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Deriving Group Velocity for Gravity Waves

  1. Oct 10, 2011 #1
    1. The problem statement, all variables and given/known data
    In a lab experiment about Gravity Waves and Dispersion, one of the preliminary questions is:

    Show that for gravity waves the group velocity is:

    C[itex]_{g}[/itex] = [itex]\frac{C_{p}}{2}[/itex][itex]\sqrt{1 + \frac{2kh}{sinh(2kh)}}[/itex]

    2. Relevant equations

    C[itex]_{g}[/itex] = [itex]dω/dk[/itex]

    ω = [itex]\sqrt{gktanh(kh)}[/itex]

    where ω is the angular frequency, and
    C[itex]_{p}[/itex] = [itex]\sqrt{\frac{gtanh(hk)}{k}}[/itex]

    3. The attempt at a solution
    By using the product rule and the chain rule I get

    [itex]d/dk[/itex](gktanh(kh))[itex]^{1/2}[/itex] = [itex]\frac{1}{2}[/itex](gktanh(kh))[itex]^{-1/2}[/itex](gtanh(kh) + [itex]gk/cosh^{2}(kh))[/itex]

    But I have no idea where to go from here.
    Any help would be greatly appreciated.
     
  2. jcsd
  3. Oct 10, 2011 #2
    Not sure if it was a mistype, but the term with cosh^2 should have an h in the denominator.

    Now try to see if there is anything you can take out of the brackets to make C_p in front of them.
     
  4. Oct 11, 2011 #3
    Yeah that was supposed to have an h in the denominator sorry. I have tried numerous ways of factoring something out but i cant seem to get it to work
     
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