## Vertical and Oblique Propagation

Before considering the behavior of a radio signal in a magne – toionic medium, we will state three theorems that relate oblique and vertical incidence propagation as depicted in Fig. 6. The first is the secant law, which relates the vertical-inci – dence frequency fv reflected at B to the oblique-incidence fre­quency fob reflected at the same true height. A typical deriva­tion of this relation is given in Ref. 5, and it is usually written as

 JfL 2n m

 ■Bn& 2.80 x 1010Bn

 fH —

 (10)

 and the angular gyrofrequency is given by le I coH = — B0 ъ 1.76 X 10nB0

 (11)

 fob — fv sec ф0

 (7)

Since electrons are much less massive than ions, the electron gyrofrequency affects the propagation of HF waves in the ionosphere more than the ion gyrofrequencies. For example, since B « 0.5 X 10-4 Wb/m2, the electron gyrofrequency is «1.40 MHz, which falls at the upper end of the medium wave band.

The Dispersion Relation. Using the recommended URSI (In­ternational Union at Radio Science) notation, the magne – toionic dispersion equation for a radio wave in a homoge­neous, partially absorbing ionized gas upon which a constant magnetic field is impressed is given by

The secant law, then, relates the two frequencies fv and fob reflected from the same true height (the distance BD in Fig. 6).

In order to determine sec ф and fob values from vertical – incidence soundings (which measure the virtual height h’), we need two more theorems. Breit and Tuve’s theorem states that the time taken to traverse the actual curved path TABCR in Fig. 6 at the group velocity vg equals the time nec­essary to travel over the straight-line path TER at the free-

 E Figure 6. Plane geometry describing vertical and oblique iono­spheric propagation.

 n2 = 1 –

 (1 – jZ) –

 2(1 – X – jZ)

 1/2

 Y 4 IT

 + Yl2

 ±

 4(1 – X – jZ)2

 where

 n = complex refractive index = (u _ jx) ш = angular frequency of the exploring wave (rad/s)

700

 Sec ^(corrected) 3.0 2.0
 16 20 24 28 32 36 40 44 48 52 56 60 70 80 Angle of departure (deg)
 Figure 7. Logarithmic transmission curves for curved earth and ionosphere, parametric in distance between transmit­ter and receiver.
 1

600

1=

is 500 ra

<d 400 sz

я

3 300 200 100 0

 1

 + Yl2

 =F I 7

 2

 (15)

 wN = angular plasma frequency wH = angular gyrofrequency = B0|e|/m (rad/s) wL = longitudinal angular gyrofrequency = (B0|e|/m) cos в wT = transverse angular gyrofrequency = (B0|e|/m) sin в X = wN/w2 Y = wH/w Yl = wL/w YT = wT/w Z = v/w в = angle between the wave-normal and the magnetic field inclination The Polarization Relation. We begin by defining the polar­ization ratio R as R = —Hy/Hx = Ex/Ey (13) Then we can write the double-valued polarization equation as

 In the upper F region of the ionosphere where the electron – ion collision frequency is very low, we may simplify the dis­persion and polarization equations by dropping the Z term (since v « 0). Equations (12) and (14) then become (for no absorption)

 If we further simplify Eq. (12) by dropping the Y terms (no magnetic field), then we obtain n2 = 1 — X, which is equiva­lent to Eq. (1).

 2 1 — X — jZ I 4 (1 — X — jZ)

 2X (1 — X)

 n2 = 1 —

 n — 1

 (14)

 (16)

 and

 Y 4 YT

 1/2-

 X

 2(1 — X) — YT2 ± [YT4 + 4YT2(1 — X)2]1/2

 According to magnetoionic theory, a plane-polarized EM wave traveling in a medium like the terrestrial ionosphere will be split into two characteristic waves. The wave that most closely approximates the behavior of a signal propagating in this medium, without an imposed magnetic field, is called the ordinary wave, and the other is called the extraordinary wave. These terms are taken from the nomenclature for dou­ble refraction in optics, although the magnetoionic phenom­ena are more complicated than the optical ones. The ordinary wave is represented by the upper sign in the polarization Eq. (14), except when the wave-normal is exactly along the direc­tion of the magnetic field. Anomalous absorption occurs for the extraordinary wave when its frequency equals the elec­tron gyrofrequency (fH = |B| e/me « 0.8 to 1.6 MHz). These frequencies lie in the medium-frequency (MF) band; conse­quently the absorption of the extraordinary wave [A « (f — fH)2] is large and the polarization of the transmitted wave is important in the determination of the fraction of the incident power that goes into the extraordinary wave. This is espe­cially true near the dip equator, where the magnetic field is nearly horizontal and the field is usually vertical. In addition to anomalous absorption effects near the elec­tron gyrofrequency, the wave may also experience significant lateral deviation. This is illustrated for vertical and oblique propagation in Sections 11.2.2 through 11.2.4 of Ref. 5. If Eq. (16) is recast as a funtion of w and we define f(e) = j(sin2 e)/cos в and wc = (B0|e|/m) f(e), then it will be seen to describe an ellipse. The quantities f(e) and wc play an impor­tant part in the description of the polarization behavior of waves in magnetoionic theory. The magnitude of wc is inde­pendent of frequency, but varies with the angle between the wave normal and the magnetic field, в, whereas the sign of wc depends on the sign of the charge e and on the direction of the magnetic field. For longitudinal propagation wc = 0, and for transverse propagation wc ^ “.In the case where X = 1, the quantity wc primarily determines the polarization of the wave. A very complete discussion of R as a function of X and of the variation of the polarization ellipse is given in Ref. 21. A more complete understanding of the behavior of EM waves in the terrestrial ionosphere may be obtained by em-
 (23)
 X
 (24)

ploying two approximations. The quasilongitudinal (QL) ap­proximation applies when the wave is propagating nearly par­allel to the geomagnetic field, and the quasitransverse (QT) approximation applies when the wave propagates in a direc­tion nearly normal to the geomagnetic field. References 21 and 22 contain extended discussions of the QL and QT ap­proximations:

QT: YT4 » 4(1 – X)2Yl2

QL: YT4 << 4(1 – X)2Y2