THEORY OF RADIO WAVE PROPAGATION IN THE IONOSPHERE

V. L. Ginzburg

CONTENTS

Introduction ........................................................................ 155
§ 1. Initial propositions of the theory of radio-wave propagation in the ionosphere ............................................................ 158
§ 2. Dielectric constant and conductivity of an ionized gas ................. 161
§ 3. Refractive and absorption indices. Approximation of geometrical optics ........................................................................ 168
§ 4. Propagation and reflection of waves from the ionospheric layer ......... 175
§ 5. Reflection of signals .......................................................... 184
§ 6. Allowance for the influence of the Earth’s magnetic field .............. 191
References ......................................................................... 200

INTRODUCTION

The possibility of establishing radio communication between the most distant points of the terrestrial globe is due to the influence exerted on the propagation of radio waves by the upper layers of the Earth’s atmosphere. Indeed, if we neglect this influence, then the problem of calculating a communication line reduces to considering the diffraction, on the terrestrial globe, of radio waves emitted by an antenna located on the Earth. In this case, as in all diffraction problems, the wave field in the region of the geometrical shadow (i.e., beyond the horizon) decreases rapidly, and at a distance of several thousand kilometers the field strength can under no circumstances attain the observed values. Physically this statement, which follows from the rigorous theory of diffraction, is quite evident, since even in the case of the longest waves (with wavelengths of thousands of meters) radio reception is effected at distances corresponding to the bending of the waves into the shadow region by thousands of kilometers, i.e., by thousands of wavelengths.

At the same time, as early as the end of the nineteenth century, in order to explain variations of the Earth’s magnetic field, the assumption was put forward that conducting layers exist in the upper strata of the atmosphere. Such layers must evidently influence the propagation of radio waves above the Earth, to which

and was pointed out by Kennelly and Heaviside in 1902, i.e. a year after Marconi first accomplished transatlantic radio communication*).

The conducting layers in the atmosphere are often called the Kennelly–Heaviside layers, and the entire region of the atmosphere with appreciable conductivity is called the ionosphere.

The presence of conducting layers leads to the fact that radio waves propagate above the earth as if in a spherical capacitor, i.e. between concentric conducting spheres (the surface of the earth and the ionosphere). As a result, the range of reception increases enormously. For short waves with wavelengths of the order of tens and hundreds of meters, the influence of the ionosphere is especially pronounced and, in particular, at short distances from the transmitter. The very existence of reflecting layers in this region can be demonstrated by the radio-pulse method proposed by Breit and Tuve (1926), which is fundamental in the study of the ionosphere. From the earth upward a radio signal of duration less than \(10^{-4}\) sec is sent. After a time of the order of \(10^{-3}\) sec the signal returns to the place from which it was sent, and its appearance can be recorded on an oscillograph. From the delay time of the reflected signal one can, obviously, judge the distance of the reflecting layer from the earth. This distance is one hundred or more kilometers.

The study of the reflection of radio waves from the ionosphere is very important for establishing the laws of radio communication on short waves and, at the same time, is the principal method for investigating the ionosphere itself, i.e. it is of great significance from the geophysical point of view. In this connection, a large number of works have been devoted to the study of the propagation and reflection of radio waves from the ionosphere; among them the works of Appleton and his collaborators are especially important, and at the present time the question may be considered essentially clarified.

The general state of the problem has been covered in a number of reviews\(^{1,2}\), where, however, the theory of radio-wave propagation in the ionosphere is presented only very incompletely and, so to speak, in a rough approximation. At the same time, further investigation of the ionosphere by radio methods, which inevitably must be directed toward more subtle questions and details, makes necessary a more thorough discussion of the theoretical side of the matter. The present review is devoted to such a systematic consideration of a number of questions in the theory of radio-wave propagation in the ionosphere, the experimental data adduced being mainly illustrative in character.

Before concluding this introduction, it seems expedient to give here the basic information about the ionosphere; the question of how

*) In accordance with the aims of the present review, we shall not dwell on the history of the question in any detail, nor shall we give references to early works. This side of the matter is covered in the review by Mimno [1].

how these conclusions are obtained from ionospheric observations is clear from what follows.

The distribution of density in the atmosphere up to heights of about 100 km is described by the barometric formula, and the composition of the atmosphere does not change with height (because of mixing). The concentration of molecules at a height of 100 km is approximately \(10^{14}\) (at sea level \(N_m=2.7\cdot 10^{19}\) molecules/cm\(^3\)). The course of the density at greater heights is unclear; apparently, here, with increasing height, the fall of the concentration is somewhat slowed, and the temperature even rises. Radio methods directly make it possible to determine only the upper limit of the concentration of molecules at heights of the order of 200–300 km (see § 2).

The influence of the upper layers of the atmosphere on the propagation of radio waves is due to the presence in this region of electrons and ions. Schematically, the electron concentration is shown in Fig. 1. In the region of the \(F\) layer, at a height of 200–350 km, the influence on radio waves is exerted by electrons whose concentration is \(N_e=N\leq 3\cdot 10^6\) el./cm\(^3\). The maximum value of the concentration and its course with height vary depending on the time of day, season, geographical latitude, and solar activity. In a number of cases the \(F\) layer splits into two layers \(F_1\) and \(F_2\). The \(E\) layer is located at a height of 100 km; its width is about 20 km.

Fig. 1.

If the influence of the Earth’s magnetic field is not taken into account, then radio methods do not make it possible to distinguish between electron and ion layers. In this case, if for an electron layer the electron concentration is equal to \(N\), then in the equivalent ion layer the ion concentration is equal to \(N_i=N\frac{M}{m}\), where \(M\) and \(m\) are the masses of the ion and the electron (see § 2; the ions are assumed to be singly charged, i.e., carrying one elementary charge \(e\)). Whereas in the case of the \(F\) layer its electronic nature has been established, in the \(E\) layer both electrons and ions affect the propagation of radio waves. In Fig. 1, therefore, the effective electron concentration is plotted; in this layer it is equal to \(N_{ef}=N+N_i\frac{m}{M}\); this concentration \(N_{ef}\leq 2\cdot 10^5\). If in fact the properties of the \(E\) layer are determined by ions, then \(N_i\leq 6\cdot 10^9\) (since the mass of nitrogen and oxygen ions is greater than the electron mass by approximately \(3\cdot 10^4\) times). The relative role of charges of both types (electrons and ions) in the \(E\) layer has not yet been definitively clarified. In addition to the two main layers \(E\) and \(F\), lower down, at a height of about 50 km, there is a weakly ionized layer \(D\),

exerting an influence mainly on long waves and very little studied.

It should also be borne in mind that there exist in the ionosphere a number of irregular phenomena which lead to a complicated picture of the reflection of radio waves from various layers.

§ 1. INITIAL PRINCIPLES OF THE THEORY OF RADIO-WAVE PROPAGATION IN THE IONOSPHERE

The ionosphere is a medium consisting of molecules, ions, and electrons, the concentration of all these particles varying with height. The task of the theory is to describe quantitatively the propagation of waves in this medium and thereby to make it possible, from observations of radio waves, to judge the composition of the ionosphere at various heights.

Since the wavelength even of ultrashort radio waves is considerably greater than the mean distance between the various particles in the ionosphere, the propagation of radio waves in it can certainly and must be considered on the basis of the ordinary phenomenological equations of electrodynamics in material media, which have the form:

\[ \begin{aligned} \operatorname{rot}\mathbf H&=\frac{4\pi}{c}\mathbf j+\frac{i\omega}{c}\mathbf D,\qquad \operatorname{div}\mathbf D=4\pi\rho,\\ \operatorname{rot}\mathbf E&=-\frac{i\omega}{c}\mathbf H,\qquad \operatorname{div}\mathbf H=0, \end{aligned} \tag{1} \]

where \(\mathbf E\), \(\mathbf H\), and \(\mathbf D\) are respectively the vectors of the electric and magnetic field strengths and the vector of electric induction, \(\mathbf j\) is the current density, and \(\rho\) is the charge density.

Equations (1) are written directly under the assumption that all variables depend on time according to a harmonic law, i.e. are proportional to \(e^{i\omega t}\); as is known, owing to the presence of dispersion, which in application to the ionosphere cannot be neglected, the more general field equations, containing time derivatives instead of the factor \(i\omega\), become meaningless; at the same time equations in the form (1) are fully applicable also in the presence of dispersion. Moreover, since in the ionosphere the magnetic permeability is practically equal to unity, no distinction is made in equations (1) between the magnetic field strength and induction.

Eliminating the field \(\mathbf H\) from system (1), we arrive at the following equation for \(\mathbf E\):

\[ \Delta\mathbf E+\frac{\omega^2}{c^2}\left(\mathbf D-i\frac{4\pi}{\omega}\mathbf j\right)-\operatorname{grad}\operatorname{div}\mathbf E=0. \tag{2} \]

In order that equations (1) or (2) may be used for solving electrodynamic problems, it is necessary to specify the relation between the vectors \(\mathbf D\) and \(\mathbf j\), on the one hand, and the electric field

$\mathbf{E}$, on the other. If the influence of the Earth’s magnetic field on the properties of the ionosphere is not taken into account, then $\mathbf{D}$ and $\mathbf{j}$ are directed in the same way as the vector $\mathbf{E}$, and are proportional to it, i.e.

\[ \mathbf{D}=\varepsilon \mathbf{E}, \qquad \mathbf{j}=\sigma \mathbf{E}, \tag{3} \]

where $\varepsilon$ is the dielectric constant and $\sigma$ is the conductivity. The relation of the quantities $\varepsilon$ and $\sigma$ to the properties of the ionosphere will be considered in § 2.

When the influence of the Earth’s magnetic field is taken into account, the vectors $\mathbf{D}$, $\mathbf{j}$, and $\mathbf{E}$ are connected with one another by a more general dependence

\[ \left. \begin{aligned} D_i &= \sum_{k=1}^{3} \varepsilon_{ik} E_k,\\ j_i &= \sum_{k=1}^{3} \sigma_{ik} E_k, \end{aligned} \right\} \tag{4} \]

where it is assumed that $D_1=D_x$, $D_2=D_y$, $D_3=D_z$, etc. Expressions for $\varepsilon_{ik}$ and $\sigma_{ik}$ will be obtained in § 2. It should be emphasized that $\varepsilon$, $\sigma$, $\varepsilon_{ik}$, and $\sigma_{ik}$ depend on the frequency $\omega$, in which the presence of dispersion is manifested. Neglecting nonlinear phenomena (the Luxembourg effect), all these quantities may be regarded as independent of the intensities of the electric and magnetic fields of the radio wave.

The inhomogeneity of the medium, if such exists, manifests itself in the fact that $\varepsilon$ and $\sigma$ or $\varepsilon_{ik}$ and $\sigma_{ik}$ are also functions of the coordinates.

In the isotropic case (3), equation (1) takes the form:

\[ \Delta \mathbf{E}+\frac{\omega^2}{c^2}\varepsilon' \mathbf{E}-\operatorname{grad}\operatorname{div}\mathbf{E}=0, \tag{5} \]

where $\varepsilon'$ is the complex dielectric constant, equal to

\[ \varepsilon'=\varepsilon-i\frac{4\pi\sigma}{\omega}. \tag{6} \]

In the general case, when $\varepsilon'$ depends on all the coordinates, further simplification of equation (5) is impossible. In the case of the ionosphere, however, the dependence of $\varepsilon'$, or (when the influence of the magnetic field is taken into account) of the corresponding tensor

\[ \varepsilon'_{ik}=\varepsilon_{ik}-\frac{4\pi\sigma_{ik}}{\omega} \]

on the height, i.e. on the distance from the Earth’s surface, is most strongly expressed. Within the limits of a relatively small region of the Earth’s surface, on which the altitude of the Sun may be considered the same, the variation of $\varepsilon'$ at a given distance from the Earth’s surface is of a random character (clouds in the ionosphere, etc.) and is superimposed on the regular pattern of the distribution of $\varepsilon'$. As is clear from what has been said, as the regular dependence of $\varepsilon'$ we...

we assume such a one in which \(\varepsilon'\) depends only on the height, i.e., if the sphericity of the earth is neglected, only on the vertically upward coordinate \(z\).

If the question of the reflection of radio waves from the ionosphere is posed in all its completeness, then it is necessary to consider the following problem. Near the surface of the earth there are a transmitting antenna \(A\) and a receiving antenna \(B\) (Fig. 2); knowing the current in antenna \(A\), it is necessary to find it in antenna \(B\). In this case the direct influence of \(A\) on \(B\) can be eliminated by the use of a pulse method, in which the reflected signal arrives at \(B\) at a time when the current in \(A\) is zero.

Fig. 2.

In a strict formulation, the indicated problem, when the finite conductivity of the earth is taken into account, is very complicated, and its solution has not been investigated (the path to finding the solution is indicated in \(^{3,4}\)). At the same time, there is no particular need for a strict solution. The point is that the distance to the ionosphere is no less than a hundred kilometers, i.e. incomparably greater than the wavelength of short radio waves, and thus the ionosphere is located far in the wave zone of a transmitter placed on the earth. Therefore the entire process of wave propagation in the ionosphere can be considered independently of the conditions in which the transmitter and receiver are found. These conditions are essential only from the standpoint of finding the transmitter’s wave field at a great height (before the beginning of the ionosphere) and determining the field strength of the reflected signal at the earth’s surface at the location of the receiver; to obtain the complete solution, i.e. ultimately the field strength at the receiving point or the current in the receiving antenna, it is of course necessary to join the solutions “below” (at the earth) and “above” (in the ionosphere).

It should be borne in mind, however, that calculating the field strength of a complicated radiator, such as a short-wave antenna placed at the earth and the objects surrounding it, is a very difficult and, above all, unreliable matter. At the same time, from the point of view of the basic method of radio investigation of the ionosphere, the field strength of the reflected signal is of no interest, since in this method only the time delay of the signal is determined. The field strength need be known only for determining the coefficient of reflection from the ionosphere. If one is not interested in determining the coefficient of reflection (see \(^{5,6}\)) and confines oneself to the most important case of normal incidence of waves on a layer, then the problem is simplified still further. We may now simply consider that a plane wave with its normal directed along the \(z\)-axis (i.e. vertically upward) is incident upward upon the ionosphere. On being reflected from the layer, the wave remains plane and returns

onto the earth with some phase shift as compared with the incident one; absorption will show itself in a decrease of the amplitude of the reflected wave. Summarizing all that has been said, we see that, for radio investigations of the ionosphere, the principal interest lies in considering the problem of the reflection of a plane wave (or a group of waves) from a medium whose properties vary in the direction of the normal to the wave. We shall confine ourselves to this problem in the present survey.

