LIGHTNING AND THE PROPAGATION OF AUDIO-FREQUENCY ELECTROMAGNETIC WAVES

Ya. L. Al’pert

1. INTRODUCTION

The propagation of ultra-long waves, whose frequencies lie at the beginning of the electromagnetic-wave scale, has until recently been little studied. We are speaking of waves with lengths from several hundreds to several tens of kilometers, i.e., of waves of audio frequency. At the same time, investigation of the field structure of these waves above the Earth’s surface is of interest from various points of view. On the one hand, the so-called long radio waves used in practice lie approximately in the range from 10 to 30 kc/s. On the other hand, the spectrum of the so-called atmospherics—electromagnetic disturbances excited by thunderstorm discharges—has high intensities already beginning at frequencies of 300–500 c/s and up to 30 kc/s and higher.

The principal difficulties in studying waves of such low frequency are as follows. Theoretically, in the general case one encounters the problem of calculating the electromagnetic field in a spherical waveguide. One of its boundaries—the Earth’s surface—may at these frequencies be regarded as an infinitely conducting wall. The structure of the other boundary of the waveguide—the ionosphere—is considerably more complicated. To obtain the correct picture it is necessary, first, to take into account its “diffuseness,” i.e., the variation of the electrical properties of the ionosphere with height, and, second, the dependence of its conductivity on frequency. A very substantial complicating circumstance for calculations is, moreover, the fact that the wavelengths $\lambda$ under consideration are comparable with the distance $h$ between the boundaries of the waveguide ($\lambda \sim 0.1 \div 10h$); this rules out the possibility of treating this problem approximately by means of geometrical optics and requires a rigorous electrodynamic solution. It should be noted

It should be noted that, for a complete investigation of the question, it is also necessary to take into account the anisotropy of the ionosphere and the “roughness” of the boundaries, i.e., the presence in the ionosphere and on the earth’s surface of both geometrical and electrical inhomogeneities in the horizontal directions. In such a general formulation the mathematical difficulties are extremely great. The consideration of simple cases, such as, for example, the spherical problem with homogeneous smooth boundaries¹, attempts to take account of the vertical stratification of the ionosphere²˒³, or other solutions⁴˒⁵˒⁶, have not led the authors to formulas more or less convenient for calculations; the approximate formulas used³˒⁶ do not give correct results.

However, for the interpretation of the observed experimental results and for obtaining correct quantitative data, it is possible to consider a simpler problem, formulated below. The solution⁷, carried through to numerical results, gave greater agreement with the results of measurements than could have been expected.

From the experimental side, the main difficulty arising in the study of the range of electromagnetic waves that interests us is due to the difficulty of creating sources of radiation. At such low frequencies, in order to obtain powers sufficient for observation, it is necessary to construct very expensive transmitting devices with cumbersome antennas. Therefore, despite the fact that the development of radio began precisely in connection with long radio waves, up to the present time only some measurements of field strength in the range 12–30 kc/s⁸ have been carried out in detail, and only isolated, more detailed studies of the structure of the field amplitude above the earth’s surface at one frequency (16 kc/s)⁹˒¹⁰; while any measurements of the phase structure or of the velocity of these waves are altogether absent.

At the same time, in nature there exists a paired source radiating electromagnetic waves in a broad spectrum of frequencies, with the greater part of the energy of its radiation lying precisely in the region of low frequencies. This source is the lightning discharge. At any point above the terrestrial globe there arrive every second, from various distances and directions, many electromagnetic disturbances excited by lightning, in the form of individual signals, called, at distances remote from their sources, atmospherics. Thus, for example, in Moscow one can record, with the average sensitivity of the corresponding apparatus, no fewer than 5–10 signals per second. It is therefore quite natural to strive to use atmospherics for studying the propagation of low-frequency electromagnetic waves. At first sight it seems incredible that in this way it is possible to obtain any sufficiently accurate results, not only quantitative but even qualitative. The chief reason for caution with respect to such experiments is as follows. It is clear that there are no fundamental difficulties in creating

equipment for a more or less undistorted recording of a single atmospheric, or for determining, with one or another degree of accuracy, the direction of its arrival and the distance to the source. However, in principle it is impossible to know exactly the time-sweep form or, in other words, the frequency characteristic of the radiated signal that produced the recorded atmospheric. Nevertheless, the results of many observations have shown that the sources of atmospherics possess universal properties, and the application of the method of complete harmonic analysis of the received single signal[^11] makes it possible, when used consistently, to obtain much important information about the properties of electromagnetic waves and thereby to carry out fairly clean experiments. Most indicative in this respect is the fact that in these experiments it has been possible to obtain not only the amplitude characteristics of the field, but, for the first time by this method, the phase velocities of electromagnetic waves in the range of audio frequencies have been measured (in essence, by very simple means) at distances varying from several hundreds to 2000–3000 km. At the same time, it is well known that measurements of the velocity of electromagnetic waves in any frequency interval belong to the class of complex and delicate experiments. A theoretical analysis of the corresponding experimental data shows that, from their processing, it will apparently be possible to obtain important data on the structure of the ionosphere.

