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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 $c=\sqrt{E/\rho}$.

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    $\begingroup$ @Harry49 No, I haven't tried. I can only think about the eigenfunctions (linear normal modes) to be the only periodic solutions. Do you have any idea of any other type of functions? $\endgroup$
    – cgyo
    Feb 27, 2018 at 17:22

1 Answer 1

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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 $\kappa L\tan\kappa L = kL$.

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