Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 10 / a

For a separated solution $y=X(x)Q(t)$, the wave equation gives $X^{\prime \prime }/X=\ddot{Q}/(v^{2}Q)$. Choose the separation constant $-k^2$. ThenThese are the oscillatory separated building blocks. Since no endpoint conditions have yet been imposed, the general solution is a superposition over the allowed continuum of $k$, rather than one product with four constants. Equivalently, combining the left- and right-travelling separated components gives the D'Alembert's formulawith arbitrary twice differentiable functions $F,G$ on the relevant characteristic intervals. Indeed, in characteristic coordinates $\xi =x-vt$, $\eta =x+vt$, the equation becomes $y_{\xi \eta }=0$, whose general solution is exactly this sum. This also includes zero-frequency and polynomial limits that would be omitted by restricting to one nonzero-$k$ sine mode.