Linearly damped string

With fixed endpoints, $y_{tt}-y_{xx}+y_t=0$ separates into sine modes whose amplitudes satisfy

For $y_{tt}+2k{y}_t=c^2{y}_{xx}$ 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 $T_n^{\prime \prime }+2k{T}_n^{\prime }+(n\pi c/l{)}^2{T}_n=0$. Here ${\omega }_n^2=(n\pi c/l{)}^2-k^2$. At critical damping, replace $\sin ({\omega }_{n}t)/{\omega }_n$ 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.

For initial sine coefficient $a_n$ and zero initial velocity, put ${\omega }_n^2=n^2{\pi }^2-1/4$. The modal amplitude is

For fixed endpoints andintegration by parts and the damped wave equation give