A Review of Current Data on Studies of Radio Wave Propagation

Ya. L. Al’pert

I. Long, Medium, and Short Waves

Contents

1. Introduction . . . 505
2. New theoretical results . . . 506
3. On the propagation of long waves . . . 536
4. Studies over the inhomogeneous surface of the Earth . . . 541
5. Influence of the troposphere—new results . . . 544
6. Studies of round-the-world echo . . . 547
7. Speed of radio waves . . . 552
8. Cross-modulation of radio waves . . . 555
   Cited literature . . . 558

1. Introduction

In the last 10 years, new and interesting results of studies of radio-wave propagation have been published which, up to the present time, have not yet been covered in review articles. Only some of these new data are contained in books published in recent years 1–5.

In the reviews published during these years, the problems of propagation of ultrashort waves6 (the meter range), data from studies of the velocity of radio waves7, and problems of ionospheric research8, 9 were set forth rather fully. However, precisely in this period of time there rapidly increased the use in technology and science of part of the ultrashort-wave range, which is now often singled out as a separate range of microwaves—centimeter and decimeter waves. Some questions connected with the data on the propagation of these waves are contained in the article by V. L. Ginzburg5a. However, a large number of new and important data;

relating to this field, had until recently been little known[^55]. In the field of the “old” radio ranges—long, medium, and short waves—new and interesting results have also been obtained; the same applies to the velocity of propagation of radio waves, to whose measurement over a wide range of radio waves several works carried out after the publication of the article by L. I. Mandelstam and N. D. Papaleksi[^7] have been devoted.

In these same years, important fundamental theoretical works were carried out in the field of the propagation of radio waves over the Earth’s surface, to a certain extent bringing to completion the almost half-century-long period of investigations by a large number of scientists in this field. These are chiefly the works of Soviet physicists on the diffraction of radio waves around the Earth with allowance for refraction in the atmosphere (V. A. Fock[^36],[^11]) and on the propagation of radio waves over an inhomogeneous Earth’s surface (G. A. Grinberg, M. A. Leontovich, E. L. Feinberg[^3]).

The present article is intended to fill, to some extent, the gap indicated above. In the first part (I) the recent new data in the field of the propagation of long, medium, and short waves are briefly presented; the second part (II) is intended to be devoted to the propagation of microwaves and, in part, ultrashort waves.

2. NEW THEORETICAL RESULTS

a) Diffraction of radio waves around the Earth with allowance for refraction in the troposphere

The propagation of radio waves over a homogeneous Earth’s surface, if the influence of the ionosphere is not taken into account, is caused by diffraction around the Earth and by refraction in the troposphere. In this case the electromagnetic field at the receiving point is produced by the so-called direct wave (or, as it is sometimes called, the ground wave).

The study of the direct wave is of great interest, since in practice conditions are very often produced under which the formulas obtained from calculations of the direct wave can be applied with a high degree of accuracy over a wide range of waves (from 1–2 m to 2000–5000 m) at distances that may amount to 1000–2000 km and more.

Many works have been devoted to constructing a mathematical theory of the propagation of the direct wave. Among them one should mention the works of B. A. Vvedensky[^10], which played a major role in creating the theory of propagation of ultrashort waves. However, the solution of this problem was fully completed only in the works of V. A. Fock[^36],[^11]. In these investigations a rigorous mathe-

classical theory of diffraction and refraction of radio waves around the Earth, and formulas were obtained giving a continuous transition from the illuminated region to the shadow region. In limiting cases one obtains the formula for a plane Earth) and the reflection formula for the illuminated region*).

For the case of taking only diffraction into account, rigorous theoretical calculations had previously been carried through to numerical calculations and graphs of the field amplitude in the works of Van der Pol and Bremmer ^13,14. Refraction in the atmosphere, before the works of V. A. Fock, had been taken into account only on the basis of ray interpretation; meanwhile, in the region of penumbra and shadow the concept of a ray generally loses its meaning, and thus the concept of the equivalent radius of the Earth, which was introduced on this basis, had no theoretical justification. V. A. Fock ^5 gave a justification of the concept of the equivalent radius of the Earth and showed that it is applicable in the region of shadow and penumbra, and that in the majority of practical cases replacing the radius of the Earth by the equivalent radius in formulas that take only diffraction into account is quite sufficient for taking account of refraction in the atmosphere.

The attenuation function (which is sometimes called the attenuation factor; since it enters as a factor in the basic formulas and determines how the modulus and phase of the components of the electromagnetic field at the receiving point change in comparison with the value that they would have in free space) is given in the above-cited works of V. A. Fock ^11,35 by the series

\[ V(x,y_1,y_2,l)=\left| e^{i\frac{x}{2}}\cdot 2\sqrt{\pi x}\sum_{s=1}^{\infty} \frac{e^{ixt_s}}{t_s-q^2}\cdot \frac{w_1(t_s-y_1)}{w_1(t_s)}\cdot \frac{w_1(t_s-y_2)^{***}}{w_1(t_s)} \right| . \tag{1} \]

The quantity entering formula (1),

\[ x=m\vartheta \tag{2} \]

is called the reduced horizontal distance from the source

*) Weyl–Van der Pol formula.

**) It was first substantiated and applied to calculations of ultrashort waves by B. A. Vvedenskii (see the interesting exposition of the limiting transitions of the diffraction formula in the article by V. A. Fock ^12).

***) It should be noted that formula (1) is convenient for calculations only in the shadow region, where it is sufficient initially to restrict oneself to two, and then only to one, terms of the series. In the penumbra region, however, allowance for a large number of terms is required, so that calculations by (1) become very cumbersome. In the illuminated region, calculations are carried out by the reflection formula. For more detail on the reflection formula, see Part II of this article.

(Fig. 1); \(r=R_0\vartheta\) is the distance between the emitter and the observation point along the surface of the Earth;

\[ y_1=\frac{k}{m}z_1 \quad \text{and} \quad y_2=\frac{k}{m}z_2 \tag{3} \]

are, respectively, the reduced heights of the observation point and the emission point; \(z_1\) and \(z_2\) are the heights of the observation and emission points above the Earth’s surface;

\[ q=im\,\frac{k}{k_2}\sqrt{1-\frac{k^2}{k_2^2}} \tag{4} \]

is a parameter characterizing the electrical properties of the Earth’s surface, since

\[ k_2^2=k^2\left(\varepsilon+i\frac{4\pi\sigma}{\omega}\right), \qquad k=\frac{\omega}{c}, \tag{5} \]

where \(\varepsilon\) and \(\sigma\) are the dielectric constant and conductivity of the Earth’s surface, and

\[ m=\sqrt[3]{\frac{kR_0}{2}} \tag{6} \]

is a large parameter of this problem (\(R_0\) is the radius of the Earth); throughout the solution, terms of order \(1/m^3\) are neglected in comparison with unity.

Fig. 1.

The quantities \(t_s\) entering into (1) are the roots of the equation

\[ w'(t)-qw(t)=0, \]

where

\[ w(t)=\frac{1}{\pi}\int_{\Gamma} e^{itz-\frac{1}{3}z^3}\,dz \tag{7} \]

is the complex Airy function, playing an auxiliary role in these computations.

For the case when either the vertical dipole or the point of observation is located on the surface of the Earth \((z_1\) or \(z_2 = 0)\), the function \(V(x, y, q)\) was tabulated by M. G. Belkina \(^{15}\) in the shadow region for a large interval of variation of the parameters \(x, y, q\), which makes it possible to calculate the field strength by the formula

\[ E_R=\frac{300\sqrt{W_\Sigma}}{r}\cdot\frac{h_e^2}{R_0^2}\frac{|V(x,y,q)|}{2}, \tag{8} \]

where \(r=R_0\vartheta\) (in km), \(W_\Sigma\) (in kW) is the radiated power, and

\[ h_e=\frac{R_0\cdot b}{C_0}\sin\vartheta \tag{9} \]

(see the notation in Fig. 1). Tables \(^{15}\) also contain the values of \(\arg V(xyq)\), which determine the additional phase and, consequently, the phase velocity of radio waves; moreover, they show how the group velocity can be calculated. The availability of data on the phase of the wave is a great advantage of these calculations in comparison with those given in \(^{11-12}\) and in other works, where these quantities either are not calculated, or the method of calculation does not make it possible at all to determine the phase of the wave.

Here the formula is given only for \(E_R\), the vertical component of the field on the surface of the Earth, since it plays the principal role in measurements, especially in the wavelength range mainly considered in this article.

In the formulas given above the radius of the Earth \(R_0\) occurs everywhere. To take account of refraction in the lower atmosphere it is necessary to replace, in the expressions for \(x, y, q\) [(2), (3), (4)], the quantity \(R_0\) by the so-called equivalent radius of the Earth \(R^*\), determined by the formula

\[ \frac{1}{R^*}=\frac{1}{R_0}+\frac{1}{2\varepsilon_0}\left(\frac{d\varepsilon}{dz}\right)_0, \tag{10} \]

where \(\varepsilon_0\) and \(\left(\dfrac{d\varepsilon}{dz}\right)_0\) are the values of the dielectric constant of the lower atmosphere and of its gradient at the surface of the Earth.

We point out here that the application of the concept of the equivalent radius of the Earth, the justification of which, as stated above, was first given in the work of V. A. Fock \(^{36}\), is, strictly speaking, permissible only for a normal atmosphere, when with a high degree of accuracy it may be assumed that the gradient of the refractive index \(\varepsilon\) does not change with height, i.e. that \(\dfrac{d\varepsilon}{dz}=\mathrm{const}\). V. A. Fock \(^{36}\) investigated the conditions under which this assumption is lawful. However, in the case of temperature inversion, when the so-called phenomenon of superrefraction arises at ultrashort waves, the quantity \(\dfrac{d^2\varepsilon}{dz^2}\) can attain a large value, and allowance for refraction requires a more

exact computations. The general solution and the attenuation function were obtained by V. A. Fock[^36] from more general assumptions concerning the refractive index of the atmosphere, so that they take into account the complicated law of variation of $\varepsilon$; however, in this case it is very difficult to obtain numerical results.

The second circumstance that should be noted here is that in V. A. Fock,[^36] as is also adopted at the present time, it is assumed that the lower atmosphere is a pure dielectric and that there is no dispersion in it. The generally accepted expression for the refractive index at the present time is[^44]

Fig. 2.

\[ n=\sqrt{\varepsilon}=\frac{79}{T}\left(p-\frac{e}{7}+4800e\right)\cdot 10^{-6}, \tag{11} \]

where $T$ is the absolute temperature, $p$ is the total atmospheric pressure, and $e$ is the pressure of water vapor in millibars (1 mm of mercury column $=1.332$ millibars), and it has been shown that this formula should not contain any frequency dependence down to short microwaves. It should, however, be borne in mind that at the present time

selective molecular absorption by water vapor and oxygen is well known (see Part II of the present article on this) in the microradio-wave region. This indicates that in the microwave range \(\varepsilon\) must depend on frequency.

In conclusion we shall present several graphs useful for practical calculations. In Figs. 2–5, for different wavelengths, curves of field strength over land \((\varepsilon=4,\ \sigma=9\cdot10^{7}\ \mathrm{CGSE})\) are plotted as a function of distance from a source of power \(1\ \mathrm{kW}\), for the

Fig. 3.

case in which the source and the point of observation are located on the surface of the Earth. In Figs. 6–9 the same is presented for the sea \((\varepsilon=80,\ \sigma=3.6\cdot10^{10}\ \mathrm{CGSE})\). These curves are borrowed from work \(^{13}\) and were calculated taking only diffraction into account. Checking these curves against the tables of M. G. Belkina \(^{15}\) gives complete agreement of the numerical results.

