Refraction of Ultrashort Radio Waves in the “Undisturbed” Troposphere

B. A. Vvedenskii and A. G. Arenberg, Moscow

Introduction

1. Experimental data concerning the propagation of ultrashort radio waves (u.s.w.) over short distances are in good agreement with theoretical data obtained under the assumption of a flat terrestrial surface. Ten years ago this could have led to the erroneous notion of a limitation on the possible range of communication on u.s.w. by the horizon line \(ACB\) (Fig. 1), to which there corresponds the well-known “horizon distance”

\[ R_0=\sqrt{2a}\,(\sqrt{h}+\sqrt{z})^{1)}. \tag{1} \]

Fig. 1.

Such a view was in fact expressed (approximately up to 1930) by many authors who studied, with comparatively primitive apparatus, the propagation of meter waves over relatively small distances \(^{2)}\). At the same time there also arose the opinion of the complete stability of communication on u.s.w., expressed in the absence of fading and in the independence of transmission from time and hydrometeorological factors.

However, already beginning in 1930, reports began to appear in the literature on communication at u.s.w. over distances exceeding the distance to the horizon. Among the earliest reports of this kind is the paper by Joua \(^{2}\), describing experiments on communication at

\(^{1)}\) Here \(R_0\), \(h\), and \(z\) are expressed in kilometers. If \(h\) and \(z\) are expressed in meters, then
\[ R_0=3.57(\sqrt{h}+\sqrt{z}). \]

\(^{2)}\) As a curiosity one may point out that similar statements are encountered up to the very latest time. Evidently this explains the emphasized insistence of van der Pol and Bremmer \(^{1}\), with which they note the absence of “breaks” in diffraction curves as they pass through the horizon.

meter waves between France and Corsica; in this article the instability of reception beyond the horizon was noted for the first time. All information concerning long-distance communications on u.s.w. up to about 1935 was, in essence, of a purely qualitative character (for the most part—aural reception) and did not provide sufficient grounds for establishing any quantitative relationships. Therefore, without entering into an analysis of these works, we shall confine ourselves merely to a reference to our book³, in which a summary of the principal observations of this period is given¹).

In recent years, in connection with the use of u.s.w. for television, directional transmissions, radio navigation, and the development of amateur radio, a considerable number of works have been published devoted to the experimental study of the propagation of u.s.w. These works were carried out with modern apparatus and therefore already provide rather substantial quantitative material, amenable to definite systematization and processing.

Fig. 2. Dependence of the voltage at the receiver output on time

\(\lambda = 7.17\ \text{m}\); transmitter height \(1\,000\ \text{m}\); reception on the ground; distance: \(a\)—\(780\ \text{km}\), \(b\)—\(200\ \text{km}\) (Ochmann and Plendl)

The corresponding materials very often relate to the propagation of u.s.w. over relatively large distances, exceeding the distance to the horizon. In analyzing them we encounter the following two facts, which do not fit within the framework of purely diffraction theories.

First of all, in radio transmissions over such distances the actually observed fields are sometimes many times greater than the fields calculated by exact diffraction formulas; furthermore, the observed fields are, as a rule, subject to very considerable and at the same time extremely irregular variations, whose duration fluctuates within very wide limits—from days and hours to minutes and seconds.

1. To illustrate these propositions, which are now already sufficiently well known, we shall cite some data from 1938 from the article by Ochmann and Plendl⁹. In Fig. 2, \(a\), the change in voltage at the receiver output as a function of time is given for transmissions on a wave of length \(7.17\ \text{m}\). The transmitter, with radiated power of about \(2.5\ \text{kW}\), was located on a hill about \(1\,000\ \text{m}\) above the average level of the terrain. The antenna was a vertical half-wave vib-

¹) See Table 18, containing the principal data of the experiments of Marconi⁴, VEI⁵, Gershenberg⁶, Lindenblad⁷, and NK Svyazi⁸.

rator. Reception near the surface of the earth took place at a distance of about 780 km, exceeding the horizon distance by a factor of 6.8. The curve shows a number of sharp, short-term field-strength enhancements, which during a considerable part of the observation time were very few.

Fig. 2, b relates to the case of reception at a distance of 200 km (1.75 times the horizon distance, the other conditions being the same), in which reception is considerably more stable. Fig. 3 shows the dependence of the reception strength on altitude during reception on an aircraft at a distance of 480 km. As can be seen, when the altitude changes from 2,500 m to 3,500 m (reception beyond the horizon), the field passes through a number of maxima, which are unstable in time. The authors indicate that when receiving at an altitude of 2,000 m a range of 800 km was obtained (with a horizon distance of 305 km).

Fig. 3. Dependence of the voltage at the receiver output on altitude

\(\lambda = 7.17\) m; transmitter height 1,000 m; distance 480 km (Oxman and Plendl)

The second transmitter, with a power of 35 W, operated on the wave \(\lambda = 4.1\) m. The height of this transmitter above the average level of the terrain was about 980 m. The antenna consisted of twenty vertical vibrators arranged in a row at a distance of a half-wave from one another. Reception on an aircraft flying at an altitude of 4,000 m was possible at distances up to 680 km (horizon 114 km).

The third transmitter, operating on the same wave and with the same power, was located on a hill 66 m high and had an antenna of sixteen horizontal vibrators (8 tiers of 2 vibrators in each, the distance between tiers being a half-wave). Reception on the ground, on an antenna of four horizontal vibrators, was possible at a distance of 120 km, i.e., beyond the horizon.

1. However, the communication ranges at ultra-short waves cited above are by no means record ones. In recent years, reports have appeared in the literature of numerous instances of reception of waves longer than 5–6 m at still greater distances, reaching several thousand kilometers.

As is known, the attainment of these ranges is explained by the fact that in 1937–1939 there was a maximum of ionization caused by the increase in solar activity. The consequence of this was an increase in the refractive capacity of the ionosphere, ensuring the passage of such short waves which in “normal” epochs of the solar cycle were not regularly refracted by the ionosphere. Relatively

of the passage of such waves in previous years there were only scattered and quite insufficient indications1.

At present it has become completely clear that reception of ultra-short waves at distances exceeding the distance to the horizon many times over is due not to ionization of the atmosphere, but to other causes connected with its inhomogeneities of an entirely different physical nature. There is now every reason to regard these inhomogeneities (changes in the density of the air, its temperature and humidity with height, etc.) as localized in the lower layers of the atmosphere.

Therefore the communication ranges on ultra-short waves that are due to these causes will hereafter be called “tropospheric ranges.” In contrast to this, communication ranges of the order of thousands of kilometers will be called “ionospheric ranges.” Finally, ranges at which the propagation of ultra-short waves is wholly determined by diffraction and the influence of the atmosphere is still practically absent will be called “diffraction ranges.”

In considering questions of radio-wave propagation in the ionosphere, the concepts of an “undisturbed” and a “disturbed” ionosphere are widely used. It seems expedient to us to transfer these concepts also to the case of propagation of ultra-short waves in the troposphere. In doing so, by the term “undisturbed troposphere” we shall mean that somewhat idealized case in which there are no sharp changes in the density, temperature, and humidity of the air either in time or in space. Taking into account the influence of such a troposphere on the propagation of ultra-short waves can provide useful material on certain average radio-transmission ranges and field strengths conditioned by such “average” refraction. By the term “disturbed troposphere” we shall mean the typical case of the presence of sharp inhomogeneities that are unstable in time, when the variability of tropospheric conditions also entails deviations of the field strength from its “average” values.

The abundance of material relating to the questions under consideration, its lack of system, and its frequent inconsistency prompted us to attempt to systematize this material. The present article is devoted to the question of taking into account the mean refraction of ultra-short waves in the troposphere. Questions connected with the fine structure of the troposphere and with the stability of communication on ultra-short waves will be considered separately.

§ 1. THE CONCEPT OF “EQUIVALENT EARTH RADII”

1. The question of the influence of the atmosphere on the propagation of radio waves was first considered by Eccles[^10], Kiebitz[^11], and Fleming[^12] as early as 1913–1914. These authors came to the conclusion that ordinary refraction of radio waves (without taking ionization into account) does not explain the very

large communication ranges on long waves. Stewart, Petry, and Willmott[^13] came to a similar conclusion in 1927; they found that the change in the refractive index of air with height, associated with the change in its density, is insufficient to explain the propagation of radio waves over great distances. Becker[^14], who in 1927 considered the question of the refraction of radio waves in the atmosphere, emphasized the importance of establishing the dependence of the refractive index of air on height. Knowledge of this dependence is necessary for determining the trajectories of radio waves in the atmosphere.

The works of the period 1935–1937 proceeded from the usual optical conception of the atmosphere as consisting of concentric layers of air whose refractive index decreases comparatively slowly with height. Electromagnetic radiation issuing from point \(A\), located in such an inhomogeneous atmosphere, passes through layers of air of different density, as a result of which the “rays” are refracted (Fig. 4), a fact already noted by Newton[^15] and later studied by Rayleigh[^16]. This curvature of the ray trajectories leads to a displacement of the horizon line from point \(C\) to point \(C_1\). The further course of the ray shows that radiation can be detected at all points both along the path of the ray (for example, at point \(B\)) and above it1.

Fig. 4. Extension of the horizon caused by refraction

To establish the relation between this distance \((R'_0)\) to the horizon and the elevation height \((h)\) of point \(A\), one usually started from the approximate formula2

\[ h = a\left(\sec \frac{R'_0}{a} - 1\right) - \rho\left(\sec \frac{R'_0}{\rho} - 1\right)\sec \frac{R'_0}{a}. \tag{2} \]

Here \(a\) is the radius of the earth and \(\rho\) is the radius of curvature of the ray.

In those cases where the second communication point (point \(B\)) is also raised above the earth to a height \(z\), expansion in a series for relatively small distances \(R'_0\) gives the well-known expression

\[ R'_0 = \sqrt{2a}\,(\sqrt{h} + \sqrt{z})\sqrt{\frac{\rho}{\rho - a}}, \tag{3} \]

which, for \(\rho = \infty\) (the case of a homogeneous atmosphere), passes into the usual expression (1) for the distance to the horizon.

