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

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.