= Elastic energy of a small-slope string
{title2=$u=\tfrac12Ty_x^2,\quad k=\tfrac12\rho y_t^2$}
A string element has extra length $[\sqrt{1+y_x^2}-1]dx\simeq y_x^2dx/2$. At fixed tension T, work done supplies elastic energy density $Ty_x^2/2$. 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 $-Ty_xy_t$. Standing waves generally exchange kinetic and elastic energy instead of maintaining equality at every instant.
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