Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 10 / e

For a mode $y_n=A_n\sin (n\pi x/L)\cos ({\omega }_{n}t)$, integrate the kinetic-energy density:Here $A_1=1$, $A_3=-1/3$ and ${\omega }_3=3{\omega }_1$, soAll other modes have zero energy. The equal modal energies for inverse-harmonic string amplitudes give each excited mode total energy $T{\pi }^2/(4L)$, even though their displacement amplitudes differ. The corresponding potential energy replaces sine squared by cosine squared, so their sum recovers the constant total energy from part (a).