Sound Waves and the Absence of Galilean Invariance in Flocks

We study a model of flocking for a very large system $(N=320\mathrm{}000)$ numerically. We find that in the long wavelength, long time limit, the fluctuations of the velocity and density fields are carried by propagating sound modes, whose dispersion and damping agree quantitatively with the predictions of our previous work using a continuum equation. We find that the sound velocity is anisotropic and characterized by its speed $c$ for propagation perpendicular to the mean velocity $〈\overrightarrow{v}〉$, $〈\overrightarrow{v}〉$ itself, and a third velocity $\lambda 〈\overrightarrow{v}〉$, arising explicitly from the lack of Galilean invariance in flocks.