Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 10 / c / Solution

The total kinetic energy and small-slope elastic potential energy areOrthogonality removes cross terms between the two distinct sine or cosine modes. A mode of amplitude $A_n$ contributes $K_n=\mu L{A}_n^2{\omega }_n^2{\sin }^2({\omega }_{n}t)/4$ and $V_n=TL{A}_n^2{k}_n^2{\cos }^2({\omega }_{n}t)/4$. For amplitudes $1,1/2$ and $n=2,4$, the prefactors agree. Writing $\Omega =2\pi v/L$,Thusindependent of time. This is another instance of equal modal energies for inverse-harmonic string amplitudes. An explicit initial amplitude $A$ multiplies every energy by $A^2$.