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Resonantly interacting solitary waves

  • John W. Miles (a1)
Abstract

Resonant (phase-locked) interactions among three obliquely oriented solitary waves are studied. It is shown that such interactions are associated with the parametric end points of the singular regime for interactions between two solitary waves. The latter include regular reflexion at a rigid wall, which is impossible for ϕi < (3α)½ (ϕ = angle of incidence, α = amplitude/depth [Lt ] 1), and it is shown that the observed phenomenon of ‘Mach reflexion’ can be described as a resonant interaction in this regime. The run-up at the wall is calculated as a function of ϕi/(3α)½ and is found to have a maximum value of 4αd for ϕi = (3α)½. This same resonant interaction also describes diffraction of a solitary wave at a corner of internal angle π − ψi, −(3α)½, and suggests that a solitary wave cannot turn through an angle in excess of (3α)½ at a convex corner without separating or otherwise losing its identity.

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References
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Chen, T. C. 1961 Experimental study on the solitary wave reflexion along a straight sloped wall at oblique angle of incidence. U.S. Beach Erosion Board Tech. Memo. no. 124.
Kaup, D. J. 1976 The three-wave interaction – a nondispersive phenomenon. Studies in Appl. Math. 55, 944.
Miles, J. W. 1977 Obliquely interacting solitary waves. J. Fluid Mech. 79, 157169.
Perroud, P. H. 1957 The solitary wave reflection along a straight vertical wall at oblique incidence. Ph.D. thesis. University of California, Berkeley.
Wiegel, R. L. 1964a Oceanographical Engineering. Prentice-Hall.
Wiegel, R. L. 1964b Water wave equivalent of Mach reflection. Proc. 9th Conf. Coastal Engng, A.S.C.E. chap. 6, pp. 82102.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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