Past exam of the mathematical methods II course of the University of Oxford / 2015 resit / 11 / b / Solution

Seek a separated normal mode $y(x,t)=X(x)Q(t)$. Substitution in the wave equation gives $X{Q}^{\prime \prime }=c^2{X}^{\prime \prime }Q$. For an oscillatory mode choose the separation constant $-{\omega }^2$:The fixed endpoints impose $X(0)=X(l)=0$. In $X=A\sin kx+B\cos kx$, the first condition gives $B=0$, and the second requires $\sin kl=0$. Nonzero displacement therefore requiresEach spatial shape $\sin (n\pi x/l)$ has fixed nodes and an oscillating overall amplitude: a standing wave. A general lossless motion is a superposition of these standing normal modes, not necessarily one single-frequency standing wave. The $n=0$ solution is trivial under both fixed-end conditions and does not supply an additional vibrational mode.