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.