On the covariance of the d'Alembert equation: the cases of sound and light

Abstract

The covariance of the d'Alembert equation for acoustic phenomena, described by mechanical waves in one or three spatial dimensions, under Galilean transformations, is demonstrated without the need to abandon the hypothesis that time is absolute in Classical Mechanics. This is true only if and only if the phase velocity of sound depends on the velocity of the observer. On the other hand, it is also shown that the same d'Alembert equation is covariant under Lorentz transformations if and only if the phase velocity of light does not depend on the observer.