= Solution
Use the inviscid, nonrotating <Boussinesq approximation>, with a stable background <mass density> $\bar\rho(z)$ and reference density $\rho_*$. Write $B=-g\rho'/\rho_*$ for the perturbation <buoyancy> and $\pi=p'/\rho_*$ for the kinematic pressure. The <buoyancy frequency> is $N_0^2=-(g/\rho_*)\bar\rho_z>0$. Dropping products of perturbations in the <Boussinesq equations> gives the <Linearized Boussinesq equations>
$$
\mathbf u_t=-\nabla\pi+B\mathbf e_z,\qquad B_t+N_0^2w=0,\qquad \nabla\cdot\mathbf u=0.
$$
The last equation expresses <incompressible flow>; the second follows by advecting the background <mass density> gradient. Neglecting rotation and <viscosity> is part of this <internal gravity wave> model.
For a <plane internal gravity wave> proportional to $\exp[i(\mathbf k_h\cdot\mathbf x_h+mz-\omega t)]$, put $\kappa=|\mathbf k_h|>0$ and $K^2=\kappa^2+m^2$. Eliminating the horizontal <velocity>, pressure and <buoyancy> from the linear equations gives
$$
\boxed{\omega^2=N_0^2\frac{\kappa^2}{\kappa^2+m^2}.}
$$
For example, the horizontal <momentum> and continuity equations give $\pi=-\omega m w/\kappa^2$; substituting $B=-iN_0^2w/\omega$ into vertical <momentum> yields the displayed <dispersion relation>. Thus an <IGW> has $0<|\omega|\le N_0$ in this model.
On the positive-frequency branch, the <phase velocity> normal to a constant-phase plane and the <group velocity> are
$$
\boxed{\mathbf c_p=\frac{\omega}{K^2}(\mathbf k_h,m),}\qquad
\mathbf c_{g,h}=\frac{N_0m^2}{K^3}\frac{\mathbf k_h}{\kappa},\quad
c_{g,z}=-\frac{N_0\kappa m}{K^3}.
$$
Consequently $(\mathbf k_h,m)\cdot\mathbf c_g=0$, and \b[phase and <group velocity> are perpendicular]. Equivalently, the <dispersion relation> is homogeneous of degree zero in the <wave vector>, so differentiating with respect to its scale proves the same orthogonality. Energy travels with the <group velocity>, along the phase planes. At the degenerate limit $m=0$, $\omega=N_0$ and the <group velocity> vanishes; the orthogonality statement then has this limiting interpretation.
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