Reflection and refraction of a transient temperature field at a plane interface using Cagniard-de Hoop approach

An instantaneous line heat source located in the medium consisting of two half-spaces with different thermal properties is considered. Green's functions for the temperature field are derived using the Laplace and Fourier transforms in time and space and their inverting by the Cagniard-de Hoop technique known in elastodynamics. The characteristic feature of the proposed approach consists in the application of the Cagniard-de Hoop method to the transient heat conduction problem. The idea is suggested by the fact that the Laplace transform in time reduces the heat conduction equation to a Helmholtz equation, as for the wave propagation. Derived solutions exhibit some wave properties. First, the temperature field is decomposed into the source field and the reflected field in one half-space and the transmitted field in the other. Second, the laws of reflection and refraction can be deduced for the rays of the temperature field. In this connection the ray concept is briefly discussed. It is shown that the rays, introduced in such a way that they are consistent with Snell's law do not represent the directions of heat flux in the medium. Numerical computations of the temperature field as well as diagrams of rays and streamlines of the temperature field are presented.