Propagation of Electromagnetic Waves in Plasma (Ionosphere)

B. N. Gershman, V. L. Ginzburg, and N. G. Denisov

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

§ 1. Propagation of electromagnetic waves in a homogeneous magnetoactive plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562

§ 2. Normal waves in a magnetoactive plasma with allowance for the thermal motion of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

§ 3. Normal waves in a magnetoactive plasma with allowance for the motion of ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

§ 4. On one feature of the field of an electromagnetic wave propagating in an inhomogeneous isotropic plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

§ 5. Behavior of the field when the influence of plasma waves is taken into account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

§ 6. Behavior of the field in an inhomogeneous magnetoactive plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

§ 7. Interaction of the ordinary and extraordinary waves in the ionosphere (the effect of “multiplication” of reflected radio signals) . . . . . . . . . . . . . . . . . . . . . . . . . 600

§ 8. Interaction of waves with absorption taken into account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

§ 9. Limiting polarization of waves emerging from the ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

Introduction

The propagation of electromagnetic waves in an ionized gas (plasma) is acquiring ever greater interest in connection with the development of radio astronomy and the study of plasma under laboratory conditions, as well as in connection with further investigations of the terrestrial ionosphere. In this field one has to deal with various cases: with plasma containing and not containing neutral particles; with homogeneous and inhomogeneous plasma; and, finally, with both isotropic and anisotropic (magnetoactive) plasma. Isotropy occurs in the absence of an external magnetic field; if this field is present, the plasma becomes magnetoactive, i.e., a gyrotropic birefringent medium. The character of wave propagation in plasma depends substantially, moreover, on the frequency of these waves. At the same time, if solid bodies are not considered, only the propagation in plasma of waves belonging to the radio range, and in some cases also of low-frequency magnetohydrodynamic and quasiacoustic waves, is of practical interest.

The question of the propagation of radio waves in plasma, especially as applied to the ionosphere, was discussed in considerable detail in Part II of the monograph¹ and earlier in². However, in the several years that have passed since their publication, a number of new results have been obtained, and certain aspects of the problem have become much clearer. Thus, as a result of taking account of the thermal motion of electrons, the question of the character of the normal waves propagating in a homogeneous magnetoactive plasma has been clarified more fully (this, in the first ...

in turn pertains to the so-called plasma wave). The role of thermal motion not only of electrons but also of ions in the case of magnetohydrodynamic waves has also been clarified. Furthermore, progress has been achieved in the study of the “interaction” of waves of different types propagating in an inhomogeneous plasma. What is meant here is an interaction connected not with the nonlinearity of the medium, but with the circumstance that normal waves in an inhomogeneous medium are close to normal waves in a homogeneous medium only in the geometrical-optics approximation. In those cases in which this approximation breaks down, the propagation of waves in an inhomogeneous medium differs essentially from that which occurs in a homogeneous or quasi-homogeneous medium. With an appropriate formulation of the problem, this difference may be associated with the presence of an interaction between waves that are normal in the regions where the geometrical-optics approximation is valid. Such an interaction of waves proves to be very substantial and practically interesting both in an isotropic and in a magnetoactive inhomogeneous plasma. It occurs in several cases (near the point of reflection from an inhomogeneous medium; for oblique incidence of a wave on an isotropic plasma—near the point \(\varepsilon(\omega)=0\), where \(\varepsilon\) is the dielectric permittivity of the plasma; for propagation of waves in a magnetoactive plasma—in the region of a small electron concentration and, under known conditions, at small angles between the direction of wave propagation and the direction of the external magnetic field).

Consideration of all the enumerated questions of the theory of wave propagation in plasma is the purpose of the present article. It is necessary, at the same time, to emphasize that in the field of the theory of wave propagation in plasma and, in particular, in the ionosphere, there are also a number of other important problems that are the subject of numerous investigations. These include nonlinear phenomena \(^{3-6}\), the propagation of long and very long radio waves \(^{7-10}\), and especially statistical phenomena connected with the diffraction of waves by plasma (ionospheric) inhomogeneities \(^{11-15}\). In addition, the study of various properties and of the dynamics of plasma \(^{16-19}\), of the physics of the ionosphere \(^{13,20}\) and of the solar corona \(^{21}\), as well as of elementary processes in an ionized gas \(^{22}\), is being carried out on a broad front, i.e., questions that, to one degree or another, are connected with the investigation of wave propagation in plasma. Recently, the consideration of the properties of a plasma situated in a magnetic field has also become of interest from the point of view of the study of semiconductors \(^{22a}\). Ideas about an electron plasma in metals also prove, under known conditions, to be quite admissible and useful \(^{52,52a}\). Finally, the study of the properties and behavior of plasma in a magnetic field is fundamental from the point of view of attempts to solve so major a problem as the industrial use of thermonuclear energy \(^{22b}\).

On all these questions, however, we shall not dwell at all, and in this respect shall confine ourselves only to the indications already made concerning the relevant recent literature (the references to the literature given are intended only for orientation and in no way claim completeness).

§ 1. PROPAGATION OF ELECTROMAGNETIC WAVES IN A HOMOGENEOUS MAGNETOACTIVE PLASMA

The question of the propagation of radio waves in a homogeneous magnetoactive plasma, neglecting the thermal motion of electrons and without taking into account the influence of ions, has been considered in detail in \(^{1}\), §§ 62, 75. Therefore here we shall dwell on this question very briefly, keeping in mind only the aims of the further exposition.

A plasma situated in a magnetic field is a doubly refracting medium and, under the restrictions indicated above, can be characterized

tensor of the complex dielectric permittivity \(\varepsilon'_{ik}\) (see\(^1\) § 62). The quantities \(\varepsilon'_{ik}\) enter into the field equations, which, in the case of plane waves of interest to us,

\[ \mathbf{E}=\mathbf{E}_0 e^{i(\omega t-\mathbf{k}\mathbf{r})}, \]

determine the relation of the frequency \(\omega\) to the wave vector \(\mathbf{k}\), as well as the relation between the different components of the amplitude \(\mathbf{E}_0\), i.e., the polarization of the waves. In this case, in a given direction \(\mathbf{k}\), for a given frequency \(\omega\) and, of course, given plasma parameters and external magnetic-field strength, two normal waves can propagate. These waves differ from one another in propagation velocity, absorption coefficient, and polarization. (If the thermal motion of the electrons and the influence of the ions in the plasma are taken into account, not two but four normal waves may propagate, as will be discussed below.)

Without loss of generality one may choose the direction of the wave vector \(\mathbf{k}\) to be the \(z\)-axis, and consider the external constant magnetic field \(\mathbf{H}_0\) to lie in the \(yz\)-plane (see Fig. 1). In this case the electric field of plane waves has the form \(\mathbf{E}=\mathbf{E}_0 e^{i(\omega t-kz)}\), and the equations

Fig. 1.

must be satisfied:

\[ \bigl[A-(n-i\chi)^2\bigr]E_{x0}+iCE_{y0}=0, \tag{1.1} \]

\[ -iCE_{x0}+\bigl[B-(n-i\chi)^2\bigr]E_{y0}=0, \tag{1.2} \]

where

\[ A=\frac{(1-is)u-(1-is)(1-is-v)^2-uv\cos^2\alpha} {(1-is)u-(1-is)^2(1-is-v)-uv\cos^2\alpha}, \tag{1.3} \]

\[ B=\frac{u(1-is-v)-(1-is)(1-is-v)^2} {(1-is)u-(1-is)^2(1-is-v)-uv\cos^2\alpha}, \tag{1.4} \]

\[ C=\frac{\sqrt{u}\,v(1-is-v)\cos\alpha} {(1-is)u-(1-is)^2(1-is-v)-uv\cos^2\alpha}. \tag{1.5} \]

In equations (1.1) and (1.2) the notation

\[ \frac{c^2k^2}{\omega^2}=(n-i\chi)^2 \tag{1.6} \]

has been introduced, where \(n\) and \(\chi\) are the refractive and absorption indices of the wave (the meaning of these quantities is clear from the very expression for the wave field)

\[ \mathbf{E} = \mathbf{E}_0 e^{i(\omega t-kz)} = \mathbf{E}_0 e^{i\left(\omega t-\frac{\omega}{c}nz\right)} e^{-\frac{\omega}{c}\chi z}. \]

Further, in relations (1.3)—(1.5),

\[ u=\frac{\omega_H^2}{\omega^2}, \qquad \omega_H=\frac{|e|H_0}{mc} \]

is the gyrofrequency for electrons (\(e\) and \(m\) are the charge and mass of the electron),

\[ v=\frac{\omega_0^2}{\omega^2}, \qquad \omega_0=\sqrt{\frac{4\pi e^2 N_e}{m}} \]

is the Langmuir frequency of plasma oscillations (\(N_e\) is the electron concentration),

\[ s=\frac{\nu_{\mathrm{eff}}}{\omega}, \]

\(\nu_{\mathrm{eff}}\) is the effective number of collisions of electrons with other particles.

As for the longitudinal component of the electric field, it can be found from the relation

\[ E_z=\frac{i\sqrt{u}\sin\alpha\,[1-(n-i\chi)^2]}{1-v-is}\,E_y. \tag{1.7} \]

For \(\alpha\ne0\), it follows from (1.7) that in the presence of an external magnetic field (i.e., when \(u\ne0\)) the longitudinal component is absent (when there is a transverse component \(E_y\)) only in the trivial case when there are no electrons in the medium \((n^2=1,\ \chi=0)\). If, however, the angle between the wave vector \(\mathbf{k}\) and the field \(\mathbf{H}\) is zero (i.e., \(\alpha=0\)), then for any values of \(u\) and \(v\) for which \(1-v-is\ne0\), the waves turn out to be purely transverse.

The equation for the quantity \((n-i\chi)^2\) is obtained from the condition for the existence of nontrivial solutions of the system (1.1)—(1.2), i.e., from the condition that the determinant of this system be equal to zero:

\[ \begin{gathered} [(1-is)u-(1-is)^2(1-is-v)-uv\cos^2\alpha](n-i\chi)^4-\\ -[2u(1-is)-2(1-is)(1-is-v)^2-uv(1+\cos^2\alpha)](n-i\chi)^2+\\ +(1-is-v)[u-(1-is-v)^2]=0. \end{gathered} \tag{1.8} \]

The solution of this equation can be written in the form

\[ (n_{1,2}-i\chi_{1,2})^2= \]

\[ =1-\frac{2v(1-v-is)} {2(1-is)(1-v-is)-u\sin^2\alpha\pm\sqrt{u^2\sin^4\alpha+4u(1-v-is)^2\cos^2\alpha}}, \tag{1.9} \]

where the index “1” and the sign “\(-\)” refer to the extraordinary wave, while the index “2” and the sign “\(+\)” refer to the ordinary wave.

Let us consider a number of consequences of (1.8)—(1.9) that are of interest for the subsequent exposition. First of all, let us note that in principle a case is possible when the indices of refraction and absorption are the same for both types of waves \(((n_1-i\chi_1)^2=(n_2-i\chi_2)^2)\), and, consequently, double refraction is absent. This occurs, as is easily seen from (1.9), if \(u\sin^4\alpha=-4(1-v-is)^2\cos^2\alpha\). For \(v=1\), from this we find a certain value of \(s\), which we denote by \(s_c\):

\[ s_c=\frac{\nu_{\mathrm{eff},c}}{\omega}=\frac{\sqrt{u}\sin^2\alpha}{2\cos\alpha}. \tag{1.10} \]

The quantity \(\nu_{\mathrm{eff},c}\) is called the “critical collision number.” Under ionospheric conditions, for short waves usually \(s\ll\sqrt{u}\), and the condition \(s=s_c\) can be fulfilled only for sufficiently small values of the angle \(\alpha\) (quasi-longitudinal propagation). Taking into account that in the case of waves shorter than approximately \(100\) m, \(s\ll1\), below in this paragraph, for simplicity, we shall neglect the influence of collisions on the character of wave propagation, assuming that \(s=0\). When \(s=\dfrac{\nu_{\mathrm{eff}}}{\omega}=0\), absorption of waves is absent; however, even in this case attenuation of the field may occur. Thus, in the isotropic case for \(s=0\) the complex dielectric constant (permittivity)

\[ \varepsilon'=\varepsilon=(n-i\chi)^2=1-\frac{4\pi e^2N_e}{m\omega}=1-v \]

and the field varies according to the law

\[ \mathbf{E}=\mathbf{E}_0 e^{\,i\left(\omega t-\frac{\omega}{c}\sqrt{\varepsilon}\,z\right)} =\mathbf{E}_0 e^{\,i\left(\omega t-\frac{\omega}{c}nz\right)-\frac{\omega}{c}\chi z}. \]

If \(\varepsilon>0\), then \(n=\sqrt{\varepsilon}\), \(\varkappa=0\), and we are dealing with a traveling wave. If, however, \(\varepsilon<0\), then \(n=0\), \(\varkappa=\sqrt{|\varepsilon|}\), and the wave is attenuated. Since absorption is absent in this case, the attenuation of the wave is associated with its complete internal reflection from the medium. Thus, for \(s=0\) it would be most consistent to introduce both quantities \(n\) and \(\varkappa\). It is more convenient, however, in this case to use a single quantity \(n^2\), which can also take negative values; moreover, it is obvious that values \(n^2<0\) correspond, in the former notation, to the case when \(\varkappa\ne0\), and \(\varkappa^2\) is equal to \(|n^2|\). An entirely analogous situation also occurs in a magnetoactive medium, and therefore for \(s=0\) we shall use the single quantity \(n^2\), which can have either sign. Let us also note that, in addition to waves of the type \(\mathbf E=\mathbf E_0 e^{i(\omega t-kz)}\), there exist, of course, waves of the type \(\mathbf E=\mathbf E_0 e^{i(\omega t+kz)}\), corresponding to the opposite direction of propagation. This circumstance may nevertheless not be taken explicitly into account, as was done above, by assuming that the values of \(k\) may have either sign. The same applies to the values of \(n\) and \(\varkappa\), since only expressions for the quantity \((n-i\varkappa)^2\) are used.

Considering equation (1.8) (for \(s=0\)), it is easy to conclude that if the coefficient of \(n^4\) vanishes, then one of the roots \(n^2\to\infty\). Thus, for finite \(v\), \(n^2\to\infty\) under the condition

\[ 1-u-v+uv\cos^2\alpha=0. \tag{1.11} \]

The value \(v_\infty\) for which condition (1.11) is fulfilled is equal to

\[ v_\infty=\frac{u-1}{u\cos^2\alpha-1}. \]

Using the expression obtained for \(v_\infty\), it is easy to see that for \(u<1\) it is always possible for one of the roots \(n_1^2\) or \(n_2^2\) to become infinite. A more detailed analysis shows that in this case \(n_1^2\to\infty\). If, however, \(u>1\), then what is essential is the value of the quantity \(u_L=u\cos^2\alpha\). For \(u_L<1\) it formally turns out that \(v_\infty<0\). However, the quantity \(v=\dfrac{\omega_0^2}{\omega^2}\), by its very meaning, must be positive; whence it follows that for \(u>1\), \(u_L<1\), and finite \(v\), the values \(n_1^2\) and \(n_2^2\) are finite. If, however, \(u_L>1\) and at the same time \(u>1\), then one can establish that at \(v=v_\infty\) the value \(n_2^2\to\infty\), while the quantity \(n_1^2\) remains finite. Thus, the presence of a pole in the function \(n^2(v)\) may be characteristic in one case of the branch \(n_1^2(v)\), and in another case of the branch \(n_2^2(v)\). There is also a possible case in which the function \(n^2(v)\) has no pole (as stated, the latter occurs if \(u>1\) and \(u_L<1\)).

From (1.8) or (1.9) it is easy to establish that for \(\alpha\ne0\), the quantity \(n_1^2=0\) if \(v=1\pm\sqrt{u}\), and \(n_2^2=0\) if \(v=1\). These well-known conditions determine the reflection levels of the extraordinary and ordinary waves propagating in an inhomogeneous plasma. If \(u>1\), then the condition \(v=1-\sqrt{u}\), owing to the positivity of the quantity \(v\), becomes fictitious.

Let us now dwell on the case of longitudinal propagation, when \(\alpha=0\). This case is exceptional, in particular, for the following reason. Putting \(\alpha=0\) in (1.9), we obtain

\[ n_{1,2}^2=1-\frac{v}{1\pm\sqrt{u}}, \tag{1.12} \]

whence it is seen that \(n_1^2=0\) for \(v=1-\sqrt{u}\), while \(n_2^2=0\) for \(v=1+\sqrt{u}\), whereas for \(\alpha\ne0\), \(n_1^2=0\) if \(v=1\pm\sqrt{u}\), and \(n_2^2=0\) for \(v=1\). It follows from this that the transition to the case \(\alpha=0\) has a special character, which is easiest to understand by comparing the curves \(n^2(v)\) for \(\alpha=0\) and for small \(\alpha\) (Fig. 2). The case of small \(\alpha\) is very interesting for the theory of wave propagation in an inhomogeneous magnetoactive plasma; its consideration will constitute the content of §§ 7 and 8.

Fig. 2.

Let us dwell on one more point connected with the polarization of normal waves in a homogeneous plasma. From relations (1.1) one can find the polarization factor \(K_{1,2}\).

\[ K_{1,2}=\frac{E_{y\,1,2}}{E_{x\,1,2}} = -\frac{2i\sqrt{u}(1-v)\cos\alpha} {u\sin^2\alpha \mp \sqrt{u^2\sin^4\alpha+4u(1-v)^2\cos^2\alpha}}, \tag{1.13} \]

where the upper sign corresponds to wave 2.

The last equality is obtained if one uses relations (1.3), (1.5), and (1.9). In the general case the polarization (in the transverse plane \(xy\)) has an elliptical character. An interesting fact here is that both waves remain elliptically polarized also in the limit as \(v\to0\), i.e. in vacuum:

\[ K_{1,2\,(v=0)} = -\frac{2i\sqrt{u}\cos\alpha} {u\sin^2\alpha \mp \sqrt{u^2\sin^4\alpha+4u\cos^2\alpha}}. \tag{1.14} \]

Here, of course, there is no contradiction, since in empty space waves of elliptical polarization may also be chosen as normal waves and, in particular, waves with polarization determined by relation (1.14). However, in the case of an inhomogeneous medium there arises the need for an additional analysis of the question, since the character of the change of polarization at small \(v\), which corresponds to propagation at the beginning of the ionospheric layer, will not obey the dependence determined by relation (1.13). This question will be considered in § 9.

In the absence of an external magnetic field the parameter \(u=\dfrac{\omega_H^2}{\omega^2}=0\), and, as is well known and is clear, for example, from (1.12),

\[ n^2=\varepsilon=1-v=1-\frac{4\pi e^2 N_e}{m\omega^2}. \tag{1.15} \]

In the presence of absorption

\[ (n-i\chi)^2=\varepsilon'=1-\frac{v}{1-is} =1-\frac{4\pi e^2 N_e}{m\omega^2\left(1-i\frac{\nu_{\mathrm{eff}}}{\omega}\right)}. \tag{1.16} \]

The indices 1, 2 are omitted here, since, with respect to polarization in an isotropic medium, degeneracy occurs (transverse waves of any polarization propagate with the same velocities and damping).

In addition to the two transverse waves with values of \(n\) and \(\chi\) determined by expression (1.16), plasma oscillations may exist in an isotropic plasma, i.e., oscillations of electrons relative to ions (these oscillations are analogous to the so-called optical or Born oscillations in a crystal lattice). The frequency of plasma oscillations, as follows directly from the field equations (see \(^{1}\) § 56), is determined from the condition that the quantity \(\varepsilon'\) be equal to zero. In the presence of collisions this frequency is complex, i.e., the oscillations are damped. Neglecting collisions, the frequency of plasma oscillations \(\omega_0\) is determined from the condition \(\varepsilon(\omega_0)=0\) and, as is clear from (1.15), is equal to

\[ \omega_0=\sqrt{\frac{4\pi e^2 N_e}{m}}=5.64\cdot 10^4\sqrt{N_e}. \tag{1.17} \]

When thermal motion is neglected, plasma oscillations that arise at one place do not depend on oscillations at another place. In other words, if one considers a longitudinal plasma wave \(E_z=E_{z0}e^{i(\omega_0 t-kz)}\), then for the given \(\omega_0\) the value of \(k\) is arbitrary. Hence it follows that the group velocity of plasma waves \(d\omega/dk\) in the approximation under consideration is zero, which gives grounds for speaking rather of plasma oscillations than of waves. Therefore, in studying plasma waves it is necessary to take into account the thermal motion of particles in the plasma, which will be done in § 2.