The graphs presented may be used practically almost up to wavelengths of \(2\text{–}5\ \mathrm{m}\). For waves of shorter length these calculations are hardly suitable, although they are useful for some estimates. This is explained by the fact that, in the case of waves shorter than \(2\text{–}5\ \mathrm{m}\), the source and the point of observation are practically never located

Fig. 4.

Visible labels in the figure:

• Left vertical axis: \(E_{\mathrm{eq}}/\mathrm{m}\)
• Right vertical axis: Decibels
• Horizontal axis: \(r,\ \mathrm{km}\)
• Annotation: \(\delta = 9 \cdot 10^{7},\ \varepsilon = 4\)
• Curve labels:
• \(20\ \mathrm{MHz}\ (10\ \mathrm{m})\)
• \(60\ \mathrm{MHz}\ (5\ \mathrm{m})\)
• \(150\ \mathrm{MHz}\ (2\ \mathrm{m})\)
• \(300\ \mathrm{MHz}\ (1\ \mathrm{m})\)
• \(750\ \mathrm{MHz}\ (0.40\ \mathrm{m})\)
• \(1500\ \mathrm{MHz}\ (0.20\ \mathrm{m})\)
• \(3000\ \mathrm{MHz}\ (0.10\ \mathrm{m})\)
• \(1\ \mathrm{MHz}\ (300\ \mathrm{m})\)

Fig. 5.

Fig. 6.

Fig. 7.

Graph labels: \(E\), \(\mu\mathrm{V}/\mathrm{m}\); \(r\), km; decibels. Curves marked: \(5\ \mathrm{Mc/s}\ (60\ \mathrm{m})\), \(15\ \mathrm{Mc/s}\ (20\ \mathrm{m})\), \(30\ \mathrm{Mc/s}\ (10\ \mathrm{m})\), \(60\ \mathrm{Mc/s}\ (5\ \mathrm{m})\), \(100\ \mathrm{Mc/s}\ (3\ \mathrm{m})\), \(150\ \mathrm{Mc/s}\ (2\ \mathrm{m})\), \(200\ \mathrm{Mc/s}\ (1.5\ \mathrm{m})\). Also indicated: \(\sigma = 3.6 \cdot 10^{10}\), \(\varepsilon = 80\), \(3 \cdot 10^{3}\).

Fig. 8.

Visible labels in the figure:

• Left vertical axis: \(E\), \(\mu\mathrm{V}/\mathrm{m}\)
• Right vertical axis: Decibels
• Horizontal axis: \(r\), km
• Inset: \(\sigma = 3.6 \cdot 10^{10}\), \(\varepsilon = 80\)
• Curve labels:
• \(30\ \mathrm{MHz}\ (10\ \mathrm{m})\)
• \(60\ \mathrm{MHz}\ (5\ \mathrm{m})\)
• \(150\ \mathrm{MHz}\ (2\ \mathrm{m})\)
• \(300\ \mathrm{MHz}\ (1\ \mathrm{m})\)
• \(750\ \mathrm{MHz}\ (40\ \mathrm{cm})\)
• \(1500\ \mathrm{MHz}\ (20\ \mathrm{cm})\)
• \(3000\ \mathrm{MHz}\ (10\ \mathrm{cm})\)
• \(10000\ \mathrm{MHz}\ (3\ \mathrm{cm})\)
• Additional marking: \(3 \cdot 10^5\), \(1\)

Fig. 9.

Visible labels in the figure:

• Left vertical axis: \(E\), \(\mu\text{V}/\text{m}\)
• Right vertical axis: Decibels
• Horizontal axis: \(r\), km
• Inset:
  \[ \sigma = 3.5 \cdot 10^{10}, \qquad \varepsilon = 80 \]
• Dashed reference curve:
  \[ \frac{3 \cdot 10^{5}}{r} \]
• Curve labels:
• \(30\ \text{MHz}\ (10\ \text{m})\)
• \(60\ \text{MHz}\ (5\ \text{m})\)
• \(150\ \text{MHz}\ (2\ \text{m})\)
• \(300\ \text{MHz}\ (1\ \text{m})\)
• \(750\ \text{MHz}\ (0.40\ \text{m})\)
• \(1500\ \text{MHz}\ (0.20\ \text{m})\)
• \(3000\ \text{MHz}\ (0.10\ \text{m})\)
• \(10000\ \text{MHz}\ (0.03\ \text{m})\)

Fig. 10.

on the surface of the Earth (illegitimately assuming \(z_1=z_2=0\)), the height of antenna suspension may reach several tens or more wavelengths, which must substantially affect the results of the calculation. In addition, in this range the roughnesses of the earth’s surface (small inhomogeneities, ripples on water, shrubs—

Fig. 11.

— and even grass) become comparable with the wavelength, which already makes illegitimate the assumption of its homogeneity.

The influence of refraction in the lower atmosphere is illustrated by Figs. 10 and 11, which give, for various wavelengths \((w_\Sigma=1\ \text{kW},\ \sigma \simeq \infty\)—sea), curves of field strength as a function of distance for \(R^*=9500\ \text{km}^{*}\) and for \(R_0=6370\ \text{km}\) (pure diffraction), calculated from the tables of M. G. Belkina\({}^{15}\). It is evident from the figures that refraction changes especially substantially

\({}^{*}\) In some works, for the middle latitudes of the European continent this value \(R^*\) is recommended; recently \(R^*=8000\ \text{km}\) has been more commonly adopted.

the magnitude \(E\) at sufficiently large distances from the source. The curves of the additional phase quantity \(\Delta\Phi\) shown in Fig. 12, calculated in the same way, show that \(\Delta\Phi\) increases as the wavelength decreases and when refraction is taken into account. In a number of cases \(\Delta\Phi\) may attain a large value; therefore, for an accurate calculation of the optical path length of the direct wave it is necessary

Fig. 12.

to take into account both diffraction around the Earth and refraction in the lower atmosphere.

In conclusion of this paragraph we shall dwell on one more question. It is of interest whether any special features appear in the character of the electromagnetic field of radio waves when passing through the horizon line, as, for example, in optics, where beyond the boundary of the shadow the intensity of the electromagnetic field falls rapidly.

Consideration of the graphs of field strength for different heights of the source and the observation point (such graphs are given in \(^{13,14}\); they can also be constructed from the formulas and tables presented in \(^{11,15}\)) shows that they do not contain any typical peculiarities. The general character of the influence of the horizon line on the intensity of the electromagnetic field is seen from Fig. 13\(^{14}\), in which, for the height of the observation point \(z=100\) m, curves are plotted, for various wavelengths indicated in the figure, of the dependence of the field strength at the earth’s surface \((z_2=0)\) on distance. The horizon line is shown by a dashed line. It is seen from the figure that only in the millimeter wave range does a more rapid decrease of the field intensity beyond the horizon line begin.

Fig. 13.

b) Propagation of radio waves over an inhomogeneous surface of the Earth

Under real conditions, the propagation of radio waves almost always takes place over an electrically inhomogeneous and uneven surface of the earth. Therefore a complete theory requires allowance for the inhomogeneity of the interface: earth—atmosphere. This is, above all, of practical importance, since in many cases, because of double-check maybe user expects no image placeholder? Need transcribe figure? Requirement do not convert figures. We can include caption. But markdown cannot show image. Since page image figure visible, translate labels? Need visible reading order. Maybe include Figure caption only, not placeholder? They asked page as markdown. For figures, probably note figure with caption. Need not transcribe graph labels? They are text visible. But "Translate Russian prose" and preserve table. Figure labels visible perhaps should translate? We can include "Fig.13" only. But title labels part of figure? Hm.

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neglect of the inhomogeneity of the Earth’s surface, a substantial discrepancy is observed between theory and measurements. At the same time one cannot count on theory being able, in each instance, to take into account all the individual features of the Earth’s surface. This would be very difficult and practically impossible, and also unnecessary. The requirements imposed on theory in the present case consist in making it possible to calculate quantitatively the structure of the electromagnetic field, at least for certain idealized cases of inhomogeneity, merely on the average reflecting the diversity of real conditions. The following typical cases may be indicated:

1. When along the propagation path there are two (or several) electrically and geometrically homogeneous, sufficiently extended strips with different values of the dielectric constant $\varepsilon$ and conductivity $\sigma$, with a rectilinear boundary of separation between them, for example: a transition from land to sea, or several strips of land with different $\varepsilon$ and $\sigma$—we shall call this the case of purely electrical inhomogeneity.

2. When along the propagation path there is a large geometrical inhomogeneity, for example in the form of a sharply elevated shore with a rectilinear boundary, or a large mountain or depression; in this case the geometrical inhomogeneity may also have other values of $\varepsilon$ and $\sigma$—this may be called the case of inhomogeneity due to relief.

3. When the propagation path can be characterized by a large number of chaotically arranged inhomogeneities of relatively small size. These may be either geometrical inhomogeneities—irregularities, hills, shrubs, trees, etc.—or electrical inhomogeneities—lakes, groups of shrubs, sandbanks, etc. In this case it may be assumed that the inhomogeneities are on the average uniformly distributed along the propagation path—we shall call this the case of a “rough” surface of the Earth. Of course, what is meant is that theory itself must determine the measure of “large” or “small” inhomogeneity.

In addition, from the standpoint of theory it is important to obtain a quantitative answer to a more general question, namely, what is the effective region of the Earth’s surface participating in the propagation of radio waves between two given points. This question is practically very important, since the answer to it should indicate to what extent and how the surrounding conditions must be taken into account in each particular case, i.e. which portions of the surface take the principal part in the formation of the electromagnetic field at the receiving point.

However, until the last decade the questions listed above remained unresolved. In the foreign literature

in most of these questions there was confusion and an incorrect interpretation of the picture of radio-wave propagation over the inhomogeneous surface of the Earth, even from the qualitative side. This erroneousness of views, like the incorrect understanding of the question of the velocity of propagation of radio waves over the Earth’s surface, was to a large extent due to the incorrect conception of the so-called Zenneck surface waves (see¹). It should be pointed out that one encounters an incorrect interpretation of these questions in the literature up to the present time.

L. I. Mandelstam long ago gave the correct qualitative picture of radio-wave propagation over the Earth’s surface, emphasizing the role of the portions surrounding the source and the point of observation (“take-off” and “landing” areas) (see³, ¹¹, ¹⁶). His point of view contradicted the view then established that along the propagation path of a radio wave the action of each of the portions is additively accumulated. However, there existed no rigorous theoretical substantiation of his views, just as there was no quantitative theory of other phenomena occurring in the propagation of radio waves over the inhomogeneous surface of the Earth. This led, for example, to the fact that for a long time there remained unknown a very curious and practically important phenomenon, first predicted theoretically by E. L. Feinberg and consisting in an increase of the field strength with increasing distance between transmitter and receiver when the path passes from land to sea, a phenomenon confirmed experimentally only recently (see on this below, section 4). In precisely this phenomenon there was manifested the above-noted role of the electrical properties of the Earth’s surface in the vicinity of the source and of the point of observation in the formation of the electromagnetic field at the receiving point.

Only in recent years, in the works of Soviet theorists, has the problem of radio-wave propagation over an inhomogeneous Earth’s surface been correctly posed and, in the main, solved. These works, as well as the studies of V. A. Fock described above, were based on the use, in solving the field equations, of the approximate boundary conditions introduced by M. A. Leontovich³ᵃ and valid for practically often encountered cases, when

\[ \left|\varepsilon+i\frac{4\pi\sigma}{\omega}\right|\gg 1. \]

Already in the work of G. A. Grinberg³ᵇ it was shown, for the case of coastal refraction, that there is no additive action of the separate portions of the surface along the propagation path. However, the solution of this problem received substantial development in the works of E. L. Feinberg³ᵍ,¹⁶, in which a general method was developed for calculating the electromagnetic field over an inhomogeneous surface of the Earth, and a number of formulas were brought to a form suitable for numerical calculations. Below the results are briefly set forth

these theoretical studies are presented, together with the corresponding formulas.

