Electromagnetic Waves in Hollow Conductors and Dielectric Rods

N. N. Malov, Moscow

In recent years radio engineering has achieved considerable success in obtaining very short (decimeter) undamped electromagnetic waves at a substantial level of generated power (tens of watts and more). In addition to a whole series of technical applications (convenient directional radio communication, high-quality television, the use of radio signals for the automation of blind flights, etc.), decimeter and centimeter waves are also of considerable interest for physics; in particular, they are widely used to study the electrical properties of molecules. It is enough to point out that the dispersion regions of water and of a number of other substances lie precisely in the centimeter and decimeter waves.

However, the very first experiments with short decimeter waves revealed a number of difficulties, comparatively little manifested at longer waves, but sharply affecting the operation of devices generating and transmitting energy at waves whose length does not exceed several decimeters.

The point is that one of the most important characteristics of oscillatory systems—the quality factor of a circuit with inductance \(L\), resistance \(R\), and natural period \(T\), equal to:

\[ Q = \frac{\pi}{\delta} = \pi \frac{2L}{RT} = \pi \frac{LJ_{\text{eff}}^{2}}{RJ_{\text{eff}}^{2}\frac{T}{2}} = \pi \frac{A_m}{A_Q}, \]

is inversely proportional to the decrement \(\delta\) and determines the ratio of the oscillatory energy \(A_m\) to the energy expended on Joule heat and radiation \(A_Q\); it falls rapidly as the wavelength decreases, which is explained chiefly by the considerable increase in the radiation resistance \(R_{\text{rad}} = \frac{P_{\text{rad}}}{J^2}\), which determines the ratio of the radiated power to the square of the current, and also by the decrease in the inductance of the circuit. The difficulties associated with the self-excitation of an oscillatory system possessing a low quality factor are considered in the article by Studenkov1, printed below; as for systems that transmit energy (for example, from a generator to an antenna), the decrease in the quality factor entails a reduction of the efficiency of these systems to practically unacceptable values.

As is known, the transmission of energy from a generator to an antenna at wavelengths measured in tens of meters is often carried out by means of a two-wire line (the Lecher system). If this system is made in the form of two parallel wires, then the radiation resistance is a function of the ratio $\dfrac{d}{\lambda}$ ($d$ is the distance between the axes of the wires, $\lambda$ is the wavelength), and for a system whose length is equal to a half-wave it is characterized by the following table:

Since the distance $d$ cannot be made too small, at very short wavelengths the use of such lines is practically impossible.

The use of shielded two-wire lines or coaxial lines, consisting of an outer hollow tube and a coaxial conductor, of course eliminates radiation losses; however, in these structures it is necessary to fasten the inner conductors on suitable insulators, which create local inhomogeneities that have a harmful effect at very short wavelengths.

All these difficulties disappear if the electromagnetic wave propagates inside a hollow metallic conductor filled with a good dielectric, for example, air. Apparently, it was these considerations that prompted American investigators to study the question of the propagation of electromagnetic waves in hollow conductors, and also in dielectric rods devoid of a metallic sheath.

It should be noted that the theoretical formulation of this problem belongs to Rayleigh$^2$; later these questions were studied by Hondros$^3$ and by Debye and Hondros$^{17}$; however, a detailed experimental study of the problem, requiring the presence of undamped oscillations, was carried out only in recent years by Southworth$^4$, Barrow$^5$, and Clavier$^{13}$.

a) Waves in hollow conductors

In the theoretical consideration of the question of the possibility of using hollow conductors for channeling electromagnetic energy, the problem is posed as follows: there is an ideal hollow conductor ($\gamma=\infty$) of constant cross section (a tube), whose axis coincides with the $X$ axis. The conductor is filled with an ideal dielectric (in what follows it is assumed that $\varepsilon=1$, $\mu=1$, $\gamma=0$); it is required to find a solution of Maxwell’s equations satisfying the boundary conditions and determining the distribution of an electromagnetic field capable of propagating along the axis of the tube without distortion. The length of the tube is assumed to be infinitely large. The desired solution may be represented in the form

\[ E' = E e^{-j\beta x + j\omega t}; \qquad H' = H e^{-j\beta x + j\omega t}, \tag{1} \]

where \(\beta\) is the propagation constant, \(\omega\) is the angular frequency of the field, \(j=\sqrt{-1}\), and \(E\) and \(H\) do not depend on the coordinate along the axis of the tube \(X\) or on the time \(t\).