In the isotropic case (3), i.e., without taking into account the influence of the earth’s magnetic field, for normal incidence of the wave on the layer, the vector \(\mathbf E\) lies in the plane \(xy\), and at the same time the value \(E\) depends only on the coordinate \(z\), along which the properties of the layer vary. Therefore equation (5) takes the following form:

\[ \frac{d^{2}E}{dz^{2}}+\frac{\omega^{2}}{c^{2}}\varepsilon'(z)E=0, \tag{7} \]

where by \(E=E(z)e^{i\omega t}\) one may understand both \(E_x\) and \(E_y\). When the influence of the magnetic field is taken into account, the equations in the case under consideration are also greatly simplified. Namely, since \(\mathbf E\) depends only on \(z\), equations (2) take the form:

\[ \left. \begin{aligned} \frac{d^{2}E_x}{dz^{2}}+\frac{\omega^{2}}{c^{2}}\left(D_x-i\frac{4\pi}{\omega}j_x\right)&=0,\\ \frac{d^{2}E_y}{dz^{2}}+\frac{\omega^{2}}{c^{2}}\left(D_y-i\frac{4\pi}{\omega}j_y\right)&=0,\\ D_z-i\frac{4\pi}{\omega}j_z&=0, \end{aligned} \right\} \tag{8} \]

where \(\mathbf D\) and \(\mathbf j\) are connected with \(\mathbf E\) by the relations (4).

For given \(\varepsilon'\) and \(\varepsilon'_{ik}\), the further investigation reduces to solving equations (7) and (8). In addition, it is necessary to express \(\varepsilon'\) and \(\varepsilon'_{ik}\) in terms of universal constants, the frequency, and the concentration of the various particles in the ionosphere. We now turn to this latter question.

§ 2. DIELECTRIC CONSTANT AND CONDUCTIVITY OF AN IONIZED GAS

The ionosphere is an ionized gas (plasma) consisting of electrons, ions, and molecules. To compute the dielectric constant and conductivity of this gas, it is necessary to consider the influence on the charges composing it of a constant electric field. It is not necessary to take into account the dependence of the field on the coordinates, because \(\varepsilon\) and \(\sigma\) are, so to speak, local characteristics of the gas, whose inhomogeneity is manifested in the change of the quantities \(\varepsilon\) and \(\sigma\) from point to point. First we shall determine \(\varepsilon\) and \(\sigma\) without taking into account the influence

of the Earth's magnetic field, i.e., when, on the particles, apart from the external electric field, there act only the forces of interaction between these particles themselves, manifested in collisions between them. Further, in practically important cases the role of the mentioned interaction in the propagation of radio waves is small, and in the first approximation it may be neglected. Therefore the equation of motion of the electrons has the form:

\[ m\ddot{\mathbf r}=e\mathbf E e^{i\omega t}, \tag{9} \]

where \(e\) and \(m\) are the charge and mass of the electron.

The forced solution of equation (9), which alone is of interest to us, has the form \(\mathbf r=-\dfrac{e\mathbf E}{m\omega^2}e^{i\omega t}\), and the polarization caused by the field \(\mathbf E\) is

\[ \mathbf P=eN\mathbf r=-\frac{e^2N}{m\omega^2}\mathbf E e^{i\omega t}; \]

but by definition \(\mathbf P=\dfrac{\varepsilon-1}{4\pi}\mathbf E\), and thus,

\[ \varepsilon=1-\frac{4\pi e^2N}{m\omega^2} =1-3.19\cdot 10^9\frac{N}{\omega^2}, \tag{10} \]

where \(N\) is the electron concentration.

The contribution to \(\varepsilon\) from the translational motion of ions is determined, evidently, by formula (10) with the electron mass \(m\) replaced by the ion mass \(M\). Thus the influence on \(\varepsilon\) of ions with concentration \(N_i\) is equivalent to the influence of electrons with concentration \(N_{ef}=N_i m/M\), as was already mentioned in the introduction. Below, for definiteness, we shall not explicitly take into account the influence of ions on \(\varepsilon\).

The influence of the interaction between the particles making up the ionosphere, i.e., the influence of collisions, on \(\varepsilon\) is not large (see below). However, the conductivity is entirely determined by these collisions, which is already clear from the fact that conductivity gives rise to absorption losses, while losses are possible only in the presence of a mechanism converting into heat the energy of the ordered motion caused by the field; such a mechanism is precisely collisions. The presence of absorption can be taken into account by introducing into the left-hand side of equation (9) a friction force \(g\dot{\mathbf r}\), and the meaning of the coefficient \(g\) can be elucidated as follows. The expression \(g\dot{\mathbf r}=g\mathbf v\) (\(\mathbf v\) is the velocity) represents the average, collision-related change of momentum per \(1\) sec; this change, on the other hand, is equal to \(m\mathbf v\nu_{ef}\), where \(\nu_{ef}\) is the effective number of collisions, since at each collision the change of momentum is, in order of magnitude, equal to the momentum \(m\mathbf v\) itself. Thus \(g=m\nu_{ef}\), and the equation of motion with collisions taken into account assumes the form

\[ m\ddot{\mathbf r}+m\nu_{ef}\dot{\mathbf r}=e\mathbf E e^{i\omega t}. \tag{9'} \]

Solving it in the same way as equation (9), we arrive at the expression for the total polarization*):

\[ \mathbf{P}_t=eN\mathbf{r}=-\frac{e^2NEe^{i\omega t}}{m(\omega^2-i\nu_{ef}\omega)} = \]

\[ =\left(-\frac{e^2N}{m(\omega^2+\nu_{ef}^{\,2})} -\frac{ie^2\nu_{ef}N}{m\omega(\omega^2+\nu_{ef}^{\,2})}\right)Ee^{i\omega t}; \]

but, by definition, in the presence of conductivity, the total current is

\[ \mathbf{j}_t=i\omega\mathbf{P}_t= \left(\sigma+i\omega\frac{\varepsilon-1}{4\pi}\right)Ee^{i\omega t}. \]

Hence we have:

\[ \varepsilon=1-\frac{4\pi e^2N}{m(\omega^2+\nu_{ef}^{\,2})},\qquad \sigma=\frac{e^2\nu_{ef}N}{m(\omega^2+\nu_{ef}^{\,2})} =\frac{1-\varepsilon}{4\pi}\nu_{ef}. \tag{11} \]

In practice (we are speaking, of course, of the practice of radio-wave propagation) usually

\[ \nu_{ef}^{\,2}\ll \omega^2 \tag{12} \]

and, thus, for \(\varepsilon\) the old expression (10) is retained, and

\[ \sigma=\frac{e^2\nu_{ef}N}{m\omega^2}. \tag{13} \]

The quantity \(\nu_{ef}\) appearing in the written formulas is still of a somewhat conventional character and requires a more rigorous definition. The latter is achieved by considering the question by the method of the kinetic equation \(^{7}\). Without dwelling on the corresponding calculations, let us indicate that as a result we obtain formulas (11), and under condition (12), of course, formulas (10) and (13). In this case, if the electrons collide with neutral molecules, which for definiteness we imagine in the form of solid spheres of radius \(a\), then

\[ \nu_{ef}=\frac{4}{3}\pi a^2N_m\overline{v}, \tag{14} \]

where \(\overline{v}\) is the arithmetic mean velocity of the electron (for \(T=300^\circ\mathrm{K}\), \(\overline{v}\simeq 10^7\ \mathrm{cm/sec}\)) and \(N_m\) is the number of molecules in \(\mathrm{cm}^3\). For the molecules \(\mathrm{O}_2\) and \(\mathrm{N}_2\)

*) The total current \(\mathbf{j}_t=\dfrac{\partial \mathbf{P}_t}{\partial t}\) is by definition equal to \(\sigma E+\dfrac{\varepsilon-1}{4\pi}\dfrac{\partial P}{\partial t}\), where

\[ \mathbf{P}=\frac{\mathbf{D}-\mathbf{E}}{4\pi}. \]

If desired, one may avoid introducing the quantities \(\mathbf{j}_t\) and \(\mathbf{P}_t\), and immediately write

\[ eN\mathbf{r}=\left(\frac{\varepsilon-1}{4\pi}+\frac{\sigma}{i\omega}\right)Ee^{i\omega t}. \]

approximately:

\[ \nu_{ef}\approx 5\cdot 10^{-16}N_m\overline{v}\;(\approx 5\cdot 10^{-9}N_m \text{ at } T=300^\circ\text{ K}). \tag{15} \]

Formula (14), if one disregards the numerical coefficient, has the usual form of the expression for the number of collisions \((\nu=\pi r^2N_m\overline{v})\). If the electrons collide with singly charged ions, then

\[ \nu_{ef}=\frac{\pi e^4}{(kT)^2}\ln\left(\frac{kT}{e^2N_i^{1/3}}\right)N_i\overline{v} \left(=10^{-3}\ln\frac{1.8\cdot 10^5}{N_i^{1/3}}\text{ at }T=300^\circ\text{ K}\right), \tag{16} \]

where \(k\) is Boltzmann’s constant, \(N_i\) is the ion concentration; the formula is valid to within a factor of order unity.

Comparing (16) with the formula \(\nu=\pi r^2N_i\overline{v}\) for the number of collisions, we see that for ions the effective cross section is equal to

\[ \pi r^2=\pi\frac{e^4}{(kT)^2}\ln\left(\frac{kT}{e^2N_i^{1/3}}\right); \]

qualitatively this result is quite understandable, since the effective radius for collisions is the distance from the center of the ion at which, when passing by, the electron substantially changes its motion. Since the kinetic energy of the electron is equal to \({}^3/_2 kT\), and its potential energy at a distance \(r\) from the ion is equal to \(\frac{e^2}{r}\), the above-mentioned effective radius is, in order of magnitude, equal to \(\frac{e^2}{kT}\), since, when passing at such a distance from the ion, the electron undergoes a change of velocity comparable with this velocity itself. For \(T=300^\circ\text{ K}\), the values of \(\nu_{ef}\) calculated from formulas (15) and (16) are given in the table.

\(N_m=N_i\)	\(\nu_{ef}\) according to formula (15)	\(\nu_{ef}\) according to formula (16)
\(10^4\)	—	\(100\)
\(10^6\)	—	\(7.5\cdot 10^3\)
\(10^9\)	\(5\)	\(5.2\cdot 10^6\)
\(10^{12}\)	\(5\cdot 10^3\)	\(2.9\cdot 10^9\)
\(10^{14}\)	\(5\cdot 10^5\)	—

We see that at \(T=300^\circ\text{ K}\) ions, with respect to the number of collisions, are more effective than molecules by a factor of \(10^6\). At \(T=1000^\circ\text{ K}\) ions are more effective than molecules by a factor of \(10^5\).

The ionosphere is quasi-neutral, i.e. the number of electrons and negative ions in it is equal to the number of positive ions (the ions are assumed to be singly charged). Therefore, if in the \(F\) layer the electron concentration is \(10^6\), then \(N_i\geqslant 10^6\), and \(\nu_{ef}\) for collisions on these ions is of the order \(10^3\)—\(10^4\); from experiment \(\nu_{ef}\) is also of the order \(10^3\)—\(10^4\) (for the experimental values of \(\nu_{ef}\), see in \(^{8}\)). In the presence of

and of molecules and ions $\nu_{ef}=\nu_{ef}$ (ions) $+\nu_{ef}$ (molecules), and on the basis of the figures given one can only say that $\nu_{ef}$ (molecules) $<5\cdot10^3$, i.e. $N_m<10^{12}$.

It is, evidently, impossible to determine the number of molecules $N_m$ by the radio method (from measurement of $\nu_{ef}$), although attempts have sometimes been made to do this. Indeed, even if we knew the temperature exactly, even then one could determine only the lower bound of $\nu_{ef}$ (ions); this lower bound corresponds to the assumption that $N_i=N$; it is enough to admit that, owing to the presence of negative ions, $N_i>N$, in order to increase $\nu_{ef}$ (ions) up to the experimental value of $\nu_{ef}$ (if this value is greater than $\nu_{ef}$ (ions) for $N_i=N$), and thus deprive us of the possibility of judging $\nu_{ef}$ (molecules).

In the $E$ layer, where usually, apparently, the value of $\varepsilon$ is determined by ions, as $\nu_{ef}$ one must take the number of collisions of ions with ions and molecules. In this case $\nu_{ef}$ is, in order of magnitude, determined by the previous formulas (15) and (16), with replacement of $\overline v$ by the mean velocity of the ions; for the ions $\mathrm{O}_2^\pm$ and $\mathrm{N}_2^\pm$ at $T=300^\circ\mathrm{K}$, $\overline v=4\cdot10^4\ \mathrm{cm/sec}$, i.e. less than the electron velocity by approximately a factor of 250.

Above, in deriving formula (10), we silently passed over one very essential and at the same time delicate question. The point is that, in the sense of the problem, in equation (9) there figures the so-called acting field $\mathbf E_d$, i.e. the field acting on an electron, which, generally speaking, is not equal to the average macroscopic field

\[ \mathbf E=\frac{4\pi}{\varepsilon-1}\mathbf P. \]

Let us recall, for example, that in compressed gases the relation

\[ \mathbf E_d=\mathbf E+\frac{4\pi}{3}\mathbf P, \]

which leads to the well-known Lorentz–Lorenz formula, is often well satisfied. The question of the value of the acting field for an ionized gas is by no means simple and has been discussed up to recent times. As a result it was found$^{9,10}$ that in the ionosphere $\mathbf E_d=\mathbf E$, and thus formula (10) is correct (for some qualifications see $^{10}$). The relation $\mathbf E_d=\mathbf E$ was also used in obtaining formulas (11). This result is preserved, as can be shown, also for ionic plasma, i.e. in the case of a field acting on ions, and not on electrons.

