Second sound

Remarks

In superfluid helium-4 below the lambda point, the two-fluid model of Landau gives the speed $u_2$ of second sound by $$u_2^2 = \frac{\rho_s}{\rho_n} \, \frac{T s^2}{c}.$$ Here $\rho_s$ and $\rho_n$ are the superfluid and normal densities, $s$ is the entropy per unit mass, and $c$ is the specific heat per unit mass.

Example: the Cattaneo–Vernotte equation $\tau T_{tt} + T_t = \alpha T_{xx}$ has the solutions $T = e^{\lambda t} \cos(\kappa x)$ with $\tau \lambda^2 + \lambda + \alpha \kappa^2 = 0$. When $4 \tau \alpha \kappa^2 > 1$, the roots are $\lambda = -1/(2\tau) \pm i \omega$ with $\omega > 0$, so the profile oscillates while it decays. This is a damped temperature wave, the behaviour of second sound.

Non-example: under Fourier's law, the same trial function gives $\lambda = -\alpha \kappa^2$. The temperature profile only decays and never oscillates, so there is no second sound.