Affiliation:
1. Department of Mathematics, University of Keele, Staffordshire ST5 5BG, UK
2. Department of Mechanical Engineering, Aalborg University, Pontoppidanstraede 101, 9220, Denmark
Abstract
This paper presents a method of analysing the dispersion relation and field shape of any type of 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. The finite products chosen must be consistent with the low-frequency, low-wavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge’s phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability theory.
Subject
General Physics and Astronomy,General Engineering,General Mathematics
Cited by
20 articles.
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