Suppose $H$ is a positive self-adjoint operator on the Hilber space $E$ and $f:\R\rar E$ is $C^1$. Then
$$
u(t)\colon=c\int_0^t H^{-1/2}\sin\Big(c(t-s)H^{1/2}\Big)f(s)\,ds
$$
solves the inhomogeneous wave equation $(c^{-2}\pa_t^2+H)u(t)=f(t)$.
Since $\sin(0)=0$ we have
$$
u^\prime(t)=c^2\int_0^t\cos\Big(c(t-s)H^{1/2}\Big)f(s)\,ds
$$
and as $\cos(0)=1$, we get:
$$
c^{-2}u^\prime(t)
=f(t)-c\int_0^tH^{1/2}\sin\Big(c(t-s)H^{1/2}\Big)f(s)\,ds
=f(t)-Hu(t)~.
$$