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Plane Wave Propogation (help understanding steps in book)

  1. Dec 3, 2006 #1
    Plane Wave Propagation (help understanding steps in book)

    Basically this problem requires that [itex] \vec H [/itex] vanishes at [itex] t = 3ms [/itex]. I don't understand how they did the last step, here is the solution:

    [tex] \vec H = \hat z 4 \times 10^{-6} \cos \left(10^7 \pi t - k_0 y +\frac{\pi}{4} \right) [/tex]

    [tex] k_0 = \omega \sqrt{\mu_0 \epsilon_0} = \frac{10^7 \pi}{3 \times 10^8} = 0.105 \,\,\,\, (rad/m) [/tex]
    [tex] \lambda = \frac{2\pi}{k_0} = 60 \,\,\,\, (m) [/tex]

    At [itex] t = 3 \times 10^{-3} \,\,\,\, (s) [/itex] we require the argument of cosine in [itex] \vec H [/itex]:

    [tex] 10^7 \pi (3 \times 10^{-3}) - \frac{\pi}{30}y +\frac{\pi}{4} = \pm n \pi + \frac{\pi}{2}, \,\,\,\,n=0,1,2,\ldots [/tex]

    [tex] y = \pm 30n - 7.5 \,\,\,\, (m) [/tex]

    Now how the hell do they get the last line? The... [itex] y = \pm 30n - 7.5 \,\,\,\, (m) [/itex] (Lets call this (1)) line.
    They drop [itex] 10^7 \pi (3 \times 10^{-3}) [/itex] why?

    My problem is that if we plug in (1) into,
    [tex] \vec H = \hat z 4 \times 10^{-6} \cos \left(10^7 \pi t - k_0 y +\frac{\pi}{4} \right) [/tex]

    and set t = 0.003, then if we feed the function values of n then the cosine doesn't drop to 0.

    Thanks in advance!
     
    Last edited: Dec 3, 2006
  2. jcsd
  3. Dec 3, 2006 #2

    OlderDan

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    The term they dropped is a multiple of pi (and in fact a multiple of 2pi). Adding a multiple of pi to an angle at which the cosine is zero is still going to give you a cosine of zero.
     
  4. Dec 4, 2006 #3
    Ahh... tricky tricky. (I mean, yeah, it's not too tricky but still, I guess I will add this to my list of things to look for subconsciously while evaluating expressions)
     
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