Physics of the Ionosphere1

H. R. Mimno, Cambridge, Massachusetts, USA

Contents

A. Introduction. B. Brief historical survey. C. Principal experimental facts. D. Elementary theory. E. Nature of the accelerating field. F. Other forces acting on the electron. G. Equations of motion of electrons. H. Analysis of the magneto-ionic double refraction. I. Friction as a result of collisions. J. Complete analysis by means of conformal representation. K. “Fine structure” of the ionosphere. L. Why stratification exists? M. Tidal effects in the ionosphere. N. Solar spots, magnetic indices, and polar lights. O. Magnetic storms and meteor showers. P. Thunderstorms and barometric effects. Q. Local clouds in the ionosphere. R. Scattering of radio waves. S. Interaction of radio waves. T. Observations during eclipses. V. Conclusion.

A. Introduction

On October 22, 1924, an eminent British engineer delivered before the Radio Society of Great Britain a lecture on “Unsolved Problems of Wireless Telegraphy.” In his address R. H. Barfield put forward the following problems:

1. Why is transmission without wires over long distances possible at all?
2. Why are signals stronger at night than during the day?
3. Why are direction-finding stations at night subject to large errors, whereas by day these errors can practically be neglected?
4. Why does the phenomenon known as “fading” occur?

During the following decade these questions and many other problems connected with them from the engineering point of view were fully resolved. In the course of the investigation, however, new engineering problems arose, which still await solution. These latter problems will be described in the appropriate places below. However, the most interesting results obtained over

over the last twelve years as a result of intensive experimental investigations, belong properly to the field of pure physics. New interesting experimental and theoretical problems confront the physicist. The development of new experimental methods and instruments makes it possible to carry out precise measurements in the field of atmospheric physics, which is not too complicated for valuable theoretical analysis. The theory is still incomplete, and experimental methods require further development; nevertheless, a satisfactory beginning in this young and vigorous branch of physical research has already been made. The new problems of the ionosphere are closely connected with a large number of other geophysical investigations, as well as investigations concerning the moon and the sun, which have attracted the attention of physicists for a long series of years. Such investigations include studies of cosmic rays, terrestrial magnetism, solar activity, auroras, the distribution of ozone in the atmosphere, thunderstorm activity, atmospheric temperatures, luminous clouds, meteor trails, motions of air masses, terrestrial electric currents, stratospheric meteorology, and studies of elastic deformations of the earth’s crust.

After a brief historical survey of the early theories and experiments on the phenomena of radio-wave propagation, I shall attempt to give a general account of a large number of investigations carried out during the last twelve years. Since, however, a complete collection of all the original papers from this active period would fill several large volumes, I shall be compelled to exclude from this survey many valuable works. The papers cited below have therefore been selected from among a much larger number, since it seems to us that they illustrate in an appropriate way various points of view. I intended to make the references representative, but by no means complete. Additional references can be found in almost all of the papers cited. In reviewing the works I have not adhered to a strict chronological sequence and have not attempted to resolve the innumerable problems of priority.

In every active branch of physics numerous, extremely controversial questions arise; many such questions may also be found in the field of the ionosphere. I believe that, when considering such questions, the reviewer should not take a conciliatory position, but should formulate his own personal impartial opinion whenever the available facts permit this to be done. In trying to proceed in this way, I have endeavored to avoid arbitrary assertions and have included references representing both disputing sides. The field that interests us is new and extensive, and one cannot expect progress in it to have taken place without inevitable accidental errors in mathematical theory, experimental technique, and interpretation of results. I shall welcome all corrections and comments.

B. Brief Historical Review

When Marconi, on December 12, 1901, first transmitted radio signals across the Atlantic Ocean, Lord Rayleigh pointed out that this result could not be explained simply by the propagation of a plane wave. Rayleigh suggested that, in order to explain how the wave follows the curvature of the Earth’s surface, one should make use of a definite diffraction theory.

Research on diffraction theory was devoted by Rayleigh ^1,^1) MacDonald ^2, Poincaré ^3, Zenneck ^4, Sommerfeld ^5, Nicholson ^6, March ^7, Rybczynski ^8, Love ^9, van der Pol ^10, and many others. These studies were briefly summarized by Frank E. Smith ^11 in the 24th Kelvin Lecture as follows: “Many years of work by a number of the most outstanding mathematicians in the world proved insufficient to bring this seemingly innocent problem to such a state that Nicholson could say that it was the one problem in the whole field of mathematical analysis about which the greatest number of divergent points of view are held.” In general, however, it is recognized that Watson ^12 in 1919 gave the correct review of the problem and found the errors that were responsible for such considerable divergence. Diffraction theory lacks a huge factor needed to explain the intensities observed at points lying considerably below the visible horizon. As was pointed out in 1924 by Larmor ^13, a wave of length 100 m on the Earth corresponds to a wave of visible light on a sphere of radius 6 cm, and therefore there is no basis for expecting any noticeable bending of the ray as the result of diffraction alone. A careful mathematical analysis only confirms this consideration. Several divergent points of view were expressed comparatively recently by Meissner ^14 and Kibitz ^15; however, Kibitz’s calculations were successfully criticized by Meni ^16.

Beginning in 1924, interest in the diffraction problem weakened, since numerous new experimental tests convincingly proved that the propagation of waves over a great distance depends mainly on entirely different factors. Nevertheless, there is no doubt that diffraction plays an important role in certain types of wave propagation over short distances, and I shall still have occasion to return to this question in Section C.

