General Analytical Solution to the Problem of Propagation of Thermal Oscillations in Semi-bounded Solids of Various Shapes

This paper presents a general analytical solution to the problem of propagation of thermal oscillations in semi-bounded bodies. The obtained results are useful for calculating and studying heat transfer during heat treatment of products, heat accumulators, ground heat exchangers, and small celestial bodies, ranging from comets and asteroids to completely solid planets. The purpose of the study is to obtain the thermal characteristics of thermal waves propagating in the body. A statement of the problem of steady oscillations in semi-bounded solids is presented. Two analytical solutions corresponding to a rectangular coordinate system are shown, and both of them could be considered as a general solution for any coordinate system. The proposed solution made it possible to express in real form the formulas of dependences on the longitudinal coordinate of the amplitude and phase of temperature oscillations. An asymptotic analysis is carried out, which shows the characteristic features and properties of the obtained solution. The calculated dependencies for some regimes are presented in order to understand the nature of heat transfer.