Mechanical energy of element of a rope with sinousoidal wave

[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.

Despite the analogies with simple harmonic motion the mechanicalenergy of the element $dm$ is not constant. [...] That's not surprising since the element $dm$ is not an isolated system and its motion is the result of the action of the rest of the rope on it.

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?

 

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