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Spherical wave

  1. May 1, 2007 #1
    Hey!

    Im quite confused about spherical waves. I mean, I understand that a spherical wave can be described by

    [tex]
    \Psi = \frac{1}{r} e^{i r},
    [/tex]

    because the intensity of such a wave decreases as [tex]1/r^2[/tex]. The intensity of such a wave is given by [tex] I = 1/r^2 [/tex] which makes sense to me. But a spherical wave can also be described by

    [tex]
    \Psi = \frac{1}{r} \cos r,
    [/tex]

    which gives a much different behaviour of the intensity because the intensity of such a wave is [tex] 1/r^2 cos^2(r) [/tex]. If these two expressions both describe a spherical wave, how come they don't have the same intensity?
     
  2. jcsd
  3. May 1, 2007 #2
    At a guess, I'd say that they're the same thing if you take time averages.

    <cos^2(kx-wt)>=1/2
     
  4. May 1, 2007 #3

    Dick

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    Those are spherical functions as in that they are angle independent. As they have no time dependence, in what sense are they waves?
     
  5. May 2, 2007 #4
    Okay, so if they both had time dependence [tex]-i \omega t[/tex] so that

    [tex]
    \Psi = \frac{1}{r} e^{i ( k r - \omega t)}
    [/tex]

    and

    [tex]
    \Psi = \frac{1}{r} \cos( k r - \omega t)},
    [/tex]

    but they still don't have the same intensity, since the intensity of the second one is an oscillating function of r and t whereas the first one takes off as 1/r^2 and is therefore not oscillating.
     
  6. May 2, 2007 #5
    Again, <cos^2(kr-wt)>=1/2 at any particular value of r and averaging over time.
     
  7. May 2, 2007 #6

    Dick

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    ???? Not oscillating? The second equation is the real part of the first.
     
  8. May 2, 2007 #7
    But isn't the intensity given by [tex]\Psi \Psi^*[/tex]? This gives an intensity equal to 1/r^2 for the wave described by a complex exponential function but an intensity equal to [tex]1/r^2 cos^2(k r - w t)[/tex] for the other one.
     
  9. May 2, 2007 #8

    Dick

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    Ah, that's what you're trying to say. Yes, the cos one has a time-dependent 'intensity' and the other doesn't. But as christianjb pointed out, <cos^2>=1/2. So in an average sense one is 1/2 of the other. Not surprising since it's also the 'real half'.
     
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