INTERFERENCE OF RADIO WAVES*

V. V. Migulin

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
I. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
II. Definition of radio-wave interference and methods for observing it . . . . . . . . . . . . . . . 362
III. Consideration of various types of radio interference . . . . . . . . . . . . . . . . . . . . . . . 371
    1. The case of a single source of radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
    2. Radio interferometers with several sources of radiation . . . . . . . . . . . . . . . . . . . 378
    3. Interference observations at variable radiation frequency . . . . . . . . . . . . . . . . . . 399
    4. Interference of modulated oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
IV. Interference measurements of the velocity of propagation of radio waves along the earth’s surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

INTRODUCTION

Electromagnetic waves excited by oscillations whose frequencies are below \(3 \cdot 10^{10}\) cycles/sec are usually called radio waves. The length of these electromagnetic waves propagating in space is correspondingly measured in quantities from \(1\ \mathrm{cm}\) and above. In practice, radio waves with wavelengths greater than \(30{,}000\ \mathrm{m}\) are not used, and this value, corresponding to a frequency of \(10^4\) cycles/sec, may be regarded as the long-wave boundary of the radio range. Radio waves with wavelengths shorter than \(1\ \mathrm{cm}\) can also be obtained, but so far they have not found practical application.

Covering such a broad portion of the frequency range, radio waves are, of course, only a particular case of electromagnetic radiation, alongside infrared radiation, X-rays, and visible light. Therefore the laws governing their behavior follow accordingly from the general laws of the electromagnetic field, just as does the theory of optical waves developed with all possible completeness. But when dealing with radio waves, we encounter such scales of temporal and spatial measurements that the phenomena differ very substantially from optical phenomena.

* This article is an abridged and somewhat revised doctoral dissertation by the author, defended by him in March 1945 at the Physics Institute of the Academy of Sciences of the USSR.

V. V. MIGULIN

As early as 1938, Academician L. I. Mandelstam, in his report at a session of the Academy of Sciences of the USSR, speaking about the application of interference methods to the study of the propagation of radio waves, drew attention to the essentially new element introduced into radio-wave theory, as compared with optics, by the difference in scales.

It was pointed out there that the colossal difference in temporal, spatial, and energy scales compels us, when considering radio waves—proceeding from the same fundamental principles of the theory of the electromagnetic field as in constructing optical theories—to take into account many factors which in optics could quite justifiably be left entirely out of consideration. The principal spatial scale—the wavelength—in the radio-wave region sometimes becomes of the same order, or even larger, than the distances and linear dimensions of interest to us. The use of radiators of considerable linear dimensions (corresponding to the wavelength), the possibility of energetically maintaining undamped oscillations of elementary vibrators, and a number of other features force the theoretical treatment of radio waves to proceed along paths other than those used in optics, and to obtain new relations, the consideration of which in optics would be devoid of physical meaning.

This applies to the theory of the propagation of radio waves, the theory of radiation, questions of the spatial distribution of radiation intensity, questions of their reception and indication, and many other questions.

The differences are very great and fully justify the emergence and existence of special branches of physics (radiophysics) and engineering (radio engineering) devoted to electromagnetic oscillations and waves of radio frequencies.

The diffraction of X-rays in a crystal lattice, their penetration through many substances opaque to optical rays—these are phenomena inherent only in the given range, only in the scales characteristic of X-ray physics.

The existence of a variable phase velocity near a radiator or near the boundary between two media—phenomena that can be readily observed only in the region of sufficiently long electromagnetic waves, i.e. radio waves, where they may also have substantial practical significance.

But alongside this one should not forget the common nature of optical and radio waves. Any phenomenon inherent in one type of electromagnetic wave must also exist in the region of other frequencies—other wavelengths—perhaps assuming only other forms or acquiring other features as a consequence of the difference in scales.

The phenomenon of optical interference was first observed by R. Boyle and R. Hooke (Newton’s rings) in the seventeenth century. The basic principles of interference had already been established by T. Young in 1802. At the beginning

observations of interference phenomena served chiefly for purposes of proving the wave nature of light. Later, and at the present time, instruments for producing interference of optical waves—optical interferometers—are used for measuring wavelengths, for the precise comparison of lengths, in spectroscopy, etc. It is obvious that in the region of radio waves as well we must encounter interference phenomena having physical foundations in common with the phenomena of optical interference.

But the distinctive method, different from that of optics, of producing radio waves, and especially the entirely new, as compared with optics, methods of observation and indication, based on the specific scales of radio waves, give radio-interference phenomena a completely special character, sharply distinguishing them from optical ones.

In recent years the phenomenon of interference of radio waves has found numerous applications both for the solution of a number of questions concerning the propagation of radio waves and for very important practical purposes. It is sufficient to point to such applications as radio range finders, phase direction finding, a number of systems of phase radio beacons, radio altimeters, and radio-interference methods in geodesy and navigation, in order to appreciate the full importance of this phenomenon for physics and practice*).

In recent years a number of works by Soviet authors of the school of Academicians L. I. Mandelstam and N. D. Papaleksi, and some works by foreign authors**), have been devoted to various scientific and practical problems of radio interference.

The cycle of works carried out under the direction of Academicians L. I. Mandelstam and N. D. Papaleksi on the development and application of radio interference advanced Soviet radiophysics to first place in world science in the study of the propagation of radio waves.

As is known, these brilliant works of L. I. Mandelstam and N. D. Papaleksi were awarded the Stalin Prize of the first degree***).

Various variants of radio interferometers proved to be the experimental means by which it was possible, quite reliably and with accuracies surpassing everything that can be achieved by other methods, to give a complete answer to the question of the content of the theory of radio-wave propagation along the earth’s surface in the part concerning the velocity and phase structure of the electromagnetic field of radio waves. Thus, for example, to a considerable extent

*) Many of these special questions are being successfully solved by Soviet investigators, but we cannot consider them here.

**) The corresponding references to individual works will be given below in the analysis of the concrete material.

***) The main results of this cycle of works are set forth in the collection Recent Investigations of the Propagation of Radio Waves, 1945, edited by L. I. Mandelstam and N. D. Papaleksi.

thanks to the application of radiointerference, one may regard as fully proven the untenability of Zenneck’s concept of plane inhomogeneous waves with respect to real radio waves excited by a radiator located on the earth’s surface.

However, these important conclusions have not yet been drawn in full measure with the use of all the quantitative data contained in our earlier publications. Therefore it seems sufficiently important, returning once again to this question, to carry out, as far as possible, a more consistent analysis of the results obtained, with the aim of establishing as completely as possible not only qualitative but also quantitative data on the question of creating a correct theory of radio-wave propagation.

In doing so, use has been made of materials obtained by the author through the application of various variants of the radiointerference method (set forth in a number of publications). But all these variants of radiointerferometers, just like the radiointerferometers described by other authors, together with a number of well-known phenomena in the field of radio waves, have one common nature—they make use of the phenomenon of interference of radio waves.

Therefore all radiointerferometers developed for various purposes are, in the final analysis, united with one another and with optical interferometers by the commonality of the very phenomenon of interference. Such an approach makes it possible, from a single point of view, to consider the operation of various radiointerferometers, establishing their kinship both with one another and with optical interferometers, and to determine the possible fields of application of each of them.

Such a general approach to radiointerference, in our opinion, should make it possible to assess more fully all the specificity of the given phenomenon, connected with the distinctive character of the methods of producing and observing it; a specificity which, just as in questions of propagation, gives every reason for singling out radiointerference into a special category of phenomena requiring special consideration and their own experimental methodology. In this connection, the consideration of the basic definitions and properties of radiointerference, as well as the methods of its observation, is of unquestionable interest.

These methods for the radio-wave range are not limited, as in optics, to measurements of intensity and, by including other methods of indication, are incomparably richer than the optical ones, as a result of which the range of phenomena having an interference nature that are studied and used is extraordinarily broadened.

This circumstance makes especially important the establishment of those principles on which the observation of radiointerference is based, since it is precisely by the peculiarities of the phenomenon of interference in the field of radio waves and by the specific methods of observing it that the feasibility of various radiointerferometers is determined; the analysis of their operation is the subject of the third chapter.

In the present article, using examples of radio interferometers already realized and only proposed, various ways of carrying out and using radio interference are considered for the solution of a number of scientific and practical problems, and they are compared with certain optical examples. A systematic analysis of the methods and results obtained in a number of works on the application of radio interference makes it possible to evaluate them from a general point of view, and also to assess the possibilities and expediency of using radio interference and individual types of radio interferometers for solving various problems.

The consideration of various radio interferometers and of the results obtained is carried out on the basis of the conception of interference as a phenomenon characterized not only by the intensity, but also by the form of the resultant oscillation formed as the result of the interaction or superposition of the initial oscillations that have traversed different paths and arrived at the point of observation with different time shifts.

In the fourth chapter an analysis is given of the question of the velocity of propagation of radio waves along the earth’s surface. This question, which is of fundamental importance in a number of problems arising in the use of radio waves for direction finding, navigation, measurement of distances, etc., could find its experimental solution only thanks to the use of radio interferometers of various types.

The use of radio interference made it possible to establish unambiguously which of the existing theoretical concepts is correct. The solution of this question is a work of very great fundamental significance and is an example of the extremely valuable application of radio interference to the solution of an essential scientific problem. Here the results obtained and, to a considerable extent, already published are analyzed; on the basis of known theoretical propositions they made it possible to give a quantitative comparison of calculated data with experimental results.

The article does not contain a detailed description of the methodology and technique of those radio-interference experiments that gave answers to the questions posed. These materials are available in the works cited in the text. It seemed essential only to give an analysis of the whole problem and to show by means of what radio-interference devices it could be solved.

I. GENERAL REMARKS

At every point of space filled by two propagating waves, the superposition of two oscillations takes place; and, when certain requirements with respect to the frequencies of both oscillations are fulfilled, one must expect that at different points of this space—

...there will be observed a weakening or strengthening of the oscillations. The phenomenon, defined in a similar way in optics, bears the name of interference.

Let us note that Rayleigh, in his work The Wave Theory of Light, already pointed out a certain inconsistency between the very term “interference” and the physical essence of the given phenomenon. Rayleigh indicates that this term presupposes the presence of interaction, mutual influence, whereas the phenomenon in question, in its pure form, requires for its existence the applicability of the principle of superposition, so that the initial wave processes may proceed completely independently of one another. In this connection, the oscillations and the corresponding waves satisfying the conditions necessary for obtaining interference are called coherent, and the sources of such oscillations, correspondingly, coherent sources. The spatial distribution of points with different intensity of oscillations or, in optics, with correspondingly different light intensity, forms what is called an interference pattern. Moreover, in order for it to be possible to speak of interference, it is necessary that the interference pattern be stationary, or at least move with a velocity permitting its observation. We give here these definitions, found in any textbook of optics, in order, proceeding from them, to subject to comparative consideration the basic properties of sources of optical and radio waves from the point of view of their application to the creation of interference phenomena.

It is obvious that the concept of coherence serves for the comparative characterization of various radiations, and if these radiations are obtained by separating, along different paths, oscillations emitted by one source, we shall automatically obtain coherent radiations, provided that certain requirements are fulfilled with respect to the difference in path lengths for the separated rays*). These latter requirements are connected with the concept of monochromaticity of each given radiation source, a concept characterizing, for the given emitter, the closeness of the oscillations emitted by it to purely harmonic ones.

It goes without saying that both the questions of coherence of various emitters and the questions of monochromaticity of each radiation separately can be analyzed only with sufficient knowledge of the mechanism underlying the radiation processes for emitters of various types.

At the basis of the classical theory, which considers the phenomena of radiation and the features of the spectral composition of light radiation, lies the elementary vibrator—the oscillating dipole.

According to this theory it is assumed that the radiating dipoles perform free oscillations, and that the energy going into the excitation—

*) It is assumed that the path lengths remain constant.

excitation of these oscillations, they receive from the electrical or thermal processes taking place in the radiating body. But even if one disregards a number of factors affecting the character of the oscillatory process of elementary vibrators, the presence of radiation is associated with damping, which, causing the form of the oscillations to deviate from a purely harmonic one, brings about the practically complete cessation of the oscillation in a time of the order of \(10^{-8}\) seconds.

Only during this interval of time could we, in dealing with two homogeneous atoms situated under identical conditions, observe a stationary interference pattern as the result of the radiation of these two individual atoms—two independent sources. However, light acquires a measurable intensity thanks to the fact that, first, a large number of vibrators radiate simultaneously and, second, the process of damping is slowly replaced again by a new excitation. These circumstances are closely connected with the statistical nature of optical light sources. In any given statistical radiator we do not have a continuous simple harmonic law of oscillations, but rather a sequence of oscillations, each time beginning with its own phase. Although each of these oscillations is very close to harmonic (the logarithmic decrement of damping is of the order of \(10^{-7}\)), the absolute lifetime of each oscillatory state is so small that, when the initial phase changes according to the law of chance, the possibility of observing interference from independent light sources is completely excluded, i.e. independent light sources are, in principle, incoherent.

Thus, the incoherence of independent sources of optical radiation is connected with the statistical character of their very nature.

One remark should be made here.

In speaking of the statistical nature of light radiators, we tacitly assume that there is no interaction between elementary radiators. Only under this condition do we have the right to regard light radiation as a statistical aggregate of separate independent processes and to apply those laws of statistics which are based on the application of the law of chance to individual elementary processes.

In the case when the interaction between individual elementary radiators can no longer be neglected, we must abandon the statistical treatment of the radiation process.

It is precisely in this that the transition consists to radiators of finite dimensions, of the type that occur, for example, in radio engineering, in which individual elementary radiators already perform coordinated oscillations, mutually connected by common electric and magnetic forces. For such radiators of finite dimensions, their individual points, which may be regarded as elementary radiators,

already emit coherent oscillations. As a result of the interference of these coherent radiations from individual points, a common radiation is formed, possessing a definite directionality characteristic of the given shape of the entire radiating body. If, for light radiation, we could speak of the fundamental incoherence of the light emitted by different points of one and the same luminous body, then in the case of dimensions of the radiating body comparable with the wavelength this assertion loses its force, and in general it becomes possible to speak of the whole body as a single radiator.

When we pass to radio waves, the first thing to which attention should be drawn is the use not of statistical, but of individual radiators. Electric charges, controlled by common electric forces, in radio-engineering radiating systems execute organized oscillations, so that, from the point of view of the macroscopic scales applied in this case, the whole system represents, as it were, a single vibrator.

The second fundamentally important circumstance consists in the fact that the excitation of the oscillations of a radio-wave radiator usually occurs not once for a long series of emitted oscillations, but during each period additional energy is supplied to the vibrator, covering the expenditure on radiation and losses. In this case the supply of energy in the stationary regime (undamped oscillations) is a continuous process in dynamic equilibrium with the expenditure of energy on radiation and losses.

Let us note that a spark station, from the point of view of the mechanism of excitation, can with a considerable approximation be likened to an optical radiator, with only the difference that in a spark radio station each oscillatory process dies out much more rapidly (on the scale of the period of oscillations), which entails an incomparably greater natural width of the corresponding emission spectral line. This circumstance excludes the possibility of observing radio interference for independent spark radiators. In what follows, however, we shall deal only with generators of undamped oscillations, so that all further considerations will refer to radiators emitting undamped radio waves, for which the question of the natural line width, caused in optics by the presence of damping of each series of oscillations as a consequence of radiation, does not arise.

If one neglects secondary factors which may cause irregular deviations from strict periodicity, then for radio waves, dealing with undamped oscillations, we obtain an infinitely narrow spectral line, accompanied by more or less sharply expressed higher harmonic components.

Therefore, in the region of radio waves we encounter only factors similar to those which in optics cause additional broadening—

broadening of lines that already have a natural width (Doppler effect, collisions, electric and magnetic fields). In the radio-frequency region these causes are mainly reduced to thermal fluctuations in tubes and oscillatory circuits, especially in cascades generating oscillations.

The question of the width of the spectral line corresponding to the radiation of oscillations generated by a self-oscillating system with a cathode tube was analyzed in the work of I. L. Bershtein,^1 who obtained for the frequency blurring a value of the order of \(10^{-9}\). Let us note that all subsequent cascades as well, owing to fluctuations of the values of the parameters of oscillatory circuits and, in particular, of resonant circuits, are capable of introducing an additional blurring of the frequency, causing rapid changes in phase shifts or in the form of the oscillations, without changing the mean value of the frequency, understood as the number of oscillations per unit time. (In this connection sufficiently long intervals of time, including a considerable number of periods, must be considered.)

The aforementioned peculiarities of radio-wave sources introduce a certain originality into the character of the radiated oscillations and make it possible to regard, for each radiator, the initial phase shift as constant and quite definite. This distinguishes radio-wave radiators in a fundamental way from optical radiators and ensures the coherence of any independent radiators when the frequencies of the oscillations emitted by them coincide.*)

However, the practical impossibility of ensuring, over a long time, an exact coincidence of the frequencies of independent radiators makes it impossible, in the radio-wave region as well, to obtain a stationary interference pattern.

It is true that the attainable accuracy of maintaining radio frequencies, when converted to time scales, makes it possible to approach so closely a prolonged coincidence of the frequencies of oscillations emitted by different sources that, in the case of radio waves, it is more correct to speak not of the fundamental impossibility of obtaining interference, but only of inconveniences connected with changes of the interference pattern in time.

In this sense it might be expedient to speak of interference only when the resulting interference pattern is stationary. The presence of this requirement would lead, for radio waves also, to the fundamental impossibility of obtaining interference from two independent radiators, since it is obvious that constancy of the initial phase shifts, with absolute coincidence of the frequency of the radiated oscillations, can be achieved only by introducing an additional synchronizing channel. This is equivalent to the optical case of separating, along two different paths, radiation emitted by a single source.

*) The case of an integral frequency ratio will be considered by us later.

Each given radiator or, more precisely, each given radiation can be characterized by the degree of monochromaticity characterizing the closeness of the form of the oscillations to a sinusoidal one.

For independent sources it is obvious that the nonideal monochromaticity of real sources is one of the principal causes responsible for the absence of coherence. For oscillations emitted by one radiator, however, the degree of monochromaticity will determine that maximum path difference which is permissible in order to obtain an interference pattern that is still observable. This circumstance, obviously, fully retains its force both for optics and for radio waves, so that the relations connecting the width of a spectral line, or the relative blurring of the frequency, with the coherence length are also applicable in the radio-frequency region. But if one estimates quantitatively those permissible path differences for radio interference which correspond to the blurring of the frequency of a radio-frequency self-oscillator, then, using the data obtained by I. Bernstein[^1], we obtain the relation

\[ L \simeq 1.95\cdot 10^{8}\lambda, \]

from which, even for the shortest radio waves, such large values of \(L\)—the coherence length—are obtained that they take us beyond the limits of feasible experiments.

II. DEFINITION OF RADIO-WAVE INTERFERENCE AND METHODS OF OBSERVING IT

The definition of interference accepted in optics—the spatial distribution of oscillatory energy in the simultaneous propagation in a given space of two or more coherent wave processes—is conditioned by the principal, if not the only, method of observation: the measurement of light intensity by measuring the magnitudes of the energy flux. For a plane monochromatic wave this energy flux is equal to

\[ |I|=|S|=cW, \]

where \(S\) is the Poynting vector and \(W\) is the energy density of the electromagnetic field, averaged over a sufficiently long interval of time.

Defining the quantity \(W\) as \(W=\dfrac{\varepsilon}{4\pi}E^{2}\), we see that the measured quantity is proportional to the square of the amplitude of the oscillation. It is precisely by starting from the nonlinearity of the indicator that the basic laws of interference are derived in optics. In doing so it is taken into account that the measurement of the energy flux is a measurement of quantities obtained as the result of averaging over a time interval sufficiently large in comparison with the period of the oscillations.

For two interfering waves the resultant intensity is

\[ I=I_1+I_2+I_{12}, \]

where \(I_{12}=2E_1E_2\) is the interference term, whose magnitude is determined by the expression

\[ I_{12}=2(a_1b_1+a_2b_2+a_3b_3)\cos\delta. \]

Here \(a_1, a_2, a_3, b_1, b_2, b_3\) are, respectively, the amplitudes of the \(x, y, z\) components of both oscillations at the given point, and \(\delta\) is the phase difference of the two oscillations*).

However, it should be borne in mind that these relations are obtained for unit monochromatic oscillations. Real light, even at the maximum theoretically attainable monochromaticity, is a statistical process—the aggregate of a number of elementary oscillations, each of which begins with an arbitrary phase.

This circumstance leads to the fact that the observed intensity is not only the result of averaging over time, but also of summation over the entire number of oscillatory processes constituting the total radiation.

In the case of radio waves, as a rule, we deal with unit oscillations, which brings the problem of amplitude indication for radio interference closer to the ideal case of monochromatic waves mentioned above, and leads to the fact that, when indication by intensity is used, such nonlinear instruments are employed (most often quadratic ones) as make it possible to average the observed effect only over time.

The simpler nature of the wave process in the case of radio waves makes it possible, when observing radio interference, to use also linear instruments as indicators, serving to measure the amplitude of the resultant oscillatory process at a given point in space, arising as the result of the simple superposition of the interfering oscillations. (As such an instrument one may use, for example, a cathode oscilloscope.) This resultant oscillatory process, as is known, in the case of the addition of harmonic oscillations has the amplitude

\[ A=\sqrt{\sum a_k^2+2\sum_{i\,k}\sum a_i a_k\cos(\theta_i-\theta_k)}, \]

where

\[ 2\sum_{i\,k}\sum a_i a_k\cos(\theta_i-\theta_k) \]

is the interference term, determining the spatial distribution of the amplitude of the resultant oscillation as a function of the [[unclear: continuation on next page]]

* See, for example, M. Born, Optics, or another textbook of wave optics.

at a point in space the phase difference \(\delta_{ik}=\theta_i-\theta_k\) between the interfering oscillations.

Let us now suppose that in the given volume there exist two coherent oscillations polarized in mutually perpendicular planes, and that the phase difference between these oscillations varies from point to point in space while remaining constant in time. In this case there will be no spatial distribution either of amplitudes or of intensities of the resultant oscillation. Interference, in the optical definition of this phenomenon, will be absent. In space there will occur only a distribution of the polarization state of the resultant oscillation, with a uniform distribution of the energy of the oscillations throughout all space.

