n g paper t o g i v e f i g u r e s 1, 2 and 3. O.Q45 = 1.54-- i r- 1 \" \u00ab~ O.f- J--^ quHD. JL.m Q, TRO'PR&tlTiON ANGLE CRRDtRtfS) S R \u2122 \u00a3 Z ' m FIG- 1 Gr'RoW\/TH (6>o) flUV ftTTtMUaitON (6 <0) 7?flT\u00a3S I l l . 2 D i s c u s s i o n o f t h e \"White N o i s e \" Growth C u r v e s 2 I n o r d e r t o i n t e r p r e t t h e graph o f '\/'e2d. v e r s u s t h e r a d i a n a n g l e o f p r o p a g a t i o n f o r b o t h passbands o f <J2 , i t s h o u l -A be u n d e r s t o o d t h a t t h e d i m e n s i o n l e s s q u a n t i t y \u20ac.2 d w i l l have a d i f f e r e n t v a l u e f o r each o f t h e two passbands. We r e c a l l 0 0 -A r d zz d T:*)dz -co - dV f o r fcz) = nTi2j and f o r t h e a p p r o x i m a t e v a l u e 5 - \/03 orn tit'2 we have d - 3\" \/0~t csm'1 f o r t h e passband I c a s e , and d - I0~n On'1 . , 2 f o r t h e passband I I c a s e , w i t h t h e above v a l u e s o f Na . E x a m i n i n g f i g u r e 1 i t i s c l e a r t h a t waves p r o p a g a t i n g w i ~ h an upward component o f d i r e c t i o n grow b u t t h a t waves w i t h a downward component o f d i r e c t i o n a r e a t t e n u a t e d , s i n c e the growth r a t e i s an odd f u n c t i o n o f t h e p r o p a g a t i o n a n g l e . T h i s b e h a v i o u r i s a l s o f o u n d i n t h e 0 t h o r d e r case where an e x p l a n a t i o n f o r i t i s g i v e n i n Appendix A. There we saw t h a t t o 0 t h o r d e r a pl-^ne wave has the form 28 For a p o l a r r e p r e s e n t a t i o n we s e t and w h i c h g i v e s the form e \/ z Now i n the 0 C6\\> d i s p e r s i o n r e l a t i o n we have a l l o w e d d\u00a3 t o be complex, i . e . and t h e mean wave has the form ^d.=t\/2 ^i[dt^ coo-ce-V) -<rt] -Xxfc\u00b0*-<e-Y>) But, s i n c e i n the \" w h i t e n o i s e \" case C'ct or then t h e mean wave form can be w r i t t e n as e 2 -e N w h i c h i m p l i e s t h a t growth o r a t t e n u a t i o n o c c u r s o n l y i f the wave has a v e r t i c a l component o f p r o p a g a t i o n ( i . e . 6 ^  o, 7T) and t h a t the growth r a t e depends on the square o f the c o r r e s p o n d i n g Oth o r d e r wave number. I n f i g u r e 1 wave number i s a \"h i d d e n v a r i a b l e \" ; as the a n g l e o f p r o p a g a t i o n approaches t h e asymptote a n g l e we f i n d t h a t t h e growth o r a t t e n u a t i o n r a t e becomes l a r g e because < t f o becomes l a r g e . So f o r a g i v e n d i s t a n c e h s h o r t e r waves w i l J row f a s t e r t h a n l o n g e r waves, p o s s i b l y t o t h e p o i n t o f b r e a k i n g , i n t r a v e l l i n g upward from the p l a n e z = 0 t o the p l a n e z = \/?. I t i s r e a s o n a b l e t h a t s h o r t e r waves s h o u l d be more a f f e c t e d by the p r e s e n c e o f t h e \" w h i t e n o i s e \" f l u c t u a t i o n s . The a n a l y s i s i s n o t u n i f o r m l y v a l i d i n the w a v e l e n g t h , as d i s c u s s e d i n Appendix C; v e r y s h o r t waves a r e e x c l u d e d from c o n s i d e r a t i o n . S i n c e d=-fa\/^a we d e f i n e J0p , a l e n g t h s c a l e d e t e r m i n e d by the v a r i a t i o n i n >^ , by A l s o l e t us d e f i n e and by a n a l o g y w i t h t h e above we d e f i n e JL^jf. by Now dejp > d so \u00a30<!if < ; i n words, t h e mean e f f e c t o f t h e random i n h o m o g e n e i t i e s i s t o s h o r t e n t h e l e n g t h s c a l e o f t h e p v a r i a t i o n s . C o n s e q u e n t l y by a n a l o g y w i t h t h e c o r r e s p o n d i n g r e s u l t o f Appendix A growth o r a t t e n u a t i o n o f t h e mean wave i s n e c e s s a r y t o p r e s e r v e k i n e t i c energy i n the p r e s e n c e o f t h e d e c r e a s e d mean d e n s i t y v a r i a t i o n s a c l e \u00a3oetf . S i m i l a r l y d e f i n i n g \/V0 efr by de* = M<8. c, we see t h i s can be i