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Waves on a 3-piece string

  1. Apr 18, 2013 #1
    1. The problem statement, all variables and given/known data
    o8rmna.png


    2. Relevant equations



    3. The attempt at a solution

    ofnkv6.jpg

    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..

    2d9pfl4.jpg

    I got the relation between A1, A2, A3 and A4 but still no A5..
     
  2. jcsd
  3. Apr 18, 2013 #2

    Simon Bridge

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    Did you use that the incoming wave has unit amplitude?
     
  4. Apr 19, 2013 #3
    Despite that, the equations 7 and 8 have a exp term, how do i get rid of that to find just the amplitude A5?
     
  5. Jun 5, 2013 #4
    I have recently given it another go:

    3347nmr.png

    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.
     
  6. Jun 5, 2013 #5
    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 travelling 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.
     
  7. Jun 5, 2013 #6

    Simon Bridge

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

    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=A3,4,5. 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.
     
    Last edited: Jun 5, 2013
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