Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 16 / b / ii / Solution

Choose $F(s)=G(s)=\frac{1}{2}\sin s$. The two travelling profiles then add by the sine addition identity:This is a standing wave. At $x=0$ and $x=\pi$ it vanishes for all $t$. At $t=0$ its displacement is $\sin 2x$, and its time derivative $-5\sin 2x\sin 5t$ is zero. Also $y_{tt}=-25y$ and $y_{xx}=-4y$, directly checking the original equation.