While the diffraction problem was being studied mathematically, various investigators also considered several other possible modes of propagation of radio waves in the terrestrial atmosphere. Stewart ^17, in 1878, suggested that certain types of periodic variations of terrestrial magnetism may be

^1) The bibliography will be given at the end of the article.

explained by the existence of conducting layers in the upper region of the atmosphere. This hypothesis was further developed in 1889 by Schuster^18. In March 1902 Kennelly^19 published a short article in which the following was said: “It is a well-known fact that the waves of wireless telegraphy, propagating through the ether and through the atmosphere above the surface of the ocean, are reflected by this electrically conducting surface.” Three months later Heaviside^20 wrote a similar brief note, which was published in December: “It is possible that in the atmosphere above there is a fairly well conducting layer. If this is so, the waves will be, more or less, so to speak, confined by it. Then the waves will be guided on one side by the sea, and on the other by one of the upper layers of the air.” The original articles by Kennelly and Heaviside have recently been republished^21,^22.

In 1912 Pierce and L. de Forest discussed in private correspondence a probable explanation of radio-signal “fading” on the basis of the idea of interference between a surface wave (“ground ray”) and a space wave (“sky ray”). Evidently the nature of the phenomenon, in broad outline, was already well understood by that time, although it was still somewhat difficult to formulate an opinion as to the exact path of the indirect ray.

In his early article Kennelly did not attempt to consider the mechanism of atmospheric conductivity, but justified his assumptions by reference to measurements of the conductivity of air in discharge tubes, due to Thomson^23. Two important advances were achieved in this connection in 1912 by Eccles^24,^25. Eccles considered the ionizing effect of solar radiation, and also gave the basic theory of ionic refraction. This phase of the theory was further developed by Zenneck^26 and van der Pol^27. The next major advance was achieved in 1924, when Larmor^13,^28 investigated the whole problem anew and attributed a large part of ionic refraction to the presence in the ionosphere of a large number of free electrons.

The possibility of refraction in the lower layers of the atmosphere as the result of the existence of a barometric gradient, a gradient in water-vapor content, or a temperature inversion was also considered by several investigators. All these effects were calculated, and it soon became evident that the maximum amount of bending that could be produced by the lower layers of the atmosphere was insufficient to explain the propagation of radio waves over long distances. This phase of the question was summarized by Fleming^29 and Larmor^28.

Meanwhile, as a basis for engineering calculations, a simple empirical equation, known as the “Austin–Cohen formula,” gained general recognition. This equation was the result of analysis of a series of experiments on the propagation of waves over water, carried out under Austin’s direction^30 in 1909 and 1910. On the cruisers Birmingham and Salem, numerous quantitative observations of the inten-

signal strengths at various wavelengths; the ship at the same time moved away from the fixed station. The greatest distances were of the order of 1200 miles. Most of the original measurements were made for wavelengths from 1000 to 3750 m.

Austin’s experimental results could be described adequately by the following empirical relation:

\[ I_R = 4.25 I_S \left(\frac{h_1 h_2}{\lambda d}\right)e^{-\alpha d^{\frac{1}{2}}}, \]

where \(I_R\) is the current strength in a 25-ohm receiving antenna of height \(h_2\) kilometers, \(I_S\) is the current strength in the transmitting antenna of height \(h_1\) kilometers, \(d\) is the distance in kilometers, \(\lambda\) is the wavelength in kilometers, and \(\alpha = 0.015\) for propagation over seawater.

Fleming \(^{31}\) showed that the first part of this equation is wholly compatible with Hertz’s original equations and represents a simple law of decrease of the energy in the wave front inversely proportional to the square of the distance. In order to satisfy the experimental curves, Austin added an exponential factor, attributing this to “atmospheric absorption.” The quantity \(\lambda^{-\frac{1}{2}}\), occurring in the exponent, was introduced after a careful analysis of Austin’s data by Cohen.

The theoretical formula derived from diffraction theory \(^{3,6,8}\) contained a somewhat similar exponential term. However, besides the large discrepancy in the numerical value of \(\alpha\), the diffraction formula contained \(\lambda^{-\frac{1}{3}}\) instead of \(\lambda^{-\frac{1}{2}}\). The experimental data seemed sufficiently accurate to rule out the possibility of this substitution. The hypothesis of a “reflecting layer” therefore received considerable additional support in 1919, when Watson \(^{32}\) derived the Austin–Cohen formula by solving the difficult mathematical problem of the propagation of waves in a medium bounded by concentric conducting spheres. Later Kennelly \(^{33}\) indicated that Watson’s result remains valid also in the case where the boundaries of the conducting spheres are not defined quite sharply.

For many years, all new experiments, which included additional frequencies, greater distances, and different types of wave propagation over the earth, seemed only to provide ever greater support for the Austin–Cohen formula and to extend its range of applicability.

Enormous sums were spent on the construction of powerful long-wave stations with a large number of lines of tower antennas reaching 800 feet in height. All wavelengths below 200 m were regarded as having practically no value for transmission over great distances. This entire region of wavelengths was consequently left to amateurs.

Under these circumstances it is not surprising that radio amateurs were at first guided by the generally accepted ideas and therefore worked

with their transmitters very close to the 200-meter wave, which represented their legal upper limit. The activity of radio amateurs in many countries was severely restricted and was entirely prohibited in others, but in the United States of America it received liberal governmental support from the Department of Commerce. The subsequent astonishing progress of amateur radio constitutes an important and most unusual chapter in the history of science. The American Radio Relay League, a strictly noncommercial organization founded in 1914 by the late Maxim, became the nucleus of a flourishing international society, which included among its active members people from almost every remote corner of the Earth. It includes men and women of all age groups and has representatives in almost every profession. The League maintains a competent technical staff, and its technical journal (QST) and handbook have become practically indispensable in professional engineering laboratories.

