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Harmonic Waves on a String

  1. Jan 20, 2010 #1
    1. The problem statement, all variables and given/known data
    Two very long strings are tied together at the point x = 0. In the region x < 0, the wave speed is v1, while in the region x > 0, the speed is v2. A sinusoidal wave is incident on the knot from the left (x < 0); part of the wave is reflected and part is transmitted.

    For x < 0, the displacement of the wave is described by...
    y(x,t) = Asin(k1x - wt) + Bsin(k1x + wt)

    For x > 0, the displacement of the wave is described by...
    y(x,t) = Csin(k2x - wt)

    where,
    w/k1 = v1
    w/k2 = v2

    a) if we assume that both the wave function y and its first spatial derivative dy/dx must be continuous at x = 0, show that:
    C/A = 2v2/(v1 + v2),
    and that:
    B/A = (v1 - v2)/(v1 + v2)

    b) Show that
    B2 + (v1/v2)C2 = A2

    2. Relevant equations
    y(x,t) = Asin(kx +- wt)
    This shows that A, B, C are the amplitudes of their portions.


    3. The attempt at a solution
    I'm really unsure how to go about his problem. I am getting stuck on the idea of getting C/A or B/A because I don't really see how you can solve for them.
     
  2. jcsd
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