In conclusion, let us note that the components of the tensor \(\varepsilon'_{ik}\), and consequently also the coefficients in equations (1.1)—(1.2), were above assumed to be independent of the field strength. Only under this condition are the field equations linear, and, by virtue of this, the principle of superposition of waves is satisfied. Nonlinear phenomena in the propagation of radio waves in plasma are also of interest and have served as the object of a number of investigations, as was already mentioned in the introduction. However, in the present article we shall restrict ourselves only to the linear approximation.

§ 2. NORMAL WAVES IN A MAGNETOACTIVE PLASMA WITH ALLOWANCE FOR THE THERMAL MOTION OF ELECTRONS

Neglect of thermal motion in a plasma is in many cases entirely justified. Thus, in the propagation of transverse waves in an isotropic plasma, allowance for thermal motion leads in the nonrelativistic case only to negligible corrections of the order of the ratio of the thermal velocity of the electrons

\[ \sqrt{\frac{\varkappa T_e}{m}} \]

to the phase velocity of the waves \(\dfrac{c}{n}\geq c\) (in this case \(n\leq 1\); see (1.15)). However, in the case of plasma waves, as was already noted at the end of § 1, one cannot dispense with consideration of thermal motion even in the first approximation (when finding the dependence of wavelength on frequency). The question of plasma waves in the isotropic case was considered long ago (see \(^{23-25}\), and also \(^{1}\) § 63). Therefore we shall not dwell here on isotropic plasma;

the corresponding dispersion equation relating \(\omega\) and \(k\) will be given incidentally somewhat below.

In the presence of an external magnetic field \(\mathbf H_0\), the separation of the normal waves into purely transverse and purely longitudinal ones is, generally speaking, impossible. However, in this case too one can immediately point to a definite, although in practice rather narrow, region in which one may expect a substantial influence of the thermal motion of the electrons. The point is that, when considering the properties of waves in the absence of thermal motion (see § 1), it was established that, for a definite value \(v=v_\infty\), for one of the waves \(n^2\to\infty\). In the region \(v=v_\infty\) the phase velocity can be comparatively small, and, consequently, one may expect that, in determining the character of the normal waves, thermal corrections will be substantial, these being characterized by the ratio of the velocity of thermal motion to the phase velocity of the wave. Moreover, from what follows it will be clear that taking account of the thermal motion of the electrons, as in the isotropic case, leads to the conclusion that instead of two normal waves there are three. The third wave arising here, alongside the ordinary and extraordinary waves, may somewhat conventionally be called the plasma wave. Several works are devoted to the study of the properties of this wave, which becomes purely longitudinal in the absence of a magnetic field; they will be discussed below.

The principal interest in plasma waves is connected with problems of gas-discharge physics, where plasma oscillations were first discovered\(^{26}\), and also with radio astronomy (see, for example, \(^{27}\)). It may be supposed—as is already indicated by the content of § 5 of the present article—that the question of plasma waves in the presence of an external magnetic field is not without interest as applied to ionospheric problems as well.

The influence of thermal motion can be taken into account most consistently by using the method of the kinetic equation. However, a sufficiently general kinetic treatment, on whose results we shall dwell below, is rather cumbersome. Therefore attempts at an approximate, but simpler, solution of the question deserve attention. A method of this kind is the method based on using equations for the mean velocities of particles, which may be called quasihydrodynamic. The question of normal waves in a magnetoactive plasma was considered by this method in works \(^{28-30}\).

In doing so one may start from the wave equation for the electric field \(\mathbf E\) (see, for example, \(^{1}\) § 63):

\[ \Delta \mathbf E-\operatorname{grad}\operatorname{div}\mathbf E-\frac{1}{c^2}\frac{\partial^2\mathbf E}{\partial t^2} =\frac{4\pi}{c^2}\frac{\partial \mathbf j_t}{\partial t}, \tag{2.1} \]

where \(\mathbf j_t\) is the total microscopic current, which, when the motion of the ions is neglected, is equal to

\[ \mathbf j_t=eN_e\mathbf v_e \tag{2.2} \]

(\(\mathbf v_e\) is the velocity of the ordered motion of the electrons). Next we use the equation of motion for the electrons and the law of conservation of particle number:

\[ m\frac{d\mathbf v_e}{dt} =e\left(\mathbf E+\frac{1}{c}[\mathbf v_e\mathbf H_0]\right) -\frac{\operatorname{grad}p_e}{N_e} -m\nu_{\mathrm{eff}}\mathbf v_e, \tag{2.3} \]

\[ \frac{\partial N_e}{\partial t}+\operatorname{div}\mathbf v_e N_e=0. \tag{2.4} \]

In equation (2.3), \(p_e\) is the partial pressure of the electrons, which can be related to the temperature of the electron gas \(T_e\); for an ideal gas \(p_e=N_e\varkappa T_e\)

(\(\chi\) is Boltzmann’s constant). Equation (2.4) was written taking into account the effect of collisions; however, in this section, for simplicity, the latter will not be taken into account. If the pressure forces are neglected (\(\nabla p_e=0\)), then from (2.1)—(2.4) one can easily arrive at the results presented in § 1. Introducing the partial pressure \(p_e\) into equation (2.3) is the simplest method of allowing for thermal motion. If the temperature \(T_e\) is regarded as constant and small deviations from the equilibrium values of the electron concentration \(N_{e0}\) are considered, putting \(N_e=N_{e0}+N_{e1}\), \(N_{e1}\ll N_{e0}\) (the value \(N_{e0}\) for a homogeneous medium does not depend on the coordinates), then, seeking the solution in the form of plane waves

\[ \mathbf{E}=\mathbf{E}_0 e^{i\left(\omega t-\frac{\omega}{c}nz\right)}, \]

one can obtain the following dispersion equation:

\[ \begin{aligned} &\beta^2(1-\mu\cos^2\alpha)n^6 -\left[1-\mu-v+\mu v\cos^2\alpha +2\beta^2(1-v-\mu\cos^2\alpha)\right]n^4 \\ &\quad+\left[2(1-v)^2-\mu(2-v-v\cos^2\alpha) +\beta^2(1-2v+v^2-\mu\cos^2\alpha)\right]n^2 \\ &\quad+(1-v)\left[\mu-(1-v)^2\right]=0. \end{aligned} \tag{2.5} \]

Here \(\beta=\sqrt{\dfrac{\chi T_e}{mc^2}}\) is the ratio of a certain mean thermal velocity of the electrons to the velocity of light in vacuum \(c\). In the nonrelativistic plasma, which alone will interest us here, the condition \(\beta^2\ll1\) is satisfied. In the absence of thermal motion (i.e. for \(\beta\to0\)) equation (2.5) passes into equation (1.8) (if in the latter one puts \(s=0\)). Equation (2.5) is an equation of third degree with respect to \(n^2\), which means that there are three types of normal waves: the extraordinary (\(n=n_1\)), the ordinary (\(n=n_2\)), and the plasma wave (\(n=n_3\)). For the last wave it is characteristic that \(n_3^2\to\infty\) as \(\beta\to0\). The separation of waves into the indicated three groups is most conveniently carried out by considering the behavior of the functions \(n^2(v)\) determined from equation (2.5). Let us note that studying the character of the curves \(n^2(v)\) is of great importance for the theory of wave propagation also in an inhomogeneous medium (see § 6).

Let us consider some features of the curves \(n^2(v)\), using, for simplicity, formulas for particular cases. For longitudinal propagation (\(\alpha=0\)), from (2.5) we obtain:

\[ \left. \begin{aligned} \frac{c^2k_{1,2}^2}{\omega^2}\equiv n_{1,2}^2 &=1-\frac{v}{1\pm\sqrt{\mu}} =1-\frac{4\pi e^2N_e}{m\omega(\omega\pm\omega_H)},\\[6pt] \frac{c^2k_3^2}{\omega^2}\equiv n_3^2 &=\frac{1-v}{\beta^2} =\frac{mc^2}{\chi T_e}\left(1-\frac{4\pi e^2N_e}{m\omega^2}\right). \end{aligned} \right\} \tag{2.6} \]

The expressions for \(n_{1,2}^2\) coincide with (1.12). In this case the thermal motion, within the framework of the method of calculation used, has absolutely no effect on the character of wave propagation. For the plasma wave one obtains the dispersion equation (i.e. the relation connecting \(\omega\) with \(k\)) corresponding to a purely longitudinal wave in the absence of the magnetic field \(H_0\) (this dispersion equation is often also written in the form \(\omega^2=\omega_0^2+\dfrac{\chi T_e}{m}k^2=\dfrac{4\pi e^2N_e}{m}+\dfrac{\chi T_e}{m}k^2\)). Thus, for \(\alpha=0\), each of the types of normal waves corresponds to its own dispersion curve \(n^2_{1,2,3}(v)\). However, the case \(\alpha=0\) is exceptional, and for \(\alpha\ne0\) it already becomes impossible to associate with each of all three types of normal waves its own dispersion curve. This is easiest to illustrate by considering transverse propagation \(\left(\alpha=\dfrac{\pi}{2}\right)\). In this case, for the ordinary wave \(n_2^2=1-v\),

and for the extraordinary and plasma waves we have the equation

\[ \beta^2 n^4+\left[(v-1)(1+\beta^2)+u\right]n^2+\left[(v-1)^2-u\right]=0. \tag{2.7} \]

Figure 3 gives typical curves \(n_{1,3}^{2}(v)\) in the region where the values of \(v\) are close to \(v_\infty\). These curves are constructed on the basis of equation (2.7). Let us note that in the lower half-plane, for \(n^2<0\), these curves are of no particular interest, since here propagation becomes impossible, because the field is exponentially damped and usually cannot penetrate into the plasma to any considerable depth (in an inhomogeneous medium the values \(n^2<0\) correspond to regions lying beyond the point of reflection). It is clear from Fig. 3 that the values \(n_3^2(v)\) are not represented by a separate curve, but, in the case when \(u<1\), constitute, as it were, a continuation of the curve \(n_1^2(v)\). In this figure that part of the curve \(n^2(v)\) (for \(n^2>0\)) where \(v<1-u\) is assigned to the plasma wave, while that part of it where \(v>1-u\) is assigned to the extraordinary wave. Such a division is based on the fact that, in the absence of thermal motion (as \(\beta\to0\)), the plasma wave must disappear and the dispersion curve (for \(n^2>0\)) is situated in the upper half-plane to the right of the vertical \(v=1-u\) (in the case under consideration, when \(\alpha=\pi/2\), the value \(v=1-u=v_\infty\), i.e., it corresponds to the point where, for \(\beta=0\), \(n_1^2=\infty\)). It should, however, be stressed that the stated classification of waves is to a certain extent conventional, which is especially characteristic of the very point \(v=1-u\), where there is no basis for assigning the values of \(n^2\) either to the extraordinary wave or to the plasma wave*). In the case under discussion, and also for other values \(\alpha\ne0\), the plasma wave, as an analysis of the character of the polarization shows, is not purely longitudinal. The cited features of the curves \(n^2(v)\) are also characteristic not only of the case \(\alpha=\dfrac{\pi}{2}\), but also of other values \(\alpha\ne0\). In Fig. 4, in schematic form,\(^{27,29}\) the place of the plasma wave (in the case \(u<1\)) at small angles \(\alpha\) is shown (for the corresponding curves without allowance for thermal motion, see Fig. 2). We emphasize that for the prop—

Fig. 3.

\[ \begin{gathered} u=0.1,\\ \beta^2=10^{-5},\\ \alpha=\frac{\pi}{2}. \end{gathered} \]

*) Thus, it appears very significant that in a magnetoactive medium plasma waves do not form an isolated branch of waves, but closely adjoin the extraordinary or ordinary waves. This circumstance had not previously been clarified, which led, for example, in § 75 of book \(^{1}\) to an insufficiently clear and correct treatment of the question of plasma waves in a magnetoactive medium.

case considered, when \(u \ll 1\), for \(\beta^2 \ll 1\) all three roots \(n^2_{1,2,3}\) of equation (2.5) are always real.

If, as \(\beta \to 0\), the pole is possessed by the ordinary wave, i.e. \(n_2^2(v_\infty)\to\infty\) (this corresponds to the region \(u>1,\ u_L>1\); see § 1), then the plasma wave is the continuation of the branch \(n_2^2(v)\).

It is significant that in the quasi-hydrodynamic approach and when collisions are neglected (i.e. when \(s=0\)) the plasma wave does not damp even if the phase velocity of the wave is comparable with the velocity of the thermal motion of the electrons. Meanwhile the kinetic treatment leads to the conclusion that in precisely this case there is a particularly substantial kind of damping, associated with the transition of the energy of the ordered motion of the wave into the energy of the thermal motion of the plasma\(^{24}\). Below it will be possible to point out a number of other discrepancies between the results of the quasi-hydrodynamic and kinetic solutions of the problem, in consequence of which the quasi-hydrodynamic treatment, although very useful for orientation, can in no way be regarded as exhaustive.

Fig. 4.

Let us therefore turn to a discussion of the results obtained in investigating the properties of normal waves in a plasma by the method of the kinetic equation. Here the question of a dispersion equation analogous to equation (2.5) is of greatest interest. In the isotropic case this question was considered in works \(^{23-25}\). As for a plasma situated in a magnetic field, here too one can point to a number of investigations \(^{28,31-36}\) based on use of the kinetic equation. However, some of these works contain a number of substantial restrictions. Thus, in works \(^{31-32}\), in considering the question of plasma oscillations, the vortex terms are neglected, i.e. it is immediately assumed that \(\operatorname{rot}\mathbf E=0\). In consequence of this, consideration of the extraordinary and ordinary waves is altogether excluded; as for the plasma wave, for it too, generally speaking, neglect of the vortex

is not legitimate. The kinetic consideration, carried out in work \(^{33}\), of the so-called transverse oscillations in a magnetoactive plasma is, in general, incorrect. In that work the condition of transversality \(\operatorname{div}\mathbf{E}=0\) is imposed on the electric field \(\mathbf{E}\), which in the general case is completely inadmissible, since normal waves have a longitudinal component of the field \(\mathbf{E}\) both when thermal motion is taken into account and in its absence. In the recently published work \(^{37}\) no new results are obtained for high-frequency waves, while in article \(^{38}\), because thermal motion is neglected, only the well-known formulas for the ordinary and extraordinary waves propagating perpendicular to the magnetic field are given.

The solution of the problem of waves in a plasma on the basis of the kinetic equation and the general system of field equations was carried out in \(^{34-35}\). In \(^{34}\), only the properties of the plasma wave for transverse propagation \(\left(\alpha=\dfrac{\pi}{2}\right)\) are considered in detail. In \(^{35}\) (see also \(^{28,36}\)) the properties of normal waves of all three types are discussed.

Starting from the kinetic equation and the electrodynamic equations with a self-consistent field, one can arrive at the following dispersion equation, analogous in form to (2.5):

\[ \beta^{2}v\left[A\sin^{2}\alpha+B\cos\alpha\sin\alpha+(1-u)C\cos^{2}\alpha\right]n^{6}- \]

\[ -\left[1-u-v+uv\cos^{2}\alpha+O_{1}(\beta^{2})\right]n^{4}+ \]

\[ +\left[2(1-v)^{2}+uv\cos^{2}\alpha-u(2-v)+O_{2}(\beta^{2})\right]n^{2}+ \]

\[ +(1-v)\left[u-(1-v)^{2}\right]=0, \tag{2.8} \]

where

\[ A=\frac{(1+3u)\cos^{2}\alpha}{(1-u)^{2}}+\frac{3\sin^{2}\alpha}{1-4u},\qquad B=\frac{4\sin\alpha\cos\alpha}{1-u},\qquad C=3\cos^{2}\alpha+\frac{\sin^{2}\alpha}{1-u}. \]

In solving the problem it is assumed that the electron distribution function \(F\) can be represented in the form \(F=f_{0}+f_{1}\), where \(|f_{1}|\ll f_{0}\) and

\[ f_{0}=N_{e0}\left(\frac{m}{2\pi\varkappa T_{e}}\right)^{3/2} \exp\left(-\frac{mv_{e}^{2}}{2\varkappa T_{e}}\right) \]

is the Maxwellian distribution. The quantities \(O_{1}(\beta^{2})\) and \(O_{2}(\beta^{2})\) in (2.8) are of order \(\beta^{2}\) and are inessential in the analysis of normal waves.

Equation (2.8) is written with the damping due to the thermal motion of the particles neglected. The calculation of this damping may be carried out, for example, when the wave number \(k\) is regarded as given and real and solutions varying in time according to the law \(e^{pt}\) are sought, where the quantity \(p=i\omega-\gamma\) is complex, with \(\omega\) determining the frequency and \(\gamma\) the damping decrement of the waves. An important problem is to establish the regions of weak and strong damping of waves. For weak damping, i.e., when \(\gamma\ll\omega\), one may in the first approximation restrict oneself to consideration of undamped waves. In the opposite case, for strong damping, when \(\gamma\gg\omega\), the very propagation of waves becomes practically impossible, since even in the absence of collisions the energy of the ordered motion is intensively transferred into the thermal energy of the plasma.

In work \(^{35}\) it is shown that any of the waves 1, 2, or 3 with a given value of \(n^{2}\) is weakly damped when the following two conditions are simultaneously satisfied:

\[ \beta^{2}n^{2}\cos^{2}\alpha\ll 1, \tag{2.9} \]

\[ \frac{\beta^{2}n^{2}}{u}\sin^{2}\alpha\ll 1. \tag{2.10} \]

The condition (2.9) is satisfied for all angles \(\alpha\) if the phase velocity of the wave \(v_\phi=\dfrac{c}{n}\) is considerably greater than the mean thermal velocity of the electrons \(\overline{v_e}\). Indeed, in the latter case
\[ \beta^2 n^2=\frac{\varkappa T_e n^2}{mc^2}\simeq \frac{\overline{v_e}^{\,2}}{v_\phi^2}\ll 1. \]
The same applies to the inequality (2.10), but only in the presence of appreciable anisotropy \(\left(\text{for example, when }u=\dfrac{\omega_H^2}{\omega^2}>1\right)\). Violation of condition (2.9) always means the presence of strong damping \((\gamma\gtrsim \omega)\). If, however, only the inequality (2.10) is not satisfied, then this also leads to appreciable damping, but only for a sufficiently strongly expressed anisotropy of the medium \((u\gg 1)\).

In papers \(^{31-32,35}\) a number of expressions are given for the damping decrement \(\gamma\), but all these expressions, for one reason or another, have a limited character. As an example, let us give the formula for \(\gamma\) obtained under the fulfillment of conditions (2.9)—(2.10) and, in addition, under the assumption that \(n^2\gg 1\) (see \(^{35}\)):

\[ \gamma= \frac{\omega}{\cos^2\alpha+\dfrac{\sin^2\alpha}{(1-u)^2}} \left\{ \frac{\sin^2\alpha\cdot\omega}{2k\cos^2\alpha} \sqrt{\frac{\pi m}{8\varkappa T_e}} \left[ \exp\left(-\frac{m(\omega-\omega_H)^2}{2k^2\cos^2\alpha\,\varkappa T_e}\right)+ \right. \right. \]
\[ \left. \left. +\exp\left(-\frac{m(\omega+\omega_H)^2}{2k^2\cos^2\alpha\,\varkappa T_e}\right) \right] + \sqrt{\frac{\pi}{8}}\, \frac{\omega^3}{k^3\cos\alpha} \left(\frac{m}{\varkappa T_e}\right)^{3/2} \exp\left(-\frac{m\omega^2}{2k^2\cos^2\alpha\,\varkappa T_e}\right) \right\}. \tag{2.11} \]

It should be stipulated that in deriving formula (2.11), in addition to the restrictions indicated, the consideration of waves with frequencies \(\omega\) close to the gyrofrequency \(\omega_H\) was excluded. Taking this remark into account, it is easy to establish that fulfillment of the inequality \(\gamma\ll\omega\) is a consequence of condition (2.9), which was used in deriving relation (2.11); condition (2.10) was already used in deriving formula (2.11).

