TECHNICAL PAPERS
Jul 1, 1999

Modeling Nonlinear Dispersive Water Waves

Publication: Journal of Engineering Mechanics
Volume 125, Issue 7

Abstract

An expository review is given on various theories of modeling weakly to strongly nonlinear, dispersive, time-evolving, three-dimensional gravity-capillary waves on a layer of water. It is based on a new model that allows the nonlinear and dispersive effects to operate to the same full extent as in the Euler equations. Its relationships with some existing models are discussed. Various interesting phenomena will be illustrated with applications of these models and with an exposition on the salient features of nonlinear waves in wave-wave interactions and the related processes of transport of mass and energy.

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References

1.
Boussinesq, J. ( 1871). “Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire.” C.R. Acad. Sci., Paris, 72, 755–759 (in French).
2.
Choi, W. ( 1995). “Nonlinear evolution equations for two-dimensional surface waves in a fluid of finite depth.” J. Fluid Mech., 295, 381–394.
3.
Choi, W., and Camassa, R. ( 1996). “Weakly nonlinear internal waves in a two-fluid system.” J. Fluid Mech., 313, 83–103.
4.
Cooker, M. J., Weidman, P.D., and Bale, D. S. ( 1997). “Reflection of a high-amplitude solitary wave at a vertical wall.” J. Fluid Mech., 342, 141–158.
5.
Daily, J. W., and Stephan, S. C. ( 1952). “The solitary wave.” Proc., 3rd Conf. on Coast. Engrg., ASCE, Reston, Va. 13–30.
6.
Ertekin, R. C., Webster, W. C., and Wehausen, J. V. ( 1986). “Waves caused by a moving disturbance in a shallow channel of finite width.” J. Fluid Mech., 169, 275–292.
7.
Green, A. E., and Naghdi, P. M. ( 1976). “A derivation of equations for wave propagation in water of variable depth.” J. Fluid Mech., 78, 237–246.
8.
Grimshaw, R. H. J., and Smyth, N. F. ( 1986). “Resonant flow of a stratified fluid over topography.” J. Fluid Mech., 169, 429–464.
9.
Hirota, R. ( 1973). “Exact N-soliton solutions of the wave equations of long waves in shallow-water and in nonlinear lattices.” J. Math Phys., 14(7), 810–814.
10.
Huang, D. B., Sibul, O. J., Webster, W. C., Webausen, J. V., Wu, D. M., and Wu, T. Y. ( 1982). “Ships moving in the transcritical range.” Proc., Conf. Behavior of Ships in Restricted Waters, Bulgarian Ship Hydrodynamics Center, Varna, Bulgaria, 2, 26/1–26/10.
11.
Kirby, J. T., and Vengayil, P. ( 1988). “Nonresonant and resonant reflection of long waves in varying channels.” J. Geophy. Res., 93, 10782–10796.
12.
Korteweg, D. J., and de Vries, G. ( 1895). “On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves.” Philos. Mag., 39, 422–443.
13.
Lax, P. D. ( 1968). “Integrals of nonlinear equations of evolution and solitary waves.” Comm. Pure Appl. Math., 21, 467–490.
14.
Lee, S. J., Yates, G. T., and Wu, T. Y. ( 1989). “Experiments and analysis of upstream-advancing solitary waves generated by moving disturbances.” J. Fluid Mech., 199, 569–593.
15.
Madsen, P. A., Banijamali, B., Schaffer, H. A., and Sorensen, O. R. ( 1996). “Boussinesq type equations with high accuracy in dispersion and nonlinearity.” Proc., 25th Int. Conf. on Coast. Engrg., ASCE, Reston, Va., 95–108.
16.
Madsen, P. A., Murray, R., and Sorensen, O. R. ( 1991). “A new form of the Boussinesq equations with improved linear dispersion characteristics, part 1.” Coast. Engrg., 15, 371–388.
17.
Miles, J. W. ( 1979). “On the Korteweg–de Vries equation for a gradually varying channel.” J. Fluid Mech., 91, 181–190.
18.
Nwogu, O. (1993). “Alternative form of Boussinesq equations for nearshore wave propagation.”J. Wtrwy., Port, Coast., and Oc. Engrg., ASCE, 119(6), 618–638.
19.
Olver, P. J. ( 1993). Applications of Lie groups to differential equations. Springer-Verlag, New York.
20.
Peregrine, D. H. ( 1967). “Long waves on a beach.” J. Fluid Mech., 27, 815–827.
21.
Schaffer, H. A., and Madsen, P. A. ( 1995). “Further enhancements of Boussinesq-type equations.” Coast. Engrg., 26, 1–14.
22.
