2006 2006 中国数学科学中国数学科学与教育发展论坛与教育发展论坛如 此 饶 趣 神 奇 的 水 波如 此 饶 趣 神 奇 的 水 波

Such interesting and marvelous water Such interesting and marvelous water waveswaves

吴耀祖吴耀祖美国加州理工学院美国加州理工学院

浙江大学数学科学研究中心浙江大学数学科学研究中心2006 2006 年 年 7 7 月 月 1 1 日日

T. Y. Wu Caltech

Capillary wavesCapillary waves

T. Y. Wu Caltech

Gravity wavesGravity waves

T. Y. Wu Caltech

Water waves of depth hWater waves of depth h

T. Y. Wu Caltech

Key parameters of water wavesKey parameters of water waves

T. Y. Wu Caltech

Dispersion relations on linear theoryDispersion relations on linear theory

T. Y. Wu Caltech

Water particle pathlines in watersWater particle pathlines in waters

T. Y. Wu Caltech

Wave packet dispersionWave packet dispersion

T. Y. Wu Caltech

Wave packet group dispersionWave packet group dispersion

T. Y. Wu Caltech

Generation of ship wavesGeneration of ship waves

T. Y. Wu Caltech

T. Y. Wu Caltech

Ship wave in shallower watersShip wave in shallower waters

T. Y. Wu Caltech

Ship wave patternShip wave pattern

T. Y. Wu Caltech

T. Y. Wu CaltechJohn Scott Russell’s 1834 chance John Scott Russell’s 1834 chance discoverydiscovery

T. Y. Wu Caltech

T. Y. Wu Caltech

Significances of the KdV evolution equationSignificances of the KdV evolution equation

T. Y. Wu Caltech

Grand discovery of the remarkable Grand discovery of the remarkable soliton soliton

T. Y. Wu Caltech

Remarkable properties of solitonsRemarkable properties of solitons

T. Y. Wu CaltechForced forward-radiation of Forced forward-radiation of solitonssolitons

T. Y. Wu Caltech

Exciting discovery of forward-radiating Exciting discovery of forward-radiating solitons at Wu’s Laboratory - 1982solitons at Wu’s Laboratory - 1982

T. Y. Wu CaltechAgreement between theory and Agreement between theory and experimentexperiment

T. Y. Wu Caltech

Comparison between experiment and theoryComparison between experiment and theory

T. Y. Wu CaltechExperimental and computational views of Experimental and computational views of forced radiation of solitary wavesforced radiation of solitary waves

T. Y. Wu Caltech

Hydrodynamic instabilities of Hydrodynamic instabilities of resonantly forced solitonsresonantly forced solitons

• Camassa, R. and Wu, T. Y. (1991a) Camassa, R. and Wu, T. Y. (1991a) Phil. Trans. R. Soc. LondPhil. Trans. R. Soc. Lond. A. A337337, 429-466 , 429-466

• --- (1991b) --- (1991b) Physica DPhysica D 5151, 295-307 , 295-307

• Having profound discussions with Having profound discussions with Cambridge Univerisity Lucasian Cambridge Univerisity Lucasian Professor Sir James LighthillProfessor Sir James Lighthill

Part IReflection for Insight on

Solitary Waves of Arbitrary Height

Wu, T. Y., Kao, J. & Zhang, J. V. Wu, T. Y., Kao, J. & Zhang, J. V. Acta Mech. Sinica 21 (1), 1-15, Acta Mech. Sinica 21 (1), 1-15,

20052005

T. Y. Wu Caltech

Sir G.G.Stokes (1880) on the solitary wave outskirtsSir G.G.Stokes (1880) on the solitary wave outskirts

, e , k x ctx t b x

22

0 cos( )

( ,0) 0

tan( )( ) 0

xx yy k x ct

y

y t

t

a e kyx

c khy h Fgh khg

• Stokes: This relation is exact!Stokes: This relation is exact!

a

T. Y. Wu Caltech

Reflective QueriesReflective Queries•1. 1. What else?What else?

•2. Can linear theory 2. Can linear theory hold for low waves of hold for low waves of diminishing amplitude?diminishing amplitude?