The determination of the radii of curvature of the rays1 was carried out according to the formula

\[ \rho=-\frac{n^2 r}{n_0 a \sin \varphi_0\left(\frac{dn}{dr}\right)} . \tag{4} \]

Here \(n\) and \(n_0\) are the refractive indices of air at height \(h\) and at the surface of the earth, \(r=a+h\), and \(\varphi_0\) is the angle between the tangent to the ray and the perpendicular to the earth’s surface (“angle of emergence”).

For \(\varphi_0=\frac{\pi}{2}\) (a ray tangent to the earth) and relatively small \(h\), this formula reduces to

\[ \rho \simeq -\frac{n}{\dfrac{dn}{dr}}=-\frac{2\varepsilon}{\dfrac{d\varepsilon}{dh}}; \tag{4'} \]

here \(\varepsilon\) is the dielectric constant of air.

1. To establish the dependence \(n=f(h)\) needed for these calculations, Smit-Rose[^19], following Fleming[^12] and other authors, used the well-known formula of L. Lorenz[^20] and H. Lorentz[^21], which relates the refractive index of a transparent medium to its density:

\[ \frac{n^2-1}{n^2+2}=B\delta,\qquad \text{whence}\qquad n \simeq 1+\frac{3}{2}B\delta, \tag{5} \]

where \(B\) is a certain coefficient characterizing the given medium.

The density of air \(\delta\), which in these calculations was assumed to be dry, was determined from the equation for the equilibrium of a gas in a gravitational field:

\[ \frac{R}{M}\left(T\frac{d\delta}{dh}+\delta\frac{dT}{dh}\right)=-\delta g . \tag{6} \]

Here \(R\) is the universal gas constant, \(M\) the mean molecular weight, \(T\) the absolute temperature, and \(g\) the acceleration due to gravity.

Integration of this equation required knowledge of the dependence \(T=f(h)\). Smit-Rose used the concept of the “international standard atmosphere,” adopted for the calibration of aircraft altimeters2, for which

\[ T=288-0.0065\,h, \]

and calculated the radii of curvature \(\rho\) for heights \(h\) from 0 to 2,000 m.

However, the results of individual measurements cited by Humphreys[^17], Wegener[^23], and other authors showed that at certain heights increases in air temperature (“inversions”) are very often observed. Taking this circumstance into account, we3 in 1936 calculated the radii of curvature of rays (tangent to the earth’s surface) for a number of values of the vertical temperature gradient \(\beta\).

The results of these calculations, presented in Fig. 5, showed that radiation could propagate rectilinearly \(\left(\frac{\rho}{a}=\pm\infty\right)\) only in the hypothetical case when the decrease of temperature

REFRACTION OF RADIO WAVES IN THE “UNDISTURBED” TROPOSPHERE

with height is determined by the value \(\beta = 0.0342\ \mathrm{deg}/\mathrm{m}\). With a more rapid decrease of temperature with height the ray turns its convex side toward the earth \(\left(\frac{\rho}{a}<0\right)\). With a slower decrease of temperature the ray turns its concave side toward the earth \(\left(\frac{\rho}{a}>0\right)\). For the “standard atmosphere” the radius of curvature of the ray is \(\rho = 37\,630\ \mathrm{km}\).

In the case of an increase of temperature with height \(\rho\) decreases

Fig. 5. Dependence of the ratio \(\frac{\rho}{a}\) on the vertical temperature gradient \(\beta\) for \(h=0\). Tangential rays

Fig. 6. Dependence of the ratio \(\frac{\rho}{a}\) on height for different vertical temperature gradients. The points on the curves (at \(h=0\)) correspond to the points shown in Fig. 5

\(1 — T = 288 - 0.02h;\quad 2 — T = 288 - 0.0065h;\quad 3 — T = 288;\quad 4 — T = 288 + 0.0065h;\quad 5 — T = 288 + 0.02h;\quad 6 — T = 288 + 0.1282h\)

and, if the value \(\beta = 0.1282\ \mathrm{deg}/\mathrm{m}\) could occur, the ray would follow the curvature of the earth \((\rho = a)^{1}\). In Fig. 6 the dependence of \(\frac{\rho}{a}\) on \(h\) and \(\beta\) is given for \(\varphi_{0}=0\) for dry air. These curves show that, for \(h\) not exceeding \(1—2\ \mathrm{km}\), the values of \(\rho\) vary within the limits from 4 to 6.

Halbert\(^{24}\) in 1935 noted that the principal cause capable of producing strong refraction of u. s. w. in the troposphere is a possible increase of temperature with height. In determining the value \(\beta\) necessary for obtaining concentric trajectories of tangential rays, Halbert proceeded from the relation\(^{2}\):

\[ \frac{\partial n}{\partial r} = -\frac{n}{r}, \tag{7} \]

where \(r = a + h\).

\(^{1}\) As Fleming\(^{12}\) showed, a similar result would also occur for \(T=T_{0}\), if the earth’s atmosphere consisted not of air but of krypton.

\(^{2}\) This relation is obtained from the condition of equality of the angular velocities of propagation of radiation along concentric circles of different radii \(r\).

For \(n=1.00027\) and \(r=a=6.38\cdot 10^8\ \text{cm}\), this relation gives \(dn=-1.57\cdot 10^{-9}\,dr\). Further, assuming that \(dn=-5.5\cdot 10^{-7}\,dT\), Halbert obtained the value of the temperature gradient \(\beta=0.290\ \text{grad}/\text{m}\). However, our calculations, carried out with allowance for the humidity of the air (see § 2), did not lead us to this value of \(\beta\). The reason for this discrepancy lies in the value of \(\dfrac{\partial n}{\partial T}\) (given by Halbert without derivation), which differs from the value obtained from generally accepted formulas.

1. Skilling, Burrows, and Ferrell \(^{25}\) indicated that the influence of refraction can also be taken into account by another method, based on comparing the curvature of the ray with the curvature of the earth. In Fig. 7,A a part of the trajectory of a ray tangent to the earth’s surface is shown, the curvature of which is \(\dfrac{1}{a}\). Let the radius of curvature of the ray be \(\rho\); let us postulate that \(\rho\) is the same along the entire length of the ray, and introduce the “relative” curvature of the ray, which we define as

\[ \frac{1}{a_e}=\frac{1}{a}-\frac{1}{\rho}. \]

Fig. 7. Transition from curvilinear ray trajectories to rectilinear ones

Let us now imagine such a new coordinate system in which the given trajectory of the ray is transformed from curvilinear into rectilinear (Fig. 7,B). Then, under the condition that the constancy of the “relative” curvature in these two systems is preserved, it should be assumed that in the new coordinate system the ratio of the “equivalent earth radius” (equal to the relative radius of curvature) to the actual radius of the earth is equal to

\[ \frac{a_e}{a}=\frac{\rho}{\rho-a}. \tag{8} \]

Since, in passing to the equivalent radius \(a_e\), it was assumed that the trajectories of the rays (tangent to the earth) are straight lines, then for these rays the refractive index of such a fictitious atmosphere must be regarded as constant. This proposition could also be extended to those rays which at height \(h\) have the exit angle \(\varphi_0=\dfrac{\pi}{2}\). In this case the equivalent radius will be different \(^{1}\). On the basis of these considerations a number of authors have made the bold conclusion that, in order to take into account the influence of refraction, it is sufficient to introduce into

\(^{1}\) Let us note that, with this, the definition of u.s.w. as “quasi-optical” waves (introduced into the radio-engineering literature by American authors) acquires a certain, though admittedly rather conventional, meaning, since, of course, optics cannot be understood as a science dealing exclusively with rectilinear propagation of light.

REFRACTION OF RADIO WAVES IN AN “UNDISTURBED” TROPOSPHERE

all calculations (including diffraction calculations) use, instead of the actual radius of the earth \(a\), its equivalent radius \(a_e\).

However, one can agree with this assertion (and even then with some reservation) only in the case when one is speaking merely of formula (3) for determining the distance to the horizon \(^{1}\), which in this case is reduced to the form

\[ R'_0=\sqrt{2a_e}\,(\sqrt{h}+\sqrt{z}). \tag{9} \]

In the more complicated case, when the corresponding points are raised above the horizon line, the trajectories of the rays (both the direct ray and the one reflected from the spherical earth) are no longer tangent to the earth’s surface. Consequently the values of \(a_e\) corresponding to different points of these trajectories depend not only on \(h\), but also on \(\varphi\). Therefore the replacement of the radius \(a\) in the corresponding “reflection” formulas for calculating the field by the radius \(a_e\), calculated for tangent rays, already gives rise to doubts.

Thus, a rigorous account of refraction in radio communication between corresponding points raised above the horizon line, contrary to the opinion of Englund, Crawford, Mumford \(^{26}\) and other authors, cannot be carried out by the indicated primitive method. Nevertheless, for not very large \(h\) and \(z\), such a replacement is still capable of bringing the calculated results somewhat closer to the experimental ones \(^{2}\). An approach to a more complete solution of this problem, based on calculating the actual ray trajectories, is given in the work of Eckart and Plendl \(^{29}\) (see § 3).

However, Eckersley \(^{30}\), Burrows, Decino and Hunt \(^{31}\), as well as Englund, Crawford and Mumford \(^{32}\), go still further in their arguments and, without proof, hold that by replacing the earth radius \(a\), entering into the diffraction formulas, by equivalent radii \(a_e\), one can take into account the influence of refraction on diffraction, and so on.

This assertion appears to us unfounded, and in general the possibility of such a justification is doubtful (see below).

1. A fundamental shortcoming of all the arguments presented is the complete disregard of diffraction phenomena, which, of course, also occur in the presence of refraction. This shortcoming can be eliminated only by solving the diffraction problem with allowance for an atmosphere whose refractive index varies according to a definite law.

We still do not have any rigorous solution of this very difficult problem \(^{3}\). The only attempt at such a solu-

\(^{1}\) These arguments are not rigorous, since the postulated independence of \(\rho\) from \(h\) does not correspond to reality (see Fig. 6).

\(^{2}\) See, for example, the curves of Englund, Crawford and Mumford \(^{27}\), reproduced in our book \(^{3}\) on p. 147. It is important to point out here that, when approaching the horizon, these curves are incorrect, since in calculating them the influence of the earth’s curvature was insufficiently strictly taken into account. On this point see the articles by Vvedenskii \(^{28}\) and by van der Pol and Bremmer \(^{1}\).

