Wave behavior in a rope with density gradient

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Wave speed changes only when medium changes. But so far, all I've seen is a definite boundary behavior where one medium abruptly ends and another one begins. What happens if there is a gradient.

For example, what happens when a wave is passed through a rope with a density gradient. It is very dense on one end and not very dense on the other. As it passes through the rope, will it constantly be reflected? Will it not be reflected? What will be the behavior of the wave and why?

Another question that ties into this is how density affects the speed of a wave. I've heard that an increase in density will result in more mass and an increase in inertia. The net result is a slower wave. But I've also heard that more dense particles are closer together, so the wave is faster. I need some clarification on how this works and how both of these factors tie in.

Thanks for any clarification.
 

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Yuqing said:
Wave speed changes only when medium changes. But so far, all I've seen is a definite boundary behavior where one medium abruptly ends and another one begins. What happens if there is a gradient.

For example, what happens when a wave is passed through a rope with a density gradient. It is very dense on one end and not very dense on the other. As it passes through the rope, will it constantly be reflected? Will it not be reflected? What will be the behavior of the wave and why?

Another question that ties into this is how density affects the speed of a wave. I've heard that an increase in density will result in more mass and an increase in inertia. The net result is a slower wave. But I've also heard that more dense particles are closer together, so the wave is faster. I need some clarification on how this works and how both of these factors tie in.

Thanks for any clarification.

Here's an intro page on gradient-index optics, in case you haven't seen it yet:

http://en.wikipedia.org/wiki/Gradient_index_optics

.
 
I've read it but I can't seem to make it connect.

Thanks for bringing it up.
 
Can anyone help me with this?