Past exam of the year 1 of the computer science course of the University of Oxford / 2005 / Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation / 4 / b / ii / Solution

The velocity contribution in D'Alembert's formula is $\frac{1}{2}{\int }_{x-t}^{x+t}1ds=t$. Hence, with arbitrary twice continuously differentiable $h$,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.