n t e r p r e t e d as an i n c r e a s e i n t h e B r u n t -V a i s a l a f r e q u e n c y o r a \" s t i f f e n i n g \" o f t h e f l u i d . U s i n g the s t a t i o n P d a t a d i s c u s s e d i n 1.4 i t i s p o s s i b l e t o e s t i m a t e t h e magnitude o f t h e change i n growth o r a t t e n u a t i o n r a t e s and the q u a n t i t i e s dcff. and \/\\\u00a3W . F o r b o t h summer and w i n t e r i n t h e 700 - 1500 m range we f i n d CO \u00a32J Tczjdz - ^ 2*IG*c*n -oo -9-1 S i n c e d i K 10 om we have rfotf^ - i = 2e*TMoS^'e\/^ ^ io'z XoZ S^yxz& Z a l k a n o b s e r v e d f o r i n t e r n a l waves i n t h e P a c i f i c t h a t >; 21T x IO~SC\/m'2 o r A 0 \u00a3. 10*<y>*\\ = 1 k>m . Thus u n l e s s B6. to'2 r a d i a n . Thus the s t o c h a s t i c e f f e c t s a r e dominant. S i n c e t h e v e r t i c a l growth r a t e i s e s s e n t i a l l y \u00a3z V z& and t h e h o r i z o n t a l growth r a t e i s ezT^o s^7^ t h e n w i t h \u00a3 l f - 2*A3?c*\u00ab we f i n d v e r t i c a l and h o r i z o n t a l growth r a t e s r e s p e c t i v e l y o f a p p r o x i m a t e l y to  SSMSGorn1 and 1 f o r a 1 km wave, and i\u00b0  Js^*i*eam'1 and ^\/o'tu^iecm1 f o r a 100 m wave. 31 If the waves are e s s e n t i a l l y h o r i z o n t a l l y propagating, as Zalkan' 8 finds, i . e . Q\u00ab 1 , e.g. Q-o.l then ^ ez and i ^ ^ e - SQ and the rates for a 100 m wave are approximately \/0~5 cm ~J and \/0 CsrrC . This represents an e-folding length of 10 wavelengths i n the v e r t i c a l and 1 wavelength i n the h o r i z o n t a l . Since A^ 2^ = de# y then with our data. Thus Ke#.\/A\/a ^ \/0b<Xo ls*\"-&f ^ -27T for 9 o.l and a wavelength of 1 km. Hence the mean e f f e c t of the random fluctuations i s to sub s t a n t i a l l y \" s t i f f e n \" the f l u i d rendering the Brunt-Vaisala frequency e f f e c t i v e l y greater. This suggests inter n a l waves i n passband I might e x i s t with frequency <r > N0 the deterministic cut-off. In fact some recently observed in t e r n a l wave frequency spectra do not ex h i b i t a sharp cut-off A\/ 5\"' 7'\" at N0. Looking at the curves of figure 1 we can account for t h e i r o v e r a l l appearance as follows: 1) Independence of \u2014 2 \u2122 { * % g j \/ o n the passband of cr 2 follows as i t i s only c 1 = (A\/*-a-i)\/((r:i-fzJ which determines the asymptotes, not the parameters i n d i v i d u a l l y . 2) Growth or attenuation requires v e r t i c a l motion of the wave; hence for 0= O,Tr we have no growth or attenuation. QR-031 (10 V* = 0.+5 . 6 5 fl.Ql) .78 (1.51) 0.9S- (l-oz) l.lo (0.63) 1.1-1 (o.os; j y i O.f 0.8 1-2. (kuRV. JL, zzr Q P R O P R G-ftTION R N & L E (7Z4D\/#A\/S) SRMF PS X, IX FIG. 2 RELRTIVE 7>H#SE SPEED CHRMGE &\/s -l)\/e*\u00a3 0 -0.5 - 0 . 6 0.+ . 0.8 1.2 i r \"i 1 6 7 . 5 5 (8.11) 0.6S (4-.53J 4-5 (2.33) l.+l (U.S) 031 (1,23) 0.7 -O.Q - 0 . 9 U G* = o.lS {10*CT = 0,37 5\u00a3c~*J FIG. 3 XELRTl\\JE PH4SE 5PEED QHRNGE E\u00b0K TRSSdAHT) 7J; \/\\\/Z < a~ * < -f* ' .' \u2022\u2022 32 3) On a g i v e n c u r v e as |el i n c r e a s e s t h e waves a r e s h o r t e r and t h u s have a h i g h e r growth o r a t t e n u a t i o n r a t e . 4) As 6fj d e c r e a s e s from c u r v e t o cu r v e t h e waves t r a v e l l i n g i n a f i x e d d i r e c t i o n 6 become s h o r t e r and t h u s have a h i g h e r growth o r a t t e n u a t i o n r a t e . I I I . 3 D i s c u s s i o n o f t h e \"White N o i s e \" Phase Speed C u r v e s I n t h i s s e c t i o n we s h a l l a t t e m p t t o i n t e r p r e t t h e gr a p h s i n f i g u r e s 2 and 3, w h i c h c o r r e s p o n d t o f o r m u l a I I I - l f o r passbr-.