1. On the effective path of a direct wave in the case of an electrical inhomogeneity. As shown in the work of E. L. Feinberg\(^{3r}\), the problem of radio-wave propagation over an inhomogeneous surface of the Earth reduces to solving the integral equation

\[ W(r)=f(s_0r)+\frac{ik}{2\pi}re^{ikr}\iint\left(\sqrt{\eta(x_1,y_1)}-\sqrt{\eta_0}\right)\frac{e^{ik(r_1+r_0)}}{r_1r_0}\times \]

\[ \times f(s_0r_0)\,W(r_1)\,dx_1\,dy_1, \tag{12} \]

which makes it possible to compute the attenuation function \(W(r)\) of the direct wave at an observation point located at a distance \(r=\sqrt{x^2+y^2}\) from the source.

In formula (12) the following notation has been introduced:

\[ \eta(x_1,y_1)= \frac{1}{\varepsilon(x_1,y_1)+i\,\dfrac{4\pi\sigma(x_1,y_1)}{\omega}} \tag{13} \]

—a parameter characterizing the electrical properties of the Earth’s surface and being a function of the current coordinates \(x_1,y_1\) [\(\eta(x_1,y_1)\) must vary little over an interval equal to the wavelength in the soil];

\[ \eta_0= \frac{1}{\varepsilon_0+i\,\dfrac{4\pi\sigma_0}{\omega}} \tag{14} \]

—an auxiliary arbitrary parameter, in which \(\varepsilon_0\) and \(\sigma_0\) may be interpreted as the electrical characteristics of some fictitious surface. Equation (12) is valid for any \(|\eta_0|\ll 1\). E. L. Feinberg\(^{3r}\) developed a method showing that, for individual types of inhomogeneities, by a proper choice of \(\eta_0\) the problem can be reduced to the calculation of certain integrals; he also considered a large number of such practically important cases.

The quantities \(r_0\) and \(r_1\) in (12) are respectively equal to:

\[ r_0=\sqrt{(x-x_1)^2+(y-y_1)^2},\qquad r_1=\sqrt{x_1^2+y_1^2}, \tag{15} \]

and \(f(sr)\) is the attenuation function of the direct wave for a homogeneous Earth’s surface. This function was first reduced to a form suitable for numerical calculations of field strength by M. V. Shuleikin (see \(^{17}\)), and then by Van der Pol. The modulus \(f(sr)\) is represented graphically by curves well known in the literature (Fig. 14) for various values of \(|\rho|=|sr|\), where \(\rho\) is the so-called

Fig. 14.

numeric distance, equal, for \(\left|\varepsilon+i\dfrac{4\pi\sigma}{\omega}\right|\gg 1\), to

\[ \rho=i\,\frac{2\pi}{\lambda}\,\frac{1}{2}\, \frac{1}{\varepsilon+i\,\dfrac{4\pi\sigma}{\omega}}\cdot r = i\cdot k\cdot \frac{1}{2}\,\eta\cdot r=s\cdot r, \tag{16} \]

and for \(\dfrac{\sigma}{\omega}\gg\varepsilon\)

\[ |\rho|=\frac{3\cdot 10^{5}}{2}\cdot \frac{\pi}{\sigma\lambda^{2}}\,r, \tag{16a} \]

where \(r\) and \(\lambda\) are expressed in km and \(\sigma\) in CGSE.

Let us also recall that for \(|\rho|\gg 1\)

\[ f(\rho)\simeq -\frac{1}{2\rho}. \tag{17} \]

Formula (12) contains \(f(s_{0}r)\) and \(f(s_{0}r_{0})\) for the value \(s_{0}=\dfrac{ik}{2}\eta_{0}\).

An essential result obtained by E. L. Feinberg\(^{3г}\) in analyzing equation (12) is the proof that this integral equation can be reduced to a single integral, i.e. that the principal role in the surface integral is played by a very limited region in the form of a highly elongated ellipse, enclosing the source and the observation point, extending beyond them by a distance of order \(\dfrac{\lambda}{4\pi}\) and having a greatest width of order \(\sqrt{\dfrac{r\lambda}{2\pi}}\). This, in general terms, resolves the question of the effective path of the direct wave.

Fig. 15. Labels in the figure: Sea \((\sigma\sim\infty)\); Observation point; Source; Land \((\eta)\); \(a_{2}\); \(a_{1}\); \(r\); \(r_{2}\); \(r_{1}\); \(\theta\); \(\theta\).

However, it is practically important to be able to calculate quantitatively the role of individual portions of the surface located in the region of the effective path, and of inhomogeneities lying to the side of the line connecting the source with the observation point. As regards the first of these questions, formulas are given below that make it possible to compute \(W(r)\) directly for several cases of variation of the electrical properties of the Earth’s surface along the propagation path. To estimate the influence of inhomogeneities lying to the side of the path, one may use the formula derived in \(^{3г}\) for the case when the source and the observation point are situated over the sea \((\sigma\sim\infty)\), while land—\(\eta\)—lies to the side of them (see all notation in Fig. 15). The resultant field is composed

in this case, from the undisturbed field and a certain addition having the character of a wave reflected from the shore. To compute the value of the field it is necessary to multiply the undisturbed field by

\[ \left(1+\frac{r e^{i k(r_1+r_2-r)}}{\sqrt{k a_1 a_2(r_1+r_2)}}\sqrt{\frac{i\eta}{2\pi}}\cdot \frac{\cos 2\theta}{2\cos\theta}\right), \]

which is valid when the conditions are fulfilled:

\[ k a_1\cos\theta \gg 1,\qquad k a_2\cos\theta \gg 1. \]

Let us now consider the formulas for the cases when the propagation path successively crosses land—sea—land or sea—land—sea; these make it possible to reveal especially clearly the influence of the intermediate regions on the value of the electromagnetic-field strength at the observation point.

In the first case: land—sea—land (for the notation see Fig. 16), when the radiator is located over land, assuming that \(\varepsilon\) and \(\sigma\), which determine the parameter \(S_c\), are identical for both land sections, the field strength is computed \(^{16}\) as follows. On the first land section the well-known formula is applicable

\[ E=\frac{300\sqrt{W_\Sigma}}{r}\,f(\rho)e^{ikr}, \tag{18} \]

where \(E\) is expressed in millivolts per meter, \(W_\Sigma\) is the source power in kW, and \(r\) is in km.

Over the sea, instead of \(f(\rho)\), the attenuation function is

\[ W(r)=-\frac{1}{2s_c r}\left\{1-\frac{2i}{\sqrt{\pi}}\sqrt{s_c(r-r_{c1})\frac{r}{r_{c1}}}\right\}, \tag{19} \]

where (19) is valid for any values of the length of the section over the sea under the condition

\[ |S_c r_c|\gg 1. \tag{20} \]

In formula (19), before the braces there stands, instead of \(f(\rho)\), its asymptotic value [see (17)].

On the second land section we have:

\[ W(r)=-\frac{1}{2s_c r}\left\{1+\frac{2}{a_1\pi}\sqrt{\frac{a_2^2-a_1^2}{1-a_2^2}}+ \frac{2}{\pi}\arcsin\frac{a_1}{a_2}\sqrt{\frac{1-a_2^2}{1-a_1^2}}\right\}, \tag{21} \]

where

\[ a_1=\sqrt{\frac{r_{c1}}{r}},\qquad a_2=\sqrt{\frac{r_{c2}}{r}}. \tag{22} \]

Formula (21) is valid under the condition

\[ |s_c r_{c1}| \gg 1, \qquad |s_c(r-r_{c2})| \gg 1. \tag{23} \]

It is curious to note that if the end sections of the path are relatively small,

\[ a_1^2 \ll 1, \qquad 1-a_2^2 \ll 1, \tag{24} \]

then instead of (21) we obtain:

\[ W(r)\sim -\frac{1}{\pi s_c \sqrt{r_{c1}(r-r_{c2})}}, \tag{25} \]

i.e., the attenuation factor depends on the geometric mean of the lengths of the two land sections. Let us note here that formula (25) is valid only when conditions (23) are fulfilled, i.e., for not very small land sections.

In the case sea—land—sea\(^{3,16}\) (see Fig. 16), over the first sea section formula (18) is valid, in which for \(\sigma\sim\infty\), \(f(\rho)\sim 1\); over land \(W(r)\) is calculated by formula (19), with the value \(r_{c1}\) replaced by \(r_{m1}\), and it is necessary that the condition

\[ |s_c r_{m1}| \gg 1; \tag{26} \]

be fulfilled; and over the second sea section we have:

Fig. 16.

\[ W(r)=f(s_c r)+\frac{i}{\sqrt{\pi s_c r}} \left[ \sqrt{\frac{r_{m1}}{r-r_{m1}}} + \sqrt{\frac{r-r_{m2}}{r_{m2}}} \right] + \frac{1}{2} \left[ 1+\frac{2}{\pi}\arcsin \frac{2a_1^2-a_2^2(1+a_1^2)} {a_2^2(1-a_1^2)} \right], \tag{27} \]

where

\[ a_1=\sqrt{\frac{r_{m1}}{r}},\quad a_2=\sqrt{\frac{r_{m2}}{r}}, \tag{28} \]

and the condition must be satisfied

\[ \left|s_c(r-r_{m2})\right|\gg 1. \tag{29} \]

For

\[ a_1^2\ll 1,\quad 1-a_2^2\ll 1 \]

one obtains

\[ W(r)=\frac{r_{m1}+(r-r_{m2})}{\pi r}, \tag{30} \]

i.e., a formula in which the influence of the land—the intermediate section—is completely absent. Let us recall that (30) is valid only when the conditions (26) and (29) are observed.

The role of the end sections of the path stands out especially clearly from Fig. 17 (see note 3), in which, according to formulas (21) and (27), for \(|s_c r|=100\), the dependence of \(\ln W\) on \(r_c/r\)—the degree to which the path is filled by land in the end sections (curve 2) or in the middle of the path (curve 1)—is plotted. It is evident from the figure that the attenuation factor is always larger, and in some cases very substantially larger, when the end sections, even if they are of insignificant size, are filled by sea.

Fig. 17.

In Fig. 18 are shown curves of field strength (see note 16), calculated respectively by formulas (21) and (27) for \(W_z=1\ \mathrm{kVt}\) and \(s_c=2\ \mathrm{km}^{-1}\) (this corresponds to: \(\lambda=51\ m\) for \(\sigma=9\cdot10^7\), \(\lambda=162\ m\) for \(\sigma=9\cdot10^6\), and \(\lambda=512\ m\) for \(\sigma=9\cdot10^5\)). From the figure it is evident that, in passing from land to sea, the value of \(E\) increases substantially with increasing distance from the source.

Formula (19), given above, already contains the case land—sea and sea—land (if \(r_1\) or \(r_{m1}\) is substituted into it). For practical purposes the case of two

Upper graph:

• Left ordinate: \(E\) in decibels.
• Right ordinate: \(E\), \(\mu\mathrm{V}/\mathrm{m}\); scale marks: \(10000\), \(1000\), \(100\), \(10\).
• Abscissa scale: \(0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100\).
• Sections along the abscissa: land — sea — land.
• Curve labels: \(\dfrac{3\cdot 10^5}{r}\ \mu\mathrm{V}/\mathrm{m}\); \(\dfrac{3\cdot 10^5}{r}\,|f_1(p)|\ \mu\mathrm{V}/\mathrm{m}\).

Lower graph:

• Left ordinate: \(E\) in decibels.
• Right ordinate: \(E\), \(\mu\mathrm{V}/\mathrm{m}\); scale marks: \(10000\), \(1000\), \(100\), \(10\).
• Abscissa scale: \(0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100\).
• Abscissa label: \(r\) (km).
• Sections along the abscissa: sea — land — sea.
• Curve labels: \(\dfrac{3\cdot 10^5}{r}\ \mu\mathrm{V}/\mathrm{m}\); \(\dfrac{3\cdot 10^5}{r}\,|f_1(p)|\ \mu\mathrm{V}/\mathrm{m}\).