In the case of a tube of circular cross section (the radius of the tube is \(a\)) it is expedient to introduce cylindrical coordinates \((\rho,\varphi,x)\). In this case the quantities \(E\) and \(H\) are functions of \(\rho\) and \(\varphi\). As calculation shows, the general solution of Maxwell’s equations splits into two particular cases.

In the first case

\[ H_{\rho}=H_{x}=E_{\varphi}=\frac{\partial H_{\varphi}}{\partial \varphi}=0;\quad H_{\varphi},\,E_{\rho},\,E_{x}\ne 0, \]

i.e., the magnetic field lies in the plane of the tube cross section and has circular symmetry, while the electric field has components along the radius and along the axis of the tube; this solution is called a wave of type \(E\) (electric).

In the second case the roles of the vectors are interchanged, so that the field \(E\) lies entirely in the plane of the tube cross section and has circular symmetry, while the field \(H\) has radial and axial components:

\[ H_{\varphi}=E_{x}=E_{\rho}=\frac{\partial E_{\varphi}}{\partial \varphi}=0;\quad H_{x},\,H_{\rho},\,E_{\varphi}\ne 0. \]

This solution is called a magnetic wave (\(H\)-wave). An essential feature of the solutions obtained is the dependence of the propagation constant \(\beta\) on the dimensions of the tube and on the frequency of the field variations, and also the existence of a “critical” frequency (depending on the radius), below which propagation proves impossible.

To emphasize this fundamental difference between waves in a tube and waves propagating in an unbounded medium (free waves), Americans have proposed for waves in tubes the term “guided waves,” which may be translated as “directed waves” (but not “oriented,” since this term has another, quite definite meaning in radio engineering).

The solution of Maxwell’s equations is, of course, obtained in this case in cylindrical functions. From the mathematical standpoint this problem is analogous to the problem of the distribution of amplitudes on the surface of a certain membrane.

We shall confine ourselves to consideration of the simplest solution. For an \(E\)-wave one obtains

\[ E_x=A J_0\left(\rho\sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}\right), \]

where \(J_0\) is a Bessel function, \(c\) is the velocity of light, and \(A\) is a constant. Taking into account that on the inner surface of the tube the field \(E_x\) must vanish, we obtain:

\[ J_0\left(a\sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}\right)=0. \tag{2} \]

Equation (2) has a series of roots, the first (smallest) being determined by the condition:

\[ a \sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}=2.405. \]

Hence, for the propagation constant one obtains

\[ \beta=\sqrt{\left(\frac{\omega}{c}\right)^2-\left(\frac{2.405}{a}\right)^2}. \]

Only real values of \(\beta\) are of physical interest, since otherwise the result is a process attenuating along the axis of the tube. Therefore the possible frequencies must not be too small; in other words, there exists a certain critical frequency

\[ \omega_{cr}=\frac{2.405}{a}c, \]

and lower frequencies are not capable of propagating through the tube. The values of the fields \(E_{\rho}\) and \(H_{\varphi}\) are determined by the equations:

\[ E_{\rho}=-jA\frac{\beta}{\sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}}\, J_0'\left(\rho\sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}\right); \]

\[ H_{\varphi}=-jA\frac{1}{\sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}}\, J_0'\left(\rho\sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}\right). \]

The phase velocity of propagation of the wave along the axis of the tube is determined by the expression:

\[ v_{ph.z}=\frac{\omega}{\beta} =\frac{c}{\sqrt{1-\left(\frac{2.405}{a}\frac{c}{\omega}\right)^2}} =\frac{c}{\sqrt{1-\left(\frac{\lambda_0}{\lambda_{cr}}\right)^2}} =\frac{c}{\sqrt{1-\Lambda^2}}>c. \tag{3} \]

The group velocity is found to be

\[ v_{gr}=\frac{c^2}{v_{ph.z}}=c\sqrt{1-\Lambda^2}<c, \tag{4} \]

as should be expected from general physical considerations; here

\[ \Lambda=\frac{\lambda_0}{\lambda_{cr}}\leqslant 1, \]

where \(\lambda_0\) and \(\lambda_{cr}\) are the wavelengths of the transmitted and critical (maximum) waves in free space.

