= Midpoint-plucked fixed-end string
{title2=$a_n=8k\sin(n\pi/2)/(\pi^2n^2),\quad\omega_n=cn\pi/L$}
A string of length $L$ with fixed endpoints, initially displaced into a triangular tent of height $k$ at its midpoint and released from rest, has a <Fourier sine series> with the coefficients in the title. Integrating the two linear halves gives $a_n$, and symmetry cancels the even modes. Each coefficient evolves as $a_n\cos(\omega_nt)$ under the <wave equation>. The displacement vanishes everywhere at $t=L/(2c)$, reverses sign at $t=L/c$, and returns after $2L/c$. A straight configuration at an intermediate instant does not mean the string is at rest: its <kinetic energy> is then nonzero. The initial slope corner gives a finite-energy <weak solution> whose corner lines propagate and reflect at the ends.
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