Below a number of results are presented from experimental studies of electromagnetic waves of audio frequency, obtained in particular from studies of atmospherics. A brief description is given of the methodology of the corresponding experiments, and the general properties of low-frequency electromagnetic waves that follow from theoretical calculations are set forth.

2. SOME RESULTS OF THEORETICAL CALCULATIONS AND MEASUREMENTS ON LONG RADIO WAVES

The general problem of the propagation of low-frequency electromagnetic waves over the earth’s surface can be substantially simplified; at the same time, as has already been indicated, the solution obtained describes, with sufficient accuracy, the principal features of the phenomena observed experimentally.

First, one may assume a priori, taking into account the high conductivity of the earth’s surface (it is taken to be infinite), and the fact that the wavelengths under consideration and the effective heights of the lower boundary of the ionosphere are small in comparison with the radius of the Earth, that the sphericity of the Earth appears merely as a geometrical factor everywhere except in the immediate vicinity of the radiator and in some neighborhood of the point antipodal to the source. It is not difficult

to show that in such a waveguide, beginning with the region where cylindrical waves have formed (distances of the order of several ionospheric heights), taking into account the fact that the total energy flux passes through a cylindrical rather than a spherical surface leads to a change in the field strength by the factor \(\sqrt{\frac{\vartheta}{\sin \vartheta}}\), where \(\vartheta\) is the angle subtended, from the center of the sphere, by the distance between the source and the point of observation. To the extent that the solution of the general spherical problem has now been analyzed, these qualitative considerations are confirmed more rigorously.\(^{6,7}\)

Secondly, in the indicated frequency range, taking into account the influence of the Earth’s magnetic field, i.e., the anisotropy of the ionosphere, does not lead, as has been shown, for example, in \(^{10}\), to any substantial change in the results of the field calculation.

Finally, the “roughness” of both the ionosphere and the Earth’s surface must undoubtedly affect the results of the calculations. However, taking them into account in the general theory is extremely difficult and is hardly necessary. The action of these factors on the propagation path is mainly averaging in character, and taking them into account should in a number of cases lead to the introduction, instead of the true values of the parameters characterizing the medium, of certain effective values, which can be calculated by solving specially problems of the corresponding type. The same applies to local field perturbations, which should be calculated theoretically for a concrete form of inhomogeneity.

Thus, the problem of interest to us can be reduced to consideration of the question of the propagation of electromagnetic waves between two plane boundaries, one of which is sharp, while the other is “diffuse” and constitutes a reflecting dispersive medium inhomogeneous in the vertical direction. In this formulation this problem has been considered theoretically and carried through to numerical results in work \(^{7}\), taking into account the concrete properties of the lower ionosphere.

In doing so, the general solution of the “plane” problem, obtained earlier in works \(^{12,13}\), was used.

In the waveguide, owing to its resonant properties, the source must excite an infinite discrete set of waves of the given frequency, whose law of variation of the wave numbers (i.e., the phase velocities and attenuation coefficients of the individual components) depends on the geometrical and electrical properties of the waveguide. In the present case, naturally, the corresponding waves must be cylindrical, i.e., their amplitudes proportional to \(\frac{1}{\sqrt{r}}\). The general solution of the problem formulated above indeed leads to an infinite sum of cylindrical waves for the magnetic and electric fields. The corresponding formulas at distances at which the Hankel functions may be replaced by their

by asymptotic representations, have the following form:

\[ \left. \begin{aligned} H&=-\frac{120\pi J z_{\mathrm{д}}}{h\sqrt{\lambda}\sqrt{r}} \sum_{n=0}^{\infty} S_n^{1/2} p(C_n) e^{-i\left(k_0 S_n r-\frac{3\pi}{4}\right)},\\ E&=\frac{120\pi J z_{\mathrm{д}}}{h\sqrt{\lambda}\sqrt{r}} \sum_{n=0}^{\infty} S_n^{3/2} p(C_n) e^{-i\left(k_0 S_n r-\frac{\pi}{4}\right)} =\\ &=\frac{120\pi J z_{\mathrm{д}}}{h\sqrt{\lambda}\sqrt{r}}\, B(\omega,r)e^{-i\Phi(\omega,r)} =\\ &=E(\omega,r)e^{-i\Phi(\omega,r)}, \end{aligned} \right\} \tag{1} \]

where \(B(\omega,r)e^{-i\Phi(\omega,r)}\) may be called the interference factor; the wave numbers are

\[ \left. \begin{gathered} k_0 S_n=k_0(S_{n1}+iS_{n2})\\ \left(C_n=\sqrt{1-S_n^2},\quad k_0=\frac{\omega}{c}\right), \end{gathered} \right\} \tag{2} \]

