= Modal energy of a localized velocity impulse on a string
{title2=$E_n/E=8L\sin^2(n\pi l/L)\sin^2(n\pi a/(2L))/(a\pi^2n^2)$}
A fixed-end string of length $L$ initially straight receives velocity $v$ on an interval of width $a$ centred at $l$, and zero elsewhere. Its sine-mode velocity coefficient is $g_n=4v\sin(n\pi l/L)\sin(n\pi a/(2L))/(n\pi)$. For linear density $\rho$, orthogonality gives mode energy $E_n=\rho Lg_n^2/4$ and total energy $E=\rho av^2/2$. Their ratio is the displayed formula. Zeros in either sine factor suppress a <normal mode> completely.
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