From (2.11), for \(\alpha=0\) we obtain the formula

\[ \gamma= \sqrt{\frac{\pi}{8}}\, \frac{\omega^4}{k^3} \left(\frac{m}{\varkappa T_e}\right)^{3/2} \exp\left(-\frac{m\omega^2}{2k^2\varkappa T_e}\right), \tag{2.12} \]

which, for \(\omega=\omega_0=\sqrt{\dfrac{4\pi e^2N_e}{m}}\), passes into Landau’s formula \(^{24}\) for the damping decrement of plasma waves in an isotropic medium. The transition, for \(\alpha=0\), from (2.11) to (2.12) with \(\omega=\omega_0\) is due to the fact that, for frequencies \(\omega\) not too close to \(\omega_H\), the condition \(n^2\gg 1\) can be satisfied only for the plasma wave. Formula (2.12) with \(\omega=\omega_0\) can be obtained both in the isotropic case and for the propagation of plasma waves in the direction of the magnetic field \(\mathbf H_0\), without the restriction imposed by the condition \(n^2\gg 1\) (see \(^{24,31-32}\)).

As \(\alpha\to \dfrac{\pi}{2}\), it follows from (2.11) that \(\gamma\to 0\). This result also follows from the consideration carried out in \(^{31-32}\).

Damping of the type under consideration in principle also occurs for transverse waves propagating in an isotropic plasma. In this case, for the damping decrement we have

\[ \gamma= \frac{\omega_0^2}{k} \sqrt{\frac{\pi m}{8\varkappa T_e}} \exp\left(-\frac{m\omega^3}{2k^2\varkappa T_e}\right). \tag{2.13} \]

However, since for transverse waves \(\dfrac{\omega}{k}=\dfrac{c}{n}>c\), such damping in a nonrelativistic gas is extremely small and has no practical significance.

The presence of damping associated with the influence of thermal motion is the most essential feature qualitatively distinguishing the results of the kinetic and the quasihydrodynamic treatments. But even when damping is neglected (i.e., when \(\gamma=0\)) there is also a number of discrepancies, which can be established by comparing the dispersion equation \((2.8)^*)\) with the quasihydrodynamic equation (2.5). The simplest example is the propagation of plasma waves in an isotropic medium (for \(u=0\)). From (2.8) (see also \(^{23-25}\)) in this case one can obtain the equation

\[ \frac{c^2 k^2}{\omega^2}=n_3^2=\frac{1-v}{3\beta^2 v}, \]

which does not coincide with relation (2.6) for \(n_3^2(v)\); the difference consists in the appearance of the factor \(1/3v\). A few more examples of this kind will be given below.

From the analysis of equation (2.8) one may arrive at the establishment of the same type of connection between the normal waves that had already been elucidated on the basis of the quasihydrodynamic equation (2.5) and illustrated in Figs. 3 and 4. Here it is only necessary to state that only the qualitative character of this connection is preserved; the detailed form of the dispersion curves in those regions where allowance for thermal motion is essential changes. Some comparison in this respect can be made by comparing the curves \(n_{1,3}^2(v)\), constructed on the basis of equation (2.8) for \(\alpha=-\dfrac{\pi}{2}\) (Fig. 5), with the analogous curves in Fig. 3.

Fig. 5.

It is necessary to point out one more new circumstance connected with the kinetic treatment. Whereas in the quasihydrodynamic treatment all three roots \(n_{1,2,3}^2(v)\) of equation (2.5), for \(u<1\) and \(\beta^2\ll 1\), were real for all values of the quantity \(v\), in the kinetic treatment two roots of equation (2.8), under the same restrictions, may in the neighborhood of the point \(v=v_\infty\) take complex values. Here we have, as it were, a more general case in comparison with the usual

\(^*)\) Since equation (2.8) was obtained without taking damping into account, it is legitimate to use not every one of the values of its roots. In considering various particular cases one must, having found one or another of the roots \(n_{1,2,3}^2\), check whether inequalities (2.9)—(2.10) are satisfied. If this is the case, then the obtained value of \(n^2\) has been found correctly.

nities \(n^2 > 0\) and \(n^2 < 0\) (see § 1). Let us recall that for \(n^2 > 0\), in the absence of collisions the field is a nonattenuating traveling wave, whereas for \(n^2 < 0\) the field decreases according to an exponential law. If, however, \(n^2\) assumes complex values and, consequently, \(n\) is also complex, then wave propagation takes place, but with a simultaneous exponential decrease of the field.

Typical dispersion curves \(n_{1,3}^{2}(\upsilon)\), for the case \(a=\pi/2\), in the presence of a region of complex values of \(n^2\), shown by hatching, are given in Fig. 6. Although in outward appearance the curves in Fig. 6 differ substantially from the analogous curves in Figs. 3 and 5, it should be emphasized that even in the presence of complex values of \(n^2\) (Fig. 6) there is a connection of the former type between the extraordinary and plasma waves. Namely, in Fig. 6, as in Fig. 3, there are also two branches for the values of the roots \(n_{1,3}^{2}(\upsilon)\), and in each of these branches there are parts corresponding to the extraordinary and plasma waves. These parts pass one into the other under a continuous change of the quantity \(\upsilon\). However, since in the neighborhood of the point \(\upsilon=\upsilon_{\infty}\) the quantity \(n^2\) is complex, the graphical representation of the connection between waves 1 and 3 becomes qualitatively different from that in the case represented in Figs. 3 and 5.

Fig. 6.

§ 3. NORMAL WAVES IN A MAGNETOACTIVE PLASMA WHEN THE MOTION OF IONS IS TAKEN INTO ACCOUNT

In an isotropic plasma, i.e. in the absence of an external magnetic field, allowance for the motion of ions can be significant from the point of view of the propagation of electromagnetic waves only in exceptional cases. Indeed, the dielectric permittivity of an isotropic plasma with allowance for the motion of ions, neglecting collisions, has the form:

\[ \varepsilon = n^2 = 1 - \frac{4\pi e^2 N_e}{m\omega^2} - \frac{4\pi e^2 N_i}{M\omega^2}, \tag{3.1} \]

where \(N_i\) is the concentration and \(M\) the mass of the ions, which for simplicity is assumed the same for all ions. From (3.1) it is evident that for \(N_e=N_i\) allowance for the motion of ions can introduce only a correction of order

\[ \frac{m}{M} < 10^{-3} - 10^{-5} \]

(for ions \(\mathrm{O}^{\pm}\), \(m/M = 3.4\cdot 10^{-5}\)). Therefore the contribution of the ions in the case (3.1) must be taken into account only for \(N_i \gg N_e\), which can occur only in the presence of a large number of negative ions.

In the presence of an external magnetic field the situation changes, and the role of ions may be large even in the case already considered, when \(N_e=N_i=N\) (for equilibrium states this condition will below be regarded as fulfilled). The point is that in a magnetoactive plasma the influence of ions on the propagation of electromagnetic waves is small only under the condition

\[ \omega \gg \Omega_H=\frac{|e|H_0}{Mc}. \tag{3.2} \]

The gyrofrequency for ions \(\Omega_H\) is usually rather low and, for example, in the terrestrial ionosphere for \(O^+\) ions in the field \(H_0 \simeq 0.5\) oersted, \(\Omega_H \simeq 300\). Therefore in the radio range the allowance for the motion of ions is immaterial, and the formulas given in §§ 1 and 2 are valid. But when the frequency is lowered and, especially, if

\[ \Omega_H \gg \omega, \tag{3.3} \]

the situation changes completely: such low-frequency electromagnetic waves in a plasma possess characteristic features and, under certain conditions, coincide with the so-called magnetohydrodynamic waves in a conducting medium (see \(^{17,39,40,41}\) and \(^{1}\) § 63).

Magnetohydrodynamic waves do not form some new branch of normal waves in a plasma, but are ordinary and extraordinary waves of low frequency. The velocity of magnetohydrodynamic waves is in a number of cases small; as a consequence, in considering them it is, generally speaking, necessary to take into account the thermal motion of the particles in the plasma. The latter is also necessary in the investigation of waves of the acoustic type. In these waves, unlike plasma waves, electrons and ions move jointly, so that in the medium, to a good approximation, no electromagnetic fields arise (what is meant is the isotropic case or the propagation of sound waves along the direction of the constant magnetic field). Acoustic, or more precisely quasi-acoustic, waves form the fourth branch of normal waves in a plasma*).

Magnetohydrodynamic and acoustic waves in a plasma are of undoubted interest both for astrophysics and for the physics of the terrestrial ionosphere.

It is most natural to carry out the consideration of low-frequency waves, as well as of high-frequency waves, on the basis of the quasihydrodynamic equations, which was first done in the papers \(^{41,42}\). If one confines oneself to the case of a strongly ionized gas, when collisions of charged particles with neutral particles may be neglected, the equations of motion for electrons and ions (\(e<0\)) are the starting point in the quasihydrodynamic approach:

\[ m\frac{d\mathbf v_e}{dt} = -\frac{\nabla p_e}{N_e} + e\left(\mathbf E+\frac{1}{c}[\mathbf v_e\mathbf H_0]\right) + m\nu_{\mathrm{eff}}(\mathbf v_i-\mathbf v_e), \tag{3.4} \]

\[ M\frac{d\mathbf v_i}{dt} = -\frac{\nabla p_i}{N_i} - e\left(\mathbf E+\frac{1}{c}[\mathbf v_i\mathbf H_0]\right) + m\nu_{\mathrm{eff}}(\mathbf v_e-\mathbf v_i). \tag{3.5} \]

Equation (3.4) almost completely coincides with equation (2.3). In the system (3.4)—(3.5), \(\nu_{\mathrm{eff}}\) is the effective number of collisions of electrons with ions; the terms \(\nabla p_e\) and \(\nabla p_i\) take into account the pressure forces arising in the electron and ion gases. The presence in equation (3.4) of exactly the same term \(m\nu_{\mathrm{eff}}(\mathbf v_i-\mathbf v_e)\) as in (3.5), but with the opposite sign, follows from the law of conservation of momentum. It is incorrect to use the system under consideration for the case of a weakly ionized—

*) It was pointed out above that plasma waves are analogous to Born oscillations in crystals. Acoustic waves in a plasma, on the other hand, are similar to acoustic waves in a solid.

of an ionized gas of the ionosphere type, since in this case equations (3.4)—(3.5) must be supplemented by terms taking account of collisions with molecules, and the whole system of equations by an equation for the motion of the molecules[^41]. However, in this case one would have to deal with very cumbersome relations, whereas the general picture can also be clarified using the example of a strongly ionized gas.

To equations (3.4)—(3.5) one must add the equation for the electric field (2.1) and the equations of conservation of the number of electrons and ions (in the case of electrons this is equation (2.4)).

Taking the plasma to be quasineutral, we shall assume that the deviations from the equilibrium values of the concentrations \(N_{e0}=N_{i0}=N\) are small. Then, seeking the solution of the resulting system of equations in the form of plane waves, one can find the dispersion equation, which is a generalization of the dispersion equation (2.5) to the case in which the motion of ions is taken into account[^28]. Without dwelling on this equation in the general case, we shall give only the results following from it for wave propagation in the direction of the constant magnetic field \(\mathbf H_0\), and also in the perpendicular direction.

For propagation of waves along the field \(\mathbf H_0\) (i.e. for \(\alpha=0\)), as in § 2, when the motion of ions was not taken into account, the normal waves can be divided into purely longitudinal and purely transverse. For transverse waves we have ([^41]–[^42], [^28] and § 63):

\[ \frac{c^{2}k_{1,2}^{2}}{\omega^{2}} =(n-i\chi)_{1,2}^{2} =1-\frac{4\pi e^{2}N} {m\omega\left(\omega-i\nu_{\mathrm{eff}}\pm \omega_{H}-\dfrac{\Omega_{H}\omega_{H}}{\omega}\right)} . \tag{3.6} \]

In the high-frequency case (3.2), this gives the expression (2.6), valid when the motion of ions is neglected (in (2.6) it is assumed that \(\nu_{\mathrm{eff}}=0\)). If, however, the “low-frequency condition” (3.3) is satisfied, then

\[ (n-i\chi)_{1,2}^{2} =1+\frac{v}{is+\sqrt{u u_i}} =1+\frac{4\pi e^{2}N} {m\omega\left(i\nu_{\mathrm{eff}}+\dfrac{\Omega_{H}\omega_{H}}{\omega}\right)}, \tag{3.7} \]

where, evidently, \(u_i=\dfrac{\Omega_{H}^{2}}{\omega^{2}}\), and the remaining notation is the same as in §§ 1 and 2.

It is characteristic that, according to (3.7), both waves 1 and 2 have identical values of \(n\) and \(\chi\); both waves are purely transverse and differ in their polarization. Waves 1 and 2 in (3.6) are circularly polarized, but with opposite directions of rotation; in the case (3.7), because of the independence of \(n\) and \(\chi\) from the state of polarization, arbitrarily polarized transverse waves may be chosen as normal waves, as occurs in the isotropic case.

Under the condition that \(\omega_H\Omega_H\gg \nu_{\mathrm{eff}}\omega\), the influence of collisions is small and, neglecting unity in comparison with the usually large quantity

\[ \frac{v}{\sqrt{u u_i}}=\frac{4\pi e^{2}N}{m\omega_H\Omega_H}, \]

for the phase velocity of wave propagation we obtain

\[ v_{\phi}=\frac{c}{n} =\frac{c}{\sqrt{\dfrac{4\pi e^{2}N}{m\omega_H\Omega_H}}} =\frac{H_0}{\sqrt{4\pi NM}} \simeq \frac{H_0}{\sqrt{4\pi\rho}}, \tag{3.8} \]

where \(\rho\simeq NM\) is the density of the plasma under consideration, containing no neutral particles.

Formula (3.8) is the well-known\(^ {39}\) expression for the velocity of magnetohydrodynamic waves and can be obtained from the system of equations of hydrodynamics of a conducting fluid and the equations of electrodynamics (see \(^ {17,39–41}\) and § 63).

For longitudinal waves, when \(a=0\), we have an equation coinciding with the equation for similar waves in the isotropic case

\[ \beta_i^2 \beta^2 (n-i\chi)^4 - \left[\beta^2+2\beta_i^2(1-v-is)\right](n-i\chi)^2 +1-v-is=0, \tag{3.9} \]

where

\[ \beta_i=\sqrt{\frac{\varkappa T}{Mc^2}}. \]

In deriving relation (3.9), small terms of order \(\frac{m}{M}\) were neglected; in addition, the plasma was assumed to be isothermal \((T_e=T_i=T)\) and, in the equilibrium state, quasineutral. Since equation (3.9) is quadratic with respect to the quantity \((n-i\chi)^2\), there are two branches of normal waves. To one of these branches, as the analysis shows, belong the high-frequency plasma waves; to the other, low-frequency acoustic waves.

Suppose that the wave frequency is so high that the condition \(\beta^2 \gg \beta_i^2 v\) is satisfied, which may be written in the form

\[ \omega^2 \gg \omega_{0i}^{\,2}, \tag{3.10} \]

where \(\omega_{0i}=\sqrt{\frac{4\pi e^2 N}{M}}\) is the frequency of ion oscillations, introduced by analogy with the natural frequency of electron oscillations \(\omega_0=\sqrt{\frac{4\pi e^2 N}{m}}\) (see (1.17)). Then, neglecting collisions, instead of (3.9) we have the equation

\[ \beta_i^2 \beta^2 n^4-\beta^2 n^2+1-v=0. \tag{3.11} \]

Taking into account that \(\beta^2 \ll 1\) and, still more, \(\beta_i^2 \ll 1\), it is easy to establish approximate values for the roots of equation (3.11). For one of them we obtain

\[ n^2 \simeq \frac{1-v}{\beta^2}, \]

i.e., we arrive at formula (2.6) for a plasma wave when the motion of the ions is neglected. For the second root we have

\[ n^2 \simeq \frac{1}{\beta_i^2}, \]

which corresponds to propagation with the phase velocity

\[ v_{\phi}=\frac{c}{n}=\sqrt{\frac{\varkappa T}{M}}, \]

which is of the same order as the mean velocity of the thermal motion of the ions and the speed of sound in the plasma. In reality, however, one probably cannot speak of the propagation of such high-frequency quasiacoustic waves, since their phase velocity is much smaller (by a factor of \(\sqrt{\frac{M}{m}}\)) than the mean velocity of the thermal motion of the electrons.

In such conditions, at least for high-frequency plasma waves, strong damping due to the thermal motion of the particles must arise (see § 2), which was not taken into account in the quasihydrodynamic treatment. Therefore an analogous result for quasiacoustic waves would be quite understandable (this supposition has not been verified by direct calculation).

Let us now consider the case of low frequencies, assuming that the inequality $\omega^2 \ll \omega_{0i}^{\,2}$, the converse of (3.10), is satisfied. In this case, neglecting collisions, from (3.9) we obtain

\[ \beta_i^2 \beta^2 n^4 + 2v\beta_i^2 n^2 - v = 0 . \tag{3.12} \]

For one of the roots of equation (3.12) we have $n^2 = \dfrac{1}{2\beta_i^2}$, which corresponds to wave propagation with velocity

\[ v_\phi = \frac{c}{n} \simeq \sqrt{\frac{2\varkappa T}{M}}, \tag{3.13} \]

which coincides with the speed of sound in an isothermal plasma; consequently the low-frequency wave under consideration may be called acoustic. For the second of the roots of (3.12), $n^2 = -\dfrac{2v}{\beta^2}$, i.e. $n^2 < 0$, and, moreover, $|n^2| \gg 1$, so that this root is of no real interest.

For propagation in the direction strictly perpendicular to the magnetic field $H_0$ (i.e. when $\alpha = \dfrac{\pi}{2}$), as also when the motion of the ions is neglected (see § 2), the dispersion equation splits into two independent equations, one of which determines the values of the refractive index of the ordinary wave $n_2^2$, and the other those of the remaining normal waves. In this case, for $n_2^2$ one obtains formula (3.1), corresponding to the isotropic case, and as a result, for this wave, allowance for ion motion introduces practically nothing new. The electric vector and the velocity of particle motion in the ordinary wave 2 at $\alpha = \pi/2$ are directed along the field $H_0$, and it is precisely for this reason that this wave propagates in the same way as transverse waves in an isotropic medium. In the low-frequency region the ordinary wave is, evidently, strongly damped ($n^2 = 1 - v \simeq -v < 0$ for $v \gg 1$).

The other part of the dispersion equation, for $\alpha = \pi/2$ and low frequency, has three roots for $n^2$ (or, when absorption is taken into account, for $(n - i\chi)^2$). Two of these roots are negative and large in absolute value, while one root is determined by the relation (it is assumed that $s = 0$ and $\dfrac{v}{\sqrt{u u_i}} \gg 1$):

\[ v\left(1 - 2\beta_i^2 n^2\right) = \sqrt{u u_i}. \tag{3.14} \]

Thus, in the low-frequency case, for $\alpha = \dfrac{\pi}{2}$, propagation of only one type of waves is actually possible, whose phase velocity, in accordance with (3.14), is determined by the formula

\[ v_\phi = \frac{c}{n} = \sqrt{\frac{2\varkappa T}{M} + \frac{H_0^2}{4\pi N M}} . \tag{3.15} \]

These waves are, as it were, both acoustic and magnetohydrodynamic at the same time. They are linearly polarized; the velocities of the ordered motion $\mathbf{v}_e$ and $\mathbf{v}_i$ are directed along $\mathbf{k}$, while the electric field $\mathbf{E}$ is perpendicular to the plane $\mathbf{k}H_0$, i.e. in our notation (see Fig. 1) is directed along the $x$ axis. Relation (3.15) can be obtained in the magnetohydrodynami-

approximation, i.e., relying on the equations of hydrodynamics of a conducting compressible fluid (see^41 and § 63).

It must be noted, however, that at low frequencies the region of angles very close to $\alpha=\pi/2$, where the condition

\[ \sqrt{\overline{u_i}}\cos^2\alpha \ll 1 \]

is satisfied, is exceptional (in this region of angles the waves propagate as was indicated above for the case $\alpha=\pi/2$). The point is that under the opposite condition $\sqrt{\overline{u_i}}\cos^2\alpha \gg 1$, i.e., at low frequencies, the waves almost always propagate in a substantially different manner. Namely, for three waves the propagation velocity is then different from zero and the relations obtained in the magnetohydrodynamic approximation are valid (see^41 and § 63), and only one wave, which may be called a plasma wave, is strongly damped (for this wave $n^2 \simeq -\dfrac{2v}{\beta_e^2}<0$).

