= Solution
Let $U(s,x)=\mathcal L_t\{u(t,x)\}$. The initial data turn the wave equation into
$$
-\frac{\partial^2U}{\partial x^2}+s^2U=f(x).
$$
Using the Green <function> from part (ii) with $m=s$ gives
$$
U(s,x)=\frac1{2s}\int_{-\infty}^{\infty}
e^{-s|x-y|}f(y)\,dy.
$$
Since
$$
\mathcal L^{-1}\!\left\{\frac{e^{-as}}s\right\}=H(t-a),
$$
we obtain the <D'Alembert formula with initial velocity>
$$
\boxed{
u(t,x)=\frac12\int_{-\infty}^{\infty}
H(t-|x-y|)f(y)\,dy
=\frac12\int_{x-t}^{x+t}f(y)\,dy}.
$$
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