-.ds I and I I r e s p e c t i v e l y . S i n c e the passband I ca s e , \/\"z< < N02, i s t he u s u a l c i r c u m s t a n c e we s h a l l g i v e more a t t e n t i o n t o i t . We n o t e t h a t when t h e o r d i n a t e (s\u00b0\/s ~ V\/G'\u00a3 >  0 t h i s means <'c o r t h e mean phase speed t o Oct*) i s l e s s t h a n t h e phase speed i n the non random c a s e . For our d a t a \u00b1^ &*\/o~6 o r ^ \/ 0 \" r and so Se\/s-1 :=2 \/0~*(' 0(1) -0(10)) on e x a m i n i n g f i g u r e 2. Thus phase speed changes a r e s m a l l , g e n e r a l l y s p e a k i n g . L e t us d e f i n e F<ej = R . H . S . o f e q u a t i o n I I I - l , t h e n F^e; i s monotonic i n c r e a s i n g i n passband I and monotonic d e c r e a s i n g i n passband I I . A l s o j- No j R 0 ) = + ' ^ i n the l i m i t i n g case 8 \u2014>'7P2 , we have <rz-l -f1 w i t h \u2022\u00a3*<< so Fco) ^ - - z \/ ^ t i n passband I , and c r 2 ? \/ 1 w i t h \/VcX so Fco)^-2\/z i n passband I I . I n passband I t h e t r a n s i t i o n case f o r w h i c h t h e phase speed 33 t o Oct*) i s nowhere g r e a t e r t h a n t h e Oth o r d e r phase speed o c c u r s when F(o)-o i . e . cr2- = ^l\/z . For fz<< No t h i s means crz - (Af*+-flJ\/z and <tg-<ry\u00abr*~fV *1 o r c 2 1 ; t h u s Qn - . On e x a m i n i n g f i g u r e 2 we see t h a t when ^ - l . ^ l c o r r e s p o n d i n g t o \/06a-1 - O.o?, t h e phase speed t o Oct1) i s g r e a t e r t h a n t h e Oth o r d e r phase speed e x c e p t f o r a s m a l l range o f S n e a r . S i n c e f o r l a r g e we have cr11 -fz i t a p p e a r s t h e i n c r e a s e i n phase speed i s a r e s u l t o f t h e r o t a t i o n . However, as 3\u2014> 6# t h e waves become s h o r t e r and a r e r e l a t i v e l y slowed down, r e f r a c t i o n e f f e c t s a p p a r e n t l y becoming dominant. The s h o r t e r t h e wave t h e more i t i s s c a t t e r e d by t h e random i n h o m o g e n e i t i e s i n t h e medium and hence the f a r t h e r i t must t r a v e l t o g e t from one p l a c e t o a n o t h e r . T h i s appears as a d e c r e a s e i n the mean phase speed. As Qff d e c r e a s e s from c u r v e t o c u r v e i n f i g u r e 2 , Cz d e c r e a s e s and, from Appendix A, t h i s means d\u20aco i n c r e a s e s f o r f i x e d 0 , i . e . t h e w a v e l e n g t h i s d e c r e a s e d . Thus t h e r e f r a c t i o n e f f e c t s become more i m p o r t a n t r e s u l t i n g i n the upper c u r v e s o f f i g u r e 2 l y i n g w h o l l y above t h e 6 a x i s . On any g i v e n c u r v e f o r O n e a r i n g &\u00bb the ends o f the c u r v e t u r n up s h a r p l y due t o r e f r a c t i o n o f the r a p i d l y s h o r t e n i n g waves. As was men t i o n e d i n t h e d i s c u s s i o n o f f i g u r e 1, :r,e s i z e o f the d i m e n s i o n l e s s q u a n t i t y \u20ac 2c? w i l l depend on t h e passband c f C*~. To i n d i c a t e w h i c h passband we a r e c o n s i d e r i n g we s h a l l use t h e s u b s c r i p t I o r I I . We have 34 ell - 3 * 10 * Cm_\/ and t h u s djL \u2014 JO 7 3 cw ~2 I t seems u n l i k e l y t h a t the r a t i o \u20ac^ \u00a3'\/Q c o u l d be s m a l l enough t o y i e l d a v a l u e o f ^jcci^ l a r g e r t h a n \u00a3 x c t x . However, i n the absence o f e x p e r i m e n t a l d a t a f o r t h e passband I I case we s h a l l c o n s i d e r t h e Oth o r d e r d i s p e r s i o n r e l a t i o n i n o r d e r t o o b t a i n some i d e a o f t h e i m p o r t a n c e o f t h e f l u c t u a t i o n s i n f o r passband I I r e l a t i v e t o passband I . Now 2 z 3. \u2022_ \/Vox ~ 0~x crx2 - \/-or - J.22 * JO~a and o r C - ~ NCJC \u2022O- ~7l ~\u2014 ,2 OJC \u2014 JO L e t Cx = c\u00a3 w i t h t h e i r common v a l u e . Now l e t us i n c r e a s e by 1 0 % i n each case and l e t A Cz be t h e i n c r e m e n t i n Cz . Then a-\/ -1.22*10-* and Now III-3 and III-4 can be s o l v e d f o r CT^ and and t h e r e s u l t s s u b s t i t u t e d i n t o t h e above t o y i e l d Arz - -$xl0~7( 1+ C2) and Hence ^2 L0 Z Thus t h e asymptotes o f the Oth o r d e r d i s