On the Propagation Speed of Radio Waves1

L. I. Mandelstam and N. D. Papaleksi

The most accurate possible determination of \(c_0\)—the “speed of light in vacuum”—is a very important problem of measurement physics. Among the so-called constants of nature the quantity \(c_0\) occupies an essential place. On the one hand, \(c_0\) is the fundamental constant of the entire theory of electromagnetism. On the other hand, \(c_0\) is the velocity of propagation of light and, in general, of electromagnetic disturbances in vacuum, i.e. the “velocity of propagation of light” in the proper sense of the word. Finally, the accurate experimental determination of a number of other fundamental physical quantities is connected with knowledge of the exact value of \(c_0\).

It may be worth pointing out that the so-called universal constants are at present determined with very different degrees of accuracy.[^1a] Thus, the accuracy of the determination of the elementary charge \(e = (4.8025 \pm 1\cdot 10^{-3})\cdot 10^{-10}\) CGS is estimated at about \(2\cdot 10^{-4}\). Planck’s constant \(h\) is known with approximately the same accuracy: \(h/e = (1.3793 \pm 2.3\cdot 10^{-4})10^{-17}\) CGS. The Rydberg constant \(R_\infty = 109\,737.30_3 \pm 0.017\ \mathrm{cm}^{-1}\) has been determined with greater accuracy, namely up to \(2\cdot 10^{-7}\). Let us note already here that the speed of light \(c_0\) is at present determined with an accuracy up to \(1.3\cdot 10^{-5}\): \(c_0 = 299\,776 \pm 4\ \mathrm{km/sec}\).

For the determination of \(c_0\), as is known, a large number of diverse methods have been used: astronomical[^2],[^3] (as methods for determining \(c_0\), now of only historical significance) and terrestrial, both direct and indirect. The indirect method of determining \(c_0\) from a comparison of electromagnetic and electrostatic units gave, in the experiments of Rosa and Dorsey,[^8] the value \(c_0 = 299\,790 \pm 30\ \mathrm{km/sec}\). The standing-wave method in the Lecher system (Mercier’s experiments[^9]) gave \(c_0 = 299\,790 \pm 20\ \mathrm{km/sec}\). At the present time the most accurate value for \(c_0\) has been obtained from the direct determination of the speed of light by methods that are essentially modernized versions of the methods of Fizeau and Foucault (rotating mirror, toothed wheel, Kerr cell).

Table 1 gives the results of the latest most accurate determinations.

Since the time of Hertz’s experiments, and especially with the development of radio, questions connected with the propagation of electromagnetic waves in the narrow sense of the word, and specifically the question of the speed of their propagation \(v\), have acquired ever greater importance. Here again two kinds of [[unclear: word continues on next page]].

...problem. One may regard the determination of the propagation velocity \(v\) of these waves in free space as a method for determining the fundamental constant of nature \(c_0\). In the range of electromagnetic—

Table 1

Velocity of light

—waves, for this purpose one can use the relation \(v=\lambda \cdot f\) (as, for example, is done in determining \(c_0\) from standing waves in the Lecher system \(^{9}\)), and \(f\)—the number of oscillations—for frequencies up to \(10^{-8}\) Hz can be determined very accurately. In the region of optical waves, as is known, we have no direct methods for determining \(f\), and the number of oscillations here is determined from the relation \(v=f\cdot\lambda\). However, at least up to now, this path has not yielded a refinement of the values of \(c_0\) obtained from direct optical experiments. Another problem that arises here, in the range of electromagnetic waves proper, consists in determining the propagation velocity \(v\) under actual practical conditions. This problem has recently acquired great importance in connection with various practical applications of radio waves: in geophysics (in the study of the ionosphere), in radio navigation, and especially in determining the distance between two points by means of radio waves (radiogeodesy).

At first glance it might seem that for the indicated applications there is no need to determine anew the propagation velocity of radio waves, and that its value may be taken as equal to the value \(c_0\) obtained from optical measurements. It is not difficult, however, to see that this is not so, and that one cannot a priori identify the sought velocity \(v\) with the “optical” one. The point here is as follows. In optical measurements we always, with the exception of certain entirely special cases, deal, in view of the smallness of the light wavelength, with a velocity of light characteristic of the given medium, or, more precisely, with that velocity with which light propagates without obstruction in an unbounded medium with the given constants. In this case, in passing from one medium to another, the Fresnel formulas, or their generalization to the case of absorbing bodies, are valid.

The situation is quite different in the case of radio waves. Owing to their immeasurably greater wavelength, the conditions of propagation of radio waves are substantially different. In particular, because usually the distances and the heights of the transmitter and receiver above the earth are comparable with \(\lambda\), in these cases one cannot speak of unobstructed propagation. It is also necessary to take into account—

to mention those restrictions on unobstructed propagation which are introduced by the presence of the ionosphere. In view of all this, in considering the problem of the velocity of propagation of electromagnetic waves, especially in radio engineering, specific questions arise, the study of which is undoubtedly of not only scientific but also great practical interest. These two sides of the question—the knowledge of how the process of radio-wave propagation proceeds under given conditions, and the practical use of this process based on that knowledge—are, of course, very closely connected with one another. It is necessary, however, to note that very many practical questions of radio engineering either do not require any knowledge of the velocity of propagation of radio waves at all, or for them it is sufficient to know its magnitude only very approximately. Such, for example, are the basic questions for radio communication concerning the intensity of reception. Although here too (for example, in fading phenomena) the velocity of propagation of radio waves plays a large role, for their practical treatment it is enough to know only approximately the magnitude of the velocity. Therefore, until very recently, in the theoretical and experimental consideration of the problem of radio-wave propagation under conditions corresponding to practice (though strongly idealized), chief attention was directed above all to questions of reception intensity, and the question of the velocity of propagation was not examined in depth.

However, as has already been mentioned above, in recent times a whole series of applications of radio waves has arisen for which the question of the velocity \(v\) is a vital one, since without exact knowledge of the quantitative relations between the quantities underlying the methods of measurement and the velocity \(v\), it is impossible to make full use of these methods in practice. Thus, exact knowledge of the wavelength \(\lambda\), and consequently of the phase lag \(2\pi D/\lambda\), which play an essential role in radio-interference methods of measuring distances, is impossible without exact knowledge of the magnitude \(v\), for \(\lambda\) is determined not directly, but from the relation: \(\lambda = v/f\).

The purpose of our report is a brief account of the state of the question of the propagation of radio waves, or, more precisely, of the question of the propagation of the so-called “direct” ray along the earth’s surface.

The first measurements under real conditions were carried out by the method of time signals in connection with precise determinations of longitudes[^11].

A special place among these methods is occupied by the method of the “round-the-world radio echo,” since here all the difficulties associated with measuring, at different points of the terrestrial globe, the times of passage of radio signals disappear.