\(E\) and \(H\) are expressed in millivolts per meter, the current \(J\) in amperes, and the effective height \(z_{\mathrm{д}}\), characterizing the radiating properties of the source; the effective height \(h\) of the ionosphere and the distance \(r\) are expressed in kilometers,

\[ p(C_n)=\left[1+i\frac{d\rho}{dc}\cdot\frac{1}{2hk_0\rho}\right]^{-1}_{(C_n)}, \]

and \(\rho(C_n,\ldots)\) is the coefficient of reflection from the ionosphere, depending on its electrical properties and on the so-called complex “cosine of the angle of incidence” \(C_n\) of a plane wave.

The determination of the wave numbers \(S_n\) constitutes the principal and most difficult part of the calculations. We do not dwell here on the details of these calculations, which reduce to the need to solve the equation of the poles of a certain function, the integral of which determines a series of computations—the discrete spectrum of waves. To solve this equation it was necessary to choose a model of the ionosphere in such a way as to take into account its reflecting properties at different frequencies. For this purpose the lower part of the ionosphere is described by the equation of a transition layer, whose maximum conductivity is characterized by the parameter

\[ \frac{N}{\nu}=10^{-4} \]

(where \(N\) is the electron concentration and \(\nu\) is the effective number of their collisions). The reflection coefficient from this layer is calculated, and then, by the “fitting method,” a new value \((N/\nu)_{\mathrm{eff}}\), depending on frequency, is determined in such a way that the reflection coefficient from a homogeneous layer with a sharp boundary, characterized by the parameter \((N/\nu)_{\mathrm{eff}}\), has, for each frequency, the same value of the reflection coefficient as that from the chosen

of the transition layer. The equation for the poles, of the form

\[ 2hk_0 C_n + i \ln \rho(C_n) = 2n\pi, \]

was solved by substituting into it the Fresnel reflection coefficient \(\rho(C_n)\) and the values \((N/\nu)_{\mathrm{eff}}\). In doing so, the effective height \(h\) of the layer was chosen from the experimental data.

As a result, values of the complex wave numbers (2) were obtained, making it possible to study the expected behavior of the amplitude and phase of low-frequency electromagnetic waves above the earth’s surface and to compare them with results known from the literature for measurements of the field strength of long radio waves. The values of \((N/\nu)_{\mathrm{eff}}\), \(S_n\), and \(p(C_n)\) for waves with numbers \(n=0\) and \(n=1\) are given in Table I.

Table I

Values of \(S_n = (S_{n1} + iS_{n2})\) and \(p(C_n)\) for \(h_{\mathrm{eff}} = 70\ \mathrm{km}\)

\(n=0\)

\(n=1\)

From Fig. 1, which shows curves of the dependence of the modulus of the interference factor \(B(\omega, r)\) and of the ratios of the mean \(\bar v\) and differential \(v\) phase velocities to \(c\) on the distance \(r\) for

Fig. 1. Dependence of the interference factor \(B(\omega, r)\) and of the ratios of the mean and differential phase velocities to \(c\) on distance at the frequency \(f = 15\ \text{kHz}\).

the frequency \(f = 15\ \text{kHz}\), it is seen that at short distances from the radiator all these quantities vary in an irregular manner. This is due to the complex interference of the various components of waves propagating with different velocities and different

with attenuation. As the distance increases, the role of waves of higher order \((n > 1,2)\), which attenuate more strongly, becomes ever smaller, which leads to a decrease in the irregularity and depth of the oscillations of the amplitude and phase of the field. And, finally, at those distances,

Fig. 2. Curves of the field strength \(E(\omega, r)\) at distances \(r = 1000, 2000\), and \(3000\) km for radiation power \(w\) of \(1\) kW
\[ \left( J_z Д \simeq \frac{10\lambda\ \mathrm{km}}{2\pi\sqrt{2}}\sqrt{w\ \mathrm{kW}} \right). \]

where only one wave of the spectrum, corresponding to the number $n=0$, is perceptible, the amplitude of the field must decrease exponentially with distance $\left(E\sqrt{r}\sim e^{-k_0 S_{02}r}\right)$, while the phase velocity $v\simeq \dfrac{c}{S_{01}}$ must not depend on distance.

