= Solution
Take positive $y$ vertically downward. For a short string element, the transverse tension resultant is $T y_{xx}\,dx$, while gravity and linear drag contribute $\mu g\,dx$ and $-2k\mu y_t\,dx$. <Newton's second law> gives
$$
\mu y_{tt}\,dx
=Ty_{xx}\,dx-2k\mu y_t\,dx+\mu g\,dx.
$$
Since $T=\mu c^2$, the <linearly damped string>[damped wave equation] is
$$
\boxed{y_{tt}+2ky_t-c^2y_{xx}=g},
\qquad
y(0,t)=y(l,t)=0.
$$
Choosing upward displacement reverses the sign of the gravity term.
Back to article page
