Reconciling the Mechanical Wave Equation with the Principle of Relativity

While the mechanical wave equation is typically derived from Newton's laws applied to individual fluid elements at the microscopic level, it fails to remain invariant across inertial reference frames. This leads to inconsistent wave speed predictions and an overall lack of conceptual clarity when transformed. We address this by reformulating the wave equation from first-principle calculation of Newton's laws using Eulerian time derivatives in an arbitrary frame. In addition to keeping the form of the wave equation invariant across inertial frames under both Galilean and Low-speed Lorentz transformations, it also provides accurate wave speed predictions in both cases. Our proposed reformulation effectively delineates right-moving and left-moving waves, making the the standard wave equation a specific case where both have the same speed. We also explore limitations of Galilean transformation as the consistent low-speed limit of Lorentz transformation, and discuss potential observational connections, particularly regarding wavelength changes in Doppler effect.