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

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 $h$.

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.

Here the velocity contribution in D'Alembert's formula isThus, with arbitrary twice continuously differentiable $h$,For the added term, $u_{tt}=e^x\sinh t=u_{xx}$ and $u_t(x,0)=e^x\cosh 0=e^x$. Its displacement at $t=0$ is zero, leaving $h$ free as required.