Find the free-vibration response of a rectangular membrane of sides \(a\) and \(b\) subjected to the following initial conditions:

\[\left.\begin{array}{c}w(x, y, 0)=0 \\\frac{\partial w}{\partial t}(x, y, 0)=\dot{w}_{0} \sin \frac{\pi x}{a} \sin \frac{2 \pi y}{b}\end{array}\right\}, \begin{aligned}& 0 \leq x \leq a \\& 0 \leq y \leq b\end{aligned}\]

Assume that the edges of the membrane are fixed.