 Midpoint-plucked fixed-end string

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 $a_n\cos ({\omega }_{n}t)$ 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.