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Consider a rope hanging from the ceiling (massive / massless irrelevant, I suppose). A wave pulse is set up on the rope. Just as the wave pulse starts propagating on this rope, the top of the rope is cut off and the rope is allowed to fall freely.

What happens to the wave pulse?

I posed this question as a homework problem for a sophomores' class on Waves, but then I realized that this was probably not as simple as the naïve answer I had.

My naïve answer was, since there is no tension in the rope once in free fall (no gravity), the velocity for transverse waves on the rope is 0, therefore the pulse will freeze relative to the rope and fall.

My colleagues pointed out to energy and momentum conservation. One of my friends concluded that the rope should do some weird sort of spiraling to conserve momentum and angular momentum about the center of the pulse.

I'd appreciate if someone could shed some light on this problem.

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1 Answer 1

At least at first, nothing unusual will happen to the pulse. It will keep traveling up the rope exactly as if it had never been cut, because it takes time for the message that the rope has been cut to travel down to where the wave is.

During this initial period where the pulse hasn't yet learned that the rope is cut, the rope's center of mass still must accelerate downward at $g$. The bottom of the rope isn't accelerating at all (except for the wave motion). That means the top of the rope is going extra fast as it accelerates down, and catches up to the bottom. Thus, the rope tends to bundle up on itself.

Once the falling part of the rope meets the upward-traveling wave, all bets are off. You basically just have a rope, which, viewed in a free-falling frame, is flapping about wildly and loosely with quite a bit of energy - whatever was stored in the wave and in the stretching before being cut.

From there, you probably want to simulate or do an experiment. Looking quickly, I found a couple of cute rope simulations online, and you can probably find more if you dig a bit more.

http://scratch.mit.edu/projects/BATzerk/42740

http://www.functionblog.com/?p=134

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Wouldn't it be a fair assumption to assume that the speed of sound is much faster than the time-scale of wave pulses on the rope, so the information travel time can be neglected? The timescales would be so far apart, that we could assume that the other end of the rope gets the message instantaneously... –  kstar Oct 19 '12 at 17:35
    
@AkarshSimha The rope doesn't support compression very well, and I'm not sure that the speed of sound is what's important here. See this video for the example of a slinky. youtu.be/eCMmmEEyOO0?t=45s –  Mark Eichenlaub Oct 20 '12 at 0:08
    
That video is just amazing! Thanks for sharing it. Yes, in this example though, the typical speed of sound is probably of the order of km/s. Whereas, for the same rope, the order of magnitude for transverse waves is probably around tens of m/s. So I'm assuming we can assume that the information passes along instantaneously. –  kstar Oct 20 '12 at 23:05

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