General solution of wave equation is \begin{align}\label{eq:sol} u(x,t) = u_0\sum_{n=1}^{\infty} \sin(n\pi x/\ell) (a_n \cos \omega_n t + b_n \sin \omega_n t ) e^{-\gamma_n t} \end{align}
The expansion coefficients are \begin{align}\label{eq:coeff} a_n &= \frac{2h(0)}{\ell} \!\!\int_{0}^{\ell} \!\! f(x) \sin(n\pi x/\ell) \,dx\,,\quad h(0) = e^{-\delta^2/\tau^2} \delta/\tau\\ b_n &= \frac{2\dot{h}(0)}{\ell\omega_n} \!\! \int_{0}^{\ell}\!\! f(x) \sin(n\pi x/\ell) \,dx\,, \quad \dot{h}(0) = e^{-\delta^2/\tau^2}\frac{1 - 2\delta^2/\tau^{2}}{\tau} \end{align}