Solving wave equation with two spatial dependancies

Question

I have an elastic wave equation $$\rho \frac{\partial^{2}}{\partial t^{2}} u(x_n) = \frac{\partial}{\partial x_n}u(x_n).$$ If I want to solve when $u = u(x_1,x_3)$, can I solve the wave equation for $u(x_1)$ and $u(x_3)$, and combine the solutions somehow?

The framework I am using makes it simple to solve when $u$ depends on one variable, but hard when there is two.

I hope my question makes sense!

Answer 1

Hint.

Using separation of variables, making $u(x_1,x_2,t) = X(x_1,x_2)T(t)$ we have

$$ X\ddot T=X_{x_1}T+X_{x_2}T\Rightarrow \frac{\ddot T}{T} = \frac{1}{X}(X_{x_1}+X_{x_2}) = \lambda $$

so

$$ \ddot T = \lambda T\\ X_{x_1}+X_{x_2} = \lambda X $$