Let us now proceed to take into account the influence of the earth’s magnetic field $\mathbf H^{(0)}$. In this case, instead of (9′), we have ($\mathbf E_d=\mathbf E$ in this case also):

\[ m\ddot{\mathbf r}+m\nu_{ef}\dot{\mathbf r} = e\mathbf E e^{i\omega t} +\frac{e}{c}\left[\dot{\mathbf r}\mathbf H^{(0)}\right]. \tag{17} \]

Hence we obtain the equation for $\mathbf P_t=eN\mathbf r$

\[ (i\omega\nu_{ef}-\omega^2)\mathbf P_t = \frac{e^2N}{m}\mathbf E + \frac{i\omega e}{mc}\left[\mathbf P_t\mathbf H^{(0)}\right]. \tag{17′} \]

Choosing the coordinate system in such a way that the field \(\mathbf H^{(0)}\) is directed along the \(z\)-axis, we easily arrive at the relations

\[ \left. \begin{aligned} P_{t,x}+iP_{t,y}&=\frac{e^2N(E_x+iE_y)} {m(-\omega^2+i\omega\nu_{ef}+\omega\omega_H)},\\ P_{t,x}-iP_{t,y}&=\frac{e^2N(E_x-iE_y)} {m(-\omega^2+i\omega\nu_{ef}-\omega\omega_H)},\\ P_{t,z}&=\frac{e^2NE_z}{m(-\omega^2+i\omega\nu_{ef})}, \end{aligned} \right\} \tag{18} \]

where \(\omega_H\) is the gyromagnetic frequency, equal to

\[ \omega_H=\frac{|e|H}{mc}. \tag{19} \]

Since

\[ j_{t,i}=i\omega P_{t,i}=\sum_{k=1}^{3} \left(\sigma_{ik}+i\omega\frac{\varepsilon_{ik}-\hat{\varepsilon}_{ik}}{4\pi}\right)E_k, \]

from (18) one can find \(\varepsilon_{ik}\) and \(\sigma_{ik}\) algebraically; we shall not dwell on this. The point is that the components of the tensors \(\varepsilon_{ik}\) and \(\sigma_{ik}\) depend on the choice of coordinate system; but the coordinate system in which the field \(\mathbf H^{(0)}\) is directed along the \(z\)-axis is inconvenient for what follows. As we have already seen, the study of radio-wave propagation is especially simple if the \(z\)-axis is directed vertically upward, i.e. along the normal to the wave. We shall choose the \(x\)- and \(y\)-axes so that the earth’s magnetic field has no component along the \(x\)-axis. We denote the projections of \(\mathbf H^{(0)}\) on the \(z\)- and \(y\)-axes respectively by \(H_z^{(0)}=H^{(0)}\cos\alpha\) and \(H_y^{(0)}=H^{(0)}\sin\alpha\), where \(\alpha\) is the angle between \(\mathbf H^{(0)}\) and the \(z\)-axis. In this coordinate system, with the aid of (17′), one can find \(\mathbf P\); we write out the corresponding expressions for the case when \(\nu_{ef}=0\), and, in addition, in accordance with the problem stated in § 6, we shall assume that \(D_z=0\):

\[ \left. \begin{aligned} D_x&=E_x+4\pi P_x= \left\{1+ \frac{\omega_0^2(\omega^2-\omega_0^2)} {(\omega^2-\omega_0^2)(\omega_L^2-\omega^2)+\omega^2\omega_T^2} \right\}E_x\\ &\quad+ \frac{i\omega_L\omega_0^2(\omega^2-\omega_0^2)} {\omega\left\{(\omega^2-\omega_0^2)(\omega_L^2-\omega^2)+\omega^2\omega_T^2\right\}}E_y,\\[0.8em] D_y&=E_y+4\pi P_y= \frac{-i\omega_L\omega_0^2(\omega^2-\omega_0^2)} {\omega\left\{(\omega^2-\omega_0^2)(\omega_L^2-\omega^2)+\omega^2\omega_T^2\right\}}E_x\\ &\quad+ \left\{1+ \frac{\omega_0^2(\omega^2-\omega_0^2-\omega_T^2)} {(\omega^2-\omega_0^2)(\omega_L^2-\omega^2)+\omega^2\omega_T^2} \right\}E_y,\\[0.8em] P_z&=i\frac{\omega\omega_T}{\omega^2-\omega_0^2}P_x,\qquad D_z=E_z+4\pi P_z=0, \end{aligned} \right\} \tag{20} \]

where

\[ \left. \begin{gathered} \omega_L=\omega_H\cos\alpha,\quad \omega_T=\omega_H\sin\alpha,\\ \omega_0^2=\frac{4\pi e^2N}{m}. \end{gathered} \right\} \tag{21} \]

Writing (20) in the form \(D_i=\sum_{k=1}^{3}\varepsilon_{ik}E_k\), we immediately find the values of \(\varepsilon_{ik}\).

Above we took into account only that part of the dielectric constant which is associated with the motion of free electrons and ions; absorption was likewise associated only with collisions of electrons (or ions) with molecules and ions. The part of the dielectric constant not associated with the oscillations of free charged particles, for a rarefied gas, is negligibly small and may indeed be neglected. In principle, however, a certain specific effect is possible, due to the influence of the earth’s magnetic field on the magnetic moment of the particles making up the atmosphere \(^{11,12}\). Apart from certain ions and atomic oxygen present in the ionosphere, oxygen molecules possess a magnetic moment; their number is \(21\%\) of the total number of molecules in the air. In the earth’s magnetic field the magnetic moment is oriented in a definite way with respect to the field, which leads to Zeeman splitting of the terms of the molecule. If the energy difference between the Zeeman sublevels of a given molecular level is \(\Delta E_i\), then it can absorb electromagnetic waves of frequency \(\omega_i=\dfrac{\Delta E_i}{\hbar}\) \((\hbar=1.04\cdot10^{-27})\). In addition to absorption, Zeeman splitting leads to the medium becoming birefringent, i.e. \(\varepsilon_{ik}\ne \varepsilon\cdot\delta_{ik}\). Since

\[ \omega_i=\frac{\Delta E_i}{\hbar}\sim\frac{\mu_0H}{\hbar}=\frac{eH}{2mc}=4.4\cdot10^6 \]

(\(\mu_0\) is the Bohr magneton, \(H\simeq0.5\) gauss is the strength of the earth’s field), the absorption frequencies lie in the radio range. However, as it turns out \(^{11}\), the indicated effect is very small and, in the best case, lies at the limit of experimental possibilities. It is of interest here that the insignificance of the effect is due to negative dispersion and absorption. Negative absorption, i.e. the stimulated emission of radiation by an excited atom in the direction of the incident wave, just like negative dispersion, can be observed only in a gas containing many excited particles. Therefore in the optical region negative absorption has not been observed at all, and negative dispersion has been studied only in the plasma of a gas discharge (Ladenburg and others). In the case considered here of radio-wave propagation, on the contrary, negative absorption and dispersion are manifested very clearly, since the number of excited molecules, i.e. molecules situated on the upper Zeeman sublevels, is only slightly smaller than the number of molecules situated on the lower—

their sublevels; the corresponding ratio is of order \(e^{\Delta E_i/kT}\simeq 1+\dfrac{\Delta E_i}{kT}\). As a result of negative absorption and dispersion, in the expressions for the absorption coefficient and for the refractive indices (or for the tensor \(\varepsilon_{ik}\)) there appears the factor
\[ \frac{\Delta E_i}{kT}\lesssim \frac{\mu_0 H}{kT}\sim 10^{-7} \]
(for \(T=300^\circ K\)), which accounts for the smallness of the corresponding effects. If negative absorption and dispersion were absent, this factor would disappear, and the character of radio-wave propagation in the atmosphere would differ radically from what is observed.

An effect analogous to that described could in principle occur owing to the influence of the electric field present in the atmosphere on polar molecules; however, the atmosphere contains only polar water molecules, for which the effect is completely absent \(^{12}\).

§ 3. REFRACTIVE INDICES AND ABSORPTION

GEOMETRICAL-OPTICS APPROXIMATION

The propagation of radio waves in a medium is described, as we have seen, by equations (7) or (8), and is completely determined by specifying the functions \(\varepsilon'(z)\) or \(\varepsilon'_{ik}(z)\). In the most general case of an inhomogeneous medium there is nothing left to do but to investigate the solutions of these equations (7) or (8). However, in the ionosphere the dependence of \(\varepsilon'\) and \(\varepsilon'_{ik}\) on \(z\) satisfies one very important condition, namely, it is slow; and therefore there opens up the possibility of a quite general consideration of the propagation process, not connected with the particular form of the functions \(\varepsilon'\) or \(\varepsilon'_{ik}\). Propagation in a medium with slowly varying properties is closely adjacent to propagation in a homogeneous medium, where \(\varepsilon'\) or \(\varepsilon'_{ik}\) do not depend on the coordinates. In this latter case, as is known, instead of \(\varepsilon'\) and \(\varepsilon'_{ik}\) one can introduce derived quantities—the refractive indices and absorption indices, which completely characterize the propagation of an electromagnetic wave. We shall now turn to this question; moreover, for convenience in what follows, up to § 6, we shall consider only the isotropic case, i.e., we shall not take into account the influence of the earth’s magnetic field.

In a homogeneous medium, equation (7) for a propagating wave has a solution of the form

\[ E=Ce^{\pm i\frac{\omega}{c}\sqrt{\varepsilon'}z} \equiv Ce^{\pm i\frac{\omega}{c}(n-ik)z} \equiv Ce^{\pm i\frac{\omega}{c}kz+i\frac{\omega}{c}nz}, \tag{22} \]

where \(C\) is a constant; the factor \(e^{i\omega t}\) has been omitted (this will be done

and so on), and

\[ (n-ik)^2=\varepsilon'=\varepsilon-i\frac{4\pi\sigma}{\omega} \tag{23} \]

The refractive indices \(n\) and \(k\), as is easily obtained from (23), are equal to:

\[ \left. \begin{aligned} n&=\sqrt{\frac{\varepsilon}{2}+\sqrt{\left(\frac{\varepsilon}{2}\right)^2+\left(\frac{2\pi\sigma}{\omega}\right)^2}},\\ k&=\sqrt{-\frac{\varepsilon}{2}+\sqrt{\left(\frac{\varepsilon}{2}\right)^2+\left(\frac{2\pi\sigma}{\omega}\right)^2}}. \end{aligned} \right\} \tag{24} \]

In the absence of absorption [see (10)]

\[ n^2=\varepsilon=1-\frac{4\pi e^2N}{m\omega^2}=1-\upsilon, \tag{10'} \]

where

\[ \upsilon=\frac{\omega_0^2}{\omega^2}=\frac{4\pi e^2N}{m\omega^2}. \]

The representation of \(n^2\) as a function of the dimensionless parameter \(\upsilon\) is especially convenient in the considerably more complicated anisotropic case (see § 6). In the isotropic case, the dependence (10′) of \(n^2\) on \(\upsilon\) is represented in Fig. 3, where the unusual choice of the direction of the axes is determined by a well-known convention. The meaning of the indices \(n\) and \(k\) in (22) is well known: the index \(k\) determines the absorption of the wave, while \(n\) determines its phase velocity, equal to \(w=\frac{c}{n}\).

Fig. 3.

For \(\upsilon>1\), as is seen from (10′), \(\varepsilon<0\), and the refractive index \(n\) is imaginary; therefore it would be more correct in this case to interchange \(n\) and \(k\), i.e. to assume that for \(\sigma=0\) and \(\varepsilon<0\), \(n=0\) and \(k^2=|\varepsilon|\). We, however, shall not do this, retaining for \(n\) and \(k\) the expressions (24) for any sign of \(\varepsilon\). In this case, for \(\sigma=0\), \(k\) is always equal to zero and \(n^2=\varepsilon\); the justification for such a choice may be seen, besides the well-known convenience, in the fact that for \(\sigma=0\) true absorption is absent, and the attenuation of the wave which occurs when \(\varepsilon<0\) is not connected with dissipation of energy.

If one does not take into account the local, small-scale inhomogeneities of the ionosphere, then the density of free charges in it may be considered smoothly and slowly varying—this is evident from the schematic

Fig. 1, which shows the altitude dependence of the effective electron concentration. Therefore the dielectric constant and conductivity, or the refractive and absorption indices equivalent to these quantities, also vary smoothly and slowly with height; the criterion for slowness of variation is here the condition that all quantities change very little over distances of the order of the radio-wave wavelength in the ionosphere. In the \(F\)-layer, extending over hundreds of kilometers, this condition of slowness, which we shall specify below, is usually very well satisfied, since in this case waves with wavelengths shorter than 100 meters are of interest. In the \(E\)-layer, whose thickness is only tens of kilometers and where the propagation of longer waves is of interest, the condition of slowness proves more stringent, but in practice it is also usually satisfied.

If, over the length of an electromagnetic wave, the properties of the medium change only slightly, then, evidently, the propagation of these waves in a small region is very close to propagation in a homogeneous medium with refractive and absorption indices corresponding to the given portion of the ionosphere. Propagation in the entire inhomogeneous medium is then equivalent to propagation in a homogeneous medium with varying constants. The mathematical description of propagation under such conditions is achieved by finding an approximate solution of the wave equation (7), and this solution, called the solution in the geometrical-optics approximation, can be obtained explicitly for any dependence \(\varepsilon'(z)\). From what has been said it is clear that the geometrical-optics approximation plays an exceptionally important role in the theory of radio-wave propagation in the ionosphere. This approximation, however, is not applicable near the place where radio waves are reflected from a layer, and here one must resort to a discussion of exact solutions of the wave equation, but not for an arbitrary layer; as we shall see below, only for the linear and parabolic ones.

Proceeding to a more detailed development of the considerations presented above, let us begin with the question of finding an approximate geometrical-optics solution as applied to equation (7). If \(\varepsilon'=\mathrm{const.}\), then the exact solution of this equation has the form (22). Therefore, if \(\varepsilon'\) varies slowly over distances of the order of \(\lambda_0=\dfrac{2\pi c}{\omega}\), then it is natural to seek the solution of equation (7) in the form of a series:

\[ E=\left(E^{(0)}+\frac{c}{\omega}E^{(1)}+\ldots\right)e^{\pm i\frac{\omega}{c}\psi}, \tag{25} \]

where \(E^{(0)}, E', \ldots\) and \(\psi\) are unknown functions of \(z\). Substituting the solution (25) into (7) and equating to zero the expressions appearing as coefficients of the different powers of \(\dfrac{\omega}{c}\), we arrive at the follow-

following system of equations*):

\[ \left. \begin{gathered} \left(\varepsilon' - \left(\frac{d\psi}{dz}\right)^2\right)E^{(0)}=0,\qquad \frac{dE^{(0)}}{dz}+ \frac{\dfrac{d^2\psi}{dz^2}}{2\dfrac{d\psi}{dz}}\,E^{(0)}=0,\\[6pt] \frac{dE^{(1)}}{dz}+ \frac{\dfrac{d^2\psi}{dz^2}}{2\dfrac{d\psi}{dz}}\,E^{(1)} = -\frac{\dfrac{d^2E^{(0)}}{dz^2}}{2i\dfrac{d\psi}{dz}}, \quad \text{etc.} \end{gathered} \right\} \tag{26} \]

The first of these equations determines the function \(\psi\), since \(E^{(0)}\ne 0\) only if

\[ \left(\frac{d\psi}{dz}\right)^2=\varepsilon', \tag{27} \]

i.e.

\[ \psi=\int \sqrt{\varepsilon'}\,dz. \tag{27'} \]

From the remaining equations (16) one can determine \(E^{(0)}\), \(E^{(1)}\), etc.

\[ \left. \begin{gathered} E^{(0)}=\frac{C}{\sqrt{\dfrac{d\psi}{dz}}} =\frac{C}{\sqrt[4]{\varepsilon'}},\\[8pt] E^{(1)}= \frac{1}{\sqrt[4]{\varepsilon'}} \int_{0}^{z} \frac{\dfrac{d^2E^{(0)}}{dz^2}\,\sqrt[4]{\varepsilon'}} {2i\sqrt{\varepsilon'}}\,dz . \end{gathered} \right\} \tag{28} \]

The geometrical-optics approximation**), of which we spoke above, is applicable if in the solution (25) one may restrict oneself to the first term, i.e. if

\[ \frac{\lambda_0}{2\pi}\,|E^{(1)}|\ll |E^{(0)}|. \tag{29} \]

*) Substitution of the solution (25) into equation (7) leads to an equation of the type

\[ A\frac{\omega^2}{c^2}+B\frac{\omega}{c}+C+D\frac{c}{\omega}+F\frac{c^2}{\omega^2}+\cdots=0, \]

where \(A,B,C\ldots\) are certain expressions containing \(E,\psi\) and their derivatives. Since the equality obtained must hold for arbitrary values of \(\omega/c\), it is satisfied only if \(A=B=C=D=F=\cdots=0\).