This remarkable progress arose from the unexpected development of the study of the short-wave region, originally made available for use by radio amateurs. Under favorable conditions, a rather unstable communication on a wavelength of 200 m could be obtained over distances lying far beyond the limits prescribed by the Austin–Cohen formula. In connection with this, systematic tests were carried out over greater and greater distances. The latest improvements in vacuum apparatus were immediately used in amateur circuits, whereas commercial progress was often hampered by patent restrictions and conservative economic policy. Finally, in 1921, American radio amateurs sent to the coast of Scotland an expedition equipped with receiving apparatus of the very latest type, and successfully carried out one-way transatlantic tests on short waves according to a previously planned program. The tests were repeated during the following winter; at times it proved possible to establish two-way communication as well. It became quite obvious that the generally accepted formula was not a reliable guide in this region of short waves. Prompted by natural curiosity and by cramped conditions near 200 m, some of the boldest radio amateurs began to investigate still shorter waves. To everyone’s surprise, transmission became still stronger and less unstable. Naturally, movement downward along the wavelength scale thereafter accelerated, and its limit was set only by the design of circuits and vacuum tubes for very high frequencies.

Below 50 m another, entirely new effect of exceptional importance was observed and partly explained. It was found that the signal strength decreased to zero at points comparatively close to the transmitter (say, about 50 miles), but at the same time, under favorable conditions, exceptionally good transmission could be maintained for thousands of miles between stations of small—

of high power. The outer boundary of the “zone of silence” or “skip zone” appeared very sharp[^31].

An analogous “zone of silence” had earlier been noted in connection with the ordinary propagation by sound waves of loud noises (produced by artillery fire or powerful explosions). The appearance of such acoustic mirages can be explained quite satisfactorily, quantitatively, by refraction caused by a temperature inversion in the stratosphere. However, the refraction of electromagnetic waves in this region is insufficient to explain the radio mirages that were observed. Numerous observations of the “skip zone” were made in 1924 by a group of radio amateurs headed by Reinartz[^35], who correctly ascribed this new phenomenon to the action of an ionized region. These amateur investigations were immediately subjected to scientific verification and were developed by Taylor and Hulburt[^36]. Rukop[^37] gave an interesting review of the early experiments and pointed out that the research departments of various commercial organizations possessed the appropriate short-wave transmitters and receivers even earlier, but had never suspected the possibility of long-distance communication with such apparatus.

Soon after the great importance of these wavelengths, which had long been neglected, was understood, a new allocation of frequencies was carried out by international agreement, and the construction of new long-wave stations was practically abandoned. In the subsequent struggle between competing countries and competing commercial interests, amateurs quickly lost about 90% of their former territory, although several “channels” were preserved for them through the friendly efforts of representatives of the American government.

With these new discoveries and with the theory of electronic refraction proposed in 1924 by Larmor, the long period of research of an accidental character came to a definite end, and a new field of atmospheric physics began to unfold. In former years physicists had paid only incidental attention to results obtained as by-products of radio communication. In the following decade, most of the progress was made by direct physical measurements, in which radio apparatus served merely as incidental research instruments. Unfortunately, this changed situation has received no recognition in the monumental structure of American governmental regulations. Academic scientific institutes have been placed in a difficult position by inflexible rules that had already become obsolete 12 years ago. Since this situation constitutes the chief obstacle to the work and cannot be compared with anything in any other field of experimental physics, the matter will have to be studied in considerable detail in Section Q. The responsible officials are aware of the present difficulties and have offered us their cooperation;

however, the action of the constraining mechanism may be accelerated by a broader knowledge of the facts.

C. Basic Experimental Facts

Under this heading I shall try to give a simple analysis of the most essential features which distinguish the various parts of the radio spectrum, leaving for a later account the detailed study of particular experiments intended to test certain special aspects of the theory of the ionosphere. In sketching this general background, the picture can be made clearer if the radio spectrum is divided into definite regions which exhibit characteristic types of behavior. A simple quantitative classification of this kind may be useful for purposes of illustration, although in it there must necessarily be, and there has been, a somewhat arbitrary definition of the boundaries between the various regions of the spectrum. With this reservation regarding the numerical data, we may adopt the following terminology:

We shall also need to consider several different concentric regions or “layers” in the ionosphere, acting somewhat differently on the various parts of the radio spectrum. Adopting Appleton’s letter notation and disregarding for the moment all “fine structure” of the ionosphere, we may distinguish in it three principal regions which control most of the observed effects: the \(F\)-region—strongly ionized, approximate height 240 km; the \(E\)-region—moderately ionized, approximate height 100 km; the \(D\)-region—weakly ionized, approximate height 50 km. After sunset the ionization in each of the regions slowly decreases; the weakly ionized \(D\) region at night is apparently relatively ineffective. The normal daily cycle of ionization and recombination is often altered by sudden and unstable increases of ionization, which may occur in any of the regions and at any hour of the day or night. Such changes are especially frequent in the \(D\) layer and in the \(E\) layer. The 11-year cycle of changes in sunspots apparently affects the mean ionization in all parts of the atmosphere. The heights indicated above are only characteristic values, taken for the purpose of illustrating the existing order of magnitudes. A more detailed consideration of measured “heights” will be given in Section K.