If the transition to the case $\alpha=\pi/2$ is made directly in the magnetohydrodynamic approximation, then for the wave which can propagate without damping, as was said, one obtains expression (3.15). But the two other waves (apart from the plasma wave) at $\alpha=\pi/2$ in the magnetohydrodynamic approximation correspond to a velocity $v_\phi=0$, whereas under the quasihydrodynamic approach it turns out that these waves are strongly damped (see above). In fact this difference at $\alpha=\pi/2$ is of little significance, since from both points of view the corresponding waves cannot propagate. Nevertheless, for an angle $\alpha$ close to $\pi/2$ such a distinction may already turn out to be substantial, and it must be remembered that the usual magnetohydrodynamic approximation is in general valid only if

\[ \sqrt{\overline{u_i}}\cos^2\alpha = \frac{\omega_H\Omega_H}{\omega^2}\cos^2\alpha \gg 1 \]

(at the same time, as is clear from what has been said, fulfillment of this condition is not essential for the wave whose velocity at $\alpha=\pi/2$ is determined by expression (3.15)).

The quasihydrodynamic treatment is only an approximate method for taking into account the influence of thermal motion on the character of propagation of low-frequency waves. Here again, as in § 2, the question arises of a more detailed allowance for thermal motion based on the method of the kinetic equation. We shall consider this question only as applied to one particular case: magnetohydrodynamic waves propagating in the direction of the magnetic field $\mathbf{H}_0$. In this case, in the quasihydrodynamic approximation, corrections due to allowance for the thermal motion of electrons and ions, as indicated by relations (3.7)—(3.8), are altogether absent. It is difficult to agree with this conclusion, in particular because in most interesting cases the velocity of magnetohydrodynamic waves

\[ v_0=\frac{H_0}{\sqrt{4\pi NM}} \]

(see (3.8)) may be comparatively small and may prove to be of the same order as, or even smaller than, the mean velocity of the thermal motion of ions. For high-frequency waves, under analogous conditions, the influence of thermal motion would be very substantial (see § 2). Therefore, without a detailed analysis it remains unclear whether a more correct account of thermal motion will not lead both to substantial corrections to the phase velocity and to additional damping, not associated with collisions, similar to that discussed in § 2. The answer to these questions may be found in the work^43 (see also^44).

Relying on the kinetic equations for electrons and ions and assuming that, in the equilibrium state, the distribution of electrons and ions is described by a Maxwellian function with a common temperature \(T\), one can, for weakly damped waves, arrive at the relations:

\[ v_{\phi,\,1,\,2}=\frac{c}{n_{1,2}}=v_0\sqrt{1\mp \frac{\bar v_i^2}{v_0^2}\frac{\omega}{\Omega_H}}, \tag{3.16} \]

\[ \gamma_{1,\,2}= \frac{\Omega_H^2}{\omega}\, \frac{v_{\phi,\,1,\,2}}{\bar v_i} \sqrt{\frac{\pi}{2}} \left( 2\mp \frac{\bar v_i^2}{v_{\phi,\,1,\,2}^{\,2}}\frac{\omega}{\Omega_H} \right)^{-1} \exp\left( -\frac{v_{\phi,\,1,\,2}}{2\bar v_i^{\,2}}\frac{\Omega_H^2}{\omega^2} \right), \tag{3.17} \]

where \(v_{\phi,\,1,\,2}\) and \(\gamma_{1,\,2}\) determine the phase velocity and the damping decrement of the waves. Here the index 1 and the upper sign in (3.16)—(3.17) refer to the low-frequency wave obtained in the limiting transition from the dispersion equation for the extraordinary wave; the index 2 and the lower sign should be assigned to the wave obtained from the equation for the ordinary wave. In formulas (3.16)—(3.17),

\[ \bar v_i=\sqrt{\frac{\varkappa T}{M}} \]

is a certain mean velocity of the thermal motion of the ions,

\[ v_0=\frac{H_0}{\sqrt{4\pi NM}} \]

is the wave velocity when thermal motion is neglected.

A characteristic result associated with taking thermal motion into account when using the kinetic-equation method is the inequality of the phase velocities of propagation of the extraordinary and ordinary waves even in the limiting transition to magnetohydrodynamic waves. In this case the phase velocity found with allowance for thermal motion depends on the frequency \(\omega\), i.e. both waves propagate with dispersion (see (3.16)).

From relation (3.17) one may draw a conclusion that is very important from the point of view of estimating the role of the damping mechanism of magnetohydrodynamic waves under consideration. The magnitude of the damping decrement in the case discussed here, \(\gamma \ll \omega\) (weak damping), is determined mainly by the expression standing in the exponent and depends not only on the square of the ratio of the phase velocity \(v_\phi\) to the thermal velocity of the slowest particles—the ions,

\[ \left(\frac{v_\phi^2}{\bar v_i^2}\right), \]

but also on the product

\[ \frac{v_\phi^2}{\bar v_i^2}\frac{\Omega_H^2}{\omega^2}, \]

which, by virtue of the “low-frequency condition” (3.3), is by definition much greater than unity. Therefore the damping may be very small even when the phase velocity of the magnetohydrodynamic wave is comparable with, or even within certain limits smaller than, the velocity \(\bar v_i\) of the thermal motion of the ions. The same also applies to the corrections to the phase velocity \(v_0\) (see (3.8) and (3.16)). However, since in (3.16) in the second term of the radicand the ratio

\[ \frac{\omega}{\Omega_H} \]

enters only to the first power, the corrections to the velocity due to thermal motion may be more substantial.

Estimates based on relation (3.16) show that, under the conditions of the terrestrial ionosphere, the thermal corrections in determining the magnitude of the phase velocity may be neglected. The damping decrement (3.17) for propagation in the ionosphere is negligibly small, so that absorption of magnetohydrodynamic waves in the ionosphere is determined exclusively by collisions.

§ 4. ON A CERTAIN FEATURE OF THE FIELD OF AN ELECTROMAGNETIC WAVE PROPAGATING IN AN INHOMOGENEOUS ISOTROPIC PLASMA

Both in the terrestrial ionosphere and in a number of other cases, one has to deal with the propagation of waves in an inhomogeneous plasma. The problem cannot be investigated with sufficient completeness unless the character of the dependence of the properties of the medium on the coordinates is specified. The most important example of such a specification is the consideration of a plane-stratified medium. In this case, to which we shall confine ourselves below, the components of the tensor \(\varepsilon'_{ik}\), or (in an isotropic medium) the complex permittivity \(\varepsilon'\), are functions of only one coordinate, which we choose to be the coordinate \(z\).

Further, it may usually be assumed that the properties of the medium vary slowly over distances of the order of the wavelength in this medium, i.e., that, when the wave propagates along the \(z\)-axis, the inequality

\[ \left|\frac{d\varepsilon}{dz}\right|\cdot \lambda = \left|\frac{d\varepsilon}{dz}\right|\cdot \frac{\lambda_0}{\sqrt{|\varepsilon|}} \ll \varepsilon, \tag{4.1} \]

is satisfied, where absorption is assumed, for simplicity, to be absent; \(\lambda_0 = \dfrac{2\pi c}{\omega}\) is the wavelength in vacuum, and \(\lambda = \dfrac{\lambda_0}{\sqrt{\varepsilon}} = \dfrac{\lambda_0}{n}\) is the wavelength in the medium. Condition (4.1) can be rewritten in the form

\[ \frac{\lambda_0}{2\pi |n|^3} \left|\frac{dn}{dz}\right| \ll 1, \tag{4.2} \]

which is applicable also to the case of an anisotropic and, in particular, magnetoactive medium (in this case \(n\) must be understood as the refractive indices \(n_1\) or \(n_2\))*.

If condition (4.1)—(4.2) is fulfilled, the medium may be regarded as quasi-homogeneous in the sense that the propagation of waves in a small region of an inhomogeneous medium takes place in the same way as in a homogeneous medium with the same parameters. The corresponding approximation is called the geometrical-optics approximation**. In the isotropic case, for normal incidence of a plane wave on a plane-stratified medium, the expression for the wave field in this approximation can be written, for example, in the form

\[ E_{x,y} = \frac{\mathrm{const}}{\sqrt{n(z)}} \, e^{\,i\left[\omega t \pm \frac{\omega}{c}\int n(z)\,dz\right]} . \tag{4.3} \]

For a homogeneous medium, when \(n(z)=n=\mathrm{const}\), the solution (4.3) coincides, of course, with the expression for the field of a plane wave in a homogeneous medium

\[ E_{x,y} = E_0 \cdot e^{\,i\left(\omega t \pm \frac{\omega}{c}nz\right)} . \]

We note that in the geometrical-optics approximation both waves, corresponding—

* In (4.2), instead of \(\lambda_0\) there appears the quantity \(\lambda_0/2\pi\). Formally this replacement, since an inequality is involved, is not essential. Practically, however, the factor \(1/2\pi\) is quite important and at the same time is obtained automatically when investigating the domain of applicability of the geometrical-optics approximation (one may say, therefore, that the properties of the medium must vary slowly over distances of order \(\lambda/2\pi\)).

** For the validity of the geometrical-optics approximation, which we identify with the possibility of regarding the medium as quasi-homogeneous, fulfillment of condition (4.2) is necessary but not yet sufficient. Since, however, conditions other than (4.2) play a secondary role, and since we shall have to encounter one such condition only in § 9, we shall not dwell on this question in greater detail (see §§ 65 and 77).

ing to the signs \(\pm\) in (4.3), are completely independent—one wave propagates without reflection in the direction of positive values of \(z\), and the other wave in the opposite direction. Both these waves are therefore entirely analogous to normal waves in a homogeneous medium.

However, in regions where conditions (4.2) are not satisfied, the geometrical-optics approximation (4.3) is invalid, and waves propagating in different directions turn out to be coupled. In this sense one may speak of the interaction of normal waves in an inhomogeneous medium, understanding by normal waves the waves of type (4.3), which are such only in the geometrical-optics approximation.

This approximation, as is clear from (4.3), is violated if the gradient \(dn/dz\) is sufficiently large and, in particular, at a sharp interface between two media. Condition (4.2) is also violated in the more interesting and more specific case for plasma when the refractive index \(n\) is small, i.e., near a point \(\varepsilon(z)\equiv n^2(z)=0\). In the region around the point \(n=0\), and also in the region of large gradients of \(n\), reflection of waves occurs, i.e., a coupling takes place between waves of different directions. The case of “reflection from the point” \(n=0\) is in this respect quite analogous to total internal reflection, well known in optics.

Up to now the discussion has concerned the normal incidence of a wave on a plane-stratified isotropic medium, when the wave vector is directed along the \(z\)-axis. The polarization of the waves then, of course, plays no role, and the solutions (4.3) apply equally to the components of the electric field \(E_x\) and \(E_y\).

For oblique incidence of waves on an isotropic medium, waves with an electric vector lying in the plane of incidence \(yz\) are independent of waves for which the field \(\mathbf E\) is perpendicular to the plane of incidence (with the coordinate choice made, in the latter case \(E_x\ne0,\ E_y=E_z=0\)). The problem of reflection of waves for which the vector \(\mathbf E\) is perpendicular to the plane of incidence differs little from the problem of normal incidence and is mathematically easily reduced to this latter one by replacing \(\varepsilon'(z)\) by \(\varepsilon'(z)-\sin^2\vartheta_0\), where \(\vartheta_0\) is the angle of incidence of the wave from vacuum onto the inhomogeneous medium (see § 67; for simplicity it is assumed that at the boundary of the medium \(\varepsilon'(z)=1\)). In the absence of absorption \(\varepsilon'=\varepsilon=n^2\), and the role of the reflection point \(n=0\) is played by the point \(n(z)=\sin\vartheta_0\) (Fig. 7).

Fig. 7.

In the case of a wave with the vector \(\mathbf E\) lying in the plane of incidence, reflection occurs from the same point \(n(z)=\sin\vartheta_0\), but, in addition, the field of the wave has a singularity at the point \(\varepsilon(z)=0\). The difference between the fields of waves with different polarizations is clear from the schematic Fig. 8 (along the ordinate is plotted the square of the modulus of the standing-wave field formed as a result of reflection of the incident wave from the region where \(n=\sin\vartheta_0\)). The peculiar behavior of the field of a wave polarized in the plane of incidence near the zero of the function \(\varepsilon(z)\) usually does not attract much attention, since in practice in the ionosphere the corresponding effect proves to be insignificant (see below). However, the question of the singularity of the field at the point \(\varepsilon=0\) is not only very interesting in principle, but may also turn out to be very important in considering the mechanism of radio-wave generation in the solar corona\(^{45}\). Therefore we shall dwell in rather great detail on the study of the field near the point where \(\varepsilon=0\).

For the first time this question, for the case of oblique incidence of radio waves on an isotropic medium with a linear dependence of \(\varepsilon\) on \(z\), was considered in the work \(^{46}\). It was shown there that the requirement that the solution of the corresponding wave equation vanish as \(z \to \infty\) (in the region of negative values of \(\varepsilon(z)\)) leads to the appearance in this solution of a term that becomes infinite at the point where \(\varepsilon(z)=0\). However, the author then evaded solving the problem, simply assuming that the function \(\varepsilon(z)\) nowhere vanishes. Subsequent investigations of this problem were the subject of papers \(^{47-49}\).

For a wave polarized in the plane of incidence, it is more convenient to consider not the electric field with components \(E_y\) and \(E_z\), but the magnetic field, which has only the component \(H_x\). Putting

\[ H_x = w(z)\cdot e^{i(\omega t+k_0 qy)}, \tag{4.4} \]

for \(w(z)\) we obtain the equation (see, for example, \(^{1}\) § 67)

\[ \frac{d^2 w}{dz^2} -\frac{1}{\varepsilon'(z)}\cdot \frac{d\varepsilon'}{dz}\cdot \frac{dw}{dz} + k_0^2(\varepsilon' - q^2)w=0, \tag{4.5} \]

where \(k_0=\dfrac{\omega}{c}\) and \(q=\sin \vartheta_0\). The components of the electric field can be found from the equation \(\operatorname{rot}\mathbf H=ik_0\varepsilon'\mathbf E\), which gives

\[ E_y=\frac{1}{ik_0\varepsilon'}\frac{\partial H_x}{\partial z},\qquad E_z=-\frac{1}{ik_0\varepsilon'}\frac{\partial H_x}{\partial y}. \tag{4.6} \]

Fig. 8.

We shall next use expression (1.16) for \(\varepsilon'(z)\), and shall assume that \(\nu_{\mathrm{eff}}/\omega \ll 1\). Then

\[ \varepsilon'(z)\simeq 1-\frac{4\pi e^2 N(z)}{m\omega^2} \left(1+i\frac{\nu_{\mathrm{eff}}}{\omega}\right) = \varepsilon(z)-i\frac{\omega_0^2}{\omega^2}\cdot \frac{\nu_{\mathrm{eff}}}{\omega}. \]

For simplicity we shall also assume that the absorption changes little with height, i.e. that \(\nu_{\mathrm{eff}}\) depends weakly on \(z\). In this case the imaginary part of \(\varepsilon'(z)\) may be regarded as constant and equal to its value at \(\omega=\omega_0=\sqrt{4\pi e^2N/m}\). As a result, for a linear layer we obtain*)

\[ \varepsilon'(z)=-az-i\frac{\nu_{\mathrm{eff}}}{\omega} =-az-is, \tag{4.7} \]

where \(a>0\), so that for values \(z>0\) the dielectric permittivity \(\varepsilon(z)<0\).

*) For an arbitrary layer, expression (4.7) is also suitable in a small neighborhood of the zero of the function \(\varepsilon(z)\), where the geometrical-optics approximation is inapplicable (see \(^{1}\) § 69).

The differential equation (4.5) is now written in the form

\[ \frac{d^2 w}{dz^2}-\frac{a}{az+is}\frac{dw}{dz} +k_0^2(-az-is-q^2)w=0. \]

Introducing the new variable \(\zeta=az+is\) and the notation \(\rho=\frac{k_0}{a}\), we obtain

\[ \frac{d^2w}{d\zeta^2}-\frac{1}{\zeta}\frac{dw}{d\zeta} +\rho^2(-\zeta-q^2)w=0. \tag{4.8} \]

The form of equation (4.8), in passing to the case \(s=0\), obviously does not change. The only difference in the problem with absorption taken into account is that in it the “mathematical” reflection point \(\zeta=-q^2\) corresponds to complex values of the coordinate \(z\). Let us also note that, for a medium with slowly varying properties, the parameter entering equation (4.8)

\[ \rho=\frac{k_0}{a}\gg 1. \]

Thus, in the \(F\) layer of the ionosphere \((a\sim 10^{-7})\), for a frequency \(\omega\sim 10^8\) the parameter \(\rho\sim 3\cdot 10^4\).

It was shown in papers \(^{47-48}\) that the solution of equation (4.8) satisfying the necessary physical requirements takes on a certain nonzero value at the point where \(\varepsilon'(z)\) vanishes. Therefore the vertical component of the electric field

\[ E_z=-\frac{1}{ik_0\varepsilon'}\frac{\partial H_x}{\partial y} =-\frac{q}{\varepsilon'(z)}\,w(z)e^{i(\omega t+k_0qy)} \tag{4.9} \]

becomes infinite at this point. The character of this singularity depends on the behavior of the function \(\varepsilon'(z)\); in particular, for a linear layer \(E_z\) becomes infinite as

\[ \frac{1}{\zeta}=\frac{1}{az+is}, \]

while the component \(E_y\) has a logarithmic singularity. These singularities lie on the real axis only when

\[ s=-\frac{\nu_{\mathrm{eff}}}{\omega}=0. \]

When absorption is taken into account, however, the maximum value of \(E_z\) will be equal to

\[ |E_z|_{z=0}=\frac{q\,|w(0)|}{s} \tag{4.10} \]

and, for sufficiently small \(s\), may be very large. In this case the magnitude of the field depends essentially on what values the function \(w(0)\) takes. This function, in turn, depends on the angle of incidence and thus determines the value of \(|E_z|_{z=0}\) over the entire range of values of the parameter \(q=\sin\vartheta_0\).

It is comparatively easy to establish the form of the function \(w(z)\) at large angles of incidence, when the reflection point \(\zeta=-q^2\) (the point \(\varepsilon'=\sin^2\vartheta_0\)) and the singular point \(\zeta=0\) (i.e., the point \(\varepsilon'=0\)) are separated from one another by a considerable distance \(^{49}\). For this purpose, instead of equation (4.8) it is more convenient to investigate the equation

\[ \frac{d^2u}{d\zeta^2} -\left[\rho^2(\zeta+q^2)+\frac{3}{4\zeta^2}\right]u=0, \tag{4.11} \]

which is satisfied by the function

\[ u(\zeta)=\frac{w(\zeta)}{\sqrt{\zeta}}. \tag{4.12} \]

The assumption that the distance between the points \(\zeta=-q^2\) and \(\zeta=0\) is large means, in the present case, that this distance is much greater than the wavelength. For a medium with slowly varying properties \((\rho\gg 1)\), this occurs even for small values \(q^2=\sin^2\vartheta_0\). Under such conditions, approxi-

the solution of equation ((4.11)), valid everywhere except for a small neighborhood of the point \(\zeta=0\) and representing, to the left of \(\zeta=-q^2\), a standing wave (Fig. 9), can be written in the form (see \(^{50}\))

\[ u=-i\sqrt{\frac{\pi\rho}{2}}\cdot e^{-i\frac{\pi}{12}}\sqrt{\frac{S}{S'}}\,H_{1/3}^{(1)}(iS), \tag{4.13} \]

where

\[ S=\rho\int_{-q^2}^{\zeta}\sqrt{\zeta'+q^2}\,d\zeta' =\frac{2}{3}\rho(\zeta+q^2)^{3/2}; \qquad S'=\frac{dS}{d\zeta} \tag{4.14} \]

and \(H_{1/3}^{(1)}\) is the Hankel function of the first kind of order \(1/3\). The constant appearing in the solution (4.13) has been chosen so that at the boundary of the inhomogeneous layer (for \(\varepsilon'(z)=1\)) the amplitude of the field of the incident wave would be equal to unity.

