The classical Doppler effect revisited by the mathematical description of the phase function

TL;DR

This paper provides a comprehensive mathematical analysis of the classical Doppler effect, deriving general expressions for frequency shift and aberration through geometric and phase-based descriptions, applicable across various reference frames.

Contribution

It introduces a unified geometric and phase-based framework for describing the classical Doppler effect, extending its validity to moving media and arbitrary reference frames.

Findings

Derived general expressions for frequency shift and aberration.

Validated the coherence between geometric and phase descriptions.

Extended the Doppler effect analysis to non-stationary media.

Abstract

This paper aims to shed some more light on one of the best known phenomena in the field of physics, the Doppler effect, in particular, on its classical version. Although, as mentioned, it is a phenomenon already described more than 150 years ago, the general case, even considering constant velocities, continues to be the subject of very recent bibliographical works which either show it incompletely or start from a not entirely accurate description of it. Here, general expressions of both frequency shift and aberration are obtained by geometrical description and, more interestingly, by describing the phase of the wave in the reference frame in which the propagating medium is at rest. The coherence between the two points of view permits to contrast the validity of the proposed results, and allows for a generalization of the resulting expressions for any reference frame, even those where the propagating medium is not still.