On the oblique interaction of a large and a small solitary wave

Abstract

The steady oblique interaction of two solitary waves on the surface of water of constant depth is considered. One wave is taken to be of arbitrary (large) amplitude and the other is small with a (non-dimensional) amplitude measured by the parameter ε. A solution is sought as an asymptotic expansion, based on ε, that assumes that in some region the solution is the sum of the two waves plus interaction terms. It is shown that this expansion is not uniformly valid close to a critical angle. This angle varies from zero (parallel waves) up to about 63°, as the amplitude of the larger wave increases from infinitesimally small to the largest-possible solitary wave. I n the limit of two small waves, the details agree precisely with the results obtained by Miles (1977a).

When the angle between the two waves is dose to the critical angle, for a given large wave, an alternative asymptotic expansion is required. In this strong-interaction case, the dominant term is just the large wave but with a phase shift that is an arbitrary function of the characteristic variable associated with the small wave. This function is determined by matching to an appropriate far field, and it turns out to be proportional to the logarithm of a hypergeometric function (which itself can be expressed in terms of the associated Legendre function P_v^−μ). The phase shift is then well-defined (finite, real) provided the angle is not very close to the critical value. When this occurs the phase shift can be infinite at a specific angle (which corresponds to the case |μ| = v > 0), and even closer to the critical angle (|μ| < v) the phase shift is undefined (no longer real). A real solution for the wave profile is still possible if negative amplitudes are allowed, but the resulting solution is unacceptable since the surface is then not undis- turbed at infinity. It is shown that the criterion |μ| < v matches exactly (for two small waves) with Miles’ criterion for the non-existence of a regular reflection.

For the strong interaction (and |μ| ≥ v i t is argued that the small wave cannot penetrate the large wave unchanged. The large wave suffers significant distortion (bending) in the interaction, but the small wave, if it penetrates at all, must have an amplitude O(ε). This is the main aspect of the problem which cannot be completely determined using the present methods. The difficulty can be traced to the large solitary wave which is not known in closed form: only the exponential behaviour in the ‘tail’ is used explicitly in this work.