## Abstract

In this paper the effects of heat conduction upon the propagation of Rayleigh surface waves in a semi-infinite elastic solid are studied theoretically in two special cases: (i) when the surface of the solid is maintained at constant temperature (case 1); and (ii) when the surface is thermally insulated (case 2). The investigation is carried out within the framework of the linear theory of thermoelasticity, a principal objective being the clarification of the relation between so-called thermoelastic Rayleigh waves and the Rayleigh waves of classical elastokinetics. The secular equation for thermoelastic Rayleigh waves is shown to define a many-valued algebraic function *μ*(*X* ) (*X* being a dimensionless frequency) the branches of which represent possible modes of surface wave propagation. Two different types of surface mode are recognized: *E*-modes, which resemble classical Rayleigh waves but are subject to damping and dispersion; and *T*-modes, which are essentially diffusive in character. Necessary and sufficient conditions for the existence of a surface wave are formulated, and questions of the existence and multiplicity of Rayleigh *E*- and T-modes in particular situations are resolved by submitting the branches of *μ*(*X*) to these requirements. The algebraic function *μ*(*X* ) has singular points at *X* = 0 and *X* = ∞, and approximations, valid at sufficiently low or at sufficiently high frequencies, to the speed of propagation *v* and the attenuation coefficient *q* of a given surface mode are obtainable from series representations of the appropriate branch of *μ*(*X*) in neighbourhoods of these singularities. The singular point *X* = 0 is associated with adiabatic deformations of the solid, and hence with classical Rayleigh waves, and the singularity at *X* = ∞ with isothermal deformations. Particular attention is devoted to the Rayleigh *E*-modes and the main conclusions reached are as follows. In case 1 there exists a single *E*-mode (mode 2) at low frequencies and two distinct *E*-modes (modes 1 and 2) at high frequencies. For mode 2, *v*/*v*_{R} = 1 + *O*(*X*^{½}), *q* = *O*(*X*^{3/2}) *X*distinct 0 (*v*_{R} being the speed of propagation of classical Rayleigh waves), and for both modes *v* and *q* approach finite limits as *X* → ∞. In case 2 the converse situation applies, there being two distinct *E*-modes (modes 1 and 2) at low frequencies and only one (mode 1) at high frequencies. For both modes, *v*/*v*_{R} = 1 + *O*(*X*^{3/2}), *q* = *O*(*X*^{2}) as *X* → 0, and for mode 1, *v* and *q* approach finite limits as *X* → ∞. Detailed numerical results referring to a medium of worked pure copper at a reference temperature of 20 °C are given. In particular the frequency dependence of the speeds of propagation and attenuation coefficients of the various *E*-modes are exhibited, and the frequencies at which mode 1 appears in case 1 and at which mode 2 disappears in case 2 are determined.

## Footnotes

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- Received November 28, 1963.

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