Wave on a string meeting a boundary between areas of different densities

In summary, the problem the respondent is working on is finding a way to solve the wave equation. They are having difficulty finding a solution and are looking for help.
  • #1
99wattr89
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0
This is the problem I'm working on: http://i.imgur.com/PBMFG.png

I'm very behind with normal modes and waves, and I need to figure out how to do this sort of question in time for my exams, so I'm hoping that you guys will be able to help me see how this can be answered.

I've answered the first part, deriving he wave equation, but for the second part I'm feeling very lost. Can someone give me a hint or nudge in the right direction for how to get started with it?
 
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  • #2
Do it as two separate problems. Solve the wave equation for x< 0 with "X" amplitude at 0, for x> 0 with "X" amplitude at 0, then determine "X" so the function is continuous at x= 0.
 
  • #3
HallsofIvy said:
Do it as two separate problems. Solve the wave equation for x< 0 with "X" amplitude at 0, for x> 0 with "X" amplitude at 0, then determine "X" so the function is continuous at x= 0.

Thank you for your reply!

I think I get the idea there, but unfortunately I'm not having much success doing it.

As I understand it, solving the wave equation means solving ∂2y/∂x∂t = 0

(I don't understand why you do that though. Am I wrong in thinking that's the way you solve the wave equation?)

To solve that, I differentiated y wrt x then t, to get ∂2y/∂x∂t = k1wAei(wt-k1x)

So then k1wAei(wt-k1x) = 0

But that would just mean that either A = 0 (which I'm pretty sure just means that there is no wave at all) or that w = 0 (which I think would mean that the wave doesn't move) or k1 = 0 (which I think would mean that the wavelength was infinite and so the wave would effectively be straight line). None of those possibilities seem like anything approaching a solution, so I think I'm doing this wrong.
 
  • #4
I've been trying other things, but it still doesn't work.

I know that the amplitude of the reflected and incident waves added will equal the amplitude of the transmitted wave, and that dy/dx at 0 will also be equal for the sum of the incident and reflected waves, and the transmitted wave. But all that gives me is;

A +Areflected = Atransmitted

and

k1(A +Ar) = k2At

How can I get from this get to the answer?
 
  • #5
Can anyone help me with this?
 

1. What is a wave on a string?

A wave on a string is a type of mechanical wave that travels along a medium, or a material substance, such as a string. The wave is created when energy is transferred to the medium, causing it to vibrate. This vibration is then passed along the string in the form of a wave.

2. How does a wave on a string meet a boundary between areas of different densities?

When a wave on a string meets a boundary between areas of different densities, two things can happen. The wave can either be reflected, meaning it bounces off the boundary and travels back in the opposite direction, or it can be transmitted, meaning it passes through the boundary and continues traveling in the same direction.

3. What is the effect of a boundary on a wave on a string?

The effect of a boundary on a wave on a string depends on the density of the medium on either side of the boundary. If the density is higher on one side of the boundary, the wave will slow down as it enters that medium, causing it to change direction. If the density is lower on one side of the boundary, the wave will speed up as it enters that medium, also causing it to change direction.

4. How does the density of a medium affect the behavior of a wave on a string?

The density of a medium has a significant impact on the behavior of a wave on a string. As the density of a medium increases, the speed of the wave decreases, and its wavelength becomes shorter. This results in a higher frequency and a higher pitch of the sound produced by the wave. On the other hand, as the density decreases, the speed of the wave increases, and its wavelength becomes longer, resulting in a lower frequency and a lower pitch of the sound produced.

5. Can a wave on a string pass through boundaries with different densities indefinitely?

No, a wave on a string cannot pass through boundaries with different densities indefinitely. The wave will eventually lose energy due to friction and other factors, causing it to dissipate. Additionally, as the wave passes through multiple boundaries, it will continue to be reflected and transmitted, resulting in a decrease in amplitude. Eventually, the wave will no longer be distinguishable from the background noise in the medium.

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