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Energy in a wave pulse

  1. Jan 5, 2014 #1
    1. The problem statement, all variables and given/known data

    one end of a fixed string is moved transversly at a constant speed u for a time τ and is moved back to it's starting point with velocity -u during the next time interval. as a result a triangular pulse is set up and moves along the string with speed v. Calculate the kinetic and potential energies associated with the pulse, and show that their sum is equal to the work done by the transverse force that has to be applied at the end of the string.


    2. Relevant equations

    Work = F x displacement
    Power = Work x time

    PE density = T/2 (dy/dx)^2
    KE density = μ/2 (dy/dt)^2

    v = (T/μ)^0.5 (T = tension)

    3. The attempt at a solution

    the triangle pulse:

    y = A f(x-vt)

    KE density = μ/2 A^2[-vAf'(x-vt)]^2
    PE density = t/2 A^2 [f'(x-vt)]^2

    using v = (T/mu)^0.5 you get that PE = KE therefore

    total energy density = μ A^2 v^2 [f'(x-vg)]^2

    work = power x time

    W = T x u x 2τ

    -TvA[f'(x-vt)] x 2τ

    I can't seem to rearrange them to get them the same. I tried using v = λ/(2τ)

    thank you
     
  2. jcsd
  3. Jan 5, 2014 #2

    mfb

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    Staff: Mentor

    I guess "g" is a typo and you mean t?
    If you replace v^2 by T/μ there, the equations get more similar.

    I don't understand where this comes from. The force required to move the line is not T.
     
  4. Jan 5, 2014 #3
    Isn't the force your putting on the end of the string the tension T?
     
  5. Jan 5, 2014 #4

    mfb

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    Staff: Mentor

    The motion is transversal, the sidewards force (=relevant for the wave generation) is different.
     
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