Another approximate solution, valid to the right of the reflection point, can be obtained by using the method proposed in Ref. \(^{50}\). We introduce a new independent variable

\[ \xi=\rho\int_{0}^{\zeta}\sqrt{\zeta'+q^2}\,d\zeta' =\frac{2}{3}\rho\left[(\zeta+q^2)^{3/2}-q^3\right]. \tag{4.15} \]

It is easy to show that the function

\[ u^*=A\sqrt{\frac{\xi}{d\xi/d\zeta}}\cdot H_{1}^{(1)}(i\xi) \qquad (A=\mathrm{const}) \tag{4.16} \]

satisfies the equation

\[ \frac{d^2u^*}{d\zeta^2} - \left[ \rho^2(\zeta+q^2) + \frac{3}{4}\left(\frac{1}{\xi}\frac{d\xi}{d\zeta}\right)^2 + \frac{5}{16(\zeta+q^2)^2} \right]u^*=0. \tag{4.17} \]

For small \(\zeta\), as is clear from (4.15), \(\xi\approx \rho q\zeta\), and equation (4.17) has exactly the same singularity at the point \(\zeta=0\) as the basic equation (4.11). In addition, for large values of the parameter \(\rho\), equations (4.17) and (4.11) generally differ very little from one another, if one excludes from consideration a certain neighborhood of the point \(\zeta=-q^2\), where the function \(\dfrac{5}{16}(\zeta+q^2)^2\) begins to grow sharply. Consequently, far from the point \(\zeta=-q^2\), appropriately chosen solutions of these equations will differ little from one another. In this case the function (4.16) approximates that solution which tends to zero as \(\zeta\to\infty\) (in the region of negative values of \(\varepsilon(z)\)).

Fig. 9.

We have thus obtained approximate solutions (4.13) and (4.16), which give the asymptotic behavior of the desired solution (for \(\rho\gg1\)) in different ranges of values of the variable \(\zeta\): to the left of \(\zeta=0\) (function (4.13)) and to the right of \(\zeta=-q^2\) (solution (4.16)). In the interval \(-q^2<\zeta<0\) both approximations are valid, which makes it possible to match these solutions so that they describe the behavior of one and the same particular solution of our

of the problem. This matching of the solutions gives, for the constant \(A\), the value\({}^{49}\)

\[ A=\sqrt{\frac{\pi\rho}{2}}\, e^{i\frac{\pi}{4}-S_0}, \tag{4.18} \]

where

\[ S_0=\rho \int_{-q^2}^{0}\sqrt{\xi+q^2}\,d\xi =\frac{2}{3}\rho q^3 . \tag{4.19} \]

On the basis of (4.16) and (4.18), the final formula describing the behavior of the function \(w(\zeta)\) in the region \(\zeta>-q^2\) may be written in the following form:

\[ w(\zeta)=\sqrt{\zeta}\,u^* =\sqrt{\frac{\pi\rho}{2}}\, e^{i\frac{\pi}{4}-S_0} \sqrt{\frac{\xi}{\xi'}}\, H_1^{(1)}(i\xi). \tag{4.20} \]

If the quantity \(\zeta\) is so small that \(\xi=\rho q\zeta\ll 1\), then, in computing the field components, we may use the expansion of the function \(H_1^{(1)}(i\xi)\) in a power series, retaining the first terms of the expansion,

\[ H_1^{(1)}(i\xi)\simeq -\frac{2}{\pi\xi}-\frac{\xi}{\pi}\ln \xi . \tag{4.21} \]

It is now easy to show that the amplitudes of the field components behave as follows. The amplitude \(H_x\), as \(\zeta\to 0\), tends to the constant value (see (4.4))

\[ |w(0)|=\frac{2}{\pi\rho q}|A| =\sqrt{\frac{2}{\pi\rho}}\,\frac{e^{-S_0}}{q}. \tag{4.22} \]

The horizontal component of the electric field \(E_y\), according to (4.6), is equal to

\[ |E_y|\simeq \frac{2q}{\pi}|A\ln \xi|, \tag{4.23} \]

i.e., it has a logarithmic singularity. And, finally, the vertical component of the electric field \(E_z\) goes to infinity according to the law (see (4.9))

\[ |E_z|\simeq \frac{q|w(0)|}{|\zeta|} =\frac{q|w(0)|}{\left|az+i\frac{\nu_{\mathrm{eff}}}{\omega}\right|}. \tag{4.24} \]

Using expression (4.22) for the quantity \(|w(0)|\), we finally write

\[ |E_z|\simeq \sqrt{\frac{2}{\pi\rho}}\, \frac{e^{-S_0}}{|\zeta|}. \tag{4.25} \]

At \(z=0\) the quantity \(|E_z|\) assumes its maximum value (in a medium with absorption \(\zeta=az+is\)):

\[ |E_z|_{z=0}=\sqrt{\frac{2}{\pi\rho}}\, \frac{e^{-S_0}}{s}. \tag{4.26} \]

Let us recall that the final formulas are applicable only for large angles of incidence; as \(q\to 0\), formula (4.22) gives an obviously incorrect result, since for \(q=0\) (normal incidence) the exact solution of the problem shows that \(E_z=0\). However, for the upper layers of the ionosphere, where \(\rho\gg 1\), the approximate formulas prove to be suitable down to angles of incidence \(\vartheta_0\) po-

of the order of \(4^\circ \div 5^\circ\), and, as is easy to verify, under these conditions the effect of the field increase near the point \(\zeta=0\) would be insignificant \((S_0\gg 1)\), even if the influence of the Earth’s magnetic field, which will be discussed below, is not taken into account. (The electric field can assume large values only for very small values of \(\nu_{\mathrm{eff}}\).) At the same time, the presence of a singularity at the point where \(\varepsilon'=0\) does not affect the behavior of the field in the region situated below the reflection point, i.e., the reflection of a wave having an \(E_z\) component occurs under these conditions in the same way as for a wave whose electric vector is perpendicular to the plane of incidence.

Formula (4.24) shows that the magnitude of the field at the point where \(\varepsilon=0\) (i.e., at the point \(z=0\)), in addition to \(\nu_{\mathrm{eff}}\), is determined by the values of the function \(q w(0)\). For normal incidence (when \(q=0\)) \(E_z=0\); for large values of \(S_0\), as is clear from (4.19) and (4.25), the field \(E_z\) decreases as \(q\) increases. Consequently, at some small angle of incidence the effect of the increase of the field at the point \(\varepsilon=0\) will be maximal. In this connection, the behavior of the function \(q w(0,q)\) for all angles of incidence is of interest. Investigation of the solutions of equation (4.8) shows\({}^{49}\) that the function \(|q w(0,q)|\) over the whole interval of values of the parameter \(q\) can be approximately represented in the form

\[ |q w(0,q)|= \frac{4\pi v(\tau^2)}{\sqrt{2\pi\rho}}\, \sqrt{\frac{v(\tau^2)}{-v'(\tau^2)}}= \frac{\Phi(\tau)}{\sqrt{2\pi\rho}}, \tag{4.27} \]

where \(v\) and \(v'\) are the Airy function and its derivative (see \({}^{51}\)), and the parameter

\[ \tau=\rho^{1/3}q=\left(\frac{k_0}{a}\right)^{1/3}\sin\vartheta_0 . \tag{4.28} \]

The dependence of the maximum value of \(|E_z|\) on the angle of incidence is thus determined by the function \(\Phi(\tau)\), whose graph is given in Fig. 10; there, in addition, parallel to the abscissa axis, the scale for the angle of incidence \(\vartheta_0\) in degrees is given for \(a=10^{-7}\,\mathrm{cm}^{-1}\) and \(\omega=2\pi\cdot10^7\) \((\lambda_0=30\,\mathrm{m})\).

Fig. 10.

It is significant that \(\Phi(\tau)\) assumes values of order unity only for a very narrow interval of angles of incidence. The maximum of the curve, equal to \(1.2\), occurs in the example considered at the angle \(\vartheta_0=1.5^\circ\), and already at \(\vartheta_0=5^\circ\), \(\Phi(\tau)\sim10^{-4}\).

Let us estimate, on the basis of formulas (4.24) and (4.27), the values which the field \(E_z\) can attain under more or less real conditions. The maximum value of \(|E_z|\) is equal to

\[ \frac{1.2}{\sqrt{2\pi\rho}}\frac{\omega}{\nu_{\mathrm{eff}}} = \frac{1.2}{\sqrt{\dfrac{2\pi\omega}{ca}}}\frac{\omega}{\nu_{\mathrm{eff}}}. \]

Therefore, in the \(E\)-layer of the ionosphere, where one may put \(a\sim10^{-6}\), for \(\lambda_0=100\,\mathrm{m}\) \((\omega=6\pi\cdot10^6)\), \(|E_z|_{z=0}\approx3.6\) for \(\nu_{\mathrm{eff}}=10^5\), and \(|E_z|_{z=0}\approx36\) for \(\nu_{\mathrm{eff}}=10^4\).

For the \(F\)-layer \((a=10^{-7},\ \omega=2\pi\cdot10^7\ (\lambda_0=30\,\mathrm{m}))\) we have, for \(\nu_{\mathrm{eff}}=10^4\), \(|E_z|_{z=0}\approx20\), while for \(\nu_{\mathrm{eff}}=10^3\), \(|E_z|_{z=0}\approx200\). Let us recall that at the lower boundary of an inhomogeneous layer \(|E|=1\) and \(|E_z|=|E|\sin\vartheta_0=q\).

Of known interest is also the effective size of the region where the field is large. From formula (4.4) it is easy to establish that \(|E_z|^2\) decreases to

of the maximum value at distances

\[ \Delta z=\frac{\nu_{\mathrm{eff}}}{\omega a} =\frac{\lambda_0}{2\pi}\cdot\frac{\nu_{\mathrm{eff}}}{ca} \tag{4.29} \]

from the point \(z=0\), where the field is maximal.

For \(\nu_{\mathrm{eff}}\sim 10^4\) and \(a\sim 10^{-7}\), it is obvious that \(\Delta z\sim \lambda_0\).

The sharp increase of the electric-field strength near the point where \(\varepsilon'(z)=0\) leads to the result that describing the field by means of the usual dielectric permittivity \(\varepsilon'(z)\) may prove impossible. The point is that the quantity

\[ \varepsilon'=\varepsilon-i\,\frac{4\pi\sigma}{\omega} \]

has a local character, relating the polarization and current at a given point to the electric field at the same point. Such a local approximation is valid if the electron mean free path \(l=\dfrac{v}{\nu_{\mathrm{eff}}}\), and also the distance \(\dfrac{v}{\omega}\) traversed by them during a quarter period (\(v\) is the electron velocity), are small in comparison with \(\Delta z\), i.e. with the dimensions of the region in which the field changes appreciably*). Taking (4.29) into account, we hence obtain the conditions for validity of the calculations carried out above:

\[ \nu_{\mathrm{eff}}\gg \sqrt{\omega va} =\sqrt{\frac{2\pi cva}{\lambda_0}};\qquad \nu_{\mathrm{eff}}\gg va . \tag{4.30} \]

The second of these conditions is always weaker than the first, since we consider the case in which \(\dfrac{\nu_{\mathrm{eff}}}{\omega}\ll 1\). This second condition, i.e. actually the condition \(\dfrac{2\pi v}{\omega}\ll\Delta z\), is nevertheless given for the following reason. If there were no collisions at all, then, when using the quantity \(\varepsilon(z)\), the field at the point \(\varepsilon=0\) would have to increase to infinity. In fact, in a sufficiently strong field an electron acquires a large velocity, so that the quantity \(\dfrac{v}{\omega}\) increases and the use of the dielectric permittivity \(\varepsilon\) becomes inadmissible. As a result, the induction \(D(z)\) differs from the quantity \(\varepsilon E(z)\) and at the point \(\varepsilon=0\) is not equal to zero, but is determined by the field \(E\) at a point displaced by a distance of the order of the amplitude of the electron oscillations \(\dfrac{v}{\omega}\). Therefore, as was noted in \(^{48}\), in a formula of type (4.24), at the point \(z=0\) (i.e. at \(\varepsilon=0\)), the denominator contains not zero, but the quantity

\[ a\int_0^t v_z\,dt, \]

where \(v_z\) is the electron velocity in the direction of the \(z\)-axis and \(t=0\) is the instant when the field passes through zero. In a strong field the velocity \(v_z\) considerably exceeds the thermal-motion velocity

\[ v\sim \sqrt{\frac{\varkappa T}{m}}\sim 10^7 \]

(for \(T\sim 300^\circ\mathrm{K}\)) and, in order of magnitude, is equal to

\[ v_z\sim \frac{eE_z}{m\omega}. \]

Hence

\[ a\int_0^t v_z\,dt \sim \frac{aeE_z}{m\omega^2} \sim 10^{-5}E_z \]

(for \(a\sim 10^{-7}\) and \(\omega=2\pi\cdot 10^7\)). Since the part depending on \(E_z\)

*) What is involved here is a requirement analogous to that used in the theory of the normal skin effect (see, for example, \(^{52}\)).

The criterion for applicability of the local approximation (4.30) given in the text is the simplest one and, perhaps, needs refinement on the basis of an analysis of the kinetic equation. For reasons that will be clear from what follows, however, we do not dwell on this question in more detail.

the quantity \(a\int_0^t v_z\,dt\) appears in the denominator of an expression of type (4.24) for \(E_z\); clearly the problem ceases to be linear. This is also understandable, since in a time-harmonic but strongly inhomogeneous field the electron no longer moves according to a harmonic law. The effect under discussion in a real medium becomes appreciable if the value of \(a\int_0^t v_z\,dt\) is comparable with the quantity \(\dfrac{\nu_{\mathrm{eff}}}{\omega}\), which appears in the denominator of expression (4.24) when absorption is taken into account.

In the \(F\)-layer, as we have seen, \(\dfrac{\nu_{\mathrm{eff}}}{\omega}\sim 10^{-4}\), and \(a\int_0^t v_z\,dt\sim 10^{-5}E_z\sim 10^{-4}\) only for \(E_z\sim 10=3000\ \mathrm{V/cm}\); but under the same conditions the field at the point \(\varepsilon=0\) is only 20 times greater than the field at the beginning of the layer (see above), and thus the nonlinear effect would have to be taken into account only in very strong fields, with which one does not have to deal in the case of the ionosphere (quite apart from the fact that in such fields the linear treatment of radio-wave propagation in a plasma becomes inadmissible not only in the neighborhood of the point \(\varepsilon=0\), but throughout the whole layer).

As for the first of conditions (4.30), then for \(a\sim 10^{-7}\), \(\omega=2\pi\cdot 10^7\), and \(v\sim \sqrt{\dfrac{\varkappa T}{m}}\sim 10^7\), it takes the form \(\nu_{\mathrm{eff}}\gg 10^4\). Such an inequality is not satisfied in the \(F\)-layer and, consequently, for a rigorous calculation of the field near the point \(\varepsilon=0\) it would be necessary to use the kinetic equation. In application to the ionosphere, however, there is no need for the corresponding investigation. The point is that above the influence of the terrestrial magnetic field was not taken into account, owing to which alone the medium could be regarded as isotropic. As we shall see below (see § 6), taking the influence of the magnetic field into account in application to the ionosphere substantially changes the picture. Therefore, when numerical estimates above referred to the ionosphere, this was of a conditional character and served, in fact, only for the purpose of choosing certain parameters of the medium. We shall proceed analogously also in § 5, where one more effect is discussed which ensures finiteness of the field at the point \(\varepsilon=0\).

§ 5. BEHAVIOR OF THE FIELD WHEN THE INFLUENCE OF PLASMA WAVES IS TAKEN INTO ACCOUNT

As was shown in § 4, taking absorption, as well as nonlinear effects, into account leads to the removal of the singularity, i.e., to a finite value of the field of the electromagnetic wave at the point \(\varepsilon=0\). The same result is also produced by taking into account the influence of plasma waves. The point is that the possibility of the appearance of these waves was ignored above. Meanwhile, at the point

\[ \varepsilon=1-\frac{4\pi e^2N}{m\omega^2}=0 \]

(we neglect absorption) the frequency of the wave \(\omega\) is just equal to the frequency of plasma oscillations

\[ \omega_0=\sqrt{\frac{4\pi e^2N}{m}}. \]

In this connection one may think that the characteristic behavior of the vertical component \(E_z\) in the neighborhood of the point \(\varepsilon=0\) is connected with the resonant properties of the plasma. The function representing the dependence of \(|E_z|^2\) on \(z\) is then a kind of resonance curve (see Fig. 9), which near the maximum \((\varepsilon=0;\ z=0)\) has the form

\[ |E_z|^2=\frac{\mathrm{const}}{(az)^2+\left(\dfrac{\nu_{\mathrm{eff}}}{\omega}\right)^2} \]

(see (4.24)). A detailed investigation of the question confirms what has been said.

In the case of an inhomogeneous medium, for the wave under consideration,

\[ \operatorname{div}\mathbf{E}=4\pi\rho=-E_z\frac{d\ln\varepsilon'}{dz}\ne 0 \]

(see\(^1\) § 67), as a result of which plasma oscillations arise whose amplitude increases as the resonance point \(\varepsilon=0\) is approached. These local oscillations are not, however, independent, since any change in the electron density in one part of the medium is transmitted to the neighboring part through the electron pressure; taking this into account leads to the appearance of plasma waves carrying with them a certain fraction of the energy of the standing electromagnetic wave. Ultimately, the energy associated with the plasma waves goes into heating the plasma.

Thus, in a sufficiently general formulation of the problem it is necessary to take into account the possibility of plasma waves arising, which leads to the removal of the singularity of the solution and to a finite value of the field at the resonance point. The corresponding analysis in the first approximation can be carried out using the quasihydrodynamic method discussed in §§ 2 and 3. Electron pressure is introduced into the equation of motion of the electrons. The corrections obtained in this way enter the field equations for a wave with components \(E_z, H_y, H_x\) (see\(^ {49}\)). Assuming, as before, that the properties of the medium depend only on the coordinate \(z\), and that the wave vector \(\mathbf{k}\) lies in the \(yz\)-plane, we seek a solution in the form (\(q=\sin\vartheta_0\), see Fig. 7):

\[ H_x=w(z)e^{ik_0qy};\qquad E_z=u(z)e^{ik_0qy}. \tag{5.1} \]

Then, taking electron pressure into account, for \(w(z)\) and \(u(z)\) we obtain a system of two coupled second-order equations:

\[ \left. \begin{aligned} \frac{d^2w}{dz^2} -\frac{1}{\varepsilon-\beta^2q^2}\frac{d\varepsilon}{dz}\frac{dw}{dz} +k_0^2(\varepsilon-q^2)w &= \frac{\beta^2q}{\varepsilon-\beta^2q^2}\frac{d\varepsilon}{dz}\frac{du}{dz}; \\ \beta^2\frac{d^2u}{dz^2} +k_0^2(\varepsilon-\beta^2q^2)u &= qk_0^2(\beta^2-1)w, \end{aligned} \right\} \tag{5.2} \]

where

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

(absorption is neglected), and, as in § 2,

\[ \beta=\sqrt{\frac{xT}{mc^2}}. \]

In order of magnitude, the parameter \(\beta\) is equal to the ratio of the thermal velocity of the electrons to the speed of light and is, under ordinary conditions, a very small quantity (for example, at \(T\sim 300^\circ\mathrm{K}\), \(\beta\sim 2\cdot 10^{-4}\)).

Thus, taking account of the thermal motion of the electrons leads to equations of higher order. The solutions of the system (5.2) describe normal waves of two types, which only in certain special cases, or outside the region where the value of \(\varepsilon\) is small, make it possible to represent the wave field as a superposition of electromagnetic and plasma waves. Thus, for \(q=0\) (normal incidence), the system (5.2) splits into two independent equations. The first of these equations coincides with equation (4.5), and its solutions describe electromagnetic waves. The second of equations (5.2) becomes the equation for plasma waves:

\[ \frac{d^2u}{dz^2}+k_0^2\frac{\varepsilon(z)}{\beta^2}u=0. \tag{5.3} \]

In the case of oblique incidence (\(q\ne 0\)), the separation of the field into electromagnetic and plasma waves is, strictly speaking, impossible. Therefore, if one chooses the normal solution which decays in the region of negative \(\varepsilon(z)\), then below a certain point of “interaction,” where \(\varepsilon=\beta^2q^2\), the asymptotic behavior of this solution will represent electromagnetic (incident and reflected) and plasma (reflected) waves.

It is easy to verify that if \(\varepsilon(z)\) has no singular points and \(\beta^2 \ne 0\), then the solutions of the system (5.2) in the region of interest to us will be analytic functions. The singularity of the solution at the point where \(\varepsilon(z)=0\) appears only as the small parameter \(\beta^2\), multiplying the highest derivative in the equivalent fourth-order system (5.2), tends to zero.

Thus, taking account of the thermal motion of the electrons does indeed lead to the removal of the singularities of the electromagnetic field. As already noted, this is connected with the fact that an electromagnetic wave incident on the layer, in the resonance region, excites a plasma wave, the energy of which is then converted into the energy of the thermal motion of the electrons. Such a mechanism of energy dissipation naturally leads to a finite value of the energy density in the resonance region.

