Consider an infinitesimal parcel of air of area dA and height dz which is embedded in a column of air in hydrostatic balance. Let p(z) and ρ(z) denote the pressure and density in the parcel and p₀(z) and ρ₀(z) the pressure and temperature in the column.

The parcel pressure and density are assumed to be equal to to those of the column at one particular height z₀. But if the parcel is moved adiabatically to a different level the pressure and density of the parcel may deviate from those of its environment in the column.

The parcel at some height z would be subject to two forces, gravity and the pressure forces from the environmental atmosphere. By Newton's Second Law

(ρdAdz)(d²z/dt²) = -(ρdAdz)g - dA(∂p₀/dz)dz
which after division by the parcel volume dAdz reduces to
ρ(d²z/dt² = -ρg - (∂p₀/dz)
 

Because the column of air is in hydrostatic balance

(∂p/dz)₀ = -ρ₀g
and thus the force balance equation
for the parcel is
ρ( d²z/dt²) = -ρg + ρ₀g
hence
d²z/dt² = -g(ρ-ρ₀)/ρ
 

At the equilibrium height of the parcel z₀, ρ and ρ₀ are equal. Suppose that the parcel always adjusts to the local environmental pressure so that buoyancy is due only to the difference in density of the parcel compared to the environment. Since by the ideal gas law

(ρ-ρ₀)/ρ = [p/RT - p/RT₀]/(p/RT)
= [1 - T/T₀]
 

But T/T₀ = θ/θ₀ since the pressures are equal. Thus

d²z/dt² = g[1 - θ/θ₀]
 

For the moment designate the RHS of the above equation as a(z). This acceleration a(z) is zero at z₀ so by a Taylor's series expansion

a(z) = (∂a/∂z)_z=z0(z-z₀)
 

The potential temperature of the parcel θ is constant because the parcel adjusts adiabatically. Therefore the change in a(z) with height is due entirely to the variation in θ₀ with height.

(∂a/∂z) = -(θ/θ₀²)(dθ₀/dz)
which at z=z₀ where θ=θ₀ reduces to
(∂a/∂z) = -(1/θ₀)(dθ₀/dz) = -d(ln θ₀)/dz
 

Thus the acceleration equation is

d²z/dt² = g[d(ln θ₀)/dz](z-z₀)
 

This is the well-known differential equation for harmonic motion about an equilibrium at z₀ and the frequency of oscillation is therefore given by N where

N² = g[d(ln θ₀)/dz]
 

N is called the Brunt-Väisälä buoyancy frequency.