We present a solution to the water-wave interaction with a submerged elastic plate of negligible thickness by the eigenfunction-matching method. The eigenfunction expansion depends on the solution of a special dispersion equation for a submerged elastic plate and this is discussed in detail. We show how the solution can be calculated for the case of normal incidence on a semi-infinite plate in two spatial dimensions and then extend this solution to obliquely incident waves, to a plate of finite length and to a circular finite plate in three dimensions. Numerical calculations showing various properties of the solutions are presented and a near-orthogonality relation for the eigenfunctions is used to derive an energy-balance relation.

We present solutions to both trifurcated and pentafurcated spaced waveguides using the mode matching (or eigenfunction expansion) method. While the trifurcated problem with mean fluid flow has been solved previously using the Wiener-Hopf technique, we solve this problem to validate and demonstrate our method. We then show how we can easily generalize the method to the pentafurcated problem that has not been solved previously. We observe that mode matching method is easier to derive and generalize than the Wiener-Hopf technique. We also investigate the numerical solution in detail for various geometries to model practical exhaust systems.

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