Simulation of wave propagation in linear thermoelastic media

Abstract

We have developed a numerical algorithm for simulation of wave propagation in linear thermoelastic media, based on a generalized Fourier law of heat transport in analogy with a Maxwell model of viscoelasticity. The wavefield is computed by using a grid method based on the Fourier differential operator and two time-integration algorithms to cross-check solutions. Because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods, first, a second-order time-splitting algorithm solves the unstable part analytically and a Runge-Kutta method the regular equations. Alternatively, a first-order explicit Crank-Nicolson algorithm yields more stable solutions for low values of the thermal conductivity. These time-stepping methods are second- and first-order accurate, respectively. The Fourier differential provides spectral accuracy in the calculation of the spatial derivatives. The model predicts three propagation modes, namely, a fast compressional or (elastic) P-wave, a slow thermal P diffusion/wave (the T-wave), having similar characteristics to the fast and slow P-waves of poroelasticity, respectively, and an S-wave. The thermal mode is diffusive for low values of the thermal conductivity and wave-like for high values of this property. Three velocities define the wavefront of the fast P-wave, i.e., the isothermal velocity in the uncoupled case, the adiabatic velocity at low frequencies, and a higher velocity at high frequencies.