**) More precisely, this is the first approximation of geometrical optics. Since, however, higher approximations of geometrical optics are never used, the condition for its applicability may be identified with the condition for the applicability of the first approximation.

It is easy to show\(^{13}\) that condition (29) is certainly fulfilled if the inequalities

\[ |\ln n| \lesssim 1,\quad \frac{\lambda_0}{2\pi}\frac{\dfrac{dn}{dz}}{n^2}\ll 1,\quad \frac{\lambda_0}{2\pi}\frac{\dfrac{d^2 n}{dz^2}}{n\dfrac{dn}{dz}}\ll 1, \tag{30} \]

are satisfied, where, for simplicity, it has been assumed that absorption is absent, i.e., that \(\varepsilon'=n^2\).

In practice, the basic condition for the applicability of geometrical optics is the second of inequalities (30); in the presence of absorption it becomes

\[ \frac{\lambda_0}{2\pi} \sqrt{\frac{\left(\dfrac{dn}{dz}\right)^2+\left(\dfrac{dk}{dz}\right)^2}{n^2+k^2}} \ll 1, \tag{31} \]

where \(n\) and \(k\) are the refractive and absorption indices, defined by formulas (24).

Under the conditions when the geometrical-optics approximation is applicable, the field \(E(z)\) has the form [see (23), (25), (27′), (28)]:

\[ E(z)=\frac{C}{\sqrt{n-ik}}\, e^{\pm \frac{\omega}{c}\int_{z_0}^{z} k(z)\,dz \pm i\frac{\omega}{c}\int_{z_0}^{z} n(z)\,dz}, \tag{32} \]

where the constants \(C\) and \(z_0\) must be determined from the boundary conditions. The two signs in (32), as in (22), correspond to waves propagating in the positive [the sign \(-\) in (22) and (32)] or negative (the sign \(+\)) direction of the \(z\)-axis. In the geometrical-optics approximation, waves traveling in both directions are completely independent. Therefore reflection from a layer, i.e. the transition of an upward-propagating wave into a downward-propagating wave, can occur only in regions where inequalities (30) are not satisfied. The latter occurs either if the refractive index is very small, or if the gradient of this index is large\(*\). The second case may occur when there are sharp inhomogeneities in the electron concentration in the ionosphere, which does not correspond to the assumptions made above (see Fig. 1), corresponding to a regular (smoothed) variation of electron concentration with height.

Thus, reflection of radio waves from the ionosphere usually occurs owing to the presence of regions where the refractive index for the given frequency is close to zero. The inapplicability of the geometrical-optics approximation in this case has a simple physical meaning. As we have already indicated, the condition for the applicability of geometrical

\(*\) We restrict ourselves to the practically interesting cases in which the second of inequalities (30) is not satisfied. Other cases that are possible in principle, as well as the effect of absorption, are of no practical interest.

of optics consists in the smallness of the change of the properties of the medium over sections of the order of the wavelength \(\lambda\) in this medium, where \(\lambda=\dfrac{\lambda_0}{n}\). As \(n\to 0\), \(\lambda\) tends to infinity, and therefore, even in the case of smoothly varying properties of the layer, near the point \(n=0\) geometrical optics is inapplicable.

When considering the region where the refractive index is very small, it is necessary to distinguish two cases. In the first of them, beyond the point where \(\varepsilon=n^2=0\), the concentration of electrons in the layer on the side of negative values of \(\varepsilon\) increases further over a considerable distance or, in other words, the region where \(\varepsilon<0\) is large (considerably greater than \(\lambda_0\)). The second case occurs if the point \(z(\varepsilon=0)\) lies near the maximum of the electron concentration in the layer, i.e. the region where \(\varepsilon<0\) is small (the conditions for the smallness of the region where \(\varepsilon<0\) will be specified below). In the first case, in the absence of absorption, complete reflection of the radio wave from the layer takes place. Physically this circumstance is clear from the following considerations. In the region where \(\varepsilon<0\), the wave is attenuated, since

\[ k=0,\quad n=i\sqrt{|\varepsilon|},\quad E\sim e^{-\frac{\omega}{c}\int \sqrt{|\varepsilon|}\,dz} \]

[see (32)], and at a distance of several wavelengths beyond the point \(z(\varepsilon=0)\) the field is practically equal to zero (if the derivative \(d\varepsilon/dz\) is not too small). Since, by assumption, true attenuation due to energy dissipation is absent, it is clear that all the energy must be reflected from the layer, so that a standing wave is formed.

The considerations given explain the fundamental fact that radio waves, even at normal incidence, can be completely reflected from the ionosphere. This reflection is analogous to the case known in optics as total internal reflection, which occurs when light passes from an optically denser medium, i.e. a medium with larger \(n\), into a less dense medium. In optics, however, total internal reflection does not occur at normal incidence, since in this case it is possible only if in the less dense medium \(n=0\). This latter case is precisely what is realized in the ionosphere, where with increasing height (until the maximum of the layer is reached) \(n\) decreases and can attain the value \(n=0\). From formula (10) it is clear that \(n=0\) if

\[ N=\frac{\omega^2}{3.19\cdot 10^9}. \tag{33} \]

The region of the layer corresponding to negative values of \(\varepsilon\) and sufficiently far removed from the point \(z(\varepsilon=0)\) has no influence on the field of the radio wave. Therefore the properties of the layer in this region are completely immaterial, and the real layer with a maximum may be replaced by a layer with a monotonically increasing electron concentration.

As is clear from what was said above, near the point \(z(\varepsilon=0)\) it is necessary to use the exact solution of the wave equation; but in a relatively small region, where \(\varepsilon\) is small and the approximation of geometrical optics is inapplicable, the dependence of \(\varepsilon\) on \(z\) may be regarded as linear and, in this way, the solution of the wave equation for a linear layer may be considered. By matching this solution with the solution of geometrical optics in the region where the latter is applicable, we shall obtain the general solution for any layer. The condition of applicability of this solution consists, evidently, in the deviation of the layer from a linear one over the interval \(\Delta z\), where the exact solution of the wave equation differs from the solution of geometrical optics, being small, i.e. in the inequality

\[ \left|\frac{d^2\varepsilon}{dz^2}\right|\Delta z \ll \left|\frac{d\varepsilon}{dz}\right|_{0}, \tag{34} \]

where the derivatives of \(\varepsilon\) are taken at the point \(z(\varepsilon=0)\).

In the second of the indicated cases, i.e. when the point \(z(\varepsilon=0)\) lies near the concentration maximum, the layer can no longer be replaced by a linear one, which is immediately evident and is manifested in the failure of inequality (34). In this case the layer may, however, be replaced by a suitably chosen parabolic layer. Near the maximum concentration of electrons in the layer, the region where \(\varepsilon<0\) cannot be large, and only partial reflection of waves from the layer takes place; part of the energy of the incident wave then passes through the layer. The frequency of the radio wave at which this passage becomes practically complete bears, as is known, the name of the critical frequency of the layer. As will be clear from what follows (see § 5), the critical frequency \(\omega_k\) may be determined with great accuracy as the frequency at which the point \(z(\varepsilon=0)\) is situated at the maximum of the layer. At higher frequencies, as is clear from (33), the point \(z(\varepsilon=0)\), where \(\varepsilon=n^2=0\), does not exist at all. If \(n=0\) at the maximum of the layer, then

\[ N_{\max}=\frac{m\omega_k^2}{4\pi e^2}=\frac{\omega_k^2}{3.19\cdot 10^9}. \tag{35} \]

This relation determines the critical frequency \(\omega_k\); usually, however, one uses not the cyclic frequency, but the ordinary critical frequency \(f_k=\dfrac{\omega_k}{2\pi}\), equal to

\[ f_k=9\cdot 10^3\sqrt{N_{\max}}. \tag{36} \]

In the \(F\)-layer \(N_{\max}\leq 3\cdot 10^6\), and thus \(f_k\leq 1.5\cdot 10^7\), and the critical wavelength \(\lambda_k=\dfrac{c}{f_k}\) is not less than 20 meters.

Of course, for oblique incidence from the ionosphere, shorter waves may be reflected than for normal incidence, since in this case the wave is reflected in a region where \(n>0\).

§ 4. PROPAGATION AND REFLECTION OF WAVES FROM AN IONOSPHERIC LAYER

Before going further, it is necessary to dwell on the question of the rigorous solution of the wave equation (7). As is known, this equation has solutions expressible in terms of known functions only in certain cases, i.e., for a quite definite dependence of \(\varepsilon'\) on \(z\). Thus, for example, in the case of a linear layer \(\varepsilon'=a+bz\) the solution is expressed in Bessel functions of order \(1/3\) \(^{14,8}\). For a parabolic layer, when \(\varepsilon'=a+bz^2\), the solution is expressed in functions of a parabolic cylinder (Weber functions) \(^{14,15}\). If \(\varepsilon'\) has the form
\[ \varepsilon'=a+\frac{e^{z}\,[\,b+(e^{z}+1)+c\,]}{(e^{z}+1)^2}, \]
where, as before, \(a\), \(b\), \(c\) are certain complex constants, then equation (7) is reducible to the hypergeometric equation and its solution can be investigated in detail \(^{16,17}\).

Some other integrable cases are given in works \(^{8,18-20}\). We shall not dwell here on the various solutions in greater detail, since, for reasons clear from what was said in § 3, in application to the ionosphere only the solution for a linear layer is of interest, and near the critical frequency—for a parabolic layer.

For a linear layer beginning at \(z=0\), in the absence of absorption,
\[ \varepsilon=1-\frac{z}{z_1}\quad (z>0;\ \text{for } z<0,\ \varepsilon=1); \tag{37} \]
and the solution is as follows:
\[ \begin{aligned} E&=A\xi^{1/2}\left\{J_{1/3}\!\left(\frac{2}{3}\xi^{3/2}\right)+J_{-1/3}\!\left(\frac{2}{3}\xi^{3/2}\right)\right\}\quad \text{for } \xi>0,\\ E&=A|\xi|^{1/2}\left\{-I_{1/3}\!\left(\frac{2}{3}|\xi|^{3/2}\right)+I_{-1/3}\!\left(\frac{2}{3}|\xi|^{3/2}\right)\right\}\quad \text{for } \xi<0, \end{aligned} \tag{38} \]
where \(A\) is a constant, \(J\) is the Bessel function,
\[ I_\nu(z)=e^{-\nu\frac{\pi}{2}}J_\nu\!\left(e^{i\frac{\pi}{2}}z\right) \]
\[ \xi=\left(\frac{\omega^2}{c^2z_1}\right)^{1/3}(z_1-z). \tag{39} \]

If \(\xi\gg 1\) (practically \(\xi>10\)), i.e.,
\[ \left(\frac{\omega}{c}z_1\right)^{2/3}\varepsilon(z)= \left(\frac{2\pi}{\lambda_0}z_1\right)^{2/3}\left(1-\frac{z}{z_1}\right)\gg 1, \tag{40} \]

then, to determine \(E\), one can use the asymptotic representation of Bessel functions. As a result, for \(\zeta>0\),

\[ E=A\,\frac{3}{\sqrt{\pi}}\,\zeta^{-1/4}\cos\left[\frac{2}{3}\zeta^{3/2}-\pi/4\right]. \tag{41} \]

We shall assume that at the beginning of the layer, at \(z=0\) [i.e. \(\zeta=\left(\frac{\omega}{c}z_1\right)^{2/3}\)], the incident wave is \(E_+=e^{i\omega t}\); the solution (41) represents a superposition of the incident and reflected waves and, consequently, at \(z=0\) (we omit the factor \(e^{i\omega t}\)):

\[ E=E_+ + E_- = 1+e^{-i\left(\frac{4}{3}\frac{\omega}{c}z_1-\frac{\pi}{2}\right)}, \tag{42} \]

whence

\[ A=\frac{2}{3}\sqrt{\pi}\left(\frac{\omega}{c}z_1\right)^{1/6} e^{-i\left(\frac{2}{3}\frac{\omega}{c}z_1-\frac{\pi}{4}\right)}. \]

The phase shift between the reflected and incident waves is equal to

\[ \varphi=\frac{4}{3}\frac{\omega}{c}z_1-\frac{\pi}{2} =2\frac{\omega}{c}\int_0^{z_1}\left(1-\frac{z}{z_1}\right)^{1/2}dz-\frac{\pi}{2} \]

\[ =2\frac{\omega}{c}\int_0^{z_1} n\,dz-\frac{\pi}{2}. \tag{43} \]

The expressions obtained, their generalization to the case when absorption is present, and also certain results for a parabolic layer will be discussed below. We may now turn to the consideration of reflection from an arbitrary layer occurring far from the maximum concentration. If the layer is everywhere linear, i.e. \(\varepsilon\) is determined by formula (37), then the exact solution of the wave equation at the beginning of the layer is given by formula (42). As is clear from (43), this solution can be written in the form

\[ E=1+e^{-i\varphi}, \]

\[ \varphi=2\frac{\omega}{c}\int_0^{z(\varepsilon=0)} n(z)\,dz-\frac{\pi}{2}, \tag{44} \]

where \(n=\left(1-\frac{z}{z_1}\right)^{1/2}\), and \(z(\varepsilon=0)=z_1\) is the point where \(\varepsilon=0\). If we now considered this case in the geometrical-optics approximation and additionally assumed that at the point \(z(\varepsilon=0)\) complete reflection of the wave occurs, then, as is clear from (32), the phase shift between the incident and reflected waves at the exit from the layer would be equal to

\[ \varphi = 2\frac{\omega}{c}\int_{0}^{z(\varepsilon=0)} n(z)\,dz. \]
Thus, the exact solution for \(\varphi\) differs from the geometrical-optics solution constructed by the method described only by the term \(-\pi/2\).

Comparing (41) with (32), it is easy to see that, as applied to a linear layer, geometrical optics is applicable provided condition (40) is satisfied, i.e., at distances from the point \(z(\varepsilon=0)\) satisfying the inequality
\[ \Delta z \gg \left(\frac{z_1 c^2}{\omega^2}\right)^{1/3} = \left(\frac{z_1\lambda_0^2}{4\pi^2}\right)^{1/3}. \tag{45} \]

From what was said in § 3 it follows that if condition (34) is fulfilled for \(\Delta z\) satisfying inequality (45), i.e., if the inequality*) is fulfilled
\[ \left|\frac{d^2\varepsilon}{dz^2}\right|_{0} \ll \left|\frac{d\varepsilon}{dz}\right|_{0}^{4/3} \lambda_0^{-1/3}, \tag{46} \]
then one can obtain the solution for an arbitrary layer by matching, at \(z-z(\varepsilon=0)\sim \Delta z\), the solution for a linear layer with the solution of geometrical optics. As a result, for any layer, if condition (46) is satisfied, at the exit from the layer (at \(z=0\)) the solution (44) holds, where \(n(z)\) is the function of \(z\) characterizing the given layer\(^{14}\). Under ordinary conditions the correction \(-\pi/2\) in (44) is much smaller than the principal first term of this expression.

