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Light Interference Wave Equation - Assumption

  1. Apr 12, 2013 #1
    1. The problem statement, all variables and given/known data
    yyi6scl2cn6c9au4g.jpg
    2. Relevant equations

    In question.

    3. The attempt at a solution

    To be clear it's part (vi) that's unclear to me.

    In order to ignore the cosine term it has to reduce to 1. This can happen, only if k(x1+x2)/2 = ωt

    Is this a correct assumption ?

    Also, it is known that k = 2∏/λ and ω=2∏/T

    However, I'm trying to think in what way these two components could be equal but I can't get it.

    Could someone give me some help, please ?
     
  2. jcsd
  3. Apr 12, 2013 #2

    mfb

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    For a long exposure, you average y^2 over time - the constant phase in the cosine does not matter for the average.
     
  4. Apr 12, 2013 #3
    I really dont understand what you mean. Could you please break it down for me ?

    I really do appreciate you taking the time out to help me.
     
  5. Apr 25, 2013 #4

    Simon Bridge

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    A photograph collects light over a period of time.
    The lightness in the photo depends on the intensity of the incoming light and the time of the exposure.

    He's telling you to find y^2, then average it over time.
    This gives you a function of position alone - which will be what the photograph shows.
    When you do this - the terms you are worried about will cancel out.

    Do you know what y=?
    Do you know how to square a function?
    Do you know how to average a function?
     
  6. Apr 25, 2013 #5
    I can do the first two but not the last one. How do I average a function over time ?

    EDIT: Why do I need to find y^2 ? Is it because intensity is directly proportional to Amplitude^2 ?
     
  7. Apr 25, 2013 #6

    Simon Bridge

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  8. Apr 26, 2013 #7
  9. Apr 26, 2013 #8

    Simon Bridge

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    Did you read the links? The second one tells how to handle the average of a periodic function - what does it say to do?
     
  10. Apr 26, 2013 #9
    I've carried out the steps for the time average for cos2x for limits 0 to T.

    I've gotten : [itex]\frac{1}{T}[/itex][[itex]\frac{1}{2}[/itex]T+[itex]\frac{1}{4}[/itex]sin2T]

    What do I choose to be T in this case ? Infinity ?
     
  11. Apr 26, 2013 #10

    Simon Bridge

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    But it is not exactly cos2x that you have to average is it? x(t)=?

    The trick with time averaging square trig-functions is to choose your interval carefully - if you let T be any integer number of quarter cycles, the average will make sense easily. I usually pick a singe period for T.
    http://hyperphysics.phy-astr.gsu.edu/hbase/math/defint.html (3rd panel down)
     
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