Consider a string vibrating (with small amplitude) against a transverse damping force b(∂ψ/∂t) that acts per unit length. Also consider the effect of a transverse driving Fy that acts per unit length. The equation of motion I got was τ0∂2xψ - b∂txψ + ∂xF(x,t) = λ0∂2tψ , where τ0 is the tension along the string and λ0 is the linear mass density. Now consider a string of length L in equilibrium under gravity. The equation of motion thus becomes τ0∂2xψ - mg = 0 . The solution I get from this is parabolic, whereas from what I remember, the form for a hanging chain/cable/wire/etc... is a hyperbolic cosine (cosh). I was wondering whether I made a mistake somewhere, or that this is a result of the small amplitude condition.