Let us illustrate the application of the formulas obtained by the example of a parabolic layer, which is very often used and which, owing to the presence of a maximum, also permits consideration of the region close to the critical frequency. In this case
\[ \varepsilon = 1-\frac{f_k^2}{f^2} \left[ 1-\left(\frac{z}{z_m}\right)^2 \right], \tag{47} \]
where \(z_m\) is the half-thickness of the layer, the coordinate \(z\) being measured, as is customary in this case, from the maximum of the layer; \(f=\omega/2\pi\) is the frequency and \(f_k\) is the critical frequency, determined from the condition that at \(f=f_k\), \(\varepsilon=0\) at the maximum of the layer, where \(z=0\). Expression (47) corresponds to choosing the electron concentration in the form
\[ N=N_{\max}\left[1-\left(\frac{z}{z_m}\right)^2\right]. \]

*) For a linear layer \(|d\varepsilon/dz|=1/z_1\), which was also taken into account in substituting (45) into (34).

For layer (47), condition (46) is satisfied if the point lies from the maximum of the layer (i.e., the point \(z=0\)) at a distance satisfying the inequality

\[ z(\varepsilon=0)\gg \sqrt{\lambda_0 z_m}\cdot \sqrt{\frac{f}{f_k}} \qquad \left(\lambda_0=\frac{c}{f}\right). \tag{48} \]

Since for waves with frequency \(f\) *)

\[ z(\varepsilon=0)=\pm \sqrt{\frac{f_k^2-f^2}{f_k^2}\,z_m}, \tag{49} \]

condition (46), or (48), is satisfied if

\[ \Delta f \gg \frac{c}{z_m}, \tag{50} \]

where

\[ \Delta f=f_k-f. \]

For the \(F\) layer, \(z_m\sim 100\) km and, thus, we obtain \(\Delta f\gg 3\cdot 10^3\), i.e., condition (46) and the formula (44) given above for \(\varphi\) are valid for frequencies differing from the critical one by more than \(3\cdot 10^4\) Hz. Since \(f_k\) in this case is of the order of \(7\)—\(8\) Mc/s, formula (44) is applicable for \(f_k/f\geq 1.005\), i.e., practically always. For the \(E\) layer, where \(z_m\sim 20\) km, condition (30) becomes \(\Delta f\gg 1.5\cdot 10^4\), and since here \(f_k\) is of the order of \(2\)—\(3\) Mc/s, the approximation considered above is applicable for \(f_k/f\geq 1.05\), i.e., also over almost the entire frequency range.

Let us note here in passing that for layer (47) **)

\[ \left. \begin{aligned} \varphi &=2\frac{\omega}{c}\int_{z(\varepsilon=0)}^{z_m} n(z)\,dz-\frac{\pi}{2} =\frac{\omega}{c}L_0-\frac{\pi}{2},\\ L_0 &=z_m\left[ 1-\frac{f_k^2-f^2}{2f_k f} \ln\frac{f_k+f}{f_k-f} \right], \end{aligned} \right\} \tag{51} \]

where we have introduced the optical path length \(L_0\) (see below).

The results presented can be extended to the case where absorption is present. For a linear layer in this case

\[ \varepsilon'=(n-ik)=\varepsilon-i\frac{4\pi\sigma}{\omega} =\left(1-\frac{z}{z_1}\right)-i\left(\alpha+\beta\frac{z}{z_1}\right), \tag{52} \]

where \(\alpha\) and \(\beta\) are real constants.

*) The point corresponding to the minus sign in (49) lies beyond the maximum of the layer and is not related to total reflection of the wave.

**) With the coordinates chosen for the parabolic layer, the point \(0\) in (44) corresponds to the value \(z=z_m\), and, owing to the opposite direction of the \(z\)-axis, the limits in the integral must be interchanged.

The solution of the wave equation (7) then still has the form (38), where

\[ \zeta=\xi-i\eta=\left(\frac{\omega}{c}\,\frac{z_1}{1+i\beta}\right)^{2/3}(n-ik), \tag{53} \]

and the conditions \(\zeta>0\) and \(\zeta<0\) must be replaced by the conditions \(\xi>0\) and \(\xi<0\).

If condition (46) is satisfied and \(\sigma\) varies slowly, i.e. the condition

\[ \frac{d\sigma}{dz}\lambda_0\ll 1, \tag{54} \]

holds, the solution of the problem can, as in the absence of absorption, be obtained for any layer.

If at the beginning of the layer, for the incident wave, \(E=1\), then the whole field at the beginning of the layer, equal to the sum of the incident and reflected waves, has the form:

\[ E=1+e^{-i\varphi}=1+\rho e^{-i\varphi}; \tag{55} \]

the phase shift \(\varphi\) and the reflection coefficient \(\rho\) are determined by the relations

\[ \varphi=\frac{2\omega}{c}\int_{0}^{z(\xi=0)} n(z)\,dz-\frac{\pi}{2} -\frac{\sqrt{2}\,\omega}{c\left|\dfrac{d\varepsilon}{dz}\right|_0} \left(\frac{4\pi\sigma(0)}{\omega}\right)^{3/2}, \tag{56} \]

\[ -\ln \rho=\frac{2\omega}{c}\int_{0}^{z(\xi=0)} k(z)\,dz+ \frac{\sqrt{2}\,\omega}{c\left|\dfrac{d\varepsilon}{dz}\right|_0} \left(\frac{4\pi\sigma(0)}{\omega}\right)^{3/2}, \tag{57} \]

where \(\left|\dfrac{d\varepsilon}{dz}\right|_0\) and \(\sigma(0)\) are the values of these quantities at the point \(z(\xi=0)\).

Expressions (56) and (57) differ from the frequently used formulas by the presence of the term

\[ \frac{\sqrt{2}\,\omega}{c\left|\dfrac{d\varepsilon}{dz}\right|_0} \left(\frac{4\pi\sigma(0)}{\omega}\right)^{3/2}, \]

which, however, is usually very small; in the real cases considered in \(^{8}\), this correction amounts to about \(5\%\) of the value of \(\ln \rho\). With respect to \(\varphi\), the correction term is entirely insignificant, since it is even much smaller than the correction \(-\pi/2\); it must be noted, however, that absorption itself affects the value of \(\varphi\), since in this case \(n\) is not equal to \(\sqrt{\varepsilon}\), but to the more complicated expression (24).

Above we have mainly dwelt on the consideration of the reflected wave at its exit from the layer, i.e. we were interested in the phase and amplitude of the reflected wave at the boundary of the layer or, essentially the same thing, far from the layer. Of known interest may also be the character of the wave field in the layer itself and, in particular, near the reflection point, i.e. the point \(z(\xi=0)\).

Far from this point the wave field has the form (32); near the point \(z(\varepsilon=0)\), if condition (34) is satisfied, the answer to the question can be obtained from consideration of a linear layer. In the absence of absorption the solution near the point \(z(\varepsilon=0)\) is determined by formula (38), where

\[ A=\frac{2}{3}\sqrt{\pi}\left(\frac{\omega}{c\left|\dfrac{d\varepsilon}{dz}\right|_0}\right)^{1/6} e^{-i\left(\dfrac{\omega}{c}\int_{0}^{z(\varepsilon=0)} n(z)\,dz-\dfrac{\pi}{2}\right)} . \tag{58} \]

Calculations using tables of Bessel functions lead to the following conclusions\(^8\). The field \(E\) vanishes at points located from the point \(z(\varepsilon=0)\) at a distance

\[ \Delta z_0=\left(\frac{3}{2}\delta\right)^{2/3} \frac{1}{\left(\dfrac{\omega^2}{c^2}\left|\dfrac{d\varepsilon}{dz}\right|_0\right)^{1/3}}, \tag{59} \]

where \(\delta\) is equal to \(2.38;\ 5.61;\ 8.64\), etc. (for the tenth zero \(\delta=30.63\)). The first maximum of \(E\) lies at the point

\[ \Delta z_m=\left(\frac{3}{2}\,0.7\right)^{2/3} \frac{1}{\left(\dfrac{\omega^2}{c^2}\left|\dfrac{d\varepsilon}{dz}\right|_0\right)^{1/3}} . \tag{60} \]

The value of \(|E|^2\) at the first maximum is\(^*\)

\[ |E_m|^2=3.6\left[\frac{\omega/c}{\left|d\varepsilon/dz\right|_0}\right]^{1/3}. \tag{61} \]

The wave field in the reflection region is represented in Fig. 4 by a solid line, with the quantity

\[ \xi=\Delta z\left(\frac{\omega^2}{c^2}\left|\frac{d\varepsilon}{dz}\right|_0\right)^{1/3} \]

plotted along the abscissa axis and the quantity \(\left|\dfrac{E}{A}\right|^2\) along the ordinate axis; for \(\xi\gg1\),

\[ \left|\frac{E}{A}\right|^2=\frac{9}{\pi}\,\xi^{-1/2}\cos^2\left(\frac{2}{3}\xi^{3/2}-\frac{\pi}{4}\right). \]

As was shown above, despite the fact that for \(n\to0\) the solution of geometrical optics is inapplicable, the phase of the reflected wave can essentially (i.e. usually with great accuracy) be computed by means of the geometrical-optics approximation, supplemented

\(^*\) In work 19 the reflection was rigorously considered not from a linear layer, but from a considerably more complicated one. However, if this layer is replaced in the reflection region by a linear layer with the same value \((d\varepsilon/dz)_0\), then the value \(|E_m|^2\), even if it differs from (61), does so only in the third digit; in 19, because of a numerical error in passing from formula (4.14) of the cited work to the last formula in § 4 of the cited work, corresponding to our formula (61), the coefficient is \(2.9\) instead of \(3.6\).

by the condition of reflection at the point \(z(\varepsilon=0)\). In this connection it is of some interest to see what the wave field looks like in the same approximation. Constructing, from the solutions (32) in the absence of absorption, a standing wave with a node at the point \(z(\varepsilon=0)\), we arrive at the expression

\[ E=\frac{2}{\sqrt{n}}\sin\left(\frac{\omega}{c}\int_0^z n\,dz\right), \tag{62} \]

where \(z\) is measured from the point \(z(\varepsilon=0)\), and the factor two has been chosen in accordance with the fact that the amplitude of the incident wave is taken to be equal to unity [as also in (42)].

Fig. 4.

Calculating the integral (62) for a linear layer, one can see that \(E\) vanishes at points differing from (59) by the replacement of \(\delta\) by \(n\pi\), where \(n\) is the number of the zero; for the first zero the difference is \(\sim 16\%\), for the second \(\sim 10\%\), and for the tenth \(\sim 1\%\). The position of the first maximum and its height are determined by replacing in (60) and (61) the coefficients \(0.7\) and \(3.6\), respectively, by \(1.5\) and \(3\).

Thus, in a number of cases formula (62) proves to be a good approximation (the values of \(\left|\frac{E}{A}\right|^2\) in this case are shown in Fig. 4 by the dashed curve). If there were no layer at all, and instead an ideal mirror were placed there, then the value of \(|E_m|^2\) would be equal to \(4\), since we take the amplitude of the incident wave to be unity. In the presence of the layer there is a certain swelling of the field, which can

characterized by the factor

\[ \gamma^2=\frac{|E_m|^2}{4}=0.9\left[\frac{\omega/c}{\left|\dfrac{d\varepsilon}{dz}\right|_0}\right]^{1/3}; \tag{63} \]

this factor is, in practice, not especially large. Thus, in the \(E\) layer, where the swelling might be of interest for nonlinear effects \(^{19}\), \(\left|\dfrac{d\varepsilon}{dz}\right| \gtrsim 10^{-7}\), and for \(\lambda_0=1000\) meters, \(\gamma^2=7.7\) (if \(\left|\dfrac{d\varepsilon}{dz}\right|=10^{-7}\)).

In the presence of absorption, the consideration of the wave field is likewise achieved by using formula (38), but with \(\zeta\) determined by means of (53). The principal change in the expression for \(|E_m|^2\), or \(\gamma^2\), now consists in the appearance, as a multiplier, of the amplitude reflection coefficient \(\rho\) [see (55)]. The numerical factors—for example, the factor 3.6 in (61)—are replaced by somewhat different ones, depending on the magnitude of the absorption. If, for example,

\[ \frac{4\pi\sigma(0)}{\omega} = \frac{1}{2} \left[ \frac{\left|\dfrac{d\varepsilon}{dz}\right|_0}{\omega/c} \right]^{2/3}, \]

then

\[ \Delta z_m \approx \left(\frac{3}{2}\cdot 0.9\right)^{2/3} \left[ \frac{1}{\dfrac{\omega}{c}\left|\dfrac{d\varepsilon}{dz}\right|_0} \right]^{1/3}, \qquad E_m \approx 6\rho \left( \frac{\omega/c}{\left|\dfrac{d\varepsilon}{dz}\right|_0} \right)^{1/3}. \tag{64} \]

Putting \(\left|\dfrac{d\varepsilon}{dz}\right|_0=10^{-7}\), \(\lambda_0=1000\) meters, and \(\rho=1/8\) (i.e. \(-\ln\rho=2\)), we obtain for \(\gamma^2\) a value equal to \(2.7\)*). From the values of \(\gamma^2\) given above it follows that the role of the “swelling” of the field reduces to the fact that the field in the ionosphere may be estimated from formulas derived for the case in which the Heaviside layer is replaced by an ideally reflecting surface. The presence of swelling in this case compensates, or even overcompensates, for the attenuation of the field due to absorption.

