Past exam of the mathematical methods II course of the University of Oxford / 2022 / 12

Use separation of variables, $y=X(x)T(t)$. A nontrivial separated solution satisfies $X^{\prime \prime }/X=T^{\prime \prime }/(c^{2}T)=-k^2$. Fixed endpoints force $X(0)=X(L)=0$, so $X=\sin (n\pi x/L)$, $n=1,2,\dots$. ThereforeHere $c>0$ is the wave speed. Zero or negative $k^2$ gives no nonzero fixed-end spatial solution. Each time coefficient is a simple harmonic oscillator.

Expand the initial velocity in the Fourier sine basis, $V(x)={\sum }_{n\ge 1}{V}_n\sin (n\pi x/L)$, where $V_n=(2/L){\int }_0^{L}V(x)\sin (n\pi x/L)dx$. Zero initial displacement removes the cosine-in-time coefficients. ThusThe trigonometric identity $2\sin a\sin b=\cos (a-b)-\cos (a+b)$ rewrites each summand asIt is therefore a superposition of travelling waves of speeds $+c$ and $-c$, whose interference enforces the fixed endpoints. For general square-integrable initial velocity, the series is interpreted in the standard finite-energy sense.

The product-to-sum identity gives $V_1=V_0$, $V_2=V_0/2$ and $V_3=2V_0/3$, with all other modal coefficients zero. If $\mu$ is the string's linear mass density, the initial potential energy vanishes and orthogonality gives the modal energy of a stringConsequently $\boxed{E_1:E_2:E_3=36:9:16}$ for $V_0\ne 0$. The fundamental mode carries the most energy, with fraction $36/61$. Comparing displacement coefficients without their frequency factors would give the wrong energy test.

Substitute the three nonzero initial-velocity coefficients into the general Fourier sine series, with ${\omega }_1=\pi c/L$:Every term satisfies the wave equation and the endpoint conditions. Differentiating in time at zero recovers the three coefficients $V_0,V_0/2,2V_0/3$, providing a direct initial-data check.

Assume $V_0\ne 0$. Linear independence of the Fourier sine basis makes a flat string equivalent to vanishing of all three temporal coefficients. The fundamental requires $\sin ({\omega }_{1}t)=0$, which automatically zeros the other two. Thus the earliest positive return isThe velocity need not have returned to its initial value at that time; full recurrence of the displacement-and-velocity data occurs at $2L/c$.

A nonzero pure fundamental profile would require $\sin 2{\omega }_{1}t=\sin 3{\omega }_{1}t=0$. The common zeros have ${\omega }_{1}t$ an integer multiple of $\pi$, so the fundamental vanishes too. A pure second or third harmonic would already require the fundamental to vanish, again making the whole string flat. The string never has a nonzero pure sine profile. If $V_0=0$, it remains identically zero and there is no earliest positive isolated return.