Free electrons in the $F$-region, set in motion by radio waves, have a comparatively large mean free path and lose little energy as a result of collisions (“friction”). In the $D$-region, by contrast, friction as a result of collisions is the determining factor.

“Long waves” obey the Austin–Cohen formula rather well, and propagation according to the scheme of conducting concentric spheres gives an acceptable and adequate explanation of their behavior. In the absence of decisive experimental facts we may try to assume that the $D$-region serves as the outer conducting sphere in the case of daytime transmission over great distances, at grazing incidence. It seems probable that, in comparison with the wavelength, the lower boundary of the layer is defined very sharply. Longer waves, therefore, do not penetrate appreciably into the ionized layer and are not absorbed as a result of the strong attenuation that would accompany their

Fig. 1. Propagation of long waves in daytime hours

propagation in the ionized region. The conducting layer acts like a simple metallic reflector, although there is no doubt that considerable absorption also takes place, resulting from a small residual ionization in the troposphere and the lower layers of the stratosphere. Some reduction in the attenuation of radio waves is observed in the case when their path lies on the unilluminated side of the Earth. Under these conditions it is natural to suppose that the upper boundary of the layer is displaced toward the $E$-region.

Fig. 1 shows continuous wave fronts extending from the ionized layer to the Earth. This distance, if measured in wavelengths, is comparatively small, and here there is no complete separation into a surface wave (“ground ray”) and a space wave (“sky ray”). Since a large part of the energy is carried to the receiver by a single ray, long waves are especially convenient for direction-finding purposes, being comparatively free from the distortions that usually accompany transmission along a complex zigzag path. The electric vector here is approximately vertical and has only a slight forward inclination, depending on the amount of energy absorbed in the intersected conducting soil or water. At these low frequencies the surface ос-

is relatively slight. The influence of the conductivity of the earth’s surface was recently considered by Dine ^38. In the case of long-wave transmission over sea water, Yokoyama and Tanimura ^39 find certain facts which permit one to think of a zigzag ray, successively reflected by the ionosphere and the sea.

The region of “long waves” provides a limited number of first-class telegraph channels free from service interruptions, with the exception of those that occur during strong magnetic storms or abnormally large “static” disturbances. The importance of “long waves” for telephony is small, since the broader frequency bands occupied by telephone channels would lead to excessive crowding. A large part of the telegraph channels is occupied by costly, extremely powerful stations that have been in continuous operation for many years. In planning new stations, the relative stability and reliability of transmission on “long waves” are usually not outweighed by the lower cost of the comparatively “short-wave” circuit.

Broadcast waves provide reliable service of high quality over a limited area in the vicinity of the transmitting station. The size and outline of this area depend on the construction of the antenna, the wavelength, geological conditions, and the cycle of variation of sunspots; as a characteristic example, however, one may cite a circle with a radius of 50 miles. Beyond this area there is a narrow zone characterized by very strong fading, which is especially distinct at night. Medium-power stations located beyond 150 miles are usually not received during the daytime, but at night they are often covered over thousands of miles, and transoceanic reception is not uncommon. Night transmission over long distances, however, is unreliable and subject to great fading.

In the daytime, almost all the energy that reaches the receiver on a “broadcast wave” is carried by the “surface wave.” The exact nature of this surface wave has been the subject of extensive investigations. According to the view of the majority, the surface wave (“ground ray”) is a true guided wave, analogous in its nature to high-frequency waves that can propagate along a single copper wire. Sommerfeld ^5,40, on the basis of Zenneck’s diffraction equations ^4, obtained a formula describing the propagation of a wave along the interface between the atmosphere and the semiconducting earth. Numerical and graphical calculations by this formula were carried out by Herschelman ^41, Ratcliffe and Barnett ^42, Rolf ^43, van der Pol ^44, Wise ^45, Eckersley ^46, Niessen ^47, and Norton ^48. A special solution by Murray ^49 was corrected by Niessen ^50; it turned out to be compatible with Sommerfeld’s formula. Eckersley considers that Sommerfeld’s formula is of importance for calculating the intensity of the field strength of the direct ray, given by waves from 60 to 2000 m at distances up to 2000 miles. Barfield ^51 used this formula to determine geological differ-

personal; however, the method he applied was criticized by Englund[^52]; Haeber[^53] admits that the surface waves studied by Zenneck and Sommerfeld are theoretically possible, but asserts that the existing methods do not permit obtaining them. In this connection it is interesting to note that recently Muskat[^54], in studying seismic waves, found that a certain modification must be introduced into the theory of the propagation of elastic waves along the surface of separation of two homogeneous elastic media.

We could regard the guided-wave hypothesis as experimentally proved if it were not known that the direct wave, propagating by the shortest path through the troposphere from the transmitting antenna, will penetrate into valleys and bend around obstacles by virtue of ordinary diffraction, and can approach the receiving points without the aid of the conductivity of the earth’s surface. This effect

Fig. 2. Propagation of broadcasting waves at night

must be amenable to calculation; however, the simple theory of diffraction by a sharp edge is not a sufficiently good approximation for it, and the whole problem apparently has not yet been solved in a satisfactory manner. For our present purpose it is enough to describe the surface wave as a stable, reliable wave which does not exhibit a noticeable diurnal variation. Its intensity decreases with distance according to an exponential law. Its range of action increases with the wavelength and with the power available to the station, and can be increased by such a construction of the antenna in which the emitted energy is concentrated within a small solid angle[^55].