When the point \(z(\varepsilon=0)\) approaches sufficiently closely the place of maximum electron concentration, replacing the layer by a linear one is inadmissible, which is also manifested in the failure to satisfy condition (34)—(46). At the same time, on approaching the maximum, the region of negative values of \(\varepsilon\) becomes ever smaller, and the wave begins to leak through the layer. So long as the transmission coefficient is very small, it can be approximately calculated from the formula

\[ D=1-R \sim e^{-2\frac{\omega}{c}\int_{z_1}^{z_2}|\sqrt{\varepsilon}|\,dz}, \tag{65} \]

where \(R\) is the reflection coefficient (the ratio of the amplitudes of the reflected

*) For the adopted values of \(\left|\dfrac{d\varepsilon}{dz}\right|_0\) and \(\lambda_0\), in the particular example considered, the value of \(\sigma(0)\) corresponds to the number of collisions \(\nu_{ef}\sim 10^4\).

and incident signals \(\rho=\sqrt{R}\)) and the integration is carried out between the points \(z_1(\varepsilon=0)\) and \(z_2(\varepsilon=0)\), i.e., in the region where the value of \(\varepsilon\) is negative [cf. (49)]. For a parabolic layer, a simple calculation of \(D\) by formula (65) leads to the following result:

\[ D=e^{-\frac{4\pi}{\lambda_k}z_m\,\frac{f_k^2-f^2}{2f_k^2}} \simeq e^{-\frac{4\pi^2\Delta f}{c}z_m}, \tag{66} \]

where \(\lambda_k=\dfrac{c}{f_k}\) and \(\Delta f=f_k-f\); the latter transformation in (66) is possible because, in the region where penetration is appreciable, \(\Delta f\ll f_k\). In the region of maximum concentration, any smooth layer can be approximated by a parabolic layer. Therefore, for a rigorous wave treatment of the question, it is sufficient to discuss the solution of the wave equation for a parabolic layer \(^{21,15}\). We shall not give here the expressions for the wave field, since they are not very transparent and, more importantly, are not of special interest. As for the reflection coefficient, for it the cited works obtained the following expression, valid for any \(R\) and under the condition \(z_m\gg\dfrac{\lambda_k}{2\pi}\):

\[ \frac{\rho^2}{1-\rho^2}=\frac{R}{1-R}=e^{\frac{4\pi^2\Delta f z_m}{c}}. \tag{67} \]

If \(R\sim 1\) (i.e., \(D=1-R\ll 1\)), then formulas (66) and (67) are identical. In this case the exact coincidence of the results is apparently accidental, since formula (65)—(66) is valid only up to the pre-exponential factor. From (66)—(67) it is evident that penetration of the signal may be neglected if

\[ \Delta f\gg \frac{c}{z_m}, \]

i.e., under condition (50), which is also the condition for the possibility of replacing the layer by a linear one. Thus there is complete harmony between the different approaches to the question of replacing the layer under consideration by a linear one (the conditions of small penetration and small deviations from linearity coincide). Using formula (66)—(67), one may see, passing to a quantitative estimate, that

\[ D\leq 10^{-3},\quad \text{if}\quad \Delta f\geq \frac{c}{6z_m}. \tag{68} \]

(Graph with vertical axis \(\rho\) and horizontal axis \(\Delta f=f_k-f\).)

Fig. 5.

Using the estimates of the value \(c/z_m\) given above for the \(F\) and \(E\) layers, one can see that for the \(F\) layer penetration is significant only for \(\Delta f<500\) cycles, and for the \(E\) layer—for \(\Delta f<2000\) cycles. The dependence of \(\rho\) on \(\Delta f\) is shown in Fig. 5.

The critical frequency \(f_k\) is called, as was said, the frequency at which the point \(z(\varepsilon=0)\) reaches the maximum of the layer. If one assumes that at the point \(z(\varepsilon=0)\) there is total reflection, and in the case when there is no such point at all (i.e. for \(f>f_k\)) total transmission, then for \(f<f_k\) \(\rho=1\) and for \(f>f_k\) \(\rho=0\). In a strict wave treatment the frequency \(f_k\) is in no way singled out, since partial reflection takes place on both sides of it. However, the great sharpness of the function \(\rho(f)\) gives the usual definition of \(f_k\) practically the same content as in the case where leakage is neglected.

In addition to the reflection coefficient, the value of the phase shift of the reflected wave relative to the incident one in the leakage region, where formula (44) is inapplicable, is also of interest. The corresponding expression for \(\varphi\) was obtained in \(^{15}\). We shall not give it here; instead, in the following paragraph we shall indicate a formula for a more interesting quantity, the so-called apparent height, obtained from \(\varphi\) by differentiation with respect to frequency.

§ 5. REFLECTION OF SIGNALS *)

Above we have discussed the propagation and reflection of a monochromatic wave. However, as is well known, in practice the propagation of a quasi-monochromatic group of waves (i.e. a pulse) is usually of interest. In order to consider this case as well, it is evidently necessary to expand the field into a Fourier integral and to carry out its investigation.

We shall represent the incident wave at the beginning of the layer in the form:

\[ E_0(t)=\int_{-\infty}^{+\infty} g(\omega)e^{i\omega t}\,d\omega, \tag{69} \]

where, by Fourier’s theorem,

\[ g(\omega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} E_0(t)e^{-i\omega t}\,dt. \tag{70} \]

In the case of a monochromatic wave with frequency \(\omega_0\), \(E_0(t)=e^{i\omega_0 t}\) and \(g(\omega)=\delta(\omega-\omega_0)\), where \(\delta\) is the delta function

\[ \int_{-\infty}^{+\infty}\delta(\omega-\omega_0)\,d\omega=1,\qquad \delta(\omega-\omega_0)=0\quad\text{for }\omega\ne\omega_0. \]

In the case of a quasi-monochromatic group of waves, by definition, the function \(g(\omega)\) is very sharp, differing substantially from zero

*) In this paragraph the exposition is more detailed than in the others. This is explained by the fact that here the author was deprived of the possibility of referring to any treatment of the question satisfactory to him.

only in a small region near the carrier frequency of the signal \(\omega_0\); in other words, the spectral width of the signal \(\Delta\omega\) in this case satisfies the inequality:

\[ \Delta\omega \ll \omega_0 . \tag{71} \]

After traversing some segment of the path, or, for definiteness, after reflection of the wave from a layer and return to the same point from which the wave was sent upward, the field has the form

\[ E(t)=\int_{-\infty}^{+\infty}\rho(\omega)\,g(\omega)\,e^{i(\omega t-\varphi(\omega))}\,d\omega, \tag{72} \]

where \(\rho(\omega)\) and \(\varphi(\omega)\) are, respectively, the reflection coefficient and the phase shift of the reflected wave.

In the absence of absorption and with complete reflection, \(\rho(\omega)=1\), which will be assumed below.

In the case of a quasi-monochromatic group of waves it is convenient to represent the field \(E_0(t)\) in the form:

\[ E_0(t)=A(t)e^{i\omega_0 t}, \tag{72'} \]

where \(A(t)\) is evidently a slowly varying function of \(t\).

According to (69), (70), (72) one may write

\[ E_0(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\int A(\eta)\, e^{i[(\omega_0-\omega)\eta+\omega t]}\,d\omega\,d\eta \tag{73} \]

and

\[ E(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\int A(\eta)\, e^{i[(\omega_0-\omega)\eta+\omega t-\varphi(\omega)]}\,d\omega\,d\eta . \tag{74} \]

For a quasi-monochromatic group, to a first approximation one may put

\[ \varphi(\omega)=\varphi(\omega_0)+\varphi'(\omega_0)\Omega, \tag{75} \]

where

\[ \varphi'(\omega_0)=\left(\frac{d\varphi}{d\omega}\right)_{\omega=\omega_0} \quad\text{and}\quad \Omega=\omega-\omega_0 . \]

In this approximation

\[ E(t)=e^{i(\omega_0t-\varphi(\omega_0))}\frac{1}{2\pi} \int_{-\infty}^{+\infty}\int A(\eta)\, e^{i\Omega(t-\eta-\varphi'(\omega_0))}\,d\eta\,d\Omega = \]

\[ = A\bigl(t-\varphi'(\omega_0)\bigr)e^{i(\omega_0t-\varphi(\omega_0))}, \tag{76} \]

by virtue of the fact that, as follows from (73),

\[ \frac{1}{2\pi}\int_{-\infty}^{+\infty}\int A(\eta)e^{i\Omega(t-\eta)}\,d\eta\,d\Omega=A(t). \]

The obtained result (76) means that, in the first approximation (75), the group does not change its shape (as a function of \(t\)), but, as a result of reflection and passage through the layer, the phase of the wave changes by \(\varphi(\omega_0)\), and the whole group is delayed in time by the “group delay time”*)

\[ \Delta t_{\mathrm{gr}}=\left(\frac{d\varphi}{d\omega}\right)_{\omega=\omega_0} =\varphi'(\omega_0). \tag{77} \]

One may also introduce the “phase delay time,” which, as is clear from (76), is equal to

\[ \Delta t_{\phi}=\frac{\varphi(\omega_0)}{\omega_0}. \tag{78} \]

In order to relate \(\Delta t_{\mathrm{gr}}\) and \(\Delta t_{\phi}\) to the group and phase velocities, as is usually done, let us consider the propagation of a pulse in a homogeneous medium. In this case, by \(\varphi(\omega)\) we shall understand the phase shift in traversing the path \(z\), i.e. \(E_0(t)\) in this case is the field at \(z=0\) and \(E(t)\) is the field at the point \(z\). Then, as is known, \(\varphi(\omega)=\dfrac{\omega}{c}n(\omega)z\) [see (22)].

In the approximation (75), i.e. using (76), we have

\[ E(t)=A\left\{t-\left(\frac{d\,\dfrac{\omega}{c}n(\omega)}{d\omega}\right)_{\omega_0}z\right\} e^{i\left[\omega_0 t-\dfrac{\omega_0 n(\omega_0)z}{c}\right]} . \tag{79} \]

From this it is seen that the phase propagates with the phase velocity

\[ w(\omega_0)=\frac{c}{n(\omega_0)}, \tag{80} \]

whereas the pulse as a whole propagates without deformation with the group velocity

\[ u(\omega_0)=\frac{c}{\dfrac{d}{d\omega}(\omega n(\omega))} =\frac{c}{\,n(\omega_0)+\omega_0\left(\dfrac{dn}{d\omega}\right)_{\omega_0}} . \tag{81} \]

Let us note that if \(n^2\) is determined by formula (10′), then

\[ u=cn. \tag{82} \]

In the ionosphere, as we have seen, formula (44) for \(\varphi(\omega)\) has a very wide range of applicability. In this case

\[ \Delta t_{\phi} =\frac{2}{c}\int_{0}^{z(\varepsilon(\omega_0)=0)} n(z,\omega_0)\,dz =2\int_{0}^{z(\varepsilon(\omega_0)=0)} \frac{dz}{w(z,\omega_0)} \tag{83} \]

*) This means, as is clear from (72) and (76), that if, for example, the incident group appears at the observation point at the moment \(t=0\), and for \(t<0\), \(E_0(t)=0\), then the reflected wave at the same point will appear at the moment \(t=\varphi'(\omega_0)\), and up to this moment \(E(t)=0\).

and

\[ \Delta t_{\mathrm{gr}}=2\int\limits_{0}^{z\,(\varepsilon(\omega_{0})=0)} \frac{dz}{u(z,\omega_{0})}, \tag{84} \]

where in (83) the vanishingly small term \(\dfrac{\pi}{2\omega_{0}}\) has been omitted.

Expressions (83) and (84), obtained in the geometrical-optics approximation [see (32) and (44)], have an obvious meaning, since for a homogeneous medium
\(\Delta t_{\phi}=\dfrac{L}{w(\omega)}\) and \(\Delta t_{\mathrm{gr}}=\dfrac{L}{u(\omega)}\), where \(L\) is the group path traversed.

Instead of the times \(\Delta t_{\phi}\) and \(\Delta t_{\mathrm{gr}}\), the lengths of the optical and group paths are very often used; these are respectively equal to

\[ L_{0}=c\Delta t_{\phi},\qquad L_{\mathrm{gr}}=c\Delta t_{\mathrm{gr}}. \tag{85} \]

Half of \(L_{\mathrm{gr}}\), i.e. the quantity

\[ h(\omega_{0})=\int\limits_{0}^{z\,(\varepsilon(\omega_{0})=0)} \frac{c\,dz}{u(z,\omega_{0})}. \tag{86} \]

is called the apparent height of the point of reflection of the radio signal from the ionosphere and, by definition, is equal to the height at which the signal would be reflected if it propagated during the time \(\Delta t_{\mathrm{gr}}\) with velocity \(c\).

For the parabolic layer (47)

\[ h=\frac{z_{m}}{2}\frac{f}{f_{k}}\ln\frac{f_{k}+f}{f_{k}-f}, \tag{87} \]

it is assumed here that the signal is sent from the beginning of the layer, where \(\varepsilon=1\).

Formulas (83), (84), (86), and (87) are applicable only if expression (44) can be used for \(\varphi\), i.e., in essence, only in the geometrical-optics approximation. Therefore, from what was said in § 4 it is clear that these formulas can no longer be used near the maximum of the layer. This fact, as should have been expected, is clearly manifested in formula (87), according to which, as \(f\to f_{k}\), \(h\) and \(\Delta t_{\mathrm{gr}}\) tend to infinity, which of course does not actually occur.

In the region \(f\sim f_{k}\), \(\varphi(\omega)\) and \(h\) must be computed using the solution of the wave equation. ^15 We shall not give the corresponding expressions, limiting ourselves to the exact formula for \(h\) at \(f=f_{k}\)

\[ h(f_{k})=\frac{z_{m}}{2}\left\{0{,}577+\ln\frac{16\pi z_{m}}{\lambda_{k}}\right\}. \tag{88} \]

To form an idea of the course of the exact value of \(h\) and the value of \(h\) given by formula (87), Fig. 6 may be of help; in it both of these quantities are presented for a parabolic layer with a semithickness \(z_m=120\) km and a critical wavelength \(\lambda_k=30\) m. From this graph it is seen that, for

\[ \frac{\Delta f}{f_k}>10^{-4}, \]

both values of \(h\) practically coincide completely, in agreement with the estimate based on inequality (50).

The principal radio method for investigating the ionosphere, consisting in determining the delay time of signals reflected from the ionosphere, leads directly to the determination of the function \(h(\omega_0)\), where \(\omega_0\) is the carrier frequency of the signal; in existing installations this frequency is varied smoothly, so that the whole function \(h(\omega_0)\) is found at once. The curve \(h(\omega_0)\), or, what is the same, \(h(f_0)\), is called the height-frequency characteristic of the ionosphere. It is shown schematically in Fig. 7. As the critical frequency of a layer is approached, the apparent height grows rapidly. This is explained by the fact that \(h\) is the greater, the larger the region of small values of the group velocity \(u\). Since \(u=cn\) [see (82)], the height is especially large if the wave has to pass through a large region with a small value of \(n\), which is precisely what occurs upon reflection near the maximum of the layer, i.e. near \(f_k\). A signal reflected from the \(F\) layer at frequencies only slightly exceeding the critical frequency \(f_k\) for the \(E\) layer undergoes a large delay in this latter layer, although it is not reflected from it. This explains the large value of the apparent height of the \(F\) layer near \(f_k\) for the \(E\) layer. A signal reflected from the \(F\) layer is split into two as a result of double refraction caused by the earth’s magnetic field; this will be discussed in § 6. The height-frequency characteristics observed experimentally are often considerably more complicated than that shown in Fig. 7, owing to the appearance of the \(F_1\) layer, the sporadic \(E\) layer

Fig. 6.

Fig. 7.

and a number of complicating circumstances; we cannot dwell on this here\(^{1,2}\).