The character of the dependence of the electromagnetic-wave amplitude on frequency is seen from Fig. 2, in which, for various fixed distances, smoothed curves of the field strength $E$ are shown. The theoretical dependence of the ratio of the mean phase velocity $v$ to $c$ on frequency for $r=1000\ \mathrm{km}$ is presented in

Fig. 3. Dependence of $\dfrac{v}{c}$ on frequency for $r=1000\ \mathrm{km}$.

Fig. 3. As in Fig. 2, the curve is smoothed in the sense that it does not reflect the shallow irregularities in its course caused by a certain influence, at this distance, in addition to the zero-order wave $(n=0)$, of waves with numbers $n=1$ and $n=2$. For this reason, for $f>5\ \mathrm{kc}$, $\dfrac{\bar v}{c}$ is everywhere greater than unity, although in fact $\dfrac{\bar v}{c}$ also assumes values less than unity.

From Figs. 2 and 3 it is seen that in the vicinity of $f\sim 2\text{--}3\ \mathrm{kc}$ (corresponding to $\lambda\sim 2h$) the field amplitude has a minimum, the depth of which increases with distance; in this same frequency region the mean phase velocity also passes through a minimum, taking values smaller than $c$, and then, increasing with increasing frequency, reaches a maximum, after which, decreasing, it tends to $c$.

Comparison of the results of measurements of the radio-wave field strength, carried out under various conditions8, up to distances of $10\,000\ \mathrm{km}$, with the theoretical curves gives agreement7 and shows that the chosen model (mean for different conditions) of the ionosphere (see Table I) is generally suitable for calculations. Good-

better agreement between the experimental and theoretical curves constructed for suitably chosen \((N/\nu)_{\mathrm{eff}}\); this is even visible in a number of details, for example, from Figs. 4 and 5, in which

Fig. 4. Comparison of measurement results (dashed line and points) and results of theoretical calculations (solid line) at a frequency of 16 kc/s up to a distance \(r \sim 800\) km.

Fig. 5. Comparison of measurement results (a) and results of theoretical calculations (b) at a frequency of 16 kc/s up to a distance \(r \sim 3600\) km.

the results are given of continuous measurements of field strength carried out in England from an aircraft at a frequency of 16 kc/s, in one experiment flown to a distance of about 800 km,\(^{9}\) and in another\(^{10}\)—in the direction of Cairo to a distance of 3640 km.

From Fig. 5 it can be seen that, beginning with \(r > 1000\) km, a periodicity is observed on the experimental curve (the distances between maxima and minima are approximately the same), which indicates that in the experiment under consideration the principal role was played by two waves (\(n = 0\) and \(n = 1\)). The corresponding theoretical treatment of curve \(^{10}\), based on this assumption, makes it possible to estimate \((S_{01} - S_{11})\), \(S_{02}\), and \(S_{12}\). The values obtained differ somewhat from the values \(S_0\) and \(S_1\) given in Table 1. Thus, from the experiment it follows that \((S_{01} - S_{11}) \simeq 0.018\) instead of 0.014; \(S_{02} \sim 5 \cdot 10^{-4}\) instead of \(5.6 \cdot 10^{-4}\), and \(S_{12} \sim 2.7 \cdot 10^{-3}\) instead of \(6.2 \cdot 10^{-3}\). If, further, on the basis of these data one determines the effective model of the ionosphere that existed under the flight conditions \(^{10}\), then the values obtained are \(h \sim 69\) km and \((N/\nu)_{\mathrm{eff}} = (40 \div 4) \cdot 10^{-6}\), whereas in the layer model used \(^{7}\) \(h = 70\) km, and \((N/\nu)_{\mathrm{eff}} = 10^{-6}\). It is evident from this that processing experimental data on the basis of theoretical calculations can apparently be used in a number of cases to determine the effective model of the ionosphere existing under the conditions of the experiment.

The experimental data cited above on the study of long radio waves in fact exhaust the principal results that can be extracted from the literature, and convincingly testify to their scarcity and incompleteness. Phase measurements, for example, are entirely absent. It was indicated above that the range of these investigations can be substantially expanded if natural sources of low-frequency electromagnetic waves, namely lightning discharges, are used. The results of such investigations obtained up to the present time are described below; they are also compared with theoretical calculations. First the research method is described in general outline.

3. LIGHTNING AS A SOURCE OF LOW-FREQUENCY ELECTROMAGNETIC WAVES

Signals—electromagnetic wave packets caused by thunderstorm discharges—arrive continuously at any point on the Earth from all possible directions and distances. In summer these signals are often produced by nearby thunderstorms. It is possible to arrange things so that only isolated signals are recorded at the receiving station, and adjustment of the sensitivity threshold of the corresponding instruments makes it possible to receive only a few atmospherics per second. Analysis of the forms of these atmospherics, i.e., of their time functions \(E(t, r)\), has shown that they are of various kinds and vary with distance. Thus, from observations in Moscow \(^{11}\) it follows that in 70–75% of cases atmospherics of a quasiperiodic smooth form are recorded, of the type shown in Fig. 6, \(a\), \(b\), or a chain of such signals—two,

three and more (approximately in 5% of cases, see Fig. 6, v); the rest of the time, atmospherics are of a more complex type, for example, of the type shown in Fig. 6, g.