The wave considered above, called the \(E_0\)-wave, is characterized by the field distribution shown graphically in Fig. 1, \(a\). The solid lines represent the distribution of the electric field, and the dashed lines that of the magnetic field.

Comparing this distribution with the field distribution in an ideal coaxial feeder (Fig. 2, \(a\)), we see that the patterns of the field distribution are very close to one another, but the conduction currents in the central conductor of the feeder are replaced in the hollow tube by axial displacement currents.

A similar solution is also obtained for the \(H_0\)-wave; in this case the field \(E_\varphi\), proportional to \(J'_0\left(\rho \sqrt{\left(\dfrac{\omega}{c}\right)^2-\beta^2}\right)\), must vanish at the walls of the tube. In Fig. 1, \(c\) shows the graphical pattern of the field (\(H_0\)-wave).

Fig. 1

Using Bessel functions of higher orders, one can obtain other possible solutions as well. Of these, the \(E_1\) wave is of interest; it satisfies the condition

\[ J_1\left(a \sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}\right)=0, \]

and its field distribution (Fig. 1, \(b\)) very closely resembles the field in a shielded two-wire line (Fig. 2, \(b\)). Here again the axial displacement currents in the tube play the role of the conduction currents in the two-wire line. Finally, the \(H_1\) wave, having the boundary condition (vanishing of the field \(E_\varphi\) at the walls)

\[ J'_1\left(a \sqrt{\left(\frac{\omega}{c}\right)^2-\beta^2}\right)=0, \]

is characterized by the field shown in Fig. 1, \(d\). For the waves \(H_0\) and \(H_1\), there apparently are no analogues in ordinary radio-engineering devices.

The limiting (maximum) wavelengths (free) capable of propagating in tubes of radius \(a\) are determined by the following table:

An exact account of the influence of the finite (though considerable) conductivity of the tube walls presents enormous difficulties. In an approximate solution of this problem it is assumed that the field configuration considered above is not substantially distorted. From the formal point of view this means that the solutions of Maxwell’s equations (1) should be written in the form

Fig. 2

\[ E' = E e^{-\alpha x + j(\omega t-\beta x)};\qquad H' = H e^{-\alpha x + j(\omega t-\beta x)}, \]

where \(\alpha\) is the absorption coefficient. However, owing to the smallness of the absorption,

\[ \alpha \ll \beta, \]

so that in the first approximation one may take

\[ \frac{\partial}{\partial x}=-\alpha-j\beta \approx -j\beta, \]

whereby the vectors \(E\) and \(H\) retain their previous values.

The flow of energy through some cross-section \(S\) is determined by the Poynting vector, and its mean value over a period is equal to

\[ A_m=\text{real part}\left[ \frac{1}{4\pi}\int_S (E_\rho H_\varphi^{*}-E_\varphi H_\rho^{*})\,ds \right], \]

where the asterisk denotes the conjugate value. The mean value of the flux is a function of the coordinate \(x\) and contains the factor \(e^{-2\alpha x}\). Therefore the decrease in energy is

\[ -\frac{\partial A_m}{\partial x}=+2\alpha A_m=Q, \]

where \(Q\) is the Joule heat released in the walls of the tube; it can be calculated using the theory of the skin effect. Knowing \(Q\) and \(A_m\), one obtains the absorption coefficient from the condition

\[ \alpha=\frac{Q}{2A_m}. \]

The calculation shows that the absorption is very small, so that the quality factor of a circuit made of a tubular conductor reaches values of the order of \(10^4\) and greater, i.e., exceeds by hundreds of times the quality factor of two-wire lines and other resonant circuits.

A detailed analysis of the question of the magnitude of the attenuation and its dependence on frequency is given in the survey theoretical work of Klave\(^{6}\).