Indeed, if \(t_1\) denotes the moment, recorded for example on an oscillogram tape, of the arrival of the direct pulse, and \(t_2\) the moment of arrival of the pulse in the same direct direction after one circuit around the terrestrial globe, then the time of the pulse’s circuit around the earth is equal to \(t_2 - t_1\). However, in order to obtain from this the velocity \(v\), it is necessary to know the path traversed by the pulse, and also to take into account the circumstance that part of the pulse’s path passes through the ionosphere. It is also necessary to bear in mind that, in traveling around the earth, we are dealing with a “closed” path and that a priori the velocity cannot in principle be regarded as “constant” along the whole path. This remark, however, applies

to all methods of measuring \(v\) along a closed path, in particular also to interference methods. Table 2 gives the results of measurements by the “round echo” method \(^{12,13}\).

Table 2

Speed \(v\) by the “round echo” method

We shall also cite the results of the latest measurements of \(v\), carried out by the time-signal method in 1933–1935 between Paris and Honolulu \(^{14}\) (Table 3).

Table 3

Speed \(v\) from time signals

The values of \(v\) obtained by these methods, despite their relatively low accuracy, are quite sufficient for introducing corrections for the propagation speed of radio signals for the astronomical time service (determination of longitudes, gravimetry, etc.); however, the conditions of radio-wave propagation here are to a considerable extent uncertain and, in any case, differ from those simplest propagation conditions that occur in a number of practical applications and for which the necessary formulas can be obtained from theory. Moreover, the very accuracy of determining \(v\) by the pulse method in the medium-wave region in its present form is insufficient both for testing the theory and for a whole series of practical applications of radio waves based on the use of formulas obtained from theory. Thus, for measuring distances of 200 km with the comparatively low accuracy desirable for hydrography, \(\pm 200\) m, it is necessary to know the value of the speed \(v\) with an accuracy not ...

less than \(1/2000\). Still greater requirements for the accuracy of the determination of \(v\) are imposed by radiogeodesy: here the question concerns accuracies of the order of one ten-thousandth and less. Such high requirements with respect to the accuracy of determining \(v\)—insofar as they exist at all—naturally make it necessary to apply more precise methods for solving this problem. Interference methods fully satisfy this condition.

In accordance with what has been said above, we shall briefly consider below first the conclusions to which the theory leads in the case of the propagation of electromagnetic waves along the earth, and then the radio-interferometric method and the results of measurements of the propagation velocity of radio waves by interference methods under real conditions.

I. THEORETICAL PART

The first attempt to give a theory of the propagation of radio waves along the surface of the earth was made by Zenneck in 1907[^15]. He posed the question of whether, along the surface of the earth, assumed to be plane and homogeneous, with constants \(\sigma\) and \(\varepsilon\), “one traveling plane wave” can propagate, and found that Maxwell’s equations and the corresponding boundary conditions at the air–earth surface are indeed satisfied by plane waves of the type of the so-called “surface waves.” The vertical \(E_z\) and horizontal \(E_x\) components of the field of such a wave are proportional to \(e^{j(\alpha z+\beta x)}\), where \(\alpha\) and \(\beta\) are complex quantities depending on \(\sigma\) and \(\varepsilon\). For \(\sigma \ne 0\) or \(\infty\), the surface wave is attenuated according to an exponential law both in the direction of propagation and in the vertical direction.

Further—and this is very essential—the phase velocity of these waves along the earth’s surface depends on the constants \(\sigma\) and \(\varepsilon\) of the soil and exceeds the velocity of light in free space. Zenneck assumed, but without foundation, that waves radiated by a transmitter located on the surface of the earth would have the character of such surface waves near the earth’s surface at a great distance from the transmitter.

Zenneck’s concept of the surface wave, erroneously supported by Sommerfeld’s authority, was until recently almost universally accepted (and not only in radio-engineering circles). It was applied to the interpretation of many anomalous phenomena observed in the propagation of radio waves, for example, to the so-called “coastal refraction.” However, comparison of Zenneck’s theory with experimental data leads to disagreement. Since, on the other hand, Zenneck’s theoretical concept completely failed to take into account the connection of the field with the source located on the earth’s surface, there is no basis for considering that it answers the question of interest to us concerning the character of the waves propagating in this case. In order to obtain the correct picture of the electromagnetic field of radiation, in this case it was necessary to give a rigorous solution of the problem of the field of a vertical dipole located on the surface of the air–earth interface. As is known, the rigorous solution of this problem was first obtained and discussed by Sommerfeld (in 1909)[^16]. This solution is given

in the form of a definite integral, which it is mathematically quite natural to represent as the sum of three terms: $P + Q_1 + Q_2$.

The term $P$ corresponds to Zenneck’s surface wave. For this reason, in the first discussion of this solution in his 1909 paper, Sommerfeld saw in it a confirmation of Zenneck’s conception. However, B. Fock $^{17}$, and then E. Nöther $^{17}$, showed that in this discussion Sommerfeld had made a substantial inaccuracy and that the term $P$ cannot be separated from the complex of waves given by the complete solution. The correctness of Fock’s and Nöther’s criticism was later acknowledged by Sommerfeld himself $^{18}$, who in 1935 formulated the state of the question as follows: “Thus waves of type $P$ and $Q$ cannot be separated from one another. Apparently, there are no conditions under which a surface wave of type $P$ arises that would constitute the principal component of the wave complex.” If, therefore, it should be considered that the character of radio-wave propagation along the earth does not correspond to Zenneck surface waves, then the question arises: what is the actual character of the field radiated by an antenna located on the surface of the earth, and, in particular, what is the propagation velocity of the radiated waves?

Although Sommerfeld’s exact solution does contain the answer to these questions for a vertical dipole, nevertheless, until very recently, despite the large number of investigations devoted to the question of radio-wave propagation and to discussions of Sommerfeld’s solution, the main attention was directed not to the propagation velocity, but to a quantity especially important for practical radio engineering, namely—the field intensity. At present, for this quantity as a function of distance, wavelength, and the constants $\sigma$ and $\varepsilon$, there exists not only a series of computational formulas, but also tables and graphs. Questions of phase velocity and, in general, of phase relations were not subjected to exhaustive analysis until very recently; nor was there any clarity concerning the phase structure of the field.

Theoretical and experimental investigations conducted since 1934 at a number of scientific institutions by us together with a large group of collaborators under our general direction have, as it seems to us, brought definiteness to these questions and have led to the establishment of quantitative dependences between the phase velocity of radio waves, the wavelength $\lambda$, and the soil constants, which can serve as the basis for calculations in the field of radio-interference measurements. Since, in particular, the results obtained are of substantial significance for determining the magnitude of the propagation velocity, we shall briefly dwell on them.