Fig. 18.

sections having two different terminal conductivities and characterized, respectively, by the parameters $s_1$ and $s_2$. E. L. Feinberg$^{3г}$ gave a method that makes it possible to obtain a formula for two or more sections of different terminal conductivity. For two sections the following formulas have been obtained$^{16}$: if $r_1 \ll r$, where $r_1$ is the length of the section corresponding to $s=s_1$, then on the second section we have:

\[ W(r)=f(s_2 r)\left\{\sqrt{\frac{s_2}{s_1}}-\frac{\sqrt{s_1}-\sqrt{s_2}}{\sqrt{s_1}}\, \frac{(1-f(s_1 r_1))}{i\sqrt{\pi s_1 r_1}}\right\}. \tag{31} \]

For

\[ |s_1 r_1|\gg 1 \]

\[ W(r)=\sqrt{\frac{s_2}{s_1}}\,f(s_2 r), \tag{32} \]

and if

\[ |s_1 r_1|\gg 1 \quad \text{and} \quad |s_2(r-r_1)|\gg 1, \tag{33} \]

then

\[ W(r)=-\frac{1}{2\sqrt{s_2s_1}\cdot r}. \tag{34} \]

The last formula shows that in the case (33) $W(r)=\sqrt{f(s_1 r)f(s_2 r)}$, i.e. it is equal to the geometric mean of two values $f(\rho_1)$ and $f(\rho_2)$ [see formula (17)], corresponding to homogeneous surfaces characterized, respectively, by the parameters $s_1$ and $s_2$.

For three sections of terminal conductivity, characterized respectively by the parameters $s_1$, $s_2$, and $s_3=s_1$, it has been obtained$^{16}$, under the condition $|s_3(r-r_2)|\gg 1$ on the third section,

\[ W(r)=-\frac{1}{2s_1 r}\left\{1-\frac{i(\sqrt{s_2}-\sqrt{s_1})}{2\sqrt{\pi s_1s_2 r}}\left[ \frac{r-2r_1}{\sqrt{r_1(r-r_1)}}+ \right.\right. \]

\[ \left.\left. +\frac{r-2(r-r_2)}{\sqrt{r_2(r-r_2)}}\right]\right\}, \tag{35} \]

where $r_1$ is the length of the first section, and $r_2-r_1$ is the length of the second section.

The formulas given above, which determine $W(r)$ for various cases of inhomogeneity, make it possible to compute at the observation point both the amplitude of the field (as was done in constructing the curves of Fig. 18) and its phase $\varphi$. The values $\varphi(x,y)$ make it possible to construct equiphase lines—the wave front—and thereby to determine at the observation point the deviation of the direction of the normal to the wave front from the radial direction, caused by the inhomogeneity of the propagation path and producing the so-called phenomenon of coastal refraction. Thus, these formulas already contain the theory of coastal refraction.

From Fig. 19 it is easy to see that the angular error is determined from the relation

\[ \tg \alpha \sim \frac{\varphi_1-\varphi_2}{AB} \]

and, thus, the exact value \(\alpha \sim \tg \alpha\) for the case of small angles \(\alpha\) considered here must be determined by the formula

\[ \alpha \sim \frac{d\varphi}{d\tau}, \tag{36} \]

where \(\tau\) is the direction of the tangent to the circle drawn through the observation point. Below are given some simple formulas for calculating the angular error of coastal refraction, obtained by E. L. Feinberg^3 from calculations of \(\dfrac{d\varphi}{d\tau}\) and suitable for a number of practical computations.

Fig. 19.

In the calculations it is assumed that there is a rectilinear boundary separating two regions of the surface with different conductivity (i.e., a rectilinear shore). In this case a transition zone “\(b\)” is introduced, determining the width, parallel to the boundary, of the region in which the transition from one set of electrical constants to another takes place. For the case when

\[ kb \gg \sqrt{n_c}, \qquad |sr_2| \ll 1, \qquad a \gg b, \qquad ka\cos\theta \gg 1, \tag{37} \]

one obtains

\[ \alpha^\circ = 1{,}1\,\frac{\tg\theta\,\sqrt{\cos\theta}}{\sqrt{10^{-7}\sigma\cdot a}}, \tag{38} \]

where \(\alpha\) is expressed in degrees and \(a\) in kilometers. By virtue of (37), the formula is not suitable for \(\theta \to \frac{\pi}{2}\), i.e., for grazing incidence on the shore.

If, however, the source and the observation point are in the sea and, moreover, according to condition (37), far outside the limits of the transition zone, then

\[ \alpha^\circ = -1.1\, \frac{\tg \theta \cos 2\theta \sqrt{\cos \theta}} {\sqrt{10^{-7}\varepsilon \cdot a}} \times \cos \left\{ 2k\left(a-\frac{b}{a}\right)\cos \theta \right\} \times \frac{\sin(kb\cos\theta)}{kb\cos\theta}. \tag{39} \]

For the case

\[ a \gg b,\qquad kb\cos\theta \ll 1, \]

\[ ka\cos\theta \ll 1,\qquad kb \gg \sqrt{\eta_c}, \tag{40} \]

which is of interest as a case of grazing incidence, one obtains

\[ \alpha^\circ = -2.4\, \frac{\sin\theta}{\sqrt{10^{-7}\varepsilon\lambda}} \times \left[ \sin^2\theta \left( 1+\frac{2}{\pi}\ln\frac{2}{\gamma ak\cos\theta} \right) + \frac{2}{\pi}\frac{\cos\theta}{ka} \right] \]

\[ (\gamma=1.78\ldots). \tag{41} \]

Fig. 20.

Formulas (38), (39), and (41) refer to the case of a plane wave. For finite \(r\), instead of (38) one obtains

\[ \alpha = -1.1\, \frac{\tg\theta\sqrt{\cos\theta}} {\sqrt{10^{-7}\varepsilon\cdot a}} \sqrt{1-\frac{r_2}{r}}, \tag{42} \]

where

\[ ka\cos\theta \gg 1,\qquad |su|\ll 1. \tag{43} \]

In Fig. 20 several curves of the angular error are shown, computed from formulas (42) and (38) for \(\lambda=300\) m, \(\sigma=5\cdot10^8\) and \(5\cdot10^5\), \(a=\frac{\lambda}{3}\), and \(\lambda\frac{r}{a}=3\) and \(\infty\). It is seen from the figure that the magnitude \(\alpha\) reaches several degrees. The magnitude \(\alpha\) increases with approach to the shore–sea boundary and with an increase of \(\frac{r}{a}\).

In conclusion, let us note one important circumstance. If the source and the observation point are interchanged, the sign of the error changes. Since in this case \(r_2\) will already be the part of the path along the sea, the quantity \(\dfrac{r-r_2}{r}\) enters formula (42) instead of \(\dfrac{r_2}{r}\). Therefore, if the maritime part of the path is considerably greater than the part of the path over land, the angular error \(\alpha\) decreases; i.e., the radio-bearing error is smaller if the coastal station is bearing-taken from a ship^{3г}.

2. Rough surface and inhomogeneity due to relief. Consideration of the case of a local inhomogeneity due to relief or of a set of irregularities leads, as shown by E. L. Feinberg^{3г}, to the solution of an integral equation of the same type as (12), in which, in the case of a single inhomogeneity, instead of \(\sqrt{\eta}\) the parameter

\[ \gamma_0 \sim \operatorname{tg}\gamma_0=\frac{\xi_0}{l}, \tag{44} \]

enters everywhere; it characterizes the slope of the geometrical inhomogeneity, where \(\xi_0\) and \(l\) are respectively the height and length of the slope (small values of \(\gamma_0\) are considered). For the case of a set of irregularities \(\eta(xy)\) is replaced by a certain \(\eta_{\mathrm{eff}}\), and it is shown precisely that the influence of sets of inhomogeneities is equivalent to a change in the electrical constants of the earth’s surface. Formulas for \(\eta_{\mathrm{eff}}\) for several cases are given in^{3г}.

Fig. 21.

Thus, in the case of an irregularity over an ideal surface of the Earth, under the condition

\[ \sqrt{\gamma_0}<\sqrt{\frac{2\pi \xi_0}{\lambda}}\ll \frac{1}{\sqrt{\gamma_0}} \tag{45} \]

one obtains

\[ \sigma_{\mathrm{eff}}=\frac{1}{4\pi A^2}\, \frac{c}{\left(\dfrac{2\pi}{l}\right)^3 \xi_0^4}, \tag{46} \]

and for

\[ \frac{2\pi \xi_0}{\lambda}\ll \gamma_0 \ll 1 \tag{47} \]

\[ \sigma_{\mathrm{eff}}=\frac{1}{4\pi B^2}\, \frac{c}{k\left(\dfrac{2\pi}{l}\right)^2 \xi_0^4}, \tag{48} \]

where \(A\) and \(B\) are parameters determined by the specific form of the relief,

where

\[ |A|\sim |B|\sim 1. \tag{49} \]

For scattered small inhomogeneities of arbitrary curvature, when \(\dfrac{s}{s_0}\ll 1\), where \(s\) is the area occupied by the inhomogeneities, and \(s_0\) the whole area, under the condition \(k r_0\ll 1\) one obtains

\[ \sigma_{\mathrm{eff}}=\frac{1}{16\pi}\frac{c}{k^{2}r_0^{2}\left(\dfrac{s}{s_0}\right)^{2}} . \tag{50} \]

Formulas (46), (48), and (50) make it possible to estimate the influence of irregularities in a number of practical cases.

Geometrical inhomogeneity is considered in the work \(3^{\mathrm{r}}\) in the form of a slope with a small inclination \(\gamma_0\) [see (44)]. Simple formulas have been obtained for three positions of the observation point: a) in front of the slope, b) on the slope itself, c) behind the slope—on an elevation (Fig. 21).