But radio-engineering devices make it possible to separate the operations of detection (reception) of electromagnetic oscillations and their indication, and in reception to receive separately oscillations having mutually perpendicular polarization. If, for the case in question, such separate reception of each of the two oscillations is applied, and the oscillations are fed to a common indicator only from the outputs of these receivers, then such a device will give a quite definite spatial distribution of the indicator readings, associated with the phase difference of the original oscillations at the observation points.

Let us note that the operation just described—separate reception of the original oscillations followed by feeding them to a common indicator—is, generally speaking, in the case of two oscillations with mutually perpendicular polarization, equivalent to reducing both oscillations to a single polarization; this can also be accomplished by means of optical instruments.

Thus, in the radio-wave region we encounter the readily realizable possibility of transforming a spatial three-dimensional oscillation into a one-dimensional oscillatory process—oscillations of current or voltage in conductors. This possibility of transforming oscillations from multidimensional to one-dimensional and back gives radio engineering the possibility of carrying out an extraordinary variety of interference experiments. In optical interference, the linear superposition of the interfering oscillations gives a spatial distribution of the amplitude of the resultant oscillation. For radio waves, a phenomenon is possible in which there is formed in space a certain distribution not of the amplitude but of the form of the oscillation (in the example mentioned above, of the polarization), and then, if the recording device can separately receive both interfering rays, the combined indicator will show different intensities when the whole device is moved from one point of the interference space to another.

In this case the process of interaction of the interfering oscillations is transferred into the recording device, in contrast to op

tics, where the indicator need only record the already existing result in space of the superposition of the interfering oscillations.

The following cases of interference of two coherent waves having identical frequencies and arriving at the point of observation along different paths may be distinguished:

1) The polarization of both interfering waves is the same. As a result of superposition there is formed a spatial distribution of the amplitude of the resultant oscillation, or, what is the same thing, a spatial distribution of intensities. In the case of radio waves this distribution can be studied by means either of a linear indicator-resonator (for example, a linear resonator or a receiver-amplifier with a cathode oscillograph), or of a nonlinear device (a thermal device, cathode voltmeter, etc.), just as is done in optics.

Phenomena of this kind, completely analogous to optical interference phenomena, are widely used in radio engineering, for example in the practice of producing various radiation patterns of complex systems of transmitting antennas.

2) Coinciding in frequency, both waves are linearly polarized in mutually perpendicular planes. In space, through every point of which there pass two such series of waves—the space of interference—there is formed a spatial distribution of different states of polarization, depending on the differences of path of the two oscillations at each point of the space of interference.

3) The intermediate case. For coinciding frequencies, with an arbitrary polarization state of both interfering waves, in the space of interference there will occur a spatial distribution both of amplitudes and of the polarization state of the resultant oscillation formed as a result of the superposition of the two waves.

The effect observed with a single nonlinear (energy) or linear indicator will be expressed the more weakly, the closer the given case approaches case 2. Separate reception of both interfering oscillations, however, also in this case makes it possible to study the spatial distribution of the features of the oscillatory state of the resultant process.

Although the usual radiators of radio waves—antennas—radiate, as a rule, linearly polarized waves, the conditions of their propagation (along the earth’s surface or in the ionosphere) on the path from radiator to receiver are such that very often the propagating wave acquires elliptical polarization, and it is quite natural to encounter the interference of two rays polarized in a similar manner. In this case one may consider separately the interference of plane-polarized components lying in two

mutually perpendicular planes, and then, summing the results, obtain the same possibilities for producing interference as for linearly polarized waves.

Let us note that a completely separate reception of two waves can be carried out only in the case where oscillations of the same frequency arrive at the point of observation either from sharply different directions, or with mutually perpendicular planes of polarization.

Up to now it has been regarded as self-evident that both interfering waves have identical frequencies. Let us now consider the case in which the frequencies of the interfering waves are different. If the oscillations have commensurable frequencies, then the difference in path differences, and the various mutual phase shifts between the oscillations at different points of the interference space caused by this, will, as a result of the superposition of the interfering oscillations, give at each point of this space its own form of the resulting oscillatory process, having a period equal to the least common multiple of the periods of both interfering oscillations. It is also appropriate to call this phenomenon interference, since here the law of the spatial distribution of the form of the resulting oscillation is determined by the path difference of the interfering oscillations. In this case, as a function of this path difference, the form of the resulting oscillation is periodically repeated.

For incommensurable frequencies of both interfering oscillations, the resulting process at any point of space will be nonperiodic, and therefore the possibility is lost of speaking about its form in the sense in which we speak of the form of a periodic oscillation. And if one can speak of the phase of each of the interfering oscillations, then the concept of phase difference loses its meaning from the point of view of the possibility of defining it as a certain quantity that determines, at a given point, the form of the resulting periodic oscillation.

For oscillations of commensurable frequencies, the form of the resulting oscillatory process is determined by the time shift of one oscillation relative to the other with respect to some arbitrary origin of time. Expressing this time shift on the scale of the period of one of the initial oscillations, one may, following E. Ya. Shchegolev, speak of the phase difference between oscillations with commensurable frequencies².

In his work Shchegolev introduces such a definition and gives a number of examples and methods for determining this “phase difference.” It would, of course, be possible to conduct observations directly on the form of the resulting oscillation at various points of the interference space (for example, with the aid of an oscilloscope with time sweep). But this is not very convenient, all the more so since the interpretation of such observations is greatly complicated by the dependence of the form of the resulting oscillation not only on the phase relations—

not only on the distances traveled by the waves, but also on the ratios of the amplitudes of the interfering waves. Since the principal parameter determining the position of the interference pattern is, for each point of the interference space, the corresponding time shifts of one oscillation relative to the other, the most effective method of observation will be one that makes it possible to measure directly this time shift—the phase difference of the interfering oscillations.

Whereas all methods based on secondary phenomena reduced either to observations of the resultant oscillation (for example, oscillographic recording of its form) or to the action of a nonlinear device under the influence of this resultant process (see, for example, the work of Lyubchenko³, in which a similar method is described for determining the phase difference of oscillations of commensurable frequencies*), direct measurement of the phase shift, which is the most desirable, requires for its implementation the separate reception of both interfering oscillations and their separate feeding to a certain phase meter. These considerations apply fully both to the case of commensurable and to the case of coincident frequencies of the interfering oscillations.

Here it is necessary to mention one method that exceptionally simplifies the measurement of the phase difference of radio-frequency oscillations. This is the replacement of measurements of the phase difference directly of the high-frequency oscillations (corresponding to the waves under investigation) by measurements of the same phase difference between oscillations of an arbitrarily lowered frequency.

This method is based on the simultaneous synchronous heterodyning of both compared oscillations in two independent mixers and on comparison of the phase difference between the obtained oscillations of difference frequency⁴˒⁵, which can be made arbitrarily low (for example, in the range of audio frequencies).

In heterodyning in converters (mixers), combination tones are separated with a total phase representing a linear combination (for example, the difference) of the complete phases of the initial oscillations and of the oscillations of the common heterodyne.

If the phases of the oscillations under investigation are $\omega t+\Phi_1$ and $\omega t+\Phi_2$, and the phase of the heterodyne oscillations is $\omega_0 t+\Theta$, then, with quadratic transformation, the separated difference tones will have the phases $\Omega t+\Phi_1-\Theta$ and $\Omega t+\Phi_2-\Theta$, where $\Omega=\omega-\omega_0$, while the phase difference of these transformed oscillations will still be $\Psi=\Phi_1-\Phi_2$.

If, in this case, $\Omega$ is so small that the frequencies $\omega$ and $\omega_0$ can simultaneously lie within the passband of the receiver, then simultaneous reception of the investigated

* Let us note that a nonlinear device, separating combination tones that are harmonic components of the resultant oscillation, makes it possible to examine the spatial distribution of their amplitudes.

oscillations and oscillations from a local heterodyne oscillator for any type of receiver suitable for the range, with subsequent extraction of the difference tone in the detector stage of the receiver.

The use of this principle makes it possible to employ highly sensitive receivers of the superheterodyne type and, while avoiding the specific difficulties associated with the need to operate at a high frequency, to use as phase meters devices that function reliably at low frequencies.

In Grosskopf’s work\(^6\), for example, a description is given of an entire device based on the indicated principle for measuring phase differences of radio waves over a broad frequency range, with automatic recording of the phase difference and of its changes over any interval of these changes\(^*\).

At present there exists a considerable number of different instruments for measuring phase differences (usually called phase meters), and it would be extremely difficult to make a detailed analysis of their types. But it is beyond doubt that the most commonly used simple and accurate systems of phase meters now in existence are based on the use of cathode oscillographs. Observation of the magnitude of the phase difference of two oscillations and of its changes by means of a cathode oscillograph is carried out either by studying Lissajous figures (see the cited work of E. Shchegolev\(^3\)), or by other methods—for example, as indicated in Nijhenhuis’s article\(^8\), where the measurement is based on circular sweep produced by one of the oscillations being compared, and modulation of the brightness of the resulting image by means of the other oscillation. In this case the arrangement of parts of the circumference with different brightness makes it possible to read off directly, in angular units, the measured phase difference on the scale of the period of the oscillation that produces the circular sweep. There also exist other phase-meter systems based on the electrodynamic principle, on the use of compensation circuits connected with phase shifters, etc.

At the same time, most of the above-mentioned circuits have been developed for the case of one and the same frequency and, in the case of different frequencies, require preliminary conversion of the oscillations to one common frequency. But any of the phase meters, when measuring the magnitude of the phase difference, can determine it only with an accuracy up to some constant quantity, which is an integral number of a certain fraction of the total period (the period of the resultant oscillation). For example, for equal frequencies the measurement of the magnitude of the phase difference can be made with an accuracy up to a value of \(\pm n 2\pi\); for a frequency ratio of two to three—with an accuracy up to \(\pm n\pi\), etc.

\(^*\) In Chaman Lal’s work\(^7\) an analogous method was used to determine the angle of arrival of a sky wave.

Let us note that the indicated ambiguity in determining the magnitude of the phase difference does not arise when determining the change in the phase difference, and here, with continuous reading of the instrument’s indications, the accuracy of the measurement is determined exclusively by the accuracy of the phasemeter itself.

Taking into account all the remarks made above, we call radio-interference phenomena those phenomena connected with the appearance of a stationary or slowly varying spatial distribution of the amplitude or form of oscillations, when several propagating electromagnetic oscillations are present simultaneously at each point of the given space.

The form of the resulting oscillation, its polarization state, or its amplitude at each given point of the interference space is determined by the magnitude of the time delay of one oscillation relative to another. This time shift, forming, on the scale of the periods of the oscillations, their phase shifts, is usually characterized in optical interference by the so-called path difference:

\[ \Delta L = L_1 - L_2 = c \int_{S_1}^{A} \frac{ds}{v(s)} - c \int_{S_2}^{A} \frac{ds}{v(s)} = \int_{S_1}^{A} n_s\,ds - \int_{S_2}^{A} n_s\,ds, \]

where, respectively, \(L_1\) and \(L_2\) are the so-called optical path lengths of the interfering oscillations as they propagate from the sources \(S_1\) and \(S_2\) (or from a single source) to the point of observation \(A\), and \(n_s = \dfrac{c}{v(s)}\) is the index of refraction along the propagation path of the given oscillation.

The optical path length is the distance that can be traversed by an oscillation propagating with velocity \(c\) during the time \(\tau\) of actual propagation from \(S\) to \(A\):

\[ L = c\tau \qquad \text{where } \tau = \int_{S}^{A} \frac{ds}{v(s)} . \]

For a harmonic oscillation propagating from some source, at the point of observation we shall have, for the oscillatory motion, an expression of the form

\[ f = u(x, y, z)\, e^{i[\omega t - \Phi(x, y, z)]}, \]

which must be a solution of the wave equation.

Without being interested in the spatial distribution of the oscillation amplitude, given by the function \(u(x, y, z)\), we shall mainly deal with the scalar function \(\Phi(x, y, z)\).

The entire expression \([\omega t - \Phi(x, y, z)]\) is called the phase of the oscillation, and we shall call it the total phase.

However, in what follows we shall be interested only in that part of the total phase which does not depend on time and characterizes the spa—

spatial distribution of the instantaneous values of the total phase of the oscillations, for which in what follows we shall retain the name phase of the oscillations at a given point (more precisely, one should speak of the value of the retarded phase).

To each point of space in which a propagating wave process exists there corresponds a definite value of the scalar function \(\Phi(x, y, z)\), so that in the given space there exists a scalar field of phases. In this case (for all points not containing sources)

\[ \oint d\Phi = 0. \]

The properties of this scalar field can be very clearly characterized by the structure of the surfaces of equal phases, whose equations are:

\[ \Phi(x, y, z)=\mathrm{const}. \]

It is evident that, in accordance with Fermat’s principle, the direction of propagation of the waves must be taken to be the direction of the vector \(\rho=\operatorname{grad}\Phi\)—the phase gradient. In its physical meaning \(\Phi(x, y, z)\) is the magnitude, expressed in angular measure on the scale of the period of oscillations, of the time lag of the oscillation; moreover, the quantity \(\Phi(x, y, z)\) at any point is determined as

\[ \Phi(x, y, z)=\omega \int\limits_{S}^{A}\frac{ds}{v(s)}, \]

and the path of integration from the radiation source \(S\) to the observation point \(A\) may be chosen arbitrarily, since \(\oint d\Phi=0\) (the initial phase \(\Phi_0\) in this expression has been taken equal to zero, which can always be achieved by a corresponding choice of the origin of time).

The magnitude of the phase velocity \(v(s)\) is a function of the direction of the path of integration and, depending on the properties of the medium in which the propagation takes place, may also depend on the frequency of the propagating oscillation.

For different directions of the path element \(ds\), \(v(s)\) may take values that differ greatly from \(c\), but it should be borne in mind that only that direction of \(v(s)\) has physical meaning which corresponds to the direction of the normal to the surfaces of equal phase, i.e., to the direction of \(\rho\).

Taking the direction \(\rho\) as the direction of phase propagation, we may write

\[ |\mathbf{v}|=\frac{\omega}{|\rho|}, \]

where the vector \(\mathbf{v}\) has the direction of the vector \(\rho\). \(\tau=\displaystyle\int_1^2 \frac{ds}{v(s)}\) is the time of propagation of the phase of the oscillation from point 1 to point 2. This expression remains valid for any conditions of propagation, both in homogeneous and in inhomogeneous media, i.e., also in the presence of dispersion and diffraction.

The presence of inhomogeneities in the medium in which propagation takes place entails a definite distortion of the geometrical structure of the surfaces of different phases and, consequently, the study of the phase structure of the wave field makes it possible to determine the features of the propagation of oscillations in the given medium.

Since the character of the interference pattern is determined by the phase structure of the field of the interfering waves, direct phase observations of interference, possible in the radio-wave region, make it possible to obtain data on the features of radio-wave propagation under the conditions being studied.

III. CONSIDERATION OF VARIOUS VARIANTS OF RADIO INTERFERENCE

Let us establish, by examples of various variants of radio-wave interference, the specific features of this type of phenomenon and clarify the possibilities they provide for solving a number of scientific and practical problems. Most of the variants considered below were either implemented or proposed for solving particular special problems. However, the systematic analysis presented here of various cases of radio interference and of the results obtained, from the point of view of the basic features of the phenomenon under study, makes it possible to draw many essential general inferences and conclusions.

The present analysis is not subordinated to a chronological principle, but is carried out on the basis of the general physical propositions on which the given group of variants, or individual variants, of radio interferometers is based.

1. Case of a single source of radiation

One of the simplest and, at the same time, chronologically earliest cases of the use of radio-wave interference is the variant proposed by Appleton and Barnett\(^{9}\) for observing the interference of radio waves reaching the point of observation along two paths: along the earth’s surface and after reflection from the ionosphere.

Schematically this experiment is shown in Fig. 1. The receiver \(I\), tuned to the frequency \(\omega\) radiated by the transmitter, receives simultaneously oscillations that have traveled along two paths: \(L_1\) and \(L_2\). The indicator

at the output makes it possible to register the amplitude of the resulting oscillation, which is determined by the value of the path difference \(\Delta L=L_2-L_1\).

The magnitude of this path difference can be determined only with the aid of additional operations, which we shall not discuss here. Let us note only that in such an implementation we have a complete analogue of a number of optical interferometers, with the sole unpleasant addition that the conditions of reflection from the ionosphere for the ray \(L_2\) are always inconstant both in amplitude and in phase. The use of an indicator based on the measurement of intensity puts the entire operation of the device in direct dependence on the constancy of the amplitudes of both interfering oscillations.

Fig. 1. Appleton and Barnett radio interferometer

Fig. 2. Radio-engineering version of Young’s interferometer.

A radio-engineering version of an interferometer, close to the case of the classical Young experiment in optics, is the one schematically shown in Fig. 2.

Here \(S\) is the source of radiation. The oscillations emitted by it reach two receiving points \(A\) and \(B\) by different paths. After reception at these points and, if necessary, after corresponding amplification or conversion, the oscillations are fed through two communication channels (\(3\) and \(4\)) to the observation point \(I\) (this observation point may also be located at one of the receiving points). In this case, oscillations already of converted frequency may be fed to point \(I\), as was indicated above.

Observations may be carried out either by the intensity of the resulting oscillation obtained at \(I\) upon superposition of the oscillations from \(S\) that have traveled different paths: \(1+3\) and \(2+4\), or by the phase difference of these oscillations with the aid of one of the phasemeters known to us.

The observed phase difference or the degree of intensity of the resulting oscillation is determined by the value of the path difference

\[ \Delta L=L_2-L_1=l_2-l_1+Q+Q_0, \]

where \(l_1\) and \(l_2\) are the optical lengths of the paths on routes \(1\) and \(2\), \(Q\) is the path difference corresponding to the difference in delay of the oscillations in communication channels \(3\) and \(4\) from receivers \(A\) and \(B\) to indicator \(I\), \(Q_0\)—

equivalent path difference arising from the difference in the delay of oscillations in the receiving-amplifying devices \(A\) and \(B\). This latter quantity \(Q_0\) can be made equal to zero by the corresponding adjustment, and in that case we arrive at a complete analogy with Young’s experiment (see Fig. 3).

Usually, in Young’s experiment the narrow slits \(A\) and \(B\) in the screen \(P\), which are virtual emitters of coherent light, are illuminated by a beam of parallel rays strictly perpendicular to the plane of the screen \(P\). Therefore the phase difference of the oscillations emitted by the virtual emitters \(A\) and \(B\) is taken to be zero. In this case the path difference producing a definite position of the interference pattern is formed only when the light travels along the paths \(L_1\) and \(L_2\) from the virtual emitters \(A\) and \(B\) to the observation point \(K\) on the surface \(R\).

Fig. 3. Diagram of Young’s experiment.

If we change the angle of incidence of the parallel rays on the screen \(P\), then at the fixed observation point \(K\) the interference fringes will shift. It is obvious that, for parallel rays incident on the screen at an angle \(\alpha\), a complete replacement of a dark interference fringe by a bright one will occur when the path difference changes by \(\lambda/2\), i.e., when the angle of incidence \(\alpha\) changes according to the condition

\[ \Delta(\sin \alpha)=\frac{\lambda}{2d}, \]

which, for small \(\lambda/2d\), where \(d\) is the distance between \(A\) and \(B\) (between the slits), gives

\[ 2d\cos\alpha\cdot \Delta\alpha=\lambda, \]

i.e.,

\[ \Delta\alpha=\frac{\lambda}{2d}\,\frac{1}{\cos\alpha}. \]

In the case of a complete shift of the interference fringe to the neighboring one, the phase difference changes by an amount \(2\pi\).

The phase difference, under the condition that the oscillations propagate in a homogeneous medium with velocity \(v\), is equal to:

\[ \Psi=\frac{\omega}{v}\,d\sin\alpha. \]

For small variations of the angle of incidence we may write:

\[ \Delta\Psi=\frac{\omega}{v}\,d\Delta\alpha\cos\alpha,\quad \text{or}\quad \Delta\Psi=2\pi\,\frac{d}{\lambda}\cos\alpha\,\Delta\alpha. \]

In carrying out such an interference experiment for radio waves, it is undoubtedly most expedient to use indi-

…not by the amplitude of the resultant oscillation formed at the observation point, but by the phase difference of both oscillations that have arrived by different channels*), all the more so since in this case \(d/\lambda\) will not always be so large that the changes in \(\alpha\) that occur could cause such changes in phase difference as would substantially affect the amplitude of the resultant oscillation.

The most substantial argument in favor of phase measurements is their practical independence of the amplitude of the oscillations, within those limits, of course, in which the phase meter operates at all.

In the work of Ross and Stoy\(^{10}\), devoted to measuring the phase velocity of radio waves along the earth’s surface, at the receiving points (\(A\) and \(B\)) there were placed two identical antennas feeding two identical receiver-amplifiers. The oscillations received and amplified at the outputs of the receivers were fed along two symmetrical feeders to a cathode-ray oscillograph, which served as a phase meter. Observation of changes in the shape of the Lissajous figure obtained on the oscillograph screen made it possible to measure the change in phase difference for a specified change in the position of the transmitter. Changing the direction to the transmitter by \(90^\circ\) changes the difference in the geometrical distances of the transmitter from both antennas by the amount \(d\).

Correspondingly, this phase meter must register a change in the phase difference by the amount

\[ \Delta \Psi = \frac{\omega}{v} d = 2\pi \frac{d}{\lambda}, \]

whence, knowing \(d\) and \(\omega\), one can determine \(v\)—the phase velocity of the radio waves.

Let us note that in the experiments described by Ross and Stoy \(d/\lambda < 1\), so that the changes in phase difference lay within \(2\pi\), which in the language of interference observations corresponds to locating within one fringe.

However, amplitude indication for radio waves also has great practical significance. When registration of the minimum of the resultant oscillation is used, which, with the corresponding balancing of amplitudes, can be equal to zero, this indication makes it possible to achieve very high accuracy in fixing one definite value of the phase shifts corresponding to a definite direction of arrival of the oscillations.

