Wave Mechanics for Ocean Eng

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Elsevier Oceanography Ser iesSeries Editor: David Halpern (1993-)F U R T H E R T I T L E S I N T H I S S E R I E SV ol um es 1-7, 11, 15, 16, 18, 19, 21, 23, 29a n d 3 2 a r e o u t o f p ri n t .

8 E. LISITZ INSEA-LEVEL CHANGES9 R.H . PARKERTHE STUDY OF BENTHIC COMMUNIT IES10 J .C.J . NIHOU L (Edi tor)M O D E L L IN G O F M A R IN E S Y S T E M S12 E .J . FERGUSON WO OD and R.E . JOHA NNE STROPICAL MA RINE POLLUTION13 E. STE EM ANN NIELSENMARINE PHOTOSYNTHESIS14 N.G.JERLOVMARINE OPTICS17 R.A . GEYER (Ed i to r)SUBME RSIBLES AND THEIR USE INOCEANO GRAPHY AND OCEAN ENGINEERING20 P .H. LEBLOND and L .A . MYS AKWAVES IN THE OCEAN22 P. DEHLINGE RMARINE GRAVITY24 F.T. BANN ER, M.B. COLLINS and K.S.MASSIE (Ed i to rs )THE NORTH -WEST EUROPEAN S HELF SEAS: THESEA BED AND THE SEA IN MO TION25 J .C.J . NIHOUL (Edi tor)MARINE FORECASTING26 H.G. RA MM ING and Z . KOW ALIKN U M E R IC A L M O D E L L IN G M A R IN EH Y D R O D Y N A M I C S27 R.A . GEYER (Ed i to r )M A R IN E E N V IR O N M E N T A L P O L L U T IO N28 J .C.J . NIHO UL (Edi tor)MARINE TURBULENCE30 A. VOIPIO (Edi tor)THE BALTIC SEA31 E .K . DU UR SM A and R. DAWSO N (Ed i to rs )M A R IN E O R G A N IC C H E M IS TR Y33 R.HEKINIANPETROLOGY OF THE OCEAN FLOOR34 J .C.J . NIHO UL (Edi tor)H Y D R O D Y N A M IC S O F S E M I-E N C L O SE D S E A S35 B. JOH NS (Edi tor)PHYSICAL OCEANOGRAPHY OF COASTAL ANDSHELF SEAS36 J .C.J . NIHOUL (Edi tor)H Y D R O D Y N A M IC S O F T H E E Q U A T O R IA L O C E A N37 W. LANGERAARSURVEYING AN D CHARTING OF THE SEAS38 J .C.J . NIHO UL (Edi tor)REMOTE SENSING OF SHELF-SEAH Y D R O D Y N A M I C S39 T.ICHIYE (Editor)O C E AN H Y D R O D Y N A M IC S O F T H E J A P A N A N DEAST CHINA SEAS40 J .C.J . NIHOU L (Edi tor)C O U PL E D O C E A N -A T M O S P H E R E M O D E L S41 H. KUNZ ENDO RF (Edi tor)MARINE MINERAL EXPLORATION42 J .C.J NIHOU L (Edi tor)MARINE INTERFACES ECOHYDRODYNAMICS

43 P. LASSER RE and J .M. MA RTIN (Edi tors)B IOGEOCHE MICAL PROCESSES AT THE LAND-S E A B O U N D A R Y44 I .P. MA RTIN I (Edi tor)C A N A D IA N IN L A N D S E A S

45 J .C.J . NIHOU L (Edi tor)T H R E E -D IM IN S IO N A L M O D E L S O F M A R IN E A N DE S T U A R IN D Y N A M IC S46 J .C.J . NIHOU L (Edi tor)SMALL-SCALE TURBULENCE AND MIX ING IN THEOCEAN47 M.R. LANDRY and B.M. HICKEY (Edi tors)C O A S T A L O C E N O G R A P H Y O F WA S H IN G T O NA N D O R E G ON48 S .R. MASS ELH Y D R O D Y N A M IC S O F C O A S TA L Z O N E S49 V .C. LAKHAN and A .S . TRENHAILE (Ed i to rs)APPLICATIONS IN COASTAL MODELING50 J .C .J . N IHOUL and B .M. JAM AR T (Ed i to rs )MESOSCALE SYNOPTIC COHERENT STRUCTURESIN GEOPHYSICAL TURBULENCE51 G.P . GLASBY (Ed i to r )ANTA RCTIC SECTOR OF THE PACIFIC52 P .W. GLYNN (Ed i to r)GLO BAL ECOLOGICAL CONSEQUENCES OF THE1982-83 EL NINO-SOUTHERN OSCILLATION53 J . DERA (Edi tor)MARINE PHYSICS54 K. TAK AN O (Edi tor)O C E A N O G R A P H Y O F A S IA N M A R G IN A L S E A S55 TAN WEIYANS H A L L O W WA TE R H Y D R O D Y N A M IC S56 R. CHARLIER and J . JUS TU SOCEAN ENERGIES, ENVIRO NMEN TAL, ECONO MICAND TECHNOLOG ICAL ASPECTS OF ALTERNATIVEPOWER SOURCES57 P.C. CHU and J .C. GASCA RD (E di tors)DEEP CONVECTION AN D DEEP WATERFORM ATION IN THE OCEANS58 P.A. PIRAZZOLIWORLD ATLAS OF HOLOCENE SEA-LEVELC H A N G E S59 T . TERAM OTO (Ed i to r )DEEP OCEAN CIRCULATION-PHYSICAL ANDCHEMICAL ASPECTS60 B. KJERFVE (Edi tor)COASTAL LAGOON PROCESSES61 P. MA LANO TTE-R IZZOLI (Edi tor)MODERN APPROACHES TO DATA ASSIMILATIONIN OCEAN MODELING62 H.W.A. BE HREN S, J .C. BORST, L .J. DROPPERT,and J .P. VAN DER MEULEN (Edi tors)O P E R A T IO N A L O C E A N O G R A P H Y

63 D. HALPERN (Edi tor)SATELL ITES, OC EANOGR APHY AND SOCIETY


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T h i s w o r k i s p r o t e c t e d u n d e r c o p y r i g h t b y E l s e v i e r S c i e n c e , a n d t h e f o l l o w i n g t e r m s a n d c o n d i t i o n s a p p l y t o it s u s e :P h o t o co p y i n gSingle pho tocop ies o f s ing le chap ters may be made fo r personal use as a l lowed by nat ional copyr igh t l aws . Permiss ion o f the Pub l i sher and payment o f a feei s requ i red fo r a l l o ther pho tocopy ing , includ ing mul t ip le o r sys temat ic copy ing , copy ing fo r adver t i s ing o r p romot ional purposes , resa le , and a l l fo rms o fdocum ent del ivery . Specia l ra t es are avai l ab le fo r educat ional ins t i tu t ions tha t wish to make pho tocop ies fo r non-pro f i t educat ional c l ass room use.Permiss ions may be sought d i rect ly f rom Elsev ier Science Global Righ t s Depar tment , PO Box 800 , Oxford OX5 1DX, UK; phone: (+44) 1865 843830 , fax :(+44) 1865 853333 , e-mai l : permiss ions@ elsev ier .co .uk . You m ay a l so contact Global Righ t s d i rect ly th rough Elsev ier ' s hom e page (h t tp : / /www.el sev ier .n l ) ,by select ing 'Ob ta in ing Permiss ions ' .In the USA , users may clear permiss ions and make paym ents th rough the Copyr igh t Clearance Center , Inc . , 222 Rosew ood Drive, Danvers , MA 01923 , USA ;phone: (978) 7508400 , fax : (978) 7504744 , and in the UK through the Copyr igh t L icens ing Agen cy Rap id Clearance Serv ice (CLAR CS) , 90 To t tenhamCourt Road , Lo ndon W IP 0LP, UK ; phone: (+44) 171 631 5555 ; fax : (+44) 171 631 5500 . Other count r i es may have a loca l rep rographic r igh t s agency fo rp a y m en t s .D e r i v a t i v e W o r k sTab les o f con ten t s may be rep roduced fo r in ternal c i rcu la t ion , bu t permiss ion o f El sev ier Science i s requ i red fo r ex ternal resa le o r d i s t r ibu t ion o f such m ater i a l .Perm iss ion o f the Pub l i sher i s requ i red fo r a l l o ther der iva t ive works , includ ing com pi la t ions and t rans la tions .Elect ron ic Sto rage o r UsagePerm iss ion o f the Pub l i sher i s requ i red to s to re o r use e lect ron ica l ly any m ater i a l con ta ined in th is work , includ ing any ch ap ter o r par t o f a chap ter .Excep t as ou t l ined above, no par t o f th i s work may be rep roduced , s to red in a re t r i eval sys tem or t ransmi t t ed in any fo rm or by any means , e l ect ron ic ,mec hanica l , pho toco py ing , record ing o r o therwise , wi thou t p r io r wri t t en permiss ion o f the Pub l isher .Address permiss ions reques t s to : E l sev ier Global Righ t s Depar tmen t , a t the mai l , fax and e-m ai l addresses no ted above.N o t i ceNo respons ib i l i ty is assumed by the Pub l i sher fo r any in ju ry and /o r damage to persons o r p roper ty as a mat ter o f p roduct s l i ab i l i ty , neg l igence o r o therwise , o rf rom any use o r opera t ion o f an y methods , p roduct s , ins t ruct ions o r ideas con ta ined in the mater i a l herein . Because o f rap id advances in the me dica l sc iences ,in par t i cu lar , indepen dent ver i f i ca t ion o f d iagnoses and drug dosages shou ld be made.

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V I I

P R E F A C E

T h e b o o k a i m s to c o v e r i n a u n i t a r y w a y b o t h t h e d e t e r m i n i s t i c a n d s t a t is t ic a l to p i c so f t h e m e c h a n i c s o f s e a w a v e s . F u r t h e r m o r e , i t a i m s to h i g h l ig h t s o m e r e c e n t p r o g r e s so n t h e d y n a m i c s o f r a n d o m w i n d - g e n e r a t e d w a v e s a n d o n l o n g - t e r m w a v e s t a t i s t i c s .F in a l l y , i t a im s t o g iv e a f r e s h ap p r o ach t o t r ad i t i o n a l co n cep t s . I n t h i s r eg a r d , s o m eo r i g in a l p r o o f s a r e g i v e n ( s e e t h e c o n c l u si v e n o t e s o f e a c h c h a p t e r ) , n e w e v i d e n c ef r o m s m a l l s ca l e f i e ld ex p e r im en t s i s u s ed t o i n t r o d u ce c r u c i a l t o p i c s l i k e w a v e f o r c e s ,a n d s o m e w i d e l y w o r k e d e x a m p l e s a r e g i v e n ( e . g . a n e x a m p l e o f 1 7 p a g e s f o r t h ec a l c u l a t io n o f t h e l a r g e s t w a v e l o a d s o n a s u b m e r g e d t u n n e l d u r i n g i ts l if e t im e ) .

T h e t e x t is i n t e n d e d f o r r e s e a r c h e r s a n d g r a d u a t e s t u d e n t s , b u t t h e s t y le is s u c h t h a tm o s t o f t h e b o o k i s s u i t a b l e f o r u n d e r g r a d u a t e s t u d e n t s . T h i s is b e c a u s e t h e v a r i o u sf o r m u l a e a r e p r o v e d f r o m t h e f u n d a m e n t a l e q u a t i o n s , a n d t h e h a r d e r c o n c e p t s a r ee x p l a i n e d w i t h b o t h e x a m p l e s a n d s o m e t i m e s a l s o w i t h s h o r t s to r ie s . S t ri c tl y s p e a k i n g ,i t is a s s u m e d t h a t t h e r e a d e r h a s k n o w l e d g e o f c a l c u lu s a n d b a s i c m e c h a n i c s ( e .g .v o l u m e s 1 a n d 2 o f C a l c u l u s b y T . M . A p o s t o l , a n d P h y s i c s I b y R . R e s n i c k , D .H a l l id a y , a n d K . S . K r a n e ) . T h e f u n d a m e n t a l s o f st r e n g t h o f m a t e r i a l s a r e n e e d e d f o r ap a r t o f c h a p t e r 1 2.C h a p t e r s 1 a n d 2 c o v e r t h e d y n a m i c s o f t h e p e r i o d i c w a v e s . C h a p t e r I s t a r ts f r o mt h e d i f f e r e n t i a l e q u a t i o n s o f m o t i o n t o o b t a i n t h e v e l o c i t y p o t e n t i a l . C h a p t e r 2 a p p l i e st h e l i n e a r m o m e n t u m e q u a t i o n a n d t h e e n e r g y e q u a t i o n t o c o n t r o l v o l u m e s w i t hs u r f a c e w a v e s . A s a c o n s e q u e n c e , i n c h a p t e r I y o u w i ll f i n d s o lu t i o n s f o r th e v e l o c i t yp o t e n t i a l o f p r o g r e s s i v e w a v e s ( w i t h a n d w i t h o u t a c u r r e n t ) , a n d o f w a v e s i n t e r a c t i n gwi th v e r t i ca l wa l l s . W h i l e , i n ch ap t e r 2 y o u w i l l f i n d t h e s o lu t i o n s t o p r o b l em s l i k et h o s e o f s h o a l in g , r e f r a c t i o n , s e t - d o w n , a n d g r o u p c e l e ri ty .