Discussion of the solution given by Sommerfeld leads to the following expression for the vertical component of the Hertz vector:

$$ \Pi = 2 f(r) e^{-j\varphi}\frac{e^{j(\omega t-k_1 r)}}{r}. \tag{1} $$

Whence, for the vertical component, we have:

$$ E_z = \frac{1}{\varepsilon}\left[\operatorname{grad.}\operatorname{div}\Pi - \frac{1}{c^2}\frac{\partial^2 \Pi}{\partial t^2}\right]; \tag{2} $$

here \(r\) is the distance to the dipole, \(k_1=\dfrac{\omega}{v}=\dfrac{2\pi}{\lambda}\) is the wave number, \(v\) is the phase velocity in the upper medium—air; \(f(r)\) is a certain real function of \(r\)—the attenuation factor; \(\varphi\) is the phase difference, additional to that which would occur in free propagation in the upper medium. For \(f(r)\) Van der Pol \(^{19}\) gives the following simple approximate expression, valid for \(\dfrac{\varepsilon f}{2\sigma}<0.2\):

\[ f(r)=\frac{2+0.3\rho}{2+\rho+0.6\rho^2}, \tag{3} \]

where \(\rho\) is the modulus of the “numerical distance” introduced by Sommerfeld

\[ S=\rho e^{i\psi}, \]

\[ \rho=\frac{\pi r}{\lambda}\, \frac{\sqrt{(\varepsilon-1)^2+\left(\dfrac{2\sigma}{f}\right)^2}} {\varepsilon^2+\left(\dfrac{2\sigma}{f}\right)^2}; \qquad \tg\psi=\frac{f(\varepsilon+1)}{2\sigma}. \tag{4} \]

For the question of phase relations in the field of electromagnetic waves propagating from a dipole and, in particular, for the question of phase velocity, the quantity of the additional phase \(\varphi\) is of primary importance. The expression for the instantaneous phase \(\Phi_M\) is equal to

\[ \Phi_M=\omega t-k_1 r-\varphi=\omega t-\frac{\omega r}{v}-\varphi. \tag{5} \]

For the differential phase velocity \(v^*\), by definition, we have \(v^*=\dfrac{dr}{dt}\), where \(dr\) and \(dt\) are related by

\[ d\Phi(r,t)=\omega\,dt-\left(\frac{\omega}{v}+\frac{\partial\varphi}{\partial r}\right)dr=0. \]

Whence

\[ v^*=\frac{v}{1+\dfrac{v}{\omega}\dfrac{\partial\varphi}{\partial r}}. \tag{6} \]

For the mean velocity \(\overline v\), from

\[ \frac{\omega r}{v}+\varphi=\frac{\omega r}{\overline v} \]

we have

\[ \overline v=\frac{v}{1+\dfrac{v}{\omega}\dfrac{\varphi}{r}}. \tag{7} \]

A detailed analysis of Sommerfeld’s rigorous solution, carried out by P. A. Razin \(^{20,21}\), under the assumption usually fulfilled in practice

ON THE VELOCITY OF PROPAGATION OF RADIO WAVES

\[ \left| \left( \frac{k_2}{k_1} \right)^4 \right| \gg 1, \]
led to the following result: in the case of propagation of radio waves directly along the surface of the earth, the additional phase \(\varphi\), which is a function of \(\frac{\omega r}{c}\), \(\omega\), \(\sigma\), and \(\varepsilon\), has the following properties.

1. At distances close to the vertical dipole situated on the boundary surface, the additional phase \(\varphi\), independently of the properties of the soils, is expressed by the same quantity

\[ \varphi = \operatorname{arctg} \frac{\frac{\omega r}{v}} {\left(\frac{\omega r}{v}\right)^2 - 1}, \tag{8} \]

as in the case of propagation of waves from a dipole in free space (or, what is the same thing, along an absolutely conducting earth).

1. Beginning with some distance close to the radiator, the additional phase \(\varphi\) is a monotonically increasing function of distance; moreover, as \(r\) increases it tends to a certain limiting value, depending both on the constants of the soil and on \(\omega\). What is essential here is the fact that \(\varphi\) does not include a term increasing linearly with distance. P. A. Ryazin gave an approximate expression for this limit, namely:

\[ \lim_{r \to \infty} \varphi = \pi - \psi . \tag{9} \]

The limiting value of \(\varphi\) lies between \(\pi\) (the case of good conductivity and long waves) and \(\pi/2\) (very poor conductivity, very short waves).

This result, which in our view is essential, may [see formulas (6) and (9)] also be expressed as follows: at sufficiently large distances from the transmitter, the phase velocity \(v\) is equal to the velocity \(c\) of propagation of radio waves in air, independently of the properties of the soil. Further, since \(\varphi \leq \pi\), for \(r \gg \lambda\) the additional phase \(\varphi\) is only a small correction to the total phase angle. This result is of very substantial importance also from the practical point of view, since it permits one, in many cases, to disregard altogether the difference in the properties of soils, which in practice is almost always difficult to take into account accurately.

In those cases where the correction has to be introduced, for its calculation it is sufficient to have an approximate knowledge of the values of the soil constants (\(\sigma\) and \(\varepsilon\)).

To illustrate the dependence of \(\varphi\) and \(v^*\) on \(r\), \(f\), \(\sigma\), and \(\varepsilon\), a number of graphs (Figs. 1 and 2), calculated by P. A. Ryazin, are given below.

Let us note that the additional phase upon moving upward from the surface of the earth must naturally decrease and, at some height above the earth, disappear. As follows from calculations made by P. A. Ryazin for this case, here, independently of the distance to the radiator, the elevation above the earth is of essential significance, and not the angle of observation (elevation) above it. A graph on which the absolute values of the change are plotted

phase \(\Delta\varphi\) (Fig. 3), illustrates these dependences. As is seen from this graph, already at a height of \(3\)—\(4\lambda\) the change in phase of the vertical component relative to the field at the surface becomes practically constant and equal to the additional phase \(\varphi\) with the opposite sign.

Fig. 1. Dependence of the additional phase \(\varphi\) of the vertical component of the electric vector of the field of radio waves on the distance from the radiating dipole in wavelengths \(\lambda=300\) m and soil conductivity \(\sigma=5\cdot10^{7}\); \(\varphi'\)—the same for the limiting case of infinite conductivity of the earth.

