Cattaneo–Vernotte equation

Remarks

Here $\alpha$ is the thermal diffusivity. To derive the second equation, differentiate the energy balance in $t$ and use $\tau \, \partial_t \mathbf{q} = -k \nabla T - \mathbf{q}$. The temperature equation is hyperbolic, and its fronts move with speed $\sqrt{\alpha / \tau}$. As $\tau \to 0$ one recovers Fourier's law and the heat equation.

Example: in one dimension, try $T = e^{\lambda t} \cos(\kappa x)$ with $\kappa > 0$. It is a solution exactly when $\tau \lambda^2 + \lambda + \alpha \kappa^2 = 0$. If $4 \tau \alpha \kappa^2 > 1$, the roots $\lambda$ are complex, so the profile oscillates in time while it decays.

Non-example: the Fourier law $\mathbf{q} = -k \nabla T$ is the case $\tau = 0$, which is excluded. For it, the same trial function gives only $\lambda = -\alpha \kappa^2$, a pure decay without oscillation.