T. Y. Wu Caltech

0.1 0.2 0.3 0.4 0.5

0.25

0.5

0.75

1

1.25

1.5

1.75

y

F2 = 1.15=0.196541

tan()

F2

2 1M

, , , ..., 2 3 M

T. Y. Wu Caltech

Reflections on Reflections on

0 0.1 0.2 0.3 0.40

0.5

1

1.5

F2

tan()

10-4 10-3 10-2 10-1 1000

0.1

0.2

0.3

0.4

0.5

M=90

70

50

30

10 5

m

• Boussinesq-Rayleigh:Boussinesq-Rayleigh:

• Asymptotic representation:Asymptotic representation:

2 tan( )F

2

1

1 tan( ) / 1

F

F

2

1

1 ( 1)M

m

m

w u iv

2 1M

T. Y. Wu Caltech

Sir G. G. Stokes (1880): Sir G. G. Stokes (1880): Corner wedge flow under gravity Corner wedge flow under gravity

= C1

= C2

= 0 x y 30

30

g

/ 6 3/ 2

exact solution:

( ) /(3 / 2)i

z x iy

f i ze F

T. Y. Wu Caltech

Formulation and analysisFormulation and analysis

-11

2

, ,/ ,

, log , =tan

11map: log1

vq u

z x iy f iw df dz u iv q w

i

f i

22 ( ) 2

2 3 ( )2

1( ) ( ) exp ( ) ( )sin( / 2)

0 ( real, 1 1); 0 ( 0)

Bernoulli's equation (on e )

( ) e ( ) -1 0

( ) e sin sin( ) 0

i

F

dx i i

i

B

G F

T. Y. Wu Caltech

Unified intrinsic functional expansion (UIFE) Unified intrinsic functional expansion (UIFE) theorytheory

• ((i) First establish a UIFE expansion for i) First establish a UIFE expansion for (() in terms of a set of ) in terms of a set of intrinsic component functionsintrinsic component functions (ICF), analytic in (ICF), analytic in , to represent all , to represent all the intrinsic wave properties in the entire flow field.the intrinsic wave properties in the entire flow field.

• (ii) The unknown coefficients in the UIFE expansion are (ii) The unknown coefficients in the UIFE expansion are determined under the given conditions by minimizing determined under the given conditions by minimizing G G and and B:B:

• (iii) The minimization of (iii) The minimization of EE22 is implemented by stepwise is implemented by stepwise optimization.optimization.

2 2

0

2 2

0

( ), ( ), (UIFE-method I)

( ), ( ), (UIFE-method II)

E G d

E B d

T. Y. Wu Caltech

The highest solitary wave• The UIFE expansion for +i of the highest solitary wave:

• Solution by UIFE-Method-I

• Solution by UIFE-Method-II

• The UIFE-Method-I solution so obtained consists of three groups of intrinsic funstions, in am0, a1n, b1n, each containing four modes.

1 1 1 12 213 2 2 2 2

1 0

( ) (log ) ( ) ( )M N

n n mmn mn

m n

a b

3(1 2 ) tan( ) 0.40134

10 20 30 40

11 12 13 14

11 12 13 14

0.456569, -0.102203, -0.158703, 0.0116986,0.253853, 0.13121, 0.0225912, -0.0077326,-0.329156, -0.0162222, 0.00144751, 0.0000421358;

0.833121, 1.29083, hst hst

a a a aa a a ab b b b

F

22 / 1. F

-70.8331990, 1.2908904, local error 2 10 hst hstF

T. Y. Wu CaltechA dwarf solitary wave -- with F =1.005 0.054873)

• Solution by UIFE-Method-I

• The UIFE-Method-I solution so obtained consists of eleven modes of

but with only one mode of a11

1 12 22 2

1 0

( ) ( ) ( )M N

n n mmn mn

m n

i a b

10 20 30 40

50 60 70 80

90 10,0 11,0 11

0.0342629, -0.0401226, 0.0218187, -0.0033137,-0.0018522, 0.00046252, 0.00032881, 0.000116193,-0.0002101, -0.000073, 0.000061, -0.00002122;

a a a aa a a aa a a a

7 2min 200.0114426; / 6.78 10 ; 2 / 0.022658reE E a F

0 ( 1,2, , ) 11; with 2 1.207ma m M M M

T. Y. Wu Caltech

Reflections on Reflections on

0 0.1 0.2 0.3 0.40

0.5

1

1.5

F2

tan()

