Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 13C (Methods)

With $\xi =x+ct$ and $\eta =x-ct$,so the wave equation becomesThus $u=F(\xi )+G(\eta )$. At $t=0$ the initial data giveSolving for $F^{\prime }$ and $G^{\prime }$ and integrating gives d'Alembert's formula>

Extend the forcing oddly across the boundary:The initial displacement $\sin x$ is already odd, so its homogeneous evolution on the line is $\sin x\cos (ct)$. Applying Duhamel's principle to the odd extension givesThe odd-reflection method makes $u(0,t)=0$. At $t=0$ the double integral vanishes together with its first time derivative, so the prescribed initial displacement and velocity are also satisfied.