= Impulsively struck fixed-end string
For $0<\xi<l$, the <wave equation> $y_{tt}=c^2y_{xx}$ with zero endpoint displacement, zero initial displacement and initial velocity $\delta(x-\xi)$ has the <distributional weak solution>
$$
y(x,t)=\frac{2}{\pi c}\sum_{n\ge1}\frac{\sin(n\pi\xi/l)\sin(n\pi x/l)\sin(n\pi ct/l)}{n}.
$$
Expand 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.
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