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

[99wattr89]

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?

 

[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.

 

[99wattr89]

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 = k₁wAe^i(wt-k₁x)

So then k₁wAe^i(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.

 

[99wattr89]

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 +A_reflected = A_transmitted

and

k₁(A +A_r) = k₂A_t

How can I get from this get to the answer?

 

[99wattr89]

Can anyone help me with this?