We encounter a similar case in radio direction finders. An ordinary loop is an example of such a system, arranged in such a way that there is a minimum indication at

*) As, for example, this is described in the previously mentioned work of Grosskopf\(^{6}\).

in the indicator when oscillations arrive perpendicular to the plane of the loop \((\alpha = 0)\).

This is achieved by the appropriate phasing of the oscillations fed to the indicator from opposite sides of the loop.

An even more distinct picture of this kind can be observed in the direction finder of the Adcock system. Since the dimensions of the antenna device (the quantity \(d\)) of direction finders are, as a rule, smaller than the wavelength, then when \(\alpha\) is varied from \(0\) to \(\pm \pi/2\) the intensity of the resultant oscillation will change from a limiting minimum (ideally, zero) to some greatest value. Usually, however, this greatest value by no means reaches the magnitude that can be obtained when both oscillations fed to the indicator (receiver) from opposite sides of the loop or from the opposite antennas of the Adcock system are in phase. In this case the greatest intensity, or in practice the received signal strength, will be the smaller the smaller the geometrical dimensions of the antenna system in comparison with \(\lambda\). This is equivalent to estimating the degree of departure from the center of the given interference fringe when the angle of incidence \(\alpha\) is varied from \(0\) to \(\pm \pi/2\).

The process of direction finding—determining direction with the aid of a loop—consists precisely in the fact that by rotating the entire antenna device the angle of incidence of the arriving oscillations on the antenna system is changed until maximum compensation of both oscillations is reached, i.e., the minimum of the resultant process, which usually occurs at an \(\alpha\) close to zero.

Let now the distance between the points of reception (\(A\) and \(B\)), which we shall call the base, be increased. Using a phasemeter as the indicator, for \(d = \lambda/4\), when the angle of incidence \(\alpha\) is varied from \(\alpha = 0\) to \(\alpha = \pm \dfrac{\pi}{2}\), we shall record a change in the phase difference by \(\pm \pi/2\), i.e., we shall have a one-to-one correspondence between angular measurements of the phase difference and the angles of incidence of the arriving waves.

A total change of the angle of incidence by \(\pi\) will correspond numerically to the same change in the observed phase difference.

For \(d = \lambda/2\), when \(\alpha\) is varied from \(-\pi/2\) to \(+\pi/2\), the phase difference changes by \(2\pi\), which optically would correspond to a complete displacement of the interference fringes.

It is quite evident that such interference experiments are impossible in optics using so close a spacing of virtual emitters of coherent light. For \(d > \lambda/2\), the path difference corresponding to different \(\alpha\) may vary by an amount exceeding \(\lambda\), and in an indicator based on the observation of intensity identical readings will be observed for different \(\alpha\).

In the case of using phase meters, accordingly, there will exist a series of values of \(\alpha\) for which the measured phase difference will differ by an integral number of \(2\pi\).

Indeed:

\[ \Psi = 2\pi \frac{d}{\lambda}\sin\alpha + \Psi_0. \]

Values of \(\Psi\) differing by \(k\,2\pi\) will occur at angles \(\alpha_n,\alpha_m\) satisfying the condition

\[ \frac{d}{\lambda}(\sin\alpha_n-\sin\alpha_m)=k, \]

where \(k\) is an integer.

Bearing in mind that none of the phase meters measures an integral number of complete cycles, we thereby obtain, for \(d>\lambda/2\), both for the intensity indicator and for the phase meter, a definite ambiguity. Increasing \(d/\lambda\), simultaneously with increasing the ambiguity, sharply increases the sensitivity of the system to changes in the angle of incidence \(\alpha\), and the introduction of special measures to eliminate this ambiguity makes it possible to use all the advantages of the high sensitivity of such a system.

For large \(d\) and small distances to the radiation source, we encounter the impossibility of neglecting \(d\) in comparison with this distance, i.e., we encounter the impossibility of considering the rays arriving at points \(A\) and \(B\) as parallel. Then it is necessary to pass to a complete measurement of the phases of the oscillations along the entire propagation path. The observed phase difference is

\[ \Psi=\Phi_2-\Phi_1+\delta_0, \]

where, as before, \(\Phi_1\) and \(\Phi_2\) are the phases of the oscillations reaching the reception points (\(A\) and \(B\)), and \(\delta_0\) is the phase difference arising as a consequence of the nonidentity of the paths supplying the oscillation from \(A\) and \(B\) to \(I\).

Let us now imagine a system of surfaces of equal phases of the radiation field of the source \(S\). It is obvious that the character of the interference pattern observed at point \(I\) (the resultant intensity or phase difference) will be determined by which equiphase surfaces pass through points \(A\) and \(B\).

If we assume that the equiphase surface \(\Phi=\Phi_0\) passes through point \(A\), then the phase-meter reading \(\Psi\) indicates that through point \(B\) there passes the equiphase surface

\[ \Phi=\Phi_0-(\Psi-\delta_0)\pm k\,2\pi. \]

The degree of possible ambiguity (\(\pm k\,2\pi\)) can be estimated from other considerations, namely from the magnitude of the ratio \(d/\lambda\), and, as was indicated earlier, for \(d<\lambda/2\), \(k=0\).

Thus, for such dimensions of the base, despite the ambiguity in interpreting the phase-meter reading, every ...

ambiguity in determining the mutual orientation of the base \(AB\) and the surface of equal phases. In this case, measurement of the phase shift \(\Psi\), as carried out in ordinary direction finders, makes it possible to determine unambiguously the direction of the base with respect to a bounded portion of the surface of equal phase.

Local diffraction or refraction effects, causing local distortions of the form of the wave surfaces—the surfaces of equal phase—can give rise to significant errors in determining the direction to the transmitter in direction finding, since it is always tacitly assumed that the direction to the transmitter coincides with the direction of the normal to the surface of equal phases.

These errors are known in the practice of radio direction finding as deviation errors caused by local objects, errors due to coastal refraction, etc.

The use of large bases for radio direction finding gives, besides an increase in accuracy, also a reduction in the role of local distortions of the surfaces of equal phase (or of the wave front) in determining the bearing. This is explained by the fact that the phase perturbations \(\delta\Phi\) for each receiving point remain constant, independently of the size of the base, while the value of each degree of phase difference decreases as it increases.

For large distances from the source, when \(r_1\) may be regarded as parallel to \(r_2\):

\[ \Psi=\frac{\omega d}{c}\sin\alpha+\delta_0\pm 2\delta\Phi =2\pi\frac{d}{\lambda}\sin\alpha+\delta_0\pm 2\delta\Phi, \]

\[ \sin\alpha=\frac{\Psi-\delta_0}{2\pi}\cdot\frac{\lambda}{d} \pm\frac{\delta\Phi}{\pi}\cdot\frac{\lambda}{d}, \]

therefore the error in determining the direction \(\alpha\) will decrease with increasing \(d/\lambda\). It should be borne in mind that the use of bases for which \(d>\lambda/2\) requires the application of special methods for eliminating ambiguity. In a homogeneous isotropic space one may take

\[ \Phi=\frac{\omega r}{v}, \]

therefore the phase difference observed in \(I\) for this simplest case may be written in the form

\[ \Psi=\frac{\omega}{v}(r_2-r_1)+\delta_0. \]

The geometrical locus of those positions of the transmitter to which the same readings of the phasemeter in \(I\) will correspond will be a family of hyperboloids of revolution with foci at the points \(A\) and \(B\), according to the equation

\[ r_2-r_1=\frac{v}{\omega}(\Psi-\delta_0). \]

Assuming that one may neglect those deviations from this ideal case which occur under real conditions, continuous observations of a moving radiator can be carried out by recording its transition from one hyperboloid of a given family to another. On the other hand, observations of the deviations of the loci of the radiator’s position, corresponding to constant readings of the phasemeter, from regular hyperboloids make it possible to estimate the distortions of the surfaces of equal phases or, more precisely, to estimate the differences in the conditions of propagation of radio waves along the paths \(SA\) and \(SB\), caused by diffraction phenomena occurring, for example, when radio waves propagate along the earth’s surface, or by the fact that along one of the trajectories the propagation velocity had a different value than along the other.

For the case of propagation of radio waves along the earth’s surface, the problem was rigorously formulated and solved by Sommerfeld \(^{11}\), and the discussion of his solution as applied to the phase structure of the field was completed in the works of Al’pert, Migulin, and Ryazin \(^{12}\) and Ryazin \(^{13}\). But even these results could be obtained only under a very far-reaching idealization of the propagation conditions. The theoretical consideration of real cases, in the majority of instances, encounters quite insurmountable difficulties, which makes the experimental study of this question especially important.

In practice, in radio engineering, the linear dimensions of diffracting inhomogeneities, propagation distances, and wavelengths are quantities of the same order, and the placement of observation and reception points in the zone of diffraction effects, i.e. at distances of a few fractions of wavelengths from the diffracting objects, is encountered extremely often in practice. All these circumstances in a number of cases may entail strong distortions of the surfaces of equal phases, and the measured optical path lengths \(L=(c/\omega)\Phi\) may differ considerably from the geometrical distances.

In the light of these considerations, a systematic study of the distortions of the hyperbolic net may provide substantial material for determining the order and character of the distortions of the surfaces of equal phases near the earth’s surface—a question that plays a role in radio direction finding.

2. Radio Interferometers with Several Sources of Radiation

It was indicated above that it is possible to obtain interference of radio waves when using separate sources of radiation. If the radiators employed operate at one frequency, then at the observation point it is practically possible to make observations only

by the intensity of the resulting oscillation, since conducting observations directly of the phase difference of oscillations arriving from different sources requires their completely separate reception, which, even with very advanced directional transmission and reception techniques or separation by polarization, is very difficult.

This problem of the separate reception of two or more interfering oscillations, so difficult at equal frequencies, is easily solved when emitters operating at commensurable frequencies are used. When different frequencies are used, separation of the oscillations at the point of reception (observation) can be carried out quite perfectly, and thus there arises the possibility of unhindered use of various phasemeters for the purpose of directly measuring phase differences between the interfering oscillations at the given point, and of observing the form of the resulting oscillation produced by the superposition of oscillations of commensurable frequencies.

To obtain a stationary interference pattern in the case of two oscillations, it is necessary to ensure the complete coherence of both interfering oscillations. This requirement leads to the need to ensure constancy of the initial phase shifts between the oscillations emitted by both sources, which is the criterion of complete synchronism of both radiations.

The initial phase shifts \(\varphi_1\) and \(\varphi_2\) enter additively into the expressions for the total phase of each of the oscillations, and their difference will also enter into the final phase difference that determines the position of the interference pattern. Therefore, obtaining a spatially stationary interference pattern requires maintaining a constant value of the initial phases.

To satisfy this requirement, it is necessary either:

1) to make both emitters subordinate to one control point; for example, simply to feed both emitters from one and the same generator;

2) of the two emitters, to have one as the main—setting—one, and the second as subordinate, so that its frequency and radiation phase would be wholly determined by the first emitter, while at the same time being subject to monitoring and adjustment.

The first of these two methods, as applied to the case of coincident frequencies in its simplest variant, is used in numerous systems of directional antennas for short waves with active reflectors. A single source of oscillations—a generator—feeds, with various phase shifts, the emitters arranged in the corresponding manner.

Let us note that the case of passive reflectors and directors belongs rather to the second type of the devices described and, possessing considerably smaller possibilities in the sense of specifying the desired phase shifts and amplitudes of the radiations, has much less

flexibility in the sense of producing various directional diagrams.

In space, a stationary distribution of the amplitudes of the resultant oscillation is then formed—an interference pattern; moreover, its character, or, as it is customary to say, the directional diagram of the antenna system, is determined entirely by the arrangement of the radiators, their mutual phasing, and also by the number of these radiators (which may be greater than two). Since the distance between individual radiators is usually of the order of \(\lambda\) or less, it may be assumed that, for distant points, the trajectories of the rays from the various radiators coincide. Therefore all these rays may be assigned one and the same value of the phase velocity \(v\). Then the phase (we are speaking of the lagging phase) of the oscillations arriving from the various radiators will be

\[ \Phi_1 = -\frac{\omega}{v} r_1 + \varphi_1;\qquad \Phi_2 = -\frac{\omega}{v} r_2 + \varphi_2;\quad \text{etc.} \]

The amplitude of the resultant oscillation,

\[ A=\sum_k A_k \cos(\omega t-\Phi_k)=B\cos(\omega t+\Phi), \]

where

\[ B^2=\left(\sum_k A_k\cos\Phi_k\right)^2+ \left(\sum_k A_k\sin\Phi_k\right)^2 \]

and

\[ \tg\Phi= \frac{\sum_k A_k\sin\Phi_k}{\sum_k A_k\cos\Phi_k}. \]

Thus, by a corresponding choice of \(r_1, r_2,\ldots,r_m\) and \(\varphi_1, \varphi_2,\ldots,\varphi_m\), it appears possible to obtain very diverse types of directional diagrams; moreover, increasing the number of radiators makes it possible to obtain sharp boundaries of the corresponding zones of different intensity, just as in optics an increase in the number of interfering rays entails an increase in the sharpness of the interference pattern.

Let us note, incidentally, the distinction between this method of obtaining directed radiation and the projector action of optical systems and of certain directional radiators of decimeter and centimeter waves. For light rays and for the indicated cases of radio waves, the principles of geometrical optics are used; these are no longer applicable to the technique of longer electromagnetic waves because of the colossal difference in scales. Conversely, for the same reason, systems of complex antennas, consisting of a set of a small number of elementary radiators, cannot be realized for microwaves.

In the case of two radiators, as follows also from the reciprocity principle, the locus of surfaces of equal phases will be a family of one-sheeted hyperboloids*). In this case, a change in the initial mutual phase shift of both radiators will cause the same change in the observed phase difference while preserving the former system of surfaces corresponding to constant phase differences. If, however, one is interested in the location of points corresponding to a certain fixed value of the phase difference, then a change in the initial mutual phase shift will, as it were, “move” our point from one hyperboloid of the family to another.

Thus it is possible, as it were, to metricize space by assigning to each hyperboloid its own value of the phase difference, for given initial phases of radiation.

If \(d\) is the distance between the radiators and \(d < \lambda/2\), then, for the given problem, surfaces all of whose points have identical phase differences will not be repeated in space.

The greatest difference of phase differences that can occur in space is

\[ \Delta \Psi = \frac{2\pi}{\lambda}\, 2d, \]

and, moreover, for

\[ d < \frac{\lambda}{2}\quad \Delta \Psi < 2\pi . \]

In observations of the intensity of the resultant oscillation, equal intensities will be observed at those points where

\[ \Phi_2 - \Phi_1 = \pm \Psi = \text{const}. \]

For equal amplitudes of the two interfering rays,

\[ B^2 = A^2\left[(\cos \Phi_1 + \cos \Phi_2)^2 + (\sin \Phi_1 + \sin \Phi_2)^2\right] = \]

\[ = 2A^2[1 + \cos(\Phi_1 - \Phi_2)] = 4A^2 \cos^2 \frac{\Psi}{2}. \]

Therefore, symmetrically with respect to the central band \((\Psi = 0)\), zones of equal intensity will be located. For \(d > \lambda/2\), the phase difference will, for some hyperboloids of the system, take values differing by \(2\pi\), i.e. indistinguishable from the standpoint of unit readings of any phasemeters. Accordingly, there will be a large number of surfaces corresponding to equal intensities of the resultant oscillation.

Such ambiguity is of no significance in the case where one measures not the phase difference itself, but its changes when the observation point is displaced. Then continuous observations of the readings

*) Here, as also in what follows, we shall regard the radiators as point-like, i.e. neglect the dimensions of the antennas in comparison with the distances.

phasemeter make it possible to count the integer number of changes in the phase difference by \(2\pi\) in the case when the observation point has moved into a region corresponding to repetitions of the previous readings. In exactly the same way, continuous observations of the intensity of the resultant oscillation make it possible to count the number of repetitions of maxima or minima in the indicator readings and thereby, as in the case of direct measurement of changes in the phase difference, to record the transition of the observation point from the initial point to another, quite definite hyperboloid of the system of surfaces of equal \(\Psi\).

However, in such observations the use of intensity indication may entail serious errors in the case when the motion of the observation point follows a complicated path. The return of the observation point into a zone already traversed may be completely indistinguishable from a transition into the next zone, and consequently false readings are entirely possible.

This circumstance undoubtedly devalues the prospects for using such a system as a high-accuracy radio beacon for \(d>\lambda/2\) when indication by intensity is used. All the more attractive becomes the use, for this purpose, of phase indication, which, when phasemeters registering the passage of whole cycles are used*), makes it possible to avoid the errors mentioned and, for a definite beginning of the motion, to determine unambiguously on which hyperboloid of the system the observation point is located. But, as was already indicated above, registration of the phase difference of two oscillations received at one point requires separate reception of these oscillations, which in practice is most conveniently accomplished by applying interference of waves of different frequencies. In this connection, the analysis of possible systems of such radio-beacon devices we shall postpone until the consideration of interference of oscillations with commensurable frequencies.

The observed change in the phase difference accompanying the displacement of the observation point may also be interpreted in the following manner. Let the total phases of the oscillations emitted by the initial radiators operating at coincident frequencies be equal to:

\[ \Theta_{10}=\omega_0 t;\qquad \Theta_{20}=\omega_0 t+\varphi_0 \]

At the receiving point we shall have at each instant the total phases:

\[ \Theta_{11}=\omega_0 t-\Phi_1;\qquad \Theta_{21}=\omega_0 t-\Phi_2+\varphi_0, \]

and the phase difference:

\[ \Psi=(\Phi_1-\Phi_2)+\varphi_0. \]

If, when the observation point is displaced, there occurs a change in the—

*) See, for example, the article by E. Shchegolev \(^{14}\).

of the measured phase difference at a rate $\zeta$, then

$$ \zeta=\frac{d\Psi}{dt}=\frac{d}{dt}(\Phi_1-\Phi_2). $$

As was indicated earlier, $\Phi=\omega_0\tau=(\omega_0/c)L$, where $L$ is the equivalent optical path length. Therefore

$$ \zeta=\frac{\omega_0}{c}\frac{d}{dt}(L_1-L_2), $$

and, from the observer’s point of view, it is of course immaterial whether the displacement of the point of observation takes place, or simply a change in $L$ for some other reasons.

As is known, the instantaneous angular frequency is $\omega=d\theta/dt$. At the receiving point

$$ \omega_1=\omega_0-\frac{d\Phi_1}{dt}=\omega_0\left(1-\frac{1}{c}\frac{dL_1}{dt}\right); $$

$$ \omega_2=\omega_0-\frac{d\Phi_2}{dt}=\omega_0\left(1-\frac{1}{c}\frac{dL_2}{dt}\right). $$

The difference of these frequencies, which are the frequencies of the primary emitters shifted by the magnitude of the Doppler displacement, will be:

$$ \omega_2-\omega_1=\frac{\omega_0}{c}\frac{d}{dt}(L_1-L_2). $$

Thus, observation of the rate of change of the phase difference $\zeta$ when the receiving point is displaced may be identified with observation of the phase relations between two oscillations arriving at the given point, whose frequencies are shifted by the amounts of the Doppler displacement, the magnitude of the rate of change of the phase difference being simply equal to the difference of these corrections to the angular frequency of each oscillation.

In observations of the intensity of the resultant oscillation, however, we shall observe changes of intensity corresponding to beats that arise under the simultaneous action of oscillations with frequencies:

$$ \omega_1=\omega_0\left(1-\frac{dL_1}{dt}\right), $$

$$ \omega_2=\omega_0\left(1-\frac{dL_2}{dt}\right). $$

In this case the beat frequency $F=\frac{\omega_2-\omega_1}{2\pi}$ would again be equal to

$$ F=\frac{1}{2\pi}\zeta. $$

Analogous general conclusions can also be reached for the second case singled out by us, when the initial oscillations are excited in

one of the radiators, while the second radiator is regulated in such a way that the oscillations it emits are exactly synchronous with the oscillations of the first and that the required constant phase shift between them is maintained. In practice this can be accomplished, for example, in the following way.

Oscillations from the primary master radiator are fed through a special communication channel to the second radiator (more precisely, to its generator) and there, after appropriate amplification, are radiated with a definite phase shift. In this case, when the second radiator operates at the same frequency as the primary—master—one, we obtain all the results discussed above, connected with the production of an interference pattern, whose character is determined by the geometrical arrangement of the radiators, their mutual phase shifts, and the emitted frequency.

But the use of a second subordinate radiator, controlled only by the master one, makes it possible to realize interference not only of identical, but also of different frequencies. The methods of synchronization and frequency transformation of nonlinear oscillatory systems*) developed in radio engineering, especially in recent years, make it possible to create devices capable of generating and radiating oscillations with a frequency rigidly determined by the frequency of the controlling oscillations—equal to it or in a rational ratio with it. In this case the phase shifts of the emitted oscillations with respect to the controlling ones may be constant and may be taken into account and regulated. The controlling oscillations, in turn, may be transmitted from the primary—master—radiator through a special channel or directly through radiation.

In the latter case we obtain a peculiar reflecting action of the subordinate station; moreover, unlike specular reflection in optics, here we acquire the possibility of controlling the amplitude, frequency, and phase shift upon “reflection.”

As already indicated above, the simultaneous radiation from two points of oscillations of commensurable frequencies creates in space a certain distribution of the form of the resultant oscillation, periodic with respect to the path difference between the oscillations of both frequencies. The possibility of completely separate reception of both oscillations makes it possible to carry out a direct measurement of the phase difference between

) The study of these questions and the development of a number of methods for solving the indicated problems were, over a number of years, the subject of numerous works of the school of Acad. L. I. Mandelstam and Acad. N. D. Papaleksi. In the work of this school the indicated questions found their most complete and rigorous solution. The results of these investigations are set forth in numerous journal articles, references to which would take up too much space. A concise survey of all these works may be found in the book New Studies of Nonlinear Oscillations* by Mandelstam, Papaleksi, Andronov, et al. (Radioizdat, Moscow, 1936), which also contains an extensive bibliography.

by both oscillations, i.e., to measure the time lag of one oscillation with respect to the other.

These measurements of phase difference can be made with the aid of various phasemeters operating either directly at the received frequencies or using their preliminary conversion*).

