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.

 

[Entropia]

I would like to first clarify that the solution to this problem may not be "too complicated," but rather it may require a deeper understanding of the concepts and principles involved. It is important to approach the problem with a clear and logical thought process.

To begin, let us first define the variables in the problem. The string has a uniform mass density of mu_0 and is stretched between x = 0 and x = a with an equilibrium tension of T_0. The equilibrium length of a piece of the string is L, and any small change in its length is denoted by deltaL. This change in length is accompanied by a change in tension, deltaT, which is related to a tension coefficient t_0.

To find the equation governing small longitudinal oscillations of the string, we can use the general wave equation:

d^2u/dt^2 = v^2 * d^2u/dx^2

where u is the displacement of the string, t is time, and x is position along the string. The velocity of the wave is denoted by v.

In this case, the tension of the string is not constant, but varies with the stretching or compressing of the string. This means that the tension coefficient t_0 is also a function of position, and can be expressed as:

t_0 = t_0(x)

Substituting this into the wave equation, we get:

d^2u/dt^2 = v^2 * (1/t_0) * d^2u/dx^2

We can rearrange this equation to get:

d^2u/dt^2 = (v^2/t_0) * d^2u/dx^2

Now, using the given relationship between deltaL and deltaT, we can substitute for t_0 and get:

d^2u/dt^2 = (v^2/L) * (deltaL/deltaT) * d^2u/dx^2

Simplifying further, we get:

d^2u/dt^2 = (v^2 * deltaT/L) * d^2u/dx^2

This is the equation governing small longitudinal oscillations of the string. We can also see that the velocity of the wave is given by:

v = sqrt(T_0/mu_0)

where T_0 is the equilibrium tension and mu_