Physical meaning of two different variable-speed wave equations From here, there are (at least) two different wave equations with variable wave speed. 
Either $c^2(x)$ is outside the Laplacian:
$$
\begin{cases}u_{tt} - c^2(x) \Delta u = 0 \quad \textrm{ in } \mathbb R \times \mathbb R^n \\
u(0,x) = f(x); \quad u_t(0,x)= g(x). \end{cases}\tag{1}
$$
or $c^2(x)$ is between the two nablas:
$$
\begin{cases}u_{tt} - \nabla\cdot(c^2(x)\nabla u) = 0 \quad \textrm{ in } \mathbb R \times \mathbb R^n \\
u(0,x) = f(x); \quad u_t(0,x)= g(x). \end{cases}\tag{2}
$$
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? 
 A: Consider for simplicity 1+1D: 
$$\rho(x) \ddot{u} ~= ~\frac{\partial}{\partial x}( Y(x) \frac{\partial u}{\partial x}).\tag{A}$$ 
Eq. (A) is the differential equation for linear deformations in a solid with non-uniform mass density $\rho(x)$ and non-uniform modulus of elasticity $Y(x)$. Two special cases of eq. (A) [uniform $Y$ vs. uniform $\rho$] yields OP's eqs. (1) & (2) in 1+1D, respectively.  
A: Both equations can describe physical phenomena. The location of $c²(x) $ represents different physical relations, i.e. they most likely hold for different $u(t,x)$.
This paper: Homogenization of the variable-speed wave equation contains valuable information on the topic. Therein you also find the following examples: The equation you namend (1) arises from Maxwell's EM equations in one dimension. Equation (2) describes among other phenomena surface water waves. 
Another important point is that the two equations are connected and can be transformed into one another. You'll find information on this in the paper I linked as well.
