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Wave formula question

  1. Aug 17, 2005 #1
    Why is the wave formula different in 3-D?
     
  2. jcsd
  3. Aug 17, 2005 #2
    Eh? Because it dependants from four values x,y,z,t and the one dimensional only on two value x,y.
     
  4. Aug 17, 2005 #3
    opps~ i think i didn't make myself clear...
    what i meant was why in 3-D does kx become k*r?
     
  5. Aug 17, 2005 #4

    Galileo

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    Because that's the natural extension. It reduces to one kx in 1D and is rotationally invariant: the predictions does not change by rotating your coordinate axes.
     
  6. Aug 17, 2005 #5
    you can rewrite k*x as k*x=kx*x+ky*y+kz*z where kx, ky and kz are the corresponding wave vectors of x, y and z.
     
  7. Aug 18, 2005 #6
    @@a
    isn't r the radius? what does that have to do with the wave vectors, x, y, z?
     
  8. Aug 18, 2005 #7

    jtbell

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    According to the three-dimensional Pythagorean formula:

    [tex]r^2 = x^2 + y^2 + z^2[/itex]

    Something that might be causing confusion here is that the formula for a three-dimensional wave depends on the "shape" of the wave. For a plane wave (whose maxima form a series of planes marching through space),

    [tex]\psi(x, y, z, t) = A \cos (\vec k \cdot \vec r - \omega t) = A \cos (k_x x + k_y y + k_z z - \omega t)[/tex]

    where the [itex]\vec k[/itex] vector and its components are constant.

    For a spherical wave (whose maxima form a series of concentric spheres spreading out from a central point, let's say the origin),

    [tex]\psi(x, y, z, t) = A \cos (kr - \omega t) = A \cos (k \sqrt{x^2 + y^2 + z^2} - \omega t)[/tex]

    At each point in a spherical wave the [itex]\vec k[/itex] vector points radially outward from the origin, so the direction is different everywhere but the magnitude [itex]k = \sqrt {k_x^2 + k_y^2 + k_z^2}[/itex] is constant.
     
    Last edited: Aug 18, 2005
  9. Aug 18, 2005 #8
    so k and r is different depending on the wave's shape?
    how many different kinds of different shapes are there?
     
  10. Aug 18, 2005 #9

    selfAdjoint

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    Infinitely many. As many as there are combinations of simple oscillators of different frequency and amplitude.
     
  11. Aug 19, 2005 #10
    thanks! :)
     
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