Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 10 / b

Assume $v\ne 0$ so that the change of variables is invertible. The chain rule gives ${\partial }_x={\partial }_r+{\partial }_s$ and ${\partial }_t=v({\partial }_r-{\partial }_s)$. Commuting the mixed partial derivatives for a twice continuously differentiable solution givesThus the wave equation is $F_{rs}=0$. Integrating once says that $F_r$ depends only on $r$; integrating again givesThe two arbitrary twice differentiable functions are waves moving in opposite directions at speed $|v|$. This is D'Alembert's formula, on a product domain in $(r,s)$ or globally on the usual full line.