<p>It is a standard practice to derive velocity addition rules for point particles from Galilean and Lorentz transformations in point (classical) mechanics, and to apply such rules to wave velocities for explaining Doppler effect. However, in such standard practice, it is never shown whether the equation for wave propagation actually transforms in a way such that the velocity addition rules get manifested through the equation itself. We address this gap in the literature as follows. We claim that the <i>velocity addition for waves</i>, being the one and only mean to explain the empirically verified Doppler effect, should be considered as an element of physical reality in accord with EPR’s completeness condition of a physical theory. Therefore, the ‘equation for wave propagation’ should manifest such velocity addition so as to be considered as a part of the respective physical theory of waves. We show that such manifestation is possible if and only if wave propagation is modeled with <i>first order partial differential equations</i>. From a historical point of view, this work settles the Doppler-Petzval debate which originated from Petzval’s demand for an explanation of Doppler effect in terms of differential equations. From the foundational perspective, this work sets the stage for a renewed focus on the mathematical modeling of wave phenomena, especially in the context of various Doppler effects.</p>