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Wave vector algebra

  1. Apr 5, 2013 #1
    1. The problem statement, all variables and given/known data

    I'm having difficulty showing that the general equation k = 2∏/λ holds from the component equations kx = 2∏/λx etc..., k = √(kx2 + ky2 + kz2), and λ = √(λx2 + λy2 + λz2). Any help? There is a photo from a textbook with the equations also below.

    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Apr 6, 2013 #2

    BruceW

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    what are you defining λ to be?

    edit: also, if you use λ in the equation k = 2∏/λ then you can see it doesn't work.
     
  4. Apr 6, 2013 #3
    Yes, I'm just tryin to derive the formular for λ from the component formulae, but obviously (as you state) it doesn't seem to work. What then, is the justification for the association between the general wavenumber and the component wavenumbers if they don't correspond in the natural way?
     
  5. Apr 6, 2013 #4

    BruceW

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    the general wavenumber does correspond in the natural way to the component wavenumbers. But the general wavenumber does not correspond in the natural way to the component wavelengths. Is this what you were thinking about?
     
  6. Apr 7, 2013 #5
    Yes, precisely. I wonder, then, how the component wavenumbers (and their corresponding component wavelengths) are related to the general wavenumber and wavelength?
     
  7. Apr 7, 2013 #6

    BruceW

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    if you think about the vector wavenumber and vector wavelength: (kx,ky,kz) (call it k) and (λx,λy,λz) (call it λ) then there is a nice equation you can write, which involves the two vectors k and λ (hint: what kind of operation can you use between two vectors?)
     
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