Let us now assume that the packet of waves constituting the received signal \(E(t,r)\) and arriving from a distance \(r\) is described by the complex spectral density

\[ Q_r=Q(\omega,r)=A(\omega,r)e^{-i\varphi(\omega,r)} =\int_{-\infty}^{\infty} E(t,r)e^{-i\omega t}\,dt. \tag{3} \]

Then the form of this function \(Q(\omega,r)\) naturally depends on the spectral density \(Q_0(\omega,0)\) of the signal emitted by the source

\[ E_0=E(t,0)=\frac{1}{2\pi}\int_{-\infty}^{\infty} Q(\omega,0)e^{i\omega t}\,d\omega = \]

\[ =\int_{-\infty}^{\infty} A_0(\omega)e^{-i[\varphi_0(\omega)+\omega t]}\,d\omega, \tag{4} \]

i.e., on the frequency characteristic \((Jz_D)\) (see formula (1)) and on the function characterizing the law of propagation of waves of different frequency. The latter may be written in the form (see (1))

\[ P(\omega,r)=E(\omega,r)e^{-i\Phi(\omega,r)}, \tag{5} \]

where \(E(\omega,r)\) is the amplitude, and \(\Phi(\omega,r)\) is the total phase of a wave of frequency \(\frac{\omega}{2\pi}\) emanating from the source. One may rewrite \(\Phi(\omega,r)\) in the form

\[ \Phi(\omega,r)=\omega\int_0^r \frac{dr}{v(\omega,r)} =\frac{\omega r}{\bar v(\omega,r)} =\frac{\omega r}{c}+\psi(\omega,r). \tag{6} \]

By definition, \(v\) and \(\bar v\) are, respectively, the differential and mean phase velocities of the wave, while \(\psi(\omega,r)\) is the so-called additional phase, characterizing the deviation of the phase and velocity of the wave from their values in free space, where the wave velocity is equal to \(c\).

Thus, the spectral density of the received signal is equal to*)

\[ Q_r=Q_0P=A_0(\omega)E(\omega,r)e^{-i\left\{\left[\omega\frac{r}{c}+\psi(\omega,r)\right]+\varphi_0(\omega)\right\}}, \tag{7} \]

*) We do not write in the following formula (7) the factor characterizing the spectral function of the receiving device, which, naturally, must be taken into account in processing the measurement results.

and if \(Q_r\) and \(Q_0\) are known, then \(E(\omega,r)e^{-i\Phi(\omega,r)}\) is determined from the equation

\[ E(\omega,r)e^{-i\left[\omega\frac{r}{c}+\psi(\omega,r)\right]} = \frac{A(\omega,r)}{A_0(\omega)} e^{-i[\varphi(\omega,r)-\varphi_0(\omega)]}. \tag{8} \]

It is important to bear in mind that, since

\[ E(t,r)=\frac{1}{2\pi}\int_{-\infty}^{\infty} A(\omega,r)e^{i[\omega t-\varphi(\omega,r)]}\,d\omega = \]

\[ =\frac{1}{2\pi}\int_{-\infty}^{\infty} A_0 E e^{i\left[\omega\left(t-\frac{r}{c}\right)-\psi+\varphi_0\right]}\,d\omega, \tag{9} \]

the form of the received signal does not depend on the linear term \(\dfrac{\omega r}{c}\), which leads only to a uniform displacement of \(E(t,r)\) along the time axis. Physically this means the self-evident fact that the form of the received signal does not depend on the propagation time \(t_0=\dfrac{r}{c}\) of the wave in the absence of dispersion, but only on differences in the times (or, what is the same thing, on differences in the additional phases) of propagation of waves of different frequency to the observation point. Therefore, from Fourier analysis one can compute the value of the phase only to within a term \(\omega\dfrac{r}{c}\), i.e.,

\[ \varphi(\omega,r)=\psi(\omega,r)+\varphi_0(\omega). \tag{10} \]

