Microradio Waves*

E. U. Condon

Introduction

In the electromagnetic theory of radiation, microradio waves are customarily understood to mean electromagnetic waves lying, roughly speaking, in the wavelength interval from \(1\ \mathrm{m}\) to \(1\ \mathrm{mm}\), which corresponds to the frequency interval from \(3\cdot 10^8\) to \(3\cdot 10^{11}\ \mathrm{cycles\ sec^{-1}}\).

On the high-frequency side this interval is limited by the fact that the technique for obtaining still higher frequencies is more “optical” than “electrical.”

On the low-frequency side the limitation is connected with the fact that, for frequencies below \(300\ \mathrm{megacycles\ sec^{-1}}\), the ordinary methods of calculating circuits with lumped constants, used in radio engineering, become entirely adequate.

The field of microradio waves is characterized by the following three basic features.

1. The methods for obtaining microradio waves are electrical, not optical. In particular, the source for obtaining microradio waves is forced oscillations in macroscopic electromagnetic systems, and not the incoherent radiation of a large number of atoms or molecules.

2. The dimensions of the devices used to obtain microwaves are usually large and, in any case, comparable with the wavelength. This circumstance makes it impossible, or at least greatly complicates, the use, for the study of microradio waves, of the ordinary methods applied to oscillatory circuits with lumped, or even distributed, constants. In any case, as yet there are no methods that would make it possible to get by without applying the methods of field theory in calculations in this region, as is done in calculating alternating currents or in ordinary radio engineering.

The growing demands of practice will undoubtedly lead to the development of such methods, but at the present stage of development it is desirable to study microradio waves precisely from the point of view of electromagnetic field theory. Evidently, engineers working in the field

* Rev. Modern Physics, 14, 341 (1942), translated by V. G. Levich.

ultrashort waves, it will be necessary to take seriously the study of the theory of the electromagnetic field.

1. Electron tubes used in the region of microwaves have the basic feature that the flight time of individual electrons in the device is not small in comparison with the period of the radiation, as is the case in the region of long waves.

Indeed, in ordinary tubes the velocities of electrons amount to from 0.01 to 0.1 of the speed of light. Therefore, over the course of 1 cycle the electrons traverse a path from 0.01 to 0.1 of a wavelength.

If the wavelength is only a few centimeters, in practice it proves difficult to construct a tube with such a short electron path that it is small in comparison with 0.01–0.1 of the wavelength.

This circumstance makes the usual concepts in the field of electron tubes inapplicable. Modern progress in the field of ultrashort waves is to a considerable extent connected with the discovery of ways of making full use of the finite (in comparison with the period of the radiation) flight time of electrons in tubes.

In other words, the finite flight time has not been a limitation for modern electronics, but has merely compelled the abandonment of traditional concepts in this field.

Historically, Hertz’s early works, in which electromagnetic waves were first obtained by artificial means, belonged to that range of wavelengths which we here call microwaves.

However, they differed from the waves considered here, on the one hand, in that their intensity was extremely small, and on the other hand, in that they were a strongly damped wave train; much more convenient for practical purposes is stationary radiation obtained with the aid of modern apparatus.

From the point of view of the use of ultrashort waves for communication, the following are the principal points.

1. New frequencies can be obtained in a dense medium.

2. Microwaves can be successfully used to obtain strictly directed radiation, since for the directionality of radiation the dimensions of the antenna must be large in comparison with the length of the emitted wave—a requirement which, obviously, is easier to satisfy in the case of short waves.

The field of microwaves is so new that there are as yet very few works in which the propagation of such waves along the surface of the earth or sea, or their behavior in the ionosphere, has been studied. Much work still has to be done in this direction. With the development of experimental technique and the improvement of equipment and instruments, physicists will receive at their disposal a new weapon for investigating regions that are still completely “open.”

It is already known that some molecules (for example, ammonia) have, in the microwave region, characteristic

frequencies, which are extremely important for understanding molecular structure. It is probable that, thanks to centimeter-wave spectroscopy, in the future we shall obtain a number of important data from other areas of physics as well.

If a ferromagnetic is placed in the field of microwave radiation, the depth of effective penetration of the field into the conductor will be of just the same order as the dimensions of the regions of spontaneous magnetization (domains). Therefore there is no doubt that the study of ferromagnetics by means of microwaves will yield much that is new for a better understanding of the nature of ferromagnetism. Further, a large number of dielectrics have an absorption maximum in the microwave frequency range. Therefore the study of the properties of such dielectrics in this frequency range should be essential for a better understanding of their structure. This is especially important for such substances as modern synthetic resins and rubbers. All these data, important for a better understanding of processes in dielectrics and ferromagnetics, constitute for the most part a contribution to the field of applied physics.

However, if we wish to find the broadest field of applications of microwaves outside the walls of the research laboratory, it immediately becomes obvious that a large part of the results achieved may find the most direct application in navigation, both marine and aerial. One should also not overlook the fact that the application of microwaves in diathermy has not yet been fully investigated, and it may turn out that they possess specific therapeutic properties absent in the long-wave radiation currently used.

Thus, this field represents a vast area for research.

Chapter I. HOLLOW RESONATORS

In the microwave region, instead of the usual antenna and capacitor as the principal elements of a resonant circuit, hollow resonators are used. By a hollow resonator we shall mean a region of space surrounded on all sides by walls made of a good conductor, which is used as an element of an oscillatory circuit. Therefore the study of microwaves should begin with a consideration of the properties of a hollow resonator.

§ 1. Maxwell’s Equations

The solution of all problems connected with the electromagnetic field is based on the use of Maxwell’s equations, which we shall write in the following form:

\[ \left. \begin{aligned} \operatorname{div}\mathbf{D} &= 4\pi\rho, & \operatorname{div}\mathbf{B} &= 0,\\ \operatorname{rot}\mathbf{E} &= -\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}, & \operatorname{rot}\mathbf{H} &= 4\pi\mathbf{j}+\frac{1}{c}\frac{\partial \mathbf{D}}{\partial t}, \end{aligned} \right\} \tag{1.1} \]

where \(\mathbf E\) is the electric-field vector in CGS units, \(\mathbf D\) is the induction vector in CGS units, \(\rho\) is the charge density in CGS electrostatic units, \(\mathbf H\) is the magnetic-field vector in gausses, \(\mathbf B\) is the magnetic induction vector in gausses, \(\mathbf j\) is the conduction-current density in abs. units/cm\(^2\).

The system of units we have chosen proves to be the most convenient and is often used in practice. However, we would not wish to fall into the common error of constantly adhering to one once-chosen system of units. When it is convenient, we shall also use other systems of units.

In ordinary media we have

\[ \begin{aligned} \mathbf B &= \mu \mathbf H,\\ \mathbf D &= \varepsilon \mathbf E, \end{aligned} \tag{1.2} \]

where \(\mu\) is the magnetic permeability and \(\varepsilon\) is the dielectric constant of the medium. The coefficients \(\mu\) and \(\varepsilon\) are most often given in handbooks in the system of units we have chosen; they are dimensionless quantities equal to unity in vacuum.

In a conducting medium the electric-field intensity \(\mathbf E\) is related to the current density \(\mathbf j\) by the relation

\[ \mathbf E=\frac{1}{\sigma}\mathbf j, \tag{1.3} \]

where \(\sigma\) is the conductivity of the substance in cm\(^{-1}\) and \(1/\sigma\) is the specific resistance. In ordinary tables the value of \(\sigma\) (or the specific resistance \(1/\sigma\)) is given in ohm·cm, which corresponds to \(\mathbf E\) being expressed in volts per centimeter and \(\mathbf j\) in amperes per square centimeter. If \(1/\sigma'\) is the specific resistance in ohms per centimeter, then

\[ \frac{1}{\sigma}=\frac{1}{30}\cdot\frac{1}{\sigma'}. \tag{1.4} \]

For copper at room temperature \(1/\sigma'=1.7\cdot10^{-6}\ \Omega\cdot\text{cm}\), \(1\cdot\sigma=5.7\cdot10^{-8}\ \text{cm}\).

At the boundary between two nonconducting media, the continuity conditions for the normal components of \(\mathbf D\) and \(\mathbf B\) and the tangential components of \(\mathbf E\) and \(\mathbf H\) must be satisfied. If there are surface charges of density \(n\) CGS units/cm\(^2\) on the interface, then the normal component of the vector \(\mathbf D\) undergoes a jump equal to \(4\pi n\).

The charge density and the current density are related to each other by the continuity equation

\[ \operatorname{div}\mathbf j+\frac{1}{c}\frac{\partial\rho}{\partial t}=0, \tag{1.5} \]

which expresses the fact that the total current flowing out of a given volume is accompanied by the corresponding decrease of charge.

in this volume. In technical handbooks the field strength is usually expressed in volts per centimeter, the current strength in amperes, the current density in amperes per square centimeter, and charge in coulombs. \(1\ \mathrm{V}=\dfrac{1}{300}\) CGS unit; \(1\) absolute unit of current \(=10\ \mathrm{A}\); \(1\) coulomb \(=3\cdot 10^9\) CGS units of charge. Power in the system of units we have chosen is expressed in absolute units of voltage multiplied by absolute units of current \(=3\ \mathrm{kW}\) \((1\ \mathrm{kW}=1\ \mathrm{V}\cdot 1\ \mathrm{A})\). Although \(\mathbf H\) is usually expressed in gausses in technical handbooks as well, nevertheless sometimes \(\mathbf H\) is expressed in amperes per centimeter. The latter corresponds to the magnetic field of an infinitely long solenoid through which a current flows, expressed in ampere-turns per centimeter \((1\ \mathrm{A}/\mathrm{cm}^2=0.4\pi\ \text{gauss})\).

Fig. 1. Relation between the vectors \(\mathbf E\), \(\mathbf D\), and \(\mathbf H\), \(\mathbf B\) in a plane wave propagating in a direction perpendicular to the plane of the drawing.

From Maxwell’s equations one can derive the general relation

\[ \operatorname{div}\mathbf S+\frac{1}{4\pi}\left(\mathbf H\frac{\partial \mathbf B}{\partial t}+\mathbf E\frac{\partial \mathbf D}{\partial t}\right)=-cjE, \tag{1.6} \]

where

\[ \mathbf S=\frac{c}{4\pi}[\mathbf E\mathbf H]\ \mathrm{erg}/\mathrm{cm}^2\mathrm{sec}. \tag{1.7} \]

The vector \(\mathbf S\) is called the Poynting vector and is interpreted as the flux of electromagnetic energy. The true value of the flux of electromagnetic energy is not determined by this or by any other relations, since to the vector \(\mathbf S\) one may add any other vector \(\mathbf S'\) whose divergence is zero, without violating the general equation (1.6). Since, however, the value of electromagnetic energy is not measured directly, but only after its conversion into mechanical or thermal energy, this indeterminacy cannot affect the values of directly observable quantities.

In those cases where \(\mu\) and \(\varepsilon\) are constant in time, the second term in (1.6) can be represented as the time derivative of the quantity

\[ W=\frac{1}{8\pi}(\mu H^2+\varepsilon E^2)\ \mathrm{erg}/\mathrm{cm}^3, \tag{1.8} \]

which is interpreted as the density of the electromagnetic energy of the field.

In the system of units chosen by us, the absolute magnitudes of the vectors \(\mathbf E\) and \(\mathbf H\) in a plane electromagnetic wave are equal to one another. For practical purposes it is necessary to bear in mind the expression for the Poynting vector in the technical system of units:

\[ \mathbf S=\frac{1}{0.4\pi}[\mathbf E\mathbf H]\ \mathrm{W}/\mathrm{cm}^2. \tag{1.9} \]

In vacuum, in these units the quantity \(\mathbf H\) in gausses is equal to \(1/300\,\mathbf E\)

in volts per centimeter, so that the absolute value of the Poynting vector is equal to \(S=(1/120\pi)\cdot E^2\). The number \(120\pi=377\) is expressed in ohms and in the literature is often grandly called “the impedance of free space.”

Returning again to equation (1.6), we see that in that part of space where currents are absent, so that the right-hand side of the equation becomes zero, it expresses the law of conservation of energy of the electromagnetic field. It also shows that a change in the energy of the electromagnetic field in any closed region, through whose boundaries electromagnetic energy cannot flow out, occurs only at the expense of the electric current flowing in the direction of the electric-field vector.

The entire content of the theory of microwaves and all the technique of their use consist in the generation, transmission, and reception of electromagnetic energy in the region of such high frequencies that the corresponding wavelengths prove to be comparable with the dimensions of the devices used for obtaining them.

In what follows we shall often have to seek the distribution of the electromagnetic field by integrating the field equations. In other branches of radio engineering one has to deal only with waves that are long in comparison with the dimensions of the apparatus. This circumstance makes it possible to avoid the field equations as a computational apparatus and to use the general theory of circuits with lumped constants, which is the basis for calculations in almost all of electrical engineering.

§ 2. Plane Waves

Before passing to the consideration of the problem of the field in a hollow resonator, it is useful to recall the solution of the field equations in the form of plane, standing, and traveling waves.

Let us suppose that the field vector is represented by the real part of the vector

\[ \mathbf{A}e^{2\pi i(\nu t-\mathbf{k}\mathbf{r})}, \tag{2.1} \]

where \(\mathbf{A}\) is a constant vector, \(\mathbf{k}\) is the wave vector, directed along the normal to the front of the plane wave in the direction of propagation of its phase and, in absolute value, equal to the number of wavelengths per \(1\ \mathrm{cm}\), and \(\nu\) is the frequency in cycles per second. In the radio-engineering literature the opposite signs in the exponent are more often chosen, but all results are independent of the choice of signs. The positive sign chosen by us for the time factor is more often encountered in other branches of electrical engineering, in particular in the theory of alternating currents, where it is assumed that all vectors in vector diagrams rotate counterclockwise.

From the two equations \(\operatorname{div}\mathbf{D}=0\) and \(\operatorname{div}\mathbf{B}=0\) it follows that

\[ \mathbf{D}\cdot\mathbf{k}=0 \quad \text{and} \quad \mathbf{B}\cdot\mathbf{k}=0, \]

i.e., that the amplitudes of the electromagnetic waves are orthogonal to the direction of propagation. We shall regard the direction of D or E as the direction of polarization of the wave.

