= Velocity-impulse Green function for a damped string
{title2=$G(x,a,t)=\frac2l\sum_{n\geq1}\sin\frac{n\pi x}l\sin\frac{n\pi a}l\,e^{-kt}\frac{\sin(\omega_nt)}{\omega_n}$}
For $y_{tt}+2ky_t=c^2y_{xx}$ with fixed endpoints, initial displacement zero and initial velocity a <Dirac delta> at $a\in(0,l)$, the <Fourier sine series> coefficient of that velocity is $2\sin(n\pi a/l)/l$. Each coefficient evolves by $T_n''+2kT_n'+(n\pi c/l)^2T_n=0$. Here $\omega_n^2=(n\pi c/l)^2-k^2$. At <critical damping>, replace $\sin(\omega_nt)/\omega_n$ by $t$; for an overdamped mode replace it by $\sinh(\sqrt{k^2-(n\pi c/l)^2}\,t)/\sqrt{k^2-(n\pi c/l)^2}$. The initial conditions hold in the sense of <distributions>.
Back to article page