Let the propagation time of one oscillation be

\[ \tau_1=\int_1^I \frac{ds}{v}, \]

and of the second,

\[ \tau_2=\int_2^I \frac{ds}{v}. \]

The time difference

\[ \tau_1-\tau_2=\int_1^I \frac{ds}{v}-\int_2^I \frac{ds}{v} \]

on the scale of angular variations of one of the frequencies \((\omega_1)\) takes the form

\[ \omega_1(\tau_1-\tau_2)=\omega_1\int_1^I \frac{ds}{v} -\frac{\omega_1}{\omega_2}\omega_2\int_2^I \frac{ds}{v} =\Phi_1-\frac{\omega_1}{\omega_2}\Phi_2=\Psi . \]

This quantity may be called the phase difference, and it will characterize the form of the resultant oscillation at the given point**). It is clear that measuring this quantity is meaningful only if

\[ \omega_1:\omega_2=m:n \]

is a rational fraction, i.e., if the overall process is periodic. In this case, the very measurement, in a definite such manner, of the phase-difference quantity is possible only within the limits of one cycle of variation of the form of the resultant oscillation and, consequently, in such measurements there always remains undetermined an integer multiple of some period, whose value is connected with the chosen frequency ratio \(m/n\).

Consider now various variants of interferometers using emitters operating at commensurable frequencies. One variant of an interferometer of this type is shown schematically in Fig. 4.

Fig. 4. Diagram of the operation of a radio interferometer using two different frequencies.

At points 1 and 2 there are located a master station and a reflecting station. The master station consists of a transmitting device radiating oscillations of frequency \(\omega_1\). For greater stability this frequency is usually—

) On various systems of phasemeters, see above.
*) See the above-cited work of E. Shchegolev\(^3\).

but is stabilized by quartz. The reflecting station consists of a transmitter emitting oscillations of frequency \(\omega_2\), and of a receiving device that receives the oscillations from point \(1\) and, after appropriate amplification and frequency transformation in the ratio \(m/n\), controls the emitted oscillations. In practice, for a number of considerations adduced in the works of Mandelstam, Papaleksi, and Shchegolev \(^{15,16,17,18}\), the most convenient frequency ratio proves to be \(m/n=3/2\).

In addition to the elements indicated above, the reflecting station must also contain a monitoring device which makes it possible to monitor the constancy of those phase shifts which arise at point \(2\) from the moment of reception to emission*).

Let us also point out that such an interferometer can be implemented by a somewhat different method, using a single control station common to both emitters, which itself does not participate in producing the interference pattern. One may have a single emitter, whose oscillations reach the subordinate emitters through some communication channels and, after the required transformation and amplification, are emitted into the space where the required interference pattern is produced.

Observations of the interference pattern formed in space are made at point \(l\) by measuring the phase difference between the oscillations reaching this point from emitters \(1\) and \(2\). For this purpose one may use, for example, the receiving device of the dispersion radio interferometer developed by us \(^{19,20,21}\). These measurements of phase difference, made with an accuracy up to an indeterminate constant value, have, however, little value, since they do not make it possible to determine the magnitude of the path difference. But such a device makes it possible to determine the change in phase difference by means of continuous observations when the position of the observation point \(l\) is changed.

The measured quantity is

\[ \Psi=\Phi_1-\frac{\omega_1}{\omega_2}\Phi_2+\delta= \]

\[ =\omega_1\int_1^l \frac{ds}{v}-\omega_1\int_2^l \frac{ds}{v}+\varphi_1-(\varphi_1+\varphi_2+\Phi_d)+\delta, \]

or

\[ \Psi=\omega_1\left[\int_1^l \frac{ds}{v}-\int_2^l \frac{ds}{v}\right]+\Psi_0. \]

Here

\[ \Psi_0=\varphi_1-(\varphi_1+\varphi_2+\Phi_d)+\delta=\delta-\Phi_d-\varphi_2, \]

where \(\varphi_1\) is the initial phase of the oscillation at \(1\), \(\varphi_2\) is the phase shift at point \(2\) (on the scale of frequency \(\omega_1\)), \(\Phi_d\) is the phase shift along the path from \(1\) to \(2\)

*) The operation of such a device is described in the above-cited works \(^{15,16,17,18}\).

(in the frequency scale \(\omega_1\)), \(\delta\) is the difference of the phase shifts in both channels of the receiving-recording apparatus.

In our papers cited above \(^{19,20,21}\), a device was described that makes it possible to measure and control the magnitude of this error—the deviation of the receiving device. A detailed description of such a “deviometer” is given in our paper \(^{12}\).

It should be pointed out that in the present receiving-recording device it is also possible to employ superheterodyning with subsequent measurement of the phase difference at an intermediate or even at a low frequency.

The geometric loci of equal phase differences, under the condition of propagation of oscillations in a homogeneous isotropic medium, again constitute hyperboloids of revolution. At large distances these hyperboloids pass over into a family of asymptotic conical surfaces.

If the value of the phase difference observed at some point is taken as the initial one, then, starting from this origin of reckoning and successively moving the point of observation, one can survey the entire spatial distribution of the phase difference. It is, of course, necessary in doing this to guarantee the constancy of the phase shifts in the apparatus, both transmitting and receiving.

In the presence of diffracting objects, in the region of diffraction phenomena there will be observed distortions of the equiphase surfaces for both oscillations and, consequently, distortions of the surfaces of equal phase differences. The presence of dispersion will also introduce certain changes into the position of these surfaces, causing the appearance of asymmetry.

If one disregards the role of diffraction, which need be taken into account only at small distances (in comparison with \(\lambda\)*) from the diffracting objects, then, as is known, such a radio-interference system can be successfully used as a radio beacon. In this case two problems can be solved:

a) homing on the beacon;

b) homing on a direction passing through the beacon and a given point, or simply homing on a given direction.

For homing on the beacon, for which the system of radiating stations is used as the beacon, a moving point equipped with a receiving-indicator device, starting from any point of space, must move in such a way that the observed phase difference remains constant. Any such trajectory (with the exception of closed curves on the surface of equal phase differences, which are unrealizable under real conditions), representing a curve on the surface of one of the hyperboloids of equal phase differences in the presence of pro-

*) As studies of recent years carried out at the Lebedev Physical Institute have shown (see also pp. 390–392).

spatial constraints, leads to one of the points lying on the base—the straight line connecting the emission points.

To obtain a direction passing through a given point, one first determines some point in the vicinity of the beacon that lies on the same surface of equal phases as the target. Starting from this control point and moving in such a way that the phasemeter reading remains constant, we shall, within the limits of measurement accuracy, move along a path passing through the target. Here it is of course necessary to replace the spatial problem by a plane one, assuming that we are dealing not with equiphase surfaces and not with surfaces of equal phase differences, but with their intersections by the earth’s surface.

In the event that the motion begins not from the control point but from some other point (whose position is precisely determined), it is necessary to move in such a way that, at first, by changing—through transition to other surfaces—the required phase difference, one obtains a change that would correspond to transition to the required curve, and only after this direct the motion along the given curve by maintaining a constant phasemeter reading.

Such a radio-navigation application of the given interferometer is close to Garms’s proposed navigation method,^22 based on producing the required interference pattern and on guidance along the directions of the corresponding lines of this interference pattern.

It goes without saying that the use of a combination of two such systems makes it possible to solve more complicated navigation problems and to carry out, for example, approach not only along a prescribed direction, but also to a prescribed point.

With fully justified simplifications (at large distances), reducing the problem to the plane case, and neglecting distortions of the equiphase surfaces, replacing the hyperbolas by their asymptotes, one can easily obtain the attainable angular accuracy of approach to a direction forming a prescribed angle with the direction of the base.

The error of the angular determination is:

\[ \delta\alpha=\frac{1}{2\pi\sin\alpha}\cdot\frac{\lambda}{d}\cdot\delta\Psi . \]

Here \(\delta\Psi\) is the error in measuring the change in phase difference by the phasemeter, \(d\) is the magnitude of the base, and \(\lambda\) is the wavelength used.

Let us give a numerical example showing the accuracies that can be achieved in real installations.

Take: \(\lambda=300\ \text{m}\), \(d=12\,000\ \text{m}\), \(\delta\Psi=5^\circ\) (the real accuracy of many phasemeters).

Then

\[ \delta\alpha=\frac{1.19}{\sin\alpha}\ \text{minutes}. \]

The attainable accuracy will accordingly depend on the angle formed by the direction to the target and the direction of the baseline. This dependence is given in Table 1, the second row of which gives, in minutes, the error in determining the angular displacement corresponding to a phase-meter accuracy of \(\pm 5^\circ\). It is obvious that, with the exception of narrow zones near zero values of \(\alpha\) (of the order of \(\pm 10^\circ\)), the accuracies obtained far exceed anything that can be achieved by ordinary radio beacons or direction finders.

Table 1

\(\alpha^\circ\)	0	10	20	30	40	50	60	70	80	90
\(\delta_\alpha'\)	\(\infty\)	6,85	3,48	2,38	2,01	1,55	1,37	1,26	1,21	1,19

The simple geometrical considerations given above were based on replacing the expressions for the phases

\[ \Phi = \omega \int_a^b \frac{ds}{v(\omega, S)} \]

by the simplified relations

\[ \Phi = \frac{\omega r}{v}, \quad \text{where } v = \mathrm{const}. \]

This is realized when the medium in which the radio waves propagate is homogeneous and isotropic, and when diffraction and dispersion are absent, which distort the simple spherical structure of the surfaces of equal phase for each of the oscillations. To study questions connected with such distortions, we developed and implemented an instrument which we called a dispersion radio interferometer.

This type of interferometer is obtained from the one just described by separating the two emitters, i.e., when \(d \ne 0\).

Fig. 5. Diagram of the operation of a dispersion radio interferometer.

Assuming that the propagation of the oscillations occurs along one path, we, by changing the distance \(r\) (Fig. 5), can obtain a change in the observed phase difference only in the case where, along this path, \(v_1 \ne v_2\) or there exists a dependence of the phase velocity on frequency, i.e., a certain dispersion takes place. Detailed descriptions of the construction and operation of the dispersion radio interferometer were given in our works cited above\(^{18, 19, 20, 21}\). In these

in the works there were also presented the principal results obtained when applying a dispersion radio interferometer to investigate the peculiarities of the propagation of radio waves along the earth’s surface.

In this case the dependence of the observed phase difference \(\Psi\) on the change of distance \(r\) was obtained experimentally. For the pure case there should exist the following simple relation:

\[ \Delta \Psi=\omega_1\left(\frac{1}{v_1}-\frac{1}{v_2}\right)\Delta r, \]

and if \(v_1\ne v_2\), with \(v_2-v_1=\Delta v\), then, when the distance is changed, a change of \(\Psi\) must be observed \((\Delta\Psi\ne 0)\). At the same time the accuracy of our measurements was such that dispersion could be detected if

\[ \frac{\Delta v}{v} \geq 3\cdot 10^{-3}. \]

In the presence of diffraction effects, the character of the perturbations of the system of equiphase surfaces at the point of observation is determined primarily by the relative dimensions of the inhomogeneity and by the distance from it to the point of observation, on the scale of the wavelength. Therefore, near diffracting inhomogeneities, the phase difference of two oscillations of different frequencies, observed in the receiving device of the dispersion radio interferometer, may undergo considerable changes.

We shall take

\[ \Phi(r)=\frac{\omega r}{v}+\varphi_0(r)+\varphi^*(r), \]

where \(\varphi^*(r)\) is the phase addition associated with the process of establishing the radiation pattern. (An analysis of this function and of its relation to the constants of the medium in which, and above which, the propagation occurs is given in our work cited above,\(^{12}\) in the article by P. A. Razin,\(^{13}\) and in his dissertation.\(^{23}\)) In this case \(\varphi^*(r)\to\mathrm{const}\) as \(r\to\infty\), and the magnitude of this constant is connected with the constants of the medium; \(\varphi_0(r)\) is the perturbation caused by the diffracting object. Obviously, \(\varphi_0(r)=0\) at a sufficient distance from the diffracting inhomogeneity:

\[ \Psi=\Phi_1-\frac{\omega_1}{\omega_2}\Phi_2= \]

\[ =\frac{\omega_1 r}{v_1}-\frac{\omega_1}{\omega_2}\frac{\omega_2 r}{v_2} +\left[\varphi_1^*(r)-\frac{\omega_1}{\omega_2}\varphi_2^*(r)\right] +\left[\varphi_{10}(r)-\frac{\omega_1}{\omega_2}\varphi_{20}(r)\right]; \]

\[ \Psi_f=\omega_1 r\left(\frac{1}{v_1}-\frac{1}{v_2}\right) +\left[\varphi_1^*(r)-\frac{\omega_1}{\omega_2}\varphi_2^*(r)\right] +\left[\varphi_{10}(r)-\frac{\omega_1}{\omega_2}\varphi_{20}(r)\right] \]

Thus, for the dependence of the observed phase difference on the distance we shall have one term which, in the presence of actual dispersion, will give a quantity proportional to the separation; a second, which, at distances from the radiator exceeding...

…amounting to \((50 \div 60)\lambda\), practically passes into a constant value and, finally, the third, containing the functions \(\varphi_{10}(r)\), \(\varphi_{20}(r)\)—diffraction perturbations of the equiphase surfaces, different owing to the difference in \(\lambda\). This last term, varying with distance, tends to zero as one moves away from the diffracting object.

Measurements made by us by the method of the dispersion radio interferometer in the presence of diffraction gave results confirming the existence of considerable distortions of the system of surfaces of equal phase in the vicinity of a diffracting inhomogeneity. Thus, Fig. 6 shows the course of the change in phase difference with change in distance when a considerable obstacle appears in the path of propagation, in the form of a rocky cape covered with forest, completely shielding the receiving device from the transmitting system. The two experimentally obtained curves have an entirely different character, which indicates the great complexity of the diffraction perturbations, so that displacement along two different paths gives a sharply different course of change in phase difference.

Fig. 6. Curves of the phase difference observed in a dispersion radio interferometer in experiments at sea in the presence of diffraction.

Fig. 7. Phase difference observed in a dispersion radio interferometer in experiments in the Moscow-region terrain.

In Fig. 7 are shown similar measurements made under conditions of the Moscow-region terrain, i.e. under conditions where along the path of propagation there are a number of diffracting objects—forest, hills, etc.

From the curves shown in this Fig. 7 and obtained with an interval of one year, with displacement of the observation point along the same path, the significant role of diffraction near the forest belt is clearly seen; it systematically affects the magnitude of the observed phase difference. In these experiments waves of length \(300\)—\(450\) m were used, and the curves presented show that distortions of the phase structure of the radio-wave field, caused by large diffracting objects, extend to distances of the order of \((10 \div 20)\lambda\).

The results of one of the experiments, carried out in 1939 from observations with the aid of a dispersion radio interferometer under conditions

even terrain (steppe in the vicinity of the city of Pugachev, Saratov oblast) and in the presence of a small (about 2 km in extent) inhomogeneity in the form of a gully and a dam, are shown in Fig. 8.

Fig. 8. Results of measurements with a dispersion radio interferometer in experiments under conditions of an even earth surface.

Here again a local perturbation of the phase structure is observed, caused by diffraction at an inhomogeneity of the earth’s surface; moreover, this perturbation, which disappears as the distance from the diffracting object increases, extends over distances of the order of \((10 \div 20)\lambda\). (In this experiment the waves used were 127.6 and 191.4 m.)

In that zone in the vicinity of the transmitting antenna in which the establishment of the phase velocity takes place, first experimentally and theoretically investigated by us,\(^{20}\) the dispersion radio interferometer must also note the presence of apparent dispersion, since the processes of establishment of phase velocity for each of the frequencies occur on the scale of its own wavelength and therefore, in a certain range of distances,

\[ \Psi = \Phi_1 - \frac{m}{n}\Phi_2 \]

will vary with distance. This circumstance made it possible, with the aid of the dispersion radio interferometer, to verify experimentally the theoretically calculated distribution of the phase of the electromagnetic field in the immediate vicinity of the transmitting antenna.

Fig. 9. Results of measurements with a dispersion radio interferometer near the transmitting antenna.

Figure 9 shows the results of one experiment from among those obtained by us in 1938 for measuring \(\Psi\) in the near zone of the transmitting antenna. In this Fig. 9 the experimental results are plotted as points. For comparison, the same plot contains the theoretically calculated curve for the real parameters of the antenna, and the exceptionally good agreement of theory with experiment confirms both the correctness of the theory and the expediency of the radio-interference method employed.

But since the zone of establishment of the phase velocity along the earth’s surface is not limited to the immediate vicinity of the radiating antenna, then even at large distances from the radiators along the earth’s surface we must observe a change in the phase difference with changing distance.

The results of similar observations, made along an even earth surface in the vicinity of the town of Pugachev, Saratov region, in 1939, are shown in Fig. 10.

Fig. 10. Curve of the phase difference, observed with the aid of a dispersion radio interferometer, as a function of distance from the radiator.

This experimental curve shows the presence of irregularities in the phase structure of the electromagnetic field of radio waves in a certain zone; moreover, at large distances these irregularities disappear and, in agreement with earlier results, at large distances \(\Psi = \mathrm{const}\).

The presence of a special law for the establishment of phase velocity along the earth’s surface is also confirmed by the results obtained with the aid of a dispersion interferometer when the receiving apparatus is raised upward from the surface of the earth. In this case there should be observed a change in \(\Psi\) corresponding to the elimination of those disturbances which are introduced by the proximity of the earth’s surface.

Fig. 11. Dependence of the phase difference on height. Observations with the aid of a dispersion radio interferometer.

Figure 11 shows the experimental results obtained in one of the experiments of this kind, carried out in 1939. In conducting these experiments the receiving apparatus of the dispersion radio interferometer was raised on an aerostat, while the transmitting installations were located on the ground.

In all these cases* we encounter a local dependence of the observed phase difference on distance, i.e. an apparent disper-

... associated with distortions of the regular structure of the surfaces of equal phases for each of the oscillations.

The experiments carried out in the summer of 1936 on the application of this version of the interferometer to the study of the ionosphere showed that, with an appropriate choice of antenna system, it is possible to achieve a situation in which, at a considerable distance of the receiving device from the transmitter, the phases of the received oscillations are determined to a considerable extent by the phase of the oscillations reflected from the ionosphere. During periods of unstable states of the ionosphere (dawn, sunset, solar eclipse), the observed phase difference undergoes considerable rapid and irregular changes. These variations indicate different changes in the optical path length for different frequencies experiencing different refractions, since in the ionosphere the refractive index

\[ n=\sqrt{\frac{\varepsilon}{2}+\frac{1}{2}\sqrt{\varepsilon^{2}+\left(\frac{2\sigma}{f}\right)^{2}}} \]

plainly depends on the frequency.

For observations of changes in the height of the ionosphere, in connection with the experiments planned during the total solar eclipse of 1941, we developed yet another modification of such an interferometer. In this version, by the appropriate design of the antenna systems (similar to what is described in the papers of N. Papaleksi \(^{54,55}\)), the radiation of the generator of frequency \(\omega_{1}\) is directed mainly upward (from a horizontal antenna), while the radiation of frequency \(\omega_{2}\) is directed mainly along the earth’s surface (a symmetrical vertical antenna). In the receiving device, reception is carried out on separate antennas: in the channel \(\omega_{1}\), on an antenna with compensation for reception of the ground ray; in the channel \(\omega_{2}\), on a vertical antenna that does not receive the horizontal component. Careful adjustment of the antenna systems makes it possible to obtain on the phasemeter of the receiving device a reading corresponding, in the main, to the phase difference between oscillations reaching the receiving device by two paths: along the earth’s surface and after reflection from the ionosphere. Assuming that the phase distribution follows elementary laws \(\left(\Phi=\dfrac{\omega r}{v_{\varphi}}\right)\), we obtain that the variations of the observed phase difference are

\[ \Delta\Psi=\frac{\omega_{1}}{c}\left[\sqrt{4(H+h)^{2}+r^{2}}-\sqrt{4H^{2}+r^{2}}\right], \]

where \(H\) is the height of the given reflecting layer,

\(h\) is the variation of this height,

\(r\) is the distance from the transmitting device to the receiving-indicating installation.

Assuming that \(h \ll H\), we obtain:

\[ h=\frac{c\sqrt{4H^{2}+r^{2}}}{4\omega_{1}H}\,\Delta\Psi . \]

INTERFERENCE OF RADIO WAVES

Observing changes in the phase difference of the two received oscillations, one can judge changes in the height of the reflecting layer of the ionosphere, i.e., judge changes in one of the propagation paths, by the change in the interference pattern at the observation point.

Here, when one of the propagation paths changes, we again encounter the possibility of interpreting the resulting changes in the phase difference as the result of a Doppler frequency shift of the “celestial ray” by the amount

\[ \Delta=\frac{d\Psi}{dt} \]

when the reflection conditions change.

A variant of the interferometer with two sources of radiation at commensurable frequencies, proposed by Academicians Mandelstam and Papaleksi and described in DAN in 1940,²⁶ acquired special significance for scientific investigations and a number of practical applications. For this variant of the interferometer it is characteristic that here the observation point is combined with the controlling emitter \((l_1=0)\). The reflecting station, as before, consists of a transmitting device and a receiving device controlling the frequency and phase of the emitted oscillation.

Fig. 12. Moving radio interferometer.

For the case of an arbitrary location of the observation point, as was already indicated earlier,

\[ \Psi=\Phi_1-\frac{\omega_1}{\omega_2}\Phi_2+\delta, \]

\[ \Psi=\omega_1\int_1^I\frac{ds}{v_1} -\omega_3\int_2^I\frac{ds}{v_2} +\varphi_1-(\varphi_1+\Phi_d+\varphi_2)+\delta = \]

\[ =\omega_1\left[\int_1^I\frac{ds}{v_1} -\int_2^I\frac{ds}{v_2}\right]+\Psi_0, \]

where

\[ \Psi_0=\delta-\Phi_d-\varphi_2 \]

Here, as before, \(\varphi_2\) is the phase shift in the apparatus of point 2, \(\Phi_d\) is the phase shift along the path \(1-2\), and \(\delta\) is the difference of the phase shifts in the channels of the recording device.

