Finding A5 in a System of Equations

[unscientific]

Homework Statement

Homework Equations

The Attempt at a Solution

The 4 main equations are equations (1), (2), (7) and (8).
But I can't seem to isolate A₅ alone as solving equations (7) and (8) will make A₅ disappear..

I got the relation between A₁, A₂, A₃ and A₄ but still no A₅..

 

[Simon Bridge]

Did you use that the incoming wave has unit amplitude?

 

[unscientific]

Did you use that the incoming wave has unit amplitude?

Despite that, the equations 7 and 8 have a exp term, how do i get rid of that to find just the amplitude A₅?

 

[unscientific]

I have recently given it another go:

I've ended up with these 4 equations. The question wants us to find A₅.

But surely A₅ must be exponential? in order to make the resultant term on the RHS of equations (3) and (4) non-exponential. Also, when I tried to express A₅ in terms of A₁, A₂, A₃ or A₄ I can't, cause equations (3) and (4) sort of make A₅ ' cancel out '

I'm not sure how to go about doing this.

 

[unscientific]

Here's the typed out question:

A semi-infinite string of mass per unit length p1 lies along the negative x-axis and is attached at x=0 to a second string of length a, with mass per unit length p2, lying along the positive x-axis. The end of this string at x=a is attached to a third (semi-infinite) string of mass per unit length p3 which lies along the positive x-axis. The combined string is under tension F.

A wave of unit amplitude, frequency w and wavelength λ1 traveling from negative towards positive x is incident at the joint x = 0. Write down expressions for the waves propagating in these three regions:

I: x<0
II: 0<x<a
III: x>a.

What are the boundary conditions that need to be satisfied at x = 0 and x = a? For the case of a = λ2, the wavelength of the propagating wave in region II, find the transmitted amplitude T3 in region III, in terms of p1, p2 and tension.

 

[Simon Bridge]

I've ended up with these 4 equations. The question wants us to find A5.

You forgot to put $a=\lambda_2 \; (=2\pi/k_2)$ in some of those equations, and I think there should be a k₁ in there somewhere. I'm also a bit concerned about the signs in there.

But surely A5 must be exponential? in order to make the resultant term on the RHS of equations (3) and (4) non-exponential.

But it's only exponential in a constant ratio, so why does that matter?

Also, when I tried to express A5 in terms of A1, A2, A3 or A4 I can't, cause equations (3) and (4) sort of make A5 ' cancel out

You can cancel A₅ that way - but only to get a relation between A₃ and A₄ ... but that does not make A₅ unfindable, you just have to use a different substitution.

It helps to change notation to something simpler ... i.e. put p,q,r in place of k1,2,3 respectively, and put z=exp[-ik₃a] or whatever that turns out to be. Then make A=A₂ and B,C,D=A_3,4,5. Now it should be easier to track the variables - notice that p,q,r and z are all constants?

Once you've checked your algebra, you have only to row-reduce the matrix of coefficients of A,B,C,D.