The dispersion relation and eigenfunction expansions for water waves in a porous structure

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The Sollitt-and-Cross model of water-wave motion in a porous structure involves a free-surface condition which contains a complex parameter. This leads to two particular difficulties when this model is used in conjunction with eigenfunction expansion techniques. First of all the roots of the dispersion relation are themselves complex and therefore difficult to locate by standard numerical methods. Secondly, the vertical eigenfunction problem is not self-adjoint and standard expansion theorems do not apply. In this paper it is shown how these two difficulties may be resolved with the aid of the theories of, respectively, complex variables and non-self-adjoint differential operators. In particular, a method is described that allows the explicit calculation of the roots of the dispersion relation, and the appropriate expansion theorem is given.

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