Impulsively struck fixed-end string

For $0<\xi <l$, the wave equation $y_{tt}=c^2{y}_{xx}$ with zero endpoint displacement, zero initial displacement and initial velocity $\delta (x-\xi )$ has the distributional weak solutionExpand the Dirac delta in the Fourier sine basis and solve each modal ordinary differential equation. Equivalently take the odd, $2l$-periodic extension $v$ of the initial velocity and use $y=(2c{)}^{-1}{\int }_{x-ct}^{x+ct}v(s)ds$. The image impulses enforce the fixed endpoints. At jump fronts the Fourier series selects the midpoint of the two one-sided values.