Before proceeding to the consideration of the experimental material obtained by us, let us make one further remark. The characteristic features of the radiation field of a vertical dipole located on the plane interface air—earth have been derived from approximate formulas obtained from the exact solution. This, of course, is the only correct justification for them. It is interesting, however, that in the case

Fig. 2. Dependence of the differential phase velocity \(\varphi^{*}\) of the electric vector on the distance from the radiating dipole in wavelengths \(\sigma=5\cdot10^{7}\), \(\lambda=300\) m and \(\varepsilon=10\)

when \(\left|\dfrac{k_2}{k_1}\right|\gg 1\), at a sufficiently large distance from the radiator on the surface of the interface we obtain an electromagnetic field,

analogous in its structure to the field of the Sommerfeld problem, if the field is calculated from the same vertical dipole, but located not at the very interface, rather at some definite height \(h\) above it, applying in this case for the calculation of the reflected wave the usual formula for the Fresnel tangents (of course, taking into account that in the case under consideration

Fig. 3. Dependence of the phase change \(\Delta\varphi\) (in degrees) of the vertical component of the electric vector on the height above the earth’s surface, expressed in wavelengths

the wave number \(k_2\), and consequently also the refractive index, is complex). For the vertical component of the Hertz vector one obtains in this case the expression

\[ \Pi_1 = (1+R)\frac{e^{i(\omega t-k_1 r)}}{r}, \]

where

\[ R=\frac{\operatorname{tg}(\alpha-\beta)}{\operatorname{tg}(\alpha+\beta)}, \]

and \(\alpha\)—the angle of incidence and \(\beta\)—the angle of refraction are related by

\[ k_1 \sin \alpha = k_2 \sin \beta . \]

Since, under the assumptions made, \(\gamma=\frac{\pi}{2}-\alpha\) and \(\beta\) are small, namely:

\[ \gamma=\frac{h}{r} \quad \text{and} \quad \beta=\frac{k_1}{k_2}, \]

then

\[ 1+R=\frac{2\gamma}{\beta+\gamma} =\frac{2}{1+\frac{k_1}{k_2}\frac{r}{h}} . \]

If we now set \(h=\left|\dfrac{k_2}{k_1^2}\right|\) and take into account that, under the assumptions made,

\[ \left|\frac{k_1^3}{k_2^2}\right|r=2\rho, \]

where \(\rho\) is nothing other than the modulus of Sommerfeld’s numerical distance [see formula (4)], then we have

\[ \Pi=\frac{2}{1+2\rho e^{-j\frac{\pi}{4}}}\,\frac{e^{j(\omega t-kr)}}{r}. \]

Putting

\[ \frac{1}{1+2\rho e^{-j\frac{\pi}{4}}}=F(r)e^{-j\varphi}, \]

we have

\[ F(r)=\frac{1}{\sqrt{1+2\sqrt{2}\rho+4\rho^2}} \]

and

\[ \varphi=2\pi-\operatorname{arctg}\frac{\rho\sqrt{2}}{1+\rho\sqrt{2}}. \tag{8*} \]

Thus, \(F(r)=1\) for \(\rho\to 0\) and \(F(r)=\dfrac{1}{2\rho}\) for \(\rho\to\infty\), i.e. in these limiting cases it behaves in the same way as the attenuation factor \(f(r)\) [cf. formula (3)]. For intermediate values of \(\rho\), \(F(r)\) differs from \(f(r)\), but by no more than tens of percent. As for the additional phase, as is seen from (8*), here too we have a monotonic behavior of \(\varphi\) with increasing \(\rho(r)\), and in this case as well \(\varphi\) tends to a definite limit \(\varphi_\infty=2\pi-\dfrac{\pi}{4}\). Thus, here too the characteristic circumstance is that at large distances from the radiator the mean and differential velocities of propagation do not depend on the properties of the soil.

Let us also note that the indicated theoretical illustration leads, for large numerical distances, to the correct value of the angle of inclination of the electric field.

II. EXPERIMENTAL PART

A. Method

For the study of various questions connected with the propagation of radio waves along the surface of the earth, and, in particular, for the measurement of the velocity of propagation, we used, as has already been indicated, the interference method. Its essence, in brief, consists in the following \(^{21,22}\).

Let the waves emitted by a transmitter located at point I, having reached point II, be reflected and return again to point I. The difference

phase \(\Phi\) at point I between the emitted and the received reflected oscillations, which is the direct object of measurement, is evidently composed of: 1) the phase difference \(\Phi_1\) on the path from point I to point II, 2) the phase difference \(\delta_2\) in the apparatus at point II upon reflection, 3) the phase difference \(\Phi_2\) on the path from point II to point I, and 4) the phase difference \(\delta_1\) in the receiving apparatus at point I. Thus, the total phase difference will be equal to

\[ \Phi=\Phi_1+\Phi_2+\delta_1+\delta_2. \]

As we have just seen, theory gives for \(\Phi_1\) the expression

\[ \Phi_1=\frac{\omega_1 D}{v}+\varphi\left(\frac{\omega_1 D}{v},\ \omega_1,\ \varepsilon,\ \sigma\right) \tag{10} \]

and an analogous one for \(\Phi_2\), and in our schematic case \(\Phi_2=\Phi_1\). Thus, putting \(\Phi-\delta_1-\delta_2=\Psi\), where \(\delta_1\) and \(\delta_2\) are constants of the apparatus, which can be measured (the so-called phase deviations of the apparatus), we arrive at the following basic formula:

\[ \Psi=\frac{2\omega_1D}{v}+\rho_\varphi, \tag{11} \]

where \(\rho_\varphi\) is a small correction term, composed of the correction terms for the direct and return paths of the radio waves.

Let us note the following. The scheme of the method considered corresponds entirely to the scheme of the well-known Michelson interferometer, used by him for the exact measurement of length standards in wavelengths of light. However, whereas in optics—owing to the smallness of the wavelengths—it is not difficult, on the one hand, to reflect the wave at point II and, on the other hand, to observe at one place (at point I) the interference pattern caused by the superposition of the direct and return rays, i.e. to obtain, by comparatively simple means, well-directed light beams and to localize the light field sharply, in the region of radio waves such a task presents very great difficulties.

First of all, the “reflection” of the waves at point II must be replaced by a “relay” action—the oscillation arriving from point I at point II sets in action a local transmitter, which plays the role of the reflector. It is further necessary that the reflected waves be “coherent” with the incident ones. This is achieved mainly by making use of the phenomenon of “entrainment.” However, the chief difficulty consists in measuring at point I the phase difference between the powerful oscillations created by its own transmitter and the weak oscillations arriving from point II. If both these oscillations have one and the same frequency, then their exact separation at point I is an extremely difficult—at the required accuracy, practically impossible—task. We have bypassed this difficulty by arranging that the oscillations arriving from point I, upon reflection at point II, are transformed in frequency in a rational ratio (usually in the ratio \(3:2\) or \(4:3\)), and this transformation must at all times take place coherently. Thus, at point I the phase difference between two oscillations with a rational frequency ratio must be measured. One of the methods of measur-

...measurement is the method of Lissajous figures, which was also widely used by us.^23

Let us now return to formula (11), which, as is easy to show, is valid in this case as well; moreover, \(\Psi\) is the total retardation (on the forward and return path), measured in periods of oscillation with frequency \(\omega_1\). From it one sees that \(\Psi\) contains, at distances large in comparison with \(\lambda\), a large number of complete phase cycles \(\theta\) (at equal frequencies \(\omega_1=\omega_2\), \(\theta=2\pi\); for \(\omega_1/\omega_2=3/2\), \(\theta=\pi\)). Since the phase angles can be measured directly only within the limits of one complete cycle of phase variation, we shall write the total phase angle \(\Psi\) in the form

\[ \Psi=z\theta+\psi . \]

Here \(z\) is an integer number of complete phase cycles, and \(\psi\) is the actually measured phase angle, representing the fractional part of the cycle. Thus formula (11) takes the form

\[ z\theta+\psi=\frac{2\omega D}{v}+\rho_z . \tag{12} \]

As is clear from this formula, it alone is insufficient for determining the propagation velocity \(v\), since by measurement one can find only the quantity \(\psi\), and not \(z\theta+\psi\). To obtain the necessary additional relation one may proceed in two ways.