The last two Maxwell equations, after substituting (2.1), give

\[ \mathbf{k}\cdot\mathbf{E}=-\frac{\nu}{c}\mathbf{B},\qquad \mathbf{k}\cdot\mathbf{H}=+\frac{\nu}{c}\mathbf{D}, \tag{2.2} \]

from which it follows that

\[ \mathbf{k}(\mathbf{k}\cdot\mathbf{E})=-\left(\frac{\nu}{c}\right)^2 \varepsilon\mu\mathbf{E} \tag{2.3} \]

and, consequently, taking into account that \(\mathbf{k}\mathbf{E}=0\),

\[ |\mathbf{k}|=\left(\frac{\nu}{c}\right)(\varepsilon\mu)^{1/2}. \tag{2.4} \]

Thus, the phase velocity of wave propagation in the medium is equal to \(c/\sqrt{\varepsilon\mu}\), and the index of refraction is \(W=(\varepsilon\mu)^{1/2}\). From (2.2) it is easy to see that the vectors \(\mathbf{E}\), \(\mathbf{H}\), and \(\mathbf{k}\) are oriented as shown in Fig. 1, and their amplitudes satisfy the equality

\[ \sqrt{\varepsilon}\cdot E=\sqrt{\mu}\cdot H. \]

In vacuum all vectors of the electromagnetic field are equal to one another. The average flux of energy carried by a plane wave is

\[ S\ \mathrm{W}/\mathrm{cm}^{2} = \frac{1}{2}\frac{1}{120\pi(\mu/\varepsilon)^{1/2}}E^{2}, \tag{2.5} \]

where \(E^{2}\) is expressed in volts per centimeter.

As was already indicated in the preceding paragraph, the coefficient in the denominator is expressed in ohms. From (2.5) we see that the medium is characterized by an impedance for a plane wave equal to \(120\pi(\mu/\varepsilon)^{1/2}\) ohms. The impedance for a plane wave in ohms may also be defined as the ratio of the strength of the electric field (volts/cm) to that of the magnetic field (ampereturns/cm). Such a definition leads to the very same numerical value of the impedance.

A standing wave arises from the superposition of two traveling waves of equal amplitude propagating in opposite directions. Suppose, for example, that one of the waves propagates in the positive direction of the \(z\)-axis and is polarized in the direction of the \(x\)-axis. Then the strengths of the electric and magnetic fields of the plane wave will have the form

\[ \left. \begin{aligned} E_x&=E_1\cos 2\pi(\nu t-kz),\qquad E_y=E_z=0,\\ H_y&=\left(\frac{\varepsilon}{\mu}\right)^{1/2}E_1\cos \pi(\nu t-kz),\qquad H_x=H_z=0. \end{aligned} \right\} \tag{2.6} \]

Analogously, for a wave polarized in the same direction but propagating in the opposite direction,

\[ \left. \begin{aligned} E_x&=E_2\cos 2\pi(\nu t+kz),\quad E_y=E_z=0,\\ H_y&=-\left(\frac{\varepsilon}{\mu}\right)^{1/2}E_2\cos 2\pi(\nu t+kz),\quad H_x=H_z=0. \end{aligned} \right\} \tag{2.7} \]

Let us now suppose that the plane \(z=0\) is an ideal conductor. On the surface of an ideal conductor the tangential component of the vector \(\mathbf E\) must vanish, and, consequently, the amplitudes of the two waves must satisfy the condition \(E_2=E_1\). The field of the two waves, the incident and the reflected, is described by the relations

\[ \left. \begin{aligned} E_x&=2E_1\sin 2\pi kz\sin 2\pi\nu t,\\ H_y&=2\left(\frac{\varepsilon}{\mu}\right)^{1/2}E_1\cos 2\pi kz\cos 2\pi\nu t. \end{aligned} \right\} \tag{2.8} \]

As is seen from (2.8), at the initial instant, for \(t=0\), the energy of the field is purely magnetic in character, and after a quarter of a cycle it is purely electric. Thus, in a standing wave the field energy does not remain completely unchanged, but pulsates, passing from magnetic into electric and back again (Fig. 2). The reflection of a plane wave by an ideal conductor is accompanied by the induction of currents in it. In Chapter IV it will be shown how the radiation is calculated for a given distribution of currents. Here we shall restrict ourselves to the assertion that the induced surface current radiates waves which are the waves reflected from the metal surface and exactly cancel the incident waves on the opposite surface of the metal.

Fig. 2. Pulsations of the energy in a standing plane wave.

The induced current on the surface of the metal can be found in the following way. The magnetic field near the boundary of the metal, in the plane \(z=0\), will be equal to

\[ H_y=2E_1\left(\frac{\varepsilon}{\mu}\right)^{1/2}\cos 2\pi\nu t\quad \text{for } z>0, \]

\[ H_y=0\qquad\qquad\qquad\qquad\quad \text{for } z<0. \]

Consequently, the line integral, taken over a closed contour of unit length, running along the surface outside the metal in the positive direction and inside it in the negative direction, will be different from zero.

From Maxwell’s equations it follows, in this case, that along the surface of the metal there flows a closed conduction current (the displacement current is equal to scant)

zero, since at the surface of the metal the tangential components of the electric field \(E\) vanish; its density is

\[ j_x=-\frac{2E_1}{4\pi}\left(\frac{\varepsilon}{\mu}\right)^{1/2}\cos 2\pi\nu t . \]

§ 3. Cavity resonators

A region of space surrounded on all sides by a good conductor can serve as a cavity resonator, or “rumbatron.” Each such resonator has an infinitely large number of resonant frequencies and corresponding resonant wavelengths.

We shall first set forth the theory of a resonator bounded by an ideal conductor with ohmic resistance equal to zero, and then take into account the influence of the finite resistance of the metal. We shall also restrict ourselves to a cavity in the form of a rectangular box, which is most convenient for our purposes because the field in it is expressed through simple trigonometric functions. Our task is to solve the field equations in such a cavity, taking account of the boundary conditions on the surface of the metal bounding the cavity: the vector \(\mathbf E\) must be perpendicular to the surface and the vector \(\mathbf H\) parallel to it.

Let us suppose that all the field vectors depend on time according to the law \(e^{2\pi i\nu t}\). Then the coordinate dependence of the field vectors can be found from the equations

\[ \left. \begin{aligned} \operatorname{div}\sqrt{\varepsilon}\,\mathbf E&=0, \qquad \operatorname{div}\sqrt{\mu}\,\mathbf H=0,\\ \operatorname{rot}\sqrt{\varepsilon}\,\mathbf E&=-i\left(\frac{2\pi n\nu}{c}\right)\sqrt{\mu}\,\mathbf H,\\ \operatorname{rot}\sqrt{\mu}\,\mathbf H&=+i\left(\frac{2\pi n\nu}{c}\right)\sqrt{\varepsilon}\,\mathbf E, \end{aligned} \right\} \tag{3.1} \]

if inside the cavity \(\varepsilon\) and \(\mu\) are constants. Here \(n\) denotes the refractive index, defined in § 2, \(n=\sqrt{\varepsilon\mu}\). We shall denote by \(k\) the factor \(2\pi n\nu/c\).

It is clear from equations (3.1) that the vectors \(\sqrt{\varepsilon}\,\mathbf E\) and \(\sqrt{\mu}\,\mathbf H\) satisfy, in a cavity filled with a medium, the same equations as the vectors \(\mathbf E\) and \(\mathbf H\) in vacuum, but with the velocity \(c\) replaced by \(c/n\). Therefore the field in a resonator filled with an ordinary medium (with constant \(\varepsilon\) and \(\mu\)) can be found without difficulty if the field of the corresponding region of empty space is known. For this reason, and especially also because in practice it is always precisely vacuum resonators that are used, in all subsequent formulas we shall put \(\varepsilon\) and \(\mu\) equal to unity.

Taking \(\operatorname{rot}\) of the third of equations (3.1) and carrying out simple transformations, we find the equation for the electric-field intensity \(\mathbf E\):

\[ \Delta \mathbf E+k^2\mathbf E=0. \tag{3.2} \]

If the solution of this equation is known, then, in order to find the magnetic-field intensity \(\mathbf H\), it is not necessary to solve the corresponding equation separately; the vector \(\mathbf H\) can be found directly from the third equation (3.2)

\[ \mathbf H=\frac{i}{k}\operatorname{rot}\mathbf E. \tag{3.3} \]

The magnetic field thus computed automatically satisfies the boundary conditions. The boundary conditions for the electric field \(\mathbf E\) may be written in the form \(\int \mathbf E\,dl=0\), where the integral is taken over any contour drawn along the surface of the metal, since on this surface the tangential components of the vector \(\mathbf E\) vanish. Therefore \(\iint \operatorname{rot}\mathbf E\,ds=0\), where the integral extends over any part of the boundary surface. From the latter relation it follows that the normal component of \(\operatorname{rot}\mathbf E\) vanishes at every point of the boundary surface; whence, in turn, it follows that the boundary conditions for the magnetic field \(\mathbf H\) will indeed be automatically satisfied [cf. formula (3.3)].

There are no general methods for solving equation (3.2) for a cavity of arbitrary shape, and its solution can be found only for a small number of cases of especially simple form. The problem of finding the electromagnetic field in a cavity is in many respects similar to the corresponding acoustic problem of finding standing sound waves in a closed cavity.

However, the electromagnetic problem is more complicated, since electromagnetic waves are vector waves, and each component of the vector \(\mathbf E\), taken separately, satisfies equation (3.2); in addition, their totality is connected by the condition \(\operatorname{div}\mathbf E=0\). In the acoustic problem, by contrast, one has to deal with a single scalar quantity, for example, the pressure in the wave.

We shall now consider the solution of equation (3.2) for a cavity having the form of a rectangular parallelepiped, the walls of which are perpendicular to the coordinate axes and cut off on them the segments

\[ 0<x<A,\qquad 0<y<B,\qquad 0<z<C. \]

If we try to seek a solution of equation (3.2) in the form

\[ E_x=E_1 \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_1x\, \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_2y\, \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_3z, \]

\[ E_y=E_2 \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_1x\, \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_2y\, \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_3z, \]

\[ E_z=E_3 \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_1x\, \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_2y\, \begin{matrix} \cos\\[-2pt] \sin \end{matrix} k_3z, \]

then the equation will be satisfied for any choice of a combination of sines and cosines, provided only that the numbers \(k_1\), \(k_2\), and \(k_3\) are connected by the relation

\[ k_1^2+k_2^2+k_3^2=k^2. \]

In order to satisfy the boundary condition—the perpendicularity of the vector \(\mathbf E\) to all the walls of the cavity—we must restrict ourselves either to sines or to cosines and impose on \(k_1\), \(k_2\), and \(k_3\) the condition

\[ k_1=\frac{l\pi}{A},\qquad k_2=\frac{m\pi}{B},\qquad k_3=\frac{n\pi}{C}, \tag{3.4} \]

where \(l\), \(m\), and \(n\) are integers.

Therefore the solution has the form:

\[ \begin{aligned} E_x&=E_1\cos\frac{l\pi x}{A}\sin\frac{m\pi y}{B}\sin\frac{n\pi z}{C},\\ E_y&=E_2\sin\frac{l\pi x}{A}\cos\frac{m\pi y}{B}\sin\frac{n\pi z}{C},\\ E_z&=E_3\sin\frac{l\pi x}{A}\sin\frac{m\pi y}{B}\cos\frac{n\pi z}{C}. \end{aligned} \tag{3.5} \]

The three amplitudes \(E_1\), \(E_2\), and \(E_3\) are not independent, but are connected by the relation

\[ \frac{l\pi}{A}E_1+\frac{m\pi}{B}E_2+\frac{n\pi}{C}E_3=0. \tag{3.6} \]

Consequently, to each set of integers \(l\), \(m\), and \(n\) there correspond two linearly independent waves. If \(k_1\), \(k_2\), and \(k_3\) are regarded as the three components of a vector, and \(E_1\), \(E_2\), and \(E_3\) as the components of another vector, then any vector \(E_1\), \(E_2\), and \(E_3\) perpendicular to the vector \(\mathbf k\) is admissible. The possible resonance frequencies satisfy the condition

\[ \left(\frac{\nu}{c}\right)^2= \left(\frac{l}{2A}\right)^2+ \left(\frac{m}{2B}\right)^2+ \left(\frac{n}{2C}\right)^2, \tag{3.7} \]

where \(l\), \(m\), and \(n\) are integers, with at least two of them simultaneously different from zero.

For waves in which one component of the vector \(\mathbf k\) is equal to zero, i.e. one of the numbers \(l\), \(m\), \(n\) is equal to zero, the electric-field vector is directed parallel to that axis along which the component of the vector \(\mathbf k\) is absent. In this case there is only one solution of equation (3.6) and only one vector satisfying condition (3.7), although, as we have just indicated, in the general case there are two linearly independent solutions corresponding to the given set \(l\), \(m\), and \(n\).

The smallest resonance frequency of waves in the box is obtained in the case when two numbers, corresponding to the axes along which the box has the largest dimensions, are equal to unity, and the third is zero. If \(A\) and \(B\) are the two large sides of the cavity, then the wave with the smallest frequency is polarized along the third side of the box, and the corresponding wavelength is

\[ \lambda=\frac{2}{\left(A^{-2}+B^{-2}\right)^{1/2}}. \]

In particular, for a cubic box the wavelength corresponding to the lowest resonant frequency is equal to the length of the diagonal of a face of the box:

\[ \lambda=\sqrt{2}\,A. \]

The number of distinct resonant oscillations increases rapidly as one moves along the frequency scale. Let us consider, for example, a box with one side much smaller than the other two (i.e. \(B=A\), \(C\ll A\)). The oscillations with the lowest frequency correspond to \(n=0\). The value of the quantity \(2A\sigma\) is determined by the expression

\[ 2A\sigma=\left[l^{2}+m^{2}+n^{2}\left(\frac{A}{C}\right)^{2}\right]^{1/2}. \]

It is easy to calculate that there exist 33 different sets of integers \(l, m, n\) leading to frequencies greater than the lowest frequency by a factor of five or less. It may also be noted that the frequencies can be grouped into series: the principal series is formed according to the law \((110)\), \((220)\), \((330)\), \((440)\); another series begins with the frequencies \((120)\) and \((210)\) and contains frequencies of the form \((240)\) and \((420)\), \((360)\) and \((630)\), etc. However, the law by which higher frequencies are formed through the formation of integral multiples is a special property of a rectangular box and is absent for cavities of other shapes.