10-4 10-3 10-2 10-1 1000

0.1

0.2

0.3

0.4

0.5

M=90

70

50

30

10 5

m

• Boussinesq-Rayleigh:Boussinesq-Rayleigh:

• Asymptotic representation:Asymptotic representation:

2 tan( )F

2

1

1 tan( ) / 1

F

F

2

1

1 ( 1)M

m

m

w u iv

2 1M

T. Y. Wu CaltechVariations of Variations of (() and F() and F() – by UIFE – I-II) – by UIFE – I-II F E VdB/Miloh

0.200000 0.111033 1.053723 2.02×10^{-6}

0.300000 0.176148 1.083661 1.6×10^{-7}

0.351018 0.21228426 1.09978834 1.2×10^{-8}

0.458885 0.2960185 1.1358535 4×10^{-7}

0.584250 0.40743022 1.18097922 3.8×10^{-8}

0.753499 0.58369035 1.24470064 3.0×10^{-8}

0.840000 0.68193652 1.27422855 7×10^{-9}

0.878825 0.7253045 1.2847658 1.8×10^{-6} 1.28472

0.900000 0.7479485 1.2892276 2.7×10^{-6}

0.910000 0.75824470 1.29091908 2.45×10^{-6}

0.940277 0.7872063 1.2939914 6.5×10^{-6} 1.29395

0.948000 0.7939264 1.2941987 3.8×10^{-6}

0.950000 0.7956161 1.294210710 3.15×10^{-8}

0.951000 0.7964528 1.294210043 2.52×10^{-8}

0.952000 0.7972840 1.29420496 3.52×10^{-8}

0.970108 0.8113862 1.2933581 1.1×10^{-6} 1.29332

0.988000 0.82395 1.29147 4×10^{-5} 1.29144

0.990000 0.82535649 1.291273345 6.17×10^{-7}

0.997500 0.83106436 1.290850281 4.22×10^{-7}

0.998400 0.83182712 1.290860312 7.64×10^{-7}

1.000000 0.83319905 1.290890430 2×10^{-7} 1.29091

T. Y. Wu Caltech

Graphical presentation of numerical Graphical presentation of numerical solutionssolutions

Full range:0<<0.833

2Extreme Waves

0.68<<0.8332

The fastest wave:fst=0.7959034Ffst=1.294211

2nd extreme – local minimum

min=0.8310643Fmin=1.290850

T. Y. Wu CaltechSample profiles of extreme solitary Sample profiles of extreme solitary waveswaves

= = hst hst (0.8331990), 0.822279, 0.811386, 0.796952;(0.8331990), 0.822279, 0.811386, 0.796952;• F F = = F F hsthst (1.290890), 1.291738, 1.293358, 1.294208. (1.290890), 1.291738, 1.293358, 1.294208.

T. Y. Wu CaltechWave profiles evaluated by UIFE-I and Wave profiles evaluated by UIFE-I and IIII

= = hst hst (0.8331990), 0.758245, 0.583690, 0.407430, 0.212284;(0.8331990), 0.758245, 0.583690, 0.407430, 0.212284;• F F = = F F hsthst (1.290890), 1.29092, 1.24470, 1.18098, 1.09979. (1.290890), 1.29092, 1.24470, 1.18098, 1.09979.

