Shock waves and second sound in a rigid heat conductor: A critical temperature for NaF and Bi

Abstract

We study shock waves propagating in a rigid heat conductor at low temperature using a generalized Maxwell-Cattaneo equation. The existence of a critical temperature \ensuremath{\theta}\ifmmode \tilde{}\else \~{}\fi{}, characteristic of the material, for which the structure of the shock changes is proved. When the unperturbed temperature ${\mathrm{\ensuremath{\theta}}}_{0}$ is less than \ensuremath{\theta}\ifmmode \tilde{}\else \~{}\fi{} the temperature ${\mathrm{\ensuremath{\theta}}}_{1}$ behind the shock wave front is such that ${\mathrm{\ensuremath{\theta}}}_{1}$>${\mathrm{\ensuremath{\theta}}}_{0}$ (hot shock), and, vice versa, if ${\mathrm{\ensuremath{\theta}}}_{0}$>\ensuremath{\theta}\ifmmode \tilde{}\else \~{}\fi{} then ${\mathrm{\ensuremath{\theta}}}_{1}$${\mathrm{\ensuremath{\theta}}}_{0}$ (cold shock. We find \ensuremath{\theta}\ifmmode \tilde{}\else \~{}\fi{}=15.36 K for NaF and \ensuremath{\theta}\ifmmode \tilde{}\else \~{}\fi{}=3.38 K for Bi. These temperatures are very close to the values at which the second sound was identified experimentally in pure NaF and Bi crystals.