Heat Propagation in a One-Dimensional Harmonic Crystal on an Elastic Foundation

Abstract

A closed system of differential equations has been derived to describe thermal processes in a one-dimensional harmonic crystal on an elastic foundation. It is shown that the evolution of thermal perturbation in such a crystal is described by a discrete unsteady-state equation, a special case of which is the hyperbolic equation of ballistic heat conduction. This equation remains valid with negative stiffness of bonds between particles of the crystal in its entire stability range. The thermal perturbation front propagates with the maximum group velocity of mechanical waves. The propagation of a short-term thermal perturbation in the crystal on the elastic foundation is determined by the equation of ballistic thermal conductivity of the same type as in the crystal without an elastic foundation. The only parameter of this equation is the maximum group velocity (in absolute value), i.e., the maximum rate of energy propagation in the crystal on the elastic foundation. This quantity is proportional to the absolute value of the half-difference of the upper and lower cutoff frequencies. The rate of heat wave propagation in the crystal on the elastic foundation with positive stiffness is always lower than that in the crystal without an elastic foundation. The obtained equation is found to be valid both for positive stiffness values and for negative ones, for which the chain stability condition is satisfied. As an example, a dynamic problem of heat distribution is solved exactly for a parabolic initial temperature profile to model heating of a one-dimensional crystal on a foundation by a short laser pulse. Due to the dispersion of mechanical waves in the chain on the foundation, their group velocity depends on the wave number and the ratio of bond stiffnesses in the chain and the elastic foundation. The thermal front propagates with the maximum possible group velocity in the system, which depends only on this ratio.