In tubes of other cross sections (elliptical\(^{7}\), rectangular\(^{10}\), etc.) the propagation conditions, in their physical content, do not differ from those considered above. Therefore we shall confine ourselves merely to pointing out that in a tube of rectangular cross section (dimension \(a\) along the \(Y\) axis, \(b\) along the \(Z\) axis), in addition to other types of waves, there can exist an \(H\)-wave determined by the vectors:

Fig. 3

\[ H_x=A\sin\frac{\pi}{b}z;\qquad H_z=-jA\beta\left(\frac{b}{\pi}\right)\cos\left(\frac{\pi}{b}\right)z; \]

\[ E_y=-jA\frac{\omega}{c}\frac{b}{\pi}\cos\frac{\pi}{b}z; \tag{5} \]

in this case the propagation constant is

\[ \beta=\sqrt{\left(\frac{\omega}{c}\right)^2-\left(\frac{\pi}{b}\right)^2} =\frac{\omega}{c}\sqrt{1-\Lambda^2} \tag{6} \]

and the limiting wave is

\[ \lambda_{\mathrm{cr}}=2b; \]

the phase and group velocities are still determined by equations (3) and (4).

The field distribution is shown graphically in Fig. 3; the solid lines represent the electric field, the dashed lines the magnetic field. We shall use this case in what follows.

The existence of a phase velocity of propagation undoubtedly indicates the presence of an interference process in the tube. Brillouin[^8] was one of the first to note this circumstance, giving an elegant physical interpretation of it.

Let us imagine a plane wave (Fig. 4) incident from the left on an ideal mirror. The reflected wave will propagate to the right. In the region \(ABC\) the two waves interfere; moreover, as is known, nodal planes parallel to the mirror are formed (they are shown by dashed lines in Fig. 4). Any of the nodal planes may, without changing the conditions of propagation, be replaced by an ideal mirror. Then, in the space between the two mirrors, multiple reflection of the wave will result; the energy, however, will propagate parallel to the mirror (Fig. 5), and the velocity of its propagation will, of course, differ from the velocity of propagation of the wave.

Fig. 4

Fig. 5

If one considers the reflection of a plane wave from two mutually perpendicular mirrors, two systems of mutually perpendicular nodal surfaces are obtained. Replacing two of these surfaces by two additional mutually perpendicular mirrors, we obtain a space bounded by four mirror walls, i.e. a tube of rectangular cross section.

These considerations of Brillouin were developed by Pledge and Adams[^9] and by Barrow and Chu[^10]. For an \(H\)-wave in a rectangular tube [equation (5)] one may write [taking (6) into account]

\[ \left. \begin{aligned} H'_x &=-j\frac{A}{2}\left\{ e^{j\left[\omega t+\left(\frac{\pi}{b}z-\beta x\right)\right]} - e^{j\left[\omega t-\left(\frac{\pi}{b}z+\beta x\right)\right]} \right\} \\ &=-j\frac{A}{2}(I-II), \\[1em] H'_z &=-j\frac{A}{2}\frac{b}{\pi}\beta\left\{ e^{j\left[\omega t+\left(\frac{\pi}{b}z-\beta x\right)\right]} + e^{j\left[\omega t-\left(\frac{\pi}{b}z+\beta x\right)\right]} \right\} \\ &=-j\frac{A}{2}\frac{\sqrt{1-\Lambda^{2}}}{\Lambda}(I+II), \\[1em] E'_y &=-j\frac{A}{2}\frac{\omega}{c}\frac{b}{\pi}\left\{ e^{j\left[\omega t+\left(\frac{\pi}{b}z-\beta x\right)\right]} + e^{j\left[\omega t-\left(\frac{\pi}{b}z+\beta x\right)\right]} \right\} \\ &=-j\frac{A}{2}\frac{1}{\Lambda}(I+II), \end{aligned} \right\} \tag{7} \]

where \(\lambda_0\) is the length of the “free” wave corresponding to the frequency \(\omega\), and \(\lambda_{cr}\) is the critical wavelength. Equations (7) can be interpreted as two waves propagating in different directions in the \((XZ)\) plane. Turning to the first, let us represent the exponent in the form:

\[ \omega t+\left(\frac{\pi}{b}z-\beta x\right)=\omega\left(t-\frac{R}{c}\right), \]

where the direction of propagation is characterized by the direction cosines:

\[ \cos(R,X)=\mu=\beta\frac{c}{\omega}=\sqrt{1-\Lambda^2};\qquad \cos(R,Z)=\nu=-\frac{\pi}{b}\frac{c}{\omega}=-\Lambda. \]

The vector of the electric field \(E\) is normal to the direction of propagation.

