I Mechanical energy of element of a rope with sinousoidal wave

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1. May 15, 2016

crick

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. May 15, 2016

wrobel

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 $\dot x^2/2+\omega^2 x/2=const$