T. Y. Wu Caltech

Integral properties of solitary wavesIntegral properties of solitary waves

212

2 212 1

mass:

potential energy:

kinetic energy:

( 1)

circulation:

( 1)

p

k

M dx

E dx

E F u dydx

F u d x

T. Y. Wu CaltechComparison with three lower-order Comparison with three lower-order theoriestheories

• First three lower-First three lower-order theories:order theories:– Kortweg/de Vries Kortweg/de Vries

1895 – 11895 – 1stst order; order;– Laitone 1960 – 2Laitone 1960 – 2ndnd_;_;– Chappelear 1962; Chappelear 1962;

Grimshaw 1971 – Grimshaw 1971 – 33rdrd__

0

1,2,3

nn x x

n

Unified Perturbation Expansion Unified Perturbation Expansion For Solitary WavesFor Solitary Waves

Solitary Wave Theory – Part 2Solitary Wave Theory – Part 2Wu, T.Y, Wang, X.L. & Qu, Wu, T.Y, Wang, X.L. & Qu,

W.D.W.D.Acta Mech. Sinica 21(12) Acta Mech. Sinica 21(12)

20052005

T. Y. Wu Caltech

Motivated reflection and queriesMotivated reflection and queries• Q2: Does the Euler model possess a Q2: Does the Euler model possess a

perturbation expansion which is perturbation expansion which is convergent?convergent?

• Effects due to change in base parameter Effects due to change in base parameter onon

• a. solution accuracy a. solution accuracy b. rate of series convergence b. rate of series convergence

T. Y. Wu Caltech

Literature contributionsLiterature contributions

• Boussinesq, J. (1871); Lord Rayleigh (1876); KdV (1895)Boussinesq, J. (1871); Lord Rayleigh (1876); KdV (1895)

• Laitone, E.V. (1960) series expansion to O( )Laitone, E.V. (1960) series expansion to O( )• Chappelear, J.E. (1962) -- series expansion to O( )Chappelear, J.E. (1962) -- series expansion to O( )• Fenton, J. (1972) numerical solution to O( )Fenton, J. (1972) numerical solution to O( )• Longuet-Higgins, M.S. & Fenton, J. (1972)Longuet-Higgins, M.S. & Fenton, J. (1972)• Wu, T.Y. (1998) --- adopting base parameter Wu, T.Y. (1998) --- adopting base parameter • Wu, T.Y. (2000) --- new variables to O( )Wu, T.Y. (2000) --- new variables to O( )• Qu, W.D. (2000) --- new variables to O( )Qu, W.D. (2000) --- new variables to O( )• Wu, T.Y., Wang, X.L. & Qu, W.D. (2005) – to O( )Wu, T.Y., Wang, X.L. & Qu, W.D. (2005) – to O( )

a =aaa

( ) ( )2 2, ; Boussinesq familya k h Oha a= = =e e

T. Y. Wu Caltech

Guiding principlesGuiding principles• 1. Simple and efficient formulation be 1. Simple and efficient formulation be

sought for exact unique solution with as sought for exact unique solution with as few unknowns and to as high order of few unknowns and to as high order of expansion as attainable.expansion as attainable.

• 2. Make comparisons between basic 2. Make comparisons between basic parameters for their effects on (a) solution parameters for their effects on (a) solution accuracy and accuracy and

(b) rate of series convergence for (b) rate of series convergence for parameters:parameters:

2 2 22

2, , , , 1a ck h F Fh gh Faa b g= = = = = -e

T. Y. Wu Caltech

A unified perturbation expansion theoryA unified perturbation expansion theory• Asymptotic Asymptotic

expansionexpansion

• High order High order equationsequations

-2 20 0

1 1 1, , 1n n n

n n nn n n

u u F F bz z e e e¥ ¥ ¥

-= = =

æ ö÷ç= = = + ÷ç ÷÷ççè øå å å

( )( )

( )

1 1 1 2 12

0 1 1 1 2 1 1 1

, ; , , ,,

, ; , , , ; , ,1,2,

n n n n n n

n n n n n n n

u P u u

u F Q u u b bn

z z z z

z z z z- - -

-- - - -

¢¢ì - =ïïïíï ¢¢- =ïïî=

LL L

L( )

( ) ( )( )

( )

