Codex @codex  0

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 $A^2$.

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).