In this case the resulting field is composed of the unperturbed field \(E_0\) and a certain increment equal to \(\beta E_0\); moreover, for the above three positions of the observation point, under the condition

\[ ka\cos\theta\gg 1,\qquad k|a-l|\cos\theta\gg 1, \tag{51} \]

one obtains:

\[ \beta_{\mathrm{a}}=-\gamma_0\sqrt{\frac{i}{8\pi}} \left( \frac{e^{2ika\cos\theta}}{\sqrt{ka\cos\theta}} - \frac{e^{2ik(a+l)\cos\theta}}{\sqrt{k(a+l)\cos\theta}} \right), \tag{52} \]

\[ \beta_{\mathrm{b}}=\gamma_0\sqrt{\frac{2}{i\pi}}\sqrt{ka\cos\theta} +\gamma_0\sqrt{\frac{i}{8\pi}}\, \frac{e^{2ik(a-l)\cos\theta}}{\sqrt{k(l-a)\cos\theta}}, \tag{53} \]

\[ \beta_{\mathrm{v}}=\gamma_0\sqrt{\frac{2}{i\pi}} \left(\sqrt{ka\cos\theta}-\sqrt{k(a-l)\cos\theta}\right). \tag{54} \]

For the angular error one obtains, respectively, the following formulas:

\[ \alpha_{\mathrm{a}}= \frac{\gamma_0\sin 2\theta}{2\sqrt{2\pi}} \left[ \frac{\cos\left\{2k(a+l)\cos\theta+\dfrac{\pi}{4}\right\}} {\sqrt{k(a+l)\cos\theta}} - \frac{\cos\left(2ka\cos\theta+\dfrac{\pi}{4}\right)} {\sqrt{ka\cos\theta}} \right], \tag{55} \]

\[ \alpha_{\mathrm{b}}=\gamma_0\frac{\sin 2\theta}{2} \left[ \frac{1}{2\sqrt{\pi ak\cos\theta}} + \frac{\cos\left\{2(l-a)k\cos\theta+\dfrac{\pi}{4}\right\}} {\sqrt{2\pi k(l-a)\cos\theta}} \right] \tag{56} \]

and

\[ \alpha_{\mathrm{v}}= \gamma_0\frac{\sin 2\theta}{4\sqrt{\pi}} \left( \frac{1}{\sqrt{ka\cos\theta}} - \frac{1}{\sqrt{k(a-l)\cos\theta}} \right). \tag{57} \]

In this case, for \(k(a-l)\cos\theta \gg 1\), instead of (57) one obtains:

\[ \alpha_{\mathrm{v}}=-\,\frac{\gamma_0\sin 2\theta}{8\pi\sqrt{2\cos\theta}}\, \sqrt{\frac{\lambda l^2}{2(a-l)^3}} . \tag{57a} \]

Still simpler formulas are obtained when

\[ k|a-l|\cos\theta \gg 1,\qquad ka\cos\theta \ll 1, \tag{58} \]

corresponding to the foot of a broad slope. In this case

\[ \alpha=\gamma_0\,\frac{\sin 2\theta}{2\pi}. \tag{59} \]

At the crest of a broad slope, when

\[ k|a-l|\cos\theta \ll 1,\qquad ka\cos\theta \gg 1 \tag{60} \]

\[ \alpha=-\gamma_0\,\frac{\sin 2\theta}{2\pi}. \tag{61} \]

Consideration of the formulas for \(\alpha\) shows that the angular error, in the case of an irregularity in the form of a rise in the terrain, generally has an oscillatory character and does not vary monotonically with the change of the angle \(\theta\) formed by the direction of the propagation path with the boundary line of the slope (say, of a raised shore), but is proportional to \(\sin 2\theta\). The magnitude \(\alpha\) is inversely proportional to the square root of the distance to the boundary of the irregularity and increases with increasing wavelength. Numerical estimates show that, for the range of applicability of these formulas \((\gamma \sim 0.1 \div 0.2)\), the value of \(\alpha\) may reach \(2—3^\circ\), and it has its greatest magnitude in front of the slope\({}^{37}\).

3. ON THE PROPAGATION OF LONG WAVES

At the present time, investigations in the field of radio-wave propagation have attained the least development for the case of long waves (approximately from \(2000 \div 3000\ \mathrm{m}\) to \(20000\ \mathrm{m}\)), despite the fact that, historically, the development of radio began precisely with radio communication on long waves. At the same time, interest in this range of radio waves has again increased in recent years in connection with its various practical applications.

From the theoretical side, such a lag is explained by the great difficulty encountered in solving the corresponding field equations\(*\). This difficulty is connected with the necessi-

\(*\) Some general considerations of this problem have been given in recent years in works\({}^{18,19}\).

by the possibility of a rigorous treatment of the problem of the propagation of radio waves between two spherical surfaces—the Earth and the ionosphere—having finite values of $\varepsilon^*$, the complex dielectric constant; moreover, in order to obtain correct results it is necessary to take into account the inhomogeneity of the ionosphere, i.e. the dependence of $\varepsilon^*$ on $z$.

When one is faced with the necessity of taking into account the influence of the ionosphere on medium (200–2000 m) and short (10–200 m) waves, the ray treatment of the question proves quite suitable—the consideration of the problem of the reflection of radio waves from an inhomogeneous medium of the ionospheric type, since in this case both the dimensions of all reflecting layers of the ionosphere (the half-thicknesses $z_m$) and the distances to them (the heights $z_0$) amount to many tens and even hundreds of wavelengths ($z_0,\ z_m \gg \lambda$). In the case of long waves, however, when $z_0$ reaches $(3 \div 5)\lambda$, and $z_m$ becomes of the order of $\lambda$, an exact solution of this waveguide problem is already required.

Another difficulty, which arises already in attempts to carry out approximate, idealized calculations, is explained by the scarcity of available data on the ionospheric $D$ layer, which plays the principal role in the transmission of long waves over great distances and is situated, beginning at heights of 50–60 km and above. For these calculations it is important to know the structure of the beginning of the layer, the dependence of the degree of ionization $N$ and of the number of collisions $\nu$ in the layer on height, which would make it possible to calculate the coefficient of reflection of long waves from the $D$ layer and would make possible certain rough estimates. At the present time, however, we generally possess no at all reliable experimental or theoretical data on the magnitudes and character of the variation of $N$ and $\nu$ in the $D$ layer. This leads, for example, to the fact that calculations of the reflection coefficient $\rho$ from the $D$ layer, in which data based on present-day ideas about it are used, give values smaller than the experimental ones by a hundred, a thousand, and more times.

Well known are the semiempirical formulas of two types, proposed long ago for calculating the field strength of long waves.

One of them—the Austin formula—is

\[ E=\frac{300\sqrt{W_{\Sigma}}}{r}\sqrt{\frac{\vartheta}{\sin\vartheta}}\,e^{-\frac{0.0015r}{\lambda^{\alpha}}}, \tag{62} \]

where $E$ is expressed in millivolts per meter, $W_{\Sigma}$ in kilowatts, $r$ and $\lambda$ in km, $\vartheta$ is the central angle (see Fig. 1), and the coefficient $\alpha$ was set in different works equal to 0.5 or 0.6.

A formula of the second type proceeds from the fact that long waves propagate between the Earth and the ionosphere as cylindrical

3 UFN, vol. XLII, no. 4

waves, and recently \(^{30*}\) it has been proposed in the form

\[ E=\frac{15\sqrt{W_{\Sigma}}}{\sqrt{r}}\cdot \sqrt{\frac{\vartheta}{\sin \vartheta}}\cdot e^{-\frac{0.003\,r}{\lambda}} . \tag{63} \]

However, comparison of the experimental data published in the literature with calculations by these formulas does not give agreement

Fig. 22.

between them. In Fig. 22 three groups of curves of the field strength in microvolts per meter are drawn, calculated for \(\lambda=5000\), \(10\,000\), and \(20\,000\) m by the diffraction formula \(^{11,15}\), by formula (62) for \(\alpha=0.5\), and by formula (63). In the same figure, the measured \(^{54}\) values of \(E\) on waves either equal or close, respectively, to these wavelengths are plotted with crosses, dots, and circles.

\(*\) It should be noted that the choice of numerical coefficients in work \(^{30}\) is no better justified than in other works. In addition, the author made an error when comparing experimental data with this formula, so that the agreement with experiment illustrated by him is only apparent.

waves. It is seen from the figure that the discrepancy with the diffraction formula reaches, at large distances, \(10^4\) times and more. It should be noted that even at distances of 500–1000 km (these data are not in the figure) the measured values of \(E\) are close to those computed from the diffraction formulas only in the daytime and then only on waves of 2000–3000 m; on still longer waves at these distances, even in the daytime, there is a noticeable discrepancy between the measured values of \(E\) and the calculated ones. As regards comparison with formula (62), in some cases the measured values of \(E\) are close to, but for the most part greater than, the values of \(E\) computed from it. The curves computed from formula (63), however, exceed the measured values.

Thus, the absence of a theory of the propagation of long waves leads to the fact that at present not only any calculations, but even a purely qualitative interpretation of a number of the principal known experimental results cited below, are very difficult*).

It was established long ago that the distribution of the field strength of long waves over the earth’s surface has an interference character**). This is manifested in the fact that the field strength varies with distance not monotonically, but has maxima and minima, which is especially clearly seen from Fig. 23, in which values of \((E \times r)\) are given according to the results of measurements carried out in recent years \(^{22}\). These measurements were made from an aircraft flying at an altitude of about 2 km, on a wave \(\lambda = 18740\) m, out to a distance of 850 km. Earlier measurements, carried out to distances of 350–400 km, are plotted in Fig. 23,a with triangles. Fig. 23,a gives the results of summer measurements made during the day. Points of different type denote measurements on different days during flights in different directions (receding or approaching). The measurements were made with an aircraft antenna or with a loop when it was placed at maximum. Fig. 23,b gives the results of one winter experiment (points denote measurements when the aircraft was moving away, crosses—when it was approaching), and Fig. 23,c gives the results of measurements with the loop placed at minimum. The fact that the interference character of the field is observed for both positions of the loop indicates that the plane of polarization of \(E\) rotates. Direct experiments had already long ago established that

*) It is interesting to note that, despite the fact that the number of published works devoted to studies of the propagation of radio waves now reaches more than 2–3 thousand, the number of works on studies of long waves which contain new experimental data scarcely reaches ten; moreover, of these only three have been published in recent years: \(^{21,22}\) and \(^{23}\).

**) We note here that theoretical calculations \(^{24}\) of the propagation of long waves between two plane ideal conductors give a similar interference pattern.

at distances up to 300–400 km an elliptically or circularly polarized wave is observed.

In recent experiments^22 with \(\lambda = 18740\) m it was established that at distances greater than 400–500 km the character of the field at long waves changes substantially. This is manifested first of all in the fact that for \(r \ge 500\) km the wave becomes linearly polarized.

Fig. 32.

Secondly, from the processing of the measurement data it follows that the reflection coefficient \(\rho^*)\) increases substantially for \(r \ge 500\) km.

Similar results were also observed earlier on the 14350 m wave. In general, from various measurements^21,22 it follows that in summer by day, for \(r < 300 \div 400\) km, \(\rho \sim 0.12 \div 0.17\), while at distances \(r \sim 500 \div 700\) km, \(\rho \sim 0.34 \div 0.42\). We note here that at these waves the value of \(\rho\) does not change substantially in the transition from day to night, whereas measurements at \(\lambda = 3000\)–4000 m showed that in winter, from night to day, \(\rho\) changes from 0.25 to 0.02, and in summer—from 0.15 to 0.003.

*) In calculations of \(\rho\), the field at the receiving point is considered in idealized form, namely as the result of interference of the direct and reflected waves.

Recently^25 results have also been reported from observations of the change in phase difference of long-wave stations received simultaneously at two receiving points 163 km apart. One radio station ($\lambda = 16300$ m) was distant from the two points by 5672 and 5614 km, respectively, and another ($\lambda = 17200$ m) by 1020 and 1048 km. The fluctuations of the phase difference reached, on the average, about $10^\circ$ for the more distant station both by day and by night, and for the nearer station only at night. By day the phase difference of the nearer station was rather stable, and often the magnitude of the change in phase difference lay within the limits of measurement accuracy.

In conclusion to this paragraph, we note that only recently, for the first time, a round-the-world echo has been observed on long waves ($\lambda \sim 16660$ m)^37. The results of these observations are given in more detail in Section 6 of the present article.

4. INVESTIGATIONS OVER AN INHOMOGENEOUS EARTH SURFACE

New experimental investigations^26–30 carried out in recent years when the path of radio-wave propagation passes from land to sea (or conversely) well confirm the theoretical results obtained earlier by E. L. Feinberg^3,16, and give full quantitative agreement (see Section 2b of this article) up to those distances for which one may confine oneself to the formulas for a flat Earth*).

Figure 24 gives the results of measurements^29 of the field strength $E$ at a wavelength of 3.9 m for the case land—sea—land; the measurements were made at both points—the mobile and the fixed points were equipped with transmitters and receivers. Crosses in the figure mark the results of measurements at fixed positions of the mobile point, and dots those obtained during its motion.

