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

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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 Recognitions: Gold Member Science Advisor Staff Emeritus 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.

 Quote by HallsofIvy 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.

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.

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

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?
 Can anyone help me with this?

 Tags oscillation, strings, wave equation, waves