The usual phonon Boltzmann equation is solved by using two mean relaxation times, ${\tau }_N$ for normal and ${\tau }_R$ for resistive processes. For a Debye solid with three polarizations, an explicit expression for the Fourier transform of the local temperature in a heat-pulse experiment is calculated. It describes hydrodynamic phenomena for $\Omega \tau \ll 1$, such as second sound and diffusive heat conduction, and heat transport by ballistic phonons for $\Omega \tau \gg 1$. In the intermediate regime, $\Omega \tau \approx 1$, we find the following results: a second-sound wave with wave vector $\overrightarrow{\mathrm{Q}}$ can only propagate if $Q{\tau }_N$ and $Q{\tau }_R$ are smaller than certain critical values, ${\left(Q{\tau }_N\right)}_c$ and ${\left(Q{\tau }_R\right)}_c$, i.e., for $T\ge T_c$, assuming the usual monotonic $T$ dependence of ${\tau }_N$ and ${\tau }_R$. The velocity $C_2$ of second sound strongly depends on these relaxation times. Its maximum value, occurring at $T=T_c$, is the larger the smaller the ratio $\frac{{\left({\tau }_N\right)}_c}{{\left({\tau }_R\right)}_c}$. Then $C_2$ decreases with rising $T$ and finally goes to zero for $\Omega {\tau }_R\lesssim 1$.

• Received 19 January 1973

DOI:https://doi.org/10.1103/PhysRevB.8.1669