# Application of eigenvalueproblems for the wave equation

I'm currently searching for a nice little application of an eigenvalueproblem and found the following for acoustics - but one part doesn't make sense for me. Consider the wave equation to find some $p(x,t)$ with

$- \Delta_x p + \frac{\partial^2 p}{\partial^2 t} = 0 \ \$ on $\Omega \times (0,\infty)$
with some initial value conditions for $p(x,0)$ and $\frac{\partial p}{\partial t}(x,0)$. As boundary values we'll choose Neumann's:
$\nabla p \cdot \overrightarrow{n} = 0 \ \$ on $\partial \Omega$

Now $|p(x,t)|$ can be interpreted as the change in pressure (sound). The script i found now makes the ansatz to find a solution with frequency $\omega \geq 0$ by setting:
$p(x,t) = e^{i \omega t} u(x)$

Which (inserted) leads to $u$ being a non-zero solution of the eigenvalue problem:
$- \Delta_xu = \omega^2u \ \$ on $\Omega$
$\nabla u \cdot \overrightarrow{n} = 0 \ \$ on $\partial \Omega$

So, if we solve this eigenvalue problem (maybe with some FEM), we know what the sound will look like for these frequencies $\omega$.
But the big problem i don't understand is the part with the ansatz. Initially we wanted a real-valued function $p$, but the ansatz-function is obviously complex-valued. And then suddendly we're back to a real-valued Laplace-Eigenvalueproblem?

It would be great if someone could help me to understand this or has some more reasonable ansatz for this kind of application.
Thank you!

If you prefer, you can consider the ansatz

$$p(x,t) = \sin(\omega t) u(x)$$

and

$$p(x,t) = \cos(\omega t) u(x)$$

By Euler's identity $\sin$ and $\cos$ span the same space as $e^{i\omega t}$ and $e^{-i\omega t}$. Observe that if you had used

$$p(x,t) = e^{-i\omega t}u(x)$$

the eigenvalue problem is the same. Complex linear combinations then gets you back to the purely real ansatz using $\cos$ and $\sin$ instead.