The apparent height is always greater than the true height of reflection \(h=z(\varepsilon(\omega_0)=0)\), since \(c/u\) in (86) is always less than unity. It can be shown\(^{22}\) that if the refractive index is expressed by formula (10), then

\[ h_u(\omega_0)=\frac{2}{\pi}\int_0^{\omega_0}\frac{h(x)\,dx}{(\omega_0^2-x^2)^{1/2}} =\frac{2}{\pi}\int_0^{\pi/2} h(\omega_0\sin\varphi)\,d\varphi, \tag{89} \]

where \(h(x)\) is determined by formula (86) with \(\omega_0\) replaced by \(x\). For a linear layer, \(h_u=\frac{1}{2}h\) (if the height is measured from the beginning of the layer).

In the approximation considered (75), the pulse propagates without any distortions. The latter, however, do occur, and to study them it is necessary to pass to the next approximation, in which

\[ \varphi(\omega)=\varphi(\omega_0)+\varphi'(\omega_0)\Omega+ \frac{\varphi''(\omega_0)}{2}\Omega^2. \tag{90} \]

Substituting (90) into (74), we have:

\[ E(t)=e^{i(\omega_0 t-\varphi(\omega_0))} \frac{1}{2\pi} \int_{-\infty}^{+\infty}\int A(\eta)\, e^{i\Omega\left(t-\eta-\varphi'(\omega_0)\right) -i\frac{\varphi''(\omega_0)}{2}\Omega^2}\,d\Omega\,d\eta . \]

Putting here

\[ \varphi''\left(\Omega+\frac{\eta-t+\varphi'}{\varphi''}\right)^2=\pi u^2, \]

i.e., replacing the variable \(\Omega\) by \(u\) and taking into account that

\[ \int_{-\infty}^{+\infty} e^{-i\frac{\pi}{2}u^2}\,du=1-i, \]

we have

\[ E(t)=\frac{1-i}{2\pi}e^{i(\omega_0t-\varphi(\omega_0))} \int_{-\infty}^{+\infty} \sqrt{\frac{\pi}{\varphi''(\omega_0)}}\, A(\eta)\, e^{\frac{i(\eta-t+\varphi')^2}{2\varphi''}}\,d\eta . \]

Replacing now \(\dfrac{(\eta-t+\varphi')^2}{2\varphi''}\) by \(\dfrac{\pi u^2}{2}\), we finally obtain:

\[ E(t)=\frac{1-i}{2}e^{i(\omega_0t-\varphi(\omega_0))} \int_{-\infty}^{+\infty} A\left[t-\varphi'(\omega_0)+\sqrt{\pi\varphi''(\omega_0)}\,u\right] e^{i\frac{\pi}{2}u^2}\,du . \tag{91} \]

If the derivative \(\varphi''(\omega_0)\) is so small that the term \(\sqrt{\pi\varphi''}\,u\) in the amplitude of the function \(A\) in (91) may be neglected, then (91) goes over into (76), as it should. In the general case, the change in the form of the signal is determined by the form of the function \(A(t)\).

In the simplest case

\[ \left. \begin{aligned} A(t)&=1 \quad \text{for } -\frac{T}{2}<t<\frac{T}{2},\\ A(t)&=0 \quad \text{for } t>\frac{T}{2}\ \text{and}\ t<-\frac{T}{2}, \end{aligned} \right\} \tag{92} \]

i.e., \(E_0=e^{i\omega_0 t}\) in the interval \(-\dfrac{T}{2}<t<\dfrac{T}{2}\) and \(E_0=0\) outside this interval. In this case\(^*\)

\[ g(\omega)=\frac{\sin(\omega-\omega_0)\dfrac{T}{2}}{\pi(\omega-\omega_0)} \tag{93} \]

and

\[ E=\frac{1-i}{2}e^{i(\omega_0 t-\varphi(\omega_0))} \left\{ F\!\left(\frac{T-\theta}{\sqrt{\pi\varphi''(\omega_0)}}\right) - F\!\left(\frac{-\theta}{\sqrt{\pi\varphi''(\omega_0)}}\right) \right\}, \tag{94} \]

where \(\theta=-\dfrac{T}{2}+t-\varphi'(\omega_0)\) is the time, reckoned from the instant \(-\dfrac{T}{2}+\varphi'(\omega_0)\) of arrival at the point of observation of the pulse without taking its spreading into account, and

\[ F=\int_0^u e^{\,i\frac{\pi}{2}u^2}\,du \tag{95} \]

is the Fresnel integral \(\bigl(F(-u)=-F(u);\ F(\infty)=\dfrac{1+i}{2}\bigr)\). Practically, usually, the signal duration \(T\gg\sqrt{\pi\varphi''(\omega_0)}\). In this case, from the point of view of the spreading of the signal, only the range of variation of \(\theta\) in which \(\theta\ll T\) is of interest. The amplitude of the reflected signal for \(\theta\ll T\) is equal to

\[ |E|=\frac{1}{\sqrt{2}}\left|F(\infty)-F(-u)\right|, \qquad u=\frac{\theta}{\sqrt{\pi\varphi''(\omega_0)}}. \tag{96} \]

Expression (96) is well known, since it determines the intensity of the wave field in Fresnel diffraction at the edge of a plane screen. A graph of the function \(|E|\) is given in Fig. 8. For \(\theta=0\), i.e. at the instant when, without taking the spreading into account, there should arrive at the point of observation

\(^*\) The spreading of the signal in the case (92) was considered somewhat differently in \({}^{23}\).

the wave front and the amplitude must change discontinuously from zero to unity

\[ |E|=\frac{1}{2}\quad \text{for } \theta=0. \tag{97} \]

The time required for the signal amplitude to become established is, in order of magnitude, equal to

\[ \tau=\sqrt{\pi\varphi''(\omega_0)}. \tag{98} \]

A more precise determination of the time of establishment can, of course, be obtained only if the required accuracy of establishment of the amplitude is specified. From Fig. 8, for example, it is seen that the time interval during which the amplitude is established to an accuracy of up to \(5\%\) for \(\theta>0\) and \(|E|<0.05\) for \(\theta<0\), exceeds \(\tau\) by a factor of 8. Estimates of \(\tau\) are given in \(^{23}\); even outside the region of the critical frequency, where the establishment time is especially large, \(\tau\sim 10^{-5}\) sec.

Fig. 8.

Taking absorption into account, a factor \(\rho(\omega)\) [see (72)] must be introduced into the expression for \(E\). In practice, over the region of the spectral width of the signal, \(\rho(\omega)\) changes very little,\(^{8}\) and thus one may put \(\rho(\omega)=\rho(\omega_0)\) and take this factor out from under the integral (72); in this case, obviously, all subsequent calculations remain unchanged. Absorption has an effect, in addition to the appearance of the factor \(\rho(\omega_0)\), on the values of \(n(\omega)\) and \(\varphi(\omega)\).

The consideration of the spreading of a wave group can be carried out not only for a quasi-monochromatic pulse, but also for pulses of any spectral width.\(^{23}\)

§ 6. ACCOUNTING FOR THE INFLUENCE OF THE EARTH’S MAGNETIC FIELD

In §§ 3–5 we did not take into account the influence of the Earth’s magnetic field on the propagation of radio waves. When this influence is taken into account, it is necessary to use, instead of equation (7), system (8). If the coordinate system is chosen as indicated in § 2 (the Earth’s magnetic field \(H^{(0)}\) then lies in the \(yz\) plane), and absorption is not taken into account, then \(D\) is expressed in terms of \(E\) by means of formulas (20). Equations (8)

take the following form:

\[ \begin{gathered} \frac{d^2 E_x}{dz^2}+\frac{\omega^2}{c^2}\left(AE_x+iCE_y\right)=0,\\ \frac{d^2 E_y}{dz^2}+\frac{\omega^2}{c^2}\left(-iCE_x+BE_y\right)=0; \end{gathered} \tag{99} \]

\[ \begin{gathered} A=\frac{u-(1-v)^2-uv\cos\alpha}{u-(1-v)-uv\cos^2\alpha},\qquad C=\frac{\sqrt{u}\cos\alpha\cdot v(1-v)}{u-(1-v)-uv\cos^2\alpha},\\ B=\frac{u(1-v)-(1-v)^2}{u-(1-v)-uv\cos^2\alpha} \end{gathered} \tag{100} \]

and

\[ u=\frac{\omega_H^2}{\omega^2},\qquad \sqrt{u}=\frac{\omega_H}{\omega},\qquad v=\frac{\omega_0^2}{\omega^2}=\frac{4\pi eN(z)}{m\omega^2}. \tag{101} \]

In the case of an inhomogeneous medium the coefficients \(A\), \(B\), and \(C\) depend on \(z\) through \(v\). If the medium is homogeneous, then equations (100) are solved without difficulty by substituting \(E_{x,y}=Ce^{\pm i\frac{\omega}{c}nz}\), where \(n\) is the refractive index. The conditions for the existence of a nontrivial solution of the resulting system of algebraic equations lead to a quadratic equation for determining \(n^2\); in implicit form this equation is, evidently, as follows:

\[ \left| \begin{array}{cc} A-n^2 & iC\\ -iC & B-n^2 \end{array} \right|=0. \]

The solution of this equation has the form:

\[ n_{1,2}^2=1-\frac{2v(1-v)}{2(1-v)-u\sin^2\alpha\pm\sqrt{u^2\sin^4\alpha+4u(1-v)^2\cos^2\alpha}}. \tag{102} \]

In the presence of absorption, i.e., taking into account in (17) the term \(m\nu_{ef}\dot r\), we obtain the following expression:

\[ (n_{1,2}-ik_{1,2})^2=1-\frac{2v\left(1-v-i\frac{\nu_{ef}}{\omega}\right)}{A\pm B}, \tag{103} \]

where

\[ A=2\left(1-i\frac{\nu_{ef}}{\omega}\right)\left(1-v-i\frac{\nu_{ef}}{\omega}\right)-u\sin^2\alpha; \]

\[ B=\sqrt{u^2\sin^4\alpha+4u\left(1-v-i\frac{\nu_{ef}}{\omega}\right)^2\cos^2\alpha}, \]

where \(n\) and \(k\) are the refractive and absorption indices.

In what follows, for simplicity, we shall not take absorption into account. In a magnetic field an ionized gas is birefringent, and waves of two types can propagate in it in a given direction, differing in their velocity, i.e., in refractive index—

... refraction and the form of the oscillations, i.e. their polarizations. In formula (102) these two solutions differ by the choice of the sign before the radical.

Choosing in (102) the upper sign before the radical, we obtain the square of the refractive index \(n_2^2\) of the “ordinary” wave; the lower sign corresponds to the “extraordinary” wave \((n^2=n_1^2)\). We take the refractive index to be \(n_{1,2}=\sqrt{n_{1,2}^2}\). The solutions \(n_{1,2}=-\sqrt{n_{1,2}^2}\) correspond to waves traveling in the opposite direction, which will be taken into account below directly in the expression for the phase of the wave.

If the magnetic field \(H^{(0)}=0\), then also \(u=0\), and, consequently,

\[ n_{1,2}^2=n_0^2=1-v, \]

as indeed it should be [see (10′)].

The refractive index \(n_2\), as well as \(n_0\), vanishes when

\[ v=v_{20}=1. \tag{104} \]

If \(u<1\), then the refractive index \(n_1^2\) vanishes*) when

\[ v=v_{10}^{\pm}=1\pm\sqrt{u}=1\pm\frac{\omega_H}{\omega} \tag{105} \]

and tends to infinity when

\[ v=v_{1\infty}=\frac{1-u}{1-u\cos^2 a}. \tag{106} \]

Fig. 9.

For \(u>1\), \(n_1^2\) vanishes only at the point \(v_{10}^{+}\), since \(v^{+}\geqslant 0\); in this case the index \(n_1^2\) does not become infinite. At the same time, if \(u\cos^2 a>1\), the index \(n_2^2\) becomes infinite; the point \(v_{2\infty}\) is then determined by equation (106).

The dependence of \(n_{1,2}^2\) on \(v\) is illustrated by Figs. 9 and 10; in Fig. 9, \(u=\tfrac14\), i.e. \(\omega=2\omega_H\) (since \(H^{(0)}=0.5\) gauss, \(\omega_H=\dfrac{eH}{mc}=8.82\cdot 10^6\) and \(\lambda_H=\dfrac{2\pi c}{\omega_H}=214\) m). For the curves of Fig. 10, \(u=1.08\) and, in the other case, \(u=4\); the angle \(a\) in both cases shown in Fig. 10 is \(20^\circ\). As is clear from (106), \(v_{2\infty}\geqslant\dfrac{1}{\cos^2 a}\), equality holding for \(u\to\infty\).

Below we shall restrict ourselves to consideration of the most important case, when \(u<1\).

*) For the time being we assume that \(a\ne 0\) (see below).

Investigation of the curves \(n_{1,2}^{2}(v)\) in various cases and, in particular, with absorption taken into account, may be found in a number of works \(^{24-26}\). Of special interest, for a number of reasons, is the still insufficiently investigated region where \(u\) is close to 1 [see \(^{10,27}\)].

The form of the oscillations of both waves propagating in a homogeneous medium is determined from (93):

\[ \frac{E_x^{(1,2)}}{E_y^{(1,2)}}= \frac{-iC}{A-n_{1,2}^{2}}= \frac{B-n_{1,2}^{2}}{iC} = i\,\frac{ u\sin^{2}\alpha \mp \sqrt{u^{2}\sin^{4}\alpha+4u(1-v)^{2}\cos^{2}\alpha} }{ 2\sqrt{u(1-v)}\cos\alpha }, \tag{107} \]

where, as before, the upper sign of the root refers to wave 2 \(\bigl(E_x^{(2)}, E_y^{(2)}, n_2\bigr)\), and the lower sign to wave 1; moreover, evidently,

\[ \frac{E_x^{(1)}}{E_y^{(1)}}=\frac{E_y^{(2)}}{E_x^{(2)}}. \]

The polarization of both waves in the general case is elliptical, the axes of the ellipses being parallel to the axes \(x\) and \(y\).

In order to form an idea of the propagation of waves in an inhomogeneous medium, it is necessary to consider the dependence of \(n_{1,2}^{2}\)

Fig. 10.

on \(v\). If the function \(N(z)\) is monotonic and, in particular, linear with respect to \(z\), then the corresponding graph of the dependence \(n_{1,2}^{2}(v)\) simultaneously describes the dependence of the refractive indices of both waves on height. If, furthermore, \(n_{1,2}^{2}\) changes sufficiently slowly with the change in \(z\), then, generally speaking, the quasi-stationary case should occur, i.e. in the neighborhood of a given point the field should be approximately the same as for a homogeneous medium with the corresponding values of \(n_{1,2}^{2}\). The justification of this intuitively clear assertion is achieved by passing to the approximation of geometrical optics.

