Wave speed on a string of non-uniform linear mass density

[JohnLCC517]

Homework Statement

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Consider a long chain of mass m and length L suspended from a tall ceiling. Like any string if one end is disturbed waves will travel along the string. However, the tension in the string is due to its own weight and is not uniform. As such the speed of the wave will be different at each point of the string. Determine the speed of the wave as a function of a location of the wave on the string.

Homework Equations

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Force (F) = mg, Linear mass density (μ) = m/L, Wave Speed (V) = √(F/μ)

The Attempt at a Solution

I began this problem by relating the three relevant equations above as follows;
Wave Speed (V) = √((mg)/(m/l)) = √(g/l), however, I am now stuck on how to relate this as a function of the location of the wave on the string. My initial thought would be to relate this to the velocity of a particle on a string in the following fashion but even this seemed off √(g/l) = -ω A sin (kx - ωt + φ₀)

 

[RUber]

Let's say you call the bottom of the chain x = 0, so that x relates to the height. μ should be a constant, mass per unit length of the chain.
Your tension (F) on the chain at height x: F(x) = x μ g .
I think the position of the wave on the string is just referring to the position x.