\(^{3}\) A certain step in this direction was made by Greenberg \(^{33}\), who solved the well-known Sommerfeld problem of the propagation of radio waves over a flat earth for the case of an optically inhomogeneous atmosphere.

tion is the work of Eckersley and Millington,^34 who applied the “phase-integral method.” Since in the field of ordinary (non-refractive) diffraction the “phase-integral method,” with the known reservations, is capable of giving the same results as rigorous diffraction theories,^1 it is in principle possible to expect from this method a sufficiently satisfactory solution of this problem as well.

However, a detailed consideration of purely diffraction calculations based on the “phase-integral method” leads to the conclusion^34, ^35 that the “phase-integral method” attains correct results only at the cost of a frank adjustment of certain fundamental constants that follow from a rigorous Watson treatment. Since, as was said above, a rigorous joint treatment of diffraction and refraction is as yet lacking, there remains a legitimate uncertainty as to the degree of approximation, to the true picture of the phenomena, of the results given by the “phase-integral method” when refraction is taken into account.

This situation is further aggravated by the circumstance that, for the sake of mathematical simplification of the exposition, Eckersley and Millington resort to the conception of a constant gradient of the refractive index, independent of height. Nevertheless, with all these reservations, the indicated attempt is still the only one in this field, and therefore we consider it necessary to present it.

Eckersley and Millington treat the question of refraction under consideration on the basis of their fundamental formulas of the “phase-integral method”:

\[ \left. \begin{aligned} &2\left(\frac{2\pi}{\lambda}\right)\int_a^{r_1} \left[1-g(r)-\left(\frac{\lambda}{2\pi r}\right)^2 n_m(n_m+1)\right]^{1/2}dr =2\pi m+\Omega+\chi,\\ &\text{and}\\ &1-g(r)-\left(\frac{\lambda}{2\pi r}\right)^2 n_m(n_m+1)=0 \quad \text{for } r=r_1, \end{aligned} \right\} \tag{10} \]

where \(g(r)\) takes account of refraction in the atmosphere; \(n_m\) are the proper values of the problem, which can be brought into close correspondence with the proper values of the diffraction problem in Watson’s treatment; \(m\) is the natural series of numbers; \(\Omega\) is an arbitrary constant which, according to the logic of the “phase-integral method,” should be equal to \(\frac{\pi}{2}\), but, for convenience in fitting the results to Watson’s formulas, Eckersley and Millington leave this constant undetermined; \(\frac{\chi}{2\pi}\) is the phase shift upon reflection from the interface (the earth’s surface).

Abstracting from real actuality, these authors restrict themselves to the case of a constant gradient of the refractive index \(n\). They introduce it in the following way. By choosing such a dependence \(n(r)\) which, in the end, would satisfy this condition and in

^1 See the article by Vvedenskii.^35

at the same time would make it possible to apply the “method of phase integrals,” they assume that

\[ n^2 = 1 - g(r) = 1 - \eta + \beta \frac{a^2}{r^2}, \]

whence

\[ \frac{\partial n}{\partial r} = -\frac{1}{n}\,\beta \frac{a^2}{r^3}; \tag{11} \]

here \(\eta\) and \(\beta\) are certain constants.

Since for the earth’s surface \(n = 1\), and for small heights \(r = a\), then

\[ \frac{\partial n}{\partial r} \cong -\frac{\beta}{a}, \]

i.e. the desired constant gradient is obtained.

Moreover, the curvature of the ray is determined as

\[ \frac{1}{\rho} \cong -\frac{1}{n}\frac{\partial n}{\partial r} \cong -\frac{\partial n}{\partial r} \]

[see (4)], and consequently, \(\rho = \dfrac{a}{\beta} = \mathrm{const}\), as indeed must be the case for a constant gradient at small \(h\).

Since Eckersley and Millington postulate the constancy of the gradient of \(n\), then on the basis of the arguments of Skilling, Burrows, and Ferrell\({}^{25}\) they take \(\rho\) to be of the order of \(5a\), whence \(\beta\) is obtained of the order of \(0.2\). In addition, at the earth’s surface \(n = 1.00029\), and therefore, on the basis of expression (11), we have

\[ 1 - \eta + \beta = 1.00058, \]

from which it follows that \(\eta\) differs very little from \(\beta\).

Therefore the basic formulas of the “method of phase integrals” (10) take the form

\[ \left. \begin{aligned} &2\left(\frac{2\pi}{\lambda}\right) \int_a^{r_1} \left[ 1-\eta+\beta\frac{a^2}{r^2} - \left(\frac{\lambda}{2\pi r}\right)^2 n_m(n_m+1) \right]^{1/2} \,dr = 2\pi m+\Omega+\chi \\ &\text{and} \\ &1-\eta+\beta\frac{a^2}{r_1^2} - \left(\frac{\lambda}{2\pi r_1}\right)^2 n_m(n_m+1) = 0, \end{aligned} \right\} \tag{12} \]

as a result of which the first equation (12) assumes the form

\[ 2\left(\frac{2\pi}{\lambda}\right)(1-\eta)^{1/2} \int_a^{r_1} \left[ 1-\left(\frac{r_1}{r}\right)^2 \right]^{1/2} \,dr = 2\pi m+\Omega+\chi. \]

Applying subsequently a series of artificial transformations and simplifications characteristic of the method of phase integrals, the authors obtain an expression for the exponential factor taking into account the attenuation of the field strength with distance in the form:

\[ \exp\left[ -\left(\frac{2\pi}{\lambda}\right)^{1/3} \left(\frac{1-\eta}{a}\right)^{2/3} \rho R \sin \frac{\pi}{3} \right]. \tag{13} \]

In this formula, unlike all the others, \(\rho\) is an expression known from diffraction theory, equal to \(\frac{1}{2}(3\pi)^{2/3}(m+1)^{1/3}\); \(R\) is the distance along the earth.

But it is obvious that the same expression can also be written differently, replacing the radius of the earth \(a\) by a certain equivalent radius

\[ a_e=\frac{a}{1-\eta}. \tag{14} \]

Thus Eckersley\(^{30}\) considers it possible to take refraction into account by replacing the true radius of the earth, entering into the diffraction formulas, by a certain equivalent radius. This position found expression in the materials of the fourth session of the International Radio Consultative Committee (CCIR) (Bucharest, 1937)\(^{36}\) and in the report of the CCIR Commission on radio-wave propagation (London, 1938)\(^{37}\). According to Eckersley, this replacement leads to a reduction of the slope of the rectilinear part of the diffraction curves (drawn on a semilogarithmic scale) in the ratio

\[ \left(\frac{a}{a_e}\right)^{2/3}=(1-\eta)^{2/3}. \tag{15} \]

This relation can be obtained from the diffraction formulas at the cost of certain assumptions, on which we shall not dwell here.

§ 2. THE INFLUENCE OF AIR HUMIDITY

1. Most of the calculations presented up to now corresponded to the case of dry air. However, in reality water vapor is always present in the air, and sometimes water in the form of droplets as well. Furthermore, since the source of the moisture present in the atmosphere is the earth’s surface and since the mean temperature of the air, generally speaking, decreases with height, it is natural that the amount of water vapor in the atmosphere and its vapor pressure should also decrease with height. Typical data for the vapor pressure of water vapor at various heights may be found, for example, in Humphreys (see \(^{17}\), pp. 76–77) and in other authors.

On the basis of observations made during flights, Söring\(^{38}\) gave an empirical formula for the dependence of the vapor pressure of water vapor on height, on the basis of which it may be shown that half of the total amount of water vapor present in the atmosphere is contained in a layer only about \(1700\ \text{m}\) high\(^{1}\). Naturally, one should suppose that the presence of water vapor in the air must affect changes in its dielectric coefficient. In the first experimental investigations of this question it was found—

\(^{1}\) In the literature there are some indications that the humidity of the air at a height of the order of \(15000\ \text{m}\), though small, is nevertheless somewhat greater than its calculated value\(^{39}\). However, analyses of stratospheric air obtained by G. Prokofiev, K. Godunov, and Birnbaum\(^{40}\) for a height of \(18800\ \text{m}\) showed the absence of water. Apparently this question still cannot be considered fully resolved.

it has been found that at certain temperatures an anomalous course is observed in the curves of the dependence of the dielectric constant of moist air on pressure, the cause of which for a long time remained unexplained. Therefore the admissibility of applying to moist air the ordinary additive law, which permits one to determine the value of the dielectric constant of a mixture as the result of summing analogous quantities for its separate parts (see below), was in doubt. Thus, Delsemme, Tansan, and Hirsch^41, who investigated the question of the effect of humidity on the dielectric constant of air (at temperatures from 15 to 25°), came to the conclusion that the additive law is inapplicable. However, the work of Zahn^42, which brought considerable clarity to this question, showed that the anomalies which occurred in experiments with pure water vapor were due to the condensation of this vapor on the plates of the measuring capacitors, and not to the association of molecules, as Iona^43 had supposed.

Therefore, on the basis of Zahn’s results, one may consider that these anomalies cannot serve as a sufficient reason (at any rate, at the frequencies of interest to us) for denying the applicability of the law of mixtures to the case of air and water vapor.

For the calculations we shall use Debye’s expression^44, which relates the dielectric constant \((\varepsilon)\) of a substance to the constants characterizing its molecular structure:

\[ \frac{\varepsilon-1}{\varepsilon+2}\cdot\frac{M}{\delta} = \frac{4\pi}{3}A\left(a_0+\frac{\mu^2}{3kT}\right). \tag{16} \]

Here \(M\) is the molecular weight, \(\delta\) the density, \(A=6.06\cdot10^{23}\) Avogadro’s number, \(a_0\) the polarizability of the molecule, \(\mu\) the dipole moment of the molecule, \(k=1.37\cdot10^{-16}\) Boltzmann’s constant, and \(T\) the absolute temperature.

Since, practically for both air and water vapor, it may be assumed that \(\varepsilon+2 \simeq 3\), then, with sufficient approximation,

\[ \varepsilon-1 = 4\pi A\left(a_0+\frac{\mu^2}{3kT}\right)\frac{\delta}{M}. \tag{17} \]

Further, treating the components of the mixture as ideal gases (see ^3, p. 260) and passing to the pressure \(p\), expressed in mm Hg, we obtain that expression (17) can be written as

\[ \varepsilon-1=K\frac{p}{T}, \]

where

\[ K=1.215\cdot10^{20}\left(a_0+\frac{\mu^2}{3kT}\right). \tag{18} \]

According to the available data, the value of \(K\) for air^1 may be taken as

\[ K_{\mathrm{air}}=2.11\cdot10^{-4}. \]

^1) It can be shown that the values of \(\varepsilon\) of air determined from formula (18) with the above value of \(K\) agree well with the values of \(\varepsilon\) determined from formula (5), into which enters the constant \(B\), adopted by us according to Magri^45 as equal to 0.15104.

