I've been working on a project to do with scattering of acoustic waves and one of the first steps towards finding my solutions was to derive the wave equation:
$$ \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2} $$
I "know" that U in this case can be any physical quantity (velocity, pressure, etc) but I don't really understand this intuitively.
I derived the equation from Navier-Stokes for pressure (and hence density) so I get those two. But why is this valid for velocity too? I'm currently using it for the velocity field for my solutions and I think missing this bit of conceptual understanding is really holding me back!