In standard practice, velocity addition rules are derived for point particles from Galilean and Lorentz transformations. Such rules are then intuitively applied for waves to explain Doppler effect, following what Doppler did himself originally. However, such application goes without any explanation in terms of equations manifesting such frame dependence of velocity. We show that wave propagation should be modeled with first order, not second order, partial differential equations so as to manifest such frame dependence of wave velocity. From a historical point of view, this work settles the Doppler-Petzval debate that was pivoted to the question of underlying differential equations for Doppler effect. From a modern theoretical physics stand point, this work ignites a foundational debate regarding what can now be considered as "equation for wave propagation".