Compare the speed and frequency of a wave in a thick rope

[YMMMA]

Homework Statement

In the attached file.

Homework Equations

No equations needed.

The Attempt at a Solution

I answered it B. Not sure,though. If it is a thick rope the frequency decreses; therefore, the speed decreases.

 

[berkeman]

Homework Equations

No equation needed.

Why are no equations needed? I would need an equation or two to solve this question. Can you say what it/they might be? And then can you apply the equation(s) to check your answer?

 

[YMMMA]

Why are no equations needed? I would need an equation or two to solve this question. Can you say what it/they might be? And then can you apply the equation(s) to check your answer?

I will search, then.

 

[YMMMA]

Here, the density increases in the thick rope causing the speed decreases. If the speed decreases, will the frequency decrease based on the rule?

 

[berkeman]

Here, the density increases in the thick rope causing the speed decreases. If the speed decreases, will the frequency decrease based on the rule?

The equation you posted is for a doubly-constrained rope, I believe, fixed at both ends. The problem in your OP is a little different. It seems to be asking about a long combination of two ropes, with the first part lighter (lower density) and the 2nd part heavier (higher linear density). The excitation at the left will generate transverse waves that will travel to the right with some frequency, velocity and wavelength. There will be a partial reflection of that traveling wave at the discontinuity between the two ropes, but you can ignore that for this question, I believe.

Now, what is the relationship between frequency, velocity and wavelength for a traveling wave? And if the left end of the light rope is being shaken up and down at some frequency, what would happen if the frequency changed at the place where the two ropes join?

After you think about that for a bit, check out this paper for information on a slightly different equation that you should consider...

http://www.cmp.caltech.edu/refael/league/wave-speed.pdf

And after reading that and thinking more about your answer, you can check out this nice summary post by @sophiecentaur in another thread about wave propagation and transitioning between different regions (in the case of that thread, it's EM waves and changing impedances of media that it is propagating through, but it still is very similar to this question of yours):

https://www.physicsforums.com/threads/refraction-of-a-wave.957400/#post-6071022

 

[haruspex]

If it is a thick rope the frequency decreses;

Think about the oscillation frequency at the join. How must this compare with the frequencies of adjacent parts?

 

[sophiecentaur]

If the speed decreases, will the frequency decrease based on the rule?

How would you think it possible for one bit of rope to be going up and down at one frequency and the adjacent bit going up and down at a different frequency? If that happened
1. Wouldn't the rope come apart?
2. What could be driving the second piece of rope at the different frequency?
Certain things have to be continuous along the length of the rope. However, the wavelength can easily change without disconnecting sections of the rope.

 

[YMMMA]

Aha, now I can clearly understand that at the join the frequency will continue while the wavelength can change. For the speed, it is proportional to the square root of the tension over the density. Therefore, the speed decreases with increasing density.
Thank you all!

 

[YMMMA]

The equation you posted is for a doubly-constrained rope, I believe, fixed at both ends. The problem in your OP is a little different. It seems to be asking about a long combination of two ropes, with the first part lighter (lower density) and the 2nd part heavier (higher linear density). The excitation at the left will generate transverse waves that will travel to the right with some frequency, velocity and wavelength. There will be a partial reflection of that traveling wave at the discontinuity between the two ropes, but you can ignore that for this question, I believe.

Now, what is the relationship between frequency, velocity and wavelength for a traveling wave? And if the left end of the light rope is being shaken up and down at some frequency, what would happen if the frequency changed at the place where the two ropes join?

After you think about that for a bit, check out this paper for information on a slightly different equation that you should consider...

http://www.cmp.caltech.edu/refael/league/wave-speed.pdf

And after reading that and thinking more about your answer, you can check out this nice summary post by @sophiecentaur in another thread about wave propagation and transitioning between different regions (in the case of that thread, it's EM waves and changing impedances of media that it is propagating through, but it still is very similar to this question of yours):

https://www.physicsforums.com/threads/refraction-of-a-wave.957400/#post-6071022

I have seen all, thanks for your help

 

[sophiecentaur]

Aha, now I can clearly understand that at the join the frequency will continue while the wavelength can change. For the speed, it is proportional to the square root of the tension over the density. Therefore, the speed decreases with increasing density.
Thank you all!

Something else that should be born in mind is the fact that, when the wave encounters a change in density (and hence wave impedance) there will be a reflection of some of the incident energy at the transition. A single pulse will show this reflection and a continuous wave will show a change in amplitude at the interface, in order that the net energy flow into the transition equals the flow out of the transition.

 

[YMMMA]

Something else that should be born in mind is the fact that, when the wave encounters a change in density (and hence wave impedance) there will be a reflection of some of the incident energy at the transition. A single pulse will show this reflection and a continuous wave will show a change in amplitude at the interface, in order that the net energy flow into the transition equals the flow out of the transition.

Understood.