As for the value of \(K\) for water vapor, Englund, Crouford, and Memford \(^{26}\), on the basis of the work of Zahn \(^{42}\), Ions \(^{43}\), Stuart \(^{46}\), Zenger \(^{47}\), and Stranathan \(^{48}\), determine it as

\[ K_{\text{water vapor}}=1.82\cdot 10^{-4}\left(1+\frac{5583}{T}\right). \]

According to another, later work by Stranathan \(^{49}\), the value of \(K\) for water vapors can be obtained from the formula

\[ \left(\frac{\varepsilon-1}{\varepsilon+2}\right)\frac{RT}{p} =4.03\pm0.39+\frac{20710\pm140}{T} \]

(here \(p\) is in dynes/\(\text{cm}^{2}\)). Expressing \(p\) in millibars, Weinik \(^{50}\), starting from this formula, obtained the expression

\[ \varepsilon-1=1.37\cdot10^{-4}\left(1+\frac{5430}{T}\right)\frac{p}{T}, \]

which gives a value of \(K\) very close to that given above.

Applying the additive law for the value of \(K\) characterizing \(\varepsilon\) of moist air, Englund, Crouford, and Memford \(^{26}\) obtain

\[ K=\left[211+\xi\left(\frac{10159}{T}-0.293\right)\right]10^{-6}; \tag{19} \]

here \(\xi=\dfrac{p_{\text{water vapor}}}{p_{\text{air}}}\cdot100\) is the percentage content of water vapor in the air, defined as the ratio of the elasticity of the water vapor to atmospheric pressure.

Some values of \(\varepsilon-1\) for moist air, calculated by formulas (18) and (19), are given in the article by Volpert \(^{51}\). Let us also note that the recently appeared work of Tredgeadge \(^{52}\) contains results of measurements of \(\varepsilon\) of water vapor at a frequency of 42 MHz. At 760 mm Hg and temperatures of 71.9, 99.8, and 147°C he obtained values of \(\varepsilon\) respectively equal to 1.0071, 1.0060, and 1.00475.

1. From expressions (4) and (18) it follows that the radius of curvature of a ray traveling in an atmosphere containing water vapor can be determined as

\[ \rho= -\frac{ M\left(\dfrac{62370}{M}K\delta+1\right)^{3/2}(a+h) }{ 31185a\sin\varphi_{0}\left(\dfrac{62370}{M_{0}}K_{0}\delta_{0}+1\right)^{1/2} \dfrac{\partial}{\partial h}(K\delta) }. \tag{20} \]

Here \(K\) is given by expression (19), \(M\) and \(\delta\) are the molecular weight and density of moist air; the zero subscripts correspond to height \(h=0\).

For the simplest case, when \(\varphi_{0}=90^\circ\) and \(h\cong0\), formula (20) is simplified and brought to the form

\[ \rho= -\frac{32.065\,M}{ 211\dfrac{\partial\delta}{\partial h} +\left(\dfrac{10159}{T}-0.293\right) \left(\xi\dfrac{\partial\delta}{\partial h} +\delta\dfrac{\partial\xi}{\partial h}\right) -\dfrac{10159}{T^{2}}\xi\delta\dfrac{\partial T}{\partial h} }. \tag{21} \]

Thus, knowledge of the vertical gradients of density, humidity, and temperature \(\left(\dfrac{\partial\delta}{\partial h},\ \dfrac{\partial\xi}{\partial h}\ \text{and}\ \dfrac{\partial T}{\partial h}\right)\) is necessary for the calculation.

Basing themselves on the mean (summer) data for the dependences of $\delta$, $\alpha$ and $T$ on height, given by Humphreys (see $^{17}$, pp. 48 and 77), Englund, Crawford and Mumford found that for the layers of air directly adjacent to the earth ($h < 2 \text{ km}$) one may take, expressing $h$ in kilometers:

\[ \delta = 0.001224 - 0.0001156\,h, \]

\[ T = 288 - 3.87\,h, \]

\[ \xi = 1.372 - 0.253\,h. \]

Substitution of these values into formula (21) gave the value of the radius of curvature $\rho = 24\,000 \text{ km}$. Comparing this result with the result obtained by us in calculating $\rho$ without allowance for humidity for the standard atmosphere, we see that this allowance leads to a decrease of the value of $\rho$ by a factor of 1.57, which is quite substantial.

There are also known results of calculations of $\rho$, carried out by Schelling, Burrows and Ferrell $^{25}$ for heights $h < 500 \text{ m}$. Table 1 gives the values obtained by them for the radii of curvature of rays tangent to the surface of the earth, and the corresponding equivalent earth radii.

Table 1

Conditions	$\rho$ in km	$a_e$ in km	$\dfrac{a_e}{a}$
Mean value for summer with allowance for humidity . . . . .	23 800	8 650	1.36
Mean value for winter with allowance for humidity . . . . .	26 500	8 420	1.32
Mean annual value with allowance for humidity $^{1}$) . . . . .	25 500	8 500	1.33

Very interesting experimental data relating to the questions under consideration are contained in the work of Englund, Crawford and Mumford $^{26}$, who describe their experiments on the propagation of ultrashort waves over the sea. This article presents a large number of curves which definitely indicate the influence of the atmosphere.

One of the most striking examples of such an influence may be the curves of the dependence of the field on distance, shown in Fig. 8, corresponding to the case of transmission at a wavelength $\lambda = 4.6 \text{ m}$ with vertical polarization.

From comparison of these curves, obtained on different days, we see that at relatively small distances (approximately not exceeding 65 km) all three curves are grouped quite well about the theoretical curve constructed according to the usual reflection formula for a spherical earth (with equivalent radius $a_e = 8500 \text{ km}$).

$^{1}$) If this “mean annual” value of $a_e$ is used, then from formula (9) we obtain

\[ R_0' = 4.12\left(\sqrt{h} + \sqrt{z}\right). \]

This corresponds to an increase of the distance to the horizon by 15.4%.

With a further increase in distance these curves diverge, and all of them lie above the curve constructed from the reflection formula. The slowest decrease of the field with distance occurred in the experiments carried out on September 27, 1933, when the water-vapor content in the air was determined as \(2.46\%\). On November 1, 1933 this content was about \(0.935\%\), and on November 20, 1933 it was only \(0.617\%\) (i.e., the air could be regarded as relatively dry)1.

If the diffraction curve calculated by Vvedenskii’s formula (see \(^{28}\) and \(^{3}\)) is also plotted here, then one can approximately estimate the additional field whose presence is explained by specific atmospheric conditions, very different from the “average” ones.

Fig. 8. Dependences of \(E\) on \(R\), obtained for reception at a height of \(333\ \text{m}\) on different days. Transmission over the sea (England, Crawford, Memford)

A series of new curves showing the dependence of the field on distance for the case of a horizontal vibrator raised above land is given in Norton’s work \(^{53}\), who used the value of the equivalent radius of the earth \(a_e = 8\,500\ \text{km}\). In the same work curves are presented illustrating the influence of the gradient \(\varepsilon\).

3. Let us now turn to an examination of phenomena connected with the condensation of water vapor present in the air. As is known, the amount of vapor required to saturate the air depends on its temperature. When certain masses of air containing water vapor are strongly cooled, the amount of moisture present in them may prove sufficient not only to produce vapor saturating the space, but also for the condensation of this vapor in the form of water droplets or ice crystals, especially in the presence of the finest dust particles as condensation centers. Large accumulations of these droplets (fog, clouds, and rain), depending on meteorological conditions, may remain fairly stable.

Some data characterizing various kinds of droplet formations encountered in the atmosphere \(^{17,54}\) are given in Table 2.

Although the simultaneous existence of fog or rain along the whole communication line, for relatively great lengths of the latter, is comparatively

Table 2

Type of droplet formation	Number of droplets in \(m^3\)	Droplet diameter in mm	Weight of a droplet in g	Mean distance between droplets in mm	Weight of water in \(1\,m^3\) of air in mg	Fall velocity of droplets in m/sec	Amount of precipitation falling in 1 hour, in mm	Relative volume of water in air
Fog	\(1.2\cdot10^7\)	0.01	\(5.2\cdot10^{-10}\)	4.3	6.2	0.003	Traces	\(6\cdot10^{-9}\)
Dense fog	\(1.1\cdot10^5\)	0.10	\(5.2\cdot10^{-7}\)	21	57.5	0.25	0.05	\(5.5\cdot10^{-8}\)
Light rain	\(2.2\cdot10^4\)	0.20	\(4.2\cdot10^{-6}\)	36	92.6	0.75	0.25	\(9.3\cdot10^{-8}\)
Gentle rain	\(2.9\cdot10^4\)	0.45	\(4.8\cdot10^{-5}\)	70	138.9	2.00	1.00	\(1.4\cdot10^{-7}\)
Moderate rain	\(5.3\cdot10^3\)	1.00	\(5.2\cdot10^{-4}\)	123	277.8	4.00	4.00	\(2.8\cdot10^{-7}\)
Heavy rain	\(4.6\cdot10^2\)	1.50	\(1.8\cdot10^{-3}\)	130	833.3	5.00	15.00	\(8.3\cdot10^{-7}\)
Very heavy rain	\(3.8\cdot10^2\)	2.10	\(4.9\cdot10^{-3}\)	138	1,851.9	6.00	40.00	\(1.8\cdot10^{-6}\)
Downpour	\(3.9\cdot10^2\)	from 3.0 to 5.0	\(1.4\cdot10^{-2}\)	137	5,401.4	7.00	100.00	\(5.4\cdot10^{-6}\)

Table 3

\(\lambda\) in cm; type of droplet formation	\(100\): \(\dfrac{q}{2}\)	\(100\): \(z\)	\(50\): \(\dfrac{q}{2}\)	\(50\): \(z\)	\(10\): \(\dfrac{q}{2}\)	\(10\): \(z\)	\(5\): \(\dfrac{q}{2}\)	\(5\): \(z\)
Downpour	\(8.86\cdot10^{-13}\)	\(2.6\cdot10^7\)	\(1.4\cdot10^{-11}\)	\(1.6\cdot10^6\)	\(8.86\cdot10^{-9}\)	\(2.6\cdot10^3\)	\(1.4\cdot10^{-7}\)	\(1.6\cdot10^2\)
Moderate rain	\(16.9\cdot10^{-16}\)	—	\(2.7\cdot10^{-14}\)	—	\(16.9\cdot10^{-12}\)	\(1.3\cdot10^6\)	\(2.7\cdot10^{-10}\)	\(8.5\cdot10^4\)
Fog	—	—	—	—	—	—	\(5.7\cdot10^{-18}\)	\(4\cdot10^{12}\)

is unlikely; however, for individual sections of the line this may be a very frequent phenomenon. Therefore it seems appropriate to us to illuminate here somewhat the question of the possibility of the influence of these factors on the propagation of ultrashort waves.¹)

The problem of the propagation of ultrashort waves in air containing suspended water droplets is completely analogous to the well-known optical problem of the colors of colloidal solutions, first posed by Maxwell Garnett⁵⁵ and analyzed in detail by Mie⁵⁶ and his successors. In the simplest case, when it is permissible to take into account only the decrease in the intensity of the transmitted waves caused solely by scattering of energy and not by absorption, the problem reduces to Rayleigh’s case⁵⁷, when the wavelength is much greater than the dimensions of the scattering particles (the well-known theory of the blue color of the sky²)).