1 10 1 12

1 1 0

2 2 1 1 1 22

2 2 1 1 1 1 22

2 2 1 1

, 01,/ 0

, / 6/ 2 / 2

4/ 3sech , 4/ 3. soliton

O uF u

u F

O u u u P

u b u u Q

P Q x b KdV

zz

z

z z

z z

z

+ = üïïï ® = ± = -ýï+ = ïïþ¢¢+ = - + =

¢¢+ = - - + == ® = = -

2

e

e

T. Y. Wu Caltech

Asymptotic reductive perturbation schemeAsymptotic reductive perturbation scheme

• 1. Reductivity results from u”1. Reductivity results from u”nn in P in Pn+1n+1=Q=Qn+1n+1 . .• 2. Reductivity chain-links all orders n=1,2, …, N. 2. Reductivity chain-links all orders n=1,2, …, N. • 3. Simplicity in using 3. Simplicity in using with (u with (u00, , ) ) (A) solves b (A) solves bnn= - C= - Cn1n1 before before

nn; (B) avoids iterations needed otherwise; (B) avoids iterations needed otherwise

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( )

2

1 11

1 0 1 1 , 1 1

1

3 -conditions : ;Two equations and yield

9 4 .by -conditions

Two complimentary solutions of 0 both vanishunder the -cond

n x

n n n n

nn n n n n n n n

n n

n

O n E x x O e xP Q P Q

L C b C Cb C E

LE

z z z

z z z z z z

z

-

+ ++

+

³ - = £ ® ¥= =

¢¢= + - = + + += -

=

L

e

( )itions.

solution is in each order .nunique OÞ e

T. Y. Wu Caltech

Exact higher-order theory of O( Exact higher-order theory of O( 18 18 ))• Cf. Wu, T.Y., Wang, X.L. & Qu, Cf. Wu, T.Y., Wang, X.L. & Qu,

W.D., ACTA Mech. Sinica 21 W.D., ACTA Mech. Sinica 21 (2005)(2005)

• With application of Mathematica With application of Mathematica 5.05.0

T. Y. Wu Caltech

Exact theory of O( Exact theory of O( 18 18 ))

T. Y. Wu Caltech

T. Y. Wu Caltech

T. Y. Wu Caltech

T. Y. Wu Caltech

T. Y. Wu Caltech

On solitary waves - part 3, On solitary waves - part 3, Wang, X.L. and Wu, T.Y.Wang, X.L. and Wu, T.Y.

Physics Letters A 350 (2006), 44-Physics Letters A 350 (2006), 44-5050

Integral convergence of the higher-Integral convergence of the higher-order theory for solitary wavesorder theory for solitary waves

T. Y. Wu Caltech

Wave Elevation and Amplitude: Divergent SolutionsWave Elevation and Amplitude: Divergent Solutions

2 3 4 5 620 104 98876 952856 10964170443 9 27 14175 70875 3898125

7 8 924765182384 12999641717276 43730337031414072383107725 79814109375 98631820546875

10467321172438850758088 162060365430895126171875 +

110159250322767988368422072683870728515625

12 131725672631559124419069681512 77900741717811450909556593136145615075935401337890625 2092258722650766591796875

3533686543760951794852970179194599837929

+

25768532067235317804459610110208757298613616

870192468105968462394953276972270094047793772852420806884765625

26O

2 2 4 284 43 9 3

2 4 6 8 3286768 39512 4928 23156212625 30375 30375 3375

26

sech sech sech

sech sech sech sech

x x x

x x x x

O

T. Y. Wu Caltech

Amplitude vs. Amplitude vs. =k=k22hh22

T. Y. Wu Caltech

Integral convergenceIntegral convergence• Extra mass, momentum, impulse, circulation, Extra mass, momentum, impulse, circulation,

energyenergy

• StarStar 、、 McCowan IdentitiesMcCowan Identities

• Series solutions for integralsSeries solutions for integrals

10, , FM dx u dx

21 13 2, 1 ,p kI FM E F M E F FM

2 3 4 58 32 3328 129536 42668032 113287331843 9 675 23625 22325625 736745625

1000797919249128927854426122239379591916866575634274476325170686477166976801177615797909162229804450549179266793077422390

M

24

6720492492735385894775390625

25O

T. Y. Wu Caltech

Integral convergence (cont.)Integral convergence (cont.)