The vector of the magnetic field is characterized by the amplitude

\[ H=-j\frac{A}{2}\sqrt{1+\frac{1-\Lambda^2}{\Lambda^2}} =-j\frac{A}{2}\frac{1}{\Lambda}=E \]

and by the direction cosines:

\[ \cos(H,X)=\xi=\Lambda;\qquad \cos(H,Z)=\eta=\sqrt{1-\Lambda^2}. \]

It is easy to see that

\[ \mu\xi+\nu\eta=0,\quad \text{i.e.}\quad E\perp H\perp R. \]

An analogous consideration is easy to carry out for wave II as well.

Thus, each of waves (I) and (II) is an ordinary plane wave. These waves meet the walls of the tube, perpendicular to the plane of propagation, at an angle \(\theta\), determined by the obvious condition

\[ \operatorname{tg}\theta=\frac{\Lambda}{\sqrt{1-\Lambda^2}}. \tag{8} \]

The character of propagation of one of the elementary waves is shown in Fig. 6. Equation (8) makes it possible to reveal the physical meaning of the existence of a critical wave. Short waves \((\lambda_0\ll\lambda_{cr},\ \Lambda\ll1)\) form small angles with the walls, i.e. they propagate almost along the axis of the tube; as \(\Lambda\) increases, the angle \(\theta\) increases; at \(\Lambda=1\), i.e. for the critical wave, \(\theta\) becomes \(90^\circ\); such a wave (if it arises) cannot propagate along the tube.

Fig. 6

These same considerations make it possible to explain\(^{11}\) the values of the phase and group velocities determined by equations (3) and (4).

Indeed, let us consider on the axis of the tube two points through which a ray passes after being reflected from the wall. To the distance \(2\frac{l}{2}\), traversed by the elementary wave with velocity \(c\), there corres—

the distance \(l_1\) by which the energy is displaced along the axis (with velocity \(v_{2p}\)). We have

\[ \frac{v_{2p}}{c}=\frac{l_1}{l}=\cos\theta=\sqrt{1-\Lambda^2}, \]

which coincides with (4).

For a visual interpretation of the phase velocity, let us take into account that, to the path \(2\frac{l}{2}\) traversed by an elementary wave, there corresponds along the tube axis the path \(l_1\), and reflection from the wall, accompanied by a jump of phase by \(\pi\), i.e., by an apparent increase of the wave path in the tube by \(\frac{\lambda}{2}\) (where \(\lambda=\lambda_0\frac{v_{\mathrm{фаз}}}{c}\) is the wavelength in the tube). Therefore one may write:

\[ \frac{v_{\mathrm{фаз}}}{c} = \frac{l_1}{l} + \frac{\lambda}{2}\frac{1}{l} = \frac{l_1}{l} + \frac{\lambda_0}{2a}\frac{v_{\mathrm{фаз}}}{c}\frac{a}{l} = \cos\theta + \frac{v_{\mathrm{фаз}}}{c}\Lambda\sin\theta = \]

\[ = \cos\theta+\Lambda^2\frac{v_{\mathrm{фаз}}}{c}. \]

Consequently,

\[ \frac{v_{\mathrm{фаз}}}{c} = \frac{\cos\theta}{1-\Lambda^2} = \frac{1}{\sqrt{1-\Lambda^2}}, \]

which coincides with equation (3).

In the practical use of hollow conductors, the question of the wave resistance of the conductor is of considerable interest. Defining the wave resistance as the ratio of the transverse potential difference (from the axis to the tube wall) to the longitudinal current, Barrow showed that the wave resistance is a function of frequency. For an \(E_0\)-wave in a circular tube the wave resistance is equal to

\[ W=48(1-\Lambda^2)\ \text{ohms}. \]

It varies from zero (at the critical frequency \(\Lambda=1\)) to 48 ohms as the frequency increases without bound (\(\Lambda=0\)). The wave resistance of resonant lines usually exceeds one hundred ohms; therefore, in those cases where, for matching with the optimum operating regime of a radio device, lines with a small wave resistance are required, hollow conductors may be used to great advantage.

