  We study a nonlinear wave equation appearing as a model for a membrane
(without viscous effects) under the presence of an electrostatic potential with
strength $\lambda$. The membrane has a unique stable branch of steady states
$u_{\lambda}$ for $\lambda\in\lbrack0,\lambda_{\ast}]$. We prove that the
branch $u_{\lambda}$ has an infinite number of branches of periodic solutions
(free vibrations) bifurcating when the parameter $\lambda$ is varied.
Furthermore, using a functional setting, we compute numerically the branch
$u_{\lambda}$ and their branches of periodic solutions. This approach is useful
to validate rigorously the steady states $u_{\lambda}$ at the critical value
$\lambda_{\ast}$.
