APPH 4200 Physics of Fluids

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1 APPH 42 Physcs of Fuds Inerna Gravy Waves (Ch 7) 1!! Revew of Surface Gravy Waves 2! Lnear gravy waves whn a connuousy srafed fud (Buoyancy!) 1

2 Rppes 2

3 Wave Equaon 4 E ' ' : () f? + ) ; 'j ( ' N '( I v ) I 11 :? ; ( ' v ' V ( V Va 'c A ( VI ' ) : X A ( C (' ' 4 (V 'k r * 3 Q :) () Q ' c 'u 'I c ) u ( () r ) C ' (4 C' 1 c: / ' Q r : / V ) Q ' U I ) ') C' u 3

4 Revew: Surface Gravy Waves : f: ' 'I '' ; r ) r ' ; 'I u ; I: ' /? ' V Q D I) I ' 1o r; Q o fa ( D I Q ' c V W ' ' f 'Q ' Q ) Q (V C V 1 V Q ) ' ' o j : u 'U ' 1 f o ' ' 4 ' ' r4 C ) ' : VI h 1: V f: ' Cb ( '1 ) 'd ( ) ' ') ' ' '! ') : ' I a: I 11 ') 4 ȳa ' ' V ' I v 1ì ( r I' ; o h '+ Q I f r ṭ z ' :) Q 4

5 Surface Gravy Wave 14 f U c V Õ C( '3 VI IL : I īi r ' o u ) E c C L 4 I VI c: 1' e 4 v ; / 1 ' ' ) v a c 4 '! ) 41 _ I: ( ' 1 'u I ') I 4 'E ' ' ( ;' v 'òf Q L e VI 'Q u Ç) ' ( v r: r â q ' ') L! '; 'w VI j c 3 'c ì 4 1 E w '): 9 r (Í r o ' $ 2 q1 ('c ' ( I 1 I: I I : 4: 3 1 o d 5

6 C: Surface Gravy Wave: The Equaons 4: ' ) ;: :' :s ' 1 I ' 'w ' ' + ' + V ' ' w j' L a ) : : I 1 '+ r( ' j I b I V' )( 1 ' r I 'u ) 'u ë ) L cu U 1 (3 ( () C '( c: + ' ' U U 1: V U a ' ') 'U ' q r f I 'U ) oy c O' c ( c c: ' u ' ' '? ( C' ) L )( r' L ' ' ( L) ' 1': 1 C C' C1 1 1 I () f ' r a :: I' ' ( + + ) : ' c ' Q ' I o I ' QI r ' 1U v ) b q f '4 rf ( C' v ( ' I ox ': ' ; I r4 ' r 1I I I f b I b o ) V r r I 1 H c?' 7 6

7 Lnear (Sma Ampude) Waves ' '11 r1m ) ṭ ' ' r: 1= è ; ' e L : 3 o of VI k ' q ) j I 1 ( I ) ' r: I G :: ' ) 7: 9 f? x 3 () r' f' C' ) + 'a ' r + 1 n I r r f :: u F (1 ' + v 1 C' O _ c Q '''' 4 4 ' ' ': 7

8 Pane Wave Souon ' W ' F Revew c (! e o ( I U ' ' ) ' fa L ' 4 (r C' ( C' ') C C' ( ( I) + 1 I r q ( ( 'C o f r ; 4f ' ( (' N (Wn ' 4 ' ao ' C '1 ' u r h ' u + r ' NI V 1 ' f I S ( ' q /' + ß + ' ( ' C' n y: f ' 4 L W 1 ' c: ' ' ': I' I '4 r d va u () î ( s ' 8

9 Souon (con) c: f I f + (' ' İI N 3 o : c /' V c: ' s g u cr r f r ' ' 3 ': L ø c: C' C o I' q ' I ' 11 I ' c: v v ) A o a a: ' A ' j r ' : ' o L _ V (y L o n co GI ( ṫ o ( 3 Revew : o f ' c : ' v 3 IV 1 I f ) ' u L ) Q Q Q (': :! 'ḷ Ie ') ' : A (! IA 9

10 Fndng he Veocy Poena Revew U 3 : ': 'C '! (' ' V ' ' 9 ( C re () r /' U ' 1 I ( v f L j 1 a ' ' ' () Q q v q + I ( I r w I ' C q ' I u I / ' ' 1 V 1 r ri! ' Q V 31 /' v 4 I ' If V ) ' r ' ' ' 1

11 Surface Gravy Waves (Con) () I Q ' 1 'e h ' y j # r I : g f r f (I u F ' r ' ) ' Q j D ' f 1; f r ' : ' c: ': / 1 1 V I! c ' 1 ' ' ' ) ' o o : () C 1 Q ( ( ( ( 1 r ' ' ' ' ' ' + I ) ) ' rp! ' V) ; I c; ' (j ' v 11

12 Sreamnes 12

13 Pahnes 13

14 Wave Moon hp://facuygvsuedu/vdecp/waveshm 14

15 Shaow Waer (kh ) 1I I ') Q Q ' I ' I V' () ( ) / ' 'I ' /' V? I + 4 ( f I' f (( ' :: : ' u ; L : ) 11 6 ù ( ' Revew 15