In this case each particle is replaced by a certain equivalent dipole, forced to oscillate with the frequency of the incident wave. The axis of this dipole is parallel to the vector of the electric field of the incident wave, and the amplitude of its electric moment is equal to

\[ M_0=\frac{\varepsilon-1}{\varepsilon+2}\,b^3E_0; \tag{22} \]

here \(b\) is the radius of the particle (droplet), \(\varepsilon\) is the dielectric coefficient of water, and \(E_0\) is the amplitude of the electric-field strength of the incident wave.

Next, assuming that a unit volume contains \(N\) identical droplets, the distances between which are large in comparison with their dimensions, for the total amount of energy scattered by all these droplets per unit time one may obtain

\[ P_{\text{scat}}=\frac{16}{3}\frac{\pi^4 cNb^6}{\lambda^4} \left(\frac{\varepsilon-1}{\varepsilon+2}\right)^2 E_0^2 . \tag{23} \]

The mean energy of the incident wave (taken to be plane) and its decrease caused by scattering along the path \(z\) may be determined from the relations

\[ P=\frac{c}{8\pi}E_0^2=P_0e^{-qz}, \]

where \(q\) is the scattering coefficient. The magnitude of this scattering coefficient, defined as the ratio of the energy scattered by a unit volume to the energy of the incident wave, is equal to

\[ q=\frac{P_{\text{scat}}}{P}= \frac{128\pi^5Nb^6}{3\lambda^4} \left(\frac{\varepsilon-1}{\varepsilon+2}\right)^2 . \tag{24} \]

Stratton⁵⁹ showed that in the case when it is desirable also to take into account the influence of absorption in the droplets themselves, one may use

¹) It is interesting to note that the amount of water contained in \(1\ \text{m}^3\) of saturated air, at \(20^\circ\) and \(760\ \text{mm Hg}\), is approximately four times greater than during rain, and 4000 times greater than during fog, if only the water in the droplet-liquid state is taken into account.

²) See, for example, the survey by I. Khvostikov⁵⁸.

by the results of the works of Mie\(^{56}\) and Jobst\(^{60}\). In these calculations each droplet must be replaced not by a single equivalent dipole, but by an entire system consisting of several electric and magnetic dipoles. However, the calculations carried out by Stratton showed that for wavelengths \(\lambda > 5\) cm the absorption in water droplets should be so insignificant that, in comparison with scattering, it may quite safely be neglected.

In Table 3, which we have borrowed from Stratton, are given the values of the coefficient \(q\) in cm\(^{-1}\) and the distances \(z\) in km corresponding to the case of a reduction of the field strength to \(0.1\) of its initial values, for some of the most interesting cases\(^{1}\).

On the basis of this table one may conclude that rain and fog should not appreciably affect the diminution of u.h.f. energy (at least within the practically used part of this range). This is confirmed by the work of Wolf and Linder\(^{54}\). They experimented with a wavelength \(\lambda = 9\) cm over distances up to 30 km (Atlantic Ocean) and found no noticeable influence of rain on the propagation of these waves\(^{2}\).

French\(^{61}\), like Stratton, believes that appreciable attenuation of the field occurs only for waves approximately shorter than 5 cm. However, the results of his work, while generally agreeing with Stratton’s data as regards scattering, are in sharp contradiction with them as regards absorption. Thus, according to French, the influence of absorption at waves of 20 cm already considerably exceeds the influence of scattering, and at millimeter waves becomes so considerable that it becomes necessary to point out the doubtful value of the practical use of these waves. Unfortunately, French’s article contains ambiguities which make clear conclusions from it very difficult.

In passing to still shorter waves (\(\lambda < 5\) cm), in addition to everything set forth above, it will perhaps be necessary also to take account of selective absorption in water. Moreover, some selective absorption is also possible in gases, for Klumb\(^{62}\) points to three rather sharp absorption bands in hydrogen at \(\lambda = 28\) cm, \(\lambda = 9\) cm, and \(\lambda = 3\) cm (Fig. 9), although in nitrogen and oxygen no such sharp bands are found. It is interesting that the positions of these bands were predicted by Grotrian\(^{63}\) from considerations of the possibility of quantum transitions in the level scheme of the hydrogen atom (Fig. 10), where the wavelengths obtained theoretically are \(\lambda = 2.74;\ 9.25;\ 27.75\) cm. There are also data on the absorption of waves of the order of a few centimeters (down to 1.3 cm) in ammonia, published by Cleeton and Williams\(^{64}\). It must be pointed out, however, that these data are almost isolated and that this whole question has as yet been little studied. In addition, Hulmes\(^{65}\) introduces into consideration an entirely new phenomenon which, in his opinion, should occur in the propagation of radio waves in moist layers of the troposphere—namely, oscillations of water molecules, the dipole moment of which experiences in the electric field of the earth the action of direct—

\(^{1}\) This table is reproduced with some corrections.
\(^{2}\) According to Wolf and Linder, the absorption caused by heavy rain at these wavelengths does not exceed \(0.06\) db per 1 km.

ing pair of forces. Calculating the natural periods of oscillation of such elementary dipoles and taking into account the effect of thermal collisions, he arrives at the conclusion that significant and, moreover, selective absorption of radio waves of all wavelengths (including, and perhaps preferably, ultrashort waves) is possible.

Fig. 9. Absorption of centimeter and decimeter waves in certain gases (Clump)

Fig. 10. Possible transitions in the atom \(H_2\) (Hori)

Thus, dispersion of radio waves is introduced also into the phenomena of tropospheric refraction, whereas previously all the corresponding arguments were entirely free of it. Although there is as yet no question of experimental confirmation of this phenomenon, the idea itself is very interesting and seems worthy of further development.

§ 3. RAY TRAJECTORIES AND RADIATION DIAGRAMS

1. In § 2 we have already indicated that a rigorous account of the influence of refraction on the propagation of ultrashort waves between corresponding points raised above the horizon line requires knowledge of the ray trajectories. Having these trajectories, one can construct radiation diagrams of antennas (raised above a spherical earth) that take refraction into account. The problem of constructing such diagrams was considered in 1938 by Eckart and Pfendle \(^{29}\). Despite a number of assumptions, inaccuracies, and even outright errors contained in this calculation, we cite it as the only one in this field \(^{1}\). Elsewhere we intend to return to this question.

Considering atmospheric air as a mixture of dry air and water vapor, subject to the additive law, the authors mentioned proceed from the formula

\[ \frac{\varepsilon - 1}{\varepsilon + 2} = \frac{4\pi}{3}\sum N_i \alpha_i, \tag{25} \]

where \(N_i\) is the number of molecules in \(1\ \mathrm{cm}^3\), \(\alpha_i\) is the total polarizability of the molecule,

\(^{1}\) We note here that Gans \(^{66}\) in 1924 calculated, for rays close to tangential ones, the distances at which these rays, under the influence of refraction, return to the earth. At present this calculation is no longer of interest.

The quantities \(N\) and \(\alpha\) are expressed as

\[ N=\frac{A\delta}{M}\quad \text{and}\quad \alpha=\alpha_0+\frac{\mu^2}{3kT} \]

(the notation is the same as in § 2).

Since \(\varepsilon \simeq 1\), formula (25) can be represented in the form

\[ \varepsilon-1=4\pi\sum N_i\alpha_i, \tag{26} \]

which is in complete agreement with formula (17).

Using Hahn’s data for the mean dependence of air density on altitude, cited by Zoringer \(^{38}\), Eckart and Plendl calculated the value of \(\varepsilon-1\). It was assumed here that the humidity of the air decreases with altitude according to a linear law, and that beginning at an altitude of \(5000\ \text{m}\) this humidity was not taken into account.

Fig. 11. Dependence of \(\varepsilon-1\) on altitude

Air: \(A\)—dry, \(B\)—of average humidity, \(C\)—of high humidity

Fig. 12. Approximating curves for \(\varepsilon-1\)

Air: \(A\)—dry, \(B\)—of average humidity, \(C\)—of high humidity

In Fig. 11 the curves \(\varepsilon-1=f(h)\) are given for the cases: \(A\)—dry air, \(B\)—average humidity, and \(C\)—high humidity.

Further, these curves are approximated by parabolas of the form

\[ \varepsilon-1=a+bh+ch^2. \tag{27} \]

The coefficients \(b\) and \(c\) are determined in the usual way so that, over a certain interval of altitudes \((0\le h\le H)\), the integral of the square of the errors is minimal. For dry air, the altitude interval from 0 to \(10\ \text{km}\) was taken. For humid air, two intervals were taken: from 0 to \(5\ \text{km}\) and from 5 to \(10\ \text{km}\).

