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3D, constant \( c \): $$ \begin{equation*} \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \end{equation*} $$
2D, variable \( c \): $$ \begin{equation} \varrho(x,y) \frac{\partial^2 u}{\partial t^2} = \frac{\partial}{\partial x}\left( q(x,y) \frac{\partial u}{\partial x}\right) + \frac{\partial}{\partial y}\left( q(x,y) \frac{\partial u}{\partial y}\right) + f(x,y,t) \end{equation} $$
Compact notation: $$ \begin{align} u_{tt} &= c^2(u_{xx} + u_{yy} + u_{zz}) + f, \tag{41}\\ \varrho u_{tt} &= (q u_x)_x + (q u_z)_z + (q u_z)_z + f \tag{42} \end{align} $$
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