In conclusion it is necessary to define several terms that will be needed later. Any frequency for which there exists a solution of the field equations satisfying the boundary conditions will be called a natural frequency. The smallest of the natural frequencies is called the fundamental frequency. If the higher frequencies are integral multiples of the fundamental frequency, they are called harmonics. A particular solution of the equations for \(\mathbf E\) and \(\mathbf H\) forms a natural oscillation in the cavity. If, at the same frequency, several natural oscillations correspond, then such oscillations are called degenerate. The degree of degeneracy is the number of distinct linearly independent natural oscillations corresponding to one and the same frequency.

Thus, in the example just considered, the fundamental frequency is nondegenerate, while all higher frequencies are twofold degenerate, since, for example, the solutions \((1,2,0)\) and \((2,1,0)\) are linearly independent and have identical frequencies. This type of degeneracy we shall call polarization degeneracy. Degeneracy arising from the symmetry of the shape of the cavity is called symmetry degeneracy. For example, the oscillations \((1,2,0)\) and \((2,1,0)\) have identical frequencies only in the case when the edges \(A\) and \(B\) are equal to one another.

A slight deviation from the condition \(A=B\), whether deliberate or connected with the imperfection of the instrument or with the nature of the inclusion of the resonator in the common circuit, leads to the result that the degenerate frequencies begin to differ slightly from one another. We shall then say that the degeneracy is removed.

A very important point is the absence of uniqueness in the solution of the wave-field equations, connected with degeneracy. For example, in the case of polarization degeneracy, any two (preferably mutually perpendicular) vectors satisfying condition (3.6) may be chosen as fundamental oscillations. Any linear combination of them is a possible natural oscillation associated with this frequency.

Similarly, in the case of symmetry degeneracy, the true natural oscillation may be a linear combination of degenerate oscillations. Consider, for example, the oscillations \((1, 2, 0)\) and \((2, 1, 0)\). According to (3.5), for both combinations only the \(z\)-component of the vector \(\mathbf E\) is different from zero, and is equal, respectively, to

\[ E_{120}=C\sin\left(\frac{\pi x}{A}\right)\sin\frac{2\pi y}{A}, \]

\[ E_{210}=D\sin\left(\frac{2\pi x}{A}\right)\sin\frac{\pi y}{A}, \]

where \(C\) and \(D\) are arbitrary amplitudes. Depending on the relative values of the two amplitudes, the most varied types of field distribution in the cavity are possible; some of them are shown schematically in Fig. 3.

Since the combination with \(C=D\) represents oscillations along the line \(y=x\), as well as along \(y=0\) and \(x=0\), it satisfies all the requirements imposed on a fundamental oscillation in the rectangular prism formed by these three lines. Thus, particular solutions can often be found for cavities of simple form which are difficult to find by other methods.

This method, however, is no longer suitable for a rectangular prism with unequal sides, since in it the natural oscillations are not degenerate and cannot be directly superposed on one another.

Example: Consider oscillations corresponding to the amplitude relation \(D=\pm iC\).

Fig. 3. Various types of field distribution corresponding to the expression of the natural oscillations (120) and (210).

§ 4. Normal coordinates*)

The solution, considered in detail in the preceding paragraph, of the problem of natural oscillations in a rectangular resonator makes it possible—

*) A large part of this paragraph may be omitted on first reading, but for what follows it is necessary to keep in mind the main results obtained in it.

enabled us to become acquainted with the basic features characteristic of problems on natural oscillations in a cavity of arbitrary form. Namely, we saw that solutions of the problem satisfying the boundary conditions exist only for a discrete series of frequencies, with each frequency corresponding to one or several natural oscillations.

Obviously, in the most general case all natural oscillations are simultaneously excited in the resonator cavity, just as, when a fixed membrane is struck, all possible natural oscillations are excited in it. For the mathematical description of the natural oscillations of the field in a cavity in the general case we shall introduce the so-called normal coordinates, which represent the totality of the amplitudes of all the fundamental electromagnetic waves in the cavity.

Vector potential. Instead of using directly the electric and magnetic field strengths \(\mathbf E\) and \(\mathbf H\), it is more convenient to use the scalar and vector potentials \(\varphi\) and \(\mathbf A\), which are defined by the relations:

\[ \mathbf E=-\frac{1}{c}\frac{\partial \mathbf A}{\partial t}-\operatorname{grad}\varphi, \qquad \mathbf H=\operatorname{rot}\mathbf A. \tag{4.1} \]

With this definition the vector potential is measured in the same units as the current.\(^1\) The equations for \(\operatorname{rot}\mathbf E\) and \(\operatorname{div}\mathbf H\), when the potentials \(\varphi\) and \(\mathbf A\) are substituted in them, will be satisfied automatically. Substituting (4.1) into the two other equations of the electromagnetic field, we find:

\[ \begin{aligned} -\Delta \mathbf A+\frac{1}{c^2}\frac{\partial^2\mathbf A}{\partial t^2} +\operatorname{grad}\left(\operatorname{div}\mathbf A+\frac{1}{c}\frac{\partial\varphi}{\partial t}\right) &=4\pi \mathbf j,\\ -\Delta\varphi+\frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2} -\frac{1}{c}\frac{\partial}{\partial t}\left(\operatorname{div}\mathbf A+\frac{1}{c}\frac{\partial\varphi}{\partial t}\right) &=4\pi\rho. \end{aligned} \tag{4.2} \]

We may dispose at our discretion of the quantity \(\operatorname{div}\mathbf A\) so as to simplify the equations obtained. If we put \(\operatorname{div}\mathbf A=-\frac{1}{c}\cdot\partial\varphi/\partial t\), we obtain the following equations for the potentials:

\[ \begin{aligned} \Delta \mathbf A-\frac{1}{c^2}\frac{\partial^2\mathbf A}{\partial t^2}&=-4\pi\mathbf j,\\ \Delta \varphi-\frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2}&=4\pi\rho,\\ \operatorname{div}\mathbf A+\frac{1}{c}\frac{\partial\varphi}{\partial t}&=0. \end{aligned} \tag{4.3} \]

These equations will be fundamental in the theory of radiation of a system of moving charges and currents, to which Chapter IV will be devoted.

\(^1\) The results—the orthogonality of the wave functions [formula (4.5)] and the dynamical equations for the amplitudes of the natural oscillations [formula (4.10)]. This paragraph is an application to the problem of the formalism of quantum electrodynamics that is of interest to us.—Ed.²

Taking \(\operatorname{div}\) of the first equation, multiplying the second by \(1/c\), and adding them, we shall find an equation for the time dependence of the quantity \(\operatorname{div}\mathbf A+\frac{1}{c}\cdot \partial\varphi/\partial t\), namely

\[ \left(\Delta-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right) \left(\operatorname{div}\mathbf A+\frac{1}{c}\varphi\right)=0. \]

On the right-hand side of the last equation there is zero because the current and charge must satisfy the equation of continuity (1.5). It shows that if we have a solution satisfying the third equation (4.3), together with its time derivatives, at \(t=0\), then thereby we satisfy it at all other times as well.

Let us now suppose that the problem of finding the eigenfrequencies and the corresponding eigenwaves in a cavity can be solved in the same way as was done for the rectangular box. This means that we know the set of quantities \(k_1,k_2,k_3\), etc., and the corresponding solutions \(\mathbf A_1,\mathbf A_2,\mathbf A_3\), etc., of the equations for the potential

\[ \Delta \mathbf A+k^2\mathbf A=0,\qquad \operatorname{div}\mathbf A=0, \tag{4.4} \]

satisfying the boundary conditions.

\(\mathbf A\) is perpendicular to the boundary of the region or vanishes. As we have seen, with each degenerate number \(k_n\) there is associated not one but several linearly independent solutions \(\mathbf A_n\). Therefore, for a complete numbering of all vectors \(\mathbf A\) it is necessary to use one more index, making it possible to distinguish the \(\mathbf A\)’s corresponding to one and the same value of \(k_n\). However, one can usually avoid introducing such a complication of notation, and number by a single index \(n\) all independent wave functions, so that in the case of degeneracy several different values of the index would correspond to identical values of \(k_n\).

Orthogonality of wave functions. The vector waves \(\mathbf A_n\) possess an important orthogonality property, which makes it possible to express other functions through them, much as is done in the theory of Fourier series. Form the expression

\[ \mathbf A_m\operatorname{rot}\operatorname{rot}\mathbf A_n -\mathbf A_n\operatorname{rot}\operatorname{rot}\mathbf A_m =(k_n^2-k_m^2)\mathbf A_n\mathbf A_m. \]

Using the known identity of vector analysis

\[ \operatorname{div}(\mathbf a\mathbf b)=\mathbf b\operatorname{rot}\mathbf a-\mathbf a\operatorname{rot}\mathbf b, \]

this equation can be written in the form

\[ \operatorname{div}(\mathbf A_n\operatorname{rot}\mathbf A_m+ \operatorname{rot}\mathbf A_n\mathbf A_m) =(k_n^2-k_m^2)\mathbf A_n\mathbf A_m. \]

Let us now integrate both sides of the last equation over the volume of the cavity. The integral on the left-hand side can be transformed into a surface integral extended over the surface bounding the cavity. However, since \(\mathbf A_n\) and \(\mathbf A_m\) are perpendicular to the sur—

surface, this integral vanishes. Therefore, we have

\[ \int \mathbf{A}_n \mathbf{A}_m\,dV=0,\quad \text{if } n\ne m. \tag{4.5} \]

In the case of degenerate values of \(k\), it is always possible to choose such a linear combination of the vectors \(\mathbf{A}_n\) that it would satisfy the orthogonality conditions even in the case when the originally found \(\mathbf{A}\)'s did not satisfy these conditions.

Since a particular solution of the equations for \(\mathbf{A}\) will remain a solution when multiplied by an arbitrary constant, we can choose constant multipliers for the various \(\mathbf{A}_n\) so as to normalize the functions, i.e., so that the equality

\[ \int \mathbf{A}_n\mathbf{A}_n^*\,dV=V, \tag{4.6} \]

holds, where \(\mathbf{A}_n^*\) is the complex-conjugate function of \(\mathbf{A}_n\), and \(V\) is the volume of the cavity filled with radiation. With such a normalization the functions \(\mathbf{A}_n\) are considered dimensionless.

Let us first consider the case in which the charge density inside the cavity is zero at all times. Then also \(\varphi=0\), and we can try to find the solution of the first of equations (4.3) in the form

\[ \mathbf{A}=\sum q_n(t)\mathbf{A}_n(x,y,z). \tag{4.7} \]

Since the vectors \(\mathbf{A}_n\) are dimensionless, the time factors \(q_n(t)\) have the dimension of absolute amperes. The time amplitudes characterizing the excitation of the corresponding electromagnetic oscillations are called the normal amplitudes of the field.

Each natural oscillation has its own amplitude, so that their total number is infinitely large.

Amplitudes of the excitation currents. Let us also expand, in the resonator cavity, \(\mathbf{j}(x,y,z,t)\) in a Fourier series in the vectors \(\mathbf{A}_n\), so that

\[ \mathbf{j}(x,y,z,t)=\sum I_n(t)\mathbf{A}_n(x,y,z). \tag{4.8} \]

For the formal determination of the expansion coefficients we shall use the orthogonality and normalization conditions of the functions \(\mathbf{A}\) and, as is usually done in the theory of Fourier series, we shall without difficulty find

\[ I_n(t)=\frac{1}{V}\int \mathbf{j}\mathbf{A}_n^*\,dV. \tag{4.9} \]

The dimension of \(I_n(t)\), like that of \(\mathbf{j}\), is \(\mathrm{A}/\mathrm{cm}^2\). We shall call the coefficients \(I_n(t)\) the amplitudes of the excitation current of the \(n\)-th oscillation. Incidentally, note that the distribution of the currents \(\mathbf{j}\) is the more effective for exciting the \(n\)-th natural oscillation, the closer its spatial distribution is to the spatial distribution of the oscillation being excited.

Equation for the field amplitudes. Substituting (4.8) and (4.7) into the first of equations (4.3) and equating the coefficients of

vectors \(\mathbf A_n\), we obtain the following equation for the field amplitudes in the resonator:

\[ \ddot q_n(t)+(ck_n)^2 q_n(t)=4\pi c^2 I_n(t). \tag{4.10} \]

This equation is completely identical with the equation of forced oscillations of a harmonic oscillator possessing the natural frequency \((ck_n)\,2\pi\).

If the amplitude of the excitation current \(I_n(t)\) is equal to zero, the corresponding amplitude \(q_n\) is a harmonic function of time, varying with frequency \((ck_n)\) and with constant amplitude. The absence of damping in free oscillations is connected with the fact that we assume the walls of the cavity to be made of an ideal conductor. The influence of the finite electrical resistance of the walls will be taken into account in § 8.

