Classical longitudinal wave on taut String

[Gianni2k]

Hi guys, this is Barton Zwiebach's Introduction to String theory question 4.2 on the longitudinal wave on a taut string. The problem is purely classical and I seem to obtain a solution which seems far too complicated for me. If anyone has the answers it would be great, if not just your help would be amazing. For people that don't have the book this is how the question goes.

"Consider a string with uniform mass density mu_0 stretched between x = 0 and x = a. Let the equilibrium tension be T_0. Longitudinal waves are possible if the string tension varies at it stretches or compresses. For a piece of this string with equilibrium length L, a small change in its length deltaL is accompaigned by a change in the tension deltaT where:

1/t_0 = (1/L)(deltaL/detaT)

where t_0 is a tension coefficient with units of tension. Find the equation governing small longitudinal oscillations of the string. Give the velocity of the waves."

many thanks.

 

[Gianni2k]

Just so you know what I did is calculate the general form of T(L):

T(L) = t_0 ln(L/a) + T_0

Then find the force on an infinitesimal stretch of the string then finally equate this to the acceleration via Newton's second law. The method is consistent for tangential waves but I have problems with longitudinal ones where the tension varies across the string.