Budden  shows the derivation of the Appleton formula. For electromagnetic field components with time behavior exp(iωt), we conclude from the derivation that a field component solution of the coupled Maxwell and Newton's second law equations for inhomogeneous electron plasma in an external magnetic field has the form
where we recast the exponent in terms of time as the independent variable. We use the summation convention for repeated subscripts. The wave vector magnitude satisfies k = ηω/c, where c is the speed of light in a vacuum and η = η′ − iη″ is the complex index of refraction, given by the Appleton formula as
where fN is the electron plasma frequency, f is the wave frequency, ν is the electron collision frequency, e is the electron charge (<0), m is the electron mass, Bj is a component of the Earth's magnetic field vector, and fH is the associated electron gyrofrequency. The contribution of ions to the index of refraction is ignored, which is justified for frequencies of interest here (f > 0.5 MHz). The plus or minus signs in (2a) are associated with ordinary (O) and extraordinary (X) modes of propagation. The O mode is the one that minimizes the magnetic effect for transverse magnetic field geometry (YL = 0). Consequently, the O mode is associated with the plus sign for X < 1 and with the minus sign for X > 1. The alternate sign choices are made for the X mode. For these sign choices, the index is continuous at X = 1 for each mode [cf. Budden, 1985]. This choice of signs is easy enough to see for no collision frequency and real index, but when we include the collision frequency, the method for computing the complex index can affect the choice of sign for O and X modes. This is worth checking. In the calculation of D in (2a), we first let ρ1 exp(iϕ1) = YT2/2(U − X) and compute ϕ1 = a tan 2((ν/ω), 1 − X), where the intrinsic function atan2(y, x) computes the actangent of y/x and returns the result in the interval (−π, π] in standard Fortran or C++ implementations. In this case, as X increases from 0 to 1, ϕ1 increases from a small positive value to π/2. Further increase of X causes ϕ1 to increase from π/2 toward π. Next, we compute the radicand in D by letting ρ22 exp(2iϕ2) = YL2 + ρ12 exp(2iϕ1). Now we see that as 2ϕ1 increases in the interval (0, 2π] with X, the computed value of 2ϕ2 = atan2(ρ12 sin(2ϕ1), YL2 + ρ12 cos(2ϕ1)) ranges from a small positive value when X is small up to π when X = 1. As X increases further, 2ϕ2 switches discontinuously from π to −π and then increases toward zero. Hence, because the calculation behaves this way, the result for DisD = U − ρ1 exp(iϕ1) ± ρ2 exp(iϕ2), where the phase angles increase from a small value up to π/2 as X increases from 0 to 1. However, as X increases further, ϕ1 increases from π/2 toward π, whereas ϕ2 increases from −π/2 toward 0. Consequently, the above sign switch for O and X modes is appropriate with or without the collision frequency in our computation scheme.
 A wave packet signal associated with a transmitter is associated with an integral of (1) over frequency, to proscribe the wave packet's temporal extent, and wave vector components, to proscribe its spatial extent. The phase of a wave packet component is seen to be
where we keep only the real part of the wave vector, associated with η′. The ray trace equations follow from the requirement that phase variation, subject to the Appleton formula constraint, within the wave packet be stationary in order for there to be appreciable field intensity. The stationary phase requirement δP = 0 is of a standard type treated in the calculus of variations, that is, variation subject to constraints [see, e.g., Arfken, 1985]. The result is that arbitrary dependent variable displacements between fixed end points in an integral of the form P = L(yi, dyi/dτ, τ)dτ, subject to a constraint G(yi, t) = 0, result in stationary P if and only if the integrand satisfies
where λ is an undetermined multiplier. Students of classical mechanics will recognize this as the derivation of Lagrange's equations with constraints. These are the ray trace equations we need if we make the following substitutions in the preceding formula: L = ω − k′i (dxi/dt) and yi = (ω, k′1, k′2, k′3, x1, x2, x3). Hence
The first equation solution for λ is substituted into the remaining six equations, which are then the six basic ray trace equations that are integrated in progressive ray path increment solutions from the transmitter to the receiver position in point-to-point homing calculations or to the receiver height in area coverage calculations, as explained by Reilly . One chooses the mode for the calculation (O or X) and assumes that the phase expression is preserved along the entire path. It is obvious from the way it was derived that ct is the group path length. It is straightforward to add rate equations for phase path length, deviative absorption, and other parameters of interest. We find it convenient and sufficient to choose G as
The collision frequency model we use provides a good analytic fit to the graphs given by Budden  and Davies :
where h is the height variable. Units of the ai are Hz and units of the αi are km–1.