As a result of these calculations the following expressions were obtained. For dry air (curve \(A\) in Fig. 12) at altitudes

\[ 0\le h\le 10\ \text{km} \]

\[ \varepsilon-1=5.9\cdot10^{-4}-0.639\cdot10^{-4}h+2.41\cdot10^{-6}h^2. \]

For air of medium humidity (curve \(B\)) at altitudes \(0 \leq h \leq 5\) km and \(5 \leq h \leq 10\) km

\[ \varepsilon - 1 = 7.425 \cdot 10^{-4} - 1.155 \cdot 10^{-4}h + 6.87 \cdot 10^{-6}h^2 \]

and

\[ \varepsilon - 1 = 5.35 \cdot 10^{-4} - 0.444 \cdot 10^{-4}h + 0.96 \cdot 10^{-6}h^2 . \]

For very humid air (curve \(C\)) at altitudes \(0 \leq h \leq 5\) km and \(5 \leq h \leq 10\) km

\[ \varepsilon - 1 = 8.91 \cdot 10^{-4} - 1.36 \cdot 10^{-4}h + 5.04 \cdot 10^{-6}h^2 \]

and

\[ \varepsilon - 1 = 5.35 \cdot 10^{-4} - 0.444 \cdot 10^{-4} + 0.96 \cdot 10^{-6}h^2 . \]

2. Determination of the ray trajectory (Fig. 13) is based on Fermat’s principle, according to which radiation always propagates along the path requiring extremal (in the present case, minimal) time. Eckart and Plendl write the corresponding condition in the form:

\[ \int_{\varphi_0}^{\varphi_1} \frac{\sqrt{r^2+\left(\dfrac{dr}{d\varphi}\right)^2}}{n(r)} \, d\varphi = \min, \tag{28} \]

where \(r = a + h;\ \varphi\) is the geocentric angle1.

Fig. 13. Notation adopted in calculating the ray trajectory

The Euler equation2, obtained when solving this variational problem, is

The criticism of the further exposition is made exceedingly confusing by the following circumstances:

a) Euler’s equation (29) is evidently written on the basis of (28), and in computing the individual partial derivatives errors in signs were made at \(\dfrac{1}{n}\dfrac{dn}{dh}\). These signs would have been correct for the proper form of equation (28), but then, of course, the denominator would not contain \(n\).

b) The subsequent approximation \(\dfrac{1}{n}=1\), fortunately, to a known extent compensates for the erroneous introduction of \(n\) into the denominator of (28) and (29). However, the neglect of \(\left(\dfrac{dh}{ds}\right)^2\) and the approximation \(a+h=a\) again introduce inadmissible and, most importantly, entirely unnecessary neglect.

It is clear that, without repeating all the derivations anew and computing new curves, it is impossible to determine quantitatively the magnitude of the resultant error made by Eckart and Plendl throughout the entire chain of inconsistencies they allowed.

We shall return to this question later (see § 4, section 2).

problem, leads Eckart and Pendy to the equation

\[ a^{2}\frac{d^{2}h}{ds^{2}}-a^{2}\left(\frac{dh}{ds}\right)^{2}\left(\frac{2}{a+h}+\frac{1}{n}\frac{dn}{dh}\right)- \]

\[ -(a+h)^{2}\left(\frac{1}{a+h}+\frac{1}{n}\frac{dn}{dh}\right)=0, \tag{29} \]

where \(s=a\varphi\) is the distance measured along the earth’s surface.

Further, these authors assume that the refractive index \(n \cong 1\), and that its gradient \(\left(\dfrac{dn}{dh}\right)\) is so small that appreciable deviations of the trajectory from a straight line are obtained only for rays whose direction differs only slightly from the direction of the tangents to the surfaces of the layers. Therefore they suppose that

\[ \left(\frac{dh}{ds}\right)^{2}=\operatorname{tg}^{2}\alpha \ll 1,\quad \text{and since } \frac{1}{n}=1 \quad \text{and} \quad h\ll a, \tag{30} \]

then, neglecting the second term of equation (29), they obtain

\[ \frac{d^{2}h}{ds^{2}}=\frac{dh}{ds}\frac{d\left(\dfrac{dh}{ds}\right)}{dh}=\frac{1}{a}+\frac{dn}{dh}. \tag{31} \]

Moreover,

\[ n=\sqrt{\varepsilon}=\sqrt{1+(\varepsilon-1)}\cong 1+\frac{\varepsilon-1}{2}, \tag{32} \]

and consequently,

\[ \frac{dn}{dh}=\frac{1}{2}\frac{d(\varepsilon-1)}{dh}. \]

Then

\[ \frac{1}{2}\left(\frac{dh}{ds}\right)^{2} =\int\left(\frac{1}{a}+\frac{dn}{dh}\right)\,dh+\frac{C_{1}}{2} = \]

\[ =\left(1.57\cdot 10^{-4}+\frac{b}{2}\right)h+\frac{ch^{2}}{2}+\frac{C_{1}}{2}, \tag{33} \]

so that

\[ s=\int_{h_{1}}^{h_{2}}\frac{dh}{\sqrt{(3.14\cdot 10^{-4}+b)h+ch^{2}+C_{1}}}. \tag{34} \]

The integration constant \(C_{1}\) is determined from (30) and (31) as

\[ C_{1}=\operatorname{tg}^{2}\alpha-2\left(1.57\cdot 10^{-4}+\frac{b}{2}\right)h_{1}-ch_{1}^{2}. \]

Formula (34) functionally relates the height of the point at which the ray leaves, \(h_{1}\) (the transmitting point), to the height of the point at which the ray arrives, \(h_{2}\) (the receiving point), and to the distance \(s\) between them, measured along the arc of a great circle of the earth’s surface. Thus, the ray trajectory is obtained in the coordinates \(h\) and \(s\).

With unrestricted variation of \(h\), there will always be found such a point of the trajectory whose distance to the center of the earth is the smallest.

At the same time, it is convenient to include in the unified conception of a beam of rays also those rays which fall on the earth’s surface at acute angles; this can be achieved by considering a fictitious continuation of the ray above the earth’s surface. At this point the smallest distance from the center of the earth corresponds to the distance from the earth’s surface \(h_{\min}\), which may be both positive and negative (Fig. 14). It is obvious that this distance \(h_{\min}\) is determined by the relation

Fig. 14. Values of \(h_{\min}\) for different trajectories

\[ \frac{dh}{ds}=\operatorname{tg}\alpha=0 \quad (\text{for } h=h_{\min}). \tag{35} \]

The trajectories of the entire beam of rays, counted from the height \(h_{\min}\), are expressed by the formula

\[ s_1=\int_{h_{\min}}^{h} \frac{dh}{\sqrt{(3.14\cdot 10^{-4}+b)h+ch^2+C_1}} . \tag{36} \]

Here \(h_{\min}\) is precisely the parameter of the beam, while \(h\) (the upper limit of the integral) is arbitrary within the limits of admissible values (with this, of course, \(h\), unlike \(h_{\min}\), cannot be negative).

Further, putting \(3.14\cdot 10^{-4}+b=B\) (here \(3.14\cdot 10^{-4}=\dfrac{2}{a}\)), Eckart and Plendl find

\[ s_1=\frac{1}{\sqrt{c}}\operatorname{arch} \frac{h+\dfrac{B}{2c}} {\sqrt{\dfrac{B^2}{4c^2}-\dfrac{C_1}{c}}} \Bigg|_{h_{\min}}^{h}. \tag{37} \]

But, since \(h_{\min}\) is determined from relation (35), which according to (33) leads to the equation

\[ Bh_{\min}+ch_{\min}^2+C_1=0, \]

then

\[ h_{\min}=-\frac{B}{2c}\pm \sqrt{\frac{B^2}{4c^2}-\frac{C_1}{c}} . \tag{38} \]

In view of this, expression (37) takes the form

\[ s=\frac{1}{\sqrt{c}}\operatorname{arch} \frac{h+\dfrac{B}{2c}} {\sqrt{\dfrac{B^2}{4c^2}-\dfrac{C_1}{c}}}, \tag{39} \]

whence

\[ h=-\frac{B}{2c}+ \sqrt{\frac{B^2}{4c^2}-\frac{C_1}{c}}\, \operatorname{ch}(\sqrt{c}\cdot s). \tag{40}\,^{1)} \]

\(^{1)}\) In the article by Eckart and Plendl \(^{29}\) there is an obvious misprint in this formula [see formula (52) of the cited article].

This is nothing other than equation (34), expressed in explicit form. Here \(C_1\) is still a function of the coordinates of some point \(h\) (for example, the point at which the ray emerges) and of the angle \(\alpha\), defined as \(\operatorname{arctg}\dfrac{dh}{ds}\). In addition, \(s\) is measured from the beginning of the segment \(h_{\min}\).

Adding to these results the value of \(\dfrac{dh}{ds}\) obtained from equation (33), namely:

\[ \frac{dh}{ds}=\sqrt{Bh+ch^2-C_1}, \tag{41} \]

we obtain the possibility of constructing two graphs by means of which, in Eckart and Plendl’s opinion, the problem is solved.

3. Indeed, first of all one can construct a family of curves (their parameter is \(h_{\min}\)) which relate the height of a point \(h\) of a given ray to the value of the angle \(\alpha\) for the same point. With the aid of this diagram (Fig. 15), constructed for air of average humidity, we are able either, given the height \(h\) and the angle \(\alpha\), to determine the parameter \(h_{\min}\), or, given \(h_{\min}\) and the height \(h\), to determine the angle \(\alpha\) of the given ray at that height.

Fig. 15. Relation between the angle of inclination of the ray \(\alpha_0\) at the point under consideration, the height of this point \(h\), and the value \(h_{\min}\).

The authors indicate, as an example, point \(a\) in Fig. 15, which corresponds to the inclination angle of the ray trajectory \(\alpha=0^\circ.9\) at the height \(h=4000\ \text{m}\). It follows from this diagram that a ray satisfying this condition is characterized by the value \(h_{\min}=3000\ \text{m}\). A check calculation, carried out by us in order to clarify the method of construc-

of the construction of these curves, found that the values of $h$ and $h_{\min}$ (indicated in Fig. 11 in meters) should be expressed in kilometers in the calculations1.

In addition to this diagram, the authors use a second diagram (Fig. 16), giving the dependence of $h$ on $s$, determined by formula (40). For calculations it is convenient to represent this dependence in the form

\[ h=-\frac{B}{2c}+\left(h_{\min}+\frac{B}{2c}\right)\operatorname{ch}(\sqrt{c}\cdot s), \]

which is obtained by substituting (38) into (40). By specifying the parameter $h_{\min}$ and various values of $s$, one can construct the corresponding curve of Fig. 162.