The figure shows curves calculated from different theoretical formulas: for the case of homogeneous land—by formula (18), and for inhomogeneous land, under the assumption that the conductivity of the sea $\sigma = \infty$, by formulas (18) and (21), and by formulas (34) and (35), which take into account the finite conductivity of the three sections, since for ultrashort waves the assumption $\sigma = \infty$ is already invalid. It is seen from the figure that complete agreement of the theoretical—

*) It should be noted that recently several articles^26,27,28 have appeared in foreign journals in which the results published earlier by E. L. Feinberg as far back as 1943^31 are obtained only partially and not always with sufficient rigor. At the same time, in these articles the theoretical results are presented as new, while the results of Soviet works^3 are ignored; moreover, it is evident that these Soviet works are known to the authors^26,28.

Fig. 24.

Fig. 25.

… of the formulas (34) and (35), with the experimental results. It should be noted here that, although (34) and (35) were derived under the condition \(|s_1 r_1| \gg 1\) and \(|s_2 r_2| \gg 1\), from comparison with experiment one may conclude that the formula is already suitable for \(|sr|\sim 1\).

At a distance of \(0.5\) km from the land–sea boundary, \(E\) has a maximum and is approximately four times greater than the value it would have over homogeneous flat land.

The results of other measurements\({}^{30}\), carried out for the land–sea case at a wavelength of \(96\) m, are shown in Fig. 25. Points of different types denote measurements over land \((+)\) and over the sea during recession \((\times)\) and approach \((\bigcirc)\). The theoretical curves were calculated respectively from formulas (18) and (19), and give overestimated values of \(E\), which is apparently connected with the need to take the curvature of the Earth into account. The correctness of this supposition is evident from the following. In Fig. 25 a theoretical curve for homogeneous land is drawn with a dashed line (with curvature taken into account\({}^{16}\)); this shows that, if the curve calculated from (19) is lowered by the difference of the ordinates between this curve and the curve calculated from (18), then the curve calculated from (19) will pass through the measured values of \(E\). In this experiment \(E\) reaches a maximum at a distance of \(25\) km from the land–sea boundary and is almost four times greater than the value it would have over homogeneous flat land.

Fig. 26.

In the same note, the results are given of measurements at a wavelength of \(269\) m on an aircraft flying at an altitude of about \(300\) m. The land section had a length of \(230\) km, and over the sea the total distance from the transmitter was \(390\) km. Over the sea, \(E\) had a maximum at a distance of about \(50\) km from the land–sea boundary, with a three- to fourfold increase of the field.

The results of measurements\({}^{21}\) for the sea–land case at a wavelength of \(268\) m are shown in Fig. 26. The same figure also shows the curve,

calculated by formula (19). The quantitative discrepancy that begins between curve (19) and the measured values of \(E\) is also, apparently, connected with the need to take account of the curvature of the earth’s surface.

5. THE INFLUENCE OF THE TROPOSPHERE—NEW RESULTS

Despite many years of research on the propagation of radio waves and the large number of works devoted to discovering the connection that might exist between the properties of the electromagnetic field of the direct wave at the receiving point and meteorological conditions, there are still no definite data on this question for the range of wavelengths considered in this article.

At the same time, the influence of the troposphere on the propagation of radio waves above the earth’s surface is now fully confirmed54, 55. This follows, first, from the fact that, in order to obtain fuller agreement between the results of measuring the field strength \(E\) of the direct wave and calculations, it is necessary to take into account normal refraction in the earth’s atmosphere. Its influence, as we saw in Section 2a, is determined by the gradient of the refractive index \(n\), and although on the average

\[ \frac{dn}{dz} \sim 0.039 \cdot 10^{-6}\ \text{m}^{-1}, \]

i.e. is a small quantity, the presence of the gradient leads to a noticeable increase in \(E\). In addition, the influence of the troposphere on the propagation of radio waves appears in the so-called phenomenon of superrefraction at ultrashort waves and in a number of other phenomena.

However, when one speaks of the connection between meteorological conditions and the propagation of radio waves, what is meant is the establishment of a specific dependence between parameters characterizing, on the one hand, the electromagnetic field at the receiving point and, on the other hand, the state of the troposphere along the propagation path: temperature, humidity, amount of precipitation, wind directions, etc. At the same time, precisely the results of investigating such connections are highly contradictory, so that often directly opposite results are reported in different works. This is apparently explained, first, by the fact that the variability of the state of the troposphere is very diverse, so that its influence on the propagation of radio waves must manifest itself under real conditions in a complex combination of circumstances that perturb the electromagnetic field in different ways. Moreover—and this is of fundamental importance, as was seen above (Section 2b)—the propagation of the so-called direct wave is essentially a spatial process, i.e. a process occurring through the atmosphere in such a way that the role of intermediate portions of the earth’s surface along the propagation path drops out. At the same time, comparison of the results of studying the electromagnetic field of a radio wave is usually carried out—

...to agree with the values of the tropospheric parameters measured only at the earth’s surface, and the character of their variation with height is not analyzed, which is precisely what plays the main role, say, in refraction in the troposphere and leads to an increase in the range of radio-wave propagation.

Secondly, this inconsistency and uncertainty of the data may also be explained by the fact that, often, because of the difficulty of taking them into account, purely local changes in the immediate vicinity of the source or the observation point are not considered; these may appear when meteorological conditions change: changes in the pattern and character of the antenna radiation, in the reflecting capacity of surrounding objects, in the depth of penetration of radio waves into the ground, etc. Thus, under real conditions the influence of the troposphere on radio-wave propagation is so varied that, without a detailed investigation of many of its manifestations, it is difficult to count on obtaining definite and unambiguous regularities. In this connection, every new investigation of this question is of interest. Recently a small number of works have been devoted to it. Their principal results are set forth below.

Fig. 27.

It has long been known that the intensity of the direct wave is subject to rather strong fluctuations on medium waves also during the daytime.

time; moreover, this is also manifested with pulse reception, when the influence of the ionosphere is completely excluded.^33 In this case the limits of variation of \(E\) and the frequency of oscillations of the values of \(E\) increase with increasing distance between the source and the observation point. The variability of the field strength \(E\) is evident, for example, from Fig. 27, which gives the results of 475 measurements of \(E\) at a wavelength of \(256\ \text{m}\), made every day at one and the same time (13 h 30 min local time) at a distance \(r = 122\ \text{km}\) from the transmitting device.^33 It is clear from the figure that on different days \(E\) had different values, with \(E_{\max} = 470\ \mu\text{V}/\text{m}\), \(E_{\min} = 175\ \mu\text{V}/\text{m}\), and \(E_{\text{av}} = 289\ \mu\text{V}/\text{m}\). These observations^33 were carried out on six waves: \(\lambda = 256\ \text{m}\) \((r = 122\ \text{km})\), \(\lambda = 350\ \text{m}\) \((r = 262\ \text{km})\), \(\lambda = 470\ \text{m}\) \((r = 417\ \text{km})\), \(\lambda = 289\ \text{m}\) \((r = 430\ \text{km})\), \(\lambda = 362\ \text{m}\) \((r = 627\ \text{km})\), and \(\lambda = 366\ \text{m}\) \((r = 900\ \text{km})\). With increasing distance the limits of variation of the values of \(E\) increased, so that for \(r = 900\ \text{km}\) \((\lambda = 366\ \text{m})\), \(E_{\max} = 26\ \mu\text{V}/\text{m}\), \(E_{\min} = 1.3\ \mu\text{V}/\text{m}\), and \(E_{\text{av}} = 5.3\ \mu\text{V}/\text{m}\).

Fig. 28.

For the terminal and intermediate points of each of the paths, the data for this same period of time were examined concerning air temperature and humidity, the amount of precipitation, atmospheric pressure, the dew point, and the pressure of water vapor. The analysis carried out showed the greatest correspondence between the course of \(E\) and the course of the temperature, as is evident, for example, from Fig. 28, which gives the curve of values of \(E\), averaged over three days, as a function of the temperature \(T\), also averaged over three days according to the mean daily values at all points of measurement. It is clear from the figure that \(E\) increases as the temperature decreases. The experiments showed that \(\dfrac{dE}{dT}\) increases with increasing \(r\).

Often, but not always, an increase in \(E\) (which persisted for several days) was observed after heavy precipitation (rains). In another study\(^{34}\), measurements at \(\lambda = 5.1\) and \(\lambda = 85\) m \((r = 145\ \text{km})\), \(\lambda = 342\) m \((r = 112\ \text{km})\), \(\lambda = 448\) m \((r = 353\ \text{km})\), and \(\lambda = 373\) m \((r = 336\ \text{km})\) established that under conditions of stable weather (anticyclones present), the value of \(E\) increased on the last three waves by 25, 40, and 33%, respectively. The reverse occurred in unstable (turbulent) weather.

6. INVESTIGATIONS OF ROUND-THE-WORLD ECHO

Recently, interesting results have been published from studies of the so-called round-the-world echo, i.e., the phenomenon in which a radio signal is observed to travel around the Earth from one to three times. This phenomenon was first observed on short waves more than 20 years ago. However, recent studies of the round-the-world echo of radio signals on short waves\(^{35,36}\) were more complete, and the measurements were carried out with high accuracy, which made it possible to obtain new data on this phenomenon. In addition, in 1948 a round-the-world echo was detected for the first time also on long waves\(^{37}\).

Fig. 29.

Beginning in December 1941 and continuing through January 1945, a German expedition carried out investigations of round-the-world echoes using apparatus specially developed for this purpose (until April 1944 in Friedrichshafen, and then in Randers, Denmark). Observations were conducted in the 10–20 Mc/s range at 47 stations located in various countries at distances from the observation point varying between 1,000 and 17,000 km. In addition, during this period five German stations, specially for these observations, transmitted signals at frequencies of 13.925 Mc/s, 15.075 Mc/s, 17.670 Mc/s, and 19.947 Mc/s at long time intervals.

In all, 785 round-the-world echoes were observed, including also cases when the echo did not traverse a complete circle around the Earth, but only a part of the great circle on which the transmitting station and the observation point were situated in the opposite direction (Fig. 29). This is the case of the so-called reverse echo. The oscillogram of one such measurement is shown in Fig. 30, in which the delay time of the reverse echo is \(t_{\mathrm{rev}}=0.07856\) sec.

The mean value \(t_{\mathrm{rt}}\)—the delay time of the round-the-world echo, calculated from all 785 values, with the distance \(r_{\mathrm{pt}}\) between the two points taken into account from the coordinate values of both points in the cases of the reverse echo—was found to be

\[ t_{\mathrm{rt}}=0.137767\ \text{sec.} \tag{64} \]

The author singles out 218 cases in which the measurements were more accurate; they gave the mean value

\[ \bar t_{\mathrm{rt}}=0.137788\ \text{sec.}, \tag{65} \]

while the scatter of the individual measurements was less than \(10^{-4}\) sec.

In these experiments it was established that the quantity \(t_{\mathrm{rt}}\) changes little and does not depend on the frequency of the received radio station, on the time of day and year, or on the position of the radio-wave propagation path.

The reverse echo signals were often clearer, and sometimes more intense, not only than the round-the-world echo but also than the direct signal propagating along the shortest path. This is apparently explained by the fact that both the direct signal and the round-the-world echo often consisted of several interfering signals traveling along different but neighboring paths. In general, the direct signal was distorted and was often weaker than distant echoes, which was usually observed at small distances (\(<1000\) km) between the two points. One such case is shown in Fig. 31, which gives an oscillogram of especially good observations (19.XI 1944), when single, double, and triple round-the-world echoes were observed.

It is very interesting that the delay time of the round-the-world echo was very stable. This indicates that the ionosphere as a whole, as a shell surrounding the entire terrestrial globe, is on the average rather stable in its properties and structure, so that it ensures the stability of the path of radio waves along all possible propagation routes. This is especially clear from the results of measurements of the values \(t_{\mathrm{rt}}\) and \(t_{\mathrm{rev}}\) for one and the same radio station, when it was possible to calculate from these values the distance \(r\) between the radio station and the observation point. Indeed, assuming that the mean propagation velocity of radio waves is the same in both cases, we have:

\[ \frac{2\pi R_0}{t_{\mathrm{rt}}} = \frac{2\pi R_0-2r}{t_{\mathrm{rev}}}, \tag{66} \]

Direct signal
0.07856
Return echo

0 20 40 60 80 100 120 msec

Fig. 30.

