ΔAmplitude of wave on rope with a change in linear density

[Nathanael]

Would it be correct to assume that if the linear density of a rope changes by a factor of [itex]k[/itex], then the amplitude of a wave traveling on the rope would change by a factor of [itex]\frac{1}{\sqrt{k}}[/itex]? (Assuming tension and frequency are fixed.)
I get this by assuming the average power of the wave is unchanged by the change in linear density (and also because the average power depends on the square the amplitude and the linear density).

This is essentially what I'm wondering:
If the linear density ([itex]\mu[/itex]) varies with the position ([itex]x[/itex]) as described by some function, [itex]\mu(x)[/itex], then would the amplitude ([itex]A[/itex]) as a function of [itex]x[/itex] be [itex]A(x)=A_0\sqrt{\frac{\mu(0)}{\mu(x)}}[/itex] ?
(Again, assuming tension and frequency are fixed.)

 

[Orodruin]

You also need to consider that part of the wave will be reflected where the medium changes density. If you think about the limiting case where the new medium has a very large mass, it will essentially act as if the lighter part was fixed in a wall. Most of the power would then go into the reflected wave.

 

[Nathanael]

You also need to consider that part of the wave will be reflected where the medium changes density.

Would the frequency of the reflected wave and that of the wave that moves along the denser part be the same? (If this is true, is the reason for it's truth related to the idea that the two rope densities have a 'common point'?)

If you think about the limiting case where the new medium has a very large mass, it will essentially act as if the lighter part was fixed in a wall.

What if the wave is traveling from a higher density to a lower density? A wave would still be reflected? Would the limiting case (as the density goes to zero) then be as if it were fixed to a frictionless ring? (A "free end" is what I think my book called it.) Then, the smaller the density of the second part, the more power would go into reflected wave?

If you think about the limiting case where the new medium has a very large mass, it will essentially act as if the lighter part was fixed in a wall. Most of the power would then go into the reflected wave.

I haven't been able to see any way of determining how the power is distributed based on the relative density. Any insight?

If I were to continuously generate a wave, and then it were to cross some point where the density changes, part of it being reflected and part of it continuing on; would I then be creating a 'partially standing' wave? That is, would there be 'semi-nodes' where the amplitude never gets bigger than some limit? (The limit of the amplitude of the "semi-node" would depend on how much power is reflected.)

 

[Nathanael]

What if the density of a rope varied linearly, [[itex]\mu (x)=\mu_0+kx[/itex]] then what would you be able to say about a wave traveling along that rope?

Would it be as if a wave of increasing power are continuously being reflected? (But if that's so, and if I was right about waves reflecting when the density suddenly decreases, then wou have something like waves reflecting back and forth everywhere. That could either drastically confuse things or drastically simplify things; either way, I'm drastically curious.)

 

[pmacias]

I cannot confirm the accuracy of the proposed relationship between the change in linear density and the change in amplitude. While it is true that the average power of the wave is dependent on the amplitude and linear density, the specific relationship between these two variables is not as straightforward as \frac{1}{\sqrt{k}}. It would depend on the specific properties of the rope, such as its elasticity and stiffness, as well as the type of wave traveling on the rope.

Furthermore, the assumption that the average power remains unchanged may not always hold true. For example, if the change in linear density results in a change in tension in the rope, then the average power may also change. In this case, the relationship between the change in linear density and the change in amplitude would be even more complex.

In general, it is not safe to make assumptions about the relationship between different variables in a scientific context without proper experimental evidence or theoretical models to support them. Therefore, it would be incorrect to assume that a change in linear density by a factor of k would result in a change in amplitude by a factor of \frac{1}{\sqrt{k}}. More research and analysis would be needed to accurately determine the relationship between these variables.