In the described variant (Fig. 12) the indicator is combined with point 1. Therefore, obviously,

\[ \int_1^I\frac{ds}{v_1}=0 \]

The value of the measured phase difference is

\[ \Psi=\omega_1\int_2^I\frac{ds}{v_2}+\Phi_d+\varphi_2+\delta \]

The transmission of the controlling oscillation from 1 to 2 also occurs by virtue of the propagation of oscillations along the path \(1—2\).

Therefore

\[ \Phi_a=\omega_1\int_1^2 \frac{ds}{v_1} \]

and then, finally,

\[ \Psi=\omega_1\left[\int_1^2 \frac{ds}{v}+\int_2^1 \frac{ds}{v_2}\right]+\Theta, \]

where \(\Theta\) denotes phase shifts associated with the apparatus.

It is evident that displacement of one of the radiators will cause a change in the interference pattern throughout all space and, in particular, in the indicator \(I\), placed at point 1.

In the case where it may be assumed that the propagation of oscillations along the path \(1—2\) proceeds according to the simple laws of propagation of oscillations in homogeneous media (for example, in the case of propagation along the sea surface or along a level earth surface at sufficiently large distances), one may use simple relations connecting the change in the distance between the stations \(\Delta r\) with the change in the observed phase difference \(\Delta\Psi\):

\[ \Delta\Psi=2\frac{\omega}{v}\Delta r. \]

For the case of arbitrary propagation conditions, when point 2 is moved to point 3,

\[ \Delta\Psi=\omega_1\left[\int_2^3 \frac{ds}{v_1(\omega,s)}+\int_3^2 \frac{ds}{v_2(\omega,s)}\right]. \]

This radio interferometer, entirely analogous to the optical Michelson interferometer, in simple cases makes it possible to determine with exceptional accuracy the change in the distance between stations on the scale of \(\lambda\):

\[ \Delta r=\frac{1}{4\pi}\lambda_1^*\cdot\Delta\Psi. \]

Here

\[ \lambda_1^*=2\pi\frac{\bar v^*}{\omega_1^*}, \]

where \(\omega_1\) is the angular frequency of the first oscillation, and \(\bar v^*\) is the phase velocity of the oscillation, in some way averaged over the frequencies \(\omega_1\) and \(\omega_2\) and over the interval \(\Delta r\).

At the same time, numerous measurements made by us\(^{12, 27, 28}\) and by other collaborators of the laboratory of Academician L. Mandelstam

and N. Papaleksi \(^{15,16,17,18,29,30}\), as well as the radiogeodetic laboratory of TsNIIGAiK \(^{31}\), showed that the quantity \(\bar v^{*}\) is very close to \(c\).

In our work cited above \(^{12}\) there are also presented results obtained by us when using the moving radio interferometer described for the study of the phase structure and phase velocity of radio waves in a zone where the structure of the equiphase surfaces undergoes distortions: near the transmitting antenna and near inhomogeneities that cause diffraction distortions.

Fig. 13. Curve of the dependence of the averaged phase velocity on the distance to the transmitting antenna in the region of small distances.

Fig. 13 gives the experimentally obtained curve of the dependence of the averaged phase velocity on distance, for distances small in comparison with the wavelength. Attention should be paid to the fact that only for radio waves is it possible to carry out interference observations at such small distances from the radiators. In optics it is at least difficult to place an indicator at distances of the order of a fraction of a wavelength from the radiation source. Only the fundamentally different order of magnitude of the wavelengths that serve as the scale of linear measurements makes it possible, in the radio-wave region, to carry out such measurements, as well as the measurements with the dispersion interferometer described above.

In Fig. 14 experimental curves are presented for the same quantity, obtained for the case of propagation of radio waves along the earth’s surface. Here one of the curves corresponds to the absence of diffracting inhomogeneities. The second curve was taken in a direction in which there was a slight inhomogeneity causing diffraction distortions of the phase structure (the same beam and dam as in the experiment that gave the curve shown in Fig. 8).

The author, together with A. Prokhorov, in 1940 near Moscow carried out experiments to study the phase structure of the field of radio waves near the edge of a forest that distorted the field structure. A moving interferometer was used, and the measured quantity was still the averaged phase velocity. The wavelengths used were 60 and 90 m.

The results obtained showed that the straight boundary of the forest causes perturbations of the phase velocity near the edge, and, as one moves away from the forest, periodic changes in the magnitude of the phase velocity occur, decreasing in amplitude, with a spatial period \(\lambda/2\).

This indicates the presence of reflection from the forest and distortion of the phase structure of the field. The equal-phase surfaces are respectively compressed in places with lower phase velocity and rarefied

Fig. 14. Curve of the dependence of the averaged phase velocity on distance for distances up to \(40\,r/\lambda\).

where the phase velocity has its greatest value. Let us note, among other things, that in a field of standing waves the phase velocity, determined as

\[ \bar{v}^{*}=\frac{\omega \Delta r}{\Delta \Psi}, \]

at all points except the nodes is equal to infinity. At the nodes \(\bar{v}^{*}=0\). Therefore, the presence of a periodic oscillation of the phase velocity with spatial period \(\lambda/2\) indicates that in the field of propagating oscillations, besides normal traveling waves, there is also a certain percentage of standing waves. The data obtained on the magnitude of the oscillations of the phase velocity and on the rate of attenuation of these oscillations make it possible to estimate the order of magnitude of the coefficient of reflection and attenuation of the reflected waves for each given case under study.

It should also be borne in mind that the quantity measured by means of a moving interferometer,

\[ \bar{v}^{*}=\frac{\omega_{1}}{\dfrac{\Delta \Psi}{\Delta r}}, \]

will depend on the direction of displacement. If the displacements \(\Delta r\) are sufficiently small that \(v_{1}\) and \(v_{2}\) may be regarded as unchanged over the interval \((r,\,r+\Delta r)\), then

\[ \frac{1}{\omega_{1}}\frac{\Delta \Psi}{\Delta r} = \left( \frac{1}{v_{1}}+\frac{1}{v_{2}} \right), \]

and the magnitude of the projection of

\[ \frac{1}{v_1}+\frac{1}{v_2} \]

onto the direction of displacement is determined. The true direction of the phase velocity for each oscillation will be determined by the direction of the vector

\[ \rho=\operatorname{grad}\Phi . \]

Therefore, final conclusions about the phase velocity can be drawn only after a sufficiently complete study of the entire structure of the phase field (equiphasic surfaces).

3. Interference observations with variable radiation frequency

Up to now we have considered such variants of radio interferometers as used one or several coherent radiators operating at certain fixed frequencies. For all these variants one can find analogues among optical interferometers. But the methods of generation, radiation, reception, and indication of radio waves, in contrast to optics, make it possible to carry out measurements with a continuous change of frequency over an arbitrary interval, practically at any rate of change.

In optics the emitted frequencies are determined exclusively by the properties of the substance emitting the light. All possible external actions (thermal, magnetic, electrostatic, and others) can cause only statistical deviations of the emitted discrete frequencies from their mean values, thereby broadening or splitting the spectral lines.

With the aid of methods known in optics it seems possible only to select one or another of the frequencies emitted by the source. One may suppose that, if a light source possessing a continuous spectrum is used and this spectrum is spread out with the aid of a suitable spectral instrument, i.e., by isolating from this continuous spectrum a narrow region, then, by moving the slit along the spectrum, one can obtain a radiation source with variable frequency. We do not know whether experiments have been carried out on the interference of light using sources with variable wavelength. It is possible that they are not feasible.

In radio engineering the possibility of changing the emitted frequency of radio waves is extremely accessible and presents no technical difficulties. Here, rather, the accurate fixing of the limits of the frequency variation—its limiting values—encounters certain difficulties.

It is evident that, when the emitted frequency is changed, the spatial interference pattern created in one way or another will change continuously, and the character of this change at every point of the interference space will be connected in a quite definite way with the law of variation of the frequency.

As has already been pointed out more than once, the retarded phase of the oscillation is equal to

\[ \Phi=\omega \int_a^b \frac{ds}{v}=\omega\tau, \]

where

\[ \tau=\int_a^b \frac{ds}{v} \]

is the propagation time of the given oscillation. When the frequency changes, the phase of the oscillation arriving at the given point \(b\) from \(a\) will change for the following reasons: first (even with \(\tau\) kept constant), owing to a change of scale; second, owing to a possible change in the propagation time, occurring either because of a change in the phase velocity of the given oscillation (dispersion), or because of a change in the propagation path (diffraction).

Let us divide the total change \(\Phi(\omega)\) into two parts: a part depending linearly on \(\omega\), and an additional part connected with \(\omega\) through the propagation time \(\tau\):

\[ \Phi(\omega)=k\omega+\varphi(\omega). \]

At some point of space, to which two oscillations of one and the same frequency arrive, we have the phase difference

\[ \Psi=\Phi_1-\Phi_2=k_1\omega-k_2\omega+\varphi_1(\omega)-\varphi_2(\omega)= \]

\[ =(k_1-k_2)\omega+[\varphi_1(\omega)-\varphi_2(\omega)]. \]

After a change of the frequency by the amount \(\Delta\omega\),

\[ \Delta\Psi=(k_1-k_2)\Delta\omega+[\Delta\varphi_1(\omega)-\Delta\varphi_2(\omega)]. \]

In a homogeneous nondispersive space \(k=\dfrac{r}{v}\), where \(v\) is the phase velocity.

For this case

\[ \Delta\Psi=\frac{\Delta\omega}{v}(r_1-r_2), \]

since, on going beyond the limits of the near zone,

\[ \Delta\varphi_1(\omega)-\Delta\varphi_2(\omega)=0. \]

In a homogeneous dispersive medium the propagation trajectories still do not depend on \(\omega\). However, owing to the dispersive properties of the medium, \(v\) depends on \(\omega\), and if

\[ \Phi(\omega)=\omega\frac{r}{v(\omega)}, \]

then the change in phase of the given oscillation when the frequency changes is equal to

\[ \Delta\Phi=r\Delta[\omega/v(\omega)]. \]

For \(\Delta v \ll v\) one may write:

\[ \Delta \Phi = 2\pi r \Delta \left( \frac{1}{\lambda} \right), \]

\[ \Delta \Psi = \Delta \Phi_1 - \Delta \Phi_2 = 2\pi \cdot \Delta \left( \frac{1}{\lambda} \right)(r_1-r_2). \]

Using the previous expression

\[ \Delta \Psi = (k_1-k_2)\Delta \omega \]

and still assuming \(\Delta \varphi_1(\omega)-\Delta \varphi_2(\omega)=0\), we put \(k=r/u\). Then we obtain

\[ \Delta \Psi = \frac{1}{u}(r_1-r_2)\Delta \omega, \]

while, on the other hand,

\[ \Delta \Psi = 2\pi \Delta \left( \frac{1}{\lambda} \right)\cdot (r_1-r_2). \]

Comparing these expressions, we find that for a homogeneous dispersive space one may use the expression

\[ \Delta \Psi = \frac{1}{u}(r_1-r_2)\Delta \omega, \]

where

\[ u=\frac{\Delta f}{\Delta \left( \dfrac{1}{\lambda} \right)}, \]

i.e., it is a quasi-group velocity in the given frequency interval from \(\omega\) to \(\omega+\Delta\omega\)*).

Let us now consider the case in which \(v\) remains constant along the entire given trajectory, and only the length of the trajectory \(r\) changes as a result of diffraction:

\[ \Delta \Phi = 2\pi \Delta \left( \frac{r}{\lambda} \right). \]

This change of phase may be written in the form

\[ \Delta \Phi = 2\pi \cdot r \cdot \Delta \left( \frac{1}{\lambda_1} \right). \]

But into this expression there will enter the changed wavelength

\[ \lambda_1=\lambda \times (1-x+x^2-\cdots), \]

where \(x=\Delta r/r\), which corresponds, as it were, to a change in the phase velocity along the former trajectory, so that the corrected value of the velocity is

\[ v_1=v(1-x+x^2-\cdots), \]

i.e., from the observer’s point of view, it will appear as though there were additional dispersion.

* This case was analyzed in the work of L. Mandelstam and N. Papaleksi (see, \(^{15}\)).

Using for \(\Delta\Psi\) the previous expression

\[ \Delta\Psi=\frac{1}{u_1}(r_1-r_2)\Delta\omega, \]

we shall have to take

\[ u_1=\frac{\Delta f}{\Delta\left(\dfrac{1}{\lambda_1}\right)}. \]

Therefore, possible interference experiments without special knowledge of the properties of the propagation path cannot make it possible to separate the consequences of dispersion and diffraction, since both of these phenomena ultimately produce one and the same phenomenon of a change in the phase difference, differing from the linear one corresponding to the case of free space.

Fig. 15. Scheme of operation of a radio interferometer with one radiation source.

Let us now consider, taking into account the conclusions drawn, what a frequency change can give when applied to the previously analyzed variants of radio interferometers.

The first of the variants considered was based on the use of one radiation source \(S\). The phase difference observed at \(I\) is

\[ \Psi=\Phi_{SA}-\Phi_{SB}+\varphi_1-\varphi_2, \]

where \(\Phi_{SA}=\omega\tau_{SA}\); \(\Phi_{SB}=\omega\tau_{SB}\); \(\varphi_1\) and \(\varphi_2\) are, respectively, the phase shifts in the channels \(AI\) and \(BI\) (Fig. 15).

In what follows we shall regard the shifts \(\varphi_1\) and \(\varphi_2\) as constant, which in practice can almost always be achieved by using the appropriate adjustment:

\[ \Psi=\omega(\tau_1-\tau_2)+\xi, \]

where \(\xi=\varphi_1-\varphi_2=\mathrm{const}\).

When the frequency \(\omega\) is changed by an amount \(\Delta\omega\),

\[ \Delta\Psi=\Delta\omega(\tau_1-\tau_2)+\omega(\Delta\tau_1-\Delta\tau_2)+\Delta\omega(\Delta\tau_1-\Delta\tau_2), \]

and it should be expected that, if \(\Delta\omega/\omega\) is small, then \(\Delta\tau/\tau\) will also be small. Moreover, if the appearance of the terms \(\Delta\tau_1\) and \(\Delta\tau_2\) is due in origin to dispersion, then \(\Delta\tau=q\cdot\Delta\omega\) (the assumptions concerning the smallness of \(\Delta\tau/\tau\) may be called into question only in the region of anomalous dispersion).

Neglecting the term \(\Delta\omega(\Delta\tau_1-\Delta\tau_2)\), we obtain:

\[ \Delta\Psi=\Delta\omega(\tau_1-\tau_2)+\omega(\Delta\tau_1-\Delta\tau_2). \]

The phasemeter at the point \(I\) makes it possible to measure the quantity \(\Delta\Psi\). The quantities \(\omega\) and \(\Delta\omega\) may be known with the accuracy with which frequency measurement is generally possible.

Therefore, by applying a frequency change in the given interferometer, in the absence of diffractional and dispersive distortions \((\Delta \tau_1 - \Delta \tau_2 = 0)\), one can determine the quantity

\[ \tau_1 - \tau_2 = \frac{\Delta \Psi}{\Delta \omega}. \]

Assuming that dispersion and diffraction are absent, we thereby suppose that

\[ \tau_1 - \tau_2 = \frac{l_1}{v} - \frac{l_2}{v}, \]

i.e.

\[ l_1 - l_2 = \frac{\Delta \Psi}{\Delta \omega}\, v. \]

Thus, under the stated assumptions, one can determine the difference of the distances \(SA\) and \(SB\).

The method proposed by Appleton and Barnett\(^9\) for measuring the height of reflecting layers of the ionosphere is based on this. Applying a prescribed frequency change in the interferometer described above (see Fig. 1), the cited authors, observing the shift of the interference pattern at the receiving point, thereby determine the value \(\Delta \Psi\) and from this value compute \(L_2 - L_1\), whence the height of the reflecting layer can easily be calculated. Of course, in this case the distance between the transmitting and receiving points \(L_1\) must be known.

If the experiment shows a deviation of the dependence of \(\Delta \Psi\) on \(\Delta \omega\) from direct proportionality, then the degree of this deviation may serve as a measure of diffractional and dispersive distortions.

When the instruments of the interferometer are arranged so that \(l_1 = l_2\),

\[ \Delta \Psi = \omega(\Delta \tau_1 - \Delta \tau_2), \]

i.e. direct reading of the phasemeter can make it possible to quantitatively estimate the degree of difference in the distortions of radio waves along the paths \(SA\) and \(SB\), determining the difference in propagation times:

\[ \tau_1 - \tau_2 = \frac{\Delta \Psi}{\Delta \omega} - \eta, \]

where \(\eta = (\omega/\Delta \omega)(\Delta \tau_1 - \Delta \tau_2)\). In this case, if diffraction is absent, the expression for the difference of the times becomes

\[ \tau_1 - \tau_2 = \frac{\Delta \Psi}{\Delta \omega} = \frac{l_1}{u} - \frac{l_2}{u} = \frac{1}{u}(l_1 - l_2), \]

where \(u\) is the quasigroup velocity.

Such an interferometer, with a prescribed frequency change, can serve not only to determine the change in the position of the transmitter (in the sense of its transition from one hyperbola of the family to another), but also to determine the position of the initial hyperboloid. Under the assumption of ideal propagation conditions:

\[ l_1 - l_2 = v\,\frac{\Delta \Psi}{\Delta \omega}. \]

When two coherent emitters are used and the phase difference between two oscillations is correspondingly observed at a common point of reception, applying a prescribed frequency variation leads to the same results as in the preceding case.

Indeed, if two emitters, 1 and 2, located at a distance \(d\) from one another (see Fig. 4), emit coherent oscillations with a definite mutual phase shift, then, as was said above, the phase difference observed at \(I\), determined on the frequency scale \(\omega_1\), will be \(\Psi=\omega_1(\tau_1-\tau_2)+\Psi_0\), where \(\Psi_0=\delta-\vartheta\), \(\delta\) is the phase error introduced by the indicator at \(I\), and \(\vartheta\) is the phase shift of the oscillation emitted by 2 relative to 1.

By varying the frequency, we may encounter two possibilities. Either the phase shift of the oscillation emitted at point 2 relative to 1 \((\vartheta)\) remains constant when the frequency is changed, which can be achieved by appropriate adjustment of the apparatus and by the central location of the common synchronizing generator, or the change in frequency may cause a certain change in \(\vartheta\).

In the first case, changing the frequency by the amount \(\Delta\omega\) causes a change in the phase difference observed at \(I\) by the amount \(\Delta\Psi\), coinciding with that determined for the case of a single emitter:

\[ \Delta\Psi=\Delta\omega(\tau_1-\tau_2)+\omega(\Delta\tau_1-\Delta\tau_2)+\Delta\omega(\Delta\tau_1-\Delta\tau_2). \]

(Here it is assumed that the phase errors introduced by the apparatus do not change when the frequency is changed.)

In this version of the interferometer, just as in the one discussed above, it appears possible, for ideal laws of propagation, to determine the quantity \(l_1-l_2\), and under real conditions—the quantity

\[ \int_{1}^{I}\frac{ds}{v^{*}(\omega_1)}-\int_{2}^{I}\frac{ds}{v^{*}(\omega_2)}, \]

where \(v^{*}(\omega_1)\) and \(v^{*}(\omega_2)\) are certain mean values of the phase velocities of propagation of the oscillations in the frequency intervals \((\omega_1,\omega_1+\Delta\omega_1)\) and \((\omega_2,\omega_2+\Delta\omega_2)\).

In the second case it is necessary to know the law according to which \(\vartheta\) varies with frequency. This question is most simply resolved for the case when one of the emitters, for example 1, is the driving one, and the phase of the oscillations emitted by the “reflecting” point 2 is determined by the phase of the oscillations that have arrived from 1 along the path \(d\), and by the instrumental shifts at point 2:

\[ \vartheta=\Phi_d+\varphi_2. \]

Then, taking all instrumental shifts to be constant, for the change in phase difference observed at \(I\) when the frequency is changed from \(\omega_1\) to \(\omega_1+\Delta\omega_1\) and from \(\omega_2\) to \(\omega_2+\Delta\omega_2\), we obtain:

\[ \Delta\Psi=\Delta\omega(\tau_1-\tau_2-\tau_d)+\omega(\Delta\tau_1-\Delta\tau_2-\Delta\tau_d)+\Delta\omega(\Delta\tau_1-\Delta\tau_2-\Delta\tau_d). \]

where

\[ \Delta \omega_1=\frac{\omega}{\omega_2}\Delta \omega_2=\Delta \omega . \]

For the simplest case, when \(\tau=l/v\) and the phase velocity does not depend on \(\omega\) (in the absence of dispersion and diffraction),

\[ \Delta \Psi=\frac{\Delta \omega}{v}\,[l_1-(l_2+d)], \]

whence

\[ l_1-(l_2+d)=\frac{\Delta \Psi}{\Delta \omega}\,v, \]

and consequently,

\[ l_1-l_2=\frac{\Delta \Psi}{\Delta \omega}+d. \]

Unlike the first case, here the quantity \(l_1-(l_2+d)\), and not \(l_1-l_2\), is determined.

If \(d\) is known, then this in fact gives the same thing. Knowledge of \(d\) is necessary so that, starting from the obtained value of \(\Delta \Psi\), it would be possible to calculate the surface (a hyperboloid, or a hyperbola in the plane case) on which the receiving-indicating device is located. Let us note that this case of an interferometer with two sources of radiation, unlike an interferometer with a single radiator, makes it possible to determine hyperboloids passing through the observation point, and not through the source of radiation. Correspondingly, the possible fields of application of these variants will also be different [for example, a system with two radiators makes it possible, with the aid of a light receiving-recording device, to solve a number of geodetic problems (“radiosonde”)].

In both cases of this variant of the interferometer, a change in the frequency of the radiation is accompanied by a change in the observed phase difference by \(l\). The same also applies to the case of a single radiator.

Here the following visual interpretation of the observed phenomenon is also possible. A change in the phase difference with velocity \(\zeta=d\Psi/dt\) is equivalent to the fact that oscillations are fed to the phase meter whose angular frequencies, on the scale of measurement, differ by the quantity \(\zeta\).

This is the quantity by which the frequency of radiation has managed to change over the difference in propagation times \((\tau_1-\tau_2)\) along different paths from the radiators to the indicator, or from a single radiator to the receiving points.

