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Superposition of two waves with different frequencies

  1. Apr 17, 2014 #1
    1. The problem statement, all variables and given/known data
    Hi all! It's a superposition question: Two waves travel through dispersive medium, with different frequencies and wave number.
    P1(t)=Acos(k1x-w1t)
    P2(t)=Acos(k2x-w2t)
    Obtain the P(t)=P1(t)+P2(t)
    2. Relevant equations
    Well I used identity:
    cosα+cosβ=2 cos 1/2(α+β)cos1/2(α-β)
    and the following:
    w(av)=(w1+w2)/2 Δw=w1-w2
    k(av)= (k1+k2)/2 Δk=K1-k2

    3. The attempt at a solution
    So, this is what I tried to do:
    P(t)=A0(2cos(1/2)((k1x-w1t)+(k2x+w2t))cos(1/2)((k1x-w1t)-(k2x-w2t)

    =2A0(cos(((k+k)/2)x)-((w-w)/2)t))cos(((k-k)/2)x)-((w+w)/2)t))

    =2A0(cos(k(av)-(1/2)Δwt)cos((1/2)Δkx-w(av)t))

    And, from here on I'm stuck: Is this all that needed? Help would be very appreciated. :)
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 17, 2014 #2

    BvU

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    I think this is all that's needed. But you have to be a bit sharper with the signs, to get k(av) and w(av) in the first and Δk and Δw in the second cosine - I think...
     
  4. Apr 17, 2014 #3
    Hmm, what do you mean?
    Thank for the help by the way :)
     
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