C h a p t e r 3 c o v e r s t h e b e a c h p r o c e s s e s . H o w l a r g e i s t h e r u n - u p ? H o w i s t h ep l a n f o r m e v o l u t io n o f a n o u r i s h m e n t p r o j e c t? H o w is t h e d e f o r m a t i o n o f a b e a c h a f t ert h e c o n s t r u c ti o n o f a d e t a c h e d b r e a k w a t e r o r a g r o in ? T h e s e p r o b l e m s a r e d e a l t w i tha n a l y ti c a ll y . T h e s t a r t in g p o i n t i s t h e f o r m a l s o l u t i o n f o r t h e b e a c h p l a n f o r m e v o l u t i o nu n d e r t h e a s s u m p t i o n o f s m a ll c u r v a t u r e o f t h e c o n t o u r l in e s.

C h a p t e r s 4 a n d 5 c o v e r t h e s h o r t - t e r m w a v e s t a t i s t i c s . C h a p t e r 4 i n t r o d u c e s t h eb a s i c c o n c e p t s ( s e a s t a t e , a u t o c o v a r i a n c e o f t h e s u r f a c e d i s p l a c e m e n t , f r e q u e n c y


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VIII Preface

s p e c t r u m , a n d s o o n ) . I t a l s o e x p l a i n s h o w t o o b t a i n t h e a u t o c o v a r i a n c e a n d t h ef r e q u e n c y s p e c t r u m f r o m t h e t i m e s e r ie s d a t a, a n d h o w t o in f e r t h e n a t u r e o f a s eas t a t e f r o m t h e s e f u n c t i o n s . C h a p t e r 5 g i v e s f o r m a l s o l u t i o n s a n d e x p e r i m e n t a le v i d e n c e f o r t h e s t a t is t ic s o f w a v e h e i g h t a n d w a v e p e r i o d i n a s e a s t at e .

C h a p t e r s 6 a n d 7 c o v e r t h e l o n g - t e r m w a v e s t a t i s t i c s . C h a p t e r 6 e x p l a i n s h o w t oo b t a i n t h e p r o b a b i l i t y o f e x c e e d a n c e o f th e s i g n i f ic a n t w a v e h e i g h t a t a g i v e n l o c a ti o n .T h e n , i t i n t r o d u c e s t h e c o n c e p t o f tr i a n g u l a r e q u i v a l e n t s t o rm : f o r e a c h a c t u a l s e as t o r m , a t r i a n g u l a r s t o r m e x i s t s w i t h t h e s a m e p r o b a b i l i t y t h a t t h e l a r g e s t w a v e h e i g h te x c e e d s a n y f i x e d t h r e s h o l d . C h a p t e r 7 a p p l i e s t h i s c o n c e p t t o g i v e t h e f o r m a ls o l u t i o n s f o r r e t u r n p e r i o d s o f se a s t o r m s w i t h s o m e g i v e n c h a r a c t e r i s ti c s . T h i s c h a p t e ra l s o d i s c u s s e s i n d e p t h t h e r e l a t i o n s h i p b e t w e e n t h e r e t u r n p e r i o d , t h e l i f e t i m e , a n dt h e p r o b a b i l i ty o f o c c u r r e n c e .

C h a p t e r s 8 , 9 , a n d 1 0 c o v e r th e d y n a m i c s o f t h e r a n d o m w i n d - g e n e r a t e d w a v e s .C h a p t e r 8 g i v e s t h e v e l o c i t y p o t e n t i a l o f a s e a s t a t e t o t h e f i r s t a p p r o x i m a t i o n i n aS t o k e s ' e x p a n s i o n . A l s o , th e c a s e s o f w i n d w a v e s i n t e r a c t i n g w i t h v e r t i c a l w a l l s a r ed e a l t w i t h in d e t ai l. T h i s c h a p t e r p o i n t s o u t s o m e v e r y b ig d i f f e re n c e s b e t w e e n w i n d -g e n e r a t e d w a v e s a n d p e r i o d i c w a v e s , i n w h a t c o n c e r n s r e f l e c t i o n a n d d i f f r a c t i o n .T h e n , c h a p t e r 9 d e v e l o p s t h e r e c e n t t h e o r y o f q u a s i - d e t e r m i n i s m o f t h e h i g h e s t w a v e si n a s e a s t a t e : i f a w a v e wi t h a g i ve n he i gh t H o c c u r s a t a f i x e d p o i n t Xo , a nd H i s ve ryl a r g e w i t h r e s p e c t t o t h e m e a n w a v e h e i g h t a t t h i s p o i n t , w e c a n e x p e c t t h e w a t e rs u r f a c e a n d v e l o c i t y p o t e n t i a l n e a r Xo t o b e v e r y c l o s e t o s o m e w e l l p r e c i s ed e t e r m i n i s t i c f o r m s .C h a p t e r 1 0 d e a l s w i t h t h e m o r e e x c it in g to p ic . T h e h i g h e s t w a v e s in a r a n d o m s e as t a te b e l o n g t o q u a s i - d e te r m i n i s t ic ( t h r e e - d i m e n s i o n a l ) w a v e g r o u p s . T h e w a v e g r o u pi s l i k e a f a m i l y a nd e a c h i nd i v i du a l wa ve is a m e m b e r o f t h i s f a m i l y , w i t h a li fe c y c le .T h e c h a r a c t e r i s t i c s o f t h e s e w a v e g r o u p s a r e d e e p l y i n v e s t i g a t e d i n v i e w o f t h ea p p l i c a t i o n s ( w a v e l o a d s o n s tr u c t u r e s ) . A n d , t h r o u g h t h i s i n v e s t i g a t i o n , w e a r r i v e a t an e w i n s i g h t i n t o r a n d o m w i n d - g e n e r a t e d w a v e s : t h e s e w a v e s a r e h i g h e r o n t h e t i m ed o m a i n t h a n o n t h e s p a c e d o m a i n a n d t h e y p o s s e ss a so r t o f g e n e t i c c o d e .C h a p t e r s 1 1 a n d 1 2 c o v e r t h e w a v e l o a d s o n o f f s h o r e s t r u c t u r e s . C h a p t e r 1 1 f i r s t l yd e a l s w i t h t h e s o - c a ll e d l a r g e b o d i e s o n w h i c h w a v e s e x e r t o n l y th e i n e r t i a f o r c e . T h e ni t c o n s i d e r s t h e s m a l l b o d i e s o n w h i c h w a v e s a l s o e x e r t a s i g n i f i c a n t d r a g f o r c e . T h ep r o b l e m o f a l o n g s t r u c t u r e ( w h a t i s t h e m a x i m u m e x p e c t e d w a v e h e i g h t o n a l o n gs t r u c t u r e l i k e a f l o a t i n g t u n n e l ? ) c o m p l e t e s t h i s c h a p t e r . T h e n c h a p t e r 1 2 i s e n t i r e l yd e v o t e d t o w o r k e d e x a m p l e s o f w a v e f o rc e e st im a t e s .

C h a p t e r 1 3 c o v e r s t h e w a v e l o a d s o n c o a s t a l s t r u c t u r e s : f i r s t , w a v e f o r c e s o nv e r t ic a l b r e a k w a t e r s ; t h e n , st a bi li ty o f t h e r u b b l e m o u n d b r e a k w a t e r s u n d e r t h e a c t i o no f s e a s t a t e s o f g i v e n c h a r a c t e r i st i c s .

C h a p t e r 1 4 d e a l s w i t h t o p i c s w h i c h c a l l f o r a n o v e r a l l o v e r v i e w o f o f f s h o r e a n dc o a s t a l e n g i n e e r i n g . T h e f i r st t o p i c is c o m p a r i s o n b e t w e e n t h e e f f e ct s o f t s u n a m i s a n do f w i n d w a v e s , f r o m t h e o p e n s e a t o th e c o a s t. T h e s e c o n d t o p i c is s m a l l s c a le m o d e l s ,a n d t h e t h i r d t o p i c i s w a v e m e a s u r e m e n t s f o r t h e v a r i o u s n e e d s .

T h e s e s u b j e c ts , w h i c h I a p p r o a c h e d i n a n o r g a n i c w a y f o r t h e f ir s t t i m e i n m y I t a l i a nb o o k I d r a u l i c a M a r i t t i m a ( U T E T , 1 9 9 7) , s e r v e as t h r e e c o u r s e s t o g r a d u a t e s t u d e n t s


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Preface IX

i n c iv il e n g i n e e r i n g o r o c e a n e n g i n e e r i n g , c o u r s e s t h a t m y a s s i st a n t s a n d I g iv e in t h eU n i v e r s i t y o f R e g g i o C a l a b r i a .

F i r s t c o u r s e ( w a v e t h e o r y a n d c o a s t a l s t r u c t u r e s )a . 1) dy na m i c s o f t he p e r i o d i c wa ve s : c h a p t e r s 1 a nd 2 , e x c e p t s e c t s. 2 .5 .4 a nd 2 . 6 .3( s e t - d o w n ) ;a . 2 ) s ho r t - t e rm wa ve s t a t i s t i c s : c ha p t e r s 4 a nd 5 ;

a . 3 ) l o n g - t e r m w a v e s t a t i s t i c s : c h a p t e r 6 e x c e p t s e c t . 6 . 3 ( d i r e c t i o n a l w a v ep re d i c t i o n ) , c ha p t e r 7 t i l l s e c t . 7 . 5 ( de s i gn s e a s t a t e f o r c o a s t a l s t r u c t u re s ) , a ndA p p e n d i x A ;a . 4 ) s t a b i l it y o f c o a s t a l s t r u c t u re s : c ha p t e r 1 3;

a . 5 ) s m a l l s c a l e m o de l s a nd wa ve m e a s u re m e n t s : s e c t s . 1 4 . 2 . 1 , 1 4 . 2 . 2 a nd 1 4 . 3 .S e c o n d c o u r s e ( a d v a n c e d w a v e t h e o r y a n d o f f s h o r e s t r u c t u r e s )b . 1 ) de s i gn wa ve f o r o f f s ho re s t ru c t u re s : f r o m s e c t . 7 . 6 t o s e c t . 7 . 8 ;b . 2) d y n a m i c s o f t h e r a n d o m w i n d - g e n e r a t e d w a v e s : c h a p t e r s 8 , 9 , a n d 1 0 , e x c e p t

t he f o rm a l p ro o f f ro m s e c t . 9 . 6 t o s e c t . 9 . 1 0 ;b . 3 ) w a v e l o a d s o n o f f s h o r e s t r u c t u r e s : c h a p t e r s 1 1 a n d 1 2 ;b .4 ) d i scuss ion on the sma l l s ca le model s : f rom sec t . 14 .2 .3 to sec t . 14 .2 .5 .T h i r d c o u r s e ( b e a c h p r o c e s s e s )c .1 ) va r i a t i o n o f t he m e a n w a t e r l e ve l f ro m o f f s ho re t o c o a s t : s e ct s . 2 .5 . 4 a nd 2 .6 .3 ;c . 2 ) wa ve a c t i o ns o n c o a s t s : c ha p t e r 3 ;c .3 ) p r e d i c t i o n o f t he w a ve he i g h t f o r g i ve n wa ve d i r e c t i o n : s e c t s. 6 .3 a nd 7 . 9;c .4 ) c o m p a r i s o n b e t w e e n e f f ec t s o f t s u n a m i s a n d e f f ec t s o f w i n d w a v e s : s e ct . 1 4.1 .T h e f i r s t c o u r s e i s i n t r o d u c t o r y t o t h e o t h e r s , s o t h a t a s t u d e n t c a n a t t e n d o n l y

c o u r s e a ) , o r a ) a nd b ) , o r a ) a nd c ) , o r a ) , b ) , a nd c ) .T h e s e c t i o n s f r o m 9 . 6 t o 9 . 1 0 a n d A p p e n d i x B g o b e y o n d t h e t h r e e c o u r s e s a n d a r e

i n t e n d e d f o r r e s e a r c h e r s o n l y. I t is a ls o a d v i s a b l e t o l i m it a f e w o f t h e m o r e a n a l y t ic a lp a r t s . S p e c i f ic a l ly , fo r se c t . 2 .1 0 ( s ho a l i ng a n d s e t - d o w n o f wa ve s o n c u r r e n t ) o n ec o u l d g i ve t he i n t ro du c t i o n ( s e c t . 2 . 1 0 . 1 ) a nd c o nc l u s i o n ( s e c t . 2 . 1 0 . 9 ) . Of c o u r s e t he rei s a l s o t he p o s s i b i l i t y t o c u t o f f a f e w p a r t s , s o a s t o s h o r t e n t he c o u r s e s , i f ne c e s s a ry . I np a r t i c u l a r , c o u r s e a ) c o u l d o m i t th e w a v e - c u r r e n t i n t e r a c t i o n ( se c ts . 1 .9 a n d 2 . 1 0 ), a n d /o r t he wa y t o o b t a i n t he c o n t i nu o u s s p e c t ru m ( f ro m s e c t . 4 . 4 . 3 t o s e c t . 4 . 4 . 6 ) . F i na l l y ,t h e r e a r e a f e w i n t r o d u c t o r y s e c t i o n s w h i c h h a v e b e e n g i v e n f o r t h e s a k e o fc o m p l e t e n e s s , a n d w h i c h c a n b e o m i t t e d i f t h e r e l e v a n t s u b j e c ts a r e t r e a t e d i n o t h e rc o u r s e s . T h e s e a r e : se c ts . 1 .1 - 2 a nd 2 .1 - 2 wh i c h g i ve t he c o nc e p t s o f fl u i d m e c h a n i c sn e e d e d t o d e v e l o p t h e w a v e t h e o r y , a n d s e c t s . 5 . 1 - 2 w h i c h i n t r o d u c e t h e r a n d o mG a u s s i a n p r o c e s s e s .

