Modal energy of a localized velocity impulse on a string ($E_n/E=8L{\sin }^2(n\pi l/L){\sin }^2(n\pi a/(2L))/(a{\pi }^2{n}^2)$)

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 $E_n=\rho L{g}_n^2/4$ and total energy $E=\rho a{v}^2/2$. Their ratio is the displayed formula. Zeros in either sine factor suppress a normal mode completely.