Direct signal

Single round-the-world echo

Double echo

Triple echo

Fig. 31.

where \(R_0\) is the radius of the Earth. From (66) we obtain, using (65) and the value \(2\pi R_0 = 40024\) km, that

\[ r = 20012\left(1-\frac{t_{\mathrm{обр}}}{0.13778}\right), \tag{67} \]

i.e., a formula determining the distance to the remote station. The results of the corresponding processing of the experiments, carried out by Hess[^36], are very instructive.

Measurements of \(t_{\mathrm{обр}}\) and \(t_{\mathrm{кр}}\) for a radio station located at a distance \(r_{\mathrm{точ}}\) gave, by formula (67), the following values:

For \(r_{\mathrm{точ}} = 6052\) km:

\[ r = 6068\ \text{km}\ (\text{with measurement error } \Delta r = +16\ \text{km}),\ 6045\ \text{km} \]
\[ (\Delta r = -7\ \text{km}),\ 6091\ \text{km}\ (\Delta r = +39\ \text{km}),\ 6046\ \text{km}\ (\Delta r = -6\ \text{km}) \]
\[ \text{and } 6057\ \text{km}\ (\Delta r = +5\ \text{km}). \]

For \(r_{\mathrm{точ}} = 5934\) km:

\[ r = 5934\ \text{km}\ (\Delta r = 0\ \text{km}) \ \text{and}\ 5947\ \text{km}\ (\Delta r = +13\ \text{km}). \]

For \(r_{\mathrm{точ}} = 6604\) km:

\[ r = 6648\ \text{km}\ (\Delta r = +44\ \text{km}),\ 6643\ \text{km}\ (\Delta r = +39\ \text{km}) \ \text{and}\ 6601\ \text{km} \]
\[ (\Delta r = -3\ \text{km}). \]

For \(r_{\mathrm{точ}} = 8171\) km:

\[ r = 8132\ \text{km}\ (\Delta r = -39\ \text{km}),\ 8147\ \text{km}\ (\Delta r = -24\ \text{km}),\ 8128\ \text{km} \]
\[ (\Delta r = -43\ \text{km}) \ \text{and}\ 8179\ \text{km}\ (\Delta r = +8\ \text{km}). \]

For \(r_{\mathrm{точ}} = 8613\) km:

\[ r = 8575\ \text{km}\ (\Delta r = -38\ \text{km}),\ 8562\ \text{km}\ (\Delta r = -51\ \text{km}),\ 8641\ \text{km} \]
\[ (\Delta r = +28\ \text{km}),\ 8603\ \text{km}\ (\Delta r = -10\ \text{km}) \ \text{and}\ 8641\ \text{km}\ (\Delta r = +28\ \text{km}). \]

For \(r_{\mathrm{точ}} = 8598\) km:

\[ r = 8579\ \text{km}\ (\Delta r = -19\ \text{km}) \ \text{and}\ 8570\ \text{km}\ (\Delta r = -28\ \text{km}). \]

For \(r_{\mathrm{точ}} = 9372\) km:

\[ r = 9363\ \text{km}\ (\Delta r = -9\ \text{km}),\ 9378\ \text{km}\ (\Delta r = +6\ \text{km}),\ 9358\ \text{km} \]
\[ (\Delta r = -14\ \text{km}) \ \text{and}\ 9372\ \text{km}\ (\Delta r = 0\ \text{km}). \]

For \(r_{\mathrm{точ}} = 10217\) km:

\[ r = 10225\ \text{km}\ (\Delta r = +8\ \text{km}),\ 10217\ \text{km}\ (\Delta r = 0\ \text{km}),\ 10244\ \text{km} \]
\[ (\Delta r = +27\ \text{km}) \ \text{and}\ 10228\ \text{km}\ (\Delta r = +11\ \text{km}). \]

For \(r_{\mathrm{точ}} = 12105\) km:

\[ r = 12088\ \text{km}\ (\Delta r = -17\ \text{km}). \]

For \(r_{\mathrm{точ}} = 16081\) km:

\[ r = 16069\ \text{km}\ (\Delta r = -12\ \text{km}). \]

Thus, from these data it is evident that precise measurements of \(t_{\mathrm{обр}}\) and \(t_{\mathrm{кр}}\) make it possible to measure distances of many thousands of kilo-

meters to the remote station with an average relative accuracy reaching \(10^{-3}\) and less.

The degree of activity of round-the-world echoes, i.e. the frequency of their occurrence and their intensity, depends on the time of day, the season, and also on the state of the ionosphere. However, the available data are still insufficient for specifying definite procedures for finding them. In general, it has been shown that a round-the-world echo propagates best of all along a great circle passing close to the line of the twilight zone.

In the case when the propagation path passed through polar regions, the form of the received echo signals often had the appearance of a modulated oscillation and changed periodically. In this case the modulation frequency varied within the limits \(0.1 \div 0.5\) Hz, and in one case was equal to \(2.4\) Hz.

The author\(^{35}\) believes that this phenomenon is the result of interference of two or several signals propagating along

Fig. 32.

different paths, one of which is reflected from a moving region of the ionosphere causing a Doppler frequency shift. Calculations from these data give velocities of motion in the ionosphere of \(10 \div 500\) m/sec.

In work\(^{35}\) it is assumed that the round-the-world echo propagates, like the so-called head or sliding wave known in acoustics, along the lower boundary of the \(F_2\) layer, so that it is obtained from \(t_{\mathrm{cr}}^{65}\), for a propagation velocity \(c = 299\,776\) km/sec, that the height of the sliding region is equal to 204 km. However, this theoretical interpretation by the author cannot be considered correct. P. E. Krasnushkin\(^{38}\) showed the illegitimacy of transferring the theory of the head wave, used in acoustics, to the case of propagation of electromagnetic waves. He also proposed the assumption that the echo signal propagates along the so-called ricocheting trajectory (Fig. 32) (analogously to the known case of the “whispering gallery,” considered by Rayleigh), and gave a general mathematical theory of radio-wave propagation along such a trajectory.

Analyzing the results of Hess’s investigations35, 36, various authors39, 40 come to the conclusion that changes in the delay times of round-the-world echoes, which give path oscillations within the limits \((1.028 \div 1.037) 2\pi R_0\), can be explained by a zigzag trajectory (Fig. 33). If, for example, it is assumed that the height of the reflecting layer varies within the limits 200–300 km, then for the angle \(\psi = 0\), formed by the normal to the wave front with the earth’s surface, one obtains a change in the length of the radio-wave path within the limits \((1.021 \div 1.031)\cdot 2\pi R_0\), and for \(\psi = 6^\circ\)—within the limits \((1.032 \div 1.043)\cdot 2\pi R_0\).

Recently37 the results were described of observations of a round-the-world echo at a frequency of 18 kc/s \((\lambda = 16660\ \mathrm{m})\) from a radio station with a power of 350 kW; the observation point was located approximately 80 km from the radio station. Round-the-world echoes were observed over the course of

Fig. 33.

four months. With an undisturbed ionosphere, \(t_{\mathrm{kr}} = 0.1373 \pm 0.0005\) sec, and on days of ionospheric disturbances \(t_{\mathrm{kr}} = 0.1365 \pm 0.0005\) sec. Round-the-world echoes were recorded throughout the observation period around the clock; moreover, the voltage at the receiver input from the direct signal was 280,000 μV, while from the round-the-world echo it varied for the greater part of the day within the limits 100–150 μV and reached a maximum, equal to \(\sim 450\ \mu\mathrm{V}\), during the period of sunset at the observation site.

7. VELOCITY OF RADIO WAVES

The question of the velocity of propagation of radio waves has long attracted great attention, and in the last 10–15 years it has acquired ever greater practical importance in connection with the various applications of radio waves for purposes of radio navigation and radiolocation.

It is well known from the literature that, in questions of the theory of the propagation of radio waves over the earth’s surface, and in particular the velocity of radio waves, for a long time there existed confusion and an incorrect understanding of the general picture even from the qualitative standpoint (see above, Section 2). Clarity in these questions and the correct

The theoretical understanding was introduced by the works of L. I. Mandelstam and N. D. Papaleksi and their pupils and collaborators[^1]. They were also the first to carry out correct and sufficiently accurate measurements of the velocity of radio waves above the earth’s surface by means of radio-interferometric methods developed by them. The results of these theoretical investigations have in recent years been fairly fully covered in various works and in the review by L. I. Mandelstam and N. D. Papaleksi[^7]. However, recently new results of measurements of the velocity of radio waves have been published. In view of the great interest which such measurements always present, both for scientific and for practical problems, it is expedient to compile a summary of the data of these measurements that have been published up to the present time1. This is the purpose of the present section.

All available results[^7],[^41]–[^46] of measurements of the velocity of radio waves are collected in Table I (see p. 554).

In the first four lines of the table (Nos. 1–4), for comparison, are given some results of measurements carried out on medium waves in the Soviet Union by the radio-interferometric method of L. I. Mandelstam and N. D. Papaleksi.

The measurements on long waves (No. 5) were carried out[^41] with the aid of a decanavigator, which is one variant of radio-interferometric methods (a phase probe), borrowed by the authors from the Soviet literature2. A circular flight was made at an altitude of 300 m around three synchronized base stations

\[ \left(1:\frac{4}{3}:\frac{3}{2}\right) \rightarrow \]

\[ \rightarrow 85 \ \text{kc}, \ 113.33 \ \text{kc} \ \text{and} \ 127.5 \ \text{kc}, \]

mutually separated by 43 km and 52 km. When an airplane crossed the two extreme points of the base, the total change in the phase difference was measured, from which the mean value of the phase velocity of the radio waves was calculated.

On ultrashort waves (No. 6), the quantity measured[^42] was already the group velocity, since a pulsed radar station was used in the experiments. The antenna was raised to a height of 61 m, so that propagation in fact took place along a line-of-sight path—purely in the atmosphere. At the same time, the authors also made special measurements over the sea and in the case where part of the path was hilly; no difference whatever in the magnitude of the velocity was found.

All the other measurements were carried out in the centimeter wave range, likewise with the aid of radar apparatus.

Table 1

Velocity of propagation of radio waves

No.	Value of the velocity and accuracy of measurement (km/sec)	Limiting distance (km)	Wavelength range (m)	Method of measurement	Experimental conditions	Observers
1	299 600 ± 100	100	300—450	Radio-interference (radio rangefinder)	Over the sea	Shchegolev, Borushko¹, ⁷ (1937)
2	299 650 ± 170	6	130—195	The same (radiobeacon)	Over land — steppe	Al’pert, Migulin¹ (1939)
3	299 500 ± 80	100	240—360	The same (radio rangefinder)	Under various land conditions	Gruzinov, Mindlin, Borushko¹ (1939)
4	299 500 ± 180	145	300—450	The same (radio rangefinder)	Over the sea	Meshcheryakov, Preobrazhenskii¹ (1940)
5	299 250 ± 40	100	2400—3500	Phase zones (Decca navigator)	Over land, with the baseline flown over at an altitude of 300 m	Mendoza⁴¹ (1947)
6	299 695 ± 50	140	5—13	Pulse method (radar)	Over the sea and land with elevated antennas	Smith, Franklin, Whiting⁴² (1947)
7	299 687 ± 25	68	0.09	The same	Over the sea — there was clear optical visibility	Jones⁴³ (1947)
8	299 713 ± 25 299 733 ± 25 299 750 ± 25	210 290 370	0.09	The same	At aircraft flight altitudes of 3000 m, 6000 m, 9000 m	Jones, Cornford⁴⁴ (1949)
9	299 792 ± 1.4 (Converted to vacuum)	550	Centimeter range	Pulse method (Shoran)	Over land in 47 directions	Aslakson⁴⁵ (1949)

Measurements at the level of the earth’s surface (No. 7) were carried out under conditions of direct visibility on various paths and at various distances. An increase in the scatter of the magnitude of the velocity with increasing distance was noted. No correspondence was established between the magnitude of the velocity of radio waves and meteorological conditions. Measurements made at different flight altitudes of an aircraft (No. 8) gave a regular increase in the magnitude of the velocity with altitude. However, the difference of the velocities at different altitudes exceeds the expected values calculated by formula (11) for the refractive index of the atmosphere, taking into account the change of temperature, humidity, and air pressure with altitude.

