Restoring Invariance to the Mechanical Wave Equation across Inertial Frames

Abstract

The mechanical wave equation, derived from Newton's laws, takes its simplest form in the rest frame of the medium, and appears to lose that form entirely once expressed in the coordinates of another inertial frame --- raising the question of whether, and how, it can be written consistently for an arbitrary inertial observer. This puzzle, often glossed over in undergraduate instruction, is typically set aside by invoking the rest frame of the medium as ``preferred,'' without clarifying what this means for the principle of relativity underlying physical law. We revisit the wave equation from first principles and demonstrate that this apparent non-invariance stems from its expression in Eulerian coordinates. We present two reformulations: a manifestly invariant form using Lagrangian derivatives applied at the level of microscopic particle force laws, and a factorized Eulerian form separating left and right-moving wave solutions. In this context, we explore limitations of Galilean transformation as the low-speed limit of Lorentz transformation, and discuss potential observational connections, particularly regarding wavelength changes in the Doppler effect. Our derivation provides both conceptual clarity and pedagogical value for intermediate and advanced undergraduate physics.