P a o l o B o c c o t t iR e g g i o C a l a b r i a , I t a l yM a y 1 9 9 9


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12/520

XI

A C K N O W L E D G E M E N T SI a m v e r y g r a t e f u l t o p r o f e s s o r s E n r i c o M a r c h i a n d G i u l i o S c a r s i w h o , n e a r l y

t w e n t y - f i v e y e a r s a g o , i n t r o d u c e d m e t o r e s e a r c h o n s e a w a v e s .T h e n ! w i s h t o e x p r e s s m y g r a t it u d e t o a s s i s t a n t p r o f e s s o r s F e l i c e A r e n a , G i u s e p p eB a r b a r o , P a s q u a l e F i l ia n o t i , a n d t o e n g i n e e r F r a n c i s C i r i a n n i fo r t h e i r r e a d i n g o f t h em a n u s c r i p t a n d s o m e u s e f u l c o m m e n t s . F . A r e n a a l s o d e s e r v e s a n a c k n o w l e d g e m e n tf o r h is h e lp w i t h t h e a n a l y s is o f N D B C b u o y d a t a a n d s a t e l l it e d a t a . T h e g r a p h i c s a r ed u e t o e n g i n e e r V i n c e n z o F i a m m a w h o d r e w t h e f i g u r e s f r o m m y f i l e d a t a a n d m ys k e t c h e s . T h e y a l l w e r e m y s t u d e n t s , a n d I a m p r o u d o f e a c h o f t h e m .

F i n a ll y , I v e r y g r a t e f u l ly a c k n o w l e d g e t h e g r e a t c o u r t e s y a n d f r u i tf u l c o - o p e r a t i o no f U T E T o f T u r i n p u b l i s h e r o f m y I t a li a n b o o k Idraulica Marittima (1 9 9 7 ) wh ich wasth e b a s i s o f t h i s n ew b o o k .P a r t o f th i s w o r k w a s d o n e w i t h t h e f i n a n c ia l s u p p o r t o f t h e I t a l i a n N a t i o n a l G r o u pf o r P r e v e n t i o n f r o m H y d r o g e o l o g i c a l D i s a s t e r s ( u n d e r c o n t r a c t C N R - G N D C I N o98.00586).

P . B .


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XII



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XIII

NOTATIONS o m e s y m b o l s u s e d i n o n l y o n e s e c t i o n a r e n o t i n c l u d e d i n t h e f o l l o w i n g l i s t .

aaa - ( a x , a y, a z)A

Cc~C F RC~, Ci~Cl, C2, C3, C4

dDDD2eeEZfE Xff - - ( fx,L,f~)FFFo, F~V -( F x , Fv, F~).S, J / / /

gGG

w a v e a m p l i t u d et r i ang l e he i gh tpa r t i c l e a cce l e r a t i ona r e aw i d t ht h r e s h o l d o f t h e s u r fa c e d i s p l a c e m e n td u r a t i o n o f t h e e q u i v a l e n t t r i a n g u l a r s t o r mt hresho l d o r spec i a l va l ue o f t he wave c r e s t e l eva t i onb e r m h e i g h tp r o p a g a t i o n s p e e ddi f f rac t ion coef f ic ientF r e s n e l i n t e g r a l w i t h i n t e g r a n d c o s i n edrag coe f f i c i en t , i ne r t i a coe f f i c i en tsa f e t y f ac t o r sw a v e c r e s t e l e v a t i o ns t i l l wa t e r dep t hd i a m e t e rd i r ec t i ona l sp r ead i ng f unc t i onm e a n p e r s is t en c edu ra t i on o f a s t o rm o r o f t he des i gn sea s t a t ep o t e n t i a l + k i n e t i c e n e r g y p e r u n i t m a s seccen t r i c i t yf r e q u e n c y s p e c t r u mm e a n w a v e e n e r g y p e r u n i t s u r f a c en o n d i m e n s i o n a l f r e q u e n c y s p e c t r u me x p e c t e d n u m b e r p e r u n i t t i m eg e n e r a l f u n c t i o nf o rce pe r un i t l eng t hg e n e r a l f u n c t i o nh o r i z o n t a l f o r c e p e r u n i t e x t e n s i o n o f u p r i g h t b r e a k w a t e rho r i zon t a l f o r ce , ve r t i ca l f o r ceforced i s t r i b u t i o n f u n c t i o n o f t h e f r e q u e n c y s p e c t r u ms p e e d d r o p f a c t o racce l e r a t i on o f g r av i t yl ongsho re d i f f u s i v i t yg e n e r a l f u n c t i o n


15/520

X I V Notation

hH/4 ,i x , i y , i zgkK~1LLmm ym oMMl'ln , N , ~ 4 ~n - - ( n x , n y , n z )

PPPPp*

QrRRR eR o , R vR vR - (ICx, R~ )SSSSSD, SuSF RJ

t h r e s h o l d o r s p e c i a l v a l u e o f t h e s i g n i f i c a n t w a v e h e i g h tw a v e h e i g h ts i g n i f i c a n t w a v e h e i g h tu n i t v e c t o r sI r r i b a r r e n n u m b e rw a v e n u m b e rn o n d i m e n s io n a l w a v e n u m b e rK e u l e g a n - C a r p e n t e r n u m b e rl e n g t h o f a s t r u c t u r ew a v e l e n g t hl i f e t i m e o f a s t r u c t u r em a s sj t h o r d e r m o m e n t o f th e f r e q u e n c y s p e c t r u mv a r i a n c e o f t h e s u r f a c e d i s p l a c e m e n t o f a s e a s t a tem o m e n t o f a f o r c e o r o f a f o r c e p e r u n i t l e n g t hd e t e r m i n a n t o f a c o v a r i a n c e m a t r i xp a r a m e t e r o f t h e s p r e a d i n g d i r e c t i o n f u n c ti o nn u m b e r b e i n g s p e c i f i e d e v e r y t i m eu n i t n o r m a l v e c t o r t o a s u r f a c ep r e s s u r ep r o b a b i l i t y d e n s i t y f u n c t i o np r o b a b i l i t y o f e x c e e d a n c ew e i g h tw e i g h t i n s t i l l w a t e rp r o b a b i l i t yf l o w r a t e p e r u n i t l e n g t hp o l a r c o o r d i n a t e ( t h e r - c u r v e s a r e s t r a i g h t l i n e s t h r o u g h t h e o r i g i n )r a d i u sr e t u r n p e r i o dR e y n o l d s n u m b e rr e a c t i o n s o f t h e f o u n d a t i o nr u n - u pr a d i a t i o n s t r e s s t e n s o rl o c a l p r o p a g a t i o n a x i sd i r e c t i o n a l w a v e s p e c t r u mv e l o c it y h e a dt o t a l u p li ft p r e s s u r e p e r u n i t le n g t h o f b r e a k w a t e rs e t - d o w n , s e t - u pF r e s n e l i n t e g r a l w i t h i n t e g r a n d s i n en o n d i m e n s i o n a l d i r e c t i o n a l s p e c t r u m


16/520

Not a t i on XV

ttoTTT *T,J

HHUu - ( G , u , , u z )~ - ( ~ , ~ , ~ )VwWWw~, w~WXXX o - ( X o , y o )XX - (X , Y )YYYY

OLOLA

AA p

t i m espec i a l t i me i ns t an tw a v e p e r i o dt i me l aga b s c i s s a o f t h e a b s o l u t e m i n i m u m o f t h e a u t o c o v a r i a n c e f u n c t i o ni n t e r a r r i v a l t i m e o f a r a n d o m p o i n t p r o c e s sv e r y l a r g e t i m e i n t e r v a lg r e a t e s t l a g u s e d f o r o b t a i n i n g t h e s p e c t r u m f r o m t h e a u t o c o v a r i a n c ed u m m y v a ri ab l ec u r r e n t v e l o c it y , w i n d s p e e dp a r a m e t e r o f t h e p r o b a b i l i t y o f H ,p a r t i c l e d i s p l a c e m e n tpa r t i c l e ve l oc i t yr a n d o m v a r i a b l ed u m m y v a ri ab l en o n d i m e n s i o n a l f r e q u e n c yp a r a m e t e r o f t h e p r o b a b i li ty o f H ,p a r a m e t e r s o f t h e p r o b a b i l i t y o f H , f o r a g i v e n w a v e d i r e c t i o nv o l u m ed u m m y v a ri ab l eh o r i z o n t a l c o o r d i n a t e a x i sf i xed po i n t o f t he ho r i zon t a l p l aneanc i l l a ry va r i ab l e r e l a t ed t o H,vec t o r whose i n i t i a l po i n t i s Xoh o r i z o n t a l c o o r d i n a t e a x i sc o m p o n e n t o f v e c t o r Xa n c i l l a r y v a r i a b l e r e l a t e d t o t h e p r o b a b i l i t y o f e x c e e d a n c e o f H ,fe tchv e r t i c a l c o o r d i n a t e a x i s w i t h t h e o r i g i n a t t h e m e a n w a t e r l e v e la n g l e b e t w e e n x - a x i s a n d d i r e c t i o n o f w a v e a d v a n c equo t i en t be t w een wave he i gh t and r . m. s , su r f ace d i sp l acem ent o f a sea s t a t eq u o t i e n t b e t w e e n w a v e h e i g h t a n d H sP h i l l i p s ' s p a r a m e t e rpo l a r coo rd i na t e ( t he f l - cu rves a r e c i r c l e s cen t r ed a t t he o r i g i n )q u o t i e n t b e t w e e n w a v e c r e s t e l e v a t i o n a n d r . m . s , s u r f a c e d i s p l a c e m e n tof a sea s ta tespec if ic w e i gh tde l t a f unc t i onv a r i a t i o n o f t h e m e a n w a t e r l e v e l d u e t o t h e w a v e m o t i o na c t u a l p r e s s u r e + p g z


17/520

XVI Notation

T]ph

q~~b = ( e~ , e , )

~u

phase anglevertical coordinate axis with the origin at the seabedsurface displacementfluctuating pressure head at points which remain always beneath thewater surfaceangle between the y-axis and the direction of wave advanceazimuth of the direction of wave advanceangle between the y-axis and the dominant direction of the spectrumslope anglescale factorcoefficient of frictionkinematic viscosity ( 1 0 . 6 m 2 s - 1 in calculations)distance offshoreratio between wave crest elevation and wave heightwater density (1030 kg/m 3 in calculations)r.m.s, surface displacement of a sea statetime lag between crest and troughthreshold of interarrival timevelocity potentialcovariance of the surface displacement and the velocity potentialnorm of q~mean energy fluxshape parameters of the JONSWAP spectrumautocovariance functionnarrow bandedness parametercovariance of surface displacementsangular frequency


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X V I I

0aAbBCCc lC FdefFF RGh

d e e p w a t e r 1w a t e r ( a q u a )t r i a n g l e h e i g h tb r e a k i n gt r i a n g l e b a s ec u r r e n tc e n t r ec l o s u r ec r e s td i f f r a c t i o ne n t e r i n gs e a b e d ( f u n d u s )F o u r i e rF r e s n e lg r o u ph i g h w a v e s

S U B S C R I P T Sinr nm ono

O b lp

p hp rRS

S Os t

S . W .VW

i n e r t i aa b sc i s sa m a x i m u m / m i n i m u mm o d e ln o u r i s h e d b e a c h , n a t u r a l , n o m i n a lh o r i z o n t a l 2o u t g o i n gp e a k 3p r e s s u r e h e a dp r o t o t y p er e a c t i o ns o l i d 4s o i ls t i l l w a t e rs a m e w a v ev e r t i c a lw a v e

1 Except ions : m0 stands for zero th orde r m om ent o f the f requenc y spectrum ; in sects. 9 .6-10 and inA pp en dix B, r/0, ~b0 stan d for r/(0), ~ (0).2 Except ion: xo, Yo, to where the subscr ip t o denotes a specia l value of x , y or t .3 Except ion: hp in sect . 7.10 where p deno tes a g iven proba bi l i ty .4 Except ions : Hs where s s tands for "s ignificant" ; v , in chap. 2 where s de notes the local propa gat ion axis .

