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Sinusodial wave motion and reflection

  1. Jan 18, 2007 #1
    1. The problem statement, all variables and given/known data

    Okay, I've got a question that's been bugging me for the longest time.

    I've got a string attached to a wall at one end (the other end is free to move, though) and it has a sinusoidal wave traveling to the right and hits the wall at x = L and reflects. I'm supposed to prove that the sum of the leftward and rightward waves are

    yR(x,t) + yL(x,t) = 2Asin(k(x - L))cos(kvt + phi)

    which I did (unless there's a mistake somewhere), which makes a nice standing wave. but now they want me to solve for phi (the phase shift) in terms of k, L, and v. No numbers are given, it's all variables.

    2. Relevant equations

    yR(x,t) + yL(x,t) = 2Asin(k(x - L))cos(kvt + phi) <--- equation of the standing wave
    yR(x,t) = Asin(k((x - L) - vt) - phi) <--- rightward wave equation (I hope)
    yL(x,t) = Asin(k((x - L) + vt) + phi) <--- leftward wave equation (I hope)
    k = 2pi / lambda
    v = lambda / T
    k = omega / v

    k = wave number
    v = velocity of wave
    lambda = wavelength
    phi = phase shift
    y = height of section of wave at time t
    T = period

    3. The attempt at a solution

    Okay, I know that at t = 0, parts of the wave actually hit the maximum height of the wave (2A). I found out that the parts of the rope hit this height are at x = L - n*lambda/4 = L - n*pi/2k, but putting this into the formula just gets rid of L completely, which is probably something I don't want to do. I did find out that phi = -kvt (set x to L - n*2pi/k and solve for phi), but I need L, not t.

    I tried substituting x = 0 and t = 0 right away, but the problem is I don't know any initial values and just get y = 2Asin(-kL)cos(phi). This leads to cos(phi) = y(0,0)/2Asin(-kL), which I dunno helps or not. Besides, setting t=0 gets rid of the v term, which I think I need.

    Differentiation gets me nowhere, so far. I tried to differentiate before adding and adding before differentiating and seeing if I can get an equation out of that, to no avail. Doing dy/dt brings out the v, but I dunno what to do with that (finding the max height means setting dy/dt to 0, which just gets rid of the constant coefficients, like v).

    I'm really stuck and have been looking at this for about two days. Help!
  2. jcsd
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