Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Dispersion relationship for internal gravity wave

  1. Dec 15, 2011 #1
    Hi, I am a postgraduate environmental science student (NOT a mathematician!) struggling through some necessary maths. Any help with the following (which I suspect will be relatively straight foward) would be very much appreciated. Please ask questions if I have not made myself clear.

    1. The problem statement, all variables and given/known data

    Given the dispersion relationship and intrinsic frequency of a 2-d internal gravity wave (in the horizontal (x dimension) and vertical (z dimension)), show that / explain why the group velocity vector is parallel to lines of constant phase and hence perpendicular to the phase speed. Note that this isn't a question I have been set, it is simply something I am struggling with from a text book.

    (I have checked and rechecked the equations below - they are definitely exactly the same as in the textbook.)

    2. Relevant equations

    Dispersion relationship:
    (ω - uk)2 (k2 + m2) - N2k2 = 0

    where ω is frequency, u is flow speed in the x direction (which is constant, i.e. does not vary in the z direction), k is wavenumber in the x direction, m is wavenumber in the z direction and N is a constant.

    Intrinsic frequency, v:
    v = ω - uk = Nk / (k2 + m2)1/2

    Horizontal phase speed, cx:
    cx = v / k

    Vertical phase speed, cz:
    cz = v / m

    Horizontal group velocity, cgx:
    cgx = ∂v/∂k = u + (Nm2) / (k2 + m2) 3/2

    Vertical group velocity, cgz:
    cgz = ∂v/∂m = -Nkm / (k2 + m2) 3/2

    In the textbook it simply says "it is easily shown from [the group velocity equations] that the group velocity vector is parallel to lines of constant phase."
    It may be easy, but not for me. Any help much appreciated.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted