From here, there are (at least) two different wave equations with variable wave speed.
Either $c^2(x)$ is outside the Laplacian: $$ \begin{cases}u_{tt} - c^2(x) \Delta u = 0 \quad \textrm{ in } \mathbb R \times \mathbb R^n \\ u(0,x) = f(x); \quad u_t(0,x)= g(x). \end{cases} $$ or $c^2(x)$ is between the two nablas: $$ \begin{cases}u_{tt} - \nabla\cdot(c^2(x)\nabla u) = 0 \quad \textrm{ in } \mathbb R \times \mathbb R^n \\ u(0,x) = f(x); \quad u_t(0,x)= g(x). \end{cases} $$ My question is, what do the differences in the location of $c^2(x)$ actually represent? What is the physical meaning distinguishing these two equations?