= Equal modal energies for inverse-harmonic string amplitudes
{title2=$E_n=T\pi^2n^2A_n^2/(4L)$}
For a uniform ideal fixed-end string, a real <normal mode> of displacement amplitude $A_n$ has total <mechanical energy> $T\pi^2n^2A_n^2/(4L)$. 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.
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