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/ρ₀)(∂p'/∂x) = 0
(∂/∂t + u(∂/∂x))w' + (1/ρ₀)(∂p'/∂z) - (g/θ)θ' = 0
∂u'/∂x + ∂w'/∂z = 0
(∂/∂t + u(∂/∂x))θ' + w'(dθ/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.,

(uk-ν)U +(k/ρ₀)P = 0
i(uk-ν)W +(im/ρ₀)P - (g/θ)Θ = 0
mW + kU = 0
(dθ/dz)W + i(uk-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	(uk-ν)	k/ρ0	0       |
| i	0	(im/ρ0)	-g/θ   |
| m	k	0	0       |
| (dθ/dz)	0	0	i(uk-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

-(uk-ν)[-m(-m(uk-ν)/&ρ₀] + (k/ρ₀)[k(-(uk-ν)² + (g/θ)(dθ/dz)]
= - [(uk-ν)²(k²+m²) - g(d ln(θ)/dz)]/ρ₀
 

If the determinant is to be zero it must be that

(uk-ν)² = gk²(d ln(θ)/dz)/K²
 

where K²=(k²+m²).

Let g(d ln(θ)/dz) be denoted as N². Then the allowable frequencies ν are given by

ν = uk ±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 = -[(dθ/dz)/(i(uk-ν))]W. From the first equation then
P=[ρ₀(uk-ν)/k]U =-[ρ₀(uk-ν)m/k²]

In summary

U = -(m/k)W
P = -[ρ₀(uk-ν)m/k²]W
Θ = i[(dθ/dz)/(uk-ν)]W
 

Taking into account the dispersion relation these reduce to:

U = -(m/k)W
P = ±[ρ₀Nm/(k(k²+m²)^1/2)]W
Θ = ±i[(dθ/dz)(k²+m²)^1/2)/(Nk)]W