Journal of Fluid Mechanics

Resonantly interacting solitary waves

John W.  Miles a1
a1 Institute of Geophysics and Planetary Physics, University of California, La Jolla

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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 [double less-than sign] 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.

(Published Online April 11 2006)
(Received August 12 1976)