I. First, one may proceed as is done in measuring length standards by means of a Michelson interferometer, namely: observe how the phase difference changes under continuous variation of the distance \(D\) between points I–II from \(D_1\) to \(D_2\). Then we shall obviously obtain:

\[ \Delta\Psi_D=h_D\theta+\psi_D=\frac{2\omega\Delta D}{v}+\Delta\rho_D, \tag{13} \]

where \(h_D\) is the observed integer number of complete cycles of phase variation under a smooth change of the distance by \(\Delta D\). This method, which fully corresponds to the method of the Michelson interferometer, received the name “method of displacement” or “method of radar.”^24

II. Secondly, observing, at unchanged distance \(D\), the change of \(\Psi\) under a smooth variation of the frequency \(\omega_1\) from \(\omega_k\) to \(\omega_a\)—a technique first applied by Appleton in his well-known method of measuring height—we shall obviously obtain:

\[ \Delta\Psi_\omega=h_\omega\theta+\psi_\omega=\frac{2\Delta\omega D}{v}+\Delta\rho_\omega . \tag{14} \]

Since this method of changing the frequency makes it possible, knowing the velocity \(v\), to determine directly the distance \(D\) between two points, it received the name of the method of the radio rangefinder.^21,^22

In our measurements of the velocity \(v\) we used both of these methods.

In the case of applying the radio range-finder method, the distance between the two points at which the measuring stations were located was determined geodetically with the greatest possible accuracy; the frequency \(\Delta \omega\) was determined as \(\Delta \omega=\omega_{1a}-\omega_{1k}\), where \(\omega_{1k}\) and \(\omega_{1a}\) were fixed by two quartz standards whose temperature was measured or kept constant by means of a thermostat.

From numerous series of observations, the mean \(\Delta \Psi\) was used to compute not \(v\), but \(\overline v\), and only when this seemed necessary from the standpoint of the accuracy attained was a correction for \(\Delta \varphi\) introduced (usually of the order of ten degrees), and \(v\) and \(v^*\) were computed.

In the case of applying the radiolog method, one of the stations was situated on a vessel or an automobile. The frequency \(\omega\) remained unchanged, and \(\Delta \Psi'_D\) was measured while moving from point \(P_1\) to \(P_2\), the distances of these points to the fixed station being measured geodetically.

It must be noted that, in measurements at sea, the chief source of errors was the difficulty of precisely determining or fixing both points I and II. Even in those cases when, in measurements by the radio range-finder method, one or both stations were located on vessels standing at anchor, small changes in the exact position of the points are possible, caused by drift or by the turning of the vessel as a result of changes in the force or direction of the wind, the current, the tide, or the ebb. Of especially great importance here is drift during the measurement itself, since it causes a change of phase because of a change in \(D\), whereas the observed change is attributed to a change in \(\Delta \Psi\) resulting from a change in frequency at unchanged \(D\). It is easy to see from (13) and (14) that, in this way, the error in determining \(D\) is increased by a factor of \(\dfrac{\omega}{\Delta \omega}\). Thus, for example, if the vessel shifted during the transition from \(\omega_k\) to \(\omega_\alpha\) by only \(3\ \mathrm{m}\), then, with \(\dfrac{\omega}{\Delta \omega}=40\), this leads to an error corresponding to a change in \(D\) by \(\pm 120\ \mathrm{m}\).

In our experiments, much attention was paid to these sources of error, and their harmful influence was eliminated as far as possible both by a more careful determination of the position of the vessel and by accelerating and increasing the number of measurements with the transition from \(\omega_k\) to \(\omega_\alpha\) and back.

Б. Experimental Results

Before proceeding to present the results of the measurements, let us note the following. In the theoretical derivation of formulas for phase relations in the radiation field of a dipole located near the surface of the earth, we proceeded from a far-reaching idealization of the actual conditions in which radio waves propagate. In reality, the surface of the earth is not a plane separating two homogeneous spaces unchanging in time. The electrical properties of the earth (\(\sigma\) and \(\varepsilon\)) are not only not constant in space, but may also vary in time (for example, depending on meteorological conditions). On the other hand, the atmosphere surrounding the earth is inhomogeneous, and its

electrical properties, especially of the upper ionized layers, are subject to changes in time. Therefore, under real conditions a change is possible both in the path length of the radio waves \(D\), and in their mean velocity \(\bar v\).

An estimate of the influence of these heterogeneous causes shows that, in a large number of practically important cases, these changes may be regarded as small; nevertheless, an experimental check of possible changes in these cases appeared absolutely necessary. It should be noted that conditions can also be created under which these changes (especially those caused by the influence of the ionosphere at night) are large. This was used by us, for example, in observations during the total solar eclipse of June 19, 1936.

In this connection, from the very beginning we were faced with the question of the extent to which formula (10) is applicable under real conditions and, above all, whether the right-hand side of formula (10) is in fact constant.

A judgment concerning constancy can be obtained on the basis of measurements of phase angles, carried out in large number and at different times when determining the velocity \(\bar v\) both by the method of frequency variation and by the displacement method. Thus, the determination of the constancy of \(\dfrac{D}{\bar v}\) is obtained as a by-product in determining \(\bar v\). A more accurate determination of the degree of constancy of \(\dfrac{D}{\bar v}\) can also be obtained directly, namely by continuous observation of the change of the phase pattern (phase difference) at one of the points while controlling the constancy of the frequency and of the phase deviations of the apparatus at both points. This method makes it possible to investigate the constancy of \(\dfrac{D}{\bar v}\) not only when there is a single path of propagation, but also in more complex cases. Moreover, it is considerably more sensitive than indirect methods, since it does not require the measurement of phase deviations, but only the control of their constancy, which can be done with great accuracy (see \(^{21}\)).

However, with the apparatus that was at our disposal, it was practically possible to follow the constancy of \(\dfrac{D}{\bar v}\) only over comparatively short intervals of time (several hours).

1. Study of the constancy of the velocity of propagation of radio waves along the earth’s surface. Let us begin with an exposition of those conclusions which can be drawn concerning the constancy of \(\dfrac{D}{\bar v}\) from the experimental data obtained in determining the velocity \(\bar v\). Although, as was said above, direct immediate methods give more accurate results, we shall nevertheless dwell briefly on these conclusions, first of all because in this case very diverse propagation conditions are encompassed, and also because in this way those peculiarities are revealed with which one has to deal when measuring the velocity \(\bar v\) itself. In addition, this material

makes it possible to form an opinion about the constancy of \(\dfrac{D}{v}\) over considerable intervals of time.

