Wave equation with Robin and Neumann boundary conditions What are the periodic solutions of the following problem:


*

*Governing equation (wave equation):
$u_{tt} - c^2 u_{xx} = 0 $, $\forall x\in[0;L]$, $\forall t\geq 0$

*Robin boundary conditions at $x=0$: 
$u_x(0,t) = k u(0,t)$, $\forall t\geq 0$

*Neumann boundary conditions at $x=L$: 
$u_x(L,t)=0$, $\forall t\geq 0$
where $k$ is a constant and $c=\sqrt{E/\rho}$. 
 A: Let us search for a solution writing as monochromatic periodic plane waves under the form $u = (A \text{e}^{\text{i}\kappa x} + B \text{e}^{-\text{i}\kappa x})\, \text{e}^{\text{i}\omega t}$. Injecting this Ansatz in the wave equation gives the dispersion relation $\kappa = \omega/c$, where $\kappa$ is the wave number. Now, the combination of both boundary conditions gives the linear system
$$
\left[
\begin{array}{cc}
\text{i}\kappa - k & -\text{i}\kappa - k \\
\text{i}\kappa\, \text{e}^{\text{i}\kappa L} & -\text{i}\kappa\, \text{e}^{-\text{i}\kappa L}
\end{array}
\right]
\left[
\begin{array}{c}
A \\
B
\end{array}
\right]
=
\left[
\begin{array}{c}
0\\
0
\end{array}
\right] .
$$
This system has non-trivial solutions $A\neq 0$, $B\neq 0$ provided that its determinant vanishes, i.e.
$$
\kappa L\tan\kappa L = kL \, ,
$$
which solutions in terms of $\kappa = 2\pi f/c$ provide the resonance frequencies. The non-trivial solutions of the linear system give the normal mode shapes $A \text{e}^{\text{i}\kappa x} + B \text{e}^{-\text{i}\kappa x}$, where the wave number satisfies $\kappa L\tan\kappa L = kL$.
