The Scattering Amplitudes and Cross Sections in the Theory of Thermoelasticity

Abstract

A plane thermoelastic wave, propagating in an isotropic and homogeneous, medium in the absence of body forces and heat sources, is scattered by a smooth, convex and bounded three-dimensional body. The body could be a rigid scatterer at constant temperature, a rigid scatterer at thermal insulation, a cavity at constant temperature, or a cavity at thermal insulation, while in all cases the Kupradze’s radiation conditions are assumed to hold at infinity. The second law of thermodynamics imposes an attenuation of the elastothermal and the thermoelastic waves, which is reflected upon the lack of symmetry of the unified differential operator governing the Biot theory of dynamic thermoelasticity. Normalized spherical scattering amplitudes are introduced for the displacement as well as the temperature field, via asymptotic analysis of appropriate integral representations. With the exception of the scattering amplitudes corresponding to the transverse elastic waves, all the other scattering amplitudes involve attenuation factors. The classical definition of the scattering cross section reduces, in practice, the thermoelastic scattering process to a consideration of the transverse incident and the transverse scattered wave alone. However, more detailed analysis demands the introduction of local units for measuring the energies carried by the elastothermal and the thermoelastic waves, which give rise to five types of scattering cross sections. They measure the total energy for each type of wave, scattered by the body, by the standards of the corresponding incident wave in the forward direction. Throughout this work the thermoelastic coupling is indicated by the well known dimensionless constant $\varepsilon $. The corresponding scattering theory for the classical dynamic elasticity is recovered in the limiting case as $\varepsilon \to 0 + $.