Expression for the energy. Substituting into the general expression for the energy of the electric field in vacuum

\[ W_e=\int {E^2\over 8\pi}\,dV={V\over 8\pi c^2}\sum_n j_n^2, \]

expressing \(\mathbf E\) through \(\mathbf A\) and using formulas (4.5), (4.6), and (4.7), we find the following expression for the energy of the electric field in a cavity with radiation:

\[ W_e={V\over 8\pi c^2}\sum_n \dot q_n^{\,2}. \tag{4.11} \]

Similarly, for the magnetic energy we have

\[ W_m=\int {H^2\over 8\pi}\,dV =\sum_{n,m}\int q_n q_m(\operatorname{rot}\mathbf A_n\,\operatorname{rot}\mathbf A_m)\,dV. \]

To simplify this expression, let us compute separately the integral

\[ \int(\operatorname{rot}\mathbf A_n\,\operatorname{rot}\mathbf A_m)\,dV =\int \operatorname{div}(\mathbf A_n\,\operatorname{rot}\mathbf A_m)\,dV +k_m^2\int \mathbf A_n\mathbf A_m\,dV \]

\[ =\int(\mathbf A_n\,\operatorname{rot}\mathbf A_m)\,ds +k_m^2\int \mathbf A_n\mathbf A_m\,dV. \]

The surface integral vanishes, since the component normal to the surface of the vector \(\mathbf A_n\,\operatorname{rot}\mathbf A_m\) is equal to zero. Consequently,

\[ \int \operatorname{rot}\mathbf A_n\,\operatorname{rot}\mathbf A_m\,dV = \begin{cases} 0, & n\ne m,\\ k_n^2 V, & n=m. \end{cases} \]

Thus, finally, for the magnetic energy we find

\[ W_m={V\over 8\pi}\sum_n k_n^2 q_n^2. \tag{4.12} \]

Since the individual natural oscillations are independent and the interaction energy between them is zero, as is seen from the orthogonality condition for the vectors \(\mathbf A_n\), the total energy of the field is simply equal to the sum of the electric and magnetic energies, i.e.

\[ W_n=\frac{V}{8\pi c^2}\left[\dot q_n^{\,2}+(c k_n)^2 q_n^{\,2}\right]. \tag{4.13} \]

The expression for \(dW_n/dt\) can be found from (4.10) in exactly the same way as the energy integral is found for a system of particles in mechanics. Simple calculations give

\[ \frac{dW_n}{dt}=V\dot q_n(t) I_n(t), \tag{4.14} \]

i.e. the rate of increase of the energy of the electromagnetic field of the \(n\)-th oscillation is equal to the rate of increase of the amplitude of this oscillation, multiplied by the excitation current corresponding to this oscillation and by the volume of the cavity filled by the field.

This expression is completely analogous to the expression for power in mechanics, equal to the product of force (in our case \(I_n\)) by velocity (in our case proportional to the rate of increase of the amplitude \(q_n\)).

Effective inductance and capacitance. For those accustomed to thinking in terms of resonant circuits with their capacitances and inductances, it will be useful to introduce certain quantities which, for a cavity resonator, play the role of effective capacitance and inductance. The ordinary inductance is related to the magnetic energy by the relation \(W_m=Li^2/2\), where \(W_m\) is expressed in ergs, \(i\) in absolute amperes, and \(L\) in centimeters.

In our case, the amplitude \(q_n\) of the \(n\)-th oscillation plays, in the magnetic energy, the role of the current, so that we may identify the coefficient of \(q_n^2\) in (4.12) with one half of the effective inductance \(L_n\) of the \(n\)-th natural oscillation:

\[ L_n=\frac{V k_n^2}{4\pi}=\frac{\pi V}{\lambda_n^2}. \tag{4.15} \]

For conversion to practical units, let us recall that an inductance of \(1\ \text{cm}\) is equal to \(10^{-9}\) henry.

We define the capacitance of the \(n\)-th natural oscillation of the field so that the product \(L_n C_n\) gives the correct resonant frequency in accordance with the known relation

\[ \lambda_n=2\pi\sqrt{L_n C_n}. \]

Hence it is easy to find the electrostatic capacitance \(C_n\) associated with the \(n\)-th natural oscillation of the field:

\[ C_n=\frac{4\pi}{V k_n^4}. \tag{4.16} \]

Example: Show that the normalized vector \(\mathbf A_4\) for the proper oscillation \((110)\) in a cubical resonator with edge \(A\) will be

\[ \mathbf A_4 = 2\mathbf k \sin \frac{\pi x}{A}\sin \frac{\pi y}{A}, \]

where \(\mathbf k\) is the unit vector directed along the \(z\)-axis. Show also that the electric field attains its maximum value on the line \(x=A/2,\ y=A/2\), and that if the amplitude \(q_{110}=1\) abs. units, then the maximum value of the electric field is equal to \(4\pi \lambda\) abs. units, where \(\lambda=\sqrt{2}A\).

Influence of volume charges. We now pass to the consideration of the more general case when, in the cavity of the resonator, both the current density and the charge density are different from zero. In this case the expansion (4.7) no longer applies, since it leads to the equality \(\operatorname{div}\mathbf A=0\), which is no longer satisfied.

The same applies also to the expansion (4.8). The necessary generalization of the relations obtained above consists in the following.

Let us suppose that the homogeneous boundary-value problem for the scalar function

\[ \begin{gathered} \Delta \varphi_m + k_m^2 \varphi_m = 0 \\ \varphi_m = 0 \quad \text{on the boundary of the region} \end{gathered} \tag{4.17} \]

has been solved, and that the corresponding eigenfunctions and eigenvalues are known. The functions \(\varphi_m\) may be regarded as orthogonal and normalized,

\[ \varphi_l \Delta \varphi_m - \varphi_m \Delta \varphi_l + (k_m^2-k_l^2)\varphi_m\varphi_l = 0 \]

or

\[ \varphi_l \Delta \varphi_m - \varphi_m \Delta \varphi_l = \operatorname{div}(\varphi_l \operatorname{grad}\varphi_m-\varphi_m \operatorname{grad}\varphi_l) = (k_l^2-k_m^2)\varphi_m\varphi_l . \]

Integrating over the whole volume of the cavity, we have

\[ (k_l^2-k_m^2)\int \varphi_m\varphi_l\,dV = \int \operatorname{div}(\varphi_l \operatorname{grad}\varphi_m-\varphi_m \operatorname{grad}\varphi_l)\,dV = \int(\varphi_l \operatorname{grad}\varphi_m-\varphi_m \operatorname{grad}\varphi_l)\,dS =0 \]

for \(l\ne m\).

The normalization condition imposed on the functions \(\varphi_m\) has the same form as (4.6):

\[ \int \varphi_m^2\,dV = V. \tag{4.18} \]

We shall further suppose that the charge density \(\rho(x,y,z,t)\) can be expanded in a series in the orthogonal functions \(\varphi_m\), i.e.

\[ \rho(x,y,z,t)=\sum_m R_m(t)\varphi_m(x,y,z) \tag{4.19} \]

and, analogously, the scalar potential \(\varphi(x,y,z,t)\)

\[ \varphi(x,y,z,t)=\sum_m \Phi_m(t)\varphi_m(x,y,z). \tag{4.20} \]

Substitution of these expansions into the equation for \(\varphi\) (4.3) leads to equations for the coefficients \(\Phi_m\), entirely analogous to (4.10):

\[ \ddot{\Phi}_m(t)+(ck_m)^2\Phi_m=4\pi c^2 R_m . \tag{4.21} \]

Into the expressions for \(\mathbf A\) and \(\mathbf j\) such additions must be introduced that their divergence does not vanish. Suitable functions for this purpose are

\[ \mathbf B_m=\frac{1}{k_m}\operatorname{grad}\varphi_m,\qquad \operatorname{rot}\mathbf B_m=0. \tag{4.22} \]

The functions \(\mathbf B_m\) are orthogonal to one another and to the functions \(\mathbf A_n\). To verify the latter assertion, we shall use the general formula

\[ \int\left(\operatorname{rot}\mathbf a\,\operatorname{rot}\mathbf b+\operatorname{div}\mathbf a\,\operatorname{div}\mathbf b+\mathbf a\,\operatorname{rot}\operatorname{rot}\mathbf b\right)dV = \int(\mathbf a\,\operatorname{rot}\mathbf b)d\mathbf S+\int(\operatorname{div}\mathbf b\cdot\mathbf a)d\mathbf S . \]

Identify the vector \(\mathbf a\) with \(\mathbf A_n\) and \(\mathbf b\) with \(\mathbf B_m\). Then the first two integrals on the left-hand side vanish, the first because \(\operatorname{div}\mathbf A_n\) is equal to zero, the second because \(\operatorname{rot}\mathbf B_m\) is equal to zero, and the third becomes

\[ -k_n^2\int \mathbf A_n\mathbf B_m\,dV . \]

On the right-hand side the first integral vanishes, since the vector \(\mathbf B_m\) is perpendicular to the surface, and the second because \(\operatorname{div}\mathbf A_n=0\). The factor \(1/k_m\) has been introduced in (4.22) so that \(\mathbf B_m\) be normalized in the same way as \(\mathbf A_n\):

\[ \int \mathbf B_m\mathbf B_l\,dV= \begin{cases} 0, & m\ne l,\\ V, & m=l . \end{cases} \tag{4.23} \]

The latter follows from the relation

\[ \int \operatorname{grad}\varphi_m\,\operatorname{grad}\varphi_l\,dV = \int \operatorname{div}(\varphi_m\operatorname{grad}\varphi_l)\,dV - \int \varphi_m\Delta\varphi_l\,dV . \]

We shall now assume that the expressions (4.7) and (4.8), in the presence of volume charges in the cavity, are generalized as follows:

\[ \left. \begin{aligned} \mathbf A&=\sum_n q_n(t)\mathbf A_n+\sum_m P_m(t)\mathbf B_m,\\ \mathbf j&=\sum_n I_n(t)\mathbf A_n+\sum_m H_m(t)\mathbf B_m . \end{aligned} \right\} \tag{4.24} \]

Substituting these expressions into the equation for \(\mathbf A\) (4.3), we find the equation for \(P_m\):

\[ \ddot P_m(t)+(ck_m)^2P_m(t)=4\pi c^2H_m(t), \tag{4.25} \]

which, together with equations (4.21) and (4.10), forms a complete system of equations for all amplitudes of the wave field. A significant complication arises when the influence of volume charges on signi-

ing capacitance. In doing this it is necessary not only to introduce the scalar potential into the energy, but also to take into account the change arising in the vector potential \(\mathbf A\).

Analogously to (4.3), one can show that if at the initial instant the equalities

\[ \dot{\Phi}_{m}-ck_m^{b}P_m=0 \]

and

\[ \frac{d}{dt}\left(\dot{\Phi}_{m}-ck_m^{b}P_m\right)=0, \]

hold, then they remain satisfied for all subsequent time. Consequently, the solutions of equations (4.10), (4.21), and (4.25) must be chosen so that they satisfy these conditions, as part of the initial conditions of the problem.

If the electric and magnetic energy of the field is computed, it turns out that no changes occur in the magnetic energy, since the curl of the terms added to the vector potential is zero; in the electric energy, however, new terms appear, and it takes the form

\[ W_e=\frac{V}{8\pi c^2}\sum_n \dot{q}_n^{\,2} +\frac{V}{8\pi c^2}\sum_m \dot{P}_m^{\,2} +\frac{V}{8\pi}\sum_m k_m^2\Phi_m^2 . \tag{4.26} \]

§ 5. Cylindrical resonator*)

By a cylindrical resonator we shall mean a resonator bounded at its ends by the planes \(z=0\) and \(z=C\), with the cross-section in any plane \(z=\mathrm{const}\) representing one and the same curve. For such a resonator the problem of the spatial distribution of the field in its most general form can be reduced to a two-dimensional problem. We shall proceed from Maxwell’s equations (3.1), in which \(\varepsilon=\mu=1\):

\[ \begin{aligned} \operatorname{div}\mathbf E&=0, & \operatorname{div}\mathbf H&=0;\\ \operatorname{rot}\mathbf E&=-ik\mathbf H, & \operatorname{rot}\mathbf H&=+ik\mathbf E. \end{aligned} \qquad\} \tag{5.1} \]

It is natural to assume that the solution depends on the coordinate \(z\) in the same way as in the case of the rectangular box, i.e. is expressed through \(\cos k_z z\) and \(\sin k_z z\).

All possible natural oscillations of the field in a cylindrical resonator can be divided into two classes: oscillations of the \(E\)-type, for which

\[ E_z\ne 0,\quad \text{and } H_z=0, \tag{5.2} \]

and oscillations of the \(H\)-type, for which

\[ H_z\ne 0,\quad \text{and } E_z=0. \]

*) The literature on this question is very scanty. Some general questions are covered in books; see \({}^{3}\).

E-type oscillations. Let us first consider oscillations of the \(E\)-type. Since for these oscillations \(H_z=0\), the equations for \(\operatorname{rot} H\) have the form

\[ \left. \begin{aligned} ikE_x&=-\frac{\partial H_y}{\partial z},\\ ikE_y&=+\frac{\partial H_x}{\partial z},\\ ikE_z&=\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}, \end{aligned} \right\} \tag{5.3} \]

and, analogously, the equations for \(\operatorname{rot} E\)

\[ \left. \begin{aligned} -ikH_x&=\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z},\\ -ikH_y&=\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x},\\ 0&=\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}. \end{aligned} \right\} \tag{5.4} \]

The components \(E_x\) and \(E_y\) can be expressed in terms of \(E_z\):

\[ \frac{\partial^2 E_x}{\partial z^2}+k^2E_x= \frac{\partial}{\partial x}\left(\frac{\partial E_z}{\partial z}\right), \]

\[ \frac{\partial^2 E_y}{\partial z^2}+k^2E_y= \frac{\partial}{\partial y}\left(\frac{\partial E_z}{\partial z}\right). \]

The left-hand side of these equations is equal, respectively, to \((k^2-k_z^2)E_x\) and \((k^2-k_z^2)E_y\) for any dependence of \(E_z\) on the coordinate \(z\). Therefore, if by \(E_s\) we denote the components of the vector \(\mathbf E\) in the cross-section of the resonator, perpendicular to the \(z\)-axis,

\[ E_s=\mathbf i E_x+\mathbf k E_y, \]

then the equations may be written in vector form

\[ (k^2-k_z^2)E_s=\operatorname{grad}_s\frac{\partial E_z}{\partial z}, \tag{5.5} \]

where \(\operatorname{grad}_s\) denotes the gradient in the transverse section.

Using next equations (5.4), in order to eliminate the magnetic field from the third equation (5.3), we obtain the basic equation describing the variation of \(E_s\) over the section:

\[ \Delta_s E_s+(k^2-k_z^2)E_s=0, \tag{5.6} \]

where \(\Delta_s\) denotes the Laplace operator for the section (equal to the ordinary Laplace operator without derivatives with respect to \(z\)).