S Y M B O L S< f ( t) > = t e m p o r a l m e a n

V = m e a n v a l u e o f t h e r a n d o m v a r i ab l e Vf = d e r i v a t i v e

f ! f ir st o r d e r i n th e S t o k e s e x p a n s i o n , o r o t h e r m e a n i n g s b e i n g s p e c if i ede v e r y t i m e ; t h e a p e x i s a l so u s e d t o d i s t i n g u is h b e t w e e n t h e d u m m yv a r i a b l e a n d t h e i n d e p e n d e n t v a r i a b l e o f a n in d e f i n i te i n t e g r a l

f l! = s e c o n d o r d e r i n t h e S t o k e s e x p a n s i o nG = q u a n t i t y r e l e v a n t t o t h e e q u i v a l e n t w a t e r v o l u m e


19/520

XVI I I

A B B R E V I A T I O N Se.t.s.1.h.s.M W Lp.d.f .RC 1990 , RC 1991 , RC 1992 , RC 1993 ,R C 1 9 9 4r . m . s .r.h.s.

e q u i v a l e n t t r i a n g u l a r s t o r ml e f t - hand s i dem e a n w a t e r l e v e lp robab i l i t y dens i t y f unc t i one x p e r i m e n t s i n t h e n a t u r a l l a b o r a t o r y o f R e g g i oC a l a b r i ar o o t m e a n s q u a r er i gh t - hand s i de


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XIX

157913161921232829343838

394246485458616570748586

L I S T OF C O N T E N T S~~~ ~ _ _ _ ~

Chapter 1Periodic w ave pattern: the app roach of di fferen tial calculus1.1 The irrotational flow, the continuity equation, the Bernoulli equation1.2 The differential equations of an irrotational flow with a free surface3.3 Introduction to wave mechanics1.4 Stokes theory to the first order1 S Analysis of the linear dispersion rule1.6 The flow field1.7 Stokes theory to the second order1.8 Non-linearity effects1.9 Wave-current interaction. Part I: velocity potential and wavelength1.10 Preliminary remarks on three dimensional waves1.11 Wave reflection1.12 Wave diffraction

Conclusive noteReferences

Chapter 2Periodic w ave pattern: the control volume app roach

2.12.22.32.42.52.62.72.82.9 The group celerity2.10 Wave-current interaction. Part 11: shoaling and set-down

The linear momentum equation for a control volumeThe energy equation for a control volumeRadiation stress, mean energy flux, mean wave energy per unit surfaceFormulae for radiation stress and mean energy flux of progressive wavesThe problem of the control volume extending from deep to shallow waterPractical consequences of the control volume problemA current associated with the wave motionWave refraction for an arbitrary configuration of the seabed

Conclusive noteReferences


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XX List of contents

8788929495100103104107111113116118

Chapter 3W a v e e f fe c t s on coa s t s

3.13.23.33.43.53.63.73.83.9

The control volume from the breaker line to the beachThe run-upThe longshore transportThe analytical approach to the problem of beach planform evolutionProblem of beach planform evolution: the case of contour lines parallel up todeep waterProblem of beach planform evolution: the case of contour lines parallel onlywithin a certain distance from the shorelinePlanform evolution of a natural shorelineStability of a nourished beachPlanform evolution of beach nourishment projects3.10 A useful simplificat ion3.11 Beach planform evolution caused by structuresC o n c l u s i v e n o t eR e f e r e n c e s

119121123126136140145150151

Chapter 4W ind ge ne ra t e d w a v e s : ba s i c conce p t s

4.14.24.34.44.54.64.7

The sea stateThe theory of the sea statesSome basic relations in the theory of the sea statesHow to obtain the input data of the theoryA mathematical form of the wind wave spectrumPossibility of testing small scale models in sea or lakesInferring the nature of waves from the bandwidthC o n c l u s i v e n o t eR e f e r e n c e s

153157159161162166167172

Chapter 5Ana lys i s o f t he sea s ta t es : t he t im e do m a in

5.15.25.35.45.55.65.75.8

Why the surface displacement represents a stationary Gaussian processJoint probability of surface displacementsRice's problemRice's logicCorollaries of Rice's problemSolved and still unsolved problemsThe period of a very high wave and the wave height probability under generalbandwidth assumptionsExperimental verification


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Lis t o f c o n t e n t s X X I

175177180181

5 .9 C h a r a c t e r i s t i c w a v e h e i g h t s5 .1 0 T h e m a x i m u m e x p e c t e d w a v e h e i g h t in a s e a st a t e o f g i v e n d u r a t i o n

C o n c l u s i v e n o t eR e f e r e n c e s

1831861911961992 0 02 0 42 0 62 06

Chapter 6T h e w a v e c lim a t e6.16 .26 .36 .46 .56 .66.7

T h e f u n c t i o n H , ( t)T h e p r o b a b i l i t y o f th e s i g n i f i c an t w a v e h e i g h tT h e p r o b a b i l i t y o f t h e s i g n i f ic a n t w a v e h e i g h t f o r a g i v e n d i r e c t i o n o f w a v e a d v a n c eP r o b a b i l i t i e s o f t h e s i g n i f i c a n t w a v e h e i g h t f o r a fe w a r e a s o f th e g l o b eT h e m a x i m u m e x p e c t e d w a v e h e i g h t i n a s t o r m w i t h a g i v e n h i s t o r yT h e c o n c e p t o f " e q u i v a l e n t t r i a n g u l a r s t o r m "S t o r m d u r a t i o n sC o n c l u s i v e n o t eR e f e r e n c e s

2 0 72112 132212 232 2 62 2 8

23 52 3 92 4 22 4 62 4 7

Chapter 7Des ig n w aves an d r is k an a lys i s7 .1 T h e r e t u r n p e r i o d o f a s e a s t o r m w h e r e t h e s i g n i fi c a n t w a v e h e i g h t e x c e e d s af i x e d t h r e s h o l d7 .2 T h e s i g n if i c a n t w a v e h e i g h t a n d it s p e r s i s t e n c e v s t h e r e t u r n p e r i o d7 .3 T h e e n c o u n t e r p r o b a b i l i ty7 .4 T h e c h a i n : l if e t i m e , e n c o u n t e r p r o b a b i l i t y ~ r e t u r n p e r i o d ~ s i g n i fi c a n t

w a v e h e i g h t7 .5 C o a s t a l s t r u c t u r e s : t h e d e s i g n s e a s t a t e7 .6 T h e r e t u r n p e r i o d o f a w a v e w i t h a h e i g h t e x c e e d i n g a f i x e d t h r e s h o l d7 .7 T h e r e t u r n p e r i o d o f a s e a s t o r m c o n t a i n i n g at l e a s t o n e w a v e h i g h e r th a n a

f i x e d t h r e s h o l d7 .8 O f f s h o r e s t r u c t u r e s : t h e d e s i g n w a v e7 .9 C a l c u l a t i o n s f o r d i f f e r e n t w a v e d i r e c t i o n s7 .1 0 C o r o l l a r y o f r i s k a n a ly s i s: a g e n e r a l r e l a t i o n b e t w e e n t h e c o n f i d e n c e i n t e r v a l

a n d t h e s a m p l i n g r a t eC o n c l u s i v e n o t eR e f e r e n c e s

2 4 9251

Chapter 8Ana lys is o f the sea s ta tes in the sp ace - t ime8 .1 T h e c o n c e p t o f h o m o g e n e o u s w a v e f ie ld

8 .2 T h e w a v e f i el d i n t h e o p e n s e a


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25 42 5 72632 672 7 02 7 42 762 792 7 9

8.38 .48.58.68.78 .88.9

T h e d i r e c t i o n a l s p e c t r u mS h o a l in g a n d r e f r a c ti o n o f th e w i n d - g e n e r a t e d w a v e sR e f l e c t i o n o f t h e w i n d - g e n e r a t e d w a v e sD i f f r a c t io n o f t h e w i n d - g e n e r a t e d w a v e sL o n g - c r e s t e d r a n d o m w a v e s : t h e l i n k b e t w e e n p e r i o d i c w a v e s a n d w i n d -g e n e r a t e d w a v e sD i r e c t p r o p o r t i o n b e t w e e n t h e m a x i m u m e x p e c t e d w a v e h e i g h t a n d t h ed i f f r a c t io n c o e f f i c i e n tS p a c e - t i m e c o v a r i a n c e sC o n c l u s i v e n o t eR e f e r e n c e s

28128 6288291

2932 9 42 963 0 03033 06

Chapter 9The th eo ry of quasi -determ in ism9.19 .29 .39.49.59.69.79.89.99.10

A s u f f ic i e n t c o n d i t i o n f o r o c c u r r e n c e o f a w a v e o f g iv e n h e i g h t v e r y l a r g eA n e c e s s a r y c o n d i t i o n f o r o c c u r r e n c e o f a w a v e o f g i v e n h e i g h t v e r y l a r g eT h e w a t e r s u r f a c e o n s p a c e - t i m e , i f a w a v e o f g iv e n h e i g h t v e r y l a r g e o c c u r s a t af i x e d p o i n tT h e v e l o c i t y p o t e n t i a l i f a w a v e o f g i v e n h e i g h t v e r y l a r g e o c c u r s a t a f i x e dp o i n tT h e o r y ' s g e n e r a l i t y a n d c o n s i s t e n c y w i t h S t o k e s ' t h e o r yF o r m a l p r o o f o f t h e n e c e s s a r y c o n d i ti o n . P a r t I : s y m b o l s a n d a s s u m p t i o n sF o r m a l p r o o f o f t h e n e c e s s a r y c o n d i t i o n . P a r t I I: c o r e o f t h e p r o o fF o r m a l p r o o f o f t h e n e c e s s a r y c o n d i t i o n . P a r t I II " t h e c e n t r a l i n e q u a l i t yF o r m a l p r o o f o f t h e n e c e s s a r y c o n d i t io n . P a r t I V : c o n c l u s i o nC o r o l l a r y : t h e c l o s e d s o l u t i o n f o r t h e w a v e h e i g h t d i s t r i b u t i o n

31131731 832 032 43263353 393 463 5 03 593 59

Chapter 10Uses and consequences of the qu asi -determ inism theo ry10.110.210.310.410.510.610.710.810.910 .10

T h e f i r s t w a y t o e m p l o y t h e t h e o r yA t h r e e d i m e n s i o n a l w a v e g r o u pT h e w a v e s a r e h i g h e r o n t h e t i m e d o m a i n t h a n o n t h e s p a c e d o m a i n !E f f e c t s o f w a t e r d e p t h a n d o f s p e c t r u m s h a p e o n t h e w a v e g r o u pS h o a l i ng a n d r e f r a c t i o n o f th e w a v e g r o u pE x p l a n a t i o n o f t h e f ir st b i g d i f f e re n c e b e t w e e n s e a w a v e s a n d p e r i o d i c w a v e sE x p l a n a t i o n o f t h e s e c o n d b i g d i f f e re n c e b e t w e e n s e a w a v e s a n d p e r i o d i cw a v e sT h e s e c o n d w a y t o e m p l o y t h e t h e o r yT h e " g e n e t i c c o d e " o f t h e s e a w a v e sT h e d e t e r m i n i s m a r i s e s f r o m w i t h i n t h e r a n d o m w a v e sC o n c l u s i v e n o t eR e f e r e n c e s


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L i s t o f c o n t e n t s XXI I I

