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 $c=\sqrt{E/\rho}$.
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 $\kappa L\tan\kappa L = kL$.