It is obvious that since the frequency of the emitted oscillations \(\omega=d\Theta/dt\), where \(\Theta\) is the total phase of the emitted oscillation, then

\[ \zeta=\int_{t+\tau_2}^{t+\tau_1} d\omega=\omega_{t+\tau_1}-\omega_{t+\tau_2} =\frac{d}{dt}\,[\Theta(t+\tau_1)-\Theta(t+\tau_2)]. \]

However, if such an interpretation for phase measurements under a sufficiently slow change of frequency is only of fundamental interest, then, when using indicators based on measuring the intensity of the resultant oscillation, it may also have practical significance for rapid frequency changes.

Indeed, when oscillations of the same frequency—whose value varies, say, according to a linear law—are emitted by two radiators, at the point of observation there will be a periodic change in the intensity of the resultant oscillation: beats will be observed, whose frequency is determined as follows.

For a linear change of frequency

\[ \omega(t)=\omega_0+\alpha t. \]

Oscillations will arrive at the point of observation along two different paths with time shifts \(\tau_1\) and \(\tau_2\), respectively:

\[ \omega(t+\tau_1)=\omega_0+\alpha(t+\tau_1);\quad \omega(t+\tau_2)=\omega_0+\alpha(t+\tau_2), \]

\[ \Theta_1=\omega_0 t+\frac{\alpha}{2}(t+\tau_1)^2,\quad \Theta_2=\omega_0 t+\frac{\alpha}{2}(t+\tau_2)^2. \]

Accordingly, the difference term singled out in the square-law indicator has the instantaneous phase

\[ \Theta_1-\Theta_2=\alpha(\tau_1-\tau_2)t+\frac{\alpha}{2}(\tau_1^2-\tau_2^2), \]

whence it follows that the beat frequency will be

\[ \xi=\alpha(\tau_1-\tau_2),\quad \text{i.e.}\quad \xi=\omega(t+\tau_1)-\omega(t+\tau_2). \]

If now, instead of a phasemeter, an appropriate frequency meter is used at the point of observation, and if the process of linear frequency change is made to repeat periodically with a sufficiently high rate of this change, we again arrive at the possibility of determining, from the observed beat frequency, the quantity

\[ \tau_1-\tau_2. \]

This principle underlies the frequency-modulated radio beacon proposed in 1940,\(^{32}\) and the radio altimeter proposed considerably earlier (1936).\(^{33-35}\) In the radio beacon proposed by Dingle, we have radiators whose frequency is modulated according to a sawtooth law, so that for the greater part of the time the law of frequency variation of both radiators, controlled from a single point, may be regarded as linear.

It is obvious that to each point in space, symmetric with respect to the plane passing through the midpoint of the straight line joining the points of emission (the base \(d\)) and perpendicular to it, there will correspond a definite beat frequency. The value of this frequency will determine the position of the point of observation on one or another hyperboloid of a family having the points of emission as its foci.

Practically, in this way the matter is reduced to determining the difference of the propagation times of the oscillations from both radiators to the point of observation, or, with a quite sufficient degree of accuracy, one may say that the difference of the distances between the indicated points is measured. It is obvious that, in the practical realization of this idea, it is necessary to take into account the relations between the required rate of change of frequency, the frequency of the beats obtained, and the period of repetition of the frequency changes (on the basis of the prescribed accuracies and the dimensions of the base).

The same idea underlies an altimeter with frequency modulation of ultrashort waves. Here the indicator is combined with the radiator of oscillations modulated in frequency in an analogous manner and records the beats between the emitted oscillations and the oscillations returning after reflection from the earth.

In this case \(\tau_2=0\) and \(\tau_1=\zeta/a\), while \(\tau_1=\tau'_1+\tau''_1=2\tau_h\), where \(\tau_h\) is the time required for the propagation of the oscillations from the airplane to the earth, \(\zeta\) is, as before, the beat frequency, and \(a\) is the rate of change of frequency.

The conditions for measuring the flight altitudes of an airplane require, for the realization of such an altimeter, the use of decimeter waves, since only in this range is it possible to obtain technically convenient instruments and to observe all the necessary relations between the rate of change of frequency, the period of this modulation, the beat frequency, and the height being measured.

When we have two sources of coherent radiation, of which one is the master and the other the subordinate—“reflecting”—one, then, by combining the point of observation with the master radiator, in observations with a change of frequency we obtain a radio range finder, first described in the work of L. Mandelstam and N. Papaleksi in 1937. \(^{15}\)*) In this instrument the observed phase difference on the scale of frequency \(\omega_1\)

\[ \Psi=-\omega_1\tau_2+\Psi_0=-\frac{\omega_1}{\omega_2}\Phi_2-\Phi_d(\omega_1)+\delta-\varphi_2, \]

where, as before (see Fig. 12), \(\Phi_2\) and \(\tau_2\) are the phase delay and the propagation time on the path from 2 to the indicator \(I\), \(\Phi_d\) is the phase shift on the path from 1 to 2, \(\delta\) is the phase distortion in the receiving-measuring device, and \(\varphi_2\) is the phase shift in the apparatus of point 2.

Since \(l_2=d\), then, disregarding the sign of the phase, we obtain:

\[ \Psi=\Phi_d(\omega_1)+\frac{\omega_1}{\omega_2}\Phi_d(\omega_2)+\varphi_2+\delta. \]

*) Let us note that by 1937 a large number of experiments and measurements had already been carried out with this instrument.

When the frequency \(\omega_1\) is changed by \(\Delta \omega\) and \(\omega_2\) by \((\omega_2/\omega_1)\Delta \omega\), we have:

\[ \Delta \Psi=\Phi_d(\omega_1+\Delta\omega)-\Phi_d(\omega_1)+\frac{\omega_1}{\omega_2}\Phi_d\left(\omega_2+\frac{\omega_2}{\omega_1}\Delta\omega\right)-\frac{\omega_1}{\omega_2}\Phi_d(\omega_2), \]

or (still assuming that \(l=\mathrm{const};\ S=\mathrm{const}\))

\[ \Delta \Psi=\Delta\omega(\tau_{\omega_1}+\tau_{\omega_2})+\omega_1(\Delta\tau_{\omega_1}+\Delta\tau_{\omega_2})+\Delta\omega(\Delta\tau_{\omega_1}+\Delta\tau_{\omega_2}). \]

For the simplest case of absence of dispersion and diffraction,

\[ \Phi_d=\frac{\omega d}{v}\quad\text{and}\quad \Delta\Psi=\Delta\omega\frac{2d}{v}, \]

whence

\[ d=\frac{v}{2}\cdot\frac{\Delta\Psi}{\Delta\omega}. \]

Numerous measurements carried out in recent years have shown*) that, for a number of cases of practical interest, one may use such simplified representations, making an error not exceeding \(10^{-4}\), which, generally speaking, lies within the limits of instrumental errors. This confirms that the role of diffraction and dispersion at sufficiently large distances is very small and, consequently, that distortions of the interference pattern have a local character, which is in good agreement with other interference observations mentioned in the works cited above.

If points \(1\) and \(2\), between which radiotelemetric measurements are made, are not fixed, so that the distance \(d\) between them changes, then carrying out radiotelemetric measurements encounters the following difficulty.

The interference pattern and the phase difference observed at \(l\) will change not only because of the change in frequency, but also because of the change in \(d\), i.e., because of the change in the retardation time of one oscillation relative to the other.

This difficulty, in the case where during the measurement the change in \(d\) may be regarded as uniform, can easily be circumvented by the following device. The change of frequency is performed twice: once in the direction of increase, and the other time in the direction of decrease. If the time spent on changing the frequency is the same in both directions, then the quantity

\[ \Delta'\Psi=\frac{\Delta_1\Psi-\Delta_2\Psi}{2}, \]

where \(\Delta_1\Psi\) and \(\Delta_2\Psi\) are, respectively, the measured changes in the phase difference in one case when the frequency is increased and in the other case when the frequency is decreased**), makes it possible to determine

\[ d_{\mathrm{av}}=\frac{v}{2}\cdot\frac{\Delta'\Psi}{\Delta\omega} \]

—the value of \(d\) at the mean instant of time of the given interval. \(\Delta_1\Psi+\Delta_2\Psi\)

*) The principal results of these extensive works are set forth in \(^{15,16,17,18,24,30,31}\).

**) It should be borne in mind that \(\Delta\Psi=\Psi(\omega+\Delta\omega)-\Psi(\omega)\) will have different signs for an increase and a decrease of frequency.

characterizes the magnitude of the change in \(d\) during the measurement time:

\[ d_2-d_1=\frac{v}{2}\frac{\Delta_1\Psi+\Delta_2\Psi}{\Delta\omega}. \]

This method of measuring distance under continuous change of the quantity being measured was proposed and first used by the author in 1936, as applied to measurements of the distance between the shore and a moving ship during expeditionary work in the Kara Gates.

An analysis of this version of the use of a radio rangefinder was given by E. Shchegolev\({}^{36}\) in 1939, as applied to the simplest (dispersionless and diffractionless) conditions of propagation of radio waves, which are especially close to real conditions precisely in measurements along the sea surface.

4. Interference of modulated oscillations

In all the variants of radio interferometers discussed above, the emission and reception of unmodulated oscillations were considered, and it was assumed that the receiving device is an instrument that makes it possible to receive and amplify the oscillations while preserving the form that they had in propagating from the point of emission to the point of reception. The sole exception is the phase-measuring device based on frequency conversion by means of heterodyning.

However, radio engineering has at its disposal the possibility of using high-frequency radiation as a carrier of certain other signals, which, during transmission, modulate the primary radiation in a definite manner and, upon reception, are again separated out as a result of the corresponding demodulation, reproducing at the receiving point the form of the primary modulating signal\({}^{*}\).

It is obvious that, using harmonic oscillations for modulation, we may in the general case, abstracting from the modulation and demodulation system, regard the process as though there were a direct propagation of the harmonic modulating oscillations from the transmitting point to the receiving point. Then, with respect to these oscillations, separated in the receiving device (or in the receiving devices), we may apply all those considerations that were set forth earlier regarding methods of producing interference. And it is perfectly clear that any of the types of radio interferometers considered above can be used for interference observations if the direct emission of coherent oscillations is replaced by the emission of oscillations of arbitrary frequencies, modulated by coherent oscillations, and observations of inter-

\({}^{*}\) In optics we likewise have the possibility of using modulation of light radiation to transmit a modulating signal.

ference at the modulation frequency after extracting the modulating oscillations at the points of observation.

In this case the requirements relating to the coherence and monochromaticity of the oscillations producing the interference will apply only to the modulating process. In the space in which the modulated oscillations propagate, there will be no stationary distribution of the amplitudes or phases or form of the resultant process, since, in the absence of definite requirements on the frequencies of the modulated oscillations and on the frequency relation between the modulating and modulated oscillations, the resultant process will be nonperiodic. In such systems the entire process of interference takes place only in the receiving devices, and it should not be confused with the case in which there is present in space the interference of some stationary distribution of oscillatory states.

Fig. 16. Simplest variant of a radio interferometer using modulated oscillations.

Let us consider the simplest variant of such a radio interferometer, shown in Fig. 16.

At points 1 and 2 oscillations \(\omega_1\) and \(\omega_2\), modulated by an oscillation of frequency \(\Omega\), are emitted.

At the receiving point both oscillations with carrier frequencies \(\omega_1\) and \(\omega_2\) are received separately. The oscillations of frequency \(\Omega\), extracted after demodulation, are fed to the indicator \(I\), which can register, on the scale of frequency \(\Omega\), the path difference formed by the difference in the propagation times of the modulated oscillations along the paths \(l_1\) and \(l_2\). In this case the indication may be carried out either by the amplitude or by the phase method, depending on the type of indicator to which the oscillations obtained after demodulation of the modulated signals received at points 3 and 4 are supplied.

Since the reading of the indicator is determined by the time shift of one of the received oscillations with respect to the other, it is clear that, as before, the velocity of transfer of the modulating process will play an essential role, alongside the geometrical difference of the propagation paths of both oscillations.

This velocity need not coincide with the propagation velocity of the modulated oscillation, and it is connected with the dispersive properties of the medium in which propagation takes place. Its magnitude will be more or less close to the group velocity of propagation in the given medium for the frequency of the carrier oscillations. The narrower the part of the spectrum occupied by the modulated oscillation,

the more exactly the velocity of transfer of the modulating process will coincide with the group velocity corresponding, for the given medium, to the frequency of the carrier oscillation.

This qualification should be borne in mind when using modulated oscillations in any of the above-considered types of radio interferometers. Accordingly, when modulated oscillations are used in any of the above-considered types of radio interferometers, the apparatus must include modulating devices at the transmitting points and demodulating devices at the receiving points.

Let us note once again that, in radio interferometers using modulated oscillations, the study of phase relations on the scale of the relatively low modulation frequency \(\Omega\) leads to a sharp decrease in accuracy, since \(\Omega \ll \omega\), the frequency of the carrier oscillation, on whose scale the phase measurements were made in the variants described earlier.

One of the first variants of a radio interferometer using modulated oscillations was described in 1934 in the work of Deko and Gally \(^{37}\)*).

These authors used a radio interferometer operating with amplitude-modulated short waves for the experimental determination of the path constant or, more precisely, of the propagation time of short waves for various communication lines (Paris—Algiers and Paris—Strasbourg). In these experiments an audio frequency (1000 cps) modulated the primary emission at point \(1\). This emission reached the second point—\(2\), was received there, and after demodulation the same audio frequency (1000 cps) modulated the return emission from point \(2\), produced at a frequency different from that of the emission from point \(1\). The audio-frequency oscillations received at point \(1\) and separated after demodulation were compared in phase with the primary—modulating—oscillation.

It is obvious that the observed phase difference is

\[ \Psi = \Omega(\tau_1+\tau_2), \]

where \(\Omega = 2\pi F\) is the frequency of the modulating oscillation, and \(\tau_1\) and \(\tau_2\) are respectively the times of transfer of the modulation from point \(1\) to point \(2\) and back. In this case, since the carrier frequencies \(\omega_1\) and \(\omega_2\) are different, it is natural to expect that \(\tau_1 \ne \tau_2\).

Deko and Gally observed the constancy of \(\Psi\), i.e. the constancy of \(\tau_1+\tau_2\), associated with the propagation conditions of the short waves used on the selected path, taking into account all factors affecting the path and the velocity of propagation of short waves—first of all, the ionosphere.

*) To this same group of instruments belongs the device proposed by Kulikovskii and Shinkovsky for measuring distances—English patent No. 302609 of 15/XII 1927.

The observed changes in \(\Psi\) confirmed the data on the nonconstancy of the conditions for reflection of short waves from the ionosphere at night and at twilight, with an insignificant role of this factor under daytime conditions.

If we assume \(\tau = r/v\), which can be done with great accuracy in the case of propagation in free space, for example in communication between airplanes, then

\[ \Psi=\frac{\Omega}{v}\,2r=\frac{4\pi F}{v}\,r . \]

Then for each \(r\) such a modulation frequency \(F\) can be chosen that \(\Psi\) assumes one and the same predetermined value \(\Psi_0\). This is the basis of the method of measuring distances proposed in 1934 by Fire\(^{38}\). In this method, which is typically radio-interferometric, two stations are used, analogous in purpose to the Decca and Gee installations. By selecting the modulation frequency, a definite phase difference is established between the oscillations modulating the first transmitter and the oscillations after demodulation of the signal received at point 1 after emission at point 2.

It is indisputable that this method cannot provide high accuracy in measuring distances; however, since here the whole measurement is reduced to measuring frequency while using an indicator of the constancy of the phase shift \(\Psi_0\), which can be made very sensitive, this method may receive a very convenient technical implementation.

Let us determine the accuracy of the method:

\[ |\Delta r|=\frac{v}{720}\,\frac{\Delta\Psi_0 F+\Delta F\cdot\Psi_c}{F^2}. \]

Setting \(F=10^3\), \(\Psi_0=\pi\), which corresponds to \(r=75\) km, we obtain:

\[ |\Delta r|_{\text{m}}\simeq 416\Delta\Psi+75\Delta F . \]

If we assume that \(\Delta F=2\) c/s, \(\Delta\Psi=0^\circ.5\), then

\[ |\Delta r|=358\ \text{m}. \]

Let us note that the method described is completely analogous to the well-known Fizeau and Foucault method used to measure the speed of light. There, too, a modulation frequency is selected, produced by means of a toothed wheel, corresponding to a definite phase shift of the modulating process \((\pi)\), and the unknown quantity is only the propagation speed \(c\), determined for a known propagation path.

It has already been pointed out above that the basic fundamental difference between the interference of modulated and unmodulated oscillations consists in the fact that the position of the resulting interfer-

... of the interference pattern in the first case is determined by the group velocity—the group delay time, and in the second by the phase velocity of the interfering oscillations.

Interferometers that use the measurement of phase relations of unmodulated oscillations make it possible to determine the group velocity only when the frequency of the radiation is changed. Only under conditions of radio-wave propagation without dispersion does this distinction lose its significance, since in this case \(v=u\), and both types of interferometers will lead to the same results.

It is possible, however, to create a radio interferometer combining both of the features mentioned.

In 1941 the author, together with A. M. Prokhorov, proposed and developed a version of a radio interferometer intended for ionospheric observations during the total solar eclipse of 1941. In this interferometer, simultaneous observation was carried out of the positions of interference patterns associated with both phase and group velocities. This was achieved by the fact that, in the given version of the interferometer—based on the radio interferometer described above (p. 394)—an additional shallow amplitude modulation of both emitted oscillations is introduced. In this case the modulation frequencies are in an integral ratio, and a constant phase shift is maintained between the two modulating oscillations. This modulation only slightly blurs the Lissajous figure on the oscillograph, by means of which the change in the phase difference of the high-frequency oscillations is indicated, only slightly reducing the accuracy of the reading. Detection of both received oscillations makes it possible to isolate the modulation frequencies. After suitable amplification these oscillations are fed to a separate phase meter.

Fig. 17. Operating diagram of a radio interferometer with modulated oscillations for observing variations in the height of the reflecting layer of the ionosphere.

On this phase meter the reading of the phase difference, made on the scale of one of the modulation frequencies, depends on the difference of the group propagation times of the two signals that have traveled along different paths.

The general scheme of such a device is shown in Fig. 17. In this figure, \(1\) and \(2\) are two combined transmitters, operating at carrier frequencies \(3\omega\) and \(2\omega\), respectively modulated

with amplitude frequencies \(2\Omega\) and \(\Omega\). The control indicators \(K_1\) and \(K_2\) make it possible to monitor the constancy of the initial phase shifts at the high and low frequencies. 3 and 4 are the corresponding receiving devices, receiving signals emitted by the transmitting devices and, in one case, having traversed the path from the transmitter to the receiver with reflection from the ionosphere, and in the other, the direct path along the earth’s surface. \(I_1\) is a phasemeter measuring the phase difference at the high frequency; \(I_2\) is a phasemeter measuring the phase difference between demodulated oscillations already at the low frequency.

Comparison of continuous observations of the readings of both phasemeters should provide information about changes not only in the path length of the ray reflected from the ionosphere, i.e. the height of the reflecting layer of the ionosphere, but also about changes in the dispersion properties of the ionosphere, which alter the value of the group velocity of radio waves upon their reflection from the ionosphere.

In addition, the reading of the phasemeter at the modulating frequency, owing to the much smaller scale of such measurements, may be useful for observing fairly rapid changes in the height of the reflecting layer of the ionosphere, when the high sensitivity of phase measurements at radio frequencies may make it impossible to indicate such changes by means of the principal phasemeter.

According to the formula already mentioned, variations in the height of the reflecting layer of the ionosphere will be related to variations in the phase difference at the high frequency by the relation

\[ h=\frac{c\sqrt{4H^2+r^2}}{4\omega_1 H}\cdot \Delta\Psi . \]

For \(H=100\) km (the \(E\) layer), taking \(r=20\) km; \(\omega_1=2\pi\cdot10^6\), we obtain

\[ \Delta\Psi=13.3\pi h\ \text{km}, \]

where \(h\), as before, denotes variations in the height of reflection. Thus, for a change in the height of reflection by \(1\) km, we shall have a change in phase difference of \(2400^\circ\).

If the observations are carried out during periods of strong ionospheric disturbances, such a change in the height of reflection may occur in a time measured by a few seconds or even fractions of a second. It is obvious that the corresponding rate of change of the phase difference will be too great for continuous recording of the phasemeter readings to be possible.

For modulating oscillations with

\[ F=\frac{\Omega}{2\pi}=10^4 \]

the same change in the optical path length will correspond to a change in the observed phase difference

\[ \Delta\Psi^*=\frac{\Omega}{\omega}\Delta\Psi=0.13\pi=24^\circ . \]

This quantity is sufficiently small not to disturb the taking of continuous readings on the second phase meter even when the reflection height changes very rapidly. Thus, the reading on the second phase meter should make it possible to carry out a rough estimate of changes in the reflection height, while the reading on the first phase meter makes it possible to refine the observed changes and to follow small variations in the reflection height.

Undoubtedly, changes in the group velocity or, more precisely, in the group propagation time of radio waves during their propagation in the reflecting layer itself can cause only insignificant variations in the observed phase difference at the low frequency, since it is to be expected that the path traversed by radio waves in the ionized medium will not exceed a few percent of the entire propagation path of the “celestial” ray. Unfortunately, tests of such a device under actual conditions could not be carried out because of the outbreak of the war. However, the laboratory experiments that were conducted gave reason to count on the effectiveness of this version of interference observations of the state of the ionosphere.

Even if one disregards the dispersion properties of the path of propagation of modulated oscillations, in this case too the use of modulation makes it possible to diversify considerably the methods of obtaining interference. Thus, with amplitude modulation by a harmonic oscillation, when the spectrum of the emitted frequencies has only three lines, \(\omega+\Omega\), \(\omega\), \(\omega-\Omega\), it is possible to carry out a separate study of these oscillations for one of the radiators of the interferometer. And if the points of emission of the carrier oscillation and of the side frequencies are spatially separated, then the phase of the low-frequency oscillation of frequency \(\Omega\), isolated after detection of the carrier and side frequencies, will be determined not only by the distance of the receiving point but also by the orientation of the observation point relative to the emission points.