During the day practically all radiation from broadcasting antennas at large angles is absorbed by the ionosphere. The frequency is sufficiently high to allow the wave to penetrate into the \(D\)-region, where it rapidly dies out owing to the friction arising as a result of collisions of electrons.

After nightfall the ionization of the \(D\)-region decreases, and broadcasting waves are strongly “reflected” from the \(E\) layer, as is shown in Fig. 2. The reduced attenuation is the result of a longer mean free path of electro-

at a greater altitude (the effects of refraction and polarization will be considered later). At points close to the transmitter, the energy of the space wave (“sky ray”) is much less than the energy of the surface wave (“ground ray”), but the fraction of energy of the space wave does not decrease rapidly with distance. As a result of the decrease in intensity in the surface wave according to an exponential law, we soon arrive at a zone where both waves produce approximately equal fields. The slightest change in atmospheric conditions will cause very strong fading, since the relation between the phases of the two waves is thereby changed. The frequently observed periodic fading indicates a progressive change in the equivalent path of the space wave.

At distant points the surface wave is entirely ineffective, while the space wave, under favorable conditions, is detected at great distances. Namba and Hiraga \(^{56}\) observed an improvement in transmission across the Pacific Ocean in years close to the minimum of sunspots. Such transmission is better in autumn than in midwinter, and worst of all in midsummer. Berkner \(^{57}\) reported the reception of numerous American broadcasting stations in the southern polar region, at distances exceeding \(12\,000\) km. Beyond the zone of the surface wave the signal is often comparatively stable, although not sufficiently reliable. Slow changes in field intensity occur as a result of changes in absorption. In other cases periodic fading is the result of the fact that propagation of the space wave takes place along a zigzag path of the type shown in Fig. 3.

Fig. 3. Radio transmission by means of multiple reflection of the sky ray

The ionized region is not an entirely linear transmitting medium. As shown in Fig. 4, propagation of the wave from point \(A\) to point \(C\) may be appreciably distorted by cross modulation by a powerful interfering station located near the midpoint on the great-circle arc connecting the transmitter and receiver. These interaction effects will be considered in Section S. This disturbance is everywhere insignificant, with the exception of the lower boundary of the frequencies of the broadcasting-wave range.

For convenience in describing the short-wave region, I have divided it into three parts. Wavelengths from 100 to 200 m, as is known, are unstable and unsatisfactory for long-distance communication. The surface

the wave attenuates so rapidly that the local area served by it is too small for effective use for broadcasting purposes. The space wave also undergoes very strong attenuation, and therefore this part of the spectrum may be regarded as a true atmospheric absorption band. Later we shall mention other theories as well, but it may nevertheless be considered almost certain that these anomalous atmospheric effects are due to the resonance frequency of the free electrons set in motion by the wave. The electrons move in circular orbits about the lines of force of the earth’s magnetic field and possess, in this region, a natural frequency.

The unstable properties of 100–200-meter waves arise partly from one additional complication. This is a transition region of frequencies, and here one may expect reflections from the \(E\) layer, or from the \(F\) layer, or simultaneously from both layers. Shorter waves in this frequency band are capable of penetrating into the \(E\) region, but their frequencies are not sufficiently high to prevent appreciable partial reflection, attenuation, polarization, and lowering of the group velocity.

Fig. 4. Interaction

In group \(B\) of short waves we may include the region extending from 50 to 100 m. These waves are especially valuable for transmission over land within the limits of a single continent, and they are widely used by aviation and the military department. With the exception of points lying within a radius of 30 miles from the transmitter, where the surface wave produces an appreciable effect, reception is determined by the propagation of the space wave. In general, the reliable space wave comes from the \(F\)-layer, although strong reflections from the \(E\)-layer are also not uncommon. It may apparently be thought that these irregular reflections are caused by relatively small dense ionic clouds entering the \(E\)-region at random. In other cases there is a general increase in ionization, which shifts the path of the wave for several hours from the \(F\)-region to the \(E\)-region. Since this shift often takes place very rapidly, transitions of this type often occur without severe fading or interruption of communication. Although group \(B\) is sometimes used in transoceanic service, the absorption at grazing incidence is here somewhat greater than in group \(C\).

Group \(C\), extending from 10 to 50 m, is characterized chiefly by a remarkable effect in transmission over long—

still greater distances between different continents and the noticeable effect of “skip distance,” which we mentioned earlier in the historical survey.

To understand this phenomenon of “skip distance,” let us first direct our attention to the nocturnal space wave reflected from the layer \(F\), or, in other words, to the echo that returns to the earth at a point near the transmitter after reflection at almost normal incidence. Since the ionization density during the night slowly decreases, the space wave (“sky ray”) gradually penetrates into the ionized region to ever greater and greater heights. The corresponding delay time of the signal—the echo—thereby gradually increases. An analogous effect can be produced artificially by increasing the frequency of the transmitter, while the ionization density remains essentially constant. In both cases a critical state may be reached in which the slow increase of the “effective height” (measured by the delay time of the radio echo) is interrupted by a sudden upward jump. At that very same moment the strength of the reflected signal rapidly decreases, and suddenly it disappears altogether.

