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Equation of a Wave

  1. Nov 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Equation of a wave is a = a°sin(ωt - kx + φ)

    where φ is the phase of a wave. if 2 waves with phases φ1 and φ2 interfere, show how the intensity I =a² varies as a function of the phase difference φ1 - φ2. Use one of the trigonometric double angle forumula or otherwise to obtain your result.



    2. Relevant equations
    The double angle formulas



    3. The attempt at a solution

    Well am I supposed to map I =a² onto the equation?

    If so then the only double angle formula is cos(2x) = cos²(x) - sin²(x)

    But I get a really stupid answer when I square the wave equation..

    What do I do?
     
  2. jcsd
  3. Nov 16, 2009 #2

    lanedance

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    show your work

    start by writing the sum of the amplitude of the 2 waves with same fequency but different phase & work from there...

    also intuitively, what do you expect will happen?
     
  4. Nov 16, 2009 #3
    I have 2 waves with a phase difference:

    a1 = a°sin(ωt - kx + φ1)
    a2 = a°sin(ωt - kx + φ2)

    If the waves combine, then an interference occurrs...

    a1 + a2 = a°sin(ωt - kx + φ1) + a°sin(ωt - kx + φ2)

    K apparently, sin a + sin b = 2cos 0.5(a - b) sin 0.5(a + b)

    So in relation:

    a1 + a2 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)

    a3 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)

    Now I =a²

    But I'm unsure of how to square this expression I have, assuming it is even right..
     
  5. Nov 16, 2009 #4

    tiny-tim

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    Hi Unto! :smile:

    I think by a "double angle formula" they mean like sin(A + B) or (sinA + sinB) etc …

    these are trigonometric identities which you must learn. :wink:
     
  6. Nov 16, 2009 #5

    lanedance

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    using a few diffenrt trigonamteric identities you can show the identity you used
    [tex] sin(a) + sin(b) = 2cos(\frac{a-b}{2})sin(\frac{a+b}{2}) [/tex]
     
    Last edited: Nov 17, 2009
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