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 / i

For the wave equation with wave speed one, D'Alembert's formula with arbitrary twice continuously differentiable initial displacement $h(x)=u(x,0)$ isThe arbitrary function $h$ is essential: only the initial velocity is prescribed. For $g=0$, this givesBoth travelling terms solve the wave equation, and their time derivatives cancel at $t=0$. Conversely every classical solution has this form for its initial displacement $h$.