The reflexion of long waves at a step

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Abstract
A train of long waves travelling along a canal of depth h2 encounters a vertical step at which the depth changes to h1. The wave-length is much greater than h1 or h2. The classical Long-Wave theory where vertical acceleration is neglected is clearly inapplicable, although a treatment has been given by Lamb. In the present paper the problem is treated rigorously by the linearized theory of surface waves. A singular Fredholm integral equation of the first kind is obtained for the horizontal velocity above the step and is converted into a regular equation of the second kind with a kernel which tends to zero as the wave-length tends to infinity. To achieve this we first take the formal limit λ = ∞ in the integral equation of the first kind. This corresponds to a fluid motion between rigid boundaries which can be solved explicitly by a conformal transformation. In this way we obtain the inverse operator corresponding to λ = ∞, which is then applied to the original equation where λ < ∞. An equation of the second kind results which has a kernel tending to zero as λ tends to infinity and which is soluble by iteration for large λ. Although the Long-Wave method of Lamb fails to describe the details of the motion correctly, his predicted reflexion coefficient appears as a first approximation.