Solution
= Solution
The velocity contribution in <D'Alembert's formula> is $\tfrac12\int_{x-t}^{x+t}1\,ds=t$. Hence, with arbitrary twice continuously differentiable $h$,
$$
\boxed{u(x,t)=\frac{h(x+t)+h(x-t)}2+t.}
$$
The added term has both second <partial derivatives> zero and has <time derivative> one, so it supplies precisely the required <initial condition> while preserving the <wave equation>.