Shen, M. C. ( 1969). “Asymptotic theory of unsteady three-dimensional waves in a channel of arbitrary cross section.” SIAM J. Appl. Math., 17, 260–271.
23.
Shuto, N. ( 1974). “Nonlinear long waves in a channel of variable section.” Coast. Engrg. Japan, Tokyo, 17, 1–12.
24.
Teng, M. H., and Wu, T. Y. ( 1992). “Nonlinear water waves in channels of arbitrary shape.” J. Fluid Mech., 242, 211–233.
25.
Teng, M. H., and Wu, T. Y. ( 1994). “Evolution of long water waves in variable channels.” J. Fluid Mech., 266, 303–317.
26.
Teng, M. H., and Wu, T. Y. ( 1997). “Effects of channel cross-sectional geometry on long wave generation and propagation.” Phys. Fluids, 9(11), 3368–3377.
27.
Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R. ( 1995). “A fully nonlinear Boussinesq model for surface waves, part 1.” J. Fluid Mech., 294, 71–92.
28.
Weidman, P. D., and Maxworthy, T. ( 1978). “Experiments on strong interactions between solitary waves.” J. Fluid Mech., 85, 417–431.
29.
Whitham, G. B. ( 1974). Linear and nonlinear waves. Wiley, New York.
30.
Wu, D. M., and Wu, T. Y. ( 1982). “Three-dimensional nonlinear long waves due to moving surface pressure.” Proc., 14th Symp. on Naval Hydrodyn., National Academy Press, Washington, D.C., 102–125.
31.
Wu, T. Y. ( 1979). “On tsunami propagation: evaluation of existing models.” Tsunamis—Proc., Nat. Sci. Found. Workshop, Tetra Tech Inc., Pasadena, Calif., 110–149.
32.
Wu, T. Y. (1981). “Long waves in ocean and coastal waters.”J. Engrg. Mech. Div., ASCE, 107, 501–522.
33.
Wu, T. Y. ( 1994). “A bidirectional long-wave model.” Methods and Appl. of Anal., 1(1), 108–117.
34.
Wu, T. Y. ( 1995). “Bidirectional soliton street—the inaugural Pei-Yuan Chou Memorial Lecture (6th Asian Congress of Fluid Mechanics, 21–26 May 1995, Singapore).” Acta Mech. Sinica, 11, 289–306.
35.
Wu, T. Y. ( 1997). “On modeling nonlinear water waves.” Proc., 12th Int. Workshop on Water Waves and Floating Bodies, Marseilles, France, 321–324.
36.
Wu, T. Y. ( 1998a). “Nonlinear waves and solitons in water.” Physica D, 123, 48–63.
37.
Wu, T. Y. ( 1998b). “On fully nonlinear water waves.” Proc., 3rd Int. Conf. on Nonlinear Mech., Wei-Zang Chien, ed., Shanghai University Press, Shanghai, 119–124.
38.
Wu, T. Y. ( 1998c). “Modeling and computing nonlinear dispersive water waves.” Proc., 3rd Int. Conf. on Hydrodyn., H. Kim, H. S. Lee, and S. J. Lee, eds., UIAM Publ., Seoul, Korea, 12–15.
39.
Wu, T. Y., and Zhang, J. E. ( 1996a). “On modeling nonlinear long waves.” Mathematics is for solving problems: a volume in honor of Julian Cole on his 70th birthday, Society for Industrial and Applied Mathematics, Philadelphia, 233–247.
40.
Wu, T. Y., and Zhang, J. E. ( 1996b). “Mass and energy transfer between unidirectional interacting solitons (a tribute to Prof. C. C. Yu in honor of his 80th anniversary).” Chinese J. Mech., 12(1), 79–84.
41.
Yih, C. S., and Wu, T. Y. ( 1995). “General solution for interaction of solitary waves including head-on collisions.” Acta Mech. Sinica, 11, 193–199.
42.
Zabusky, N. J. ( 1967). “A synergetic approach to problems of nonlinear dispersive wave propagation and interaction.” Nonlinear partial differential equations. Academic Press, New York.
43.
Zakharov, V. E. ( 1968). “Stability of periodic waves of finite amplitude on the surface of a deep fluid.” Z. Prik. Mek. Tekh. Fiziki, 9, 86–94.
44.
Zhu, J. L., Wu, T. Y., and Yates, G. T. ( 1986). “Internal solitary waves generated by moving disturbances.” Proc., 3rd Int. Symp. Stratified Flows, E. J. List and G. H. Jirka, eds., ASCE, Reston, Va., 74–83.

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Go to Journal of Engineering Mechanics
Journal of Engineering Mechanics
Volume 125Issue 7July 1999
Pages: 747 - 755

History

Received: Sep 24, 1998
Published online: Jul 1, 1999
Published in print: Jul 1999

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Authors

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Theodore Yaotsu Wu
Prof. Emeritus of Engrg. Sci., California Inst. of Technol., Pasadena, CA 91125.

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