If the quantity \(q=\sin \vartheta_0\) is not small, then the interaction between the electromagnetic and plasma waves is insignificant. Solving the system of equations (5.2) by the method of successive approximations, one can then show that the value of \(E_z\) at the resonance point, in order of magnitude, is equal to

\[ \left. |E_z| \right|_{z=0} \sim \frac{q\,|w(0,q)|}{\left(\dfrac{\beta}{\rho}\right)^{2/3}} . \tag{5.4} \]

This result may also be obtained by means of the following simple reasoning, to which we shall confine ourselves in the present case. Let us write the equation of motion of the electrons under the action of the component \(E_z\), taking account of collisions and of the pressure gradient (see (2.3)),

\[ -\omega^2 N m r + i\omega \nu_{\mathrm{eff}} N m r = e N E_z - \chi T \frac{\partial N}{\partial z}, \tag{5.5} \]

where, for simplicity, the motion is at once assumed to be harmonic (all quantities are proportional to \(e^{i\omega t}\)) and, evidently, \(r=v/i\omega\) is the electron displacement. If, in addition, one takes into account the continuity equation

\[ \frac{\partial N}{\partial t} + i\omega \operatorname{div} N\bar r = 0 \]

and sets

\[ \frac{\partial N}{\partial z} \simeq kn,\qquad \frac{\partial N}{\partial t}=i\omega n,\qquad \operatorname{div} N r \simeq k N r, \]

where \(n\) is a small deviation of the electron concentration \(N\) from its equilibrium value, and \(1/k\) is a certain dimension characterizing the wave field, then equation (5.5) may be written in the form

\[ -\omega^2 N m r + i\omega \nu_{\mathrm{eff}} N m r = e N E_z - \chi T k^2 N r. \tag{5.6} \]

From this equation it is clear that allowing for the pressure gradient (the term \(\chi T k^2 N r\)) is analogous to allowing for collisions (the term \(i\omega \nu_{\mathrm{eff}} m N r\)). Consequently, one may think that, in order to allow for the pressure gradient (i.e., for plasma waves) while neglecting collisions, it is necessary, in the formula obtained earlier with collisions taken into account,

\[ |E_z| \simeq \frac{q\,|w(0,q)|} {\left|az+i\dfrac{\nu_{\mathrm{eff}}}{\omega}\right|} \tag{4.24} \]

to replace \(\nu_{\mathrm{eff}}/\omega\) by

\[ \frac{\chi T}{m\omega^2} k^2. \]

As for the characteristic dimension \(1/k\), it is natural to choose for it the distance \(\Delta z\) over which the quantity \(|E_z|^2\)

decreases, say, by a factor of two. Obviously, the latter will occur if

\[ a \Delta z \sim \frac{a}{k} \sim \frac{\varkappa T}{m \omega^2} k^2 \]

(see (4.24), where the quantity \(\nu_{\mathrm{eff}}/\omega\) is replaced by \((\varkappa T/m\omega^2)k^2\)). Hence

\[ k \sim \left( \frac{m\omega^2 a}{\varkappa T} \right)^{1/3}, \]

and the value of the field \(E_z\) at the resonance point \(z=0\), in order of magnitude, will be equal to

\[ |E_z|_{z=0} \sim \frac{q\,|w(0,q)|}{\dfrac{\varkappa T k^2}{m\omega^2}} = \frac{q\,|w(0,q)|}{\left(\dfrac{\beta}{\rho}\right)^{2/3}}, \]

which completely coincides with (5.4).

Thus, the influence of plasma waves in our problem may be compared with the analogous influence of absorption if one introduces a certain equivalent collision number

\[ \frac{\nu_{\mathrm{eq}}}{\omega} = \left( \frac{\beta}{\rho} \right)^{2/3} = \left( \frac{\varkappa T}{m} \right)^{1/3} \left( \frac{a}{\omega} \right)^{2/3}. \tag{5.7} \]

For the layer \(E\) \((\beta \sim 2\cdot 10^{-4};\ a=10^{-6};\ \lambda_0=100\ \text{m})\), \(\nu_{\mathrm{eq}}\simeq 10^3\). For the layer \(F\) \((\beta \sim (2 \div 4)\cdot 10^{-4};\ \lambda_0=30\ \text{m})\), \(\nu_{\mathrm{eq}}\simeq 4\cdot 10^2\) (for \(a=10^{-7}\)) and \(\nu_{\mathrm{eq}}\simeq 2\cdot 10^3\) (for \(a=10^{-6}\)).

From these figures it is clear that taking account of the influence of plasma waves in some cases \((a\sim 10^{-6})\) could be just as significant as taking account of collisions (for \(\nu_{\mathrm{eff}}\sim 10^3\)). However, under ionospheric conditions, even if one abstracts from the influence of the Earth’s magnetic field (see § 6), absorption would nevertheless play the dominant role.

The effect discussed of the interaction of a transverse and a plasma wave is of great interest from the standpoint of the theory of sporadic radio emission from the Sun (see \(^{45}\)).

§ 6. BEHAVIOR OF THE FIELD IN MAGNETOACTIVE INHOMOGENEOUS PLASMA

An external magnetic field, generally speaking, strongly affects the propagation of waves in plasma. For a homogeneous medium this circumstance has already been emphasized repeatedly in §§ 1—3. If, however, the plasma is inhomogeneous, then the differences between the isotropic and magnetoactive cases become still more varied and pronounced. In regions where geometrical optics is applicable, wave propagation in an inhomogeneous plasma is in general analogous to what occurs in a homogeneous plasma—the situation here is the same as in the isotropic case (see § 4). Indeed, in the geometrical-optics approximation the polarization of the traveling (normal) waves at each point of the inhomogeneous medium turns out to be the same as for a homogeneous medium with parameters corresponding to that point. The amplitude of the wave is also determined by the properties of the medium at the point under consideration, and the expression for the phase

\[ \varphi_{1,2}=\frac{\omega}{c}\int n_{1,2}\,dz \]

has an obvious physical meaning. Finally, waves of different types (ordinary and extraordinary) and waves of a given type but traveling in opposite directions propagate independently of one another. Under such conditions it is especially convenient and natural to consider the propagation of waves in the medium with the aid of the graph of the functions \(n_{1,2}^2(z)\) (see Fig. 2).

The parameter \(v=\dfrac{\omega_0^2}{\omega^2}=\dfrac{4\pi e^2 N(z)}{m\omega^2}\) characterizes here the distance from the beginning of the inhomogeneous layer, and for a linear layer \((N(z)=az)\) is simply proportional to this distance.

At the beginning of the layer the incident wave splits into an ordinary (index 2) and an extraordinary (index 1) wave, which then propagate independently of one another “along the curves” \(n_2^2(v)\) and \(n_1^2(v)\). For \(u=\dfrac{\omega_H^2}{\omega^2}<1\) (for definiteness we shall have in mind precisely this case) wave 1 reaches the point

\[ v_1^{-}=1-\sqrt{u}=1-\frac{\omega_H}{\omega}, \tag{6.1} \]

where \(n_1(v)=0\), and total reflection must take place (absorption is neglected; normal incidence of the wave on the layer is considered). At the point \(n_1(v)=0\), or, more precisely, in the region near this point, geometrical optics is of course not applicable to a wave of type 1. But for a wave of type 2 the geometrical-optics approximation is applicable in this region*) and, consequently, a wave of type 2 is not reflected here. The behavior near the point \(v_1^{-}\) of a wave of type 1 proves, naturally, to be quite analogous to the behavior of a wave of any polarization in an isotropic medium near the point \(\varepsilon(z)=n^2(z)=0\) (we are speaking of normal incidence). As for a wave of type 2, for it \(n_2(v)=0\) at

\[ v=v_2=1, \tag{6.1′} \]

i.e., at the same place where the refractive index vanishes in the isotropic case, when \(\varepsilon(v)=n^2(v)=1-v\). At the point (6.1′) a wave of type 2 is reflected; a wave of type 1 in this region propagates independently of wave 2 and, moreover, cannot get there from the region \(v<1-\sqrt{u}\) (for details see \({}^{1}\), Ch. V).

In a magnetoactive medium, however, geometrical optics may prove inapplicable not only in the two cases indicated (near the point \(v_1^{-}\) for wave 1 and near the point \(v_2\) for wave 2), in which the picture is similar to that occurring in an isotropic medium. Under certain conditions the geometrical-optics approximation in a magnetoactive medium may also break down at the very beginning of the layer, where the values of the parameter \(v=\dfrac{4\pi e^2 N}{m}\) are very small. This case will be analyzed in § 9. In addition, geometrical optics is also inapplicable in a region embracing the point \(v_2=1\), simultaneously for waves of both types, if the angle \(\alpha\) between the magnetic field and the \(z\)-axis (the direction of the wave vector) is sufficiently small. As a result, for small angles \(\alpha\ne0\), the ordinary and extraordinary waves (waves of types 2 and 1) interact with one another in the sense indicated at the beginning of § 4. The presence of this interaction leads to a peculiar effect of “multiplication” of signals reflected from the ionosphere (see \({}^{53}\) and \({}^{1}\), § 79). We shall dwell on new results concerning the interaction of the ordinary and extraordinary waves in §§ 7 and 8. For the moment we turn to a discussion of the question of the influence of the magnetic field on the peculiar “swelling” of the field near the point \(\varepsilon(v)=n^2(v)=0\), which was considered in §§ 4 and 5 only as applied to an isotropic medium.

*) It is assumed that the layer is gradual (i.e., \(\varepsilon_{ik}\) vary smoothly as functions of \(z\)) and, in addition, that the reflection point is not in the immediate vicinity of the maximum of the layer.

As was clarified in § 5, the singularity of the field at the point \(\varepsilon=0\), which occurs for oblique incidence of a wave with a nonzero component \(E_z\), is connected with the resonant properties of the plasma. It is therefore no accident that at the point \(\varepsilon=0\) the frequency \(\omega\) of the incident transverse electromagnetic wave coincides with the plasma frequency \(\omega_0\). In this connection one may think that, in the presence of a magnetic field, the “swelling” of the electric field of the wave will also occur in the case of coincidence of the frequencies of the incident and plasma waves. But in a magnetic field weakly damped plasma waves can exist only near the point

\[ v_\infty=\frac{u-1}{u\cos^2\alpha-1} =\frac{\omega_H^2-\omega^2}{\omega_H^2\cos^2\alpha-\omega^2}, \]

at which the refractive index for the extraordinary wave \(n_1(v)=\infty\) (see § 1; we assume that \(u<1\)). Thus we arrive at the conclusion—which is confirmed as a result of a more detailed analysis (see \(^{54}\) and below)—that in a magnetoactive medium the field of the wave will have a singularity (or, if absorption is taken into account, a maximum) at the point \(v_\infty\). Here, for values \(u=\dfrac{\omega_H^2}{\omega^2}<1\), when \(n_1(v_\infty)=\infty\), the extraordinary wave will have the singularity, while for \(u>1\), \(u_L>1\), the ordinary wave will have the singularity (in the case \(u>1\), \(u_L>1\), as is known, \(n_2(v_\infty)=\infty\); in the region \(u>1\), \(u_L<1\) neither wave has a singularity; see § 1). This result is, of course, in agreement with everything said earlier as applied to an isotropic medium, since as the magnetic field tends to zero \((u\to 0)\), \(v_\infty\to 1\), and thus the field singularity occurs precisely at the point \(\varepsilon=1-v=0\).

The general field equations for oblique incidence are very complicated. We shall therefore restrict ourselves to the special case when the external magnetic field \(\mathbf H_0\) is directed along the \(x\)-axis, and the normal to the wave lies in the \(yz\)-plane (the properties of the medium vary only as a function of \(z\))*). Under such conditions (transverse propagation, i.e. angle \(\alpha=\dfrac{\pi}{2}\)), the field equations split into two independent equations for the ordinary and extraordinary waves. In the ordinary wave the nonzero components are \(E_x\), \(H_y\), \(H_z\), and it propagates in the same way as in the absence of a magnetic field. This is quite understandable, since the electric field of this wave is parallel to the external magnetic field \(\mathbf H_0\), and the latter does not affect the forced motion of electrons under the action of the field \(\mathbf E\). The field of this wave has no singularities.

For the extraordinary wave the nonzero components are \(H_x\), \(E_y\), and \(E_z\). Assuming in this case (the notation is the same as in § 4)

\[ H_x=w(z)e^{ik_0qy}, \]

for \(w(z)\) we have

\[ \frac{d^2w}{dz^2} -\frac{1}{\mu}\frac{d\mu}{dz}\frac{dw}{dz} +k_0^2\left(\mu-q^2-\frac{q\mu}{k_0}\frac{df}{dz}\right)w=0. \tag{6.2} \]

Further,

\[ \left. \begin{aligned} E_y&=\left[iqf(v)\,w(z)-\frac{i}{k_0\mu(v)}\frac{dw(z)}{dz}\right]e^{ik_0qy},\\ E_z&=\left[-q\frac{w(z)}{\mu(v)}+\frac{f(v)}{k_0}\frac{dw(z)}{dz}\right]e^{ik_0qy}. \end{aligned} \right\} \tag{6.3} \]

*) In order that the notation correspond more closely to that adopted in §§ 4 and 5, we direct the field \(\mathbf H_0\) along the \(x\)-axis, in contrast to the condition adopted in § 1 (in § 1 the field \(\mathbf H_0\) was assumed to lie in the \(yz\)-plane).

In expressions (6.2)—(6.3) the following notation has been introduced \(\left(v=\dfrac{\omega_0^2}{\omega^2}=\dfrac{4\pi e^2N(z)}{m\omega^2},\ u=\dfrac{\omega_H^2}{\omega^2}\right)\):

\[ \mu(v)\equiv n_1^2(v)=1-\frac{v(1-v)}{1-u-v};\qquad f(v)=\frac{\sqrt{u}\cdot v}{(1-v)^2-u}. \tag{6.4} \]

The function \(\mu(v)\) is nothing other than the square of the refractive index for the extraordinary wave in the case \(\alpha=\dfrac{\pi}{2}\) (see\({}^{1}\) § 75, or formula (1.9) for \(\alpha=\dfrac{\pi}{2}\)). Under normal incidence the zeros of the function \(\mu(v)\equiv n_1^2(v)\) are “reflection” points; their coordinates are \(v_1^{\pm}=1\pm\sqrt{u}\) (Fig. 11). Plasma waves for \(\alpha=\dfrac{\pi}{2}\), however, can exist near the point

\[ v_{\infty}=1-u=1-\frac{\omega_H^2}{\omega^2}. \tag{6.5} \]

In form, equation (6.2) differs only slightly from the corresponding equation (4.5) for the isotropic case, when \(u=0\). Therefore one might think that here one should expect the appearance of a singularity of the field at the point \(v_1^{-}=1-\sqrt{u}\), since the expressions for \(E_y\) and \(E_z\) (see (6.3)) contain factors \(1/\mu(v)\) and \(f(v)\), which become infinite at this point (we neglect absorption). However, investigation of the solutions shows that the field at this point remains finite\({}^{54}\). This conclusion is physically clear after what was said above: the singularity of the field should be expected not at the point \(v_1^{-}\), but at the point \(v_{\infty}\), where the frequency of the external field coincides with the natural frequency of plasma oscillations. Analysis of equation (6.2) also confirms this latter conclusion. It is interesting to note that singularities of the field now exist also for \(q=\sin\vartheta_0=0\), i.e. also in the case of normal incidence. This fact is explained by the fact that for the extraordinary wave, even under normal incidence, the electric vector has a longitudinal component \(E_z\), which excites plasma oscillations near the point \(v_{\infty}=1-u\). Therefore in what follows we shall restrict ourselves to considering only normal incidence.

Fig. 11.

In the case of normal incidence it is convenient first to determine the behavior of the component \(E_y\), which obeys the equation

\[ \frac{d^2E_y}{dz^2}+k_0^2\left(1-\frac{v(1-v)}{1-u-v}\right)E_y=0, \tag{6.6} \]

and to determine the remaining components from the relations

\[ E_z=i\,\frac{\sqrt{u}\cdot v}{1-u-v}\,E_y;\qquad H_x=\frac{1}{ik_0}\,\frac{\partial E_y}{\partial z}. \tag{6.7} \]

Assuming the function \(v(z)\) to be linear \(\left(\dfrac{dv}{dz}=a\right)\) and making in (6.6) the change of variable \(1-u-v=-\zeta\), we obtain

\[ \frac{d^{2}E_y}{d\zeta^{2}}+\rho^{2}\left(2u-\zeta+\frac{u(1-u)}{\zeta}\right)E_y=0;\qquad \left(\rho=\frac{k_0}{a}\right). \tag{6.8} \]

For very small \(\zeta\) (neglecting the terms \(2u\) and \(\zeta\) in comparison with \(\dfrac{u(1-u)}{\zeta}\)) we obtain an approximate solution of this equation in the form

\[ E_y=\sqrt{\zeta}\,H_1^{(1,\,2)}\!\left(2\rho\sqrt{u(1-u)\zeta}\right), \tag{6.9} \]

where \(H_1^{(1,\,2)}\) are Hankel functions of the first order. It can then immediately be seen that, as \(\zeta\to 0\), the component \(E_y\) remains finite, since

\[ H_1^{(1,\,2)}(\sqrt{\zeta})\sim \frac{1}{\sqrt{\zeta}}. \]

Consequently, the component \(E_z\), according to (6.7), tends to infinity according to the law

\[ |E_z|=\frac{\mathrm{const}}{|\zeta|}. \tag{6.10} \]

Moreover, by simple differentiation one can show that the component

\[ H_x=\frac{1}{ik_0}\frac{\partial E_y}{\partial z}\sim H_0^{(1,\,2)}\!\left(2\rho\sqrt{u(1-u)\zeta}\right) \]

has a logarithmic singularity, since \(H_0^{(1,\,2)}(\sqrt{\zeta})\sim \ln\zeta\) as \(\zeta\to 0\).

When absorption is taken into account, the field will be finite at every point. Indeed, in this case, in formulas (6.6), (6.7), in place of the function \(1-u-v\) there will stand \(1-u-v-i\dfrac{\nu_{\mathrm{eff}}}{\omega}(1+u)\) (see \({}^{1}\), § 75; it is taken into account that \((\nu_{\mathrm{eff}}/\omega)\ll 1\) and \(v\simeq v_\infty=1-u\)), and, consequently, for small \(\zeta\), instead of formula (6.10) we obtain

\[ E_z=\frac{\mathrm{const}} {\left|\zeta+i\frac{\nu_{\mathrm{eff}}}{\omega}(1+u)\right|}. \tag{6.11} \]

In the isotropic case the quantity \(E_z\) depends on the angle of incidence. Analogously to this, in the case of normal incidence of a wave on an ionized medium in the presence of a magnetic field, the component \(E_z\) depends on the magnitude of the magnetic field or, more precisely, on the quantity \(u=\dfrac{\omega_h^2}{\omega^2}\). The maximum effect will evidently occur for some small value of the parameter \(u\), when the zeros of the functions \(n_1^2(v)\) approach one another. For \(u=0\) the field singularity, of course, disappears (we are speaking of normal incidence).

As estimates show \({}^{54}\), under ionospheric conditions the effect of field growth is insignificant, since the value of the Earth’s magnetic field is relatively large and the reflection point \(v_1=1-\sqrt{u}\) for optimal frequencies lies considerably below the resonance point \(v_\infty=1-u\). Thus, allowance for the influence of the Earth’s magnetic field does not in fact permit the results of §§ 4 and 5 to be used in the case of the terrestrial ionosphere, and also, in general, leads to the insignificance of the effect of a sharp “swelling” of the field under ionospheric conditions. This

circumstance, however, in no way deprives this effect of great significance, both in principle and in application to other cases (for example, to the solar corona).

Let us note that the infinite increase of the field at the resonance point \(v_\infty\) is eliminated not only in the presence of collisions, but also when the possibility of the appearance of plasma waves is taken into account. For this purpose, as in the isotropic case (see § 5), it is necessary to take into account the thermal motion of the electrons. As a result, as was shown in § 2, the pole of the function \(n_1^2(v)\) disappears. Nevertheless, the function \(n_1^2(v)\) and the function \(n_3^2(v)\) adjacent to it still consist of two branches \(A\) and \(B\), shown in Fig. 12 (see also Figs. 3 and 5).

Fig. 12.

