# Longitudinal wave in a string

1. Jun 12, 2009

### Iamu

$$\Delta$$1. The problem statement, all variables and given/known data
Consider a string with mass density $$\mu$$0 stretched between x=0 and x=a. Let the equilibrium tension be T0. Longitudinal waves are possible if the tension of the string varies as it stretches or compresses. For a piece of this string with equilibrium length L, a small change $$\Delta$$L of its length is accompanied by a small change $$\Delta$$T of the tension where

1/$$\tau$$0$$\equiv$$(1/L)*($$\Delta$$L/$$\Delta$$T)

Here $$\tau$$0 is a tension coefficient with units of tension. Find the equation governing the small longitudinal oscillations of this string. Give the velocity of the waves.

2. Relevant equations

v=$$\sqrt{F/\mu}$$ (for tranverse waves, but does this apply to longitudinal waves?)
y(x,t)=Acos(kx-$$\omega$$t)

3. The attempt at a solution

I'm not sure how to manipulate the given equation; I've only worked with tranverse waves before. I guess I'm looking for T(x, t), as that would seem to be the analog of vertical displacement, but then again maybe I should be looking for L(T, t). L is only a piece of the string, so maybe L=dx? If I treat L as a variable and integrate to find T(L), I get a term with ln|L| that doesn't seem right. I wish I could make a better try at this, but I feel like I'm just pounding my head against a wall.