Why does a vibrating string under gravity give a parabolic shape, not a catenary?

[Gear300]

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 F_y that acts per unit length.

The equation of motion I got was
τ₀∂²_xψ - b∂_txψ + ∂_xF(x,t) = λ₀∂²_tψ ,
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∂²_xψ - 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.

 

[AlephZero]

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.

If you assume the loading on the cable is uniform in the horizontal direction, the solution is a parabola.

If you assume the loading comes from the weight of the cable, the loading per horizontal distance is NOT uniform, because it depends on the slope of the cable at each point. For that case, the solution is a catenary.

http://mathworld.wolfram.com/Catenary.html

 

[Gear300]

If you assume the loading on the cable is uniform in the horizontal direction, the solution is a parabola.

If you assume the loading comes from the weight of the cable, the loading per horizontal distance is NOT uniform, because it depends on the slope of the cable at each point. For that case, the solution is a catenary.

http://mathworld.wolfram.com/Catenary.html

Thanks. I just realized that might have been the case. My force was not done correctly.