In a homogeneous medium the waves belonging to the branches \(A\) and \(B\) do not interact. However, when waves propagate in an inhomogeneous medium, a phenomenon will occur analogous to that which we encounter in the study of the interaction of the extraordinary and ordinary waves under conditions of quasi-longitudinal propagation (see \({}^{1}\) § 79 and below § 7). The simplest consideration of this problem in the quasihydrodynamic approximation shows that the field equations of a wave having a longitudinal component of the electric field \(E_z\), already for \(\alpha=\pi/2\), are written in the form of two coupled second-order equations. In the medium there may exist waves of type \(A\) and \(B\), which interact with one another in the region of sharp variation of the function \(n_{13}^2(v)\). When a wave \(A\) propagates, waves of type \(B\) arise in the medium: one of them, to the right of the interaction region \((v\sim 1-u)\) (see Fig. 12), travels toward the zero of the function \(n_{1B}^2(v)\), being completely reflected from it; the other propagates toward large values of the refractive index. The latter represents, in this region, a plasma wave, which in the final analysis is completely absorbed. It is quite clear from this picture that in this case there will be no singularities of the field.

Let us finally note that in a medium with slowly varying properties \(\left(\rho=\dfrac{k_0}{a}\gg 1\right)\) the field of the extraordinary wave, reflected at the level \(v_1^-=1-\sqrt{u}\), reaches the resonance point \(v=1-u\) with a very small amplitude, provided only that the parameter \(u\) is not close to zero. Under these conditions the interaction of the waves will be so insignificant that the existence of the wave \(B\) may be neglected altogether and the reflection of the extraordinary wave from the point \(v_1^-\) may be considered by ordinary methods. Appreciable interaction may be expected only for small \(u\), when the curves \(n_A^2(v)\) and \(n_B^2(v)\) approach each other (see Fig. 12*). Let us note that in the limit when \(u=0\) (there is no magnetic field), the functions \(n_A^2(v)\) and \(n_B^2(v)\) degenerate into two straight lines (see (2.6) and Fig. 4,b for \(u=0\)). In this case one may speak of a complete separation of the field into electromagnetic wav—

* The indices \(n_A\) and \(n_B\) are the refractive indices for the branches \(A\) and \(B\) shown in Fig. 12. Here, evidently, \(n_A=n_1\) for \(v<1-u\) and \(n_A=n_3\) for \(v>1-u\); similarly, \(n_B=n_3\) for \(v<1-u\) and \(n_3=n_1\) for \(v>1-u\) (the case \(\alpha=\pi/2\) is meant).

...waves with refractive index \(n_1(v)\) and plasma waves with index \(n_3^2(v)\) (waves of type 2 propagate independently also for \(u \ne 0\); recall that what is meant is the case \(\alpha=\dfrac{\pi}{2}\) and \(u \ll 1\)).

In conclusion, let us dwell—independently of the investigation of the field singularities—on the behavior of waves in the region where the refractive index becomes infinite. Investigation of the behavior of the field in a small neighborhood of the pole of \(n^2(v)\) cannot give an exhaustive answer to the question of what changes arise in an electromagnetic wave passing through a resonance region. For example, in some cases it turns out that, even in the absence of damping (collisions), the wave is partially or completely absorbed in this region.

For transverse propagation \(\left(\alpha=\dfrac{\pi}{2}\right)\), the solution of this problem is connected with the investigation of exact solutions of equation (6.8) (for a linear layer). Since obtaining the general solution of this equation is difficult, we shall confine ourselves to the investigation of exact solutions for simplified forms of the function \(n^2(v)\). For this purpose let us consider the following equations:

\[ \frac{d^2E}{d\zeta^2}+\rho^2\,\frac{b}{\zeta}\,E=0, \tag{6.12} \]

\[ \frac{d^2E}{d\zeta^2}+\rho^2\left(b_1+\frac{b_2}{\zeta}\right)E=0 \tag{6.13} \]

\[ \left(\zeta=az\ \text{is a dimensionless coordinate and}\ \rho=\frac{k_0}{a}\right), \]

where in an isotropic medium \(E\) may be understood as any of the components \(E_x\) or \(E_y\) (for an isotropic medium the longitudinal components of the field are zero). If, however, we are interested in the case of transverse propagation in a magnetoactive medium, then \(E\) must be understood as the component \(E_y\), while the remaining components are to be found from relations of the type (6.7).

Equation (6.12) describes the propagation of waves in a medium whose refractive index has no zeros and becomes infinite at the point \(\zeta=0\) (Fig. 13). This equation can be obtained, for example, from equation (6.8), by neglecting the terms \(2u\) and \(\zeta\) in comparison with \(\dfrac{u(1-u)}{\zeta}\). As the investigation\({}^{55}\) of exact solutions of equation (6.12) shows, a wave incident on such a layer from the side of positive \(\zeta\) (positive values of \(n^2(\zeta)\)) is completely absorbed in the region where \(n^2(\zeta)\) becomes infinite (complete absorption occurs even if \(\nu_{\mathrm{eff}}=0\)). It is clear that the energy density in this region increases to infinity.

Fig. 13.

Equation (6.13) describes the propagation of waves in a layer whose dielectric permittivity has one zero at the point \(\zeta=-\dfrac{b_2}{b_1}\) and becomes ...

infinity at the point \(\zeta=0\) (Fig. 14). If one chooses

\[ b_1=\sqrt{u}\,(1-u);\qquad b_2=u(1-u), \]

then the character of the singularity and one of the zeros \(\zeta=-\sqrt{u}\) of the function \(n^2(\zeta)\) in equations (6.8) and (6.13) will be the same. An investigation of the rigorous solutions for this case shows the following.^55 If the wave is incident from the side of positive values of \(\zeta\), then reflection does not occur, and its penetration into the region lying to the left of the zero \(\zeta=-\dfrac{b_2}{b_1}\) is characterized by the transmission coefficient

\[ D=e^{-\frac{\pi}{2}\frac{b_2}{\sqrt{b_1}}} =e^{-\frac{\pi}{2}u^{3/4}\sqrt{1-u}}. \tag{6.14} \]

Fig. 14.

The relative fraction of the energy that is absorbed in the resonance region is equal to

\[ A^2=1-D^2=1-e^{-\pi u^{3/4}\sqrt{1-u}}. \]

For a wave incident from the left, i.e., from the side of negative values of \(\zeta\), the reflection and transmission coefficients are equal to

\[ R=1-e^{-\pi\frac{b_2}{\sqrt{b_1}}}, \tag{6.15} \]

\[ D=e^{-\frac{\pi}{2}\frac{b_2}{\sqrt{b_1}}}. \tag{6.16} \]

The fraction of the energy that is absorbed is in this case also not equal to zero \((A^2=1-R^2-D^2>0)\), and the energy density in the neighborhood of the point \(\zeta=0\) increases without bound*).

Let us note that an analogous effect of energy absorption as \(v_{\mathrm{eff}}\to 0\) was considered earlier in the case of interaction of the ordinary and extraordinary waves at small values of \(\alpha\) (see 1 § 79).

§ 7. INTERACTION OF THE ORDINARY AND EXTRAORDINARY WAVES IN THE IONOSPHERE (THE EFFECT OF “MULTIPLICATION” OF REFLECTED RADIO SIGNALS)

As was already mentioned in § 6, in an inhomogeneous magnetoactive medium, even when the electron concentration varies smoothly with height (coordinate \(z\)), geometrical optics becomes inapplicable in a certain region for both waves 1 and 2 (the extraordinary and the ordinary) in the case of small angles \(\alpha\) between the normal to the wave and the direction of the external magnetic field. For normal incidence of a wave on a layer, as will be assumed below, the region of inapplicability of geometrical optics is located near the points \(v=1\) and \(v_\infty\), which at small angles \(\alpha\) are close to one another (recall,

*) Let us note that the problem of absorption of waves in the resonance region \(v\simeq 1-u\) for transverse propagation in a magnetoactive plasma has not yet been solved in the general case. It is evident, however, that when reflection from the second zero of the function \(n_1^2(v)\) is taken into account (see Fig. 11), the absorption increases.

that at normal incidence the angle \(a\) is the angle between the direction of the magnetic field and the \(z\)-axis). It is interesting that for \(a=0\) (longitudinal propagation) the region near the point \(v=1\) for both waves 1 and 2 is in no way distinguished, while the longitudinal plasma wave 3, which can exist in this case at \(v=1\), is completely independent of waves 1 and 2. This is also understandable, since wave 3 propagates along the magnetic field, which does not affect the motion of particles in the same direction. As for small angles \(a \ne 0\), the noted violation of the conditions of applicability of geometrical optics is already clear in this case from the corresponding curves \(n^2_{1,2}(v)\), shown in Figs. 2 and 15.

The inapplicability of geometrical optics to both waves 1 and 2 leads to the phenomenon of “interaction.” The mathematical description of this interaction consists in finding asymptotic representations of the exact solution of the wave equation, belonging to different regions separated by the region of interaction. This interaction leads to the fact that an ordinary wave incident from the region \(v<1\), in the region of interaction (\(v\simeq 1\)), is partly reflected and partly transformed into extraordinary waves propagating to the right and to the left of the point \(v=1\) (see the right branch of the function \(n_1^2(v)\) in Fig. 15). In application to the ionosphere such an interaction leads to the effect of “multiplication” of reflected radio signals, considered in \(^{53}\) and then in more detail in \(^{1}\S 79\) (works \(^{56-62}\) are also devoted to this question, as well as to related problems). Below we shall dwell only on new results in this field which were not reflected in § 79 of monograph \(^{1}\).

Fig. 15.

Fig. 16.

The entire investigation is carried out on the basis of phenomenological (macroscopic) equations as applied to a plane-stratified magnetoactive medium. In the coordinate system shown in Fig. 16, these equations take the form (all quantities are proportional to \(e^{i\omega t}\), the properties of the medium vary along the \(z\)-axis, and normal incidence is considered):

\[ \left. \begin{aligned} \frac{d^2 F_1}{dz^2} + k_0^2(C+B)F_1 &= -ik_0^2 A F_2,\\ \frac{d^2 F_2}{dz^2} + k_0^2(C-B)F_2 &= ik_0^2 A F_1, \end{aligned} \right\} \tag{7.1} \]

where

\[ F_1 = E_x + iE_y;\qquad F_2 = E_x - iE_y, \tag{7.2} \]

and the following notation is used:

\[ \left. \begin{aligned} C&=1-\frac{(1-is-\varepsilon')\left[\omega_y^2-(1-is)\varepsilon'\right]} {2(1-is)\omega_y^2-\left[(1-is)^2-\omega_z^2\right]\varepsilon'},\\[4pt] A&=\frac{-\left[(1-is)-\varepsilon'\right]\omega_y^2} {2(1-is)\omega_y^2-\left[(1-is)^2-\omega_z^2\right]\varepsilon'},\\[4pt] B&=\frac{(1-is-\varepsilon')\omega_z\varepsilon'} {2(1-is)\omega_y^2-\left[(1-is)^2-\omega_z^2\right]\varepsilon'},\\[4pt] \omega_x&=\omega_y=\frac{|e|H_{0r}}{mc\omega} =\frac{1}{\sqrt2}\frac{\omega_H}{\omega}\sin\alpha,\\[4pt] \omega_z&=\frac{\omega_H}{\omega}\cos\alpha;\qquad s=\frac{\nu_{\mathrm{eff}}}{\omega};\qquad k_0=\frac{\omega}{c}. \end{aligned} \right\} \tag{7.3} \]

(\(\alpha\) is the angle between the \(z\)-axis and the direction of the external magnetic field \(H_0\); the equality \(\omega_x=\omega_y\) is connected with the nature of the chosen coordinate system). The functions \(A(z)\), \(B(z)\), \(C(z)\) depend on the coordinate \(z\) through the parameter

\[ \varepsilon'=1-is-\frac{4\pi e^2N(z)}{m\omega^2}, \]

which, for a nonabsorbing medium \((\nu_{\mathrm{eff}}=0;\ s=0)\), is the dielectric permittivity of an ionized gas in the absence of an external magnetic field and in this case is equal to

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

Systems of two coupled equations of the second order, similar to the system (7.1), have been studied by various authors and have been written in very different forms. Their comparison is carried out in paper \(^{56}\).

We shall assume that the function \(\varepsilon(z)=1-v\), in the neighborhood of the point where \(\varepsilon(z)=0\), can be represented in the form

\[ \varepsilon=az \qquad (a=\mathrm{const}). \]

Introducing the new variable \(\zeta=az-is\) and eliminating the function \(F_2\) from system (7.1), we obtain the following fourth-order equation:

\[ \frac{d^4F_1}{d\zeta^4} -\frac{2A'}{A}\frac{d^3F_1}{d\zeta^3} +\left\{-\left(\frac{A'}{A^2}\right)'A+2\rho^2C\right\}\frac{d^2F_1}{d\zeta^2} +2A\rho^2\left(\frac{C+B}{A}\right)'\frac{dF_1}{d\zeta} +\left\{\rho^2\left(\frac{C+B}{A}\right)''A+\rho^4(C^2-B^2-A^2)\right\}F_1=0 \tag{7.4} \]

\[ \left(\rho=\frac{k_0}{a},\ \text{primes denote differentiation with respect to the variable } \zeta\right). \]

The solution of the equation obtained, in the approximation of geometrical optics, can be written in the following form \(^{58}\):

\[ F_1= \left\{ \left( \frac{c_2}{n_2^{1/2}}e^{i\rho\int n_2\,d\zeta} + \frac{d_2}{n_2^{1/2}}e^{-i\rho\int n_2\,d\zeta} \right) e^{-\frac12\int\frac{d\eta}{\sqrt{1+\eta^2}}} + \left( \frac{c_1}{n_1^{1/2}}e^{i\rho\int n_1\,d\zeta} + \frac{d_1}{n_1^{1/2}}e^{-i\rho\int n_1\,d\zeta} \right) e^{\frac12\int\frac{d\eta}{\sqrt{1+\eta^2}}} \right\} e^{-\frac12\int\frac{\eta\,d\eta}{1+\eta^2}} . \tag{7.5} \]

Here \(c_{1,2}\) and \(d_{1,2}\) are arbitrary complex constants, while the expressions

\[ n_{1,2}^2= \frac{ (1-is)(\omega_y^2-\varepsilon'^2)+(\omega_z^2+\omega_y^2)\varepsilon' \pm(1-is-\varepsilon')\omega_z \sqrt{\varepsilon'^2+\frac{\omega_y^4}{\omega_z^2}} } {2(1-is)\omega_y^2-\varepsilon'\left[(1-is)^2-\omega_z^2\right]} \tag{7.6} \]

are the complex squares of the refractive indices of the ordinary (2) and extraordinary (1) waves (considering these quantities as functions of the complex variable \(\zeta\), it is convenient also in the case of an absorbing medium to retain the notation \(n^2_{1,2}\), instead of the notation \((n-i\varkappa)^2_{1,2}\) introduced in § 1). Further,

\[ \eta=-\frac{B}{A}=-\frac{\omega_z}{\omega_y^2}\varepsilon'=-i\,\frac{\varepsilon'}{s_c}, \tag{7.7} \]

where \(s_c=\dfrac{\nu_{\mathrm{eff},c}}{\omega}\); \(\nu_{\mathrm{eff}}=\dfrac{\omega_y^2}{\omega_z}\) is the so-called critical collision frequency; see (1.10).

Approximation (7.5) is an asymptotic representation of the exact solution as \(\rho\to\infty\), and approximates it sufficiently well (for \(\rho\gg1\)) in certain regions of the complex \(\zeta\)-plane. The functions entering this solution describe traveling waves of different types; it is known that in the indicated approximation they are independent. Consequently, the description of the phenomenon of interaction of waves of different types must be based on the additional use of certain properties of the exact solution.

This interaction, as was indicated, is reflected in the existence of a connection between the various asymptotic forms of the solution, valid in different regions adjacent to the region of interaction. In our case the region of interaction is the region surrounding the points

\[ \zeta_{\pm}=\pm i s_c \quad (\eta=\pm i). \]

At these points \(n_1=n_2\). Solution (7.5), as is readily seen, breaks down in this region.

We shall take the absorption to be zero (for the effect of absorption see § 8). Then the functions \(n_1^2(\zeta)\) and \(n_2^2(\zeta)\) are determined by formulas (7.6) with \(s=0\). The qualitative behavior of these functions as functions of the parameter \(\vartheta-1=-\zeta\) is shown in Fig. 15 for the case when \(u<1\) and the angle \(\alpha\) is small. In the limit as \(\alpha\to0\) (\(s_c\to0\)) these functions reduce to two straight lines (see Fig. 2). Especially strong interaction is observed at small angles \(\alpha\) (when the points \(\zeta_{\pm}=\pm i s_c\) approach the real axis) in that region where \(n_1^2(\zeta)\) and \(n_2^2(\zeta)\) take values close to one another (see Fig. 15).

The points of “interaction” \((\zeta_{\pm}=\pm i s_c)\), as well as the zeros of the functions \(n^2_{1,2}(\zeta)\) (the points of “reflection”), are branch points of the functions \(n_1(\zeta)\) and \(n_2(\zeta)\). However, the refractive indices for the different types of waves can be represented as a single-valued function on a four-sheeted Riemann surface. On the basis of the geometrical-optics approximation, by an appropriate choice of the path of integration on the Riemann surface one can find the reflection coefficients for one or another wave. Investigations of the interaction phenomenon from this point of view are contained in works \(^{59-62}\). It was precisely along this path that in \(^{60}\) it was possible to establish that absorption substantially affects the interaction of the ordinary and extraordinary waves (see also § 8 and § 79).

The interaction phenomenon that interests us here, as has already been mentioned, reduces to the following: an ordinary wave incident on an inhomogeneous layer from the side of positive values of \(n_2^2(\zeta)\), in the interaction region, is partly reflected in the form of ordinary and extraordinary waves, and partly penetrates into the region of negative values of \(n_2^2(\zeta)\) in the form of an extraordinary wave, where the refractive index for this latter wave takes real values. Of primary interest here are the coefficients of reflection and transmission of waves through the region of interaction. Their values are easily obtained by arranging the asymptotic representations of the exact solution on both sides of the region of interaction. For this purpose

it is sufficient, obviously, to establish the relation that exists between the coefficients entering into solutions of type (7.5) and referring to different intervals of the variable \(\xi\), adjacent to the interaction region.

Such a relation can be found by using the method of phase integrals, widely applied in the study of the asymptotic form of solutions of second-order equations \(^{63-65}\). A generalization of this method to the case of fourth-order equations was carried out in work \(^{63}\), devoted to the study of a quantum-mechanical problem. In that work it was shown that the points where \(n_1=n_2\left(\xi_{\pm}=\pm i s_c\right)\) (in the terminology of our problem) possess properties similar to those that we encounter in studying the asymptotic behavior of solutions of second-order equations

\[ \frac{d^2 E}{dz^2}+k_0^2 \varepsilon(z)E=0. \]

Fig. 17.

When the zero of the function \(\varepsilon(z)\) is encircled in the complex \(z\)-plane, the so-called Stokes phenomenon is observed: on certain rays issuing from the point where \(\varepsilon=0\), the coefficients in the asymptotic expansion of the solution change discontinuously (see, for example, \(^{64,65}\)).

In exactly the same way, for solutions of equation (7.4), when the interaction region is encircled, on certain rays issuing from the points \(\xi_{\pm}=\pm i s_c\), the coefficients in solution (7.5) change discontinuously. For solutions of equation (7.4), these points and the Stokes lines issuing from them are arranged as shown in Fig. 17. The equations of these lines can be written in the form

\[ \arg\left(\pm i\int_{\pm i s_c}^{\xi}\frac{(n_1-n_2)}{2}\,d\xi'\right)=\pi+m\cdot 2\pi;\qquad m=0,1,2,\ldots \]

Studying the change in the asymptotic representation of the solutions when going around the points \(\xi_{\pm}=\pm i s_c\), one can establish the law of change of the constants entering into solution (7.5), and thus connect the approximate solutions referring to different regions separated by the interaction region. Such a calculation was carried out in work \(^{58}\), and its results, also given in \(^{57}\), reduce to the following. An ordinary wave incident on the layer produces in the interaction region the appearance of a reflected wave of the same type with reflection coefficient

\[ |R_2|=1-e^{-2\delta_0}, \tag{7.8} \]

while its leakage into the region where \(n_1^2>0\) is determined by the coefficient

\[ |D_1|=e^{-\delta_0}. \tag{7.9} \]

In formulas (7.8)—(7.9) the real quantity \(\delta_0\) is determined by the integral

\[ \delta_0=-i\rho\oint_L\frac{n_1-n_2}{4}\,d\xi = 2i\rho\int_A^B\frac{n_1-n_2}{4}\,d\xi. \tag{7.10} \]

The contour \(L\) encloses the two singular points of the integrand at which \(n_1=n_2\) (see Fig. 17). In the second integral the integration is carried out along a half-loop enclosing the point \(-i s_c\). It is assumed that the ordinary wave

falls on the side of positive \(\zeta\) (i.e. on the side of small values of
\(\upsilon=\dfrac{4\pi e^2 N}{m\omega^2}\)).

