Propagation of Rayleigh waves along isothermal and insulated boundaries

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.