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.(adsbygoogle = window.adsbygoogle || []).push({});

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}(k^{2}+ m^{2}) - N^{2}k^{2}= 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 / (k^{2}+ m^{2})^{1/2}

Horizontal phase speed, c_{x}:

c_{x}= v / k

Vertical phase speed, c_{z}:

c_{z}= v / m

Horizontal group velocity, c_{gx}:

c_{gx}= ∂v/∂k = u + (Nm^{2}) / (k^{2}+ m^{2})^{3/2}

Vertical group velocity, c_{gz}:

c_{gz}= ∂v/∂m = -Nkm / (k^{2}+ m^{2})^{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.

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# Homework Help: Dispersion relationship for internal gravity wave

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