3 613 6 43 6 73 7 33 8 33 8 73 9 23 9 2

Chapter 11Ana lys i s o f t he wave f o r ces on o f f shore s t r uc t u r es

1 1. 1 W a v e f o r c e s o n g r a v i t y o f f s h o r e p l a t f o r m s1 1 .2 L o c a l p e r t u r b a t i o n o f t h e fl o w f i el d a t a n o f f s h o r e s t r u c t u r e1 1.3 W a v e f o r c e s o n s u b m e r g e d t u n n e l s1 1 .4 T h e d i f f r a c t i o n c o e f f i c i e n t s o f t h e f o r c e s1 1 .5 W a v e fo r c e s o n s p a c e f r a m e s t r u c t u r e s1 1.6 T h e l o n g - s t r u c t u r e p r o b l e m


3 9 33 9 74 0 24 1 24 1 3

Chapter 12Ca lcu la t i on o f t he wave f o r ces on o f f shore s t r uc t u r es

1 2.1 C a l c u l a t i o n o f t h e w a v e f o r c e s o n a g r a v i t y o f f s h o r e p l a t f o r m1 2 .2 C a l c u l a t i o n o f t h e w a v e f o r c e s o n a s p a c e f r a m e s t r u c t u r e1 2 .3 D e s i g n o f a s u b m e r g e d t u n n e l . I : c a l c u l a t i o n o f t h e w a v e f o r c e s1 2 .4 D e s i g n o f a s u b m e r g e d t u n n e l . I I : t h e e f fe c t o f c u r r e n t s1 2 .5 D e s i g n o f a s u b m e r g e d t u n n e l . I I I: t h e ri s k o f r e s o n a n c e

4 1 94 2 74 3 04 3 64 3 84 4 14 4 54 4 5

Chapter 13Stability analysis of coastal structures1 3 .1 W a v e p r e s s u r e o n a w a l l1 3 .2 F o r c e s o n a v e r t ic a l b r e a k w a t e r1 3 .3 D e s i g n o f v e r ti c a l b r e a k w a t e r s1 3 .4 F u r t h e r v e r i f ic a t io n s o f t h e v e rt i c a l b r e a k w a t e r s1 3.5 T h e J a p a n e s e p r a c ti c e1 3.6 T h e p r o b l e m o f t h e r u b b l e m o u n d b r e a k w a t e r s

C o n c l u s iv e n o teR e f e r e n c e s

4 4 74 5 04 5 54 6 14 6 1

Chapter 14T op i c s c a l l i n g f o r a n ov e r a l l o v e r v i e w o f o f f s h o r e a n d c oa s t a lengineering

1 4.1 A c o m p a r i s o n b e t w e e n t s u n a m i a n d w i n d w a v e s f r o m t h e o p e n s e a t o t h ec o a s t

1 4 .2 S m a l l s c a le m o d e l s1 4.3 W a v e m e a s u r e m e n t s

C o n c l u s iv e n o t eR e f e r e n c e s


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X X IV L i s t o f co n t en t s

4 6 34 6 44 7 04 7 3

Appendix AA p p e n d i x t o c h a p t e r s 6 a n d 7 : u s e o f w a v e h i n d c a s t a n d w a v emeasu remen ts f r om sa te l l i t esA . 1 L o n g t e r m w a v e s t a ti s ti c s f r o m s a t e ll i te d a t a

A . 2 W a v e h i nd c a stA . 3 T r e n d i n t h e w a v e c l im a t e a n d i ts e f f e ct s o n e n g i n e e r i n g

R e f e r e n c e s

4 7 54 8 04 8 24 8 54 8 5

Appendix BA p p e n d i x t o c h a p te rs 9 a n d 1 0 : t he w a v e g r o u p o f th e m a x i m u mex p ec t ed c r es t e leva t io n , an d t h e w av e g r o u p o f t h e m ax im u m ex p ec t edc r e s t - t o - t r o u g h h e i g h t

B . 1 T h e fi rs t v e r s i o n o f t h e q u a s i - d e t e r m i n i s m t h e o r yB . 2 C o r o l l a r i e s o f t h e f i rs t v e r s i o nB . 3 T h e r e l a t i o n s h i p b e t w e e n t h e t w o v e r s i o n s o f t h e t h e o r y



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Chapter 1P E R IO D IC W A V E P A T T E R N :T H E A P P R O A C H O F D I FF E R EN T I A L C AL C U L U S

1 .1 The irrotat ional f low, the cont inui ty equat ion , the Bernoul l i equat ionT h e c o n d i t i o n

Ov~ _ O, Ovy = O, (1 .1 )Oy ozw h e r e re, v~ a r e t h e v e l o c i ty c o m p o n e n t s , is n e c e s s a r y a n d s u f f i c ie n t f o r a t w o -d i m e n s i o n a l y - z m o t i o n o f a r ig i d b o d y t o b e i r ro t a t i o n a l . S u c h a c o n d i t i o n is o n l ys u f f i c i e n t f o r a t w o - d i m e n s i o n a l y - z w a t e r f lo w . I n d e e d a g i v e n w a t e r m a s s d o e s n o th a v e a f o r m , a n d h e n c e i t c a n u n d e r g o s u c h a d e f o r m a t i o n t h a t c o n d i t i o n ( 1 .1 ) is n o ts a t i s f ie d a n d t h e f l o w i s s ti ll i r r o t a t i o n a l [ se e fi g. 1 .1 ]. T h e n e c e s s a r y a n d s u f f i c i e n tc o n d i t i o n f o r a t w o - d i m e n s i o n a l y -z m o t i o n o f a sm a l l v o l u m e o f w a t e r d x d y d z n o tt o b e r o t a t i o n a l is

Ov~ Ov~= . (1 .2 )Oy Oz

dz

8 5

IiI1

Fig. 1.1 Poss ible deform at ion of a smal l wate r mass in an i r rota t ional f low.


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2 Chapter 1

S u c h a c o n d i t i o n g u a r a n t e e s t h a t a c lo c k w i s e r o t a t i o n 6c~ o f f a c e d x d z is b a l a n c e db y a n e q u a l a n t i c l o c k w i s e r o t a t i o n 8c~ o f f a c e d x d y ( o r v i c e v e r s a a n a n t i c l o c k w i s er o t a t i o n o f d x d z is b a l a n c e d b y a n e q u a l c l o ck w i s e r o t a t i o n o f d x d y ) .

W e s h a l l s e e t h a t w a v e m o t i o n s a r e g e n e r a l l y a s s u m e d t o b e ir r o t a t io n a l , a n d t h a ti n a v e r y c o m m o n c a s e th e o r b i t s o f t h e p a r t i c l e s a r e c i rc u l a r . O b v i o u s l y t h e t w of a c t s ( i r r o t a t i o n a l m o t i o n a n d c i r c u l a r o r b i t ) d o n o t c o n t r a d i c t e a c h o t h e r . T ou n d e r s t a n d th is p o i n t w e c a n t h i n k o f t h e F e rr is w h e e l i n a n a m u s e m e n t p a r k . T h eh e a d o f t h e m a n i n f ig . 1. 2 m o v e s a l o n g a c i r c u la r t r a j e c t o r y o f r a d i u s R , a n d h i s f e e ta l so m o v e a l o n g a ci r cu l a r t r a j e c t o r y o f r a d i u s R . N e v e r t h e l e s s t h e m a n d o e s n o tr o t a t e , a n d i n d e e d h is h e a d a l w a y s r e m a i n s a b o v e h i s f e et . T h e o r b i ts o f th e v a r i o u sp a r t ic l e s f o r m i n g t h e m a n a r e c ir c u la r , b u t th e m a n d o e s n o t r o t a t e s i m p l y b e c a u s et h o s e o r b i t s h a v e d i f f e r e n t c e n t r e s . W e s h a l l s e e t h a t t h e m o t i o n o f w a t e r p a r t i c l e s i na w a v e is s i m i l a r t o th e m o t i o n o f t h e m a n ' s p a r t i c l e s o n t h e w h e e l .

Fig. 1.2 Th e head and feet of the ma n on the Ferris wh eel mov e along circular trajectories withthe same radius and different centres.

I f c o n d i t i o n ( 1 .2 ) is s a t i s f ie d , t h a t i s t o s ay , if t h e m o t i o n is i r r o t a t i o n a l , a f u n c t i o n05 (y , z , t ) d oe s exis t , for w hich :

0q5 0q5 (1 .3 )Vy= Oy, Vz-- OZ"W i t h t h i s f u n c t i o n , n a m e l y " v e l o c i t y p o t e n t i a l " , t h e c o n t i n u i t y e q u a t i o n t a k e s t h ef o r m 0 2 40 2 ( ~ t = 0 , ( 1 .4 )Oy2 Oz 2a n d t h e B e r n o u l l i e q u a t i o n , t h e f o r m

O~ 1 [ ( O ~ 2 + ( O (~ 2 ] _ f ( t)p + p g z + p - - ~ -+ -~ p \O y / k, Oz J J ' (1 .5 )


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Periodic wa ve pattern : the app roac h of differential calculus 3

w h e r e f ( t ) is a n a r b i t r a r y f u n c t i o n o f t i m e . T h e B e r n o u l l i e q u a t i o n s a ys : th e 1 .h.s. o f( 1 . 5 ) c a n v a r y w i t h t i m e , b u t a t a n y f i x e d t i m e i t d o e s n o t v a r y f r o m o n e p o i n t t oa n o t h e r .L e t u s b r i ef l y re c a ll t h e r e a s o n i n g l e a d i n g t o th e c o n t i n u i t y e q u a t i o n a n dB e r n o u l l i e q u a t i o n . H a v i n g t o d e a l w i t h a y-z f lo w t h e a b s t r a c t i o n o f t w o -d i m e n s i o n a l s p a c e i s c o n v e n i e n t , a n d h e n c e w e s h a ll r e s o r t to a s m a ll " v o l u m e "d y d z in o r d e r t o p r o v e e q u a t i o n s ( 1 .4 ) a n d ( 1 .5 ) . N a t u r a l l y , i f o n e p r e f e r s t h e m o r ec o n c r e t e t h r e e - d i m e n s i o n a l a p p r o a c h , o n e c a n r e s o r t t o a s m a l l v o l u m e d x d y d z . I nt h a t c a s e , a l l t e r m s i n t h e f o l l o w i n g e q u a t i o n s m u s t b e m u l t i p l i e d b y d x , w h i c h d o e sn o t m o d i f y t h e f i n a l r e s u l t .T o p r o v e t h e c o n t i n u i t y e q u a t i o n w e s h a l l w r i t e t h a t t h e w a t e r m a s s e n t e r i n g t h es m a l l v o l u m e d y d z i n t h e s m a l l ti m e i n t e r v a l d t is e q u a l t o t h e w a t e r m a s s g o i n g o u to f t h e s a m e v o l u m e i n th e s a m e t i m e i n t e r v a l. T h e r e s u lt is

(vy ) (v~ Ov ~ dz )OVy dy dz dt + p d y dt -P Oy 2 Oz 2

- p 4 ~ y y d z d t + p - t O zz d z d y d t , (1 .6 )

w h e r e t h e 1 . h . s . g i v e s t h e e n t e r i n g m a s s a n d t h e r . h . s , g i v e s t h e o u t g o i n g m a s s [ s e ef i g . 1 . 3 a ] . E q u a t i o n ( 1 . 4 ) p r o c e e d s s t r a i g h t f o r w a r d l y f r o m ( 1 . 6 ) o n c a n c e l l i n g a f e wt e r m s a n d u s i n g d e f i n i t i o n ( 1 . 3 ) .

I V z -t- ~ ~

i! dy _y Oy 1

Ov z dzOz 2.y,z

Vy + ~ ~

p + ~ ~pdzOz 2

o p [ Io y 2 p - o , 2 [ 1 p + ~ ~ dyay 2

Ovz dzOz 2 apdzOz2( a ) ( b )

Fig. 1.3 (a) Reference scheme for the continuity equation. (b) Referenc e scheme for theBernoull i equation.


