Dispersion relationship for internal gravity wave

[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.

Homework Statement

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 textbook.

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

Homework 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, 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 + (Nm²) / (k² + m²) ^3/2

Vertical group velocity, c_gz:
c_gz = ∂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.

 

[Nate810]

The Answer From the equations for the horizontal and vertical phase speed, we can see that the phase speed is a function of both k and m. That is, the phase speed is not constant, but varies with changes to both the horizontal and vertical wavenumbers. Therefore, lines of constant phase form a surface in the k-m-space.Now, let's look at the group velocity equations. For the horizontal group velocity, we can see that it is a function of both k and m, like the phase speed. However, for the vertical group velocity, we can see that it is only a function of k. That is, the vertical group velocity does not vary with changes to the vertical wavenumber (m). This implies that the group velocity vector is parallel to lines of constant phase, since the vertical component of the group velocity vector does not vary with changes to m, and thus it is always perpendicular to the surface of constant phase in the k-m-space.