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

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.

Fixed endpoints select sine normal modes with $k_n=n\pi /L$ and ${\omega }_n=v{k}_n$. The product identity $\sin z\cos z=\frac{1}{2}\sin 2z$ decomposes the held initial profile into $n=2$ and $n=4$, with amplitudes $1$ and $1/2$. Zero initial velocity selects cosine time dependence. ThereforeBoth endpoints vanish at every time. At $t=0$ the product-form displacement is recovered and $y_t=0$. The implicit leading displacement amplitude is one in the source's chosen length units; including a physical amplitude $A$ multiplies the whole solution by $A$.

The total kinetic energy and small-slope elastic potential energy areOrthogonality removes cross terms between the two distinct sine or cosine modes. A mode of amplitude $A_n$ contributes $K_n=\mu L{A}_n^2{\omega }_n^2{\sin }^2({\omega }_{n}t)/4$ and $V_n=TL{A}_n^2{k}_n^2{\cos }^2({\omega }_{n}t)/4$. For amplitudes $1,1/2$ and $n=2,4$, the prefactors agree. Writing $\Omega =2\pi v/L$,Thusindependent of time. This is another instance of equal modal energies for inverse-harmonic string amplitudes. An explicit initial amplitude $A$ multiplies every energy by $A^2$.