Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 2

Set $y(x,t)=X(x)T(t)$ in the wave equation. A nonzero product obtained by separation of variables obeysThe fixed-end Dirichlet boundary conditions require $X(0)=X(L)=0$. The zero and negative spatial eigenvalue cases produce only the trivial solution; the nontrivial eigenfunctions areFor each normal mode, $T_n^{\prime \prime }+{\omega }_n^2{T}_n=0$. By superposition and completeness of the Fourier sine basis, the general finite-energy solution has the expansionIf $f(x)=y(x,0)$ and $g(x)=y_t(x,0)$, orthogonality gives $A_n=(2/L){\int }_0^{L}f(x)\sin (k_{n}x)dx$ and $B_n=2{\int }_0^{L}g(x)\sin (k_{n}x)dx/(L{\omega }_n)$. Individual products are separable; their sum is the general solution.

Assume the struck interval lies within the string, so $0\le \ell -a/2<\ell +a/2\le L$. The initially stationary straight string has $A_n=0$. Write its initial velocity as a Fourier sine series $g(x)={\sum }_n{g}_n\sin (k_{n}x)$. ThenThus $y(x,t)={\sum }_n(g_n/{\omega }_n)\sin ({\omega }_{n}t)\sin (k_{n}x)$. Immediately after the strike all the energy is kinetic energy:The orthogonality integral ${\int }_0^L\sin (k_{n}x)\sin (k_{m}x)dx=(L/2){\delta }_{nm}$ assigns energy $E_n=\rho L{g}_n^2/4$ to mode $n$. Since the ideal modes evolve independently, their total kinetic-plus-elastic energies remain constant. Therefore the modal energy of a localized velocity impulse on a string givesThe centre factor selects modes having significant motion at the impact location; the width factor accounts for cancellation across the finite hammer footprint. Parseval's identity ensures ${\sum }_{n\ge 1}{E}_n/E=1$. The velocity discontinuities are understood in the mean-square sense and do not alter the energy integrals.

With the specified ratios, the two factors in the modal amplitude become $\sin (n\pi /5)$ and $\sin (n\pi /14)$. Hence a normal mode is unexcited precisely when one of these factors is zero:That includes all multiples of five or fourteen, with multiples of seventy counted once. The mode $n=0$ is not a fixed-end string mode.