The first substantial experimental material for judging the degree of constancy of the velocity \(v\) in propagation over considerable distances under actual conditions was obtained during the expeditionary work\(^{22,23}\) carried out in 1934 by the Laboratory of High-Frequency Physics of the TsRL jointly with TsNIGAIK in the region of Pyatigorsk (North Caucasus) at five different distances:

\[ \begin{aligned} &\text{Mashuk--Dzhutsa}\ .\ .\ .\ .\ .\ .\ D = 10\,424\ \text{m}\\ &\text{Mashuk--foot of Dzhutsa}\ .\ .\ D = 11\,416\ \text{m}\\ &\text{Mashuk--Kaban}\ .\ .\ .\ .\ .\ .\ D = 27\,894\ \text{m}\\ &\text{Post--Kaban}\ .\ .\ .\ .\ .\ .\ D = 24\,995\ \text{m}\\ &\text{Post--gorge beyond Kaban}\ .\ .\ D = 26\,620\ \text{m} \end{aligned} \]

on waves \(\lambda_1 = 230\ \text{m},\ \lambda_2 = 345\ \text{m}\).

If the results of these measurements, of which about 1,000 were made, are represented graphically, one obtains pictures similar to Figs. 4 and 5\(^{22}\). In these figures are presented the results of measuring phase differences made for one and the same distance on different days; moreover, for clarity, along the ordinate axis are plotted not the full phase angles \(\Psi_k\) and \(\Psi_a\), but only the directly measured fractional parts of the phase cycles \(\psi_k\) and \(\psi_a\) [see (12)]. As is clear from these figures, despite the considerable scatter of the observed values of \(\Psi_a\) and \(\Psi_k\), all of them differ from the mean by no more than \(\delta_\psi = \pm 25\). Analogous results were also obtained at the other distances.

Fig. 4

Fig. 5

Hence one can draw the following important conclusion. Since it is completely improbable that \(\Psi_a\) and \(\Psi_k\) would always change by jumps of an integral number of phase cycles, it is clear that from observation to observation \(\Psi_a\) and \(\Psi_k\) changed only by \(\delta_\psi\), i.e. by a part of one phase cycle. This conclusion was also supported by the fact that during the entire period of measurements there was not a single spontaneous change of phase (Lissajous figures), not caused by retuning or by a change in the state of the apparatus, from which it would have been possible to conclude that the phase angle could change gradually or abruptly by an integral number of cycles. This result makes it possible to obtain an estimate of the constancy of \(\Psi_a\); since from the observations it was found that

\[ \dfrac{D}{v} \simeq 0.4 \cdot 10^{-5}\ \text{sec.}, \]

then, knowing \(f_a = 1\,304\,886\ \text{Hz}\), we find that

\[ \Psi_a \simeq 90\,000 \quad\text{and}\quad \dfrac{\delta_\psi}{\Psi_a} \simeq 3 \cdot 10^{-4}, \]

i.e. that \(\dfrac{D}{v}\) is constant with an accuracy of \(3 \cdot 10^{-4}\).

When observing in Pyatigorye the greatest distance was only about 28 km. It should be noted that, since T-shaped vertical antennas were used in this case, the influence of rays reflected from the ionosphere was, naturally, insignificant here. It therefore seemed important to carry out investigations of the constancy of the phase difference at points located at a more considerable distance (100 km and more) from one another, in order to determine the limits of applicability of interference methods to the practical problem of measuring distances and the degree of influence upon them of ionospheric rays. In addition, the very first measurements, made over comparatively short distances, showed that their accuracy was considerably lower than that which, in principle, the interference method can provide. At the same time it became clear that one of the apparently most substantial causes limiting the accuracy of the measurements is the insufficient precision of the measuring apparatus. Therefore, in order to clarify the important question of the constancy of the propagation process itself and the question of the extent to which the degree of constancy can limit the accuracy of radio-interference measurements, it was expedient to proceed not along the path of repeated measurements, but along the second path indicated above. For this purpose it was necessary to arrange the experiment in such a way that, while guaranteeing the greatest possible invariability of the apparatus and constancy of the frequency, it would be possible directly to observe changes in the phase difference caused by changes in the propagation conditions of radio waves. The results of such experiments ^21,22,26, carried out by us in 1936 and 1937, are presented in Table 4.

Table 4

Observation of the constancy of \(\dfrac{\omega D}{v}\)

From these observations one may conclude that, in the wave range \(\lambda \geq 300\) m, the constancy of \(\dfrac{D}{v}\) is preserved during propagation over a sea surface at distances up to 150—200 km with relativel-

with an accuracy not less than \(10^{-5}\). Hence it also follows that, with appropriate antennas, under these conditions one may neglect the influence of “sky rays.”

As measurements with radiodalnometers over land in the region of Pugachev at a distance of about \(100\) km on waves \(\lambda = 300\)—\(450\) m have shown, no influence of sky rays is observed at night even in the predawn hours (November 1939).\(^{27}\)

Thus, it may be regarded as established that the constancy of the optical path length of radio waves in propagation along the earth’s surface over distances of the order of \(100\) km and even greater is so high that from this side there are no obstacles to the use of radio-interference methods for an accurate determination of the velocity \(v\) of radio waves.

As for still greater distances, at present there is not yet sufficiently extensive experimental material, especially with the sensitive interference methods described above.

Investigations of the constancy of \(\dfrac{D}{v}\) over large distances were published in 1934 by Déco and Gallé.\(^{28}\) The experiments were carried out on a wave \(\lambda = 349\) m between Paris and Strasbourg (a distance of about \(350\) km) and on waves of about \(24\) and \(33\) m between Pontoise, near Paris, and Algiers (a distance of about \(1500\) km). The method they used consisted in the following: from point I (Pontoise, near Paris) waves of frequency \(f_1\) (\(\lambda_1 = 24.15\) m), modulated by an audio frequency \(N\) (a 1000-cycle current from a tuning-fork generator), were sent out; these were received by point II (Algiers or Strasbourg), and after detection they modulated the waves, emitted by the transmitter at point II, of a slightly different frequency \(f_2\) (\(\lambda_2 = 24.65\) m). These waves were received at point I, and the 1000-cycle current obtained after detection was compared in phase with the original modulating current, the phase measurements likewise being made by Lissajous figures. It is easy to see that the method of Déco and Gallé is considerably (with respect to \(\dfrac{f}{N}\dfrac{u}{v}\), i.e. approximately 1000 times, since \(u \sim v\)) less sensitive than the interference method used by us, for in the experiment of Déco and Gallé we are dealing with changes essentially not of the quantity

\[ 4\pi f \frac{D}{v}, \]

as in the radiodalnometer, but of the quantity \(4\pi N \dfrac{D}{u}\), where \(u\) is the group velocity.

