General solution of 3d wave equation as a superposition of plane/spherical waves

You can solve it using Fourier transforms. If you have never seen them, the following could be a bit confusing. However, basically what one can do is write your unknown function as a infinite sum of waves with a spatial frequency $k$, solve it for each value of $k$ and then go sum it back to the original function. This procedure, as the wave equation is relatively easy to solve for a single wave, often really helps!

If you have the wave equation

$${partial^2 y over partial t^2} = c nabla ^2 y$$

where $nabla^2={partial^2 over partial x^2}+{partial^2 over partial y^2}+{partial^2 over partial z^2}$

you can use a Fourier transform to attempt solving it, i.e. a sort of infinite sum of plane waves.

In case you don't know Fourier transforms, briefly you can write every function in 3D $y(textbf{r}=(x, y, z))$ (provided they satisfy some very general conditions) as a sum of plane waves of the form $e^{itextbf{k}cdottextbf{r}}$, with $textbf{k}=(k_x, k_y, k_z)$, as

$$y(textbf{r}) = A int hat{y}(textbf{k}) e^{itextbf{k}cdottextbf{r}}dtextbf{k}$$

where $A$ is a constant and $hat{y}(textbf{k})$ is the so called Fourier transform and represents the continous coefficients of the series.

By substituting this expression in the wave equation we get an equation for $hat{y}(textbf{k})$ which is

$${partial^2 hat{y}(textbf{k}) over partial t^2} = -c k^2 hat{y}(textbf{k})$$

To prove it, just plug the expression I wrote up there in the wave equation and use the fact that $i^2=-1$ and ${partial e^{itextbf{k}cdottextbf{r}} over partial x} = i k_x e^{itextbf{k}cdottextbf{r}}$ and similar.

This equations is often easier to solve (it is very close the equation for a harmonic oscillator) but still requires the boundary conditions for $y$ to be completely solvable. Once it is solved, using the expression for $y(textbf{r})$, you can "invert" the Fourier transform to get the solution of your original functions.

You could do the same for spherical waves by finding an appropriate expression for $nabla$ and for the Fourier transform.