= Solution
Transform in time and use the zero interior <initial conditions>. The <Laplace transform of a derivative> formula turns the <wave equation> into $c^2Y_{xx}=p^2Y$. Its transformed endpoint conditions are $Y(0,p)=a/p$ and $Y(L,p)=0$. A <homogeneous solution> satisfying the right condition is a multiple of $\sinh[p(L-x)/c]$; the left condition fixes that multiple. Thus, for $\operatorname{Re}p>0$,
$$
\boxed{Y(x,p)=\frac{a\sinh[p(L-x)/c]}{p\sinh(pL/c)}.}
$$
Here $L,c>0$ and the denominator has no zeros in this half-plane. The initial-boundary corner at $(0,0)$ is deliberately discontinuous: the applied step launches propagating fronts, so one should not demand a globally smooth solution across them.
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