Finally, with the aid of (5.4) one can express the magnetic field in terms of the electric field

\[ H_x=\frac{ik}{k^2-k_3^2}\frac{\partial E_z}{\partial y},\qquad H_y=-\frac{ik}{k^2-k_3^2}\frac{\partial E_z}{\partial x}. \tag{5.7} \]

or, in vector form,

\[ \mathbf H=-\frac{ik}{k^2-k_3^2}\,\mathbf k\,\operatorname{grad}_{s} E_z, \tag{5.8} \]

where \(\mathbf k\) is the unit vector in the direction of the \(z\)-axis.

The boundary condition for the electric field \(\mathbf E\) is the normality of the vector \(\mathbf E\) to the boundary surfaces of the resonator. To satisfy this condition at the ends of the cylinder in \(E_z\), one must take the dependence on \(\cos k_3 z\), and not on \(\sin k_3 z\). The condition on the walls of the cylinder leads to the fact that admissible are only those solutions of equation (5.6) which vanish on the boundary of the domain.

Let us denote by \(\psi_n(x,y)\) and \(k_n\) the eigenfunctions and eigenvalues of the two-dimensional boundary-value problem

\[ \Delta_s^2\psi_n(x,y)+k_n^2\psi_n(x,y)=0, \tag{5.9} \]

\[ \psi_n(x,y)=0\quad \text{on the boundary of the domain.} \]

In mathematics the solution of this problem is well known for the most varied boundaries, since problems of this type are often encountered in other branches of mathematical physics.

Then for the natural oscillations of \(E\)-type we finally have:

\[ \begin{gathered} E_z=A\psi_n(x,y)\cos k_3 z,\\ k^2=k_n^2+k_3^2,\\ \mathbf E_s=-\left(\frac{k_3}{k_n^2}\right)A \left(\frac{\partial\psi_n}{\partial x}\mathbf i+ \frac{\partial\psi_n}{\partial y}\mathbf k\right)\sin k_3 z,\\ \mathbf H_s=-i\left(\frac{k}{k_n^2}\right)A \left(-\frac{\partial\psi_n}{\partial y}\mathbf i+ \frac{\partial\psi_n}{\partial x}\mathbf j\right)\cos k_3 z. \end{gathered} \tag{5.10} \]

For the vector potential we have

\[ \mathbf A=B\left\{\psi_n(x,y)\cos k_3z\cdot \mathbf k -\frac{k_3}{k_n^2}\operatorname{grad}_{s}\psi_n\sin k_3z\right\}, \tag{5.11} \]

where \(B\) is a normalization constant chosen so that condition (4.6) is fulfilled. Namely, we have:

\[ \int A^2\,dV = B^2\frac{C}{2} \left[ \iint \psi_n^2\,dx\,dy + \frac{k_3^2}{k_n^4} \iint (\operatorname{grad}\psi_n)^2\,dx\,dy \right]. \]

Since

\[ (\operatorname{grad}\psi)^2=\operatorname{div}(\psi\operatorname{grad}\psi)-\psi\Delta\psi, \]

this condition reduces to

\[ \int A^{2}dV = B\,\frac{C}{2} \left[1+\frac{k_z^2}{k_n^2}\right] \int \psi_n^2\,dx\,dy, \]

so that, if \(V\) denotes the volume of the resonator, then

\[ B^{2}=\frac{2Vk_n^{2}}{Ck^{2}\int \psi_n^{2}\,dx\,dy}. \tag{5.12} \]

Oscillations of \(H\)-type. The theory of natural oscillations of \(H\)-type is completely analogous to the theory of \(E\)-type oscillations just set forth. Instead of (5.3) and (5.4) we have

\[ ikE_x=\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}, \qquad ikE_y=\frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}, \]

\[ 0=ikE_z=\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}, \]

\[ -ikH_x=-\frac{\partial E_y}{\partial z}, \qquad -ikH_y=\frac{\partial E_x}{\partial z}, \qquad -ikH_z=\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}. \]

These equations make it possible to express \(H_x\) and \(H_y\) through \(H_z\) and to obtain, for the components of the field lying in the cross-section of the resonator, the equation

\[ (k^{2}-k_z^{2})H_s=\operatorname{grad}_s\left(\frac{\partial H_z}{\partial z}\right). \tag{5.13} \]

For \(H_z\) we have, analogously to (5.6),

\[ \Delta_s H_z+(k^{2}-k_z^{2})H_z=0 \tag{5.14} \]

and, finally,

\[ E_s=\frac{ik}{k^{2}-k_z^{2}}\,k\,\operatorname{grad}_s H_z, \tag{5.15} \]

analogously to (5.8).

Since the boundary conditions require that \(\mathbf H\) be parallel to the wall, the factor \(\sin k_z z\) must be taken in \(H_z\). From (5.13) we see that, in order for the vector \(\mathbf H_s\) to be tangential to the surface of the cylinder, the normal gradient of \(H_z\) must vanish on the walls.

Let \(\varphi_m(x,y)\) and \(k_m\) denote the eigenfunctions and eigenvalues of the two-dimensional problem

\[ \Delta_s\varphi_m+k_m^{2}\varphi_m=0, \tag{5.16} \]

\[ \partial\varphi_m/\partial n \quad \text{on the boundary}, \]

where \(\partial/\partial n\) denotes differentiation along the normal to the surface. The difference in the boundary conditions (5.9) and (5.16) leads to the fact that the \(H\)-type problem has a different set of eigenvalues and eigenfunctions.

For natural oscillations of the \(H\)-type we have

\[ \left. \begin{aligned} H_z &= A\varphi_m(x,y)\sin k_z z,\\ k^2 &= k_m^2+k_z^2,\\ \mathbf{H}_s &= \left(\frac{k_z}{k_m^2}\right)A\,\operatorname{grad}_s\varphi_m\cos k_z z,\\ \mathbf{E}_s &= i\left(\frac{k}{k_m^2}\right)A\left(-\frac{\partial \varphi_m}{\partial y}\mathbf{i} +\frac{\partial \varphi_m}{\partial x}\mathbf{j}\right)\sin k_z z. \end{aligned} \right\} \tag{5.17} \]

For the vector potential of these oscillations one may put

\[ \mathbf{A}=B\left(-\frac{\partial \varphi_m}{\partial y}\mathbf{i} +\frac{\partial \varphi_m}{\partial x}\mathbf{j}\right)\sin k_z z, \]

where \(B\) is a normalizing factor, equal, as calculation shows, to

\[ B^2=\frac{2V}{Ck_m^2\displaystyle\int \varphi_m^2\,dx\,dy}. \tag{5.18} \]

Let us note that among the oscillations of the \(E\)-type there is an oscillation with \(k_z=0\), for which the resonance frequency does not depend on the height of the cylinder, whereas oscillations of the \(H\)-type exist only for \(k_z\ne0\).

The permissible values of \(k_z\) are, of course, only the values

\[ k_z=\frac{n\pi}{C}\qquad (n\text{—an integer}). \tag{5.19} \]

We shall need special designations for both kinds of natural oscillations of a cylindrical resonator. The most convenient designations for oscillations of the \(E\)- and \(H\)-types are respectively \(E(n,l)\) and \(H(m,l)\). For cylinders with special cross-sectional shapes the indices \(n\) and \(m\) will in some cases be replaced by more convenient indices.

Resonator with double walls. If the cross-section of a cylindrical resonator is bounded on the outside by the curve \(C_1\), and on the inside by the curve \(C_2\), as indicated in Fig. 4, then the cavity of the condenser is no longer a simply connected region. This means that an arbitrary closed contour inside the region cannot be contracted to a point by a continuous deformation within the region. The multiply connected nature of the region entails certain important consequences.

Fig. 4. Diagram of the cross-section of a resonator with double walls.

In practice the curves \(C_1\) and \(C_2\) are usually concentric circles, but, as we shall now see, the general properties of resonators with double walls do not depend on the form of the boundary curves.

The two most important features of such resonators are: 1) that a purely magnetic static field can exist in the cavity of the resonator, and 2) that in them the existence of

systems of natural oscillations with a frequency depending only on the length, but not on the transverse dimensions of the cylinder, for which the terms \(E_z\) and \(H_z\) vanish simultaneously. We shall call such oscillations oscillations of a coaxial cable.

If \(E_z\) and \(H_z\) are simultaneously equal to zero, then the system of Maxwell equations (5.3) and (5.4) has the form

\[ ikE_x=-\frac{\partial H_y}{\partial z},\qquad ikE_y=\frac{\partial H_x}{\partial z}, \]

\[ 0=\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}; \]

\[ -ikH_x=-\frac{\partial E_y}{\partial z},\qquad -ikH_y=\frac{\partial E_x}{\partial z}, \]

\[ 0=\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}. \]

From the expressions for the \(z\)-components of the equations for the electric and magnetic fields it follows that \(\mathbf E\) and \(\mathbf H\) can be represented as the gradient of a certain scalar function \(U(x,y,z)\).

Namely, put

\[ \begin{aligned} E_s&=-\operatorname{grad}_s U(x,y,z);\\ ikH_s&=k\operatorname{grad}\frac{\partial U}{\partial z}. \end{aligned} \tag{5.20} \]

Then from the first two equations we have

\[ \frac{\partial^2}{\partial z^2}\left(\frac{\partial U}{\partial x}\right) +k^2\left(\frac{\partial U}{\partial x}\right)=0, \]

\[ \frac{\partial^2}{\partial z^2}\left(\frac{\partial U}{\partial y}\right) +k^2\left(\frac{\partial U}{\partial y}\right)=0, \]

and from the third we find

\[ \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right) \left(\frac{\partial U}{\partial z}\right)=0. \]

The boundary conditions state that \(E_s\) vanishes for \(z=0\) and \(z=C\) and on the side walls of the vessel. To satisfy these conditions, put

\[ U(x,y,z)=u(x,y)\sin\frac{n\pi z}{C}, \]

where

\[ \Delta U=0 \]

and \(u\) becomes constant on the boundary curves \(C_1\) and \(C_2\). If we were dealing with a simply connected domain bounded by a single curve \(C_2\), then, as is known from potential theory, the function \(u\), satisfying Laplace’s equation inside the domain and constant on its boundary, would be constant everywhere. Therefore the electric and magnetic field inside the domain are identically and

would vanish. This shows that in a simply connected cavity the existence of an electromagnetic field with simultaneously vanishing \(z\)-components \(E_z\) and \(H_z\) is impossible.

The situation is different, however, in a doubly connected region between the curves \(C_1\) and \(C_2\). Here we can satisfy the boundary conditions by putting \(u=u_1\) on the curve \(C_1\) and \(u=u_2\) on the curve \(C_2\), where \(u_1\) and \(u_2\) are different constants. In this case the function \(u(x,y)\) will no longer be constant, but will turn into a function coinciding with the electrostatic potential satisfying the indicated conditions. Since the wave number \(k\) does not enter the boundary conditions on the lateral surfaces of the cylinder, it is completely determined by the conditions on the end surfaces, i.e. \(k=n\pi/C\). It follows from this that, for any form of the curves \(C_1\) and \(C_2\), in a resonator with double walls there may exist natural oscillations with wavelength \(\lambda=2C/n\), where \(n\) is an integer.

If we pass to the limit \(k\to0\), then the electric field will evidently turn to zero [since \(U(x,y,z)=u(x,y)\sin kz\) tends in this case to zero]. However, the magnetic-field strength will be different from zero and equal to

\[ \mathbf{H}_s=k\,\operatorname{grad}u. \]

Thus, in the resonator there may exist a constant magnetic field produced by the circulation of a constant current flowing along the inner and outer walls of the cylinder. Such static fields always exist in a resonator whose interior is a multiply connected region.

Use of function theory. Since the function \(u(x,y)\) satisfies Laplace’s equation in two dimensions, a number of results can be obtained with the aid of function theory.

For what follows, put \(z=x+iy\) (not to be confused with the coordinate \(z\) along the axis of the cylinder). Let, further,

\[ w=f(z)=u(x,y)+iv(x,y) \tag{5.21} \]

be an analytic function of \(z\). In order that \(w\) have a single-valued and continuous derivative \(f'(z)\), the Cauchy–Riemann conditions must be satisfied:

\[ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}. \tag{5.22} \]

From (5.19) it follows that

\[ E_x=-\left(\frac{\partial u}{\partial x}\right),\qquad E_y=-\left(\frac{\partial u}{\partial y}\right), \]

\[ iH_x=-\left(\frac{\partial v}{\partial x}\right),\qquad iH_y=-\left(\frac{\partial v}{\partial y}\right), \]

if one abstracts from the sinusoidal or cosinusoidal dependence of \(E\) and \(H\) on the coordinate \(z\), directed along the axis of the cylinder.

The formulas (5.22) may be written briefly in vector form as

\[ \mathbf E-i\mathbf H=-\operatorname{grad} w. \tag{5.23} \]

Let us now consider an arbitrary function \(w=f(z)\) such that the equation \(u(x,y)=\mathrm{const.}\) determines a family of closed curves, each of which encloses all the preceding ones. Any two of these curves may be chosen as the boundary curves \(C_1\) and \(C_2\) of a hollow resonator with double walls; consequently, each of the functions \(u\) may represent a solution of the field equations for the whole family of hollow resonators.