Fig. 16. Relation of the distance $s$ (measured along the earth from the point corresponding to the height $h_{\min}$ to the point corresponding to the height $h$) to the values $h$ and $h_{\min}$

Eckart and Plendl note that the concept of an “equivalent” earth radius, introduced by Skilling, Burrows, and Ferrell[^25], can be obtained from the considerations they develop by neglecting the coefficient $c$ in expansion (27) for $\varepsilon-1$. For this equivalent earth radius they obtain the expression

\[ a_e=\frac{2a}{2+ba}>a\quad(\text{for } b<0), \]

which, under the given linear approximation of $\varepsilon$, can be obtained directly from formulas (4′) and (8)3. To bring these values of $a_e$ into agreement with the values obtained by other authors (see § 2), Eckart and Plendl

REFRACTION OF RADIO WAVES IN THE “UNDISTURBED” TROPOSPHERE

one has to use coefficients \(b'\), differing somewhat from \(b\).

1. We shall now apply the curves obtained to determine the path difference of two rays (the direct ray and the one reflected from the earth) in the presence of refraction. For this purpose consider the ray \(SP_1\) with angle \(a=-a_0\), which touches the earth at the point \(P_1\) (Fig. 17). For this point \(P_1\), obtained without taking refraction into account, the path difference between the direct and reflected rays is equal to zero, since both these rays coincide with the tangent ray \(SP_1\). However, in fact, as a consequence of refraction, the ray tangent to the earth will be another ray \(SP_2\), leaving at the angle \(a_1\) and touching the earth at the point \(P_2\).

If we compute the vertical radiation diagram for the given distance1, then, because of the presence of refraction, zero field strength will correspond not to the angle \(a_0\), but to the angle \(a_1\). Thus, refraction causes a certain correction to the angles \(a_0\).

Fig. 17. Decrease of the angle of emergence of the tangent ray \(\Delta a_0\), caused by refraction

Fig. 18. Path differences of the direct and reflected rays \(\Delta r\), computed for a spherical earth without taking refraction into account

In Fig. 18 are given the path differences of the direct and reflected rays \((\Delta r)\), computed by Eckart and Phelps without taking refraction into account, for different angles \(a\). In these calculations it was assumed that both rays are parallel to one another. The angle \(a\) is measured from the perpendicular to the earth’s radius at the point \(S\) (line \(SN\) in Fig. 17). For rays directed away from the earth, it is assumed that \(a>0\); for rays directed toward the earth, \(a<0\).

The use of these curves in the presence of refraction requires a corresponding change in the angles \(a\). For the angle \(a=a_0\) this refraction correction is equal to \(\Delta a_0=a_1-a_0\) (Fig. 17). The values of the angle \(a_1\) are determined from equation (41) as

\[ a_1=\frac{180}{\pi}\sqrt{Bh_s+c} \]

(for the tangential ray \(C_1=0\)). These values are given on the curve \(h_{\min}=0\) (Fig. 15). For the angle \(a_0\), on the basis of Fig. 19, we have

\[ \cos a_0=\frac{a}{a+h_s}, \]

whence, approximately, we obtain that

\[ 1-\frac{a_0^2}{2}=1-\frac{h_s}{a}, \quad \text{or} \quad a_0=-\frac{180}{\pi}\sqrt{\frac{2h_s}{a}}. \]

Further, Eckart and Plendl rely on the assumption already stated earlier, that noticeable deviations of ray trajectories from straight lines are obtained only for angles \(a\) close to zero. On this basis they consider that, for angles \(a \ge 1^\circ\!.5\), the influence of refraction is practically already so weakened that it may simply be neglected. Therefore, resorting to linear interpolation (Fig. 20), they obtain that the correction to the angle \(a<a_1\) is approximately equal to

Fig. 19. On the determination of the angle \(a_0\) of the tangential ray in the absence of refraction

Fig. 20. Determination of the refraction correction \(\Delta a\) by linear interpolation

\[ \Delta a=\Delta a_0\frac{1^\circ\!.5-a}{1^\circ\!.5-a_0}. \tag{42} \]

Taking this correction into account, we are able to determine (from the curves of Fig. 18) the path difference of the rays when the influence of refraction is present. To facilitate these computations, Eckart and Plendl provide auxiliary graphs.

1. Eckart and Plendl construct the radiation diagrams according to the reflection formula\({}^{1}\)

\[ E=\frac{120\pi l}{\lambda r}\sin\left(\frac{\pi \Delta r}{\lambda}\right), \tag{43} \]

\({}^{1}\) This formula is easily obtained from formula (3.2,8) of our book\({}^{3}\).

where \(\Delta r\) is the path difference of the direct and reflected rays, obtained taking refraction into account, \(I\) is the current in amperes, \(l\) is the effective length of the vibrator in meters, \(\lambda\) and \(r\) are in kilometers, \(E\) is in \(\mu\mathrm{V}/\mathrm{m}\).

As an example, they give a calculation of the radiation pattern for a half-wave vibrator \(\left(l=\dfrac{\lambda}{\pi}\right)\) with current strength \(I=1\,\mathrm{A}\), raised to a height \(h_s=1000\,\mathrm{m}\). The construction is carried out for distances up to \(400\,\mathrm{km}\).

First we assign a series of values of \(h_{\min}\) and, knowing \(h_s\), with the aid of Fig. 15 determine the inclination angles of these rays (Table 4).

Table 4

Ray No.	1	2	3	4	5	6	7	8
\(h_{\min}\) in m	1000	800	600	400	300	200	100	0
\(\alpha\)	\(0^\circ\)	\(-0^\circ,372\)	\(-0^\circ,525\)	\(-0^\circ,64\)	\(-0^\circ,69\)	\(-0^\circ,738\)	\(-0^\circ,78\)	\(-0^\circ,86\)

Then, from the curves of Fig. 16, we find the distances (along the earth’s surface) between the points corresponding to the heights \(h=h_s\) and \(h_{\min}\).

To illustrate the results obtained we use a special scale; it is explained by Fig. 21, a, which gives a section of the earth with concentric circles of height. The tangent \(SOP\) is the horizon line (without allowing for refraction), and the rays \(SP_1\) and \(SP_2\) are straight lines. Fig. 21, b is given in a new scale, in which \(SOP\) is still a tangent, while the circumference of the earth and the circles of equal heights have been transformed into parabolas, with all distances from the earth being measured not along radii, but along ordinates. Distances along the earth are measured along the tangent \(SOP\). In Fig. 22 the trajectories of the selected rays, constructed in this scale, are given. It follows from Fig. 16 that a ray tangent to the earth \((h_{\min}=0)\) reaches the height \(h=1000\,\mathrm{m}\) at a distance of \(140\,\mathrm{km}\); therefore, leaving a transmitter situated at the height \(h_s=1000\,\mathrm{m}\) at an angle of \(-0^\circ,82\), it will reach the earth at the same distance.

Fig. 21. Transition from ordinary coordinates to conventional ones

Another ray, for example, for \(h_{\min}=400\,\mathrm{m}\), will have a height of \(1000\,\mathrm{m}\) at a distance of \(108\,\mathrm{km}\) from its lowest point; thus, leaving at an angle of \(-0^\circ,64\) from a transmitter situated at a height of \(1000\,\mathrm{m}\), it will pass through a point with a height of \(400\,\mathrm{m}\) at a distance of \(108\,\mathrm{km}\). The subsequent path of each of these rays is deter-

are plotted along the same curves of Fig. 16 for each value of \(h_{\min}\). In doing this, having set a definite distance along the ground, one must subtract from it the distance between the points corresponding to \(h_{\min}\) and \(h_s\) (for the tangent ray No. 1 equal to 140 km, for ray No. 4 equal to 108 km, etc.), and find from the curve the height \(h\) at which the given ray will pass at the indicated distance. In this way all the rays of the table are constructed.

Fig. 22. Trajectories of non-refracted rays
\(h_s = 1\,000\) m

Let us now return to Fig. 18 for the path differences of the rays computed without allowance for refraction. As an example, let us determine the values of \(a\) corresponding to the “first zero,” the “first maximum,” and the “second zero” of the radiation diagram for \(\lambda = 4.1\) m. Calculating, on the basis of elementary considerations, the necessary values of these differences, from the curve for \(h = 1000\) m we find the departure angles of the rays corresponding to the maximum field values (Table 5).

The refraction corrections to these angles are calculated as follows. First of all we determine the quantity \(\Delta a_0 = a_1 - a_0\).

Since for the tangent ray allowing for refraction the angle \(a_1 = -0^\circ.86\), while for the tangent ray not allowing for refraction (Fig. 19) the angle \(a_0 = -1^\circ.04\), then \(\Delta a_0 = 0^\circ.18\). Using this value of \(\Delta a_0\), we determine, by formula (42), the refraction corrections \(\Delta a\) to the remaining angles. The last row of Table 5 gives the values of the angles corresponding to zero and maximum fields, obtained with allowance for refraction.

Table 5

Field	First “zero”	First max	Second “zero”
\(\Delta r\)	0.00	2.05	4.1
\(a_0\)	\(-1^\circ.04\)	\(-0^\circ.65\)	\(-0^\circ.5\)
\(\Delta a_0\)	\(0^\circ.18\)	\(0^\circ.16\)	\(0^\circ.15\)
\(a_0 + \Delta a_0\)	\(-0^\circ.86\)	\(-0^\circ.49\)	\(-0^\circ.35\)

By assigning intermediate values of \(\Delta r\), one can similarly find the values \(a + \Delta a\), and, using formula (43), compute

field strengths. Figure 23 gives a radiation diagram constructed by the indicated method¹). The left-hand graph is constructed without taking refraction into account.

Fig. 23. Vertical radiation diagrams
\(\lambda = 4.1\ \mathrm{m};\ h_s = 1\,000\ \mathrm{m};\ R = 350\ \mathrm{km}\) (Eckart and Pfendle).
1—line of the first “zero,” 2—line of the first max, 3—line of the second “zero”—with refraction; 4—line of the first max, 5—line of the second “zero”—without refraction.