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4 Chapter 1

A s t o th e B e r n o u l l i e q u a t i o n , i t p r o c e e d s o n a p p l y i n g N e w t o n ' s s e c o n d l a w toa n y s m a l l v o l u m e d y d z u n d e r i d e a l f lo w a s s u m p t i o n s , t h a t i s u n d e r t h e a s s u m p t i o ntha t the shea r s t ress i s negl ig ib le [ see f ig . 1 .3b] . The resu l t i sOp d Y )d z_ ~ + Op d__yy)dz_pdydzay (1 .7a )@ 2 O y 2

@ OP dZ ) dY - @ - ~ OP dz)Oz 2 (1 .7b)A c c e l e r a t i o n a o f t h e p a r t i c le b e i n g a t p o i n t y , z ( t h e c e n t r e o f t h e s m a l l v o l u m e )a t t ime t i s g iven by

Vp (t + d t) - Vp (t)a - - d tw he r e vp de n o te s the ve loc i ty o f the spe c i f ic pa r t i c l e ( c a l le d P ) w h ic h a t t ime t i s a tpo in t y , z . T he r e f o r e vp ( t ) c o inc ide s w i th the ve loc i ty v a t po in t y , z a t t ime t ; w h i l evv(t + d t ) is t he ve loc i ty a t t ime t + d t a t t he po in t b e ing oc c up ie d by pa r t i c l e P .S inc e a t time t + d t pa r t i c l e P i s a t po in t

y + vydt, z + vzdt,w e h a v e O v O v O vve(t + dt) - v (y + Vy dt, z + Vz dt, t + dt) - v + -~y v, dt + -~z Vz dt + - ~ dt,a n d h e n c e

O v O v O va -- -~ yV , + --~z Vz + 0--7-" (1.8)W i t h t h i s f o r m u l a a n d d e f i n i t i o n ( 1 . 3 ) , t h e e q u a t i o n s ( 1 . 7 a - b ) a r e r e d u c e d t o

0 1 ( 0 q 5 ) 2 + 1 ( O0"]2+p - 0o y p + 2 P \ o y / T P \ Oz J --g - '

[ +10 1 ( 0 4 ~ 2 + 1 ( O0~ 2+p - 0Oz P + Pgz + -2 P \ oy J -2P \ Oz J --& 'f r o m w h i c h f o l l o w s t h a t t h e s u m

1 ( 0 4 ) ~ 2 + 1 ( O 0 ~ 2 + p _ _P + Pgz + -2 P \ oy ) -2 P Oz ) Ot

doe s no t va r y i f w e m ove a long y f o r a f ixe d z a nd t , no r i f w e m ove a long z fo r af ixe d y a nd t ; w h ic h c a n be e xp r e s se d in the f o r m ( 1 .5 ) .


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Periodic wave pattern: the app roach of differential calculus 5

B e f o r e c o n c l u d i n g t h i s i n t r o d u c t o r y s e c t i o n , w e n e e d t o e x p l a i n t h e r e a s o n f o rt h e u n u s u a l y -z r e f e r e n c e f r a m e in p la c e o f t h e m o r e c o m m o n x-z . T h e r e a s o n i st h a t, t h r o u g h o u t t h e b o o k , y w ill b e t h e d i r e c ti o n o f w a v e p r o p a g a t i o n ( a t l e a stw h e n e v e r t h is d i r e c t i o n is c o n s t a n t ) , a n d c l e a r l y if t h e w a v e t r a v e l s a l o n g t h e y - a x is ,t h e f l o w i s t w o - d i m e n s i o n a l y-z .

1 .2 T h e d i f f e r e n t i a l e q u a t i o n s o f a n i r r o t a ti o n a i f l o w w i th a f r e e s u r f a c e

1 . 2 . 1 The free surface equationL e t u s a s s u m e t h a t t h e t w o - d i m e n s i o n a l i r r o t a t i o n a l f l o w h a s a f r e e s u r f a c e , a n d

l e t u s c a ll ~( y , t ) t h e v e r t i c a l d i sp l ac e m e n t o f t h is f r e e su r f ac e . O u r f i rs t g o a l is t o p u ti n m a t h e m a t i c a l f o r m w h a t w e h a v e j u s t s a id , n a m e l y " ~ ( y , t) is t h e s u r f a c ed i s p l a c e m e n t " .

I n o r d e r t o e x p r e s s i n m a t h e m a t i c a l f o r m t h a t ~ ( y , t) is t h e v e r t i c a l e l e v a t i o n o ft h e a i r - w a t e r b o u n d a r y , w e s h a l l r e s o r t t o t h e s m a l l c o n t r o l v o l u m e o f f ig . 1 .4 .S p e c i f i c a l l y w e s h a l l s a y t h a t t h e w a t e r m a s s e n t e r i n g t h i s v o l u m e i n a s m a l l t i m ei n t e rv a l d t is e q u a l t o th e s u m o f t h e w a t e r m a s s l e a v i n g t h e v o l u m e a n d o f t h e w a t e rm a s s p i l i n g u p i n t h e v o l u m e i n t h e s a m e i n t e r v a l d t . D o i n g s o , w e s h a l l i m p l i c i t l ysp e c i f y t h e p h y s i c a l m e a n i n g o f ~ /( y, t ).

T h e w a t e r m a s s e n t e r i n g t h e s m a l l v o l u m e i n a s m a l l ti m e i n t e r v a l d t is

I " 0 ~m~ - p dz dt- d - ~ Y "T h e w a t e r m a s s g o i n g o u t o f t h e s m a l l v o l u m e is

to o. - p + d d z d t w i t h d r / - O r / d y . ( 1 .9 )~, O y 2 O y

f ~

~ l y ~

Fig. 1.4 The sm all volume used for obtaining equation (1.13).


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6 Chapter 1T h e w a t e r m a s s p i l i n g u p i s

A m - p ( - - ~ d O d y .

In t h es e eq u a t i o n s , ~7 an d 0q5 a r e , r e s p ec t i v e ly , t h e s u r f ace d i s p l a cem en t an d t h e0yh o r i z o n t a l p a r t i c l e v e lo c i t y a t t h e l e ft s i d e o f t h e co n t r o l v o lu m e .E q u a t i o n ( 1 . 9 ) c a n b e r e w r i t t e n i n t h e f o r m

I ' O ( - - ~ y O 2 O d y ) [ O ( - - ~ - y ) ( O 2 ~ ) d y ] O r l d y d tm o u - p + d z d t + p + OY2- d O Y 2 = 7 = 7 - ~ Y ' (1 .10)

s o t h a t t h e e q u a t i o n

y i e ld sm o , - m e + A m = 0

J ~ 02q5 dz + (-~Y)z Or/ Or/ _ 0 (1 .11 )Oy 2 + ot 'ap a r t f r o m s o m e n eg l ig ib l e t e r m s . L e t u s co n s id e r t h e i n t eg r a l o n t h e 1 .h .s. o f t h i se q u a t i o n . T h a n k s t o t h e c o n t i n u i t y e q u a t i o n ( 1 . 4 ) , w e h a v e

] ~ 0 2 ~ d z - - J ~ 0 2 ~ d z - - ( - ~ z l ~ +Y 2 - d O z = = - e ' ( 1 .1 2 )

w h e r e t h e t e r m ( O ~ / O z ) z = _ d is z e r o b e c a u s e i t r e p r e s e n t s t h e v e r t i c a l p a r t i c lev e l o c i t y a t t h e b o t t o m , w h i c h is z e r o i f t h e b o t t o m is h o r i z o n t a l. T h e r e f o r e e q u a t i o n(1 . 1 1 ) i s r ed u ced t o

0(__0~_zz 0(__~_yz Or/ OI7 (1 .1 3 )= , - = , ~ + O tw h i c h i s t h e g e n e r a l e q u a t i o n o f t h e f r e e s u r fa c e .

In the proof we have assumed the b ot tom to be horizontal. B ut the same resul t holds evenif the bot to m is s loping. In this more general case d denotes the bot tom depth at the lef t s ideof the small contro l volume, so that the bo t tom depth at the r ight side is d + (d d/dy ) d y .Therefore, the lower limit in the integral on the r .h.s, of equation (1.10) changes, and (1.11)takes a new term:

J ' O C h d z + O (_~yz O rl O(_~yz d d O r lOy- T - - += _ . d y O t = 0 . (1.1 4)


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Periodic w ave pattern : the approa ch of differential calculus 7Moreover the boundary condit ion becomes

O(__~Z) __--- O(__~y)~ d d. . . . / . . . . t dY

which taken together with (1.12) and (1.14) leads us again to the equation (1.13) of the freesurface.1 .2 .2 T h e s y s t e m o f e q u a t i o n s

O n the w ho le , i n a n i r r o t a t iona l tw o- d ime ns iona l f low w i th a f r e e su r f a c e ,func t io ns 0 and ~7 m ust sa t i s fy the fo l lo wing sys te m of equ a t ion s :

(_ _~)~ 1 [ ( 0~ ) 2 + ( O q ; ) 2] _ i f ( t ) (1 15a )+2Lk0yj \ O z J l - '='q z =r lO(__~Z~ O(__~y Or/ & / (1 15b): , , - 0 - - 7 '

02-~- f- 02~ = O, (1 .15c )O y O z

O(-~z ~ = d - O. (1 .15d )T h e f i rs t one sa ys tha t t he p r e s su r e is ze r o on the f r e e su r f a c e ( a tm osp he r i c p r e s su r eb e i n g t a k e n a s r e f e r e n c e p r e s s u r e ) ; i t p r o c e e d s s t r a i g h t f o r w a r d l y f r o m t h e B e r n o u l l ie q u a t i o n . T h e s e c o n d o n e i s t h e g e n e r a l e q u a t i o n o f t h e f re e s u r f ac e , w h i c h w e h a v eju s t ob ta ine d . T he th i r d a nd f ou r th one s a r e , r e spe c t ive ly , t he c on t inu i ty e qua t iona n d t h e c o n d i t i o n o f t h e s o li d b o u n d a r y w h i c h h a v e b e e n r e w r i t t e n i n o r d e r t o g e t ag e n e r a l o v e r v i e w o f t h e w h o l e s y s t e m o f e q u a t i o n s . A s t o t h e s o li d b o u n d a r yc ond i t ion , i t ha s be e n spe c i f i e d f o r a ho r i zon ta l bo t tom, w h ic h i s t he c a se be inge xa mine d in the r e s t o f t h i s c ha p te r .1 .3 In t r o d u c t i on t o w a v e m e c h a n i c s

A ve r t i c a l p l a t e sw ing ing in a pe r iod ic w a y a t one e nd o f a c ha nne l ge ne r a t e sw a ve s on the f r e e su r f ac e . I f w e t a ke a ph o to o f the w a te r su r f a c e w e ge t a p i c tu r e o fthe su r f a c e d i sp la c e m e n t ~7 a s a f unc t ion o f a b sc i s sa y a long the p r o pa g a t io n a x is( the c han ne l ' s ax is ). Fun c t io n ~7 (Y) a t a f ixed ins ta nt r ep res en ts the w aves on thespa c e d om a in [ see fig. 1.5a ]. I f w e r e c o r d the su r f a c e d i sp la c e m e n t a t a f ixe d po in tas a func t ion of t ime t , we ge t the waves on the t ime domain [see f ig . 1 .5b] .

F r om f igs . 1.5a-b o f t h e w a v e s o n th e s p a c e d o m a i n a n d o n t h e t im e d o m a i n , w eg e t t h e d e f i n it i o n s o f t h e b a s ic p a r a m e t e r s : w a v e h e i g h t H w h i c h r e p r e s e n t s t h ed i f f e r e n c e i n h e i g h t b e t w e e n c r e s t a n d t r o u g h ; w a v e l e n g t h L w h i c h r e p r e s e n t s t h e


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8 C h a p t e r 1

(a)

Z

. L~ _ Y

~ \ I_ 0' t 1

(b )

F ig . 1 .5 ( a ) W a v e s o n th e s p a c e d o m a i n . ( b ) W a v e s o n th e ti m e d o m a i n .

d i s ta n c e b e t w e e n t h e tw o e x t r e m e z e r o s o f t h e w a v e ; w a v e p e r i o d T w h i c hr e p r e s e n t s t h e t im e l a g b e t w e e n t h e t w o e x t r e m e z e r o s o f t h e w a v e . B e s i d e s t h e s et h r e e p a r a m e t e r s , i t i s c o n v e n i e n t t o d e f i n e

( i ) t h e w a v e a m p l i t u d e

( i i ) t h e a n g u l a r f r e q u e n c y

( i i i ) t h e w a v e n u m b e r

a - H I 2 ,

aJ = 2 7r / T ,

k :_ 2 7 r / L .T h e s c h e m e o f fig . 1 .6 s h o u l d b e u s e f u l to u n d e r s t a n d t h e w a v e m o t i o n . E a c h

p o i n t i n t h e f ig u r e m o v e s a l o n g a c i rc u l a r o r b i t, w i t h c o n s t a n t s p e e d . T h e t i m e t a k e nt o c o v e r t h e o r b i t ( c i r c u m f e r e n c e ) i s T , a n d t h e f i g u r e s h o w s t w o i n s t a n t p i c t u r e st a k e n a t a t i m e i n t e r v a l o f T / 4 f r o m e a c h o t h e r . T h e l i n e c o n n e c t i n g t h e p o i n t sr e p r e s e n t s a w a v e . W e s e e t h e w a v e a d v a n c e o f L / 4 i n a t i m e i n t e r v a l o f T / 4 , a n dt h is m e a n s t h a t t h e p r o p a g a t i o n s p e e d ( c e le r i ty ) o f t h e w a v e i s

c = L / T .T h e s p e e d v o f e a c h p o i n t i s g e n e r a l l y d i f f e r e n t f r o m c , i n d e e d

v = 2 7 r R / T( R i s t h e r a d i u s o f t h e c i r c u l a r o r b i t s ) . T h e p a r t i c l e s p e e d v i s n o t o n l y g e n e r a l l yd i f fe r e n t f r o m 4 h e p r o p a g a t i o n s p e e d c o f t h e w a v e , b u t w e c a n e v e n v a ry va r b i t r a r i l y w i t h o u t m o d i f y i n g c : i t s u f f i c e s t o f i x p e r i o d T a n d l e n g t h L ( w h i c hd e p e n d s o n l y u p o n t h e d i s t a n c e b e t w e e n t h e c i r c u m f e r e n c e s ) a n d l e t r a d i u s R v a r y .


