The reflection of a solitary wave by a vertical wall

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Abstract
In this paper we consider the head-on collision of two equal solitary waves this being equivalent, in the absence of viscosity to the reflection of one solitary wave by a vertical wall. The perturbation expansion of the Euler equations, which lead to the Boussinesq equation at lowest order, is recast to obtain two weakly coupled KdV equations. We show analytically that the amplitude of the solitary wave after reflection is reduced. This change in amplitude is shown to be fifth order in ε, the amplitude of the wave. It is also shown that the experimentally observed transient loss of amplitude can be explained by the presence of the third-order dispersive tail.