In the solution obtained, moreover, there is an extraordinary wave arising as a result of the interaction, which propagates “downward,” toward the pole of the function \(n_1^2(\zeta)\). The coefficient characterizing the “reflection” of this wave from the region of interaction turns out to be equal to

\[ |R_1|=e^{-\delta_0}\sqrt{1-e^{-2\delta_0}}. \tag{7.11} \]

As is readily verified, in accordance with the requirement of the law of conservation of energy,

\[ |R_1|^2+|R_2|^2+|D_1|^2=1. \tag{7.12} \]

It is interesting to note that, despite the asymptotic character of the solutions obtained, formulas (7.8)—(7.11) are applicable for arbitrary values of the angle \(\alpha\) (see\(^{58}\)). Indeed, for \(\delta_0\gg 1\) (\(\alpha\) not very small) the interaction is small. For this case, in\(^{53}\) and in\(^{1}\) § 79, by variational methods, the leakage coefficient \(|D_1|\) was obtained. This result coincides with formula (7.9), if in (7.10) the path of integration is contracted to the line connecting the points \(\pm i s_c\). Investigation of formulas (7.8), (7.9) shows that they do not lose their significance also in the transition to small angles \(\alpha\), i.e. under conditions of quasi-longitudinal propagation, when \(|R_2|\ll 1\), and the coefficient \(|D_1|\simeq 1\). Thus, for strong leakage (\(\delta_0\ll 1\)) formulas (7.8)—(7.9) give

\[ |R_2|\simeq 2\delta_0;\quad |D_1|\simeq 1-\delta_0, \tag{7.13} \]

and calculation of the integral (7.10) under the indicated conditions shows\(^{58}\) that these values of the coefficients coincide completely with the corresponding formulas obtained in\(^{1}\) § 79 by another method, in which the assumption of the smallness of \(|R_2|\) is used from the very beginning.

Expressions (7.13), valid for very small angles \(\alpha\), were obtained in\(^{1}\) without taking into account the possibility of reflection of the extraordinary wave. It then turned out that the values of the coefficients (7.13) do not satisfy the law of conservation of energy, but are related by the relation

\[ |R_2|=2(1-|D_1|). \]

This fact was associated in\(^{1}\) § 79 with the existence of additional losses in the resonance region (i.e. in the neighborhood of the point

\[ \upsilon_\infty=\frac{u-1}{u_L-1}=\frac{\omega_z^2+2\omega_y^2-1}{\omega_z^2-1}, \]

where

\[ \varepsilon_\infty=1-\upsilon_\infty=\frac{2\omega_y^2}{1-\omega_z^2} \]).
]

With a more detailed description, as we have seen, it is possible to isolate the extraordinary wave which propagates toward the pole of the function \(n_1^2(\zeta)\). Taking this wave into account (its amplitude can be determined from (7.11)) precisely leads to fulfillment of the law of conservation of energy (see (7.12)).

The solution describing this wave, when the pole \(\varepsilon=-\varepsilon_\infty\) is bypassed, is damped. The wave is not reflected, and the energy must, therefore, accumulate in the resonance region. Here we have, obviously, a situation analogous to that considered in\(^{55}\) and at the end of § 6. When absorption is taken into account, the energy of the field passes into heat in accordance with the interpretation indicated in\(^{1}\) § 79 (see above).

From formula (7.11) it is clear that for \(\delta_0\gg 1\) the absorption associated with the appearance of a traveling “downward” extraordinary wave is very small. For

strictly longitudinal propagation \(R_1=0\) (in this case \(\delta_0=0\)). Consequently, for some small angle \(\alpha\) the absorption will attain a maximum. From formula (7.11) one can see that the coefficient \(|R_1|^2\) has a maximum value equal to \(1/2\) at \(e^{-2\delta_0}=1/2\), and, thus, at \(2\delta_0=\ln 2\), \(1/4\) of the entire energy of the incident ordinary wave remains in the medium. This energy remaining in the medium (the wave \(I\), traveling “downward,” cannot leave the medium), as was already indicated, is converted into heat. The mechanism of such a transition is twofold. Absorption of the wave is caused primarily by the presence of collisions of the electrons with other particles. But, in addition, one must bear in mind that when the thermal motion of the electrons is taken into account the pole of the function \(n_1^2(v)\) disappears, and its branch passes directly into the plasma wave (see § 2). Only at \(\alpha=0\) (longitudinal propagation) do the curves \(n_1^2(v)\), \(n_2^2(v)\), and \(n_3^2(v)\) separate. The character of the corresponding limiting transition is clear from Fig. 4. As for the plasma wave 3, into which wave 1 passes, it is absorbed (i.e., its energy is converted into heat) not only as a result of the action of collisions, but also by means of the peculiar mechanism discussed in § 2.

Let us note that under ionospheric conditions, as can be shown, in the region of interaction the inclusion of thermal motion changes the form of the functions \(n_1^2(v)\) and \(n_2^2(v)\) only very insignificantly. Therefore the solution of the interaction problem given above without allowance for thermal motion retains its significance.

In conclusion, we emphasize that, in view of the interaction discussed, the reverse transition of plasma waves into the ordinary wave (or, for \(u>1\), \(u_L>1\), into the extraordinary wave) is of course also possible. Such an effect is of great interest from the standpoint of the theory of sporadic solar radio emission and, in general, the theory of generation of radio waves in magnetoactive plasma (see \(^{27}\)).

§ 8. INTERACTION OF WAVES WITH ABSORPTION TAKEN INTO ACCOUNT

In investigating the interaction of waves in an inhomogeneous magnetoactive medium, we did not take absorption into account above in the concrete calculations. At first glance it may seem that the introduction of small absorption should not substantially change the picture of wave interaction. However, in some cases this is not so, and allowance for absorption leads to essential changes in the interaction effect (see \(^{60,1}\) § 79). The point is that in a medium with absorption the conditions for the transition to quasiloongitudinal propagation depend on the number of collisions \(\nu_{\mathrm{eff}}\). This transition sets in at \(\nu_{\mathrm{eff}}\gg \nu_{\mathrm{eff},c}\), where \(\nu_{\mathrm{eff},c}\) is the critical number of collisions introduced in § 1. In this case it is seen from (1.10) that the value of \(\nu_{\mathrm{eff},c}\) is sufficiently small precisely at small angles \(\alpha\). Hence it is clear that, in the presence of the corresponding absorption, the appearance of a strong interaction and, consequently, of the effect of “multiplication” of signals may also be expected at latitudes not very close to the magnetic pole. The method set forth in § 7 makes it possible without difficulty to solve the problem of wave interaction also in the presence of absorption.

We shall assume that the absorption is small, i.e. \(\dfrac{\nu_{\mathrm{eff}}}{\omega}\ll 1\), and does not depend on the coordinates. Then the changes connected with the introduction of the collision frequency can be reduced merely to a change of the parameter \(\varepsilon'=\varepsilon-is\) (in the absence of absorption \(\varepsilon'=\varepsilon\)). The other changes in formulas (7.6) for the squares of the refractive indices reduce to the appearance of the quantity \(1-is\), which can still be regarded as equal to unity.

Thus, from the formal point of view, with the introduction of the variable $\zeta=\varepsilon-is=az-is$ we arrive at the very same equations that were investigated in the case $s=0$. Their only difference consists in the fact that, when absorption is taken into account, the material values of the coordinate $z$ are located on a straight line parallel to the real axis $\xi$ at a distance equal to $is$ (Fig. 18).

The most interesting point in the problem under discussion is the determination of the transmission coefficient $|D_1|$. This value is easily obtained by specifying the field in the region $n_2^2(\zeta)>0$ ($\zeta>0$) in the form of incident and reflected ordinary waves and a reflected extraordinary wave, and by seeking the solution to the left of the points $\pm is_c$ (Fig. 18). Applying the rule of traversal along a loop enclosing the point $-is_c$, one can obtain the following value for the transmission coefficient:

\[ |D_1|=e^{-\delta}, \tag{8.1} \]

where

\[ \delta=iq\int_{A'}^{B'} \frac{n_1-n_2}{2}\,d\zeta, \tag{8.2} \]

and the integral is taken over a loop enclosing the point $-is_c$ from below. Formula (8.2) for $s=0$ goes over into formula (7.10), obtained without allowance for absorption.

Fig. 18.

A very significant consequence of the results obtained is the fact that the quantity $\delta$ can, for a certain value of the absorption, become zero. Then, obviously, $|D_1|=1$, and in this case the ordinary wave incident on the layer passes entirely through the region of interaction. It is easy to see that this occurs for $s\ge s_c$. For $s<s_c$ the transmission coefficient is given by formula (8.1), in which $\delta\to 0$ as $s$ approaches $s_c$ (at the same time, in formula (8.2), the length of the integration loop is shortened). The value $|D_1|=1$, which is reached at $s=s_c$, remains unchanged also for $s>s_c$, since in the latter case the integration path passes below the point $\zeta=-is_c$ and the integral (8.2) remains equal to zero. All this is in good agreement with the known facts indicating that the transition to quasi-longitudinal propagation is determined by a critical number of collisions. As a result, for values $s>s_c$ the conditions of quasi-longitudinal propagation are fulfilled, when the ordinary wave freely passes through the region of interaction, being reflected not from the point $v_2=1$, where its refractive index is equal to zero, but from the overlying level $(v_1^+=1+\sqrt{u})$, where the function $n_1^2(v)$ vanishes. A detailed investigation of formulas (8.1) and (8.2) is contained in works $^{60}$ and $^{1}$ § 79.

In conclusion, we note that the effect of interaction of waves in an inhomogeneous magnetoactive medium was considered above only for normal incidence. Meanwhile, it is physically obvious that also for oblique incidence the interaction of waves must be preserved to one degree or another (it is sufficient to say that the interaction, of course, cannot be absent for very small, though nonzero, angles of incidence of waves on the layer). Unfortunately, the corresponding investigation of the interaction of waves for oblique incidence has not yet been carried out.*)

*) The works $^{66-68}$ apparently bear some relation to the question of the interaction of waves for oblique incidence. In these works, however, no investigation of the wave equations was performed, and the problem was not even clearly formulated.

§ 9. LIMITING POLARIZATION OF WAVES EMERGING FROM THE IONOSPHERE

In an inhomogeneous medium and, in particular, in a magnetoactive medium, geometrical optics is inapplicable also for small anisotropy. In the case of the ionosphere the latter occurs at the beginning of the layer, where the electron concentration \(N\), and consequently also the parameter \(v=\dfrac{4\pi e^{2}N}{m\omega^{2}}\), may be sufficiently small.

As \(v\to 0\) the medium approaches vacuum, where, as in any isotropic medium, a “polarization degeneracy” occurs, consisting in the fact that waves with any polarization may be chosen as normal waves. On the other hand, even for \(v\to 0\) the polarization of normal waves in a homogeneous anisotropic medium is not arbitrary, but is quite definite (see (1.14)). It follows from this that, in a weakly anisotropic but inhomogeneous medium, electromagnetic waves will not, generally speaking, be close to normal waves in a homogeneous medium. Indeed, the polarization of a wave propagating in the medium for \(v\to 0\) will obviously change very little, while the polarization of the normal (ordinary and extraordinary) waves may, even in this case, change quite appreciably*).

The situation occurring at the beginning of the layer (for \(v\to 0\)) may be characterized by saying that in this region there is an interaction of normal waves in the sense already used by us earlier. This interaction leads to the fact that an ordinary or extraordinary wave, arriving “from above” from the region of large values of \(v\), at the beginning of the layer gives rise to a wave of the other type, as a result of which the polarization of the wave field as a whole changes not as for a normal wave. The interaction here occurs mainly in the region where

\[ \left|n_{2}-n_{1}\right|\sim \frac{1}{k_{0}}\left|\frac{dn_{1,2}}{dz}\right|. \tag{9.1} \]

This condition is approximate in character and can be refined on the basis of wave theory (see condition (9.8)). With increasing \(v\), geometrical optics becomes already applicable (for the corresponding criterion see \(^{1}\) § 77).

The effect of wave interaction at the beginning of the layer was investigated in works \(^{69-73}\). Below we shall present the results of the most complete of these investigations \(^{71}\), where the following system of equations, equivalent to system (7.1), is used as the starting point:

\[ \begin{aligned} \frac{d^{2}\Pi_{1}}{dz^{2}}+\left(k_{0}^{2}n_{1}^{2}-\psi^{2}\right)\Pi_{1} &=\Pi_{2}\frac{d\psi}{dz}+2\psi\frac{d\Pi_{2}}{dz},\\ \frac{d^{2}\Pi_{2}}{dz^{2}}+\left(k_{0}^{2}n_{2}^{2}-\psi^{2}\right)\Pi_{2} &=-\Pi_{1}\frac{d\psi}{dz}-2\psi\frac{d\Pi_{1}}{dz}. \end{aligned} \tag{9.2} \]

Here \(\Pi_{2}=E_{x,2}\cdot\sqrt{1-K_{2}^{2}}\), \(\Pi_{1}=E_{x,1}\cdot\sqrt{1-K_{1}^{2}}\); \(n_{1}, n_{2}\) and \(K_{1,2}=\dfrac{E_{y,1,2}}{E_{x,1,2}}\) are the refractive indices and polarization coefficients of the ordinary and extraordinary waves (the coordinate system is chosen so that the magnetic field \(\mathbf H_{0}\) lies in the \(yz\) plane; see § 1).

*) It is enough to say that for \(v=0\) the polarization of the normal waves depends on the parameters \(u=\omega_{H}^{2}/\omega^{2}\) and \(\alpha\), whereas in vacuum \((v=0)\) no change in these parameters can, of course, affect the electromagnetic field.

The so-called coupling parameter enters equation (9.2)

\[ \psi=\frac{i}{2}\frac{d}{dz}\ln\frac{K_2-1}{K_2+1} =\frac{i}{4}\frac{d}{dz}\ln\left(\frac{\varepsilon-is+is_c}{\varepsilon-is-is_c}\right) \tag{9.3} \]

\[ \left(\varepsilon=1-v=1-\frac{4\pi e^2N}{m\omega^2}\right). \]

The parameter \(\psi\) is small in regions where the interaction between the ordinary and extraordinary waves is small. If \(\psi=0\) (a homogeneous medium), then system (9.2) splits into two independent equations: the ordinary and extraordinary waves do not interact.

It is not possible to obtain an exact solution of system (9.2). If, however, one uses the assumption that the ionospheric parameters vary slowly with height, then the indicated interaction can be described as follows.

For a medium with slowly varying properties the solution can be written in the geometrical-optics approximation. We choose solutions describing the ordinary and extraordinary waves traveling from above downward:

\[ \begin{aligned} \Pi_1&=A_1\Pi_{11}=A_1 n_1^{-1/2}\exp\left\{ik_0\int n_1\,dz\right\},\\ \Pi_2&=A_2\Pi_{21}=A_2 n_2^{-1/2}\exp\left\{ik_0\int n_2\,dz\right\}. \end{aligned} \tag{9.4} \]

If there is no coupling between the ordinary and extraordinary waves, then the coefficients \(A_1\) and \(A_2\) remain constant. When the coupling is taken into account, \(A_1\) and \(A_2\) are slowly varying functions.

Using the method of variation of constants and making certain simplifying assumptions, for \(A_1\) and \(A_2\) one can obtain the following equations \(^{71}\):

\[ \begin{aligned} k_0\frac{dA_1}{dz}&=-i\psi A_2\Pi_{11}^{*}\frac{d\Pi_{21}}{dz},\\ k_0\frac{dA_2}{dz}&=i\psi A_1\Pi_{21}^{*}\frac{d\Pi_{11}}{dz}. \end{aligned} \tag{9.5} \]

\(\Pi^{*}\) is obtained from \(\Pi\) by replacing \(i\) by \(-i\) in the exponent (see 9.4). The change of variables

\[ \begin{aligned} U_1&=A_1\exp\left\{\frac{ik_0}{2}\int_{z_0}^{z}(n_2-n_1)\,dz\right\},\\ U_2&=A_2\exp\left\{-\frac{ik_0}{2}\int_{z_0}^{z}(n_2-n_1)\,dz\right\} \end{aligned} \tag{9.6} \]

(\(z_0\) is the point where \(n_1=n_2\)) makes it possible to reduce (9.5) to one second-order equation

\[ \frac{d^2U}{dz^2} +\left\{ \psi^2+\frac{k_0^2}{4}(n_2-n_1)^2 -\frac{1}{2}ik_0\frac{d}{dz}(n_2-n_1) \right\}U=0, \tag{9.7} \]

where \(U\) denotes \(U_1\) and \(U_2\).

Investigation of the solutions of this equation shows the following. If

\[ |\psi|^2\gg \left| \frac{k_0^2}{4}(n_2-n_1)^2 -\frac{ik_0}{2}\frac{d}{dz}(n_2-n_1) \right|, \]

then a wave with arbitrary polarization will propagate in the medium without a change of polarization.

If, however, the reverse inequality is satisfied,

\[ |\psi|^2 \ll \left| \frac{k_0^2}{4}(n_2-n_1)^2 - \frac{i k_0}{2}\frac{d}{dz}(n_2-n_1) \right|, \]

then the normal waves in the geometrical-optics approximation (9.4) propagate independently of one another; the polarization of the normal waves is then easily determined on the basis of the usual formulas (see § 1).

It follows from this that the region of interaction, which determines the limiting polarization of waves after leaving the layer, is characterized by the fact that in it

\[ |\psi|^2 \simeq \left| \frac{k_0^2}{4}(n_2-n_1)^2 - \frac{i k_0}{2}\frac{d}{dz}(n_2-n_1) \right|. \tag{9.8} \]

For high frequencies (in the case of the ionosphere, for \(f > 1\) Mc/s), this condition can be written approximately in the form

\[ |\psi|^2 \simeq \frac{k_0^2}{4}|n_2-n_1|^2, \]

whereas for long waves (\(f < 500\) kc/s), in the form

\[ |\psi|^2 \simeq \frac{k_0}{2}\left|\frac{d}{dz}(n_2-n_1)\right|. \]

An investigation of the solutions of equation (9.7) for a certain model of the ionosphere

\[ \left(\frac{k_0}{2}(n_2-n_1)=Me^{az}; \quad M=\mathrm{const}; \quad a=\mathrm{const}\right) \]

shows that the limiting polarization of short waves emerging from the ionosphere is determined by the formulas

\[ K_{1,2}=\frac{K_{p1,2}(F-G)-(F+G)}{(F-G)-K_{p1,2}(F+G)}, \tag{9.9} \]

where

\[ \frac{F}{G}=-2^{-2r}r^{2r}e^{i\pi r}\frac{\Gamma(1-r)}{\Gamma(1+r)};\qquad r=i\frac{\psi}{a}, \]

\(\Gamma\) is the gamma function, and \(K_{p1,2}\) are the polarization coefficients, computed by the usual formulas into which one must substitute the value of the electron concentration at the level where

\[ \psi^2=\frac{k_0^2}{4}(n_2-n_1)^2. \]

As the investigation of the formulas obtained shows, for high frequencies (\(f>1\) Mc/s), the quantity \(F/G\simeq -1\), and, thus, we obtain

\[ K_{1,2}\simeq K_{p1,2}. \]

Since in the region of interaction \(\varepsilon=1-v\simeq 1\) and \(s=\dfrac{\omega_{\mathrm{eff}}}{\omega}\ll 1\), the limiting polarization is determined only by the Earth’s magnetic field and, consequently, its experimental determination cannot serve as a source of new information about the ionosphere.

In conclusion, we would like once again to emphasize that, within the scope of the present article, it has proved possible to dwell on only a part of the questions relating to the theory of wave propagation in plasma. At the same time, the problems considered above are interconnected and, taken together, form a distinct and at the same time very important and interesting field of investigation of the properties of plasma. Therefore we hope that the appearance of the present article will prove justified.

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