Circular coaxial cable. The simplest of the applications of the general method is the solution for a circular coaxial cable. It is given by the function

\[ w=\log z \tag{5.24} \]

or

\[ e^{u+iv}=z, \]

whence

\[ e^u=|z|=r, \]

so that

\[ u=\log r \]

and

\[ v=\arg z=\varphi, \]

where

\[ \log z=\log r+i\varphi. \]

Consequently, the lines \(u=\mathrm{const.}\) represent circles \(r=\mathrm{const.}\), and the lines \(v=\mathrm{const.}\) are rays drawn at an angle \(\varphi=\mathrm{const.}\)

The electric and magnetic fields, according to (5.23), will be

\[ \mathbf E-i\mathbf H = -\operatorname{grad} w = -\left(\frac{1}{r}\right)\mathbf r_0 -i\left(\frac{1}{r}\right)\boldsymbol{\varphi}_0 . \tag{5.25} \]

The electric field is directed along the radius vector, the magnetic field is circular, and both decrease inversely proportionally to the radius. If the radius of the inner cylinder is \(a\), and that of the outer is \(b\), then the density of the current flowing over the surface of the inner cylinder will be equal to \(1/4\pi a\) abs. units/cm, and that on the outer to \(1/4\pi b\). The total currents flowing over both surfaces are equal to one another and amount to \(1/2\) abs. A; the line integral of the field vector within the limits from \(r=a\) to \(r=b\) is equal to \(\log(a b)\) abs. units. Therefore, if the amplitude of excitation of the resonator is such that the maximum amplitude of the current (at the points where \(\cos \pi z/C=\pm 1\), i.e. \(z=0\) or \(v=\pm C/m\)) is equal to \(1\) A, the maximum amplitude of the field strength (at the points where \(\sin \pi z/C=\pm 1\)) is equal to \(60\log b/a\). This is often expressed in words: “the impedance of a circular coaxial cable is equal to \(60\log b/a\) ohms.”

If the boundary curves are not circles, but more complicated curves, then the solution may be obtained in the following way: suppose that the coordinates \((x,y)\) are periodic func-

we \(v\); without restricting generality, one may assume that the period is \(2\pi\) and that the function of a complex variable satisfying this condition is the Fourier expansion

\[ z=\sum_{m=-\infty}^{\infty} A_m e^{m(u+iv)}; \tag{5.26} \]

the term \(A_0\) may be excluded from the expansion, since if it were different from zero, then by shifting the origin of coordinates in the \((x,y)\)-plane in the corresponding manner, one could make it vanish.

A circular coaxial transmitter is obtained for \(A_1=1\), \(A_m=0\) (for \(m\ne 1\)).

Elliptic coaxial cable. An interesting case is when

\[ A_1=A_{-1}=\frac{f}{2}, \qquad A_m=0 \quad \text{for } m\ne \pm 1. \]

Formula (5.26) gives

\[ z=f(\cos hu\cos v+i\sin hu\sin v). \]

It follows from this that the curves \(u=\mathrm{const.}\) are confocal ellipses

\[ \left(\frac{x}{f\cos hu}\right)^2+ \left(\frac{y}{f\sin hu}\right)^2=1 \]

with foci at the points \((x,y)=(\pm f,0)\).

Thus, in this case we obtain the theory of a resonator bounded by two confocal elliptic cylinders.

If the inner and outer cylinders have semimajor axes \(a\) and \(b\), respectively, \((a,b>f)\) on the inner and outer walls, then

\[ \cos hu_1=\frac{a}{f} \quad \text{and} \quad \cos hu_2=\frac{b}{f}. \]

The linear integral, taken from the inner to the outer wall, will be equal to

\[ u_2-u_1=\cos h^{-1}\left(\frac{b}{f}\right)-\cos h^{-1}\left(\frac{a}{f}\right). \]

The magnetic field at the point \((u,v)\) is equal to \(-\operatorname{grad} v\), so that the axial current per unit length on each of the walls is \(1/4\pi\,\operatorname{grad} v\) abs. units. The total current along the wall is equal to \(\frac{1}{2}\) abs. units, since the integral of \(\operatorname{grad} v\) over the surface is equal to \(2\pi\). Consequently, determining the impedance in the same way as for a circular coaxial transmitter, we find for the impedance of the transmitter

\[ 60\left[\cos h^{-1}\frac{b}{f}-\cos h^{-1}\frac{a}{f}\right] = 60\lg\frac{b+(b^2-f^2)^{1/2}}{a+(a^2-f^2)^{1/2}}. \tag{5.27} \]

We can finally obtain a general expression for the impedance of a resonator bounded by two cylinders of arbitrary shape.

Independently of the shape of the surface, the total current flowing along the inner conductor is equal to \(\frac{1}{2}\) abs. units, and the line integral of the electric-field vector, taken along a path from the inner to the outer conductor, is equal to \(u_2-u_1\) abs. units, if the conductors are at potentials \(u_1\) and \(u_2\).

Consequently, the impedance of the resonator, independently of the shape of the bounding cylinders, is equal to

\[ 60(u_2-u_1). \tag{5.28} \]

§ 6. Circular Cylinder

The general conclusions of the preceding paragraph may be illustrated by the example of a circular cylinder of radius \(R\), which is of considerable practical interest. Instead of Cartesian coordinates \(x,y\), it is convenient to introduce polar coordinates.

Equations (5.9) and (5.15) will be satisfied by solutions of the type

\[ I_m(kr)e^{im\varphi}, \tag{6.1} \]

where \(I_m(x)\) is a Bessel function with integral index \(m\). The boundary conditions for oscillations of \(E\)-type will be satisfied if \(k_a\) is chosen so that

\[ I_m(k_a R)=0. \tag{6.2} \]

The latter condition leads us to the necessity of replacing the index \(a\) in \(k\) by two indices \(m\) and \(p\), where \(m\) is the order of the Bessel function and \(p\) is the number of the root of the corresponding Bessel function. Some of the roots of Bessel functions are given in Table 1. The roots

Table 1

Values of the roots \(X_{mp}\) of the equation \(I_m(X_{mp})=0\)

\(p\)	\(m=0\)	1	2	3
1	2.405	3.832	5.135	6.379
2	5.520	7.016	8.417	9.760
3	8.654	10.173	11.620	13.017
4	11.792	13.323	14.796	16.229

are arranged in order of increasing numerical value. Thus, we shall denote each \(E\)-oscillation by \(E=(n,m,p)\), and the corresponding wave number by \(k_{Enmp}\).

The wave number of the oscillation satisfying the boundary conditions will be

\[ k^2_{E nmp}=\frac{X^2_{mp}}{R^2}+\frac{n^2\pi^2}{C^2}. \tag{6.3} \]

The \(E\)-type oscillation of the lowest frequency will be \(E(001)\). Its wave number

\[ k_{E001}=2.405/R \quad \text{or wavelength} \quad \lambda_{E001}=2.61R. \tag{6.4} \]

The next oscillation in the symmetric series \(m=0\) will be \(E(002)\). Its wave number and wavelength are, respectively,

\[ k_{E002}=5.520/R \quad \text{and} \quad \lambda_{E002}=1.14R. \tag{6.5} \]

We see that the frequency of this oscillation exceeds the lowest natural frequency of the resonator by more than a factor of two.

Similarly, for \(H\)-type oscillations the boundary conditions require that the wave number \(k_b\) satisfy the equation

\[ I'_m(k_bR)=0. \tag{6.6} \]

The roots of this equation are given in Table 2.

The frequencies of \(H\)-type oscillations will therefore be determined by the equation

\[ k^2_{H nmp}=\frac{Y^2_{mp}}{R^2}+\frac{n^2\pi^2}{C^2}. \tag{6.7} \]

Table 2

Values of the roots \(Y_{mp}\) of the equation \(I'_m(Y_{mp})=0\)

\(p\)	\(m=0\)	\(1\)
1	3.832	1.840
2	7.016	5.335
3	10.173	8.535
4	13.323	11.705

From Table 2 we see that \(Y_{11}\) is smaller than any of the \(X_{mp}\). However, since for \(H\)-type waves the value \(n=0\) is forbidden, it is easy to see that the frequency of the oscillation \(E(001)\) is lower than the frequency of \(H(111)\) only if \(C<1.15R\). If, however, \(C>1.15R\), then, conversely, the frequency of the oscillation \(H(111)\) is found to be lower than the frequency of the oscillation \(E(001)\).

It is easy to generalize the results obtained to the case of a resonator having the form of a sector of a circular cylinder. Suppose that the sector is bounded by the planes \(\varphi=0\) and \(\varphi=a\), where \(a<2\pi\). For \(E\)-type oscillations, on these planes the equality \(E_z=0\) must be satisfied. This condition may be satisfied by putting, instead of (6.1),

\[ \psi_a(r,\varphi)=I_{\frac{m\pi}{a}}(k_ar)\sin\left(\frac{m\pi\varphi}{a}\right). \tag{6.8} \]

Boundary conditions require that, for \(r=R\), the Bessel functions of the proper index \(m\pi/\alpha\) in (6.8) vanish. This condition determines the allowed values of the frequencies.

Similarly, for oscillations of the \(H\)-type we must put

\[ \varphi_b(r,\varphi)=I_{m\pi/\alpha}(k_b r)\cos \frac{m\pi\varphi}{\alpha}, \tag{6.9} \]

where the allowed wave numbers \(k_b\) are determined from the requirement that the derivative of the Bessel function vanish for \(r=R\).

Example: Show that if the resonator is a sector (with aperture angle \(\alpha\)) of concentric circular cylinders of radii \(A\) and \(B\), then the \(E\)-oscillations have the form

\[ \psi=\bigl[CI(k_a r)+DN(k_a r)\bigr]\sin \frac{m\pi\varphi}{\alpha}, \]

where \(I\) and \(N\) are two conjugate Bessel functions of fractional order \(m\pi/\alpha\). Consider the dependence of the fundamental frequency on \(\alpha\) and on the ratio \(A/B\).

§ 7. Resonator Having the Shape of a Solid of Revolution

In practice, resonators having the form of a solid of revolution are often used. In considering such resonators it is convenient to use cylindrical coordinates \(r,\varphi,z\), with their cylindrical axis coinciding with the axis of symmetry of the resonator.

In a resonator of this form there may exist symmetric oscillations, for which \(E_\varphi=0\) and \(H_\varphi\) does not depend on the angle \(\varphi\).

In this section the theory of oscillations of this type will be developed. In cylindrical coordinates, the pair of Maxwell equations for \(\operatorname{rot} E\) and \(\operatorname{rot} H\) will have the form:

\[ \left. \begin{aligned} ikE_r&=\frac{1}{r}\frac{\partial H_z}{\partial\varphi}-\frac{\partial H_\varphi}{\partial z},\\ ikE_\varphi&=\frac{\partial H_r}{\partial z}-\frac{\partial H_z}{\partial r},\\ ikH_z&=\frac{1}{r}\frac{\partial}{\partial r}(rH_\varphi)-\frac{1}{r}\frac{\partial H_r}{\partial\varphi},\\ -ikH_r&=\frac{1}{r}\frac{\partial E_z}{\partial\varphi}-\frac{\partial E_\varphi}{\partial z},\\ -ikH_\varphi&=\frac{\partial E_r}{\partial z}-\frac{\partial E_z}{\partial r},\\ -ikH_z&=\frac{1}{r}\frac{\partial}{\partial r}(rE_\varphi)-\frac{1}{r}\frac{\partial E_r}{\partial\varphi}. \end{aligned} \right\} \tag{7.1} \]

Suppose now that \(H_r=H_z=0\) and that \(H_\varphi\) does not depend on \(\varphi\). Then the equations reduce to

\[ \begin{aligned} a.\quad & ikE_r=-\frac{\partial H_\varphi}{\partial z},\\ b.\quad & ikE_\varphi=0,\\ c.\quad & ikE_z=\frac{1}{r}\frac{\partial}{\partial r}(rH_\varphi),\\ d.\quad & 0=\frac{1}{r}\frac{\partial E_z}{\partial \varphi},\\ e.\quad & -ikH_\varphi=\frac{\partial E_r}{\partial z}-\frac{\partial E_z}{\partial r},\\ f.\quad & 0=-\frac{1}{r}\frac{\partial E_r}{\partial \varphi}. \end{aligned} \tag{7.2} \]

Equations \((d)\) and \((f)\) are consequences of equations \((c)\) and \((a)\), since \(H_\varphi\) is independent of \(\varphi\) by assumption. Eliminating from \((e)\) the components of the electric field with the aid of \((a)\) and \((c)\), we find the following equation for \(H_\varphi\):

\[ \frac{\partial^2 H_\varphi}{\partial r^2} +\frac{1}{r}\frac{\partial H_\varphi}{\partial r} +\frac{\partial^2 H_\varphi}{\partial z^2} +\left(k^2-\frac{1}{r^2}\right)H_\varphi=0. \tag{7.3} \]

The boundary conditions for the vector \(\mathbf H\) impose no restrictions on the solution of equation (7.3). However, restrictions are imposed by the boundary conditions for \(\mathbf E\).

From \((a)\) and \((c)\) we have

\[ \mathbf E_s=E_r\mathbf r_0+E_z\mathbf z_0 =\frac{i}{kr}\,\boldsymbol\varphi_0\times \operatorname{grad}(rH_\varphi). \tag{7.4} \]

Let the boundary surface be the curve \(f(r,z)=0\), whose normal will be

\[ \mathbf n=\operatorname{grad} f. \]

The boundary condition for \(\mathbf E\) says that the tangential component of the vector \(\mathbf E\) vanishes on the surface of the body of revolution, so that

\[ \mathbf n\cdot \mathbf E_s=0. \]

This boundary condition can be written in scalar form as

\[ \frac{\partial f}{\partial r}\frac{\partial}{\partial r}(rH_\varphi) +\frac{\partial f}{\partial z}\frac{\partial}{\partial z}(rH_\varphi)=0 \tag{7.5} \]

on the curve \(f(r,z)=0\).

For the solution of (7.4) it is convenient to introduce the quantity \(u\), defined by

\[ u=H_\varphi r. \tag{7.6} \]

The differential equation for \(u\) will be

\[ \frac{\partial^2 u}{\partial r^2}-\frac{1}{r}\frac{\partial u}{\partial r} +\frac{\partial^2 u}{\partial z^2}+k^2u=0; \]

\[ u=0 \text{ for } r=0,\quad \frac{\partial u}{\partial r}=0 \text{ on the curve } f(r,z)=0. \tag{7.7} \]

As a simple example let us consider a coaxial cable bounded by cylinders of radii \(r=A\) and \(r=B\) and of length \(0\le z\le C\). Put \(u(r,z)=v(r)w(z)\).