p e r s i o n r e l a t i o n a r e much more a f f e c t e d by changes i n t h e B r u n t - V a i s a l a f r e q u e n c y i n passband I t h a n i n passband I I . T h i s a n a l y s i s s u g g e s t s t h a t t h e change i n phase speed o f t h e mean wave f o r passband II w i l l be n e g l i g i b l e , so we s h a l l n o t c o n c e r n o u r s e l v e s w i t h i n t e r p r e t i n g f i g u r e 3, w h i c h i s i n c l u d e d f o r c o m p l e t e n e s s . A d i f f e r e n c e i n t h e b e h a v i o u r o f waves o f passbands I and n i s n o t s u r p r i s i n g when i t i s r e c a l l e d t h a t t h e two c l a s s e s o f i n t e r n a l waves have e s s e n t i a l l y d i f f e r e n t d e p e n d e n c i e s on t h e media t h r o u g h w h i c h t h e y p r o p a g a t e . Passband I waves cannot e x i s t 36 i n an u n s t r a t i f i e d ocean and so a r e l i k e l y t o be more s e n s i t i v e t o f l u c t u a t i o n s i n t h a n passband I I waves, w h i c h a r e i n e r t i a l and can e x i s t i n an u n s t r a t i f i e d ocean. BIBLIOGRAPHY 3 7 1 . B a t c h e l o r , G.K., An I n t r o d u c t i o n t o F l u i d Dynamics, C.U.P., 1 9 6 8 . 2. B h a r u c h a - R e i d , A.TV, On t h e t h e o r y o f random e q u a t i o n s , P r o c . Symp. A p p l . Math., XVI, p p . 4 0 - 6 9 , 1 9 6 4 . 3. Boyce, W.E., Random e i g e n v a l u e problems, P r o b a b i l i s t i c . Methods i n A p p l i e d M a t h e m a t i c s , I , p p . 1 - 7 3 , E d i t e d by A.T.Bharucha-Reid, Academic P r e s s , New Y o r k , 1 9 6 8 . 4. C a r r i e r , G.F., Krook, M., and P e a r s o n , C.E., F u n c t i o n s o f a Complex V a r i a b l e : Theory and Technique, M c G r a w - H i l l , 1 9 6 6 . 5\u201e F a f o n o f f , N.P., R o l e o f th e NDBS i n f u t u r e n a t u r a l v a r i a b i l -i t y s t u d i e s o f th e N o r t h A t l a n t i c , P r o c . F i r s t S c i e n c e  A d v i s o r y M e e t i n g , N a t i o n a l Data Buoy Development P r o j e c t , U.S. C o a s t Guard, p p . 5 0 - 6 1 , 1 9 6 9 . 6 . F r i s c h , U., Wave p r o p a g a t i o n i n random media. P r o b a b i l i s t i c Methods i n A p p l i e d M a t h e m a t i c s , I , p p . 7 5 - 1 9 8 , E d i t e d by A.T.Bharucha-Reid, Academic P r e s s , New Yo r k , 1 9 6 8 . 7 . G a r r e t t , C. and Munk, W., I n t e r n a l wave s p e c t r a i n t h e p r e s e n c e o f f i n e - s t r u c t u r e , J . Phys. 0 c e a n o q r 0 , I , pp. 1 9 6 - 2 0 2 , 1 9 7 1 . 8 o K e l l e r , J.B., Wave p r o p a g a t i o n i n random media, P r o c . Symp.  A p p l . Math., X I I I , p p o 2 2 7 - 2 4 6 , 1 9 6 2 . 9 . K e l l e r , J.B., S t o c h a s t i c e q u a t i o n s and wave p r o p a g a t i o n i n random media, P r o c . Symp. A p p l . Math., XVI, p p . 1 4 5 - 1 7 0 , 1 9 6 4 . 1 0 . K e l l e r , J.B., The v e l o c i t y and a t t e n u a t i o n o f waves i n a random medium, E l e c t r o m a g n e t i c S c a t t e r i n g , p p . 8 2 3 - 8 3 4 , E d i t e d by R.L.Rowell and R . S . S t e i n , Gordon and B r e a c h S c i e n c e P u b l i s h e r s , New Y o r k , 1 9 6 7 . 1 1 . K e l l e r , J.B. and V e r o n i s , G., Rossby waves i n t h e p r e s e n c e o f random c u r r e n t s , J . Geophys. Res., 7 4 , 8 , p p . 1 9 4 1 - 5 1 , A p r i l 1 5 , 1 9 6 9 . 1 2 c L e B l o n d , P \u201e H o , On the damping o f i n t e r n a l g r a v i t y waves i n a c o n t i n u o u s l y s t r a t i f i e d ocean, J . F l u i d Mech. , 2 5 , I , pp. 1 2 1 - 1 4 2 , 1 9 6 6 . 38 13. P a r z e n , E., S t o c h a s t i c P r o c e s s e s , Holden-Day I n c . , San F r a n c i s c o , 1964. 14. P h i l l i p s , O.M., The Dynamics o f t h e Upper Ocean, C.U.P., 1966. 15. T a y l o r , A.E., Advanced C a l c u l u s , B l a i s d e l l Pub. Co., T o r o n t o , 1955. 