These changes will be produced on the scale of the high frequency and will be additively added to the phase of the low-frequency oscillation.

In this way one can create a very diverse distribution of the phase of the given low-frequency oscillation and realize the most varied types of interference patterns. Correspondingly, the types and purpose of radio interferometers using modulated oscillations can be exceptionally diverse.

IV. INTERFERENCE MEASUREMENTS OF THE VELOCITY OF PROPAGATION OF RADIO WAVES ALONG THE EARTH’S SURFACE

In a number of cases in the practical use of radio waves we deal with radiation sources and observation (receiving) points located near the earth’s surface.

The use of radio waves with wavelengths from tens to thousands of meters has made the mathematical idealization entirely legitimate

of this problem, which reduces to the assumption that the radiator and the observation points are located directly on the interface between air and earth.

The second assumption, adopted in the mathematical treatment of the given problem, consists in taking the interface to be a plane and the earth itself to be homogeneous. This assumption imposes definite limitations on those distances to which the conclusions obtained on the basis of the analysis of the plane case are applicable, and makes it necessary to introduce into consideration certain averaged constants of the lower medium—the earth. However, even under the conditions of such a far-reaching idealization, the problem of the propagation of radio waves along the earth’s surface proves sufficiently difficult, not to mention cases of complex relief or of abrupt variation in the electrical constants of the earth.

One can undoubtedly proceed to the solution of such more complicated problems only when one has a satisfactory solution of the question of the propagation of radio waves along a plane homogeneous surface of the earth.

In the course of developing this question, primary attention was directed first of all to circumstances connected with the law of distribution of the amplitude of oscillations propagating along the interface from a radiator situated on this surface. This was explained by the demands of radio-communication engineering, for which amplitude relations were decisive. However, the development of a number of special applications of radio waves (radio direction finding, radio beacons, radio range finders, etc.) required a more complete analysis also of questions connected with the velocity of propagation of radio waves. On the other hand, these same technical means placed in the hands of investigators new methods for the experimental verification of various theoretical constructions.

Let us note that the measurement of the distance between two points on the earth’s surface reduces to determining the time of propagation of a signal between these two points, and consequently the calculation of the distance between these two points from the data obtained requires knowledge of the velocity of propagation of radio waves under the conditions of measurement. Knowledge of the laws determining the magnitude of the velocity of radio waves along the earth’s surface may make it possible to determine the paths of propagation of signals, to determine the true track and its deviations from the straight line connecting the point of radiation with the point of observation, and hence also the errors that may arise in direction finding and in the use of radio beacons (apart from effects connected with the action of the ionosphere). These brief remarks emphasize the importance of establishing the actual laws of propagation of radio waves along the earth’s surface, and among them—the laws determining the velocity of radio waves. In this connection, a plane homogeneous earth and a radiator situated at the boundary between earth and air are the principal

the initial case, in which the picture of the phenomenon under study can first of all be established.

The velocity of propagation of radio waves is unambiguously determined from those mathematical expressions which describe the processes of propagation under the conditions being studied, and therefore the experimental study of the velocity of radio waves is one of the ways of testing those concepts which are made the basis of one theory or another.

In a homogeneous unbounded space, plane waves propagate with the phase velocity:

\[ v=\frac{c}{\sqrt{\varepsilon}\sqrt{\dfrac{1}{2}+\dfrac{1}{2}\sqrt{1+\left(\dfrac{2\sigma}{\varepsilon f}\right)^2}}}. \]

This phase velocity \(v\) is a constant quantity, depending on the constants of the medium (\(\varepsilon\) and \(\sigma\)) and on the frequency \(f\), and not connected with the coordinates of the point of observation.

At the same time there is dispersion

\[ D=\frac{\partial v}{\partial f}\ne 0. \]

And only for those cases when \(2\sigma/\varepsilon f \ll 1\), i.e. when the conduction currents are much smaller than the displacement currents, can the given medium for the corresponding frequencies \(f\) be regarded as a dielectric and \(v=c/\sqrt{\varepsilon}=\mathrm{const}\) for all frequencies greater than that starting from which the quantity \(2\sigma/\varepsilon f\) may be neglected.

In considering an elementary radiator in a homogeneous medium we obtain different laws of variation of the phase of the wave at different distances from the radiator. At sufficiently large distances from the radiator—in the wave zone—the spherical wave propagating from it has a phase velocity coinciding with the expression given above for a plane wave.

As we approach the radiator we shall have an increase of the phase velocity, and in the immediate vicinity of the radiator the phase velocity will be infinitely large. In what follows we shall agree to take as the phase of the wave the phase of that component of the electric field which is situated in the plane passing through the axis of the dipole and is directed along the tangent to the wave.

Let us write the expression for the corresponding oscillation in the form

\[ E=A(r)\cdot \sin[\omega t-\Phi(r)] \]

and introduce, as before, two quantities for consideration:

\[ \overline{v}=\frac{\omega r}{\Phi(r)} \quad \text{and} \quad v=\frac{\omega}{\partial \Phi/\partial r}; \]

\(\overline{v}\)—the mean velocity—defines a quantity which makes it possible, using the simple relation \(\Phi(r)=\omega r/\overline{v}\), to find the total

Fig. 18. Theoretical curves of the additional phase \((\varphi^*)\) for radiation in a homogeneous medium for various constants of the medium.

change of phase along the entire path from \(r=0\) to \(r\); \(v\) is the differential phase velocity at the point corresponding to the given \(r\).

It is self-evident that the time delay of the oscillation along the path \(r\) will be

\[ \tau=\frac{r}{\overline{v}}=\int_0^r \frac{dr}{v} =\frac{1}{\omega}\int_0^r \frac{\partial\Phi}{\partial r}\,dr =\frac{1}{\omega}\Phi(r). \]

For waves radiated by a dipole in a homogeneous space,

\[ E_v=A(r)\cos[\omega t-ar-\varphi^*] =A(r)\cos[\omega t-\Phi(r)], \]

where

\[ \Phi(r)=ar+\varphi^*, \]

\[ a=\frac{\omega}{c}\sqrt{\varepsilon}\, \sqrt{-\frac{1}{2}+\frac{1}{2}\sqrt{1+\left(\frac{2\sigma}{\varepsilon f}\right)^2}}, \]

\[ b=\frac{\omega}{c}\sqrt{\varepsilon}\, \sqrt{-\frac{1}{2}+\frac{1}{2}\sqrt{1+\left(\frac{2\sigma}{\varepsilon f}\right)^2}}, \]

\[ \tg\varphi^* = \frac{\dfrac{a}{r}+2ab} {a^2-b^2-\dfrac{b}{r}-\dfrac{1}{r^2}}. \]

Graphs of the additional phase \(\varphi^*(r)\) for various values of \(\varepsilon\) and \(\sigma\) at \(f=10^6\) are shown in Fig. 18. Here

\[ \varepsilon=4,\ \sigma=0 \quad \text{and} \quad \sigma=5\cdot10^5 \quad \text{and} \quad \varepsilon=1,\ \sigma=0. \]

The latter case corresponds to the radiation of a dipole in free space.

Hence, for the differential phase velocity,

\[ v=\frac{\omega}{\dfrac{\partial \Phi}{\partial r}} =\frac{\omega}{a+\dfrac{\partial \varphi^*}{\partial r}} \]

for

\[ \varepsilon=1,\quad \sigma=0 \]

and

\[ \varepsilon=4,\quad \sigma=0 \]

we obtain the curves shown in Fig. 19.

Let us note that, defining the mean phase velocity by the expression

\[ \bar v=\frac{\omega r}{\Phi(r)}, \]

we, for \(\Phi(r)_{r\to 0}\to \Phi_0\), at small \(r\), determine a quantity that does not have the physical meaning of a phase-propagation velocity.

If, by analogy with the expression \(\Theta=\omega t-kr=\omega(t-r/v_0)\), we represent the general expression \(\Theta=\omega t-\Phi(r)\) in the form

\[ \Theta=\omega\left(t-\frac{r}{\bar v}\right), \]

then

\[ \bar v=\frac{\omega r}{\Phi(r)} \]

can be identified with the mean velocity of phase propagation only when

\[ \lim_{r\to 0}|\Phi(r)|=\lim_{r\to 0}\left|\frac{r\omega}{\bar v}\right|=0. \]

Fig. 19. Curves for the differential phase velocity near a radiator situated in a homogeneous medium.

This will be all the more exact the smaller \(\Phi_0\) is in comparison with \(\Phi(r)\), i.e., the farther we move away from the radiator. Otherwise \(\bar v\) does not have such a clear physical meaning as the quantity \(v_0\).

The quantity of the differential phase velocity,

\[ v=\frac{\omega}{\dfrac{\partial \Phi}{\partial r}}, \]

has a quite definite physical meaning, giving the velocity of phase propagation at the given point \(r\).

Let us note once again that the primary quantity in all these constructions remains the expression for the total \((\Theta)\) and the retarded \([\Phi(r)]\) phase of the oscillation; from consideration of these one can obtain all the [[unclear: word continues on next page]].

necessary information on the propagation of oscillations and, in particular, the time lag of the oscillation

\[ \tau=\frac{1}{\omega}\Phi(r). \]

As follows from the expressions and curves presented, even near the radiator in a homogeneous space there is a complicated law of phase distribution, which determines the nonconstancy of the value of the phase velocity. Let us note, moreover, that the phase velocity (we shall henceforth speak of the differential phase velocity), for all times beginning with \(r>\lambda/4\), has a value greater than its limiting value as \(r\to\infty\), determined solely by the constants of the medium in which the propagation takes place.

A theoretical analysis of the problem of radio-wave propagation along an interface requires the successive solution of the equation

\[ \nabla^2\mathbf{\Pi}+k^2\mathbf{\Pi}=0, \]

where \(\mathbf{\Pi}\) is the “amplitude” of the Hertz vector, from which any components of the electromagnetic field can be obtained, with allowance for the corresponding boundary conditions at the interface surface, at infinity, and at the point of radiation.

The first solution of this problem was the solution given by Zenneck in 1907[^39].

According to Zenneck, the solution may be plane inhomogeneous waves propagating along the air–earth interface with a constant phase velocity (determined by the constants of the earth) and undergoing absorption along their path depending on the conductivity of the soil.

This entirely rigorous solution of the equations of the electromagnetic field in no way provides for the presence of a radiator and in fact represents one of the possible forms of waves that can propagate along the interface surface of two media.

The simplicity and visual clarity of this solution, as well as the circumstance that, in his first work (1909) on waves excited by a vertical vibrator located on the surface of the earth, Sommerfeld[^11] obtained, as one of the components of his solution, waves of the Zenneck type and came to the erroneous conclusion that these waves could exist independently at large distances from the radiator, led to the concept of Zenneck surface waves being universally accepted as fundamental in considering the problem of radio-wave propagation along the earth’s surface.

This circumstance was further reinforced by the fact that numerous measurements of the polarization state of waves propagating along the earth’s surface gave quite satisfactory agreement with the data of Zenneck’s theory.

Although subsequently Fock and Noterov \(^{40,41}\) indicated, and later Sommerfeld himself acknowledged, the erroneousness of the notion of the independent physical reality of plane waves of the Zenneck type within the radiation of a vibrator located on the surface of separation, the conclusions of Zenneck’s theory concerning the phase structure of the field and the velocity of radio waves were, until very recently, widely used in considering those questions in which this quantity plays an essential role. This interpretation was also used in those cases where various anomalous phenomena occur that are connected with distortion of the wave surfaces (coastal refraction, etc.).

According to Zenneck, the phase velocity of plane inhomogeneous waves propagating along the boundary of separation air \((\varepsilon_0=1,\ \sigma=0)\)—earth \((\varepsilon>1,\ \sigma>0)\), is determined by the wave number \(s\), where

\[ \frac{1}{s^{2}}=\frac{1}{k_{0}^{2}}+\frac{1}{k^{2}}, \]

where

\[ k_{0}=\frac{\omega}{c} \quad \text{and} \quad k=\frac{\omega}{c}\sqrt{\varepsilon+j\frac{2\sigma}{f}}. \]

Hence

\[ v=c\,\frac{\sqrt{(1+\varepsilon)^{2}+\eta^{2}}}{\sqrt[4]{(\varepsilon^{2}+\varepsilon+\eta^{2})^{2}+\eta^{2}}}\cdot\frac{1}{\cos \dfrac{\zeta}{2}}, \]

where

\[ \eta=\frac{2\sigma}{f} \quad \text{and} \quad \zeta=\operatorname{arctg}\frac{\eta}{\varepsilon^{2}+\varepsilon+\eta^{2}}. \]

This constant quantity \(v\), independent of the coordinates of the point of the surface of separation, in the limiting cases \(\sigma=0,\ \eta=0\) becomes

\[ v=c\sqrt{\frac{1+\varepsilon}{\varepsilon}}; \]

for \(\sigma=\infty,\ \eta=\infty\), it becomes \(v=c\).

Thus, throughout the entire range of values of \(\sigma\) from \(0\) to \(\infty\) and \(\varepsilon>1\),

\[ v>c. \]

This very fundamental result, together with the dependence of the phase velocity on the frequency in the region where \(\eta\) and \(\varepsilon\) are of the same order, is very important for all those questions in which the velocity of radio waves plays an essential role.

In Fig. 20 curves are shown for \(v\) as a function of \(\eta\) for various \(\varepsilon\), calculated from the formula given above.

Using these curves, one can determine the deviations of \(v\) from \(c\) over the entire practically interesting range of frequencies and values of \(\varepsilon\) and \(\sigma\). However, in recent years considerable experimental material has accumulated that does not fit within the framework of these notions, and a number of attempts to use Zenneck’s theory to explain specific phenomena occurring in direction finding likewise ended—

Such a result completely contradicts the Zenneck conception and substantially changes all constructions based on the assumption of the constancy of the phase velocity, its independence of distance, and the connection of its magnitude with the soil constants; it also substantially changes very many consequences of generally accepted propositions.

In view of this, it appeared extremely important to obtain experimental data quantitatively confirming a similar picture of the propagation of radio waves along the earth’s surface.

Fig. 21. Curve for $v/c$, calculated by Sommerfeld’s formulas (after Ryazin).

Let us now consider in somewhat greater detail certain quantitative relations following from the calculations set forth above as applied to the phase structure of the field and to the phase velocity of radio waves excited by a vibrator located on the plane interface between air and earth. Comparison of these calculations with experimental results may give essential indications as to the validity of one or another theoretical construction.

Zenneck’s solution of the problem of waves propagating along the interface, in principle, does not take into account the mechanism of radiation. In its derivation, the question of the manner of creating the electromagnetic field corresponding to the established process was not considered, and, accordingly, no conditions were introduced at the point of radiation. Therefore, in Zenneck’s solution, corresponding to plane inhomogeneous waves, one cannot look for features inherent in processes taking place at small distances from the radiator.

In this region, the rigorous treatment of the problem, leading to the Sommerfeld solution mentioned above, is the only suitable one, and here it can only be a matter of comparing the calculated data of this theory with the results of experiment.

For large distances from the radiator, when the Zenneck conception and the corresponding Sommerfeld conception of surface waves were considered applicable, calculations by Zenneck’s formulas and subseq—

tative discussion of Sommerfeld’s solution lead to different results.

This circumstance makes it possible, by comparing the results of measurements carried out at sufficiently great distances from the transmitter with calculated data obtained for the measured quantity according to either theory, to establish which of them corresponds to the actual course of the phenomenon. For this purpose the most suitable quantity is the propagation velocity of radio waves, or the phase structure of the field that determines it. Their study, thanks to the application of various variants of the radio-interference method, can be carried out with an accuracy exceeding that of measurements of all other quantities characteristic of propagating wave processes.

As was indicated above, according to Zenneck \(v\) is always greater than \(c\), approaching this value as \(\eta=2\sigma/f \to \infty\).

Thus, according to Zenneck’s formulas, in a certain range of frequencies, \(\varepsilon\) and \(\sigma\), dispersion must take place. According to Sommerfeld, however, at sufficiently great distances from the transmitter \(v\) is arbitrarily close to \(c\), independently of the soil constants, approaching the limiting value \(v=c\) from values \(v<c\).

The experiments carried out at considerable distances from the transmitter were of two types. On the one hand, the presence of dispersion (the dependence of \(v\) on \(f\)) was studied by the method of a dispersion radio interferometer; on the other hand, the mean phase velocity for sufficiently large distances was determined by the method of a radio rangefinder. Let us note that in such measurements the quantity obtained represented a certain averaged velocity for the two frequencies used.

Experiments with the dispersion interferometer were carried out at sea and under conditions of steppe terrain with dry soil. From these experiments, which showed, within the accuracy of the measurements, the absence of dispersion, an upper limit for the possible value of this quantity could be determined.

In the first case (over the sea), for \(\sigma\) one should take the values \(\sigma=6\cdot10^9 — 6\cdot10^{10}\) CGSE, and for the frequencies used \((f_1=10^6,\ f_2=0.666\cdot10^6\ \text{Hz})\)

\[ \eta_1 \geqslant 1.2\cdot10^4;\qquad \eta_2 \geqslant 1.8\cdot10^4 . \]

For such values of \(\eta\), according to Zenneck *), with very great accuracy, \(v=c\), and within the frequency interval used dispersion should have been absent.

The experiments carried out also gave \(\Delta v/c=0\) with an accuracy exceeding \(30\cdot10^{-4}\). Here \(\Delta v=v_1-v_2\), where \(v_1\) and \(v_2\), respectively—

) See, for example, Zenneck and Rukop, Drahtlose Telegraphie (1925); Van der Pol, Phil. Mag., 36*, 88 (1918).

For a radio range finder, \(\Delta f=10^5\), and with \(\Delta v/\Delta f=1.35\ \text{cm/sec}\cdot\text{sec}\), for \(f=10^6\) we obtain:

\[ \overset{*}{u} = \frac{v}{1-\dfrac{\Delta v}{\Delta f}\cdot\dfrac{f}{v}} = \frac{v}{1-\dfrac{1}{\lambda}\dfrac{\Delta v}{\Delta f}} = \frac{v}{1-0.45\cdot 10^{-4}}. \]

Thus, on the basis of measurement data obtained with a dispersion radio interferometer, one may identify, with great accuracy, the quasigroup velocity measured by means of a radio range finder with the mean value of the phase velocity along the given path.

Numerous experiments with radio range finders, references to which were given above \(^{12, 15, 16, 17, 18, 29—31}\), have shown that this quantity has a value of the order \(v=299\,600\ \text{km/sec}\), which is cited in the article by L. Mandelstam and N. Papaleksi published in 1943 \(^{52}\).

At the same time, according to Zenneck, for \(v\) we have the values shown on the curves in Fig. 20.

Comparing these figures with the data of individual measurements (each of which is the result of averaging a large number of readings), we obtain the following table:

Table IV

Locality	Soil	Soil constants CGSE	Results of measurements of propagation velocity in km/sec	Calculated value, according to Zenneck (km/sec)
Pyatigorsk (Mashuk—Kaban)	air	\(\varepsilon=1\) \(\sigma=0\)	\(298\,300\pm1200\)	\(299\,670\)
Odessa—sea	sea	\(\varepsilon=80\) \(\sigma=6\cdot10^9—6\cdot10^{10}\)	\(299\,500\pm860\)	\(299\,670\)
Lake Ilmen	fresh water	\(\varepsilon=8\) \(\sigma=9\cdot10^6—9\cdot10^7\)	\(299\,400\pm1500\)	\(301\,468\)
Kara Gates	sea	\(\varepsilon=80\) \(\sigma=6\cdot10^9—6\cdot10^{10}\)	\(299\,700\pm600\)	\(299\,670\)
Pugachev	steppe	\(\varepsilon=4—10\) \(\sigma=10^6—10^7\)	\(299\,500\pm80\)	from \(329\,637\) to \(300\,569\)
Kara Sea	sea	\(\varepsilon=80\) \(\sigma=6\cdot10^9—6\cdot10^{10}\)	\(299\,500\pm180\)	\(299\,670\)

Of these results, obtained with the aid of a radio range finder, those especially important for us are the ones that correspond to measurements over the surface of dry soil or fresh water, i.e., to the case of low conductivity of the lower medium. There are two such measurements. One—over fresh water \((\varepsilon=80,\ \sigma=9\cdot10^6—9\cdot10^7)\), which gave \(\bar v=299\,400\pm\)

\(\pm 1500\) km/sec. The second, carried out under the same terrain conditions as the measurements with the dispersion interferometer, gave

\[ \overline{v} = 299\,500 \pm 80 \text{ km/sec}. \]

The values obtained for \(\overline{v}\) differ substantially from those expected according to Zenneck, which are, for the first case, \(301\,468\) km/sec and, for the second, \(329\,637—300\,569\) km/sec.

All this confirms the inadmissibility of applying Zenneck’s theory to the given cases, since it leads to values of the phase velocity that considerably exceed \(c\).

Thus it is evident that precisely where, according to Zenneck’s theory, we should expect \(v\) to exceed \(c\) and should observe appreciable dispersion, experiments carried out with the utmost care do not reveal the expected effects and show, in agreement with Sommerfeld’s theory, that at sufficiently large distances from the radiator the phase velocity is constant, close to \(c\), and does not depend on the soil constants.

Let us note that, as has repeatedly been pointed out in the publications cited earlier, other, non-interference methods of measuring the velocity of propagation of radio waves (for example, pulse methods) have so far not made it possible to measure this quantity with an accuracy permitting a conclusion to be drawn as to the applicability or inapplicability of Zenneck’s theory (all the more so since pulse methods lead to the determination of the group velocity).

Let us now consider the question of the extent to which calculations performed on the basis of Sommerfeld’s solution agree with experimentally obtained data.

