One dashpot can terminate a string without an echo

A string on $0<x<L$ has tension $T$, line density $\mu$, and transverse displacement $y(x,t)$. Its right endpoint is attached to a viscous dashpot $b\ge0$ fixed to ground; no endpoint mass or spring is present. The left endpoint may be taken fixed. Start from force balance at $x=L$, rather than guessing a sign. Derive wave speed, characteristic impedance, endpoint condition, and exact continuum energy balance. For a harmonic wave incident from the left, use $e^{-i\omega t}$ and calculate the complex displacement reflection coefficient. Obtain reflected and absorbed energy fractions and prove the unique passive no-reflection value of $b$. Then consider an initially right-moving pulse supported strictly inside the string. Explain, in the matched case, when it has completely left and why no reflected pulse reaches the fixed end. For $T=80.0\,\mathrm N$, $\mu=0.0200\,\mathrm{kg/m}$, and $L=3.00\,\mathrm m$, calculate $c$, impedance, and transit time. If $b$ is half its matched value, compute signed amplitude reflection and both energy fractions. Distinguish amplitude from flux and explain why $b<0$ is not a passive absorber.

The string pulls the right endpoint with $-Ty_x$; balance it with $-by_t$.