The finite product method in the theory of linear wave propagation

This paper presents a method of analysing the dispersion relation and field shape of any type of linear wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. Details of the method are presented for several non-trivial examples, that of symmetric/anti-symmetric elastic waves in a layer and in a thin plate. In each case, the method gives a sequence of polynomial approximations to the dispersion relation of remarkable accuracy over a broad range of frequencies and wave numbers. These polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are perfectly suited for computations.