(see § 3 and below). Under the condition that this approximation is valid, knowledge of \(n_{1,2}^2(z)\) is indeed sufficient for a qualitative description of wave propagation in an inhomogeneous medium. In particular, total internal reflection of waves should occur near the points where \(n_{1,2}=0\) (reflection from the region near the point \(n_1 \to \infty\) we leave aside for the moment; as will be clear from what follows, generally speaking, it does not occur).

Besides \(N(z)\), or the dimensionless parameter

\[ v=\frac{4\pi e^2 N(z)}{m\omega^2}, \]

the value of \(n_{1,2}^2\) depends on \(u\) and \(a\). In the special cases when \(a=0\) (propagation along the field) and \(a=\frac{\pi}{2}\) (propagation perpendicular to the field), the expression for \(n_{1,2}^2\) is greatly simplified:

\[ a=0:\quad n_{1,2}^2=1-\frac{v}{1\mp \sqrt{u}}, \tag{108} \]

\[ a=\frac{\pi}{2}:\quad n_1^2=1-\frac{v(1-v)}{1-u-v},\quad n_2^2=n_0^2=1-v. \tag{109} \]

For \(a=0\) both waves are circularly polarized; for \(a=\frac{\pi}{2}\) they are linearly polarized, and for wave 2 the electric vector is parallel to the magnetic field (the \(y\)-axis), i.e. \(E_x=E_z=0\); for wave 1 in this case \(E_y=0\), but the component \(E_z\), as well as \(E_x\), is not equal to zero.

It is important to note that at any nonzero angle between the wave vector and the magnetic field, \(n_1^2\) vanishes at two points (105) and \(n_2^2\) at one point (104*); the positions of these points do not depend on \(a\) at all. If, however, \(a=0\), then, as is clear from (108) or directly from (102), both \(n_1^2\) and \(n_2^2\) vanish only at one point, and the position of the root of the equation \(n_2^2=0\) for \(a=0\) jumps from \(v=1\) to \(v=1+\sqrt{u}\). This “jump” of the roots \(n_2^2\) is clear from Fig. 9.

Thus, at very small angles \(a\) there is a peculiarity which becomes clear from considering the curves \(n_{1,2}^2\) for small values of \(a\) (see Fig. 11). We see that, formally speaking, there is no continuous transition to the case \(a=0\). Physically, however, it is clear that such a transition must occur, and the following will be observed. For \(v<1\), wave 1 (the “extraordinary” wave) in all cases propagates only up to the point \(v_{10}\) (see Fig. 9), where it undergoes total reflection. As long as \(a\) is sufficiently large, wave 2 (the “ordinary” wave) is totally reflected at the point \(v_{20}\) (more precisely, in the region near this point). As \(a\) decreases, the properties of wave 2

*) Let us recall that we restrict ourselves to the case where \(u<1\).

in the region \(v<1\) approach ever more closely the properties of a wave of type 1, capable of propagating for \(v>1\). Indeed, for small \(\alpha\), \(n_2^2\) at values \(v<1\), but close to 1, approaches \(n_1^2\) for \(v\) greater, but also close to 1; furthermore, in this case the form of the oscillations of waves 2 and 1 at the points \(v_1\) and \(v_{\mathrm{II}}\) (see Fig. 11) approach one another; the latter is connected with the fact that, as is clear from (107), the ratio

\[ \frac{E_x^{(1,2)}}{E_y^{(1,2)}} \]

changes sign when \(v\) passes through unity. In view of this

\[ \left(\frac{E_x^{(2)}}{E_y^{(2)}}\right)_{v_1} \simeq \left(\frac{E_x^{(1)}}{E_y^{(1)}}\right)_{v_{\mathrm{II}}}. \]

Fig. 11.

It is clear from this that it is sufficient for wave 2 to seep through to the point \(v_{\mathrm{II}}\), which in the wave treatment to a certain extent always takes place, for it to be able again to propagate upward to the point \(v_{10}^{+}\), where it is finally reflected (here it is assumed, of course, that Figs. 10 and 11 describe the variation of \(n^2\) with height, i.e., for example, \(v=az\)). Thus, near the point \(v=1\) there must occur partial reflection of wave 2 and its partial penetration into the region with \(v>1\). The reflection coefficient depends very sharply (exponentially) on \(\alpha\), as is usually the case in problems of this type. Therefore, in practice, only after \(\alpha\) becomes smaller than some angle \(\alpha_0\), which must be calculated (see below), will partial reflection occur. For \(\alpha>\alpha_0\) the reflection will be complete, and for \(\alpha=0\) it must be equal to zero.

All these arguments, generally speaking, remain valid for an upward-propagating radio signal, i.e. a quasi-monochromatic group of waves. For sufficiently small angles between the direction of the earth’s magnetic field and the direction of the normal to the front of the signal traveling vertically upward, i.e. the direction of the plumb line,* the signal must be reflected from the Heaviside layer not at two points \((v_{10}^{-}, v_{20})\), as is the case for \(\alpha>\alpha_0\), but at three \((v_{10}^{-}, v_{20}, v_{10}^{+})\); the intensity

* The effect of “tripling” of the signal can occur only if the signal propagates vertically upward, since otherwise its reflection is effected for \(n_{1,2}^{2}>0\) and the anomalous region near the point \(v=1\) is never reached. Therefore the angle \(\alpha\) may be small, and at the same time the “tripling” effect may be observed, only near the magnetic poles of the earth.

of the third reflection (from the point \(v_{10}^{+}\)) must, as \(a\) decreases, grow at the expense of the intensity of the reflection from the point \(v_{20}\), until, at \(a=0\), this latter reflection disappears completely. The difference in the delay times of the two signals will in this case be considerably greater than for \(a>a_0\).

A rigorous solution of equations (99), even for the simplest dependences of \(N\) on \(z\), has not yet been carried out. Although the possibility of a rigorous discussion of these equations is by no means excluded, it is clear that this problem is considerably more complicated than the solution of the wave equation (7). In this connection, the application to the system (99) of approximate methods of treatment, and first of all of the method of geometrical optics, acquires special significance.

The geometrical-optics approximation can also be obtained, as in the isotropic case [see (25)], by representing the solution of system (99) in the form:

\[ \mathbf{E}=\left(\mathbf{E}^{(0)}+\frac{c}{\omega}\mathbf{E}^{(1)}+\ldots\right)e^{\pm i\frac{\omega}{c}\psi}. \tag{110} \]

As a result, in the first approximation we arrive at the relations \(^{13}\):

\[ \mathbf{E}_{(1,2)}=\mathbf{E}_{(1,2)}^{(0)} e^{\pm i\frac{\omega}{c}\psi_{1,2}}, \]

\[ \left(\frac{d\psi}{dz}\right)_{1,2}^{2}=n_{1,2}^{2}, \tag{111} \]

\[ E_{x;\,1,2}^{(0)}= \frac{\mathrm{const}} {\sqrt{\left(\dfrac{d\psi}{dz}\right)_{1,2}\left(1-K_{1,2}^{2}\right)}} , \tag{112} \]

\[ \frac{E_{y;\,1,2}^{(0)}}{E_{x;\,1,2}^{(0)}}= \frac{A-\left(\dfrac{d\psi}{dz}\right)_{1,2}^{2}}{-\,iC} = \frac{iC}{B-\left(\dfrac{d\psi}{dz}\right)_{1,2}^{2}} = K_{1,2}. \tag{113} \]

The phase of the waves of both types 1 and 2 is thus determined by the expression

\[ \psi_{1,2}=\int_{z_0}^{z} n_{1,2}(z)\,dz . \tag{114} \]

The first approximation of geometrical optics (111)—(114) is applicable only when a number of inequalities are satisfied, which we shall not present here, since under the condition of a smooth and monotonic dependence of \(N\) on \(z\) all these inequalities reduce practically to the single relation

\[ \frac{\lambda_0}{2\pi} \left| \frac{\dfrac{dn_{1,2}}{dz}}{n_{1,2}^{2}} \right| \ll 1, \tag{115} \]

coinciding with the most important of inequalities (30), which refer to the isotropic case. In the geometrical-optics approximation, the waves traveling upward and downward are completely independent.

It follows from (115) that as \(n_{1,2}\to 0\) the geometrical-optics approximation is inapplicable, which is also clear immediately, since in this case \(\lambda=\dfrac{\lambda_0}{n_{1,2}}\to \infty\). Therefore, in the region of the points \(n_1=0\) and \(n_2=0\), the different solutions of the geometrical-optics approximation are not independent. In the isotropic case, as a result, in the region near the point \(n=0\) reflection takes place, i.e. a wave \(\sim e^{-i\frac{\omega}{c}\psi}\) gives rise to a wave \(\sim e^{+i\frac{\omega}{c}\psi}\). If the region where \(n^2<0\) extends sufficiently far, then the reflection is total; and although geometrical optics near the point \(n=0\), strictly speaking, is inapplicable, the corresponding solution, taking into account the presence of total reflection, is qualitatively valid. For a sufficiently smooth variation of \(N\) with \(z\), this applies (see § 4) both to the calculation of the phase of the reflected wave \(\varphi\), and to the value of the wave amplitude near the point \(z\,(n^2=0)\).

In the case of an anisotropic medium the situation is completely analogous.

If \(a=0\) or \(a=\dfrac{\pi}{2}\), then the variables in equations (99) separate, i.e. the functions \(E_{\pm}=E_x\pm iE_y\), in the case \(a=0\), and the functions \(E_x\) and \(E_y\), for \(a=\pi/2\), turn out to satisfy independent second-order equations of the type of equation (7); in these cases, therefore, all the conclusions of § 4 remain fully valid. For arbitrary \(a\) the variables do not separate, since in this case the ratio \(\dfrac{E_x^{(1,2)}}{E_y^{(1,2)}}\) depends on \(v\), i.e. on \(z\). However, even in this general case, undoubtedly, the qualitative picture remains unchanged and, in particular, the phase \(\varphi\) is determined by the formula \(\varphi_{1,2}=\displaystyle\int_0^{z(n_{1,2}^2=0)} n_{1,2}\,dz\), although the added term \(-\pi/2\) in formula (44) may change.

Leaving this question aside, let us examine what is connected with the presence, in the anisotropic case, of two types of waves. From what has been said it is clear that in the region where \(n_1\to 0\) geometrical optics is, generally speaking, applicable to wave 2, and, conversely, where \(n_2\to 0\), it is applicable to a wave of type 1. It follows from this that reflection of wave 1 “from the point” \(v_{10}\) (where \(n_1=0\)) should not be accompanied by the appearance of waves of type 2. The inapplicability of geometrical optics leads only to coupling of waves of type 1 traveling in different directions; the same applies to the reflection of wave 2.

Above it was assumed that, in the region where one of the waves cannot be considered by the method of geometrical optics, the other wave is applicable.

this method is applicable. There is, however, a case when the two waves turn out not to be independent. This occurs precisely under the conditions in which a “trebling” of the signals should be observed, i.e. for small values of \(\alpha\). As is seen from Figs. 9 and 11, in this case near the point \(v_{20}\), i.e. near the value \(v=1\), \(n_2\to 0\) and \(\dfrac{dn_1}{dz}\to \infty\). Thus, in this case (for \(v\to 1\)) the geometrical-optics approximation is indeed inapplicable to waves of both types [see (115)].

The solution of the problem of partial reflection from the point \(v_{20}\) for \(\alpha\to 0\) must be based on equations (99). The difficulties arising here are very great. A known simplification can be achieved if equations (99) are solved only for the region of \(v\) between \(v_{\mathrm I}\) and \(v_{\mathrm{II}}\) (see Fig. 11), where penetration and partial reflection are precisely “played out.” In the regions where \(v<v_{\mathrm I}\) and \(v>v_{\mathrm{II}}\), one may either regard the medium as homogeneous or apply the geometrical-optics approximation; at the boundary of the regions, i.e. at the points \(v_{\mathrm I}\) and \(v_{\mathrm{II}}\), the solutions must be joined in the proper way.

The approximate solution of the problem of partial reflection leads to the following result \(^{13}\). The coefficient of penetration, i.e. the ratio of the intensity of the wave that has penetrated through the region \(v\sim 1\) to the intensity of the wave incident on this region, is equal to

\[ D\sim e^{-\frac{2\pi}{\lambda_0}\,\beta\sqrt{u}\left(\frac{N}{dN/dz}\right)_{v=1}\sin^2\alpha} = e^{-\gamma}, \tag{116} \]

where \(\beta\) is a factor of order unity depending on \(u\), and the values of \(N\) and \(dN/dz\) are taken at the point \(v=1\). If, for example, \(u=1.4\) (i.e. \(\omega=2\omega_H=1.76\cdot 10^7\)), then \(\beta=0.6\) and

\[ \gamma=17.5\,\frac{\sin^2\alpha}{\left(\dfrac{dN}{dz}\right)_{v=1}} . \tag{117} \]

The drawback of formula (116) is the circumstance that an undetermined pre-exponential factor remains in it. If this factor is taken to be equal to unity, then for a real value \(\left(\dfrac{dN}{dz}\right)_{v=1}=0.1\), \(\gamma=2\) (i.e. \(D=0.135\)) for \(\alpha\sim 6^\circ\). The effect of “trebling,” as far as we know, has not yet been observed. From the estimate given, which is not definitive, it follows that this effect can be observed only in the immediate vicinity of the Earth’s magnetic poles. Let us also note here that formula (116) is applicable only so long as the coefficient \(D\) is small. As \(\alpha\to 0\), \(D\to 1\), and the intensity of the wave reflected from the point \(v=1\) is small.

In conclusion let us recall that in an anisotropic medium one must distinguish between the direction of the normal to the wave and the direction of the ray, i.e. the direction of the energy flux. If the problem of propagation of the wave front in the geometrical-optics approximation has been solved, then one can

determine the trajectory of the ray; this can also be done directly, using Fermat’s principle. When the wave propagates vertically upward, as we have seen, the wave normal also remains directed upward the whole time; the Poynting vector in this case has another direction, as is clear from the fact that then \(E_z \ne 0\) [see (20)].

Upon reflection the components \(E_x\) and \(E_y\) change sign, i.e., a standing wave is formed; in this process the component \(E_z\) also changes sign. Therefore, if a certain portion of a plane wave is sent upward, then after reflection back to the ground this portion will return to the very same place (see Fig. 12). A real transmitter sends upward not a portion of a plane wave, but a more complicated wave field; however, since the inhomogeneous layer is located far from the transmitter, this case reduces to the preceding one. From what has been said it is clear that the noncoincidence of the direction of the ray with the \(z\)-axis does not lead to any substantial changes in the propagation process discussed above, apart from the fact that the downward-coming signal is reflected not from the region of the ionosphere situated directly above the observation point, but from a somewhat displaced region; in this respect, for waves of types 1 and 2 these regions are different (see Fig. 12). Incidentally, for \(\alpha = 0\) the ray is directed along the \(z\)-axis, and for small \(\alpha\) the deviation of the ray from the normal is smaller than in the other cases.

Wave type 2
Wave type 1
Heaviside layer
Wave surface of the incident signal
reflected

Fig. 12.

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