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Images of standing waves often show two people using a jump rope. The shape of the rope goes from straight to waves to straight, then a sine wave the other way and then straight again as it repeats. During those straight moments, it seems the rope takes a shorter path then when it is extended into the crest and trough.

Theories:

  1. the rope does not expand or contract but instead moves out from the drawing in 3d to fit its length. But that doesn't seem to be what I experience when I do these things in real life.

  2. the rope secretly conspires to generate smaller and smaller peaks and trough along the rope during the transition. What seems a straight rope line is actually a series of very small waves, which continue to break down until they can flip.

  3. The diagram is false. The people holding the rope must move their hands closer and farther apart at double the frequency of the standing wave.

These are contrived and probably wrong. Any ideas how the rope compresses to a straight line for standing waves?

Update:

If the wave must push the particles into a dimension orthogonal to any 2d perspective, the distance correcting waves could would be small in comparison to the overall wave being observed.

For now, I assume there are no completely defined 1 dimensional waves in a medium having fixed distances.

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  • $\begingroup$ I thin kit might be best to think of this as a standing wave where the nodes might move closer together when the waves are at trough/peak than when they are not. I don't know, though... Good question. $\endgroup$
    – Basican
    Sep 27, 2022 at 2:24
  • $\begingroup$ Thanks RukiyaMeria. I seems like you are saying that the rope is like a bunch of points the move closer sort of like a pressure within the rope. But what if the rope was a long bicycle chain with links locked in place at specific distance. What would it mean for the node when the points within it can not compress together? $\endgroup$
    – peawormsworth
    Sep 27, 2022 at 2:36

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Imagine an idealised situation in which a standing wave is formed on a rope which is held at each end by immovable (sic) objects.
In such a case the length of the rope changes as does the tension in the rope. Most often when such a system is analysed assumptions made are that the changes in the length of the rope is small compared with the length of the rope and the changes in the tension of the rope is small compared with the tension in the rope.

In terms of two people holding the rope I would suggest it is very difficult to say with any certainty what happens.
It is highly likely that the end of the rope are no longer stationary but whether that is due to a change in the length/tension of the rope or just a natural movement of the hands/arms/bodies of those holding is almost impossible to say. It is probably a combination of both effects.

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  • $\begingroup$ This question is harder if we think of the rope as inelastic, so we probably should. $\endgroup$
    – Basican
    Sep 27, 2022 at 18:35

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