16. Wang, Y.C., The i n t e r a c t i o n o f i n t e r n a l waves w i t h an u n s t e a d y n o n - u n i f o r m c u r r e n t , J . F l u i d Mech., 37, IV, pp. 761-771, 1969. 17. Webster, T.F., L e c t u r e s , Second C o l l o q u i u m on t h e Hydrodynam-i c s o f t h e Ocean, L i e g e U n i v e r s i t y , C a h i e r s de mechanique mathematique, 26, pp. 20-53, 1970. 18. Z a l k a n , R.L., H i g h f r e q u e n c y i n t e r n a l waves i n the P a c i f i c Ocean, Deep-Sea R e s e a r c h , 17, pp. 91-108, Feb. 1970. APPENDICES 39 A- The Oth Order D i s p e r s i o n R e l a t i o n F o r a p l a n e wave p r o p a g a t i n g i n a s t r a t i f i e d f l u i d w i t h c o n s t a n t B r u n t - V a i s a l a . f r e q u e n c y A 4 t h a t i s u n i f o r m l y r o t a t i n g about t h e v e r t i c a l ( z ) a x i s , t h e d i s p e r s i o n r e l a t i o n i s where Cz = (tf-<r\\)\/(o-x-fx) , <t = rtZ\/3 , \u2022\u00a3 = c o r i o l i s p a r a meter, X. = h o r i z o n t a l d i s t a n c e , t = t i m e . P u t t i n g H^fx+i^x w i t h JK, r e a l and r e q u i r i n g k r e a l (as t h e r e i s no p h y s i c a l r e a s o n f o r growth o r a t t e n u a t i o n o f the wave a l o n g t h e h o r i z o n t a l d i r e c t i o n ) g i v e s 4 r = - ^ and c * f = (A-l) A p l a n e wave now has t h e form e Q . Waves p r o p a g a t i n g w i t h an upward component o f d i r e c t i o n grow, and waves w i t h a downward component o f d i r e c t i o n a r e a t t e n u a t e d . Now h a v i n g \/V - N0 , a c o n s t a n t , i m p l i e s {>cc2)\u00b0<e . The k i n e t i c energy o f a wave i s p r o p o r t i o n a l t o $>A2 where A i s t h e wave a m p l i t u d e . Hence t h e f a c t o r Qdi\/,z i s seen t o be n e c e s s a r y t o p r e s e r v e t h e k i n e t i c energy o f t h e wave. The e q u a t i o n e.*k*-J* = c o r r e s p o n d s t o a d i s p e r s i o n d i a g r a i r f c o n s i s t i n g o f a r e c t a n g u l a r h y p e r b o l a o p e n i n g t o w a r d l a r g e I M v a l u e s , f o r f i x e d 0~ . We i n t r o d u c e a p o l a r c o o r d i n a t e 40 r e p r e s e n t a t i o n o f t h e r e a l wave numbers. S e t (*j = fa (tote, SlsriQ) S u b s t i t u t i n g i n t o A - l , we f i n d f o r 9 6 (-Qf,,Qfl) K}(7r-e^jlTi-Qf,) where 6^ , t h e a n g l e a t w h i c h t h e asymptotes t o t h e h y p e r b o l a a r e i n c l i n e d t o t h e k - a x i s , i s g i v e n by Thus i t i s c l e a r t h a t an i n c r e a s e i n C c o r r e s p o n d s t o an i n c r e a s e i n 0# or a s p r e a d i n g o f t h e a s y m p t o t e s . I n a d d i t i o n e q u a t i o n A-2 i m p l i e s t h a t f o r f i x e d 6 an i n c r e a s e i n Cz r e s u l t s i n a d e c r e a s e i n d&0 \u2022 B- D e r i v a t i o n o f t h e Stream F u n c t i o n E q u a t i o n L e t Z) be a p o i n t i n a r i g h t - h a n d e d system o f C a r t e s i a n c o o r d i n a t e s r o t a t i n g u n i f o r m l y about t h e 2 - a x i s , w h i c h i s v e r t i c a l and p o s i t i v e upward. L e t f = zQ. ^ be the C o r i o l i s p arameter, where X~L i s t h e magnitude o f t h e e a r t h ' s r o t r .'on v e c t o r and f i s the l a t i t u d e . Then t h e system r o t a t e s w i t h a n g u l a r f r e q u e n c y -f\/z . If $ i s t h e magnitude o f the e f f e c t i v e g r a v i t a t i o n a l a c c e l e r a t i o n , a n t i - p a r a l l e l t o t h e B - a x i s , (u,i\/,ur) a r e the f l u i d v e l o c i t y components, ^ i s the f l u i d d e n s i t y and o^. i s t h e p r e s s u r e , then t h e momentum c o n s e r v a t i o n 41 e q u a t i o n s a r e W-t + UVX + . + usu\u00b1 +fu + ^ = O art + uusK + yur^ + usur^ + g + ly ^  = 0 f o r an i n v i s c i d f l u i d . The e q u a t i o n o f mass c o n s e r v a t i o n i s + ( f i L ) x -h (\u00a3(Sjv + (^ur)% - o and t h e i n c o m p r e s s i b i l i t y c o n d i t i o n i s f t + w p x + i- \"Sfe - 0 I n i t a l l y s e t t i n g ( U, l\/, UJ) - ( 0 , 0, 0) g i v e s w i t h ^ = ^ 0 ( H j \/ f ^ - - f 9 ( i ) t h e b a s i c s t a t e o f h y d r o s t a t i c e q u i l i b r i u m I n t r o d u c i n g p e r t u r b a t i o n s such t h a t ^cx, y, 2,-e; = ^>0<rz) +Q (x,\\\/.