Then equation (7.7) will be satisfied if

\[ v''-\frac{1}{r}v' + k_a^2 v=0, \]

\[ w''+k_3^2 w=0, \]

where

\[ k=k_a^2+k_3^2. \tag{7.8} \]

The solution of the equation for \(w\) satisfying the boundary conditions is

\[ w(z)=\cos k_n z,\quad \text{where } k_n=\frac{4\pi}{C}. \]

Fig. 5. Schematic distribution of currents and magnetic field for the natural oscillation with zero frequency of a coaxial cable.

We obtain the simplest solution for \(v\) by putting \(k_a=0\) and \(v=1\). This solution and the value \(n=0\) correspond to the lowest frequency of the natural oscillations, namely a frequency equal to zero. Thus, in the resonator there can exist a purely magnetic static field which is induced by constant currents circulating along the metallic walls of the resonator, as shown in Fig. 5. The first oscillation with a frequency different from zero is the oscillation with \(n=1\). Its frequency does not depend on the radii \(A\) and \(B\) and is obtained from the condition that the resonator length \(C\) be equal to half the wavelength of the standing oscillations. The following waves of this series have wavelengths equal to \(C=n\lambda/2\) and \(v=1\).

The equation for \(v(r)\) for \(k_a\ne0\) can be satisfied by putting

\[ v(r)=rZ_1(k_a r), \tag{7.9} \]

where \(Z_1\) is the general integral of the Bessel equation of the first order:

\[ Z_1(x)=aI_1(x)+bN_1(x). \tag{7.10} \]

The ratio \(a,b\) and the parameter \(k_a\) should be chosen so that \(v'(r)=0\) for \(r=A\) and \(r=B\). Thus, the frequencies of natural oscillations of higher orders can be calculated*).

These boundary conditions determine a sequence of values \(k_a\), each of which may correspond to any of the values \(n\), leading to a definite frequency of natural oscillations.

Quarter-wave coaxial resonator. In practice one often uses the so-called quarter-wave coaxial resonator. It is obtained by rotating about the vertical axis the figure shown in Fig. 6.

An exact theory of such a resonator is lacking; for an approximate treatment**) we divide the resonator into three regions I, II, and III and consider each of them separately.

In region I, especially for \(z\) close to zero, the field may be regarded as very close to the field of the coaxial cable considered earlier. Therefore, we shall assume that in this region

\[ u_{\mathrm{I}}=a\cos kz \qquad (0<z<C). \tag{7.11} \]

Here \(k=k_3\), since \(k_a'=0\).

Just as is done in the theory of circuits with lumped constants, we may assume that at the boundary of region I, at \(z=C\) or \(z=D\), there is a node in the voltage wave, provided that the capacitance of region II is sufficiently large. In region II we put

\[ u_{\mathrm{II}}=bJ_1(kr) \tag{7.12} \]

(\(k=k_a\), since here \(k_3=0\)). In most practically important cases the length \(B\) is small in comparison with a quarter of the wavelength, i.e. \(kB\ll \pi/2\). Therefore in this region \(rJ_1(r)\) is practically equal to the first term of the expansion of this expression in a power series in \(r\), so that to a good approximation one may put

\[ u_{\mathrm{II}}=\frac{bkr^2}{2}, \]

which corresponds to the existence in region II of a uniform axial field \(E_z=-ib\). Consequently, the line integral from the point \(z=C\) to the point \(z=D\), at a distance \(r=A\) from the axis, is equal to

Fig. 6. Cross-sectional diagram of a quarter-wave coaxial resonator.

*) This problem was examined in detail in the work of Borgnis\(^{4}\).

**) A more exact theory is given in the work of Hansen\(^{5}\); some interesting experimental results are given by Barrow and Mieher\(^{6}\).

— \(ib(C-D)\). On the line \(z=C\), for \(A\leq r\leq B\), we have, approximately,

\[ E_r=-\left(i\frac{a}{r}\right)\sin kC, \]

so that the line integral along this path is equal to

\[ -ia\sin kC\ln\left(\frac{B}{A}\right). \]

If we neglect the flux passing through region III, the line integral of the field strength \(E\), taken over a contour enclosing this region, must vanish and, consequently,

\[ a\sin kC\ln\frac{B}{A}=b(D-C). \tag{7.13} \]

The magnetic field at the boundary of the two regions remains continuous. Therefore, equating at the point \(r=A,\ z=C\) the magnetic fields of the two regions, we have

\[ a\cos kC=\frac{bkA^2}{2}. \tag{7.14} \]

In our approximate theory, with the same degree of success one could have equated the values of the magnetic fields at any other point in region III. This point has been chosen simply because there \(u_I\) and \(u_{II}\) are probably better approximated by the expressions written above than deeper in region III.

Dividing (7.13) by (7.14), we have

\[ \tg kC=\frac{2(D-C)}{A^2 k\ln\frac{B}{A}}. \tag{7.15} \]

The last expression determines the value of \(k\) through the characteristic dimensions of the resonator.

The preceding, very rough, analysis may naturally not satisfy the reader. However, it corresponds to results obtained by means of the usual engineering calculation in transmission theory, as will be shown in § 18. The latter method has the advantage that the approximations made are seen in it more clearly.

The ratio \(a/b\) may be obtained from (7.13) or (7.14) and turns out to be equal to

\[ \frac{a}{b}=\frac{D-C}{\sin kC\ln\frac{B}{A}}=\frac{kA^2}{2\cos kC}. \tag{7.16} \]

The preceding theory may be illustrated by the following numerical example.

Let us take the frequency to be 150 megacycles, which corresponds to a wavelength of 200 cm. Suppose that we wish to use a resonator with dimensions \(D-C=3\) dm, \(A=6\) dm, and \(B=10\) dm. What then will be the permissible value of \(C\)? From (7.15), since \(k=1/12.5\), we have

\[ \tan kC = 4.07 \quad \text{or} \quad kC = 76.2^\circ, \]

whence we find the value of \(C\) to be

\[ C=\frac{76.2}{90}\,\frac{\lambda}{4}=16.6\ \text{dm}. \]

Consequently, in this case the resonator must be 15% shorter than a quarter wavelength. From (7.16) we obtain for the ratio \(b/a\):

\[ \frac{b}{a}=0.0647. \]

Thus, if the excitation is such that the amplitude of the magnetic field at the point \(z=0,\ r=A\) is equal to one gauss, the amplitude of the electric field on the \(z\)-axis in region II is \(b=0.986\) abs. units (since \(A=15.26\) and \(a=15.26\)). The line integral of the electric-field vector, taken along the straight line \(r=0\) from \(z=C\) to \(z=D\), is equal to \(2260\) V.

Sometimes, for symmetric oscillations of resonators having the form of a surface of revolution, another method of calculation is used. If \(\psi\) is any function satisfying the scalar wave equation

\[ \Delta \psi + k^2 \psi = 0, \tag{7.17} \]

then, as is easy to verify, the vector

\[ \mathbf{A}=\mathbf{C}\cdot \operatorname{grad}\psi, \tag{7.18} \]

where \(\mathbf{C}\) is a constant vector, satisfies the vector wave equation

\[ \operatorname{rot}\operatorname{rot}\mathbf{A}-k^2\mathbf{A}=0. \tag{7.19} \]

Suppose that we are given particular solutions of (7.17) independent of the angle \(\varphi\), and we choose as the vector \(\mathbf{C}\) the unit vector \(\mathbf{k}\), directed along the axis of rotation. Then the vector \(\mathbf{A}\) will be oriented along the vector \(\boldsymbol{\varphi}_0\), and it is convenient to identify it with the magnetic field of symmetric oscillations. Putting

\[ \mathbf{H}=\mathbf{k}\operatorname{grad}\psi=\frac{\partial \psi}{\partial r}\,\boldsymbol{\varphi}_0, \tag{7.20} \]

we have for the electric field

\[ ik\mathbf{E}=\operatorname{rot}\mathbf{H} = -\frac{\partial^2\psi}{\partial z\partial r}\,\mathbf{r}_0 +\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \psi}{\partial r}\right)\mathbf{k}. \tag{7.21} \]

The boundary conditions state [cf. (7.5)] that the normal derivative
\(u=r_0(\partial\psi/\partial r)\) vanishes on the surface of rotation and \(u=0\) for \(r=0\). The solutions (7.17) that remain finite on the axis of rotation and do not depend on the angle \(\varphi\) have the form

\[ \psi=J_0(\alpha r)e^{i\beta z}, \qquad \alpha^2+\beta^2=k^2 . \]

Therefore the general solution is

\[ \psi(z,r)=\int_{-k}^{+k} g(\beta)J_0\!\left(\sqrt{k^2-\beta^2}\,r\right)e^{i\beta z}\,d\beta, \]

where \(g(\beta)\) is an arbitrary function. Since \(J_0'(x)=-J_1(x)\),

\[ u=r\frac{\partial\psi}{\partial r} = r\int_{-k}^{+k} h(\beta)J_1\!\left(\sqrt{k^2-\beta^2}\,r\right)e^{i\beta z}\,d\beta, \]

where \(h(\beta)\) is a new arbitrary function.

§ 8. Skin Effect

To study the question of losses in a hollow resonator caused by the finite conductivity of its enclosing metallic walls, it is first necessary to consider the process of propagation of electromagnetic waves in good conductors.

From the field equations (1.1) for a medium with constant \(\varepsilon\), \(\mu\), and \(\sigma\), one can find the wave equation describing the coordinate dependence of the field vectors, if their time dependence has the form \(e^{i\omega t}\). Namely, the equations for the field amplitudes can be written in the form

\[ \left. \begin{aligned} \operatorname{rot}\operatorname{rot}\mathbf E &= k^2\mu\left(\varepsilon-\frac{2i\lambda}{\sigma}\right)\mathbf E,\\ \operatorname{rot}\operatorname{rot}\mathbf H &= k^2\mu\left(\varepsilon-\frac{2i\lambda}{\sigma}\right)\mathbf H, \end{aligned} \right\} \tag{8.1} \]

where, as always, \(k=\omega/c\). These equations have the same form as in a nonconducting medium, but the medium is now characterized by the complex refractive index

\[ n^2=\mu\left(\varepsilon-\frac{2i\lambda}{\sigma}\right). \tag{8.2} \]

Since the electrical conductivity of metals \(\sigma\) is of the order of \(10^{-8}\ \mathrm{cm}\), the imaginary part of \(n^2\), even in the optical region—and still more so in the wavelength region that interests us—turns out to be much larger than the real part. Therefore the quantity \(\varepsilon\) in (8.2) may be neglected. Physically this means that displacement currents may be neglected in comparison with conduction currents. Then for the refractive index we have

\[ n=\sqrt{\frac{\mu}{\varepsilon}}\,(1-i). \tag{8.3} \]

The general solution of equation (8.1) can be represented in the form of plane waves

\[ \mathbf{E}=\mathbf{E}_0 e^{i\omega t-kz}=\mathbf{E}_0 e^{-z/\delta}\cos\left(\omega t-\frac{z}{\delta}\right). \tag{8.4} \]

The quantity \(\delta\), equal to

\[ \delta=\frac{1}{2\pi}\sqrt{\frac{\delta\lambda}{\mu}}, \tag{8.5} \]

is called the penetration depth. It determines the depth of penetration into the metal of a rapidly attenuating electromagnetic field. Values of \(\delta\) for several wavelengths for copper are given in Table 3. In particular, at a frequency of 60 cycles/sec. the value of \(\delta\) for copper is \(0.85\ \text{cm}\).

Table 3

Penetration depth \(\delta\) for copper

\(\lambda\ \text{cm}\)	\(\delta\ \text{cm}\)
1	0.368
3	0.670
10	1.22
100	3.86
1000	12.2

If electromagnetic waves propagate along the \(z\)-axis into the depth of the metal in such a way that the vector of the electric field is directed along the \(x\)-axis, and the vector of the magnetic field along the \(y\)-axis,

\[ H_y=H_y^0 e^{-z/\delta}\cos\left(\omega t-\frac{z}{\delta}\right), \tag{8.6} \]

then the conduction current induced in the metal is, as is known, equal to

\[ \mathbf{j}=\frac{1}{4\pi}\operatorname{rot}\mathbf{H}. \]

Using (8.6) we have

\[ j_y=j_z=0, \]

\[ j_x=-\left(\frac{H_y^0}{4\pi\delta}\right)e^{-z/\delta} \left[ \cos\left(\omega t-\frac{z}{\delta}\right) -\sin\left(\omega t-\frac{z}{\delta}\right) \right]. \tag{8.7} \]

The amount of energy converted into Joule heat per unit time in \(1\ \text{cm}^3\) of metal is equal to \(\sigma j^2\ \text{erg}/\text{cm}^3\cdot\text{sec}\); therefore the total power loss throughout the entire thickness of the metal, referred to unit area and averaged over a cycle, will be

\[ \sigma c\,\frac{(H_y^0)^2}{4\pi\delta^2} \int_0^\infty e^{-2z/\delta}\,dz = \frac{\pi\delta}{\lambda}\, \frac{c\mu(H_y^0)^2}{8\pi}. \tag{8.8} \]

In the last expression the factor \(\mu(H_y^0)^2/8\pi\) represents the energy density of the magnetic field.

Owing to the fact that the penetration depth \(\delta\) is very small, in all practically encountered cases (with the exception of very thin wires) one may regard the surface of the metal as plane and use expression (8.8) for the absorbed energy. Then the total power loss in the walls of a hollow resonator will evidently be equal to

\[ \frac{c\delta}{8\pi}\int \mu \mathbf{H}^2\,dS, \tag{8.9} \]

where the integration is carried out over the surface of all the walls of the resonator, and \(\mathbf H\) is the component of the magnetic-field intensity vector tangential to the boundary surfaces.

Because of the finite conductivity of the walls, the electric-field intensity vector is not exactly perpendicular to the boundary surface. From the expression for \(j_x\), taken at \(z=0\), we find

\[ E_x^0=\delta j_z=\frac{\sqrt{2\delta}\,H_y^0}{4\pi\sigma}\sin\left(\omega t-\frac{\pi}{4}\right). \tag{8.10} \]

From (8.10) we see, however, that the tangential component of the electric field is smaller than the corresponding component of the magnetic field in the ratio \(\sqrt{2\delta}/4\pi\sigma\).