The sudden increase in the echo delay time is not difficult to interpret as evidence of a corresponding decrease in the group velocity of the wave when it enters the ionized \(F\)-layer and passes through a considerable part of its thickness. Similar effects are often produced by the \(E\)-region when the ionization density is close to the critical value, which will be just sufficient to produce reflection at the given frequency. Such phenomena are predicted by the mathematical theory; moreover, it is natural to expect a corresponding decrease in the strength of the signal, caused by attenuation.

By analogy with the behavior observed under similar circumstances in the \(E\)-region, one may suppose that the complete disappearance of the signal corresponds to the complete penetration of the signal through the \(F\)-layer and to the resulting loss of the signal in interstellar space. This is the usual hypothesis; many authors have regarded it as a self-evident fact. Eckersley, however, inclines toward the opposite idea of complete absorption of the wave within the \(F\)-layer. We shall return to this question in section K.

In accordance with both hypotheses, a signal lost in this way can be restored by increasing the angle of incidence beyond a certain critical value. We must, therefore, expect the pattern of the path of the ray shown in Fig. 5.

At points lying at a small distance (say, up to 15 miles) from the transmitter, the surface-wave signal is noticeable, but it is entirely imperceptible at distant points because of very strong weakening. At points lying beyond this small zone of the surface wave, no reliable signal is observed until we reach the sharply outlined boundary curve, where the usual downward-going pro-

spatial wave (“sky ray”). The main receiving zone lies beyond this boundary.

Through careful investigation, these skip-distance phenomena can be detected at night in the wavelength range from 50 to 100 m, but in general such skip distances are short, and the effect is strongly masked by the propagation of the surface wave. However, in the wavelength band 10–50 m the skip distances may be measured in thousands of miles. Here the effect becomes the determining factor in the choice of the best frequency for use in a circuit designed for a specified great distance, at some hour of the day, season, and year.

The “skip zone” is not completely dead. Strong signals give a peculiar reverberating echo, easily detected by an experienced ear in speech transmission or in telegraph transmission. These scattered signals have still not been fully explained; for further details on this question we refer to section R.

Fig. 5. Skip distance

When the frequency of the wave is increased, the boundary of the skip zone moves outward until, finally, the skip zone covers the entire Earth and transmission by reflection from the ionosphere becomes no longer possible. Under ordinary conditions this limit is reached near 10 m. Consequently, wavelengths somewhat greater than 10 m are of great value for daytime transmission over large distances, whereas wavelengths somewhat less than the critical value are, in this respect, entirely useless. The ionization density of the F-layer, however, varies considerably from day to day. Departures from the mean value not infrequently make it possible to obtain unusual transmission on such a wavelength as 8.5 m; it has also been definitely established that when the ionization is exceptionally high, 5-meter signals can be obtained over great distances.

In general, however, in the quasi-optical region, extending from 1 to 10 m, one absolutely cannot rely on transmission by means of

of the ionosphere and transmission by means of a guided surface wave. This region, which seems narrow if measured in wavelengths, covers an enormous range of frequencies and is therefore especially attractive for future television applications. These waves are also particularly convenient for two-way communication with patrolling police cars, since a short vertical rod, which can easily be carried by a moving car, serves well as an undoubtedly effective quarter-wave transmitting antenna. The reliably served area is in practice limited by the territory lying within the optical horizon, and these waves are useful mainly for communication within a single district of a capital city.

Unfortunately, the area over which interference affects quasi-optical transmission is much larger in extent than the reliably served area. Even at the comparatively low power used in modern experimental installations, very strong 5-meter signals are often received considerably below the visible horizon, at points more than 100 miles from the transmitter. This effect is definitely connected with meteorological conditions in the lower layers of the troposphere and is apparently a simple mirage phenomenon caused by temperature inversions, which often occur in the atmosphere at heights of several thousand feet. This type of transmission appears capable of being used as a new meteorological instrument in studying the distribution and motion of air masses. Preliminary experiments carried out by the American Radio Relay League jointly with Harvard University revealed an astonishingly close connection between quasi-optical transmission on the Boston—Hartford path and the corresponding meteorological temperature gradients (“falling speeds”), measured at the Boston airport and at Mitchel Field, near New York. By means of simultaneous field experiments at wavelengths of 1.25, 2.50, and 5.00 m we are at present studying the possible effect of diffraction at hilltops on quasi-optical transmission. It appears that a simple guided wave following the surface of the earth, owing to exceptionally strong attenuation at such high frequencies, is completely ineffective. Mumford believes that the large dipole moment of water-vapor molecules may contribute significantly to the bending of quasi-optical waves. Gradients in water-vapor content usually accompany temperature inversions.

Microwaves, extending from 0.1 to 1.0 m, will probably exhibit features reminiscent of quasi-optical waves. For investigations in this region it is necessary to construct special types of radio tubes. The power output and efficiency obtained up to now are very low, and as yet little is known about the transmission itself. However, two-way communication across the English Channel has been successfully established on waves 17 cm long; other experiments are also being developed.

D. Elementary Theory

A detailed study of the action of the ionosphere requires a complex mathematical formulation, but the basic facts can be explained in a very simple way.

Let us consider the action of a single electron placed between the plates of a capacitor. If the frequency of the alternating potential difference applied to the plates of the capacitor is low, and if the motion of the electron is constrained by elastic forces, then the displacement of the electron in phase essentially coincides with the applied potential difference; the oscillating charge produces an alternating current in phase with the Maxwell displacement current through the otherwise empty space. In other words, the (negative) electron approaches the positive plate of the capacitor at the peak of the positive half-cycle. It therefore neutralizes part of the charge on the plates of the capacitor and allows the potential difference present to drive a larger charging current through the external circuit. In such cases we usually say that the presence of the bound electron has increased the dielectric constant of the region penetrated by the electric field.

