Please help me understand how transverse waves reflect at a boundary

[constant_puzzlement]

Hello, I am a student who is trying to learn some physics independently so I apologize in advance if I am not making sense. I have studied physics a bit in school but nothing very rigorous and it is a subject that I have trouble with, especially waves.

This is what I have been reading: https://www.physicsclassroom.com/class/waves/Lesson-3/Boundary-Behavior
The explanation for why the wave inverts at the hard boundary is: "As the last particle of medium A pulls upwards on the first particle of medium B, the first particle of medium B pulls downwards on the last particle of medium A." But I do not understand this explanation because if B pulls on A the same amount as A pulls on B then wouldn't B stop pulling on A when A gets to the midpoint? Then you would end up with a flat string.

Something like this question (https://www.physicsforums.com/threads/reflection-of-waves-and-formation-of-standing-waves.844915/) I think kind of gets at the question I am asking but it is unanswered and has been for a while. I have been looking around for a physical explanation for this behavior (preferably with something like force diagrams) but I have not found any. Am I going about this the wrong way? Is it not possible to use this kind of physical intuition for waves/is the wave equation the only way to predict the behavior of waves?

 

[A.T.]

...wouldn't B stop pulling on A when A gets to the midpoint?

Yes, but that doesn't stop A immediately at the midpoint because of inertia.

 

[tech99]

Summary:: Confused about mechanism by which waves flip polarity with a hard boundary (visualizing with force diagrams)

Hello, I am a student who is trying to learn some physics independently so I apologize in advance if I am not making sense. I have studied physics a bit in school but nothing very rigorous and it is a subject that I have trouble with, especially waves.

This is what I have been reading: https://www.physicsclassroom.com/class/waves/Lesson-3/Boundary-Behavior
The explanation for why the wave inverts at the hard boundary is: "As the last particle of medium A pulls upwards on the first particle of medium B, the first particle of medium B pulls downwards on the last particle of medium A." But I do not understand this explanation because if B pulls on A the same amount as A pulls on B then wouldn't B stop pulling on A when A gets to the midpoint? Then you would end up with a flat string.

Something like this question (https://www.physicsforums.com/threads/reflection-of-waves-and-formation-of-standing-waves.844915/) I think kind of gets at the question I am asking but it is unanswered and has been for a while. I have been looking around for a physical explanation for this behavior (preferably with something like force diagrams) but I have not found any. Am I going about this the wrong way? Is it not possible to use this kind of physical intuition for waves/is the wave equation the only way to predict the behavior of waves?

Maybe look first at a longitudinal wave. I think we can say that the last particle hits the wall and bounces back. So its velocity reverses and it exerts a large force at the point of collision. Maybe the last particle of the transverse wave does the same.

 

[constant_puzzlement]

Yes, but that doesn't stop A immediately at the midpoint because of inertia.

Oh okay, so B exerts a downwards force on A, and then A continues traveling downwards because of inertia and by doing so it starts the reflected wave? I think I am still a bit confused though because it seems like when the reflected wave is formed, wouldn't A pull on B again causing B to exert an upwards force on A which would make another reflected wave? This doesn't happen so I must be misunderstanding something.
Thank you for helping me out.

Maybe look first at a longitudinal wave. I think we can say that the last particle hits the wall and bounces back. So its velocity reverses and it exerts a large force at the point of collision. Maybe the last particle of the transverse wave does the same.

Oh, I hadn't thought of that. I'll have to think about this some more, thank you.

 

[zoki85]

Hopefully , the following visualization with balls connected with string will help:

 

[A.T.]

I think I am still a bit confused though because it seems like when the reflected wave is formed, wouldn't A pull on B again causing B to exert an upwards force on A which would make another reflected wave?

A is pulled up by B at some point, but when it starts to move up it is also pulled down by the 2nd last particle, which now lower. So A doesn't go above the neutral level.

Here is another video with some explanations:

 

[PeroK]

Oh okay, so B exerts a downwards force on A, and then A continues traveling downwards because of inertia and by doing so it starts the reflected wave? I think I am still a bit confused though because it seems like when the reflected wave is formed, wouldn't A pull on B again causing B to exert an upwards force on A which would make another reflected wave? This doesn't happen so I must be misunderstanding something.
Thank you for helping me out.Oh, I hadn't thought of that. I'll have to think about this some more, thank you.

