(a) Find the solution to the following PDE:

\[\begin{aligned}& \frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}} \\& u(0, t)=u(L, t)=0 \\& u(x, 0)=0 \\& \frac{\partial u}{\partial t}(x, 0)=f(x)\end{aligned}\]

where \(L\) is a constant and \(f\) is a given function.

(b) Find the solution when \(f(x)=\sin x\).