-t) (U.W, us) = <ut ,u},urt) f- - f-J?) + fcC^V.-t) and l i n e a r i z i n g t h e e q u a t i o n s i n t h e q u a n t i t i e s s u b s c r i p t e d w i t h 1 we o b t a i n f\u201eu^ +$9 1- ft? = 0 (S-3) UIX f i\/ly + ur^ = o (B-4-) Pit f = 0 _ \u2022 \u2022 (B-5) D r o p p i n g t h e s u b s c r i p t 1, B - l , B-2 y i e l d fotU - -fxx\u00b1 -ff.? ^oLW - --p-^ +ff-x. where Thus and A p p l y i n g p0L t o B-4 y i e l d s S u b s t i t u t i n g B-6, B-7 g i v e s or\" f \u00ab z t +f-wit - (%\u00b0Lwi)i - 0 (B-8) E q u a t i o n B-3 g i v e s f o ^ t t -f-^69 + fit -  0 U s i n g e q u a t i o n B - 5 or fo u^. + -fit -9%* LLT - O 43 Thus and S u b s t i t u t i n g t h i s i n t o B-8 produces A l t e r n a t i v e l y , + e<*\/fo <\"htt +fX?\u00b0Vpur* = 0 (13-9) Put A\/2= - y f 0 * ^ and dy = 0 , t h e n B-9 r e d u c e s t o Now UK -h usi - O i m p l i e s where 3 i s a stream f u n c t i o n ! Thus th e e q u a t i o n f o r \u00a7 i s seen t o be , t h e same as e q u a t i o n 1-1 o f the t e x t . C- K e l l e r ' s Method 44 We now g i v e a b r i e f a c c o u n t o f K e l l e r ' s method f o r d e r i v i n g t h e d i s p e r s i o n r e l a t i o n f o r t h e mean wave i n a random medium. A more g e n e r a l t r e a t m e n t w i l l be f o u n d i n K e l l e r ' s paper i n E l e c t r o m a g n e t i c S c a t t e r i n g 1 \" I f c<f i s an i n v e r t i b l e random l i n e a r o p e r a t o r and X i s a known f u n c t i o n , t h e n t h e e q u a t i o n i m p l i e s $ i s a random p r o c e s s . A p p l y i n g X t h e i n v e r s e o f oC, t o b o t h s i d e s o f e q u a t i o n C - l we o b t a i n T a k i n g ensemble a v e r a g e s , denoted by < > , t h i s becomes I n v e r t i n g <\u00a3~ 2> g i v e s <^\"1>\"1<$> = X (C-2) Now i f \u00ab*f i s s t a t i s t i c a l l y homogeneous and ~X.-Q e q u a t i o n C-2 has as e i g e n d i f f e r e n t i a l s t h e p l a n e wave s o l u t i o n s g i v e n by w h i c h obey t h e d i s p e r s i o n r e l a t i o n ^\u00abtx+J?z-crt)<^l>-i^(kxi-J>z-(r-L) = Q ( c _ 3 ) S i n c e Z\u00a3-cM. \u2022+<\/\/' and J\\T : i s \" s m a l l \" compared t o JA. t h e b i n o m i a l e x p a n s i o n i s used t o g e t 45 X'1 -<M~2-M-'JfM'1* M-'JiTM'1^^'1 w h i c h i s v a l i d i f . II <M i ^ H < 1 . A v e r a g i n g t h i s e q u a t i o n w i t h K.tA\/'y \u2014 0 , one o b t a i n s <X~2> =M'~L+<M-1<Jf<M-W><M-t- . . . . T h i s i s i n v e r t e d t o y i e l d w h i c h i s c o r r e c t t o second o r d e r i n JV* , and hence \u00a3 . S u b s t i t u t i n g C-4 i n t o C-3 g i v e s e \" \" \" - \" ^ - ^ ^ * \" ' ' ^ o fc-s; as the d i s p e r s i o n r e l a t i o n o f t h e i n f i n i t e s i m a l a m p l i t u d e mean wave c o r r e c t t o 0((3). E q u a t i o n C-5 i s i d e n t i c a l w i t h I I - 3 . I n Chapter I we have \u00a7cx.t.t) = e\" t i r t Thus i n t h i s case C-5 r e d u c e s t o e-i(\"'*\"'{j(-<w IJf>}e''\"0\"'*' = o w h i c h c o i n c i d e s w i t h e q u a t i o n 1-3. The f o r e g o i n g a n a l y s i s i s dependent f o r i t s v a l i d i t y on t h e n o t e a s i l y a p p l i c a b l e c o n d i t i o n t h a t \\\\Ji~2<J\/'ll be s m a l l . I n o r d e r t o a c h i e v e a n o t h e r p e r s p e c t i v e on t h e v a l i d i t y o f e q u a t i o n C-5 we p r e s e n t an a l t e r n a t i v e d e r i v a t i o n o f i t . P r o c e e d i n g d i r e c t l y from (M+Jf)$ - o ( C - 6 ) we have <M$ - - (c-7) A p p l y i n g uU'1 t o b o t h s i d e s o f C-7 we o b t a i n \u00a7 = -M^JV^ (C-8) Now C-8 i s an i n t e g r a l e q u a t i o n and the f i r s t s u b s t i t u t i o n y