Insert \(u(x,t) = u_0 e^{i(k x -\tilde{\omega} t)}\) in wave equation
Consider boundary conditions \begin{align} u(0,t) &= u(\ell,t) = 0\,,\qquad u_{xx}(0,t) = u_{xx}(\ell,t)=0 \end{align} yielding the dispersion relation \begin{align}\label{eq:freq} \omega_n &= \sqrt{(n \pi c/\ell)^2 + (EI/\rho)(n\pi/\ell)^4 -\gamma_n^2}\,, \end{align} where \(\gamma_n(\tilde{\omega}_n) = R(\tilde{\omega}_n)/2\rho\) and \(c = \sqrt{T/\rho}\)