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Group Velocity in terms of Wavlength and velocity

  1. Apr 17, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that the group velocity
    vg=dω/dk
    can be written as
    vg=v-λ*dv/dλ

    where v = phase velocity

    2. Relevant equations

    n=n(k)=c/v
    k=2∏/λ
    ω=2∏f=kv
    fλ=c

    3. The attempt at a solution
    dω/dk = d(kv)/dk= v+k(dv/dk)= v+ck(d(n^-1)/dk) =v-(ck/n^2)(dn/dk)
    =>v-(vk/n)(dn/dk) = v-(ω/n)(dn/dk) = v- (2∏f*v/c)(dn/dk)
    => v(1-(1/λ)(dn/d(1/λ))
    I'm not sure where to go from here, I've been working at this a while and I'm not sure how I could get the 1/λ to become just a λ. I'm also not sure how to get my n in dn/d(1/λ) to be a v since say if I multiply that term by c/c then I get

    v(1-(c/λ)d(1/v)/d(1/λ)) where c/λ is really just f
     
  2. jcsd
  3. Apr 17, 2013 #2

    rude man

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    Homework Helper
    Gold Member

    Don't mess with n.
    Write w as a function of k and v where w = 2pi f.
    Then take dw
    The rest is just messing around with k = 2pi/lambda, eliminating k.
    You will also need dv/dk. Hint: chain rule.
     
  4. Apr 17, 2013 #3
    Got it, thanks that was way easier than I was making it
     
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