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

TL;DR

This paper explores the covariance properties of the d'Alembert equation for sound and light, showing it is covariant under Galilean transformations if sound's phase velocity depends on observer velocity, and under Lorentz transformations if light's phase velocity is observer-independent.

Contribution

It demonstrates the conditions under which the d'Alembert equation remains covariant for acoustic and optical phenomena without abandoning classical mechanics assumptions.

Findings

Covariance under Galilean transformations requires sound's phase velocity to depend on observer velocity.

Covariance under Lorentz transformations requires light's phase velocity to be observer-independent.

The same equation's covariance depends on the nature of the wave's phase velocity.

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.