The most accurate measurements of the velocity were carried out in Aslakson’s work^45 (No. 9). The data given in the work are averaged over the results of numerous measurements and recalculated for the propagation velocity in vacuum. A detailed account of these experiments has not yet appeared in the literature.

It should be noted that two more articles^46, ^47 were also devoted to the question of the velocity of radio waves; they are only of a review character and do not contain any new data on the measurement of the velocity of radio waves in free space.

8. CROSS-MODULATION OF RADIO WAVES

In the range of medium waves, the so-called Luxembourg–Gorky effect^48 is well known, when cross-modulation of radio waves is manifested, namely: on a receiver tuned to the received radio station, another powerful radio station is heard—the interfering one (see Fig. 34), operating on another wave, far from the receiver tuning. Investigations of this phenomenon are of interest for the analysis of phenomena occurring in the ionosphere—measurement of the number of collisions of electrons (with neutral molecules or ions), whose velocities are changed under the action of the powerful interfering radio station, which causes a change in the number of collisions at the place of reflection

Fig. 34.

Ya. L. Al'bert

Fig. 35.

Fig. 36.

of the received radio station. On the other hand, these investigations also have practical significance, since they must determine the conditions under which this interference to radio reception manifests itself, which is important especially in broadcasting.

Recently a number of studies of cross-modulation \(^{49,53}\) have been carried out over a wide range of wavelengths, some of whose results it seems of interest to present in this article.

Cross-modulation, as already indicated above, is observed in the medium-wave range; moreover, the dependence of the cross-modulation depth \(M_0\) on the wavelength \(\lambda\) of the interfering radio station has a rather complicated character. In a first approximation it has been established that \(M_0\) increases with increasing \(\lambda\). However, on the long-wave side the greatest effect was observed only up to \(\lambda \cong 1800\ \text{m}\), after which \(M_0\) began to decrease rapidly, so that in the range \(\lambda \cong 3000 \div 5000\ \text{m}\) \(M_0\) had negligibly small values. On the other hand, an increase of \(M_0\) is observed in the range close to the gyroscopic frequency \(f_H\) (\(\lambda \sim 200\ \text{m}\)); moreover, the frequency at which the maximum effect is reached was shifted somewhat relative to \(f_H\) at the place of observation. At shorter waves this effect weakens, so that in investigations at \(\lambda = 75\ \text{m}\) cross-modulation was not detected at all. The magnitude \(M_0\), as follows from the theory, is directly proportional to the power \(W\) of the interfering station, which is clearly seen, for example, from Fig. 35, which gives the results of measurements on a wave of \(391\ \text{m}\) with the wavelength of the interfering station \(\lambda = 1800\ \text{m}\) and modulation frequency \(\lambda = 400\ \text{cps}\).

Experiments show that cross-modulation manifests itself most strongly when the midpoint of the path between the received station and the receiver is separated from the interfering station by a distance \(r_0 \cong 100\text{—}200\ \text{km}\). In this case the maximum value \(M_0 = 5\%\) is obtained if the results of the experiments are reduced to \(W_{\Sigma} = 100\ \text{kW}\) and \(r_0 = 100\ \text{km}\), for reception of radio stations in the range \(300\text{—}3700\ \text{m}\) and interfering stations operating in the range \(\lambda = 1500 \div 1800\ \text{m}\). For practical calculations for nighttime reception in the range \(300\text{—}2000\ \text{m}\) up to distances of \(500\ \text{km}\), the following formula is recommended \(^{51}\):

\[ M = M_0 \frac{W}{100}\left(\frac{140}{r_0}\right)^2 \frac{M_m}{100}, \tag{68} \]

where \(M_0\) varies within the limits \(2 \div 5\%\) (\(W_{\Sigma}\) is expressed in kilowatts, \(r_0\) in kilometers, and \(M_m\) is the modulation depth of the interfering station in percent).

The dependence of the cross-modulation depth on the modulation frequency \(F\) of the interfering station is evident from Fig. 36 and is described well, as

as experiments show, by the theoretical formula

\[ M=\frac{M_0}{\sqrt{1+\left(\frac{2F}{G\bar{\nu}}\right)^2}}, \tag{69} \]

where \(M_0=M\) for \(F=0\), \(G\) is the mean relative magnitude of the amount of energy lost by an electron in a collision, and \(\bar{\nu}\) is the mean number of collisions of electrons with neutral particles.

From different experiments one obtains \(G\bar{\nu}=1140 \div 2860\).

The depth of cross-modulation decreases monotonically with sunrise, as is seen from Fig. 37, which gives the results

Fig. 37.

of observations during reception on a wavelength of \(286\ m\) and with the wavelength of the interfering station \(\lambda=1500\ m\).

CITED LITERATURE

1. Recent Studies of Radio-Wave Propagation, edited by L. I. Mandelstam and N. D. Papaleksi, Gostekhizdat (1944).
2. Ya. L. Alpert, Propagation of Radio Waves in the Ionosphere, Gostekhizdat (1947).
3. Recent Studies of Radio-Wave Propagation, a) M. A. Leontovich, p. 5, b) V. A. Fok, p. 40, c) G. A. Grinberg, p. 69, d) E. L. Feinberg, pp. 97–215, Publishing House of the Academy of Sciences of the USSR (1948).
4. B. A. Vvedenskii and A. G. Arenberg, Questions of the Propagation of Ultrashort Waves, Part One, Publishing House “Soviet Radio” (1948).
5. V. L. Ginzburg, Theory of the Propagation of Radio Waves in the Ionosphere, Gostekhizdat (1949).

5a. V. L. Ginzburg, UFN 34, 469–480 (1948).
6. B. A. Vvedenskii and A. G. Arenberg, UFN 25, 273–309 (1941), 26, 1–44 (1943).
7. L. I. Mandelstam and N. D. Papaleksi, UFN 26, 144–168 (1944).
8. V. L. Ginzburg, UFN 28, 155–201 (1946).
9. Ya. L. Al’pert, UFN 34, 262–302 (1948), 36, 1–29 (1948), 38, 303–337 (1949).
10. B. A. Vvedenskii, ZhTF 6, 163–176 (1936); 6, 1837–1847 (1936); 7, 1647–1657 (1937).
11. V. A. Fok, Diffraction of Radio Waves around the Earth’s Surface, Publishing House of the Academy of Sciences of the USSR (1946); ZhETF 19, 916–929 (1949).
12. V. A. Fok, UFN 36, 308–327 (1948).
13. Balth van der Pol, H. Bremmer, Phil. Mag. 24, 141–176, 825–863 (1937); 25, 817–834 (1938); 27, 261–275 (1939).
14. H. Bremmer, Terrestrial radio Waves, Elsevier Publishing Company, Inc., Amsterdam (1949).
15. M. G. Belkina, Tables for Computing the Electromagnetic Field in the Shadow Region for Various Soils, “Sovetskoe radio” Publishing House (1949).
16. E. L. Feinberg, Radiotekhnika 5, 3–16 (1950).
17. M. I. Ponomarev, Izv. AN SSSR, OTN No. 9, 1191–1194 (1947).
18. G. A. Grinberg, Izv. AN SSSR (ser. fiz.) 7, 99–113 (1943).
19. O. E. Rydbeck, On the propagation of radio waves, Trans. Chalm. University, Göteborg (1944).
20. O. Zinke, Frequenz 1, 16–22 (1947).
21. Conferences: Proc. Phys. Soc. 63, 141–150 (1950); Journ. Geophys. Res. 54, 281–294 (1950).
22. K. Weeks, Proc. Inst. El. Eng. 97, Part III, 100–107 (1950).
23. M. V. Wilkes, Proc. Roy. Soc. 189, 130–147 (1947).
24. P. A. Ryazin and L. M. Brekhovskikh, Izv. AN SSSR (ser. fiz.) 10, 285–305 (1946).
25. P. G. Redgment, Journ. Inst. El. Eng. 94, Part IIIa, 803–898 (1947).
26. G. Millington, Proc. I. E. E. 96, Part III, 53–64 (1949).
27. A. L. Kirke, Proc. I. R. E. 37, 489–496 (1949).
28. P. G. Glennon, Nature 165, 107 (1950).
29. G. Millington, Nature 163, 128 (1949).
30. G. Millington, N. E. Elson, Nature 164, 114–116 (1949).
31. E. L. Feinberg, Izv. AN SSSR (ser. fiz.) 7, 168 (1943); 8, 132 (1944).
32. W. J. Beynon, Nature 164, 711 (1949).
33. F. R. Gracey, Proc. I. R. E. 37, 360–363 (1949).
34. D. W. Heightman, Nature 163, 527 (1949).
35. H. A. Hess, Zeits. f. Naturforsch. 1, 499–505 (1946); 2a, 528–533 (1947).
36. H. A. Hess, Proc. I. R. E. 36, 981–992 (1948); 37, 986–989 (1949).
37. J. N. Brown, Journ. Geophys. Res. 54, 367–372 (1949).
38. P. E. Krasnushkin, The Method of Normal Waves as Applied to the Problem of Long-Range Radio Communications, Moscow State University Press (1947).
39. F. Hamburger, K. Rawer, Zeits. f. Naturforsch. 2a, 521–527 (1947).
40. H. Lassen, Funk und Ton No. 8, 420–424 (1948).
41. E. B. Mendoza, Journ. I. E. E. 94, Part III, 396–398 (1947).
42. R. A. Smith, E. Franklin, F. B. Whiting, Journ. I. E. E. 94, Part III, 391–396 (1947).
43. F. E. Jones, Journ. I. E. E. 94, Part III, 399–401 (1947).
44. F. E. Jones, E. C. Cornford, Proc. I. E. E. 96, Part III, 447–452 (1949).
45. C. I. Aslakson, Nature 164, 711–712 (1949).
46. R. L. Smith-Rose, Proc. I. R. E. 38, 16–20 (1950).

1. L. Essen, Nature 165, 582 (1950).
2. R. V. Lvovich, Radiotekhnika 2, 5—14 (1937).
3. L. G. Huxley, H. G. Foster, C. C. Newton, Proc. Phys. Soc. 61, 134—161 (1948).
4. J. A. Ratcliffe, T. T. Shaw, Proc. Roy. Soc. 193, 311—343 (1948).
5. L. G. Huxley, J. A. Ratcliffe, Proc. I. E. E. 96, part III, 433—440 (1949).
6. L. G. Huxley, Proc. Roy. Soc. 200, 486—511 (1950).
7. C. C. Newton, F. J. Hyde, H. G. Foster, Proc. Phys. Soc. 63, 616—622 (1950).
8. A. I. Shchukin, Propagation of Radio Waves, Svyazizdat (1940).
9. Collected papers, Propagation of Radio Waves in the Troposphere, IIL (1950).

1. See also G. Rosenberg, “Recent measurements of the velocity of light and microwaves in vacuum,” UFN, 42, 485 (1950). ↩

2. It should be surprising that the corresponding references and indications are not contained in article[^41]. At the same time, earlier in this same English journal the priority of Soviet scientists in this field had repeatedly been noted. ↩