J.R. Holton in his *An Introduction to Dynamic Meteorology* (p.199) gives the linearized version
of the governing equations for a two-dimensional internal gravity
wave system as:

(∂/∂t + __u__(∂/∂x))u' + (1/ρ_{0})(∂p'/∂x) = 0

(∂/∂t + __u__(∂/∂x))w' + (1/ρ_{0})(∂p'/∂z) - (g/__θ__)θ' = 0

∂u'/∂x + ∂w'/∂z = 0

(∂/∂t + __u__(∂/∂x))θ' + w'(d__θ__/dz) = 0

The solution for w' in this system of equations
was found by Holton to be of the form w'=Re[We^{iφ}] where
W is a complex constant and φ=kx+mz-νt. The system of
equations is homogeneous so the solutions for u', p' and θ'
must be of the same functional form as that of w'; i.e.,

u' = Re[Ue^{iφ}]

p' = Re[Pe^{iφ}]

θ' = Re[Θe^{iφ}]

where U, P and Θ are complex constants.

When the forms, Ue^{iφ}, Pe^{iφ} and
Θe^{iφ} are substituted into the equation system
the result is a set of linear complex equations in U, P and Θ;
i.e.,

(__u__k-ν)U +(k/ρ_{0})P = 0

i(__u__k-ν)W +(im/ρ_{0})P - (g/__θ__)Θ = 0

mW + kU = 0

(d__θ__/dz)W + i(__u__k-ν)Θ = 0

This system of linear homogeneous equation has only the trivial
solution unless the determinant of the coefficient matrix is zero.
The coefficient matrix is

| 0 | (__u__k-ν) |
k/ρ_{0} | 0 | |

| i
| 0 |
(im/ρ_{0}) | -g/__θ__ | |

| m | k | 0 | 0 | |

| (d__θ__/dz) | 0 | 0 |
i(__u__k-ν) | |

The evaluation of a fourth order determinant seems formidable but there
are enough zero elements to make the evaluation quite easy. The value
of the determinant D is given by

-(__u__k-ν)[-m(-m(__u__k-ν)/&ρ_{0}] + (k/ρ_{0})[k(-(__u__k-ν)^{2} + (g/__θ__)(d__θ__/dz)]

= - [(__u__k-ν)^{2}(k^{2}+m^{2}) - g(d ln(__θ__)/dz)]/ρ_{0}

If the determinant is to be zero it must be that

(__u__k-ν)^{2} = gk^{2}(d ln(__θ__)/dz)/K^{2}

where K^{2}=(k^{2}+m^{2}).

Let g(d ln(__θ__)/dz) be denoted as N^{2}.
Then the allowable frequencies ν are given by

ν = __u__k ±Nk/K

When the determinant condition is satisfied the solutions for
U, P and Θ can be expressed in terms of W.

From the third equation it is found that

U = -(m/k)W.

From the
fourth equation

Θ = -[(d__θ__/dz)/(i(__u__k-ν))]W.

From the first equation then

P=[ρ_{0}(__u__k-ν)/k]U
=-[ρ_{0}(__u__k-ν)m/k^{2}]

In summary

U = -(m/k)W

P = -[ρ_{0}(__u__k-ν)m/k^{2}]W

Θ = i[(d__θ__/dz)/(__u__k-ν)]W

Taking into account the dispersion relation these reduce to:

U = -(m/k)W

P = ±[ρ_{0}Nm/(k(k^{2}+m^{2})^{1/2})]W

Θ = ±i[(d__θ__/dz)(k^{2}+m^{2})^{1/2})/(Nk)]W