2006 中国数学科学 与教育发展论坛

1. 2006 中国数学科学与教育发展论坛 如 此 饶 趣 神 奇 的 水 波 Such interesting and marvelous water waves 吴耀祖 美国加州理工学院 浙江大学数学科学研究中心2006 年 7 月 1 日

2. Capillary waves

3. Gravity waves

4. Water waves of depth h

5. Key parameters of water waves

6. Dispersion relations on linear theory

7. Water particle pathlines in waters

8. Wave packet dispersion

9. Wave packet group dispersion

10. Generation of ship waves

12. Ship wave in shallower waters

13. Ship wave pattern

15. John Scott Russell’s 1834 chance discovery

17. Significances of the KdV evolution equation

18. Grand discovery of the remarkable soliton

19. Remarkable properties of solitons

20. Forced forward-radiation of solitons

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

22. Agreement between theory and experiment

23. Comparison between experiment and theory

24. Experimental and computational views of forced radiation of solitary waves

25. Hydrodynamic instabilities of resonantly forced solitons • Camassa, R. and Wu, T. Y. (1991a) Phil. Trans. R. Soc. Lond. A337, 429-466 • --- (1991b) Physica D51, 295-307 • Having profound discussions with Cambridge Univerisity Lucasian Professor Sir James Lighthill

26. Part IReflection for Insight on Solitary Waves of Arbitrary Height Wu, T. Y., Kao, J. & Zhang, J. V. Acta Mech. Sinica 21 (1), 1-15, 2005

27. Sir G.G.Stokes (1880) on the solitary wave outskirts a • Stokes: This relation is exact!

28. Reflective Queries • 1. What else? • 2. Can linear theory hold for low waves of diminishing amplitude?

29. F2 = 1.15 =0.196541 tan() F2 

30. Reflections on • Boussinesq-Rayleigh: • Asymptotic representation:

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

32. Formulation and analysis

33. Unified intrinsic functional expansion (UIFE) theory • (i) First establish a UIFE expansion for () in terms of a set of intrinsic component functions (ICF), analytic in , to represent all the intrinsic wave properties in the entire flow field. • (ii) The unknown coefficients in the UIFE expansion are determined under the given conditions by minimizing G and B: • (iii) The minimization of E2 is implemented by stepwise optimization.

34. The highest solitary wave • The UIFE expansion for w(z)=t +i q 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.

35. A dwarf solitary wave -- with F =1.005 (m = 0.054873) • Solution by UIFE-Method-I • The UIFE-Method-I solution so obtained consists of eleven modes ofbut with only one mode of a11

36. Reflections on • Boussinesq-Rayleigh: • Asymptotic representation:

37. Variations of () and F() – by UIFE – I-II

38. Graphical presentation of numerical solutions Full range: 0<<0.8332 Extreme Waves 0.68<<0.8332 2nd extreme – local minimum min=0.8310643 Fmin=1.290850 The fastest wave: fst=0.7959034 Ffst=1.294211

39. Sample profiles of extreme solitary waves •  = hst (0.8331990), 0.822279, 0.811386, 0.796952; • F = F hst (1.290890), 1.291738, 1.293358, 1.294208.

40. Wave profiles evaluated by UIFE-I and II •  = hst (0.8331990), 0.758245, 0.583690, 0.407430, 0.212284; • F = F hst (1.290890), 1.29092, 1.24470, 1.18098, 1.09979.

41. Integral properties of solitary waves

42. Comparison with three lower-order theories • First three lower-order theories: • Kortweg/de Vries 1895 – 1st order; • Laitone 1960 – 2nd_; • Chappelear 1962; Grimshaw 1971 – 3rd_

43. Unified Perturbation Expansion For Solitary Waves Solitary Wave Theory – Part 2 Wu, T.Y, Wang, X.L. & Qu, W.D. Acta Mech. Sinica 21(12) 2005

44. Motivated reflection and queries • Q2: Does the Euler model possess a perturbation expansion which is convergent? • Effects due to change in base parameter on • a. solution accuracy b. rate of series convergence

45. Literature contributions • Boussinesq, J. (1871); Lord Rayleigh (1876); KdV (1895) • Laitone, E.V. (1960) series expansion to O( ) • Chappelear, J.E. (1962) -- series expansion to O( ) • Fenton, J. (1972) numerical solution to O( ) • Longuet-Higgins, M.S. & Fenton, J. (1972) • Wu, T.Y. (1998) --- adopting base parameter • Wu, T.Y. (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( )

46. Guiding principles • 1. Simple and efficient formulation be sought for exact unique solution with as few unknowns and to as high order of expansion as attainable. • 2. Make comparisons between basic parameters for their effects on (a) solution accuracy and (b) rate of series convergence for parameters:

47. Asymptotic expansion High order equations A unified perturbation expansion theory

48. Asymptotic reductive perturbation scheme • 1. Reductivity results from u”n in Pn+1=Qn+1 . • 2. Reductivity chain-links all orders n=1,2, …, N. • 3. Simplicity in using  with (u0, )  (A) solves bn= - Cn1 before n; (B) avoids iterations needed otherwise

49. Exact higher-order theory of O( 18 ) • Cf. Wu, T.Y., Wang, X.L. & Qu, W.D., ACTA Mech. Sinica 21 (2005) • With application of Mathematica 5.0

50. Exact theory of O( 18 )