I'm looking for a two dimensional wave-like equation (or the best that can be had) for the following situation:
Roughly we have a fluid resting not over a fixed flat surface but over a substrate of variable height.
In more detail, imagine I have a solid impermeable substance (think "rock") whose height over a bounded rectangle is given by a function $r$. Above the surface of the "rock" is a surface (think "water") given by a function $w$ with $w \geq r$. This surface then evolves as one would expect water to flow over a solid subsurface of variable height.
Pardon the lack of formality, I'm not familiar with the terminology. What I'm fishing for here is better terminology, references, and numerical simulation of this situation.
Here are the shallow water equations, translated into your notation and ignoring Coriolis and viscosity terms. They involve two extra variables, $u$ and $v$, representing the velocity of the fluid in the $x$ and $y$ directions. $$\begin{aligned} {\frac {\partial w}{\partial t}}&+{\frac {\partial }{\partial x}}{\bigl (}(w-r)u{\bigr )}+{\frac {\partial }{\partial y}}{\bigl (}(w-r)v{\bigr )}=0,\\ {\frac {\partial u}{\partial t}}&+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=-g{\frac {\partial w}{\partial x}},\\ {\frac {\partial v}{\partial t}}&+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=-g{\frac {\partial w}{\partial y}}, \end{aligned}$$ where $g$ is acceleration due to gravity.
If you want something that looks more like the wave equation, you can proceed as follows. Assuming the velocities are relatively small, and $w$ does not vary significantly relative to its mean value $W$, we obtain the linearized shallow water equations by neglecting the quadratic terms: $$\begin{aligned} {\frac {\partial w}{\partial t}}&+{\frac {\partial }{\partial x}}{\bigl (}(W-r)u{\bigr )}+{\frac {\partial }{\partial y}}{\bigl (}(W-r)v{\bigr )}=0,\\ {\frac {\partial u}{\partial t}}&=-g{\frac {\partial w}{\partial x}},\\ {\frac {\partial v}{\partial t}}&=-g{\frac {\partial w}{\partial y}}. \end{aligned}$$ Taking the time derivative of the first equation and substituting in the second and third, this is equivalent to $${\frac {\partial^2 w}{\partial t^2}}=g{\frac {\partial }{\partial x}}\left ((W-r){\frac {\partial w}{\partial x}}\right )+g{\frac {\partial }{\partial y}}\left ((W-r){\frac {\partial w}{\partial y}}\right ).$$