If now, by means of harmonic analysis of the signal \(E(t,r)\), one determines the dependence of the modulus \(A(\omega,r)\), reduced to some frequency \(\omega_0\), on frequency and (to within \(n\pi\)) the behavior of the argument \(\varphi(\omega,r)\) of the spectral density of the signal, then, provided that \(A_0(\omega)\) and \(\dfrac{\Delta\varphi_0}{\Delta\omega}\) are known, one can obtain directly from the experimental data (see (6) and (8)) the values of the relative field amplitude

\[ \frac{E(\omega,r)}{E(\omega_0,r)} = \frac{A(\omega,r)}{A(\omega_0,r)} \frac{A_0(\omega_0)}{A_0(\omega)} \tag{11} \]

and of the mean phase velocity

\[ \bar{v}= \frac{c}{1+\dfrac{\Delta\varphi-\Delta\varphi_0}{\Delta\omega}\,\dfrac{c}{r}}. \tag{12} \]

From the preceding it is clear that, in principle, harmonic analysis of an atmospheric makes it possible to study, under various conditions, the amplitudes of the electromagnetic waves comprising its spectrum, as well as

measure their mean phase velocity as a function of frequency and distance. For this, however, it is first of all necessary to know the modulus and the derivative of the argument \(\dfrac{\Delta \varphi_0}{\Delta \omega}\) of the spectral density of the source of atmospherics (i.e., the lightning discharge), and the distance \(r\) to it. Thus, the accuracy of such measurements depends essentially on the correctness of the choice of the form of the lightning discharge and on its variability from experiment to experiment.

A number of studies \(^{11,14\text{--}17}\) have shown that in the immediate vicinity of lightning discharges the form of the signals excited by them has predominantly the form similar to that shown in the oscillograms of Fig. 7. Here \(\tau_1\) varies within the limits of several tens of microseconds, and \(\tau_2\) within several milliseconds. Such signals are well approximated by an expression of the form

\[ E_0(t) \simeq e^{-\alpha t} - e^{-\beta t}, \tag{13} \]

where, according to observational data under various conditions,

\[ \begin{aligned} \alpha &\sim (0.5 - 1)\cdot 10^3,\\ \beta &\sim (0.5 - 20)\cdot 10^4. \end{aligned} \tag{14} \]

When the coefficients \(\alpha\) and \(\beta\) vary within the indicated limits, the modulus of the spectral density \(A_0\) of signal (13) changes considerably. The derivative of the argument of the spectral density that is of interest to us, \(\dfrac{\Delta \varphi_0}{\Delta \omega}\), fluctuates only within the limits \((5\text{--}7)\cdot 10^{-5}\). Thus, the majority of lightning discharges possess sufficiently similar properties, and one may choose for processing the measurement results a “standard” source characterized by expression (13) with values \(\alpha = 10^3\) and \(\beta = 10^5\). The amplitude and phase characteristics of such a signal are shown in Fig. 8. With averaging of a sufficiently large number of measurement results, in the processing of which the characteristics of the standard source are used, a high accuracy of the quantities obtained can be achieved; from various experiments it is seen that even single measurements often give quite accurate results.

From the foregoing it is clear that the method described for investigating the field of electromagnetic waves is based on a complete harmonic analysis of an atmospheric \(E(t,r)\). At the same time, while for obtaining its amplitude characteristic \(A(\omega,r)\) it is in practice quite sufficient to know the values of the corresponding quantities at a number of discrete points (for example, at frequency values corresponding to the harmonics of a Fourier series), in order to construct the phase characteristic \(\varphi(\omega,r)\) it is necessary to have phase values at a considerably larger number of points, since in passing from one frequency value to another an unknown number of \(\pi\)’s may be missed, which will lead to

Fig. 7. Photographic oscillograms of nearby atmospherics, \(E(t,0)\).

Fig. 6. Photo-oscillograms of distant atmospherics \(E(t, r)\) of various types.

distortion of the curve \(\varphi(\omega, r)\) and to an incorrect determination of \(\dfrac{\Delta\varphi}{\Delta\omega}\). It is possible, however, to obtain the behavior of \(A(\omega, r)\) and \(\varphi(\omega, r)\) with any desired degree of detail in the following way.

Let the signal \(E(t, r)\) occupy on the oscillogram (within the accuracy of reading the values of \(E(t)\)) a time interval \(T\); then, using

Fig. 8. The modulus \(A_0(\omega)\) and argument \(\varphi_0(\omega)\) of the spectral density of the “standard” nearby atmospheric—the lightning discharge.

one or another method of Fourier analysis, one can determine, for a finite number of fixed frequency values \(\omega_k = \dfrac{2\pi k}{T}\), the Fourier coefficients for the signal

\[ E(t,r)=\sum_{k=0} a_k \cos \omega_k t+\sum_{k=1} b_k \sin \omega_k t \tag{15} \]

and, correspondingly,

\[ A(\omega_k)=\frac{2}{T}\sqrt{a_k^2+b_k^2}. \tag{16} \]

and with accuracy up to \(n\pi\)

\[ \varphi(\omega_k)=\operatorname{arctg}\frac{b_k}{a_k}. \tag{17} \]

It goes without saying that the series \(A(\omega_k)\) and \(\varphi(\omega_k)\) terminates for those indices \(k\) for which \(a_k\) and \(b_k\) lie within the limits of the measurement accuracy.

