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I Mechanical energy of element of a rope with sinousoidal wave

  1. May 15, 2016 #1
    I'm confused about energy driven by a wave. Consider a sinousoidal wave moving in a rope.

    Each element ##dm## of the rope follows a simple harmonic motion in time. That means that the mechanical energy ##dE=dK+dU## of the element ##dm## is constant.

    Nevertheless on Halliday-Resnik-Krane I found this explanation.

    I really do not see how this can be possible. Is this related to the fact that the energy of a wave is not concentrated in a single point but somehow spread in all the rope continuously?

    I would really appreciate some suggestion on this topic. Is the mechanical energy of ##dm## really not constant? If so, what can be an explanation for that?
     
  2. jcsd
  3. May 15, 2016 #2
    I believe that the authors should explain their thought detailed. Indeed, the equation of harmonic oscillator ##m\ddot x+k^2 x=0## has the first integral ##m\dot x^2/2+kx^2/2## this is a trivial mathematical fact and it does not depend on environment.
    Or in other words if ##x(t)=C_1\cos\omega t+C_2\sin\omega t## then ##\dot x^2/2+\omega^2 x/2=const##
     
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