Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 3

For one sine mode, $y_{xx}=-k_n^2{A}_n\sin (k_{n}x)\cos ({\omega }_{n}t)$ and $y_{tt}=-{\omega }_n^2{A}_n\sin (k_{n}x)\cos ({\omega }_{n}t)$. The wave equation on a string therefore holds whenThe superposition principle gives the same equation for the sum, assuming enough regularity to differentiate termwise, or interpreting finite-energy data in the usual modal sense. Fixed endpoints require $\sin (k_{n}L)=0$, so $k_n=n\pi /L$. Every cosine time factor has zero time derivative at $t=0$, giving the held-at-rest initial condition.

The coefficients must represent the actual held profile $f(x)$ through its Fourier sine series, $A_n=(2/L){\int }_0^{L}f(x)\sin (n\pi x/L)dx$ for $n\ge 1$. Thus the series accommodates the displacement initial condition once these coefficients are chosen; it does not select an unspecified initial shape by itself. The printed $n=0$ term is identically zero because $\sin 0=0$, so the nontrivial fixed-end modes start at $n=1$.

In the small-slope string model, the additional elastic potential energy is $V=(T/2){\int }_0^L{y}_x^{2}dx$. For the fundamental mode, $y_x=A_1{k}_1\cos (k_{1}x)\cos ({\omega }_{1}t)$. Since ${\int }_0^L{\cos }^2(\pi x/L)dx=L/2$,Using $T{k}_1^2=\mu {\omega }_1^2$ from the dispersion relation givesThe expression vanishes as the string passes through its straight equilibrium shape and is maximal at the turning shapes, when the corresponding kinetic energy is zero.