16 Deep Waer (kh ) r' V I f ( L ' ( 9 CL ' r I ( o ' ' ' A L : '! I ri I ' c ; _ : a ' f: / / ca 3 r ' o : ( f r 4ø ') ) r N (C ' ') (( r ' c:! Y VI 'U :: :: : _ ; r; Q a 1 Revew 16

17 Wave Energecs a Q c v ' :: '( 1 N 'Q r N ; ' f: ' ' n _ : ' L f' I W' :: () c I ' 11 () L f ': ' N L ( ) 1 U LU '1 ; cì ' '1 I e ' a: If /' IV r : ' c( /' 1 c r _ : c ' 1: Revew : (( : V' vḷ ' r r ù T ) G p ' I VI V () 4 c ' : f O f: ' r: ( / ' ' r ' ' á ' + ; r c 'I 1 ' À /' : rr ( ' I' 4I ' r r' ' UA 'N r' I '1 I' C!) 1 fv G () 1 17

18 Gravy Waves n a Connuousy Srafed Fud 1 c 1 /' 'Q 1 1 q v rl (u ') : w ' : ' 3 ( v ' q a Q ' ) r( CV :) 1 Q 1 c : () Q L (f c L 11 ' ' f + b a Q v () I ' ' '1 + I ) ' () ': U ; I 1 v ') ' I : : Q L ' ( VI 1 I : ' ') U VI j Q ) ( r q; : QS 3 ) ': & * ; ;: 9 18

19 Lnearzed Gravy Waves ( ) h I S C) ) + y c (1 ) :s a ri C' ' ) I ' '/ I q ) ' (! c S4S j c ') f1 ( I íî Q D Q 4 Q Q çff C 1 I 11 C I D r: C q 1 : ) o ;Q V ' '! r ) ' r L )( ' ( V 3 v I ( C4 ': ': ' ' q Q ' c Ioj ' ') oi L/ V Q I 'j ' :: 19

20 PaneWave Equaons for Compex Wave Ampude o :: + () 51 N c a ' Ie : r 1ì Q c(ô ) a: : ( a I ' ' I ': c C' Q2 1 1: f' 1ì Q ) r6 C ' ': f Q ( ( : r & r ( r : ' r ; 'I r ( 1 I? F :: U ( ( r c r 11 C' I f) p ' 1 ' u 'Û ṫ ' v C)Q Æ N o Q! N c '1 T 'C V n r () 1: u 3 I ( ' o ;L ' f' 3 ': x Q j u e ' I : 2

21 Dsperson Reaon for Inerna Gravy Wave L Q ') ' 1 E r Q 3 1 ( ' ; I I f I C' 6 'r'4' ) r C; : ) Q I: ') I Q qo ' N Y )' r + '; ' h v ' ( co ( 'X e ; ) ) I c L 4 4 ' r : N : ( Q 'C ) ; ) ' (I ) 'n ' ' ( ) 1: ; :) : 1 21

22 Properes of Inerna Gravy Wave ) r :: ) ' :r r' ) r r + 1 I I oi ri u I 1'3 x n ōo ' 1 '' )( *' ' 'U U + )( u n ( ' : b * f 1 e ' A G r ; v ) U q Q ' a '( F 22

23 Group Veocy/Phase Veocy Az : ri r c= N Y v ' (I V 3 '1 r: v Q Iị + ) (' ) ' 3 o ' U 1 ' 3 v ' '(N '* I ' I I' ' ( a ) ( : + 3 ( ') ( k )( ý fb(l I ' : 'n ( ' 'I 1 q : I S 'N ' 3 ') ' 23

24 Measuremen of Inerna Waves 24

25 hp://wwwamoswashngonedu/durrand/ anmaons/sand55/sandng1psp 25

26 Wave Energy Densy b G ':1 1 (' n 1 o 1 : :) W a D 1': r :: c(: f :s X b : Q r ' f j o ' u : ' o ')! ( ) ' 1 ' : a: E :c r Cb r' r q ( : ' r 7: N Q N r U r $ v ( s ) I' ' o 'Ñ C1 C ' ' '1 I I ' C) ( ' ': d C' ' C f' ( 'ù I 1 ) ( V) 1 )j 1 Q Ì ' ' ( 1 26

27 I Wave Knec Energy Densy ) N ' 'I ' ; C ( ' (' f ' ) r IQ rv Q ' I ' I I 1 CO v ' ' r ' ) :: n (' V c f C' n + (: s ' ' + L ' v f' c c O:(1 ( ' ' U ( : () r ' Q L v L L 4N r: r I U 1 rl U 1 : :r C ' Ie c 27

28 Wave Energy Fux r Q 1 )( Q 11 ( I ': ' ) r 14 3 G Q I' ' o ) ') E= ' r f I 1 ' r f I L1 ( ' ' ' c j 'I Q' fc c: I N G #' c; I= I $: ( O ( I L f' V Q _ I CO ( ' I v I D 31') ' r I' N (V ') ' :: '3 Me( :: :;If I ') r '' ' P) f I': cc :s '; f: ' I / :: (I 28

29 Summary Surface waves Inerna gravy wave descrbed by: Lnearzed fud dynamca equaons Assume panewave souon:!! cos(k x x + k z z ) Buoyancy frequency Ω b s maxmum frequency Phase and group veoces perpendcuar Conservaon of wave energy densy 29

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