If we now substitute into the Fourier integral the series (15) with the known values \(a_k\) and \(b_k\), then it is not difficult to obtain formulas determining \(A(\omega)\) and \(\varphi(\omega)\) at any point of the frequency range \(\omega\) of interest to us, and not only at the values \(\omega_k\). Taking into account that the given function \(E(t,r)\) occupies a finite interval \((0—T)\) and is described by a finite number of terms \(n_0\), from the integral

\[ Q(\omega,r)=\int\limits_{0}^{\infty} E(t,r)e^{-i\omega t}\,dt = A(\omega)e^{-i\varphi(\omega)}, \tag{18} \]

we obtain that

\[ \begin{aligned} A^2(\omega)=& \left\{ \sum_{k=0}^{n_0} a_k \int\limits_{0}^{T}\cos \omega_k t \cos \omega t\,dt + \sum_{k=1}^{n_0} b_k \int\limits_{0}^{T}\sin \omega_k t \cos \omega t\,dt \right\}^{2} \\ &+ \left\{ \sum_{k=0}^{n_0} a_k \int\limits_{0}^{T}\cos \omega_k t \sin \omega t\,dt + \sum_{k=1}^{n_0} b_k \int\limits_{0}^{T}\sin \omega_k t \sin \omega t\,dt \right\}^{2}, \end{aligned} \tag{19} \]

\[ \operatorname{tg}\varphi= \frac{ \displaystyle \sum_{k=0}^{n_0} a_k \int\limits_{0}^{T}\cos \omega_k t \sin \omega t\,dt + \sum_{k=1}^{n_0} b_k \int\limits_{0}^{T}\sin \omega_k t \sin \omega t\,dt }{ \displaystyle \sum_{k=0}^{n_0} a_k \int\limits_{0}^{T}\cos \omega_k t \cos \omega t\,dt + \sum_{k=1}^{n_0} b_k \int\limits_{0}^{T}\sin \omega_k t \cos \omega t\,dt }, \tag{20} \]

and

\[ \varphi(\omega)=\operatorname{arctg} \frac{ \displaystyle -\omega(\cos \omega t-1)\sum_{k=0}^{n_0}\frac{a_k}{\omega^{2}-\omega_k^{2}} + \sin \omega t\sum_{k=1}^{n_0}\frac{b_k\omega_k}{\omega^{2}-\omega_k^{2}} }{ \displaystyle \omega\sin \omega t\sum_{k=0}^{n_0}\frac{a_k}{\omega^{2}-\omega_k^{2}} + (\cos \omega t-1)\sum_{k=1}^{n_0}\frac{b_k\omega_k}{\omega^{2}-\omega_k^{2}} }; \tag{21} \]

as \(\omega\to\omega_k\), (18) and (21) naturally pass into (16) and (17).

4. SOME RESULTS OF HARMONIC ANALYSIS OF THE FORMS OF ATMOSPHERICS

In order to ensure undistorted reception of atmospherics, broadband amplifiers are used; the output of the amplifiers is fed to an oscillograph, from which the signals are photographed on motion-picture film. The sweep of the oscillograph is “triggered” by the received signal, which simultaneously blocks the installation, thereby ensuring registration of a single “triggering” atmospheric. At the moment of photographic recording, the received signal also controls the oscillograph of the cathode direction finder, which determines the direction of arrival of the signal at the observation point, and simultaneously, with the aid of an auxiliary radio station, two more cathode direction finders are triggered, situated at two other vertices of a triangle with sides of several hundred kilometers, which likewise register the bearing values of the atmospheric. From the three bearing values the location and distance to the source of the atmospheric are determined.

In the figure: ordinate \( \dfrac{A(\omega,r)}{A(\omega_0,r)} \); abscissa \( \dfrac{\omega}{2\pi}=f \) in kilocycles; curves marked \(r\simeq1000\) km and \(r\simeq2000\) km.

Fig. 9. Averaged relative values of the amplitudes \( \dfrac{A(\omega,r)}{A(\omega_0,r)} \) of atmospherics at distances \(r\simeq1000\) and \(2000\) km.

Some results of harmonic analysis of atmospherics observed in Moscow of the type shown in Fig. 6, \(a, b\), are given in Fig. 9.

The averaged experimental curves for distances of 1000 and 2000 km characterize the variation of the relative amplitude \( \dfrac{A(\omega,r)}{A_0(\omega_0,r)} \)

spectra received by atmospherics. If the ordinates of these curves are divided by the relative values \(\dfrac{A_0'(\omega)}{A_0(\omega_0)}\) (see Fig. 8), then the desired curves of the relative field amplitude \(\dfrac{E(\omega,r)}{E(\omega_0,r)}\) are obtained.