T. Y. Wu Caltech

Domb-Sykes PlotDomb-Sykes Plot

T. Y. Wu CaltechNear-source devastation of the 1946 Near-source devastation of the 1946 Tsunami in Aleutian IslandTsunami in Aleutian Island

T. Y. Wu Caltech

Tidal bore on the Qian Tang RiverTidal bore on the Qian Tang River

T. Y. Wu Caltech

若论水波非杭莫属颂若论水波非杭莫属颂西湖水波独漪涟西湖水波独漪涟钱塘惊涛冲破天钱塘惊涛冲破天薄雾烟锁池塘柳薄雾烟锁池塘柳露滴平湖晃南山露滴平湖晃南山

2006.7.1 2006.7.1 吴耀祖敬志吴耀祖敬志

ENDEND

T. Y. Wu Caltech

Water waves of finite depthWater waves of finite depth

T. Y. Wu Caltech

Contributing LiteratureContributing Literature• Boussinesq, J. 1871 Boussinesq, J. 1871 • Byatt-Smith, J.G.B. 1971, 1976 Byatt-Smith, J.G.B. 1971, 1976 • Chappelear, J.E. 1962 Chappelear, J.E. 1962 • Cole, J. D. 1968 Cole, J. D. 1968 • Daily, J. W. Stephan, S. C. 1952 Daily, J. W. Stephan, S. C. 1952 • Fenton, J. 1972 Fenton, J. 1972 • Friedrichs, K. O. Hyers, D. H. 1954 Friedrichs, K. O. Hyers, D. H. 1954 • Grant, M. Grant, M. • Grimshaw, R. H. J. 1971 Grimshaw, R. H. J. 1971 • Hui, G. W. H. 1988 Hui, G. W. H. 1988 • Korteweg, D. J. de Vries, G. 1895 Korteweg, D. J. de Vries, G. 1895 • Laitone, E. V. 1960 Laitone, E. V. 1960 • Lenau, C. W. 1966 Lenau, C. W. 1966 • Longuet-Higgins, M. S. Fenton, 1974, Longuet-Higgins, M. S. Fenton, 1974,

19771977• Longuet-Higgins, M. S. Fox, M. J. H. Longuet-Higgins, M. S. Fox, M. J. H.

1978, 1996 1978, 1996 • Longuet-Higgins, M. S. Tanaka M. Longuet-Higgins, M. S. Tanaka M.

1997 1997 • McCowan 1891 McCowan 1891 • Miles 1980 Miles 1980

• Milne-Thomson 1964, 1968 Milne-Thomson 1964, 1968 • Nekrasov, A. I. 1921 Nekrasov, A. I. 1921 • Nwogu, O. 1993 Nwogu, O. 1993 • Qu, W.-D. 2000 Qu, W.-D. 2000 • Rayleigh, Lord 1876 Rayleigh, Lord 1876 • Starr 1947 Starr 1947 • Stokes, G. G. 1880 Stokes, G. G. 1880 • Tanaka 1985, 1986 Tanaka 1985, 1986 • Teng, M. H. Wu, T. Y. 1992 Teng, M. H. Wu, T. Y. 1992 • Vanden-Broeck, J.-M., and Miloh, T. Vanden-Broeck, J.-M., and Miloh, T.

1995 1995 • Wehausen, J. V. Laitone, E. V. 1960 Wehausen, J. V. Laitone, E. V. 1960 • Wei, G., Kirby, J. T. 1995 Wei, G., Kirby, J. T. 1995 • Wei, G., Kirby, J. T., Grilli, S. T. Wei, G., Kirby, J. T., Grilli, S. T.

Subramanya, R. 1995 Subramanya, R. 1995 • Weidman, P. D. Maxworthy, T. 1978.Weidman, P. D. Maxworthy, T. 1978.• Williams, J.M. 1981 Williams, J.M. 1981 • Wu, T. Y. 1998, 2000 Wu, T. Y. 1998, 2000 • Wu, T.Y. Zhang, J. E. 1996.Wu, T.Y. Zhang, J. E. 1996.• Yamada, H. 1957.Yamada, H. 1957.• Zakharov, V. E. 1968 Zakharov, V. E. 1968