You have two extreme cases. First, if the second medium is the same as the first, then the wave continues unchanged . At the other extreme, if the second medium is so dense it is essentially fixed, then you have a fully reflected inverted wave. All other cases lie between these two extremes. Any explanation, therefore, must involve an element of how the relative density apportions the energy to the transmitted and reflected waves.

PS if you study a simple elastic collision between two objects, where A impacts B.

If A is lighter, then it rebounds and the proportion of energy transmitted to B depends on the ratio of their masses. If they have the same mass then A stops and transmits all its energy to B.

I suggest something analagous is happening with these waves.

 

[A.T.]

You have two extreme cases. First, if the second medium is the same as the first, then the wave continues unchanged . At the other extreme, if the second medium is so dense it is essentially fixed,

One could also say the extreme cases are:
density ratio (medium2/medium1) -> 0 (string end free to move)
density ratio (medium2/medium1) -> infinity (string end fixed)

And your first case is in the middle:
density ratio (medium2/medium1) = 1 (string doesn't end)

In both extreme cases there is total reflection. In the middle case there is no reflection at all.

 

[constant_puzzlement]

A is pulled up by B at some point, but when it starts to move up it is also pulled down by the 2nd last particle, which now lower. So A doesn't go above the neutral level.

Here is another video with some explanations:

Thank you! I think I am starting to get it now. One last thing, please let me know if I have come to the right conclusions. In the fixed end case A's velocity reverses and in the free end case A moves to twice the original height of the wave, so that should mean at the boundary it experiences a force that is 2 times stronger than the initial force needed to start the wave, I think. But I don't think that there is any way to predict this from looking only at forces which is why some people trying to explain the behavior of a wave on a string do so in terms of energy conservation... As far as I can tell the wave equation based explanations that people give are all based on first stating the behavior of the wave and then finding a solution that fits that observed behavior which is not quite what I was looking for in terms of an "explanation".

You have two extreme cases. First, if the second medium is the same as the first, then the wave continues unchanged . At the other extreme, if the second medium is so dense it is essentially fixed, then you have a fully reflected inverted wave. All other cases lie between these two extremes. Any explanation, therefore, must involve an element of how the relative density apportions the energy to the transmitted and reflected waves.

PS if you study a simple elastic collision between two objects, where A impacts B.

If A is lighter, then it rebounds and the proportion of energy transmitted to B depends on the ratio of their masses. If they have the same mass then A stops and transmits all its energy to B.

I suggest something analagous is happening with these waves.

I see, thank you for giving me another way to think about this system, it is really helpful! :)

 

[A.T.]

But I don't think that there is any way to predict this from looking only at forces...

Of course there is. Some of the animations you find of this are actually simulations based on a mass-spring system. This is predicting the behavior based on forces.

 

[constant_puzzlement]

Of course there is. Some of the animations you find of this are actually simulations based on a mass-spring system. This is predicting the behavior based on forces.

Oh... Well in that case would you happen to know of any such simulations that allow you to view the code?
Sorry, it's very difficult for me to wrap my head around this for some reason.

 

[A.T.]

Oh... Well in that case would you happen to know of any such simulations that allow you to view the code?
Sorry, it's very difficult for me to wrap my head around this for some reason.

A mass-spring system is a set of masses that are connected by elements generating distance dependent forces, and simulated by integration of Newton's Laws.

 

[sophiecentaur]

in that case would you happen to know of any such simulations that allow you to view the code?

I can sympathise with you here. The problem is that "the code" and a simulation don't show the pattern in the same way that the equations do. To make sense of "the code" you would have to reverse-engineer it and get the equations in any case. There are many Google hits for a search about reflection coefficient and boundary. This one shows the result near the end.

 

[A.T.]

I can sympathise with you here. The problem is that "the code" and a simulation don't show the pattern in the same way that the equations do. To make sense of "the code" you would have to reverse-engineer it and get the equations in any case.

If I understand @constant_puzzlement correctly, the aim is to play around with the simulation, and look at the forces at specific time points, to get a better understanding on how the string parts are interacting. For this I would recommend building the string in a simple simulation tool like Algodoo. It allows you to visualize the vectors for force, acceleration, velocity for each particle.

http://www.algodoo.com/

Here is an example of a string build with Algodoo:

Here a different simulation with more complex cases:



More info in this paper:
http://www.sbfisica.org.br/rbef/pdf/334306.pdf