The observations of Déco and Gallé showed the following: on short waves by day the phase pattern in general remained stable; only from time to time was a slight rocking of the Lissajous figure observed, with a period of the order of several minutes, the maximum change of phase angle observed by day being of the order of \(90^\circ\), which corresponds to

\[ \delta \frac{2D}{u} \simeq 2.5 \cdot 10^{-4}\ \text{sec}. \]

At night, on the contrary, the Lissajous figure was continuously deformed and at times so rapidly that it was impossible to follow its changes. Hence it follows that the group time \(\dfrac{D}{u}\) changed by not less than \(10\%\).

At long wavelengths ($\lambda = 349\ \text{m}$), over a shorter distance (Paris—Strasbourg, 350 km) the phase pattern was more stable by day. At night the same phenomena as on short waves were observed qualitatively, but they were expressed considerably more weakly.

It must be noted that the relatively low stability of the phase pattern in the experiments of Deko—Gallet, despite the significantly lower sensitivity of their method, is explained by the substantial role which sky rays undoubtedly played in their observations. In particular, this occurred in the experiments between Paris and Strasbourg, owing to the weakening of the direct ray because of the great distance over land.

2. Measurement of the velocity of propagation of radio waves.

We now turn to presenting the results of our measurements of the propagation velocity $v$. As was already indicated above, our task was not only to determine the mean velocity, but also to check to what extent the formulae obtained from the solution of Sommerfeld’s problem for a strongly idealized case are applicable under real conditions. Therefore it was very essential to choose suitable terrain conditions for the measurements; an important additional condition was the possibility of an accurate geodetic determination of $D^{1})$. The first measurements were made, as already indicated above, between points situated on mountain summits, rising in isolation to a sufficient height above the plain (the Pyatigorye region). Further measurements were made over the surface of bodies of water (sea and lake) and over land.

Table 5 brings together the results of the principal measurements.

As is seen from Table 5, despite the comparatively low accuracy of most of these measurements, the values obtained for $\overline{v}$ agree comparatively closely with the magnitude $c$ of the velocity of light in air, taking into account the correction for humidity$^{6}$ ($c = 299\,670\ \text{km/sec}$), as indeed should have been expected on the basis of theory.

From all these measurements one may derive for the velocity of radio waves over the sea the mean value

\[ \overline{v} = 299\,600\ \text{km/sec}. \]

This formally obtained mean differs somewhat from the value of the velocity of light in air (with correction for humidity$^{6}$), obtained by Michelson ($c = 2.99670 \cdot 10^{10}\ \text{cm/sec}$). At present it is still difficult to say with complete certainty whether this difference may be ascribed any real significance. However, the following must be noted.

In computing $\overline{v}$ in all the cases given in Table 5 the additional phase was not taken into account, i.e. $\overline{v}$ was computed from

\[ \overline{v} = \frac{4\pi \Delta f D}{\Delta \Psi_{\omega}} \quad \text{or from} \quad \overline{v} = \frac{4\pi f \Delta D}{\Delta \Psi_{D}}. \]

$^{1})$ It should be noted that great assistance in this respect was rendered to our work by the Central Scientific-Research Institute of Geodesy, Aerial Survey and Cartography, in the person of Professor O. G. Diti and engineer A. I. Gruzinov, and by the Hydrographic Administration of the Northern Sea Route.

Table 5

Speed of radio waves

Meanwhile—and this is very important—the additional phase in formula (10) is always positive. This, apparently, may also explain the circumstance that the mean value \(\bar v\) turned out to be somewhat smaller than the value of \(c\) according to Michelson. Indeed, the order of magnitude of the correction, as is easily verified, fully agrees with this supposition. Let, for example, \(D=114\) km, \(\lambda=300\) m, \(\varepsilon=80\), and \(\sigma=10^{10}\), which approximately corresponds to the case of the measurements in the White Sea on September 4–9, 1937, between Ry-Nawolok Island and Ploskie Ludy Island.

Then, using the formulas derived by N. A. Ryazin,\(^{20,21}\) we obtain \(\Delta \varphi_\omega=4^\circ\!.1\). Since \(\Delta \Psi_\omega \cong 13\,700^\circ\) and

\[ v=\frac{\bar v}{1-\dfrac{\Delta \varphi_\omega}{\Delta \Psi_\omega}} \]

[see formula (7)], the correction

\[ \Delta v=\bar v\,\frac{\Delta \varphi_\omega}{\Delta \Psi_\omega}\cong 90 \text{ km}, \]

and consequently \(v\) is almost exactly equal to the value \(c\) obtained in air, with allowance for the humidity correction. Of course, such close agreement must, in view of the insufficient accuracy of the measurements (about \(1/2\,000\)), be regarded as accidental. Nevertheless, it may not be without interest that, by introducing the correction for the additional phase, we not only obtain a correction of the magnitude of the velocity in the required direction, but also arrive at a numerical value of \(c\) that is close to the latest, most accurate values obtained by Michelson and other investigators.

Very instructive in this respect is also the result obtained in measuring the velocity of propagation along a flat land surface in the vicinity of the town of Pugachev in November 1939.\(^{27}\)

The value obtained by the displacement method, \(\Delta \Psi_D=7\,001^\circ\), when the distance was changed by \(\Delta D_1=1\,263.5\) m from \(D_1=1\,350.5\) m to \(D_2=2\,614\) m, gives

\[ \bar v=\frac{720^\circ f\Delta D}{\Delta \Psi_D}=(298\,650\pm170)\ \text{km/sec}. \]

Here \(f=2\,298\,360\pm20\) Hz. If we now introduce the correction for the additional phase \(\Delta \varphi_D\), then, taking \(\sigma=3\cdot10^6\), \(\varepsilon=4\) (dry soil), we obtain \(\Delta \varphi_D=24^\circ\), whence we get: \(v=299\,700\) km/sec.

It should be noted that the correction \(\Delta \varphi_D\), and consequently also the correct allowance for the constants \(\sigma\) and \(\varepsilon\), plays an especially large role for comparatively small \(\Delta \Psi_D\), i.e., for small \(\Delta D\). Thus, for \(\sigma=2\cdot10^6\) and \(\varepsilon\) from 3 to 5, \(\Delta \varphi_D\) is already somewhat different, namely \(\sim 20^\circ\), so that \(v=299\,500\) km/sec. For \(\sigma\) and \(\varepsilon\) corresponding to dry soil, larger values of \(\Delta \varphi_D\), and consequently also \(v>c\), would have been obtained.

