A Wronskian method for elastic waves propagating along a tube

Abstract

A technique involving the higher Wronskians of a differential equation is presented for analysing the dispersion relation in a class of wave propagation problems. The technique shows that the complicated transcendental-function expressions which occur in series expansions of the dispersion function can, remarkably, be simplified to low-order polynomials exactly, with explicit coefficients which we determine. Hence simple but high-order expansions exist which apply beyond the frequency and wavenumber range of widely used approximations based on kinematic hypotheses. The new expansions are hypothesis-free, in that they are derived rigorously from the governing equations, without approximation. Full details are presented for axisymmetric elastic waves propagating along a tube, for which stretching and bending waves are coupled. New approximate dispersion relations are obtained, and their high accuracy confirmed by comparison with the results of numerical computations. The weak coupling limit is given particular attention, and shown to have a wide range of validity, extending well into the range of strong coupling.

A Poisson scaling approach to backward wave propagation in a tube

“…This paper shows that the method of Wronskian identities developed in [1] is powerful enough to describe backward wave propagation in practically unlimited analytical detail, without any loss of mathematical rigour. The crucial point is that the series expansions we have presented, including all displayed coefficients, are exact, in that they are consequences of the dispersion relation (2.8), derived from the linear elastic equations without any kinematic assumptions about the shape of the wave field.…”

“…Here, we give the definitions in [1] required for the dispersion relation. The elastic displacement satisfies Navier’s equation with wave speeds c1={1−νfalse(1+νfalse)false(1−2νfalse)}1/2c01emand1emc2=c0falsefalse{2false(1+νfalse)falsefalse}1/2,in which ν is Poisson’s ratio and c0=false(E/ρfalse)1/2, where E is Young’s modulus and ρ is the density.…”

“…These polynomials are exact; that is, they are not truncations of infinite series, and they may rapidly be determined from (2.8). This is the original contribution of [1]. The coefficients of the polynomials contain ν as a parameter, but in a simple way: each coefficient is either an integer or a ratio of polynomials in ν with integer coefficients.…”

“…Here, we give the definitions in [1] required for the dispersion relation. The elastic displacement satisfies Navier's equation with wave speeds…”

“…Regarded as a function of ω and k, this is the dispersion relation for the problem we are considering. Details of the steps omitted here may be found in [1].…”