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r o t a t i o n

F i g . 1 .6 E a c h p o i n t m o v e s a l o n g a c i r c u l a r o r b i t o f r a d i u s R in a t i m e T ; t h e l in e c o n n e c t i n g t h ep o i n t s i s a w a v e w h o s e p r o p a g a t i o n s p e e d i s LIT.F i n a ll y , a l a st it e m b e f o r e p a s s i n g t o t h e m a t h e m a t i c a l t r e a t m e n t o f t h e w a v e

m o t i o n s . L e t u s i m a g i n e t h a t w e m a k e a b l o b o f t w o c o l o u r s ( e .g . r e d a n d b l a c k )w i t h in t h e w a t e r m a s s i n a w a v e . I f w e p u t t h e r e d o n t o p o f t h e b l a c k , g e n e r a l l y w es h o u l d s e e t h a t t h e r e d r e m a i n s o v e r t h e b l a c k , j u s t as w e s e e t h a t t h e m a n ' s h e a dr e m a i n s a b o v e h i s f e e t o n t h e w h e e l i n f i g . 1 . 2 . I n o t h e r w o r d s , w e s h o u l d s e e t h a tt h e b l o b d o e s n o t r o t a t e , w h i c h d e n o t e s i r r o t a t i o n a l f l o w .

1 . 4 S t o k e s ' t h e o r y t o t h e f i r s t o r d e r1 .4 .1 A physical approach

L e t u s fix t h e p e r i o d a n d t h e s w i n g in g a m p l i t u d e , a n d l e t u s s e t t h e w a v e m a k e r i nm o t i o n . W a v e s w i t h a h e i g h t H~ w i ll f o r m .

L e t u s s t o p t h e w a v e m a k e r , a n d l e t u s s e t t h e e n g i n e i n a d i f f e r e n t m a n n e r : s a m ep e r i o d , s m a l l e r s w i n g i n g a m p l i t u d e . T h e n l e t u s s t a r t a g a i n . W a v e s w i t h a h e i g h t H 2s m a l l e r t h a n H I w i ll f o r m . T h e w a v e p e r i o d w i ll b e t h e s a m e a s b e f o r e . I n d e e d t h ew a v e p e r i o d p r o v e s t o b e th e s a m e a s t h e s w i n g in g p e r i o d o f t h e w a v e m a k e r .

L e t u s r e p e a t t h e p r o c e s s m a n y t im e s : e a c h t i m e w i t h t h e s a m e p e r i o d T a n d w i t hw a v e h e i g h t s s m a l l e r a n d s m a l l e r . D o i n g s o , i n t h e f ir st w a v e g e n e r a t i o n s , w h i c h a r et h e o n e s w i t h t h e g r e a t e r h e i g h t s , w e s h a l l n o t e a s y m m e t r y b e t w e e n t h e w a v e c r e s ta n d t r o u g h : t h e c r e s t w il l b e s t e e p e r t h a n t h e t r o u g h . T h e n , w e s h a l l f i n d t h a t ag r a d u a l l o w e r i n g o f t h e w a v e h e i g h t , u n d e r t h e s a m e p e r i o d , l e a d s t o w a v e s w i ths m a l l e r a s y m m e t r y : th e w a v e a p p r o a c h e s a s i n u so i d w i th a w a v e l e n g t h w h i c hd e p e n d s o n d a n d T .

S u m m a r i z i n g :a s H 0 ( d an d T f i x e d ) r / (y t) H 2 ( T 27r t~, - - ~ c o s y (1 1 6)2 \ L T / ' "

w h e r e , fo r t h e m o m e n t , w a v e l e n g t h L is u n k n o w n . F u n c t i o n ( 1 .1 6 ) r e p r e s e n t s ap e r i o d i c w a v e o f l e n g t h L o n t h e s p a c e d o m a i n a n d i t r e p r e s e n t s a p e r i o d i c w a v e o fp e r i o d T o n t h e t im e d o m a i n . T h e n e g a t i v e s i g n i n t h e c o s i n e im p l i e s th a t t h e w a v et r a v e l s a l o n g t h e y - a x i s ( w i t h a p o s i t i v e s i g n t h e w a v e w o u l d t r a v e l i n t h e o p p o s i t ed i r e c t i o n ) .


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10 Chapter 1Ge ne r a t i ng wave s w i th sm a l l e r and sm a l l e r he igh t s ( and a f i xe d pe r iod ) we sha l la l so no t e t ha t t he ve loc i t y c om pone n t s , a t any f i xe d de p th , w i l l t e nd t o f l uc tua t e i nspac e- t im e l ike r /(y , t) : Vy in phase wi th r/, and Vz wi th som e pha se angle . M ore ov er ,t he pa r t i c l e spe e d wi l l p r ove t o be p r opo r t i ona l t o t he wave he igh t . In t h i s , t heph e n om e n on is s im i la r t o the on e de p ic t e d i n f ig . 1 .6 . In t ha t f i gu r e the wa ve he igh ti s equ al to 2R, an d the par t ic le spee d is equ al to 27rR / T . Th ere f ore , i f we re du ce Rwi th t he sam e T , t he wave he igh t and the pa r t i c l e spe e d a r e r e duc e d in t he sam e

w a y .Fr om the se obse r va t i ons on pa r t i c l e ve loc i t y , we c an d r aw the fo l l owing ide n t ik i to f the ve loc i ty po tent ia l :as H ~ 0 (d a nd T fix ed) : q5 (y, z, t) =2(_L_= Hfl (z; d, T, L )c os Y T~ t + e ) + f 2 ( t ) , ( 1 . 1 7 )

wh e r e f l (z ; d , T , L ) de no te s a func t ion o f z, i n wh ic h pa r a m e te r s d , T and L m a y bep r e se n t . Fo r t he m o m e nt , func t ions f l and f2 and pha se an g le e a r e unk now n.

1 .4 .2 The system of linear equationsTh e e qua t ions (1 .16) and (1 .17) show that bo t h r / an d q5 are inf in i tes imal o f o rd erH. In par t icu lar , the fac t tha t r/ is inf in i tes imal e nab les us to rewr i te equ at io ns

(1 .15a-b) in the formg~7+ ( -& )z + ( 02qS)z 1 [ ( 0 q ~ 2 + (0q5~2]

:0 OzOt : 0 r / + 2 \ Oy J k, Oz J ]1 { 0 [ ( & b ~ 2 + ( 0 q S ~ 2 ] } z 1+ 7 -& z kOyJ 20 zJ J =0 r l - 7 f ( t ) ,

+Z=0

(1 .18a)

(--~ z) ( 0 2 c ~ ' ~ [ O ( - - f f f - y ) z ( 0 2 c ) ) z ] &7 0rl (1.1 8b ): 0 + \ O z 2 ) z : 0 - : 0 + OzOy 0 - 7 'w here the value of a func t ion a t z = r /has b een expre ssed as: [va lue of the func t ion a tz = 0] + [value of the der iv at ive with respec t to z , at z = 0] x r/ . Ne glect in g the term sof order s smal le r tha n o r equal to H 2, equa t ions (1 .18a-b) are r edu ced to

( & ) z 1O - - - g ~ 7 + - p - f ( t ) , ( 1 . 1 9 a )=0

0 ( _ ) _ 0 7 / . ( 1 . 1 9 b )\OZ/z=o Ot


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Periodic wav e pattern: the appro ach of differential calculus 11

A t t h is p o i n t , t h e p r o b l e m is h o w t o o b t a i n f u n c t i o n s f l , f 2, l e n g t h L a n d p h a s e c( p r o v i d e d t h e y e x i s t ) i n s u c h a w a y t h a t t h e t w o f o r m s ( 1 . 1 6 ) a n d ( 1 . 1 7 ) o f r / a n d q 5s a t i s f y e q u a t i o n s ( 1 . 1 9 a - b ) a s w e l l a s ( 1 . 1 5 c - d ) . T h e p r o b l e m h a s o n l y o n e s o l u t i o n ,a s w i l l b e e v i d e n t f r o m t h e f o l l o w i n g a n a l y s i s .

1 . 4 . 3 S o l u t i o n f o r c )S u b s t i t u t i n g t h e f o r m s o f r / a n d q5 i n ( 1 .1 9 a ) , w e h a v e

d H 1H f l (0 ; d , T , L ) co s in(ky -co t + c ) + -d~- f2( t ) - - g - ~ c o s ( k y - c o t) + p f ( t ) ,

a n d t h e o n l y w a y f o r t h e f u n c t i o n o f y a n d t o n t h e 1 .h .s. o f t h i s e q u a t i o n t o b e e q u a lt o t he f u n c t i o n o f y a n d t o n t h e r . h . s ., f o r e ve ry y a n d t, i s t ha t1 -1)q (0; d, T, L) - -- ~ - gco , (1.2 0a )

rc (1.20b)c - 2 ,

f2(t) - P f (t dt ' . (1 .20c)

( C l e a r l y , a n a r b i t r a r y c o n s t a n t c a n b e a d d e d t o f 2 (t ), b u t s u c h a c o n s t a n t p r o v e s t ob e w h o l l y i n s i g n i f i c a n t f o r w h a t f o l l o w s . ) T h a n k s t o ( 1 . 2 0 b - c ) t h e i d e n t i k i t ( 1 . 1 7 ) o fq5 b e c o m e s m o r e p r e c i s e: 1 I i ' )(y, z, t) - - H f~ (z; d, T, L) sin (ky - cot ) + P f ( t at ' . (1 .21)T h e m o r e g e n e r a l f o r m o f f~ s o t h a t t h i s f u n c t i o n o f ~ sa t is f ie s ( 1 . 1 5c ) i s

f~ (z; d, T , L) = A ex p ( k z ) + B e xp ( - k z ) ,w i t h A a n d B arbi t rary cons tan ts . T h e v a l u e s o f t h e s e c o n s t a n t s c a n b e o b t a i n e dt h a n k s t o c o n d i t i o n ( 1 . 2 0 a ) a n d e q u a t i o n ( 1 . 1 5 d ) . T h e r e s u l t i s

A - - 1 gco- ' ex p (kd) B - - - - 1 gco-1 e x p ( - k d )2 exp (kd) + e x p ( - k d ) ' 2 ex p (kd) + e x p ( - k d ) '

s o t h a t ( 1 . 2 1 ) b e c o m e sH -1 co sh [k (d + z)] sin (k y - cot) + -7 f (t dt ' .05 (y, z , t ) - g- ~ c o co sh (kd) {1.22)


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12 Chapter 11 .4 .4 T h e l in e a r d i s p e r s i o n r u le

At th i s s t a ge the r e i s on ly e qua t ion ( 1 .19b ) to be sa t i s f i e d , a nd on ly onepa r a m e t e r i s l e ft unkno w n : L o r k - 27r / L . T h e u se o f f unc t ions ( 1 .16 ) a nd ( 1 .22 ) ine qua t ion ( 1 .19b ) y i e lds 2k t a nh ( k d ) - w , (1 .23)gtha t i s, i n t e r m s o f L a nd T r a th e r th a n o f k a nd w ,

L - gT 22rra n h ( 2 rrd )L (1 .24)w h i c h r e l a t e s L t o d a n d T . S u c h a r e l a t i o n i s m o s t c o m m o n l y u s e d i n o c e a ne ng ine e r ing , bo th in the f o r m ( 1 .23 ) a nd in the f o r m ( 1 .24 ) , a nd i t i s c a l l e d the l ineard i s p e r s i o n r u l e .