On the other hand, if the elastic coupling acting on the electron is zero, or if the frequency is so high that the force of inertia predominates, the phase relation will be opposite, and the oscillating charge may lower the effective dielectric constant of the medium to a value less than unity. This effect is well known to us in optics, since it provides the generally accepted elementary “resonance” explanation of the anomalous dispersion observed on the high-frequency side of an absorption line in the optical spectrum. Friction arising as a result of collisions, if present, introduces a resistance term which diminishes the magnitude of the change in the dielectric constant, but cannot change its sign.

Exactly the same phenomena determine the basic conditions governing the propagation of a radio wave in the ionosphere. As a result of the action of free electrons set in motion by the electromagnetic field of the wave, the dielectric constant and the corresponding refractive index of the ionized region prove to be less than unity. The ionized layer is, consequently, a medium optically less dense than the non-ionized layers of air lying below it. If the boundary between the media is sufficiently sharp (compared with the wavelength of the electromagnetic radiation), then total internal reflection occurs at it: rays meeting the layer at angles greater than the critical angle will be strongly reflected back toward the earth.

If, however, the boundary between the two media is not sharp, then the incident ray is gradually refracted, deviating farther and farther from the normal and describing a curve that depends on the rate of increase of the density of the free electrons. If the total increase in the density of the free electrons proves sufficient, then the ray

will in the end assume a horizontal direction, and then will follow the symmetric path going downward, which brings it back to the earth. If the maximum density is insufficient, the ray will penetrate through the layer. It is obvious that for a layer with a given maximum density (and for a wave of a given frequency) there is a definite critical angle of initial incidence, which determines whether or not the given ray will return to the earth. The study of this type of refraction may be regarded as a detailed investigation of the mechanism responsible for the reflection of an electromagnetic wave from a conducting surface. When the boundary of the conducting layer is not sharply defined, it is sometimes advantageous to consider the actual curved path of the ray. For many purposes, however, it is sufficient to replace the curved path

Fig. 6. Relation between the actual and equivalent path of the ray

\(ABCDE\) by the broken equivalent path \(ABC'DE\), shown in Fig. 6, and to speak of the “equivalent height” of a certain imaginary reflector. We may use the expression “actual height,” referring to the height at which the maximum density of free electrons is encountered. In the particular case of a ray which meets the layer at the critical angle, this height coincides with the maximum height of the curved ray. All other completely reflected rays will be situated below this height.

Breit and Tuve \(^{58}\) drew attention to the fact that the time required for a definite radio signal to traverse the path \(ABCDE\) through the actual ionized medium is exactly equal to the time required for it to traverse the path \(ABC'DE\) in empty space. Consequently, the equivalent height, determined geometrically in Fig. 6, is measured quite correctly by direct observations of echo-delay times, without corrections for the reduction of the group velocity, which retards the signal in the ionized layer. The equivalent heights measured by this method therefore coincide with the equivalent heights determined geometrically by measuring the angle of arrival of the returning ray.

The actual height of the layer and its ionization gradient obviously cannot be determined directly on the basis of any single observation. Conclusions about these quantities must be drawn indirectly from a series of observations of equivalent height, in which different frequencies or different paths, or both, are used. There is reason to think that the lower boundary of the \(E\)-layer is expressed relatively sharply and that the measured equivalent height exceeds the actual height by only a few percent. The boundary of the region is more diffuse, and the actual heights for it are still not well known. Preliminary data seem to indicate that the equivalent heights may exceed the actual heights by at least \(25\%\). In the literature, in cases where this is not explicitly indicated, the data for the height of the layer almost always refer to equivalent heights determined from direct experimental observations.

Neglecting for the moment the changes introduced by the terrestrial magnetic field and by friction resulting from electron collisions, we can easily derive a simple formula for the refractive index of an ionized medium. A more detailed analysis will be given in Section H.

Pedersen \(^{59}\) proposed the following simplified exposition, based on elementary considerations and giving the well-known equation of Eccles \(^{24}\) and Larmor \(^{13}\).

Consider a capacitor with plates of unit area, separated in vacuum by a distance of \(1\ \mathrm{cm}\). A unit capacitor, provided with the appropriate guard rings, has a capacitance of \(\dfrac{1}{4\pi}\) absolute units.

Let us now introduce into the volume of \(1\ \mathrm{cm}^{3}\) enclosed between the plates of the capacitor \(N\) electrons, each with charge \(e\) and mass \(m\), and apply to the capacitor a variable potential difference with instantaneous value \(v\), angular frequency \(\omega\), and amplitude \(V\). The electrons will be set in motion by the electric field; their velocity will lag in phase by a quarter period behind the applied potential difference; its maximum value will be

\[ U=\frac{e}{\omega m}V. \]

But \(N\) charges, each of magnitude \(e\), moving all together with velocity \(u\), are equivalent to a current element \(1\ \mathrm{cm}\) long with strength

\[ i_e=Ne\cdot u. \]

The amplitude of this lagging alternating current is therefore

\[ I_e=NeU=\frac{Ne^{2}}{\omega m}V=\frac{V}{\omega \dfrac{m}{Ne^{2}}}, \]

and the oscillating electrons give the same effect as an equivalent self-inductance of \(\dfrac{m}{Ne^2}\) absolute units, shunting the condenser.

