If we're given a string with mass density $\rho$ in units $\frac{M}{L^3}$ with constant cross-section $A$, tension $T$ in units $\frac{F}{L^2}$, and whose length is $L$; and then we assume that the vertical displacement is small, the slope of the string during displacement is also small (which probably means that we can approximate tension to be constant); then I think it would be correct to say that the governing equation in this case is $$u_{tt}=c^2u_{xx}$$
where $c^2 = \frac{T}{\rho}$. (Please correct me if I'm wrong).
Now a dissipation term must be added, which should be directly proportional to the mass and velocity of the string.
Would it be correct to specify the dissipation term as follows: $-b\underbrace{\rho A L}_\text{mass} (u_t)^2\frac{1}{u}$ (where $b$ is some dimensionless proportionality constant)?
If this is correct, then the equation would become $$u_{tt}=c^2u_{xx}-bm(u_t)^2\frac{1}{u}$$
where $m$ is mass.