# Wave motion on a string

#### jojimt

I have a rather elementary question. Let's say I have an arbitrarily long string, tied at both ends, with enough tension on the string so it is a medium for wave motion from one end to the other. I inject a disturbance at one end, to initiate a set of waves towards the other end. Almost simultaneously, I cut the other end removing any elasticity in the string. In this scenario, how long will the original wave travel before it dies down? What's the best way to model this?

#### studiot

I have a rather elementary question. Let's say I have an arbitrarily long string, tied at both ends, with enough tension on the string so it is a medium for wave motion from one end to the other. I inject a disturbance at one end, to initiate a set of waves towards the other end. Almost simultaneously, I cut the other end removing any elasticity in the string. In this scenario, how long will the original wave travel before it dies down? What's the best way to model this?
Cutting the string does not change the characteristic material property of elasticity.
What it does do is create a disturbance in the tension of the string which propagates at the characteristic velocity of a longitudinal wave in the string, since tension is a longitudinal phenomenon.
Thus it propagates from the cut point towards the still fixed end at the speed of sound in the string.

The disturbance wave you are 'injecting' is a transverse wave with a different characteristic velocity so if injection and cut are simultaneous the two disturbances will meet at a point along the string determined by the ratio of the two velocities.

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#### Woody

I would expect the speed of sound (longitudinal wave) in the string to be vastly higher than the transverse wave transmission speed.

You would have to be looking very carefully to notice that the transverse wave does continue a short distance before it meets the longitudinal wave going (much faster) the other way.

#### studiot

I would expect the speed of sound (longitudinal wave) in the string to be vastly higher than the transverse wave transmission speed.

You would have to be looking very carefully to notice that the transverse wave does continue a short distance before it meets the longitudinal wave going (much faster) the other way.
Agreed, but who is to say that jo jim is not a very careful observer?

#### jojimt

Ha ha. @studiot and @woody thanks for the answers. Really appreciate it.

That makes sense. Couple more questions: 1) I think that also means there is no interaction between the two waves at all. Is that correct? 2) Would this behavior be somewhat analogous to the behavior of waves on an expanding space?

#### jojimt

Never mind. I found this: Light Waves. Thanks again. The model is clearer to me now.

#### studiot

Ha ha. @studiot and @woody thanks for the answers. Really appreciate it.

That makes sense. Couple more questions: 1) I think that also means there is no interaction between the two waves at all. Is that correct? 2) Would this behavior be somewhat analogous to the behavior of waves on an expanding space?
1) The advancing front of the longitudinal disturbance divides the string into two parts at any instant.

I will call the part not yet reached the 'stiff part' which is still capable of sustaining tension and therefore a transverse wave.

And the part where the longitudinal disturbance has relaxed the tension the floppy part.

The transverse wave cannot progress from the stiff part into the floppy part across the boundary formed by longitudinal disturbance and could not in any case propagate in the floppy part as there is no tension there.

So in that sense they interact.

2) I don't see what this has to do with expanding space? The only effect of such expansion I can think of would be to make the wave non linear by adding extra terms since each cycle would be subject to constantly changing wavelength. A bit like frequency modulation Have you seen pictures of this?
It would not, however fail to support wave motion, like the floppy part of the string.

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#### Woody

Waves in Expanding Space.

If I turn your question around a bit...

What would be the equivalent of the tension of the string in these waves?
What would be the equivalent of releasing that tension?

From this I hope you can see that you can't build a straight analogy between waves in a string and waves in space-time.

There is always a question on how far can an analogy be pushed before it fails.

#### jojimt

OK, I did push the analogy. What I meant was: can the expansion be thought of as a longitudinal wave(lets ignore the breaking of the string -- that was over-simplification on my part) which would travel faster than light? I guess then we should at least observe the change in wavelength. But we do see the doppler effect.

#### studiot

OK, I did push the analogy. What I meant was: can the expansion be thought of as a longitudinal wave(lets ignore the breaking of the string -- that was over-simplification on my part) which would travel faster than light? I guess then we should at least observe the change in wavelength. But we do see the doppler effect.
How can it?

Wave motion involves oscillation about a mean, as in a pulsating sphere.

You are talking about one way continued expansion are you not?

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