Interrupting a standing wave at a node

[Zorodius]

Suppose you have a taut string, and you pinch it firmly at the middle, so that the middle is immobile but the string has the same tension throughout.

Then you start vibrating one end of the string. I would think that the wave could not propagate past the middle of the string, because you have fixed that point. The second half of the string won't move for that reason.

So instead, you stop pinching the string, and allow the vibrations to occur at such a frequency as to activate the second mode of oscillation of your string.

With this mode in action, you once again pinch the string in the middle. The standing wave will continue uninterrupted, because this point was a node - it was fixed anyway.

This is a very unintuitive result to me. I would intuitively think that pinching the string would somehow isolate the two halves from one another, and make it impossible for energy to be supplied from the initial end to affect the opposite end, but this is not so. The best explanation that has come to me is that, although the displacement and velocity of the infinitesimal pinched segment is held to be zero, its tension oscillates, and this enables energy to be transmitted through it. Is this the explanation, or is there something else to it?

 

[mathman]

This is a very unintuitive result to me. I would intuitively think that pinching the string would somehow isolate the two halves from one another, and make it impossible for energy to be supplied from the initial end to affect the opposite end, but this is not so.

After you let go of the pinch, energy got redistributed. After the second pinch both parts would continue to vibrate, but at an amplitude lower than at the half you started with before unpinching.

I don't see a problem, unless I'm missing something.

 

[Gokul43201]

This is a very unintuitive result to me. I would intuitively think that pinching the string would somehow isolate the two halves from one another, and make it impossible for energy to be supplied from the initial end to affect the opposite end, but this is not so.

Why should pinching prevent energy from traversing the midpoint? All that the pinching ensures is that the displacement be zero there.

 

[cesiumfrog]

It doesn't look to me like energy is being transmitted across the pinch. (Gokul, pinching should also prevent tensions on either side being coupled.)

 

[berkeman]

Remember what a standing wave is. It's a left-travelling wave and a right-travelling wave. You have to have both in place before you get a standing wave. When you pluck the string to start the 2nd harmonic vibration, there is a transient period before the node forms in the middle, right?

Does that help to answer the question? The displacement in the middle is only zero after the 2nd traveling wave comes back and the system settles.

 

[Claude Bile]

I think cesiumfrog hits the nail on the head, if you pinch a section of the string, the tension from one half is no longer "connected" or coupled to the second half, which results in the wave being unable to propagate past the "pinch".

In short - pinching at the node has a drastic effect on the system, it is not as passive as it may first appear.

Claude.

 

[AlephZero]

Why should pinching prevent energy from traversing the midpoint? All that the pinching ensures is that the displacement be zero there.

That depends what the OP means by "pinch" and whether we are talking about an idealised thought experiement or a real experiment.

You could interpret "pinching" the string (between finger and thumb) as enforcing zero displacement and zero slope, compared with "touching" the string which enforces zero displacement but not zero slope.

With that interpretation, pinching would stop energy transmission in a transverse wave, but touching would not.

But in practice you would not completely stop rotation of something like a vibrating guitar string by pinching with your fingers, so there would still be some energy tranmission.

 

[Gokul43201]

It doesn't look to me like energy is being transmitted across the pinch. (Gokul, pinching should also prevent tensions on either side being coupled.)

I believe this is only true for a medium with zero stiffness. Most real materials, (guitar strings, electrical cables, plumbing lines) however, will transmit energy past a point that is rigidly anchored.