Interchange of modal properties in the propagation of harmonic waves in heat-conducting materials

Abstract

A study is made of the secular equation governing the propagation of plane harmonic waves of small amplitude in a continuum which is able to conduct heat. This equation defines an algebraic function, called the modal function, whose regular branches specify the slownesses of the possible modes of harmonic wave propagation as functions of the frequency. At each extreme of the frequency range one mode is diffusive in type and the others wave-like, and we suppose here that there is a single wave-like mode which produces changes of temperature. In this case the mode which is diffusive in type at low frequencies is wave-like or diffusive in the high-frequency limit according as the thermoelastic coupling constant (a dimensionless measure of the strength of thermo-mechanical interaction in the continuum) does or does not exceed unity. This property is shown to have a simple interpretation in terms of a Riemann surface of the modal function. The results obtained are quite general, referring to principal longitudinal waves in a homogeneously deformed isotropic heat-conducting elastic material, to dilatational disturbances of a stress-free configuration of such a material, and to acoustic waves in a heat-conducting inviscid fluid.