i e l d s T a k i n g ensemble a v e r a g e s < \u00a7 > = M ~ l<JfM 'lu4r\u00a7 > Thus M<$> = <Jf<M-W$> (C-?) We now make t h e s o - c a l l e d c l o s u r e a s s u m p t i o n 3 , 6 KJfM-'Jfgy = <JTM'1J\/,X\u00a7> (C-10) U s i n g C-10 i n C-9 g i v e s and a g a i n < 3? > has the p l a n e wave . s o l u t i o n s i so the approximate d i s p e r s i o n r e l a t i o n becomes i d e n t i c a l w i t h e q u a t i o n C-5\u201e The c r u c i a l p o i n t i n t h e a n a l y s i s i s t h e a p p r o x i m a t i o n C-10, On p. 45 W.E.Boyce g i v e s an a c c o u n t o f R . C . B o u r r e t ' s a t t e m p t t o j u s t i f y t h i s a p p r o x i m a t i o n by an argument somewhat s i m i l a r t o t h e f o l l o w i n g o I n t h e term <\"J\/Ui'tV\u00a7>it i s q u i t e c l e a r t h a t t h e s t a t i s t i c s o f $ cannot be i n d e p e n d e n t o f t h o s e o f uV i n v i e w o f e q u a t i o n C-6. However, i f t h e s c a l e s o f v a r i a t i o n o f J\/1 and \u00a7 a r e g r e a t l y d i f f e r e n t t h e n some j u s t i f i c a t i o n f o r C-10 can be g i v e n . I f t h e random p r o c e s s e s yU. and ^ a r e assumed t o have t h e e r g o d i c p r o p e r t y , i . e . space and time a v e r a g e s a r e e q u i v a l e n t t o ensemble a v e r a g e s , t h e n i t seems c l e a r t h a t i f t h e s c a l e s o f v a r i a t i o n o f Jf and $ a r e g r e a t l y d i f f e r e n t i t w o u l d be a good a p p r o x i m a t i o n t o r e g a r d t h e more s l o w l y v a r y i n g o f J\/'M.'1^ and $ as c o n s t a n t w h i l e a s m a l l s c a l e space o r time average i s made o f t h e o t h e r . T h i s average c o u l d be r e g a r d e d as an ensemble aver a g e . Then a l a r g e s c a l e a v e r a g e c o u l d be t a k e n o f the r e m a i n i n g p r o c e s s w i t h t h e p r e v i o u s l y a v e r a g e d q u a n t i t y b e i n g r e g a r d e d as a c o n s t a n t . T h i s second average c o u l d a l s o be i d e n t i f i e d as an ensemble average i n view o f the a s s u m p t i o n o f e r g o d i c i t y . When we used t h e Oth o r d e r s o l u t i o n , e q u a t i o n 1-9, i n the Oct1) terms and e x p e c t e d t h e f i r s t i t e r a t i o n o f e q u a t i o n 1-8 t o g i v e a good a p p r o x i m a t i o n f o r d\u20ac i t was n e c e s s a r y t o assume t h a t the Oct2) terms be s m a l l , o r e q u i v a l e n t l y t h a t be n o t g r e a t l y d i f f e r e n t f r om 0 . T h i s a s s u m p t i o n e n a b l e d us t o use t h e b i n o m i a l a p p r o x i m a t i o n t o o b t a i n e q u a t i o n s 1-11 and 1-12. s i m i l a r 48 remarks a p p l y t o t h e d e r i v a t i o n o f e q u a t i o n I I - 7 from I I - 6 . Now i f we r e s u b s t i t u t e 2]]Clccr<C'6 -Sun'6 i n t o e q u a t i o n s I-11 and 1-12, i t w i l l be c l e a r t h a t t h e a s s u m p t i o n t h a t t h e Oct') terms must be s m a l l i m p l i e s t h a t t h e q u a n t i t y C*c<xi.'G-svn*e cannot approach 0 a r b i t r a r i l y c l o s e l y , i . e . 6 cannot approach , d e f i n e d i n Appendix A, a r b i t r a r i l y c l o s e l y , and i n o t h e r words t h e a n a l y s i s c annot be e x p e c t e d t o h o l d f o r v e r y s h o r t waves. That t h e Oct3-) p e r t u r b a t i o n s cease t o r e m a i n s m a l l as the waves become s h o r t e r and s h o r t e r i s i n d i c a t e d i n f i g u r e s 1, 2 and 3 f o r \" w h i t e n o i s e \" yu(2). E x c l u d i n g v e r y Shockwaves' 6m the a n a l y s i s i s . c o n s i s t e n t ' w i t h k e e p i n g o n l y second o r d e r terms i n \u20ac i n e q u a t i o n C - 4 . T h i s i s so because h i g h e r powers o f J\/ w o u l d add h i g h e r powers o f the wave number t o the d i s p e r s i o n r e l a t i o n and t h e s e c o u l d n o t be ex p e c t e d t o be n e g l i g i b l e f o r v e r y s h o r t