Fig. 10. Comparison of theoretical curves \(\dfrac{E(\omega,r)}{E(\omega_0,r)}\) (dashed line) with experimental ones, obtained from the analysis of atmospherics for distances \(r=1000\) and \(2000\) km.

(see Fig. 10). Comparison with the theoretical values \(E(\omega,r)\) (dashed line in Fig. 10) shows that the experimental and theoretical data are, in general, in good agreement. At the distances under consideration, the relative amplitudes \(\dfrac{E(\omega,r)}{E(\omega_0,r)}\) depend little on \(r\). Since here the field is formed mainly by one wave (number \(n=0\)), the value reduced to equal power \(W\) is

\[ \frac{E(\omega,r)}{E(\omega_0,r)} \sim e^{-\frac{r}{c}\left(S_{02\omega}\cdot\omega-S_{02\omega_0}\cdot\omega_0\right)}, \]

and since \(S_{02\omega}\dfrac{r}{c}\ll 1\), then

\[ \frac{E(\omega,r)}{E(\omega_0,r)}\sim \mathrm{const}. \]

Therefore, analysis of experimental curves of this type is of little use for determining the effective parameters of the ionosphere, especially since it gives only the values of differences of the products \(\omega S_{02}\) at various pairs of fixed frequencies.

5. THE VELOCITY OF ELECTROMAGNETIC WAVES OF AUDIO FREQUENCY

Analysis of the phase characteristics of atmospherics makes it possible, as was shown above, to determine the mean phase velocities of electromagnetic waves and also, apparently, to obtain data on the mean effective parameters of the lower part of the ionosphere.

The values of the mean velocity obtained as a result of processing the results of some experiments by the method described above are given in Table II.

Table II

Values of the mean phase velocity \(\bar v\) in km/sec

It is seen from the table that the velocity \(\bar v\) varies irregularly in the vicinity of the value \(c\). With increasing distance and frequency the amplitude of the oscillations of \(\bar v\) decreases. In general, \(\bar v\) deviates from \(c\) by 1–3%. The average of all the values of \(\bar v\) differs from \(c\) by approximately 0.5%.

6. THE FORM OF ATMOSPHERICS

In conclusion it is appropriate to dwell on one more question. Described above were the method and some results of studies of electromagnetic waves of audio frequency, based on a complete harmonic analysis of the time function \(E(t, r)\)—the form of atmospherics. The results of these experiments, as well as of studies of the propagation of long radio waves, showed that the experimental results agree quite well with the theoretical data. The discrepancy between them is apparently mainly the result of an inexact correspondence between the values of the ionospheric parameters used in particular calculations and their values under the real conditions of the experiment. In this connection it is of interest to check theoretically what the expected forms of atmospherics are for various ionospheric parameters and to what extent the general characteristic features of the function

Figure 11. Signal calculated theoretically (b) for \(r \simeq 1000\) km, and photo-oscillogram of an atmospheric (a), received from a distance \(r \simeq 1200\) km.

\(E(t,r)\) are sensitive to changes in these parameters. The problem is thus reduced to the “synthesis” of the formulas obtained above for calculating the field and to the investigation of the dependence of the function

\[ E(t,r)=\int_{-\infty}^{\infty} E(\omega,r)\,e^{i[\varphi(\omega)+\omega t]}\,d\omega = \]

\[ =\frac{120\pi}{h\sqrt{r}}\int_{-\infty}^{\infty}\frac{J z_{\Pi}}{\sqrt{\lambda}}\sum_{n=0}^{\infty} S_n^{\frac{3}{2}}\,p(c_n)\,e^{-i\left(k_0 S_n r-\frac{\pi}{4}\right)} e^{i\omega t}\,d\omega \tag{22} \]

on \(S_n\left(\omega,\dfrac{N}{\nu},h\right)\) and \(r\). As an illustration we shall give one example of such a calculation of the expected waveform of an atmospheric for the theoretical values used for the effective ionospheric parameters (Table I).

Figure 11, b shows the curve \(E(t,r)\), calculated for the distance \(r=1000\) km by means of formula (22). The same figure gives an oscillogram of one of the atmospherics (Fig. 11, a), received at approximately the same distance from the source. The waveform of this atmospheric is very close, even in its details (signal duration, course of the envelope, values of the periods, number of zero crossings), to the theoretically calculated signal. Similar atmospherics were observed in a number of cases (but not predominantly); this indicates that, under the conditions in which such signals were received, the state of the ionosphere was close to that selected in the calculations.

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