The question of exciting one or another type of wave in tubes is, up to the present time, solved purely experimentally. One attempts to create at the beginning of the tube a field configuration more or less close to the theoretical one. Southworth\({}^{12}\) uses for this purpose electrodes of various shapes, placed on the end face of the tube; Barrow\({}^{5}\) introduces into the tube an axial conductor (rod) like that shown in Fig. 7. Rod 1 serves to obtain an \(E_0\)-wave, the mechanism of whose formation is explained by the schematic Fig. 8, where the distribution of the electric field in the tube is shown for four successive instants of time. To save space, Fig. 8 shows the field in one half of the tube (between the rod and the wall), while

symmetric pattern in the lower half of the tube is omitted. Rods 2 and 3 (Fig. 7), placed in the cross-section of the tube, serve to obtain \(H\)-waves. As experiment has shown, under these conditions a comparatively pure wave is obtained at a distance of several half-waves from the place of excitation.

Fig. 7

Klav’e and Altovskii\(^{13}\), in a detailed experimental study, investigated the conditions for exciting various types of waves. They showed, in a number of elegant experiments, that by placing in the tube “volume filters,” representing a system of conductors of suitable configuration, it is possible to isolate a very pure type of wave, and also to transform one type of wave into another. Since the propagation velocities of waves of different types are, generally speaking, different, by placing in the tube receivers oriented in such a way that they respond to one or another type of wave, and by moving a reflecting piston along the tube, they were able, by taking resonance curves, to detect various types of waves and directly prove the absorption of some of them by suitable filters.

Fig. 8

1—tube wall, 2—rod, 3—wave in a two-wire line, 4—transition regime, 5—wave in the tube

Analogous investigations, which confirmed the theoretically predicted dependence of the propagation velocity on frequency and the existence of a critical frequency below which transmission is altogether impossible, were carried out by Barrow and Southworth\(^{12}\).

The practically important question of the rational design of an output device ensuring the maximum output of energy from the tube into the external space has been intensively developed in recent times\(^{5,14}\).

The correct choice of the output device plays an enormous role. Thus, when an \(H\)-wave (Fig. 3) was excited in a tube of rectangular cross-section, the author carried out the following easily realizable, instructive experiment. If the output end of the tube is closed by a metallic piston in which a narrow slot is made parallel to the electric vector of the wave, then practically no output of energy from the slot is observed. If, however, the same slot is placed perpendicular to the electric vector, then a very considerable amount of energy is radiated from the tube.

Fig. 9

To reduce the reflection coefficient at the end of the tube, output devices resembling acoustic horns have to be used. Some of the designs proposed by Barrow\(^{5}\) are shown in Fig. 9. The two horns \(A\) and \(B\) serve for the radiation of \(E\)-waves; horn \(C\), for the radiation of an \(H\)-wave (from a circular tube); the last device \(D\) is intended for a coaxial two-wire ...

line. With proper selection of the horn, it is possible to considerably improve the energy output of the generator and to create a more or less directional radiated beam. Similar devices are also used for receiving energy by means of hollow conductors.

Fig. 10

Since some types of waves possess stability, i.e., can propagate without noticeable distortion even in a slightly bent tube (this question was analyzed by Brillouin[^15]), a hollow tube can be used as a feeder and radiating antenna, for example, when it is necessary to raise the radiating device to a considerable height (television, etc.). An approximate construction of such an antenna, fed from below, is shown in Fig. 10, borrowed from Barrow.

Neiman[^16] pointed out the possibility of using a hollow tube of special construction as a resonance line for meter waves; moreover, the dimensions and efficiency of such a line prove better than those of an ordinary coaxial line.

b) Waves in dielectric rods

Of no less interest is the study of the conditions of wave propagation in a rod made of a dielectric having a dielectric constant considerably greater than that of the surrounding medium (for example, in a column of water or in a rubber cord).