waves, as has been assumed i n u s i n g C-5. As a f i n a l n o t e we p o i n t o u t t h e f a c t t h a t t h e a n a l y z e d e x p e r i m e n t a l d a t a c o n s i d e r e d i n Chapter I i n d i c a t e d jjid) was \"whit e n o i s e \" t o a good a p p r o x i m a t i o n ; and the a n a l y s i s cannot be e x p e c t e d t o be v a l i d f o r v e r y s h o r t waves, i . e . waves whose s c a l e o f v a r i a t i o n approaches the s c a l e o f v a r i a t i o n o f ydd) . T h i s i s i n agreement w i t h t h e d i s c u s s i o n o f the a l t e r n a t i v e d e r i v a t i o n o f e q u a t i o n C-5 i n v o l v i n g the c l o s u r e a s s u m p t i o n C-10. D- The L i m i t i n g Case L << Xa 49 W i t h r e f e r e n c e t o f o r m u l a s 1-19 and 1-20 we d e f i n e -co I 3 = Lf\u00b0r<L2*) c<**(2*L-z*)dz* and Now o<= de0i*nd and we d e f i n e t h e n Ii (i = 1,2.3.+) and we want t o t a k e JjLvrx II (&) (C - 1, Z. 3. 4-J We have J-2<f) - LJ r(Lz*) sun dz* \u2014 CO r , f f ) = Lf ra.?:*) c<xL($i*)di* and L e t and Hi*) = I 2*r(Li*)\\ Now t h e i n t e g r a l s and - CO converge as P i s an a u t o c o v a r i a n c e f u n c t i o n , i t i s easy t o show t h a t t h e f u n c t i o n s svn(piV\/qi* , s^cp?*) , aacpzV and [l-caufpiVj\/pz* -are a b s o l u t e l y bounded by 1 ind e p e n d e n t o f \u00a3J . Hence \u00b1 he z*; and Then by Theorem V I I I p. 667 o f T a y l o r 1 5 t h e improper i n t e g r a l s o f the f i r s t k i n d U ' = i , 2 , 3 . - f ) above a r e u n i f o r m l y converge! f o r a i n t-co,coJ . Hence we may t a k e JU*n under t h e (3-->0 2nt i n t e g r a l s i g n s t o o b t a i n J^n It(B) = 2L2 ft*r(Li*) (3-f>0 1 _Joo 51 a n d I2(&) - O Changing v a r i a b l e s 2 if\/**(\"(Live!= 2 f l r ( i ) d z and CO Hence e q u a t i o n s 1-21 and 1-22 f o l l o w . E- The L i m i t i n g Case X 0 << L Here we d e r i v e f o r m u l a s 1-23 and 1-24 f o r t h e l i m i t i n g case and prove we have o b t a i n e d a t r u e a s y m p t o t i c r e s u l t . We have o C = d f o ^ n e = d u \/ n 6 \u2014 = 2frsunQ D e f i n e J d,i J d* Then T - T\u00b0<ir<\u00b1) L2\u00ab* . I - J ~dT'  e \u2014 CO W i t h 2 = L?* .a x J d^  e cdt* 52 L e t t i n g ^ = ^irsutiej^o\/^ we can o b t a i n t h e a s y m p t o t i c e x p a n s i o n 4 \" o f I as (3 \u2014>oo a l o n g t h e r e a l a x i s by r e p e a t e d i n t e g r a t i o n by p a r t s . I t i s n e c e s s a r y t o assume P i s s u f f i c i e n t l y d i f f e r e n t i a b l e and t o r e s t r i c t & so i t i s n o t n e a r & - o Q r 0 = 77~ . L e t t i n g 1 cU*\" -co 2 * and -oo as r i s an a u t o c o v a r i a n c e f u n c t i o n . We o b t a i n a f t e r N i n t e g r a t i o n s by p a r t s where To prove t h a t t h i s i s t h e v a l i d a s y m p t o t i c e x p a n s i o n o f I _ t i l . C oC^rcL^j as ^ _ > o o a l o n g t h e r e a l a x i s i t i s n e c e s s a r y t o prove t h a t f o r f i x e d M. = O ( E - i ) r -co Now t h e i n t e g r a l ' i n E-2 i s j u s t t h e sum o f two e x p r e s s i o n s p r o p o r t i o n a l t o t h e F o u r i e r s i n e and c o s i n e t r a n s f o r m s o f t h e ( N + l ) t h d e r i v a t i v e o f an a u t o c o v a r i a n c e f u n c t i o n and hence must d i e o u t a t i n f i n i t y i n any p h y s i c a l system. Thus e q u a t i o n E - l i s pr o v e d and we have o b t a i n e d the t r u e a s y m p t o t i c e x p a n s i o n o f I as ^ - > \u00ab . Ke e p i n g o n l y t h e Oth o r d e r i n ^\/$ g i v e s fi-yOO J Of 2 I - C O and Hence i n t h i s l i m i t i n g case t h e f o r m u l a s 1-11 and 1-12 r e d u c e t o -f and 2* l&o where \u00a3e0-&\u00b0\/d and d = d J VcDdi. v a l i d f o r Q n o t near 6 - o -\"CO o r Q - IT . These a r e f o r m u l a s 1-23 and 1-24 r e s p e c t i v e l y o f t h e t e x t . 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