1 .4 .5 T h e e f f e c t o f f u n c t i o n f (t)S inc e f ( t ) i s a n a r b i t r a r y f unc t ion o f t ime , the ve loc i ty po te n t i a l ( 1 .22 ) isind e te r m ina te . B u t the f unc t ions w h ic h a r e o f in t e r e s t , t ha t i s t o s a y v ( y, z , t ), a ndp (y ,z , t ) , p r o v e t o b e i n d e p e n d e n t o f f (t) a n d t h u s t h e y a r e d e f i n it e . I n p a r t i c u l a r ,t h e c o m p o n e n t s o f v e c t o r v p r o c e e d t h r o u g h ( 1. 3) a n d p r o v e t o b e

H -1 cos h [k (d + z) ] t ) , (1 .2 5a)~ w k c o s ( k y - wVy(y, z, t) - g 2 co sh ( k d )n -1 s inh [k ( d + z )] t ) . ( 1 .25b )~ w k s i n ( k y - wVz (y, z, t) - g 2 co sh ( k d )

A s t o t h e p r e s s u r e , i t is o b t a i n e d b y m e a n s o f t h e B e r n o u l l i e q u a t i o n ( 1. 5) . T h er e su l t i s

H cos h [k (d + z) ] cos ( k y w t )p (y, z , t ) - -p g z + pg ~2 cosh (kd)( w h e r e t h e t e r m s o f o r d e r s s m a l l e r t h a n o r e q u a l t o H 2 h a v e b e e n n e g l e c t e d ) , a n dh e n c e t h e f l u c t u a t i n g p r e s s u r e h e a d p r o v e s t o b e

rlph (Y, Z, t) -- H cos h [k (d + z)]2 c osh ( kd ) co s ( k y - w t ) . (1 .26)Cl ea r ly the for m ula e for p and ~ph a re v a l id for z _< ~ , and the for m ula for ~/ph

r e q u i r e s i n a d d i t i o n z t o b e sm a l l e r t h a n o r e q u a l t o z e ro . I n d e e d ( 1 .2 6 ) p r e s u p p o s e stha t t he s t a t i c p r e s su r e i s - p g z . T o t h i s p u r p o s e , n o t e t h a t , i n t h e w h o l e t e x t , Zlphd e n o t e s t h e f l u c t u a t i n g p r e s s u r e h e a d o n l y a t p o i n t s w h i c h a r e a l w a y s b e n e a t h t h ew a te r su r f a c e , w he r e ( 1 .26 ) i s o f c ou r se va l id .


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Periodic wave pattern : the appro ach of differential calculus 13

1 .5 A n a l y s i s o f t h e l i n e a r d i s p e r s io n r u le1.5 .1 Ca lcu la t io n o f L f ro m d a nd T

I t i s c onve n ie n t t o de f ineL0 z

a nd r e w r i t e ( 1 .24 ) in the f o r m

g T 22 7 1 -

L - Lo t an h ( 2 L d )

(1 .27)

(1.28)

Since the range of tanhx is (0 ,1 ) , i t fo l lows tha tL < L 0 ,

w i t h L a p p r o a c h i n g L 0 o n d e e p w a t e r .T h e c a l c u la t i o n o f L c a n b e d o n e t h r o u g h a n i t e r at i v e a p p r o a c h , b y m e a n s o f

Li - L0 tan h (27r dL~_~ J ' (1.2 9)f o r i - 1 , 2 , 3 a nd so on. I n th i s w a y the f o l low ing ine q ua l i t i e s a r e ob ta ine d :{ L~ < L i f i i s an od d n um be r ,L ; > L i f i i s an even n um be r .T o ve r i f y the se ine qua l i t i e s , one ha s s imp ly to a na ly se the quo t i e n t L i / L t h a tproceeds f rom (1 .28) , (1 .29)"

LiL = t a n h ( 27 r d ) / t a n h ( 2 L d )L i _ ~

I nd e e d , be ing L 0 > L , i t f ol low s : L 1 < L a nd c on se qu e n t ly L2 > L w hic h in tu r nimp l i e s L 3 < L , a nd so on . S e q ue n c e ( 1 .29 ) c onve r ge s , t ha t i s t he d i f f e r e nc e}Lz- Li_~] a pp r oa c he s ze r o a s i g r ow s .1 .5 .2 C a lc ula tio n o f d / L f r o m d / L o

Fr om ( 1 .28 ) i t f o l low s tha td ta n h rc - . (1.30 )L Lo

T h e v a l u e o f d / L w hic h sa t i s f i e s th i s e qua t ion i s e qua l t o the spe c ia l va lue 2 f o rw h i c h t h e f u n c t i o nf (x) - x ta n h (27r x) (1.31)


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14 Cha pter 1i s e q u a l t o t h e g i v e n v a l u e d/Lo . F u n c t io n (1 . 3 1 ) i s s t r i c t l y i n c r ea s in g , i t b e in g t h ep r o d u c t o f tw o f u n c t i o n s , x a n d t a n h ( 27 rx ), w h i c h a r e s t ri c tl y i n c re a s i n g . T h e r e f o r e ,o n l y o n e a b s c i s sa c o r r e s p o n d s t o a g iv e n o r d i n a t e [ c f . f i g .l .7 ] . T o o b t a i n 2 , o n ec a l c u l a t e s f ( x ) a t r e g u l a r i n c r e m e n t s A x u n t i l f ( x ) > d/Lo i s f o u n d . A t t h i s p o i n t ,o n e g o e s b a c k a s t e p Z Xx, t h e n f ix e s a s m a l l e r A x , a n d c o n t i n u e s u n t i l th e a b s c i s s a 2is r e a c h e d w i t h t h e d e s i r e d d e g r e e o f p r e c i s io n .

f(x)

.f=xa__I-,o

XdLFig. 1.7 Inpu t: d / L o . Output: d / L .

I n th e f o l lo w i n g t h e r e a r e a n u m b e r o f p a r a m e t e r s w h i ch d e p e n d o n k d, a n d w es h a l l r e p r e s e n t t h e s e p a r a m e t e r s a s f u n c t i o n s o f d/Lo. T h i s i s b e c a u s e f o r e v e r yk d - 2 7 r d / L t h e r e i s e x a c t l y o n e d/Lo a n d v i c e v e r s a .

1 .5 .3 T h e c o n c e p t o f d e e p w a t e rF r o m ( 1 . 2 9 ) w e h a v eL1 - 0 .996L0

w h i c h y i e l d s

d 1i f ~ =L o 2 '

d 10 .996Lo < L < Lo i f Lo 21 t h e d i f f e r e n c e b e t w e e n L a n d L 0 is s m a l l e r t h a nh i s m e a n s t h a t , f o r d / L o - - ~ ,0 . 4 % . T h e d i f f e re n c e b e t w e e n L a n d L 0 g e ts e v e n s m a l l e r fo r d/Lo > -- .1 F or this1 2r e a s o n , t h e w a v e s o n w a t e r d e p t h d > ~ L 0 a r e c a l l e d w a v e s o n d e e p w a t e r .


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Periodic wa ve pattern : the app roac h of differential calculus 15

1 . 5 . 4 A s i m p l i f i c a t io n v a l id f o r w a v e s o n d e e p w a t e rT h e m o d e s o f a t t e n u a t i o n f r o m t h e f re e s u r f a c e t o t h e s e a b e d a r e

A 1 ( z ) - cos h [k (d + z) ]c o s h ( k d )

A 2 ( z ) - sinh [k (d + z) ]c o s h ( k d )T h e f i rs t o n e i s t h e m o d e o f t h e h o r i z o n t a l v e l o c i t y a n d t h e p r e s s u r e f l u c t u a t i o n ( cf .e q u a t i o n s 1 . 25 a a n d 1 .2 6 ); th e s e c o n d o n e i s t h e m o d e o f t h e v e r t i c a l v e lo c i t y (c f.e q u a t i o n 1 . 2 5 b ) .

T h e t w o m o d e s c a n b e r e w r i t t e n i n t h e f o r mA 1 ( z ) - e x p ( k d ) e x p ( k z ) + e x p ( - k d ) e x p ( - k z )

e x p ( k d ) + e x p ( - k d )A 2 ( z ) - e x p ( k d ) e x p ( k z ) - e x p ( - k d ) e x p ( - k z )

e x p ( k d ) + e x p ( - k d )f r o m w h i c h , b e a r i n g i n m i n d t h a t Ik z l < k d , w e g e t

l im A 1 ( z ) - e x p (k z ) - O, l im A 2 ( z ) - e x p ( k z ) - O .k d ~o c k d ~e c

T h e r e f o r e , f o r h i g h v a l u e s o f k d , b o t h A 1 a n d A 2 c a n b e a s s u m e d t o b e e q u a l t oe x p ( k z ) . S u c h a n a s s u m p t i o n l e a d s t o e r r o r s o n A 1 a n d A 2 w i t h i n 0 .0 4 i f d / L o - 0.5,w i t h i n 0 .0 2 i f d / L o - 0 . 6 , a n d w i th i n 0 .0 1 i f d / L o = 0 . 7 . I n a p p r a i s i n g t h e s e e r r o r s ,b e a r i n m i n d t h a t A 1 a n d A 2 r a n g e b e t w e e n 0 a n d 1 .

1 .5 .5 A n a l y t i c a l e x e r c is e : g e t a n e x p l ic i t f o r m f o r th e w a v e l e n g t hFrom the def ini t ion of wave ce ler i ty (c - L / T ) and the def ini t ion of L0, we have

c 2 L / L og d 27r d / L '

f rom which, using the l inear dispersion ru le , we get

Sincec _ / t a n h ( k d )V

t a n h x - x - ~ -3X O ( x s ) a s x ~ 0 ,

( 1 . 3 2 )

(1 .33 )


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1 6 C h a p t e r 1

w h e r e O m e a n s of the order, a n d s i n c ex / l + x - 1 + l x + O ( x 2)

2i t f o l l ows tha ta s x ~ 0 ,

x 2ta----x-x = 1 - ~ + O (x 4) as x --+ 0,

w h i c h e n a b l e s u s t o r e w r i t e ( 1 . 3 2 ) i n t h e f o r mc - 1 (kd)2- ~- 0 (k d) 4 as k d -+ O. (1.34)v / g d 6

A t t h i s s t age , l e t u s exp res s k d i n t e r m s o f d/Lo. To th i s end l e t u s r ewr i t e ( 1 .30 ) i n t he f o rmk d t a n h (kd) = 2re d/L o ,

f r o m w h i c h , u s i n g ( 1 . 3 3 ) , w e g e t(kd) 2 = 27r ( ) 2 dd + O d a s - + 0L0 L0 L0

t h a t e n a b l e s u s t o r e w r i t e ( 1 . 3 4 ) i n t h e f o r md= 1 2 rr d + -O a sx /~ d 6 L0 L0m --+ 0 .

H e n c e , t h e f o r m u l a

a n d c o n s e q u e n t l y(1 2re d)v/-gd,c --- 6 L0

L - - - ( 1 2rc6 Lod ) v /-g-dT " (1 .35)F o r d/Lo


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Periodic wave pattern: the appr oach of differential calculus 17

n e g a t i v e u n d e r a w a v e t r o u g h . T h e v e r t i c a l p a r t i c l e v e lo c i t y is p o s i t i v e b e t w e e n ac r e s t a n d t h e f o l l o w i n g t r o u g h , w h i l e i t i s n e g a t i v e b e t w e e n a t r o u g h a n d t h ef o l l o w i n g c r e s t . T h i s i s c o r r e c t g i v e n t h a t t h e w a t e r s u r f a c e i s r i s i n g b e t w e e n a c r e s ta n d t h e f o l l o w i n g t r o u g h : t h e t r o u g h h a s j u s t p a s s e d a n d t h e c r e s t is a r r iv i n g .

T h e s u m m a r y is r e - p r o p o s e d i n f i g .l .8 b , o n t h e t i m e d o m a i n , a t a f i xe d p o in t . W es e e t h a t t h e h o r i z o n t a l p a r t i c l e v e l o c i t y i s p o s i t i v e w h e n a w a v e c r e s t p a s s e s o v e r t h ef i x e d p o i n t , a n d i s n e g a t i v e w h e n a w a v e t r o u g h p a s s e s . W e a l so s e e t h a t t h e v e r t i c a lv e l o c i t y i s n e g a t i v e o n t h e t i m e i n t e r v a l b e t w e e n a c r e s t a n d t h e f o l l o w i n g t r o u g h ,a n d i n fa c t d u r i n g t h i s i n t e r v a l t h e w a t e r s u r f a c e a t t h e f i x e d p o i n t is g o i n g d o w n . O nt h e c o n t r a r y , t h e v e r t i c a l v e l o c i t y is p o s i t iv e d u r i n g t h e i n t e r v a l b e t w e e n a t r o u g ha n d t h e f o l l o w i n g c r e s t , a n d i n f a c t d u r i n g t h i s t i m e i n t e r v a l t h e w a t e r s u r f a c e a t t h ef i x e d p o i n t m u s t r i s e .

(a )I: , ' : : : : /: , ~ : : : : : 7 : : ~ : 7 i : } . : ' : 7 : ! ) . > : i . : , : : , { ? i : : : / ! 5 . , : - , : , . : = : : : : i f : i : : " ,' , . " . Z - < - : : : 7 . : .

(b )

n d v z

Fig. 1.8 D irectio n o f vy and vz" (a) at a fixed time insta nt; (b) at a

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