Debye and Hondros[^17] showed that under certain conditions waves may arise whose electromagnetic field does not extend to infinity, but is localized (practically) in a comparatively thin layer of the surrounding medium. Fig. 11 schematically represents such a rod in which an $E$-wave propagates. The distribution of the electric field is shown by curves, while the magnetic-field lines are represented by circles (the section of the magnetic-field lines by the plane of the drawing). It is especially interesting that, when the ratio of the wavelength $\lambda$ to the radius of the rod $r$ decreases, the share of the energy propagating in the surrounding medium falls sharply. The relative field intensity in the rod ($\varepsilon = 81$) and in the surrounding medium ($\varepsilon = 1$) as a function of the value of the ratio

Fig. 11

$\dfrac{\lambda}{r}$ is shown in Fig. 12. The solid curves correspond to the electric field, the dashed curves to the magnetic field. Along the abscissa axis is plotted the distance from the axis of the rod, expressed in fractions of the radius of the rod. Already at $\dfrac{\lambda}{r} \approx 10$ almost all the energy proves to be concentrated inside the dielectric rod (the letter $n$ in Fig. 12 denotes the index of refraction: $n^2 = \varepsilon$).

The physical meaning of this interesting result is as follows: a complex wave propagating in a rod may be represented as the result of the superposition of elementary waves propagating at a certain angle to the axis of the rod. These waves meet the boundary of the rod at a certain angle; under suitable conditions this angle may prove to be greater than the limiting one, as a result of which total internal reflection will occur, and practically all the energy of the wave will be concentrated inside the rod.

Experimental verification of the conditions for wave propagation in a water column enclosed in a thin Bakelite tube (with thin walls and a small value of the dielectric constant of Bakelite, its influence plays no appreciable role), carried out by Southworth^12, gave results in agreement with the theory.

Fig. 12

As regards the possibility of practical use of dielectric rods, it is necessary to point out the difficulties associated with finding a material having a high dielectric constant and such small dielectric losses that it would be capable of competing with metallic tubes, where the losses are very insignificant. Detailed literature data on this question are lacking; however, it should be noted that a number of patents have already been filed for transmitting devices of this kind^18.

The use of hollow conductors and dielectric rods as oscillatory systems and as energy-channeling devices in the radio engineering of decimeter and centimeter waves is still in the initial stage of its development. Nevertheless, there is every reason to expect that these systems will find broad application, as a result of which not only will the external design of radio devices change, but, probably, prospects will open up before the radio engineering of the near future of which it was impossible even to think when using the former apparatus. As for the physics of these processes, here too much interesting work remains to be done, since, first, a whole series of physical questions has still been insufficiently clarified (for example, the conditions for the most advantageous excitation of waves and withdrawal of energy from tubes, the influence of dielectric

losses, waves in tubes and rods of variable cross-section2, etc.), and secondly, the use of hollow resonant devices possessing very small decrements will probably make it possible to develop new methods for investigating the electrical properties of matter, based on the study of changes in the decrement of a certain hollow conductor when the substance under investigation is introduced into it. Since the existing methods for measuring dielectric permittivity and electrical conductivity at very high frequencies are not especially reliable, the development of new measuring methods will be a major achievement.

In addition, hollow resonators with a small decrement may prove very useful in the development of measuring instruments (in particular, wavemeters with exceptionally sharp tuning).

References

1. Studenkov, Uspekhi fizich. nauk, 23, 417, 1940.
2. Rayleigh, Phil. Mag., 43, 125, 1897.
3. Hondros, Ann. d. Phys., 30, 905, 1909.
4. Sauthworth, Bell System Techn. Journ., 15, 284, 1936.
5. Barrow, PJRE, 24, 1298, 1936.
6. Clavier, Electr. Communic., 17, 276, 1939; Bull. Soc. Franç. d’Electr., 5 (VIII), 355, 1938.
7. Chu, J. Appl. Physics, 9, 583, 1938.
8. Brillouin, Rev. Gen. d’Électricité, 40, 227, 1936.
9. Page and Adams, Phys. Rev., 52, 647, 1937.
10. Barrow and Chu, PJRE, 26, 1520, 1938.
11. N. Malov, J. Phys. URSS (in press).
12. Sauthworth, PJRE, 25, 807, 1937.
13. Clavier and Altovsky, Electr. Communic., 18, 81, 1939.
14. Barrow and Green, PJRE, 26, 1498, 1938.
15. Brillouin, Electr. Communic., 16, 350, 1938.
16. Neiman, Izvestiya elektrom. slab. toka, No. 2, 1940.
17. Debye and Hondros, Ann. d. Phys., 32, 465, 1910.
18. British Patent, No. 420447, 462804; U.S.A. Patent, No. 208749.

2. In the visible text this footnote marker refers to reference 11. ↩