Relativistic acoustic Doppler effect

The formulation of the relativistic acoustic Doppler effect is considered in detail. Two approaches were developed, both leading to the same result. The first method parallels the derivation of the classical formula, using Lorentz velocity transformation and time dilation effects when required. The second method is an intuitive approach done in the frame of the medium, clearly showing that the relativistic correction terms arise solely from time dilation effects; the rest of the derivation is entirely classical. It is demonstrated that the relativistic deviation is exactly zero when the source and observer speeds are equal, even if the two motions are oppositely directed. Finally, a numerical estimate reveals that the correction terms amount to only a few parts in 1015 for ordinary acoustic velocities.