Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 1 / 17B (Methods)

The fixed-end wave equation has spatial modes $\sin (n\pi x)$ and temporal factors $\cos (n\pi t),\sin (n\pi t)$. The zero initial velocity removes every temporal sine term. Expand the initial displacement in a Fourier sine series, whose coefficients areTwo integrations by parts give the last equality; the endpoint terms from $x(1-x)$ vanish and the remaining derivative endpoints distinguish even from odd $n$. ConsequentlyThis solves the initial-boundary problem in the finite-energy sense, with convergent first-derivative series. At $t=0$ the energy in the question is ${\int }_0^1(1-2x{)}^{2}dx=1/3$. At any time, sine/cosine orthogonality gives the contribution of mode $n$ as $(n\pi {)}^2{b}_n^2/2$: its kinetic and elastic pieces have factors ${\sin }^2(n\pi t)$ and ${\cos }^2(n\pi t)$ summing to one. ThereforeThe square summability of the derivative coefficients justifies the orthogonality calculation and its limit.