How Do Longitudinal Waves Behave on a Taut String?

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The discussion centers on a problem from Barton Zwiebach's Introduction to String Theory regarding the behavior of longitudinal waves on a taut string. The user is struggling to derive a solution that seems overly complex, specifically in determining the governing equation for small longitudinal oscillations and the wave velocity. They have calculated the general form of tension as T(L) = t_0 ln(L/a) + T_0 and attempted to apply Newton's second law to find the force on an infinitesimal stretch of the string. However, they encounter difficulties when accounting for the variation of tension across the string. Assistance or insights from others familiar with the problem would be appreciated.
Gianni2k
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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.
 
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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.
 
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