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Longitudinal wave in a string

  1. Jun 12, 2009 #1
    [tex]\Delta[/tex]1. The problem statement, all variables and given/known data
    Consider a string with mass density [tex]\mu[/tex]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 [tex]\Delta[/tex]L of its length is accompanied by a small change [tex]\Delta[/tex]T of the tension where

    1/[tex]\tau[/tex]0[tex]\equiv[/tex](1/L)*([tex]\Delta[/tex]L/[tex]\Delta[/tex]T)

    Here [tex]\tau[/tex]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=[tex]\sqrt{F/\mu}[/tex] (for tranverse waves, but does this apply to longitudinal waves?)
    y(x,t)=Acos(kx-[tex]\omega[/tex]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.
     
  2. jcsd
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