Thermal wave reflection and refraction at a plane interface: Two-dimensional geometry

A theoretical study of thermal waves generated by a line time-periodic heat source in a medium composed of two half-spaces of different thermal properties is presented. The standard method of solution to this problem consists in the application of a Fourier transform with respect to the variable parallel to the interface, which is then inverted numerically. We propose to derive modified solutions using the deformation of the integration contour in the complex plane of the transform variable. The modification of the contour is adopted from the Cagniard-de Hoop method which is used in elastic wave propagation. The main advantage of the obtained solutions is the feasibility of their physical interpretation. In particular, it is shown that geometrical ray propagation and Snell's law can be obtained in the high-frequency approximation, but not as a general rule. It is also found that when the heat source is located in a less heat-conductive medium the structure of the temperature fields changes dramatically beyond the critical angle of incidence. This suggests a thermal analogy with the total reflection phenomenon. The analytical formulas derived for the temperature fields and the heat fluxes allow easy numerical computations, and can be used for the interpretation of experimental data.