As for the total current in the external circuit, this combination is equivalent to a condenser with capacitance

\[ C'=\frac{1}{4\pi}-\frac{Ne^2}{\omega^2 m}, \]

or, consequently, to a unit condenser with dielectric constant

\[ \varepsilon=1-\frac{4\pi Ne^2}{\omega^2 m}. \]

In this equation \(\omega\) and \(N\) may be regarded as independent variables, although in an experimental setting only \(\omega\) can be varied.

Obviously, the theory also predicts an additional decrease of the dielectric constant as a result of the presence of heavier ions of either sign, and we could easily introduce additional terms of analogous form in order to take this effect into account. Since, however, the ion mass occurs in the denominator of the term that is subtracted, a single electron is more effective than a huge number of heavy ions. Nevertheless, some physicists believe that hydrogen ions can produce a noticeable effect in the ionosphere. This possibility will be considered in one of the later sections. It is also clear that the increase in the dielectric constant that takes place during the night may be the result, chiefly, of the neutralization of electrons by positive ions, or else, chiefly, the result of the simple capture of electrons by heavier neutral atoms or molecules.

Since the number \(N\) may become very large, it is clear that the dielectric constant may decrease to zero and even change its sign to the opposite one. Such a phenomenon frequently occurs in the ionosphere in the frequency ranges used in radio communication practice. It is therefore of interest to investigate the effect of this change of sign of the dielectric constant on the refractive index.

Pedersen points out that the usual approximate expression \(n=\sqrt{\varepsilon}\) is valid only for positive values of \(\varepsilon\). This is a very serious limitation, since many authors have assumed that a negative dielectric constant necessarily implies an imaginary refractive index. In a non-magnetic conducting medium the correct relation is the following:

\[ n=\left[\frac{\varepsilon}{2}+\left\{\frac{\varepsilon^2}{4}+\left(2\pi\frac{c^2\sigma}{\omega}\right)^2\right\}^{\frac{1}{2}}\right]^{\frac{1}{2}}. \]

There, where the conductivity may be neglected, this expression reduces to the following:

\[ n=\left[\frac{\varepsilon}{2}+\left(\frac{\varepsilon^{2}}{4}\right)^{\frac{1}{2}}\right]^{\frac{1}{2}}, \]

which gives

\[ \begin{aligned} n&=\sqrt{\varepsilon}\quad &&\text{for } \varepsilon \ge 0,\\ n&=0 &&\text{for } \varepsilon \le 0. \end{aligned} \]

We may also note that the phase velocity \(V_p\) of a wave in the ionosphere is given by the expression

\[ V_p=\frac{c}{\left(1-\frac{4\pi Ne^{2}}{\omega^{2}m}\right)^{\frac{1}{2}}}, \]

where \(c\) is the speed of light.

The velocity \(V_p\) (exceeding the speed of light) is a quantity that cannot be measured directly. The speed of propagation of the observed radio signal is measured by the group velocity \(V_g\), where

\[ V_g=\frac{V_p}{\left(1-\frac{\omega}{V_p}\cdot\frac{dV_p}{d\omega}\right)} = c\left(1-\frac{4\pi Ne^{2}}{\omega^{2}m}\right)^{\frac{1}{2}}. \]

The presence of free ions in the medium therefore slows the signal.

In the case of the ionosphere, these considerations have a direct application to the most important experimental facts encountered in ordinary radio communication; later, however, we must consider the changes caused by the Earth’s magnetic field and by the friction arising as a result of collisions.

When the refractive index of the ionized layer approaches zero, the same thing also occurs for the critical angle. We may expect that, in the limiting case, total internal reflection will occur even at normal incidence. We may express this prediction in another language by saying that the jump zone, for a sufficiently high electron density in the layer or for a sufficiently low transmitter frequency, shrinks to zero. These expectations are well confirmed by experiment.

It is therefore customary to say that a radio signal directed vertically upward penetrates into the region of increasing density of free electrons and propagates with ever smaller group velocity until, finally, it reaches such a height where the electron density is just sufficient to reduce the refractive index of the ray to zero. At this point, owing to total internal reflection, the direction of the ray is reversed, and the signal propagates downward with increasing velocity, reaching the speed of light when the ray emerges—

will emerge from the ionized layer. If the concentration of free electrons in the densest part of the layer is not quite sufficient to reduce the refractive index to zero, then the vertical ray passes through the entire ionized region, and the signal emerges above it with the velocity of light. At a somewhat greater height it may then encounter a denser layer, which will return it back to the Earth; in this case the ray travels to the receiving antenna through the lower layer a second time. When the maximum electron density in the lower layer is only slightly less than the critical value, the signal is strongly delayed by this roundabout journey through the lower layer; the equivalent height of the upper layer may, as a result, prove to be abnormally increased by at least 100%. Such effects, however, are readily detected and yield to interpretation when they occur in a whole series of experimental observations.

Although this elementary explanation of reflection at vertical incidence in a diffuse medium appears quite natural and agrees well with experimental observations, it cannot be regarded as a rigorous and complete theory of the phenomenon in question. We have no obvious right to generalize the simple laws of total internal reflection and to apply them so confidently to a region possessing continuously varying optical properties. Various attempts to overcome this difficulty will be considered in Section G; however, this entire theoretical problem as a whole is difficult and has not yet been solved by a method that would be satisfactory in all respects.

(Conclusion in the next issue)

1. Rev. Mod. Phys., 9, No. 1, Jan. 1937. Translated by P. N. Uspensky. ↩