Particle scattering in a sonic analogue of special relativity

We investigate a simple toy model of particle scattering in the flat spacetime limit of an analogue-gravity model. The analogue-gravity medium is treated as a scalar field of phonons that obeys the Klein-Gordon equation and thus admits a Lorentz symmetry with respect to $c_\mathrm{s}$, the speed of sound in the medium. The particle from which the phonons are scattered is external to the system and does not obey the sonic Lorentz symmetry that the phonon field obeys. In-universe observers who use the exchange of sound to operationally measure distance and duration find that the external particle appears to be a sonically Lorentz-violating particle. By performing a sonic analogue to Compton scattering, in-universe observers can determine if they are in motion with respect to their medium. If in-universe observers were then to correctly postulate the dispersion relation of the external particle, their velocity with respect to the medium could be found.