# Homework Help: Dispersion relationship for internal gravity wave

1. Dec 15, 2011

### danhall24

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.