Carrying out similar calculations for various distances from the transmitter, one can construct curves of equal fields. Figure 24 shows

Fig. 24. Curves of equal fields for a half-wave vibrator
\(l = 1\ \mathrm{A};\ h_s = 1\,000\ \mathrm{m}\) (Eckart and Pfendle)

such curves \(E = \mathrm{const}\) for the lower lobe of the radiation diagram for waves of length \(4.1\ \mathrm{m}\) and \(7.17\ \mathrm{m}\).

¹) The small difference in the angles is explained by the fact that, in calculating this diagram, Eckart and Pfendle assumed that the influence of refraction ceases to have an effect at angles \(\alpha > 1^\circ\), whereas the calculations given above were based on values \(\alpha \geq 1^\circ.5\).

All the curves presented correspond to conditions of average air humidity. To estimate the effect of humidity, Eckart and Ploendl calculated the trajectories of the tangent ray under various conditions. In Fig. 25 the straight line \(a\) is constructed without taking refraction into account, \(b\)—for dry air, \(c\)—for air of average humidity, \(d\)—for very high humidity. From the height curves it is clear that, at a distance of \(300\ \text{km}\), the presence of humidity causes the ray trajectory to be lowered by almost \(2\ \text{km}\). This cannot fail to affect the magnitude of the field strength (see also Fig. 8, taken from the paper by Englund, Crawford, and Mumford).

Fig. 25. Effect of refraction on the trajectory of tangent rays

§ 4. SOME EXPERIMENTAL DATA, CRITICAL REMARKS, AND CONCLUSIONS

1. Let us now consider some experimental data relating to the above calculation of radiation diagrams, and supplement the criticism of this calculation. In Fig. 26 vertical radiation diagrams are given from the work of Eckart and Ploendl for a half-wave vibrator raised to a height \(h_s = 1\,000\ \text{m}\) at a wavelength \(\lambda = 7.17\ \text{m}\). The left-hand graph is constructed without taking refraction into account, the right-hand one with refraction in air of average humidity. Curves for the wavelength \(\lambda = 4.1\ \text{m}\) are given in Fig. 23.

The observations were carried out in an airplane. Points corresponding to reception maxima are shown by circles, those corresponding to minima by crosses. The authors attribute the deviations of individual points from the theoretical curves to variations in meteorological conditions.

A number of experimental points are plotted in Fig. 24. Circles lying on the lower ray (the first “zero”) correspond to disappearance of reception, crosses to the second zero of the radiation diagram.

In discussing the influence of the radiated power, these authors start from a field of \(200\ \mu\text{V}/\text{m}\), quite sufficient for reception.

From Fig. 24, for \(\lambda = 7.17\ \text{m}\), it follows that for a current \(I = 1\text{A}\) at a distance of \(300\ \text{km}\) such a field occurs when reception is at a height of \(2\,000\ \text{m}\). Quadrupling the power gives a doubling of the field, i.e. the field of \(200\ \mu\text{V}/\text{m}\) will now be at a distance of \(315\ \text{km}\) (at this distance, for the first value of the power, the field had a magnitude of \(100\ \mu\text{V}/\text{m}\)). The authors point out that at a distance exceeding \(377\ \text{km}\) (the refractive horizon), no increase in power is any longer capable of ensuring reception, which at a height of \(2\,000\ \text{m}\) can be achieved only under a special state of the weather.

Fig. 26. Vertical radiation diagrams
\(\lambda = 7.17\ \text{m};\ h_s = 1000\ \text{m};\ R = 350\ \text{km}\) (Eckart and Plendl). 1—line of the first “zero,” 2—line of the first max., 3—line of the second “zero” — with refraction; 4—line of the first max., 5—line of the second “zero” — without refraction

Let us now cite some experimental data from the already mentioned article by Oxman and Plendl \(^{9}\). A transmitter operating at a wavelength \(\lambda = 4.1\ \text{m}\), with a radiated power of 35 W, was installed on a hill 66 m high. The antenna consisted of eight bays, each with two horizontal vibrators of length \(0.5\lambda\). In Fig. 27 are given the results of determinations of the maximum receiving ranges obtained in flights at various altitudes.

The results of analogous experiments with another transmitter, installed at an altitude of 980 m, are given in Fig. 28. This transmitter, operating at the same wavelength and with the same power, had an antenna of twenty vertical vibrators arranged in a row. As we see, the region of possible reception extends considerably beyond the boundary of the refraction-free horizon, which is quite natural, since reception at low altitudes is explained not only by inhomogeneities of the troposphere, but also by the presence of diffraction, which these authors completely fail to mention. The article gives a number of further curves confirming the dependence of refraction on the state of the troposphere.

Fig. 27. Maximum receiving ranges at various altitudes
\(\lambda = 4.1\ \text{m};\ h_e = 66\ \text{m}\) (Oxman and Plendl).

However, upon consideration of all these experimental curves it turns out that they do not confirm the developed theory with sufficient persuasiveness. Indeed, it is easy to see that through the experimental points in Figs. 23, 24, and 26 one can draw entirely different curves, or even simply straight lines (see the dotted line with two points drawn by us). The authors’ assertion that reception

Fig. 28. Maximum reception distances at various heights
\(\lambda=4.1\ \mathrm{m};\ h_s=980\ \mathrm{m};\ +\)—night experiments, \(o\)—day experiments (Oxman and Plendl)

of u.h.f. beyond the refractive horizon is independent of diffraction, ignoring the influence of diffraction altogether, is, of course, erroneous, and should be regarded simply as a misunderstanding.

2. We have presented, with sufficient completeness, the reasoning and results of Eckart and Plendl, despite the unquestionable errors and unjustified omissions contained in them, since this work is nevertheless the only one in this field and has been set forth without criticism in the recently published books of Lassen4 and Beckmann5.

However, even if one disregards the question of the errors made in the formulation of equation (29), the inaccuracies in its integration, and the assumption of parallelism of the direct and reflected rays, there still remains a considerable number of objections that must be made concerning this method (see above).

Thus, in calculating ray trajectories Eckart and Plendl[^29] assert not only that for \(a \geq 1.5\) there is no refraction of the direct ray, but also that under these conditions the path difference between the direct rays and the rays reflected from the earth is likewise not changed by refraction. The latter assertion seems incomprehensible to us, for a ray that is to be reflected in the required direction reaches (if one is to rely on these authors’ graphs) the surface of the earth at the point of reflection situated considerably farther away than it would have

place for the corresponding ray when refraction is in fact absent.

In order not to be unfounded, we shall give an example. Let us consider a ray tangent to the earth \((h_{\min}=0)\), and let the angle be specified as \(a=-1^\circ.5\). From the curves of Fig. 15 we find that for this ray \(h=3000\ \text{m}\). Further, from Fig. 16 for these \(h\) and \(h_{\min}\) we obtain \(s=238\ \text{km}\). On the other hand, the distance to the horizon, determined by formula (1), which does not take refraction into account, is equal to \(195\ \text{km}\). As we see, for this ray a difference in distances equal to \(43\ \text{km}\) is obtained. For comparison, let us indicate that this same distance, determined by formula (9), which takes account of the mean refraction precisely for tangent rays, is equal to \(226\ \text{km}\), which is considerably closer to the \(238\ \text{km}\) obtained from the Eckart and Pendl curves.

Let us consider still another case, the angle \(a=-2^\circ.5\), for which, at \(h_{\min}=0\), from the curves of Figs. 15 and 16 we have \(h=7900\ \text{m}\) and \(s=380\ \text{km}\), whereas by formula (1) we obtain \(318\ \text{km}\), and by formula (8) we have \(365\ \text{km}\).

Consequently, applying the graph of the very same authors who assert the absence of the influence of refraction at angles \(a \geq 1^\circ.5\), we, contrary to this assertion, obtain an influence of refraction for tangent rays no smaller at all than when using the generally known formula (8). Therefore there are no grounds whatever for supposing that, for rays falling on the earth at distances smaller than those indicated, the influence of refraction will not make itself felt.

Thus, all the arguments with linear interpolation of the refraction correction (Fig. 20) do not seem sufficiently convincing to us, which, of course, is very essential for the whole set of conclusions.

The reflection formula (43), used by the authors in calculating the field, naturally leads to the complete absence of the field in the region below the level of the tangent ray (the refractive horizon), which, of course, does not correspond to reality. Moreover, even at the refractive horizon itself the field turns out to be equal to zero, which, of course, is incorrect. Strictly speaking, the values of the reflection coefficients themselves, adopted by Eckart and Pendl as equal to \(-1\), are also not sufficiently rigorous.

However, as we have already indicated above, a rigorous solution of the problem that would eliminate these shortcomings has not yet been given by anyone. In addition, it is still unknown how great the distorting influence of these errors and inaccuracies is. Therefore the work of Eckart and Pendl nevertheless is of interest, and it is possible that, in comparison with works based on the concept of an equivalent radius, it may be regarded as a certain further approximation.

1. All the works considered have dealt with taking account of the mean influence of the troposphere, from which considerable deviations, both short-term and prolonged, are possible. What has been set forth is to a considerable degree analogous to the ionospheric conditions of an “undisturbed” ionospheric day. This analogy goes quite far, for in tropospheric propagation there are also observed fadings that are quite similar

with ionospheric ones, although the causes of these fading phenomena, of course, do not coincide. Examples of such fading we saw above in Fig. 2, a and b.

The intensity of the fading also increases with distance. This indicates the presence of changing interference of rays propagating along slightly different paths. We shall also point to attempts to establish a correlation of the indicated phenomena with meteorological conditions.

We intend to devote a separate article to the totality of these questions.

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1. It should be noted that, when the sphericity of the earth is taken into account, the concept of a vertical radiation diagram no longer has such a definite meaning as in the case of a plane earth surface. ↩↩↩↩↩↩↩

2. Checking three points lying on the curve $h_{\min}=3\ \text{km}$ gave good agreement. ↩↩↩↩

3. Indeed, assuming that $\varepsilon=\varepsilon_0+bh$, from formula (4), for small $h$ we obtain $\rho\cong-\dfrac{2}{b}$, whence, according to (8), we have $\dfrac{a_e}{a}=\dfrac{2}{2+ab}$. The same result can also be obtained from formulas (34) and (41) for $\alpha=0$ and $h_{\min}=0$ (a ray tangent to the earth), for in this case the integration constant $C_1=0$. ↩↩