Expressing Sommerfeld’s solution for the “amplitude” of the Hertz vector in the form

\[ \Pi = \frac{2}{r} e^{-j k_0 r} J(\rho), \]

where

\[ k_0 = \frac{\omega}{c}, \qquad \rho = j \frac{s k_0^2}{2k^2}\, r, \]

\[ k = \frac{\omega}{c}\sqrt{\varepsilon + j\frac{2\sigma}{f}} \qquad \text{and} \qquad \frac{1}{s^2} = \frac{1}{k_0^2} + \frac{1}{k^2}, \]

we may specify \(J(\rho)\) in the form

\[ J(\rho) = |J|\cos\varphi + j|J|\sin\varphi = a + j \cdot b \]

and, using the expression given by Wise\(^{47}\), pass to the amplitude of the vertical component of the electric field at the interface surface:

\[ E_z = \frac{2}{r} e^{-j k_0 r} \left\{ \frac{a+jb}{1+\dfrac{k_0^2}{k^2}} + \frac{1}{1-\dfrac{k_0^4}{k^4}} \left[ \frac{1}{j k_0 r} + \frac{1}{(j k_0 r)^2} \right] \right\}. \]

Hence, taking into account that for practically interesting cases

\[ \left|\frac{k_i^4}{k^4}\right| \ll 1, \]

we obtain an expression for \(E_z\)

\[ E_z=-\frac{2}{M\cdot r}e^{-jk_0r} \left\{ a\left[\varepsilon(\varepsilon+1)+\eta^2\right]+b\eta -\frac{M}{4\pi^2}\left(\frac{\lambda}{r}\right)^2 + j\left[\eta a-b\varepsilon(\varepsilon+1)-b\eta^2-\frac{M\lambda}{2\pi r}\right] \right\}, \]

where

\[ \eta=\frac{2\sigma}{f}, \qquad M=(\varepsilon+1)^2+\eta^2. \]

Substituting \(E_z\) in explicit form, taking into account the harmonic dependence on time, we obtain

\[ E_z=B(r,\omega,\sigma,\varepsilon)\cos\left[\omega t-\left(\frac{\omega}{c}r+\varphi^*\right)\right], \]

where

\[ \operatorname{tg}\varphi^* = - \frac{ \eta a-\left[\varepsilon(\varepsilon+1)+\eta^2\right]\,b-\dfrac{M}{2\pi}\dfrac{\lambda}{r} }{ \left[\varepsilon(\varepsilon+1)+\eta^2\right]\,a+\eta b-\dfrac{M}{4\pi^2}\left(\dfrac{\lambda}{r}\right)^2 }. \]

Carrying out the corresponding numerical calculations, we obtain the curves for \(\varphi^*\) shown in Fig. 22. Here the quantities

\[ a=|J|\cos\varphi, \qquad b=|J|\sin\varphi \]

were computed by means of the series given in the works cited above by Ryazin\(^{13}\) and Al’pert, Migulin, and Ryazin\(^{12}\):

\[ a= \sqrt{\frac{\pi x}{2}}\, e^{-\frac{x}{2}\cos\psi} \sin\left[\frac{x}{2}\sin\psi-\frac{\psi}{2}\right] +1-x\cos\psi + \frac{x^2}{1\cdot3}\cos2\psi - \frac{x^3}{1\cdot3\cdot5}\cos3\psi+\ldots, \]

\[ b= \sqrt{\frac{\pi x}{2}}\, e^{-\frac{x}{2}\cos\psi} \cos\left[\frac{x}{2}\sin\psi-\frac{\psi}{2}\right] -x\sin\psi + \frac{x^2}{1\cdot3}\sin2\psi - \frac{x^3}{1\cdot3\cdot5}\sin3\psi+\ldots, \]

or, for large \(x\), by means of the series:

\[ a=-\frac{1}{x}\cos\psi-\frac{1\cdot3}{x^2}\cos2\psi-\frac{1\cdot3\cdot5}{x^3}\cos3\psi-\ldots, \]

\[ b=\frac{1}{x}\sin\psi+\frac{1\cdot3}{x^2}\sin2\psi+\frac{1\cdot3\cdot5}{x^3}\sin3\psi+\ldots, \]

where

\[ x=\frac{2\pi}{\sqrt{(\varepsilon+1)^2+\eta^2}}\cdot\frac{r}{\lambda}, \qquad \operatorname{tg}\psi=\frac{f(\varepsilon+1)}{2\sigma}=\frac{\varepsilon+1}{\eta}. \]

From the curves of Fig. 22 one can, using the relation

\[ v=\frac{c}{1+\frac{c}{\omega}\frac{d\varphi^*}{dr}} =\frac{c}{1+\frac{1}{2\pi}\frac{d\varphi^*}{d\left(\frac{r}{\lambda}\right)}} , \]

obtain the dependence of the phase velocity \(v\) on \(r/\lambda\). For the particular case \(\varepsilon=5\) (for which the curves in Fig. 21 were also calculated), we obtain the dependence of \(v\) on \(r/\lambda\) shown in the graphs of Fig. 23.

Fig. 22. Curves of the additional phase as a function of distance for various soil constants, obtained from Sommerfeld’s solution.

As was indicated above (Fig. 14), the experiments carried out by us under steppe-terrain conditions rather convincingly confirmed the character of the variation of the differential phase velocity following from Sommerfeld’s solution and illustrated by the curves of Figs. 21 and 23. However, the question of exact quantitative agreement between the calculations and the results of experiments, both for small and for large distances, can be resolved only in the case where those values of the soil constants \((\varepsilon,\sigma)\) are known which must be adopted in the calculations for the given experimental conditions. In doing so, one must also take into account the possibility that, at various distances from the radiator, in the process of wave propagation along the surface of the boundary, different thicknesses of earth may take part. And since in deriving Sommerfeld’s formulas the earth was assumed to be homogeneous, it follows that the formulas contain certain effective values of \(\varepsilon\) and \(\sigma\), corresponding, for different distances, to different thicknesses

of the working layer of the earth’s surface. All this greatly complicates the task of establishing a one-to-one correspondence between calculations and experimental data. Moreover, we have no data at our disposal that would make it possible to determine exactly the values of these soil constants for the conditions under which the experiments were carried out. It is therefore expedient to perform calculations according to Sommerfeld for various \(\varepsilon\) and \(\sigma\), choosing them from literature data obtained from measurements by other methods. Comparison of the resulting group of calculated data with the results of experiments may give an idea of the degree of possible errors associated with the application of the above formulas when using indicative values of \(\varepsilon\) and \(\sigma\).

Fig. 23. Curves for the differential phase velocity at various soil constants, obtained on the basis of Sommerfeld’s solution.

These considerations apply both to the case of small and to the case of considerable distances from the radiator. At the same time, the results obtained in experiments with a dispersion radio interferometer, used by us in checking the applicability of Zenneck’s theory, can successfully be used also for a quantitative check of calculations according to Sommerfeld. These measurements, covering a large range of distances, are therefore distinguished by great accuracy. In this case, that pair of values \(\varepsilon\) and \(\sigma\) which corresponds to the agreement of calculated and experimental data for the given method at large distances must, when used in formulas following from Sommerfeld’s solution, also give satisfactory agreement with the results of measurements made by other methods at small distances. Of course, conclusions on the applicability of the calculations performed can be made only if agreement between calculations and experimental data is achieved for reasonable values of the soil constants \((\varepsilon,\sigma)\). We note that the experimental data at our disposal (although quite reliable) refer only to one set of soil conditions; consequently, it is necessary to approach especially carefully the question of choosing the values of \(\varepsilon\) and \(\sigma\) and the question of agreement of the results ob-

... obtained when applying various variants of the interference method with the results of calculations.

The data given above, obtained by means of a dispersion radio interferometer, show that under conditions of steppe terrain the changes in the quantity

\[ \Psi=\frac{3}{2}\varphi_2^*-\varphi_1^* \]

when the distance changes from \(r_1=2900\ \text{m}\) to \(r_2=13300\ \text{m}\) did not exceed \(+40'\).

Thus

\[ \Delta\Psi= \left(\frac{3}{2}\varphi_2^*-\varphi_1^*\right)_{r_2} - \left(\frac{3}{2}\varphi_2^*-\varphi_1^*\right)_{r_1} \leq 1^\circ 20', \]

where

\[ r_1=2900\ \text{m}=22.75\,\lambda_1=15.13\,\lambda_2, \]

\[ r_2=13300\ \text{m}=104.2\,\lambda_1=69.5\,\lambda_2. \]

Carrying out calculations of the quantities \(\varphi_1^*\) and \(\varphi_2^*\) and of the \(\Delta\Psi\) determined from them for the specified interval of distances, we obtain, by the formulas indicated above and for various soil constants, the values given in Table V.

Table V

\(\varepsilon\)	\(\sigma=10^6\)	\(\sigma=5\cdot 10^6\)
2	\(1^\circ 22'\)	\(10^\circ 4'\)
4	\(4^\circ 00'\)	\(11^\circ 13'\)
6	\(8^\circ 56'\)	—

From these figures it follows that agreement of the calculated values of \(\Delta\Psi\) with the experimental results occurs only for \(\sigma=10^6\) and \(\varepsilon=2\). For larger \(\sigma\), the calculated values of \(\Delta\Psi\) for the given distances from the radiator are much greater than those obtained experimentally. In this case there is a slower approach of \(\varphi^*\) to its limiting value \((\varphi_\infty^*)\). Therefore, for the specified interval of values (from 2900 to 13300 m), when \(\sigma=5\cdot 10^6\) and higher, the total change in \(\Psi\) turns out to be considerably greater than that which could have occurred in the experiments.

Of course, for very large values of \(\sigma\), the rate of increase of \(\varphi^*\) will be so small that we shall again obtain a practical absence of dispersion and agreement of the quantities \(v_1\) and \(v_2\) with each other and with the speed of light. But the values of \(\sigma\) required for this (of the order of \(5\cdot 10^8\)

CGSE and higher) take us beyond the limits of possible soil conductivities under the conditions in which the measurements were carried out (dry steppe, in autumn).

For smaller distances from the radiator, as can be seen from the curves in Figs. 22 and 23, we shall have different values of \(v(r)\) for different frequencies, and consequently, in the given zone a definite dispersion will be observed, as well as a change with distance in the phase difference observed on the indicator of the dispersion radio interferometer.

The experiments carried out by us in 1939 confirmed these considerations, giving the curves presented above, in Fig. 8.

As one approaches the radiator further, the quantity \(\Psi^{*}\) observed on the indicator of the dispersion radio interferometer begins to change in accordance with the law of change of the “disturbance” of the phase \(\varphi^{*}\) of the component \(E_z\) in the intermediate and near zones. In this case, for small distances, analysis of Sommerfeld’s solution indicates the possibility, for \(r \ll \lambda\), of replacing the soil of finite conductivity by an ideal conductor. In this case (practically for \(r < \lambda/5\)) we obtain from Sommerfeld’s rigorous solution the same regularities for \(\varphi^{*}\) and \(v(r)\) as for a dipole in free space. However, at such small distances it is necessary to take into account the finite dimensions of the radiating antenna. The question of the phase relationships in the near zone, as was already indicated above, was investigated by us specially, both theoretically and experimentally. The results of this investigation were set forth in our paper\({}^{19}\), and one of the graphs illustrating these results is given in Fig. 9. The excellent agreement of the theoretical calculations with the experimental data for this case indicates the correctness of the theoretical treatment of the problem and the success of the method of the dispersion radio interferometer applied in this case\({}^{*}\).

It is appropriate here to mention the division into zones which we use in the present work. The character of the curves shown in Fig. 23 makes it possible to distinguish three characteristic zones: a) the near zone, in which the law of variation of the phase and of the phase velocity with distance coincides with that for the case of a radiator situated in free space. In this zone, which extends approximately to \((0.2 \div 0.1)\lambda\), the soil constants have practically no effect on the character of the radio-wave field.

b) The intermediate zone, in which the phase velocity, deviating from the values corresponding to the case of a dipole in vacuum, decreases and, becoming less than \(c\), reaches a certain minimum value.

c) The zone of establishment, in which \(v\) increases monotonically and asymptotically tends to \(c\). Beyond the limits of the zone of establishment the quantity

\({}^{*}\) These questions subsequently served as the subject of the candidate’s dissertation of Ya. L. Al’pert\({}^{53}\).

\(v\) practically does not depend on the distance or on the frequency and the constants of the soil, since under these conditions dispersion is absent and \(v=c\). It is obvious that the dimensions of the intermediate zone and of the zone of establishment, and the position of the boundary between them, will be different for different soil constants and frequencies.

The data, used by us in checking Zenneck’s theory and presented earlier in Fig. 8, obtained with the aid of a dispersion radio interferometer, gave only qualitative confirmation of the existence of the zone of establishment.

For studying the character of the establishment of the phase velocity and for checking the theoretical results following from Sommerfeld’s theory, we used another variant of the radio interferometer—a moving radio interferometer.

Fig. 24. Curves of the differential phase velocity of two oscillations and of the experimentally determined quantity \(\bar v^*\) for \(\varepsilon=5;\ \sigma=1.15\cdot 10^6\). The triangles show the results of the experiments.

The results of the most characteristic experiments carried out with the moving radio interferometer were presented earlier (see Figs. 13 and 14) in describing the principle of operation of this variant of the radio interferometer. Here we shall compare these results with quantities calculated from the formulas following from Sommerfeld’s solution.

Taking, in accordance with the results of experiments with the dispersion radio interferometer, \(\eta=\dfrac{2\sigma}{f}\) of order 1, we obtain for \(f=2.29836\cdot 10^6\ \mathrm{Hz}\), \(\sigma \simeq 1.15\cdot 10^6\).

Using the calculations already carried out for \(\varepsilon=5\), one can compute the quantities \(v(r)\) for \(f_1\) and \(f_2=2/3\,f_1\), for which respectively \(\eta_1=1\) and \(\eta_2=3/2\). The corresponding curves are shown in Fig. 24 (curves \(a\) and \(b\)). However, as a result of the experiment a somewhat different quantity is determined:

\[ \bar v^*=\frac{2\omega_1}{\dfrac{\Delta\varphi}{\Delta r}}, \]

which represents a certain conditional averaging of the differential phase velocities over the interval \(\Delta r\) and the frequencies \(\omega_1\) and \(\omega_2\).

This quantity

\[ \vec{v}^{*} = \frac{2\omega_{1}}{2\frac{\omega_{1}}{c}+\frac{\Delta\left(\varphi_{1}^{*}+\frac{3}{2}\varphi_{2}^{*}\right)}{\Delta r}} = \frac{c}{1+\frac{\lambda}{4\pi}\frac{\Delta\left(\varphi_{1}^{*}+\frac{3}{2}\varphi_{2}^{*}\right)}{\Delta r}} \]

can be directly calculated for various \(r\).

Taking \(\lambda_{1}=c/f_{1}=130.4\ \mathrm{m}\) \((c=2.99670\cdot 10^{8}\ \mathrm{m/sec})\) for the values \(\varepsilon=5\), \(\eta_{1}=1\), \(\eta_{2}=1.5\), we obtain the theoretical curve shown in the same Fig. 24 (curve \(c\)).

The experimental data plotted in the form of triangles give a sufficiently convincing confirmation of the validity of the calculations performed. It is likely that a more careful selection of the quantities \(\varepsilon\) and \(\sigma(\eta)\) could provide a more exact agreement of the theoretical curve with experiment. However, the results obtained confirm not only the qualitative law of the establishment of the phase velocity with increasing distance from the radiator, but also the quantitative data concerning the order of magnitude of \(\varepsilon\) and \(\sigma\), the law of variation of \(\vec{v}^{*}\), and, consequently, also \(v_{1}\) and \(v_{2}\) for each of the oscillations separately.

Summarizing all the material set forth in the present chapter, one may draw the following main conclusions:

1. In considering the propagation of radio waves along the surface of the interface from a radiator located on this surface, it is necessary to use the results obtained from Sommerfeld’s rigorous solution.

This conclusion, confirmed quantitatively and qualitatively by a number of experiments, undoubtedly excludes the application to the given problem of Zenneck’s concept of plane inhomogeneous waves.

1. For experimental verification of theories of radio-wave propagation and for the study of processes taking place in the field of radio waves under real conditions, in all cases where phase relations, velocities, and the phase structure of the field are involved, various versions of radio interferometers are the most suitable experimental means.

CONCLUSION

Radio engineering permits an easy transformation of multidimensional oscillations into one-dimensional ones and back again (the transition from electromagnetic field oscillations to oscillations of currents or voltages in the circuits of radio devices and conversely), and also provides methods for direct measurements of phase shifts of coherent oscillations. This makes it possible to extend the concept of interference to the case of radio waves in comparison with optics.

By radio interference one may understand not only a stationary periodic spatial distribution of the amplitude of the resultant oscillation, but also the distribution of the form of the resultant-

of the oscillatory process formed in space or in the indicator as a result of the simultaneous action of interfering waves.

Such a definition of radio interference also includes the interference of oscillations with commensurable, and not only equal, frequencies. In addition, the possibility of transferring interference from space to the indicator makes it possible, for radio waves, also to consider the interference of modulated oscillations. All these circumstances make it possible to obtain an exceptional variety of radio-interference phenomena.

The consistent application of the phenomenon of interference in the field of radio waves, the various variants of radio-interference methods for scientific investigations and practical applications, as well as the formulation of a large cycle of investigations connected with this problem, are the result of the most fruitful work of Academicians L. I. Mandelstam and N. D. Papaleksi during a number of the last years of their life and activity.

These remarkable works, which laid the foundation for a new direction in radiophysical research, yielded many very valuable scientific and technical results, some aspects of which are elucidated in the present work.

CITED LITERATURE

1. I. Bernstein, ZhTF 11, 305 (1941).
2. E. Shchegolev, ZhTF 4, 191 (1934).
3. I. Lyubchenko, ZhTF 8, 1548 (1938).
4. V. Yuzvinsky, ZhTF 11, 55 (1941).
5. Friis, P. I. R. E., 13, 683 (1925).
6. Grösskopf, T. F. T., 11, 334 (1940).
7. Chaman Lal, Ind. Journ. of Phys., No. 15, 289 (1941).
8. N. Nijenhuis, Philips Techn. Rundsch., H. 7, July (1940).
9. E. V. Appleton and Barnett M., Proc. Roy. Soc., 113, 450 (1928).
10. Ross and Slow, Nature, 139, 571 (1937).
11. A. Sommerfeld, Ann. d. Phys., 28, 664 (1909).
12. Ya. Alpert, V. Migulin and P. Ryazin, ZhTF 11, 7 (1941).
13. P. Ryazin, Izv. AN SSSR. Physical Series, 4 No. 3, 434 (1940).
14. E. Shchegolev, ZhTF, 11, 44 (1941).
15. L. Mandelstam and N. Papaleksi, ZhTF, 7, 561 (1937).
16. L. Mandelstam and N. Papaleksi, ZhTF, 7, 764 (1937).
17. E. Shchegolev, ZhTF, 7, 579 (1937).
18. E. Shchegolev, Izv. AN SSSR, Physical Series, issue 4, 551 (1938).
19. Ya. Alpert, V. Migulin and P. Ryazin, DAN, 18, 635 (1938).
20. Ya. Alpert, V. Migulin and P. Ryazin, ZhTF, 9, 824 (1939).
21. Ya. Alpert and V. Migulin, DAN, 26, 878 (1940).
22. Harms, German patent (D.R.P. No. 546000), published 8/III 1932.
23. P. Ryazin, Doctoral dissertation. Proceedings of FIAN, 3, issue 2, 45 (1946).
24. N. Papaleksi, Izv. AN SSSR, Physical Series, No. 4, 539 (1938).
25. N. Papaleksi, Proceedings of the expedition for the observation of the total solar eclipse of 19/IV 1936, vol. I, p. 115.

1. L. Mandelstam and N. Papaleksi, DAN, 26, 182 (1940).
2. J. Alpert, V. Migulin and Ryazin, Journ. of Phys., 4, 13 (1941).
3. Ya. Alpert and V. Migulin, Izv. AN. Physical Series, 4, 453 (1940).
4. L. Mandelstam, Izv. AN. Physical Series, No. 4, 539 (1938).
5. C. Viller and E. Schegolev, Techn. Phys. USSR, 4, 87 (1937).
6. A. Gruzinov, L. Mindlin, Izv. AN SSSR. Physical Series, 4, 468 (1940).
7. E. N. Dingley, Communications, p. 5. January (1940).
8. Sadahira Matsuo, Journ. I. E. E. Japan., No. 71, February (1936).
9. Sadahira Matsuo, P. I. R. E., 26, 845, July (1938).
10. L. Espenschied and Newhose, Bell. Syst. Techn. Journ., 18, 222 (1939).
11. E. Schegolev, Journ. of Phys. USSR, 1, No. 5–6, 339 (1939).
12. B. Decaux et L. B. Gall C. R., 193, 2239 (1934).
13. P. Fayard, L’onde Electrique, 13, 359 (1934).
14. J. Zenneck, Ann. der Phys., 23, 846 (1907).
15. Frank and Mises, Differential and Integral Equations of Mathematical Physics, p. 937 (1937).
16. Frank und Mises, Differential und integral gleichungen der mechanik und physik, I, p. 932 (1935).
17. Hörschelmann, Jahr. d. Draht. Tel. u. Tel., 5, 14, 188 (1911).
18. H. Weyl, Ann. d. Phys., 63, 481 (1919).
19. B. van der Pol, Zeits. f. Hochfr Techn., 37, 152 (1931).
20. B. van der Pol und Niessen, Ann. d. Phys., 6, 273 (1930).
21. K. Niessen, Ann. d. Phys., 16, 810 (1933).
    K. Niessen, Ann. d. Phys., 18, 813 (1933).
    K. Niessen, Ann. d. Phys., 28, 209 (1937).
    K. Niessen, Ann. d. Phys., 37, 522 (1938).
22. W. Wise, Bell. Syst. Techn. Journ., No. 8, 652 (1929).
23. C. Burrows, Bell. Syst. Techn. Journ., 16, 45 (1939).
24. K. Norton, P. I. R. E., 25, No. 9, (1937).
25. B. Vvedensky, Foundations of the Theory of Radio-Wave Propagation. ONTI (1934).
26. B. Vvedensky and A. Arenberg, Propagation of Ultrashort Waves. Svyaztekhizdat (1939).
27. L. Mandelstam and N. Papaleksi, Izvestiya AN SSSR. Physical Series, 7, No. 5, 145 (1943).
28. Ya. L. Alpert, Proceedings of FIAN, 3, issue 2, 3 (1946).