Wave Equation is Linear

Theorem
Let $\phi_1$ and $\phi_2$ be particular solutions to the wave equation.

Then:
 * $a_1 \phi_1 + a_2 \phi_2$ is also a particular solution to the wave equation.

Proof
Hence:

and it is seen that $a_1 \phi_1 + a_2 \phi_2$ also satisfies the wave equation.