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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!

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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.

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