For large distances the value of the correction \(\Delta \varphi\), and consequently also the influence of the soil constants, becomes smaller. In this respect, the results obtained in the same locality near Pugachev at the same time at distances of the order of 100 km are indicative. Here it was found that

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

Thus, not only in propagation over the sea, but also in propagation over a sufficiently even land surface (level steppe expanses, meadows, etc.), we obtain for the velocity of electromagnetic waves a value close to \(c\). At the same time, the validity of formula (10) is confirmed, as is the fact that the velocity \(v\) in it is indeed the velocity of light in air. It should be noted, however, that at present there are still not enough data, either experimental or theoretical, concerning the influence of inhomogeneities of the soil, unevenness of the earth’s surface, etc., on the results of measurements of the propagation velocity of radio waves. Therefore it still cannot be said definitively to what extent the results given above, obtained under definite conditions, are indicative for measurements on land in general. Nevertheless, it seems to us that it may still be considered proven that, also in measurements over land on waves of the medium-wave range with umbrella or T-shaped antennas, the influence of ionospheric rays may be neglected at distances of several tens (and even hundreds) of kilometers.

The results we have obtained allow us, as it seems to us, to draw the following conclusions.

1. In the propagation of radio waves of the medium-wave range along the surface of the earth at distances up to one hundred kilometers, the optical path length is constant, when antennas of the corresponding type are used, to an order of \(10^{-5}\). It follows from this that under these conditions the action of ionospheric rays need not be taken into account.

2. The total phase difference \(\Phi\) between two points separated by a distance \(D\) near the surface of the earth in the electromagnetic field of a harmonic radiator (vertical dipole) located on the surface of the earth can be represented in the form

\[ \Phi = \frac{2\pi f D}{v} + \varphi(f, D, \sigma, \varepsilon), \tag{15} \]

where \(v\) is equal to the velocity of light in air, and the additional phase \(\varphi(f, D, \sigma, \varepsilon)\), with increasing \(D\), depending on \(f\), \(\sigma\), and \(\varepsilon\), tends to a limit lying between \(\pi/2\) and \(\pi\).

1. In propagation over the sea up to distances of the order of 200 km, the numerical value of the velocity \(v\) is obtained, with an accuracy not worse than \(5\cdot 10^{-4}\), equal to the velocity of light in air*).

2. In propagation along land there exist conditions (for example, level steppe expanses, meadows) under which formula (10) is applicable with an accuracy not worse than \(6\cdot 10^{-4}\).

In conclusion, let us touch briefly on measurements of the velocity of propagation of radio waves along the earth’s surface, carried out in recent years (beginning in 1936) in America and England by other methods. The occasion for these measurements was the data on the velocity \(v\), published by Colwell, Hell, and Hill at the end of 1936,^29 according to which the magnitude

*) For the three wavelengths which we used, and at the indicated distances, the influence of the curvature of the earth’s surface, within the limits of this accuracy, does not yet affect the determination of \(v\).

the velocity \(v\) differed from the velocity of light by one third or more. The method used by these investigators was based on measuring (by means of a sweep on the screen of a cathode oscilloscope) the time interval between the moment of sending a pulse, of duration of the order of \(10^{-4}\) sec, from one point \(A\), and the moment of arrival of an identical pulse from another point \(B\), sent from there at the moment when the pulse from \(A\) arrived there. Adjustment of the moment at which the pulse was sent by point \(B\) was carried out by means of a phase regulator installed in the alternating-current circuit common to both points. The frequencies of the oscillations emitted by the two points were, respectively, \(f_1 = 2\,398\) kHz (\(\lambda_1 \simeq 60\) m) and \(f_2 = 1\,614\) kHz (\(\lambda \simeq 145\) m). From the observations of Colwell, Helm, and Hill it followed that \(v\) is considerably less than \(c\), amounting on the average to about \(2/3\,c\), sometimes falling to one half (54%), and they observed considerable fluctuations in the values of \(v\) from day to day and even from experiment to experiment. These results naturally aroused great distrust of their correctness, especially in view of the fact that, because of the small distance (about 20 km) between points \(A\) and \(B\), any appreciable influence of ionospheric rays had to be regarded as excluded here. It was very probable that, as we had already stated in a note to our first article\(^{22}\), which was already in press when the article by Colwell, Helm, and Hill appeared, the matter lay in an inadequate estimate and incorrect allowance by the authors for the relaxation times of the apparatus, whose exact values in their method had to play a decisive role, since they were of the same order as the measured times.

The article by Colwell, Helm, and Hill led to a series of new experimental investigations to determine the value of \(v\) for free propagation along the earth, which showed the incorrectness of these authors’ results. Thus, already in April 1937, Ross and Slow\(^{30}\), measuring with the aid of a cathode oscilloscope the phase difference of oscillations induced in two vertical antennas placed along the direction of the radio waves at a distance of 34.9 m from one another, obtained for the phase velocity \(v\) at various frequencies from 2.5 to 15 MHz an average value equal to 295,000 km/sec with an accuracy estimated by the authors at 5%, with extreme values of 310,000 and 275,000 km/sec.

In order to bring complete clarity to this question, Farmer and Morgan\(^{31}\) in 1939 repeated, at frequencies of the order of \(1.8 \cdot 10^6\) Hz, the experiments of Colwell, Helm, and Hill, taking every possible measure to eliminate sources of error. The chief such errors were: a) error of the oscilloscope time scale, b) difference in the transit time through the receiving devices of signals of different intensity, c) error due to uncertainty of the reference point on the envelope of the pulse observed on the oscilloscope screen, d) instability of the power-line conditions owing to changes in load.

The experiments gave for \(v\) a value equal to \(0.95\,c\), with a probable error of 5–7%, and thus established the erroneousness of the measurements of Colwell and his collaborators.

Finally, the erroneousness of the results obtained in 1936 by Colwell follows from a work published in 1942 by Colwell\(^{32}\), who pro-

conducted, together with his collaborators—Wood, Bailey, and Marjem—a large series of measurements on waves with frequencies \(f_1 = 3492.5\ \mathrm{kHz}\) and \(f_2 = 2398\ \mathrm{kHz}\), using a somewhat modified pulse method. In these measurements, in order to eliminate the relaxation time in the apparatus, observations were first made when point \(B\) was at a distance of \(0.79\ \mathrm{km}\) from point \(A\), and then when point \(B\) was moved to a distance of \(3.67\ \mathrm{km}\) from \(A\), and the velocity was determined from the difference of the pulse delay times obtained in the two cases. As the mean of 180 measurements, the following was obtained:

\[ v = 298\,500\ \mathrm{km/sec}. \]

Thus, all these measurements show that there is no reason to consider the velocity of radio waves \(v\), when propagated along the earth, to differ from the velocity of light in air \(c\) by more than \(3\text{--}5\%\).

As was indicated above, the results of our measurements, carried out with much greater accuracy (as, incidentally, is noted in the work of Moganti and Farmer\(^{31}\) and in the review by Smith-Rose\(^{6}\), made by him on October 7, 1942, at the annual meeting of the Radio Section of the English Electrical Society), show that the velocity \(v\) cannot differ from the velocity of light \(c\) by more than \(5\cdot 10^{-4}\).

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1. Report delivered on 16 February 1943 at a meeting of the Council on Radiophysics and Radio Engineering in Kazan. ↩