= Viscous termination of a stretched string
A semi-infinite <wave equation on a string> with <linear mass density> $\rho$, <wave speed> $c$, and <tension> $T=\rho c^2$ has positive characteristic <mechanical impedance> $Z=\rho c$. At an end attached to a massless viscous damper of coefficient $\alpha>0$, displacement and transverse-force balance give the displacement <reflection coefficient>
$$
r=\frac{Z-\alpha}{Z+\alpha},\qquad
\mathcal R=|r|^2,\qquad \mathcal A=\frac{4\alpha Z}{(\alpha+Z)^2}.
$$
Here $\mathcal A$ is the fraction of mean incident <power> dissipated. A prescribed $0<\mathcal R<1$ gives two positive damper coefficients, $\alpha/Z=(1\pm\sqrt{\mathcal R})/(1\mp\sqrt{\mathcal R})$, with opposite reflected <phases>. Matching $\alpha=Z$ removes reflection.
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