Dispersion curves for thermomechanical waves

Abstract

When mechanical deformation and heat conduction are both accounted for, two coupled partial differential equations (PDEs) describe wave propagation in a medium. For one‐dimensional wave propagation in a medium without mechanical dissipation, these PDEs yield a dispersion relation of complex quadratic form, which shows that two kinds of waves can propagate. Each wave has a mechanical and a thermal component. The speeds and the attenuations of such waves are illustrated in two cases. First, it is shown that the “classical” form of the thermomechanical PDEs provides an interesting but nonphysical description of wave propagation. Second, a Vernotte thermal relaxation time is introduced into the PDEs. When the value of the Vernotte parameter is set by applying the Debye theory of specific heats in a rudimentary way, it is shown that a physically plausible description of wave propagation can be obtained. In this latter connection, the consequences of using a “phonon‐gas” model of a thermomechanical medium is considered briefly. [Work supported by ONR.]When mechanical deformation and heat conduction are both accounted for, two coupled partial differential equations (PDEs) describe wave propagation in a medium. For one‐dimensional wave propagation in a medium without mechanical dissipation, these PDEs yield a dispersion relation of complex quadratic form, which shows that two kinds of waves can propagate. Each wave has a mechanical and a thermal component. The speeds and the attenuations of such waves are illustrated in two cases. First, it is shown that the “classical” form of the thermomechanical PDEs provides an interesting but nonphysical description of wave propagation. Second, a Vernotte thermal relaxation time is introduced into the PDEs. When the value of the Vernotte parameter is set by applying the Debye theory of specific heats in a rudimentary way, it is shown that a physically plausible description of wave propagation can be obtained. In this latter connection, the consequences of using a “phonon‐gas” model of a thermomechanical medium is co...