Implications of a nonlinearity in the theory of second sound in solids.

Abstract

The phenomenological relations usually employed to describe second sound in pure nonmetallic solids at temperatures \ensuremath{\theta} near that at which the thermal conductivity attains its maximum value were recently found to imply a quadratic dependence of the internal energy density e on the magnitude of the heat flux q, i.e., e=${e}_{0}$(\ensuremath{\theta})+a(\ensuremath{\theta})${q}^{2}$. The coefficient a(\ensuremath{\theta}) can be calculated from measurements of the temperature dependence of the speed \^U(\ensuremath{\theta}) of second-sound pulses in media for which the unperturbed temperature field is uniform. The studies of second-sound pulses in NaF crystals by Jackson, Walker, and McNelly and in Bi crystals by Narayanamurti and Dynes yield a(\ensuremath{\theta})>0 and da(\ensuremath{\theta})/d\ensuremath{\theta}0. The theory of pulse propagation along temperature gradients is examined here in detail. For a(\ensuremath{\theta})>0 the theory implies that a small pulse propagating in a body conducting heat will travel more slowly in the direction of heat flow than in the opposite direction. The magnitude of the effect is estimated for NaF and Bi crystals.