 Equal modal energies for inverse-harmonic string amplitudes

For a uniform ideal fixed-end string, a real normal mode of displacement amplitude $A_n$ has total mechanical energy $T{\pi }^2{n}^2{A}_n^2/(4L)$. This follows by integrating the squared sine velocity and cosine slope over the length; each squared trigonometric function integrates to $L/2$. Thus finitely many excited modes with $|A_n|$ proportional to $1/n$ have equal total modal energies. Their instantaneous kinetic energies need not agree, because their time factors oscillate at different frequencies.