Long surface waves incident on a submerged horizontal plate

Abstract

A train of surface gravity waves of wavelength λ in a channel of depth H is incident on a horizontal plate of length l that is submerged to a depth c. Under the assumption that both λ and l are large compared with H, the method of matched asymptotic expansions is used to show that, to first order, the reflexion coefficient R and the transmission coefficient T are given by \[ R = \chi \left\{\frac{\sigma l}{(gH)^{\frac{1}{2}}}\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}-2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\} \] and \[ T =\chi\left\{2i\left[\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\right]\right\} \] where \begin{eqnarray*} \chi &=& 1\left/ \left\{2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)+\frac{\sigma l}{b}\bigg(\frac{H}{g}\bigg)^{\frac{1}{2}}\bigg(1+\frac{c}{H}\bigg)\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}\right.\right.\\ &&\left. +2i\bigg(\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\}, \end{eqnarray*} σ is the angular frequency and g the acceleration due to gravity.