Elastic energy of a small-slope string ($u=\frac{1}{2}T{y}_x^2,k=\frac{1}{2}\rho y_t^2$)

A string element has extra length $[\sqrt{1+y_x^2}-1]dx\simeq y_x^{2}dx/2$. At fixed tension T, work done supplies elastic energy density $T{y}_x^2/2$. Transverse velocity supplies kinetic-energy density $\rho y_t^2/2$. For a single travelling profile $y=f(x\pm ct)$ with $c^2=T/\rho$, the two densities agree pointwise. Their sum obeys a local energy balance with flux $-T{y}_x{y}_t$. Standing waves generally exchange kinetic and elastic energy instead of maintaining equality at every instant.