As will be shown in the next paragraph, the losses in a hollow resonator during one cycle are small in comparison with the energy stored in the resonator. Therefore, in order to find the losses in the resonator, one may use an approximate method of calculation based on the fact that the field distribution inside the resonator is written as though the conductivity of the walls were infinitely large, and then, from the field distribution thus specified, the magnitude of the losses is found.

§ 9. Losses in the Resonator

In radio engineering it is customary to characterize losses in oscillatory systems by the quality factor of the circuit, a quantity reciprocal to the damping coefficient, \(Q\), defined by the law of decrease of the amplitude:

\[ e^{-\omega t/2Q}. \tag{9.1} \]

In this case the total field energy in the system changes according to the law

\[ W=W_0e^{-\omega t/Q}. \tag{9.2} \]

It is not difficult to see that

\[ Q=2\pi\frac{\text{total energy of the oscillatory system}}{\text{energy loss in 1 cycle}}. \tag{9.3} \]

With the aid of the formulas obtained in the preceding paragraph, the expression for \(Q\) may be written in the form

\[ Q=\frac{2}{\delta_\mu}\, \frac{\displaystyle \iiint H^2\,dV}{\displaystyle \iint H^2\,dS}. \tag{9.4} \]

For a rough estimate of the magnitude of \(Q\), let us note that, since \(\mathbf H\) has a bulge at the surface, the average value of \(H^2\) over the surface is approximately twice as large as the average value in the volume, and therefore

\[ Q\sim\frac{V}{\delta_\mu S}, \tag{9.5} \]

where \(V\) and \(S\) are the volume and the area of the resonator walls. For resonators whose linear dimensions are large in comparison with the penetration depth \(\delta\), one may expect that, in order of magnitude, the losses will be equal to the ratio of the linear dimensions to the penetration depth. Since the ratio of the two integrals in formula (9.4) has the dimension of centimeters and \(\delta\) is proportional to \(\sqrt{\lambda}\), the losses \(Q\) of two geometrically similar resonators are related as the square roots of the linear dimensions, or as the square roots of the wavelengths of the different natural oscillations.

In practice, the losses of hollow resonators in the microwave region are characterized by the value \(Q>1000\), and, consequently, the field distributions in resonators differ little from those calculated on the basis of the assumption of ideally conducting walls. Therefore, when formula (7.4) is used to find the losses, instead of the true value of the magnetic field one may substitute its value calculated under the assumption that the resonator walls are ideally conducting.

As an example, let us calculate the value of \(Q\) for \((0,m,n)\) oscillations in a rectangular resonator with edges \(A, B, C\), considered in § 3. It is not difficult to calculate that in this case

\[ Q=\frac{1}{\delta\mu}\, \frac{ABC} { BC+2AC\,\frac{\left(\frac{m}{B}\right)^2}{\left(\frac{m}{B}\right)^2+\left(\frac{n}{C}\right)^2} +2AB\,\frac{\left(\frac{n}{C}\right)^2}{\left(\frac{m}{B}\right)^2+\left(\frac{n}{C}\right)^2} }. \tag{9.6} \]

In particular, for a square prism \(B=C\),

\[ Q=\frac{B}{\delta}\, \frac{\frac{A}{B}}{1+2\frac{A}{B}}. \]

For a cube \(A=B=C\), and

\[ Q=\frac{A}{3\delta}. \]

The wavelength of the oscillation with the lowest frequency is \(\lambda=2\sqrt{2}\,A\), and therefore

\[ Q=\frac{\lambda}{3\sqrt{2}\,\delta\mu}. \]

The value of \(Q\) for a cube with copper walls, filled with a medium with \(\mu=1\), is therefore

\[ Q=5920\sqrt{\lambda}=7040\sqrt{A}. \]

For \(\lambda=10\ \text{cm}\), \(Q=18800\).

Let us now see how the equation for the normal coordinates \(g(t)\), introduced in § 4, changes when the losses in the resonator are taken into account. To take account of the losses in the resonator, it is necessary to introduce into (4.10) a term with damping. It is not difficult to verify that the damping will be

properly taken into account if this equation is rewritten in the following form:

\[ \ddot q_a+\left(\frac{\omega_a}{Q_a}\right)\dot q_a+\omega^2 q_a=4\pi c^2 I_a(t). \tag{9.7} \]

Multiplying by \(V/4\pi c^2 q_a\), we find an equation for \(W_a\), which replaces (4.14):

\[ \dot W_a=V I_a(t)q_a(t)-V\frac{\omega_a}{4\pi c^2 Q_a}q_a^2(t). \tag{9.8} \]

The second term on the right, which represents the losses in the resonator, is, obviously, essentially negative. In the stationary state, when \(q_a(t)\) is expressed by harmonic functions of time, the mean rate of conversion of the energy of the electromagnetic field into Joule heat is equal to

\[ P=\left(\frac{V\omega_a}{4\pi c^2 Q_a}\right)\frac{\dot q_a^{\,2}}{2} =\frac{2\pi^2 Vc}{\lambda_a^3 Q_a}\frac{q_a^2}{2}, \]

where \(q_a\) denotes the amplitude of the normal coordinate \(q_a(t)\). We shall call the coefficient of \(q_a^2\) the resistance of the resonator for the \(a\)-th oscillation, \(R_a\):

\[ R_a=\frac{60\lambda^2 V}{Q_a\lambda_a^3}. \tag{9.9} \]

For a cubic resonator with cube edge equal to \(A\), and copper walls, for the \((0,1,1)\)-oscillation \(R_a=0.0112\ \mathrm{ohm}\).

Shunt resistance. Another quantity which often proves more convenient for expressing losses is the so-called shunt resistance of the resonator. For certain calculations in the theory of electron-tube generators containing field resonators, we shall need to determine the amplitude of the line integral of the electric-field intensity vector along a certain path in the resonator. We have

\[ \mathbf E=-\frac{1}{c}\mathbf A_a \dot q_a. \]

Table 4

Values of the quantities entering into the expression for the shunt resistance \(S_p\)

\(p\)	\(X_{0p}\)	\(J_1(X_{0p})\)	\(\sqrt{X_{0p}}J_1^2(X_{0p})\)
1	2.4048	\(+0.5191\)	0.417
2	5.5207	\(-0.3403\)	0.272
3	8.6337	\(+0.2705\)	0.216
4	11.7915	\(-0.2325\)	0.186
5	14.9309	\(+0.2065\)	0.165

Therefore the amplitude of the electric field, expressed in volts per centimeter, is equal to \(30k_a q_a A_a\), if \(q_a\) is expressed in amperes. The linear integral of \(\mathbf E\)

\[ V_a=30k_a q_a\int \mathbf A_a\,dl. \tag{9.10} \]

We can now define the shunt resistance of the resonator as that resistance such that, if a potential difference \(V_a\) were shunted by it, the power dissipated in this resistance would be equal to the power actually dissipated in the resonator. The shunt resistance of a resonator depends not only on the resonator and the type of oscillation, but also on the path along which \(V_a\) is calculated. Denoting the shunt resistance by \(S_a\), we have

\[ \frac{V_a}{2S_a}=\frac{60\pi^2 V}{Q_a l_a^3}\,\frac{q_a^2}{2}, \]

and, consequently,

\[ S_a=60Q_a l_a\frac{\left(\int A_a\,dl\right)^2}{\iiint A_a^2\,d\sigma}. \tag{9.11} \]

Substituting for \(Q_a\) its expression from (9.4) and using the formula of vector analysis cited earlier [the formulas preceding (4.12)], one can transform this expression to another form:

\[ S_a=\frac{480\pi^3}{\lambda_a^2\delta}\, \frac{\left(\int A_a\,dl\right)^2}{\iint H^2\,dS}. \tag{9.12} \]

Thus, for example, for a cubic resonator with copper walls and edge length \(A\) cm, choosing as the path of integration a straight line passing through the point of intersection of the lines \(x=A/2,\ y=A/2\) from \(z=0\) to \(z=A\), for the \((110)\)-oscillation \(S_a\) is equal to \(105\,600\,A^{5/2}\).

\(E(00p)\)-oscillations of a circular cylinder. As an example let us calculate \(Q_a\) and \(S_a\) for \(E(00p)\)-oscillations in a circular cylinder.

As was shown in § 6, in such a resonator the electric-field intensity is expressed by the formula

\[ E_z=AJ_0(kr)\ \text{abs. units/cm}, \]

where \(kR=x_{0p}\), the \(p\)-th root of the equation \(J_0(x)=0\). Consequently, the potential drop along the path on the \(z\)-axis from \(z=0\) to \(z=C\) is equal to

\[ V=AC\ \text{abs. V}. \]

According to (5.8), the magnetic-field intensity, to within a time factor which is of no interest for this calculation, has the form

\[ \mathbf H=AJ_1(kr)\varphi_0\ \text{gauss}. \]

From (8.9), for the power released on the end faces of the cylinder, we have

\[ P_E=A^2\left(\frac{\pi c\delta\mu}{4\lambda}\right)\int_0^R J_1^2(kr)\,r\,dr =A^2\left(\frac{\pi c\delta\mu}{4\lambda}\right)\frac{R^2}{2}J_1^2(X_{0p}). \]

and for the power released on the lateral surfaces of the cylinder,

\[ P_C=A^2\left(\frac{\pi c\delta\mu}{4}\right)RCJ_1^2(X_{0p}), \]

so that the total power converted into Joule heat is

\[ P_C+2P_E=A^2\left(\frac{\pi c\delta\mu}{4}\right)(R^2+RC)J_1^2(X_{0p}). \]

The total energy reserve in the resonator is

\[ W=\int \frac{H^2}{8\pi}\,dV=\frac{A^2}{8}CR^2J_1^2(X_{0p}). \]

Consequently, by definition (9.3), for \(Q\) we have

\[ Q_p=\left(\frac{2\pi R}{\rho\mu}\right)^{1/2}\frac{C}{R+C}(X_{0p})^{1/2}. \tag{9.13} \]

It is interesting to note the influence of a change in \(C\), for a given \(R\), on the quality factor of the natural oscillations in the resonator. A change in \(Q\) does not affect the frequency of the natural oscillations of the resonator, since it depends only on \(R\). When \(C\) is small in comparison with \(R\), the value of \(Q\) is also small; as \(C\) increases, the value of \(Q\) at first increases, but for \(C\gg R\) it tends to a constant limit.

The shunt resistance \(S_p\) can be obtained by equating \(V^2/2S_p\) to the total power loss. This gives

\[ S_p=120\left(\frac{2\pi R}{\rho\mu}\right)^{1/2} \frac{C^2}{R^2+RC} \frac{1}{(X_{0p})^{1/2}J_1^2(X_{0p})}\ \text{ohms}. \tag{9.14} \]

Comparing the formula for \(S_p\) with the formula for \(Q_p\), we see that both quantities are related to one another by the simple relation

\[ S_p=120\left(\frac{C}{R}\right)\left[\frac{1}{X_{0p}}J_1^2(X_{0p})\right]Q_p. \tag{9.15} \]

For finding numerical values of \(S_p\), Table 4 may be used. (\(X_{0p}\) in this table denotes the \(p\)-th root of the equation

\[ J_0(x)=0.) \]

Oscillations of a coaxial cable. Another example of great practical importance is a coaxial cable bounded by internal and external cylinders of radii \(r=a\) and \(r=b\), respectively, and by the planes \(z=0\) and \(z=C\). The field in such a resonator was considered in § 5 and is described by formula (5.25). Substituting into (5.25) the dependence on the coordinate \(z\), not explicitly written there, we have

\[ \mathbf{H}=\frac{A}{2}\cos\frac{n\pi z}{C}\,\varphi_0,\ \text{gauss}. \]

Using (8.9) we find the losses at both ends of the resonator:

\[ P_E=A^2\left(\frac{\pi c^2\rho\mu}{4\lambda}\right)\frac{b}{a} \]

and at the inner and outer walls

\[ P_i=A^2\left(\frac{\pi c^2\rho\mu}{4\lambda}\right)\frac{C}{2a}, \]

\[ P_0=A^2\left(\frac{\pi c^2\rho\mu}{4\lambda}\right)\frac{C}{2b}. \]

The total power loss in the resonator will be

\[ P_i+P_0+2P_E = A^2\left(\frac{\pi c^2\rho\mu}{4\lambda}\right) \left(2\lg\frac{b}{a}+\frac{C}{2a}+\frac{C}{2b}\right). \]

The energy stored in the resonator is

\[ W=\frac{A^2}{8}\,C\lg\frac{b}{a}. \]

Therefore for \(Q\) we obtain the following expression:

\[ Q_n= \left(\frac{2C}{\rho\mu}\right)^{1/2} \frac{\lg\frac{b}{a}} {4\lg\frac{b}{a}+\frac{C}{a}+\frac{C}{b}} \sqrt{n}. \tag{9.16} \]

The most natural path with respect to which the shunt resistance can be determined is the straight line from the point \(r=a\) to the point \(r=b\), drawn in such a plane \(z=\mathrm{const}\) that \(\sin n\pi z/C=\pm 1\), i.e. in which there is a voltage-wave antinode. Along such a path

\[ V=A\lg\frac{b}{a}\ \text{abs. units}. \]

Calculating from this expression and from the expression for the total power loss the shunt resistance \(S_n\), we have

\[ S_n= 120\left(\frac{2C}{\rho\mu}\right)^{1/2} \frac{2\left(\lg\frac{b}{a}\right)^2} {4\lg\frac{b}{a}+\frac{C}{a}+\frac{C}{b}} \sqrt{n}. \tag{9.17} \]

By analogy with (9.15) one may write the relation connecting \(Q_n\) with \(S_n\):

\[ S_n=\left(120\,\frac{Q_n}{n\lambda}\right)\lg\frac{b}{a}\ \text{ohms}. \]

Example. Consider a resonator for which \(C=150\ \text{cm}\), and consequently the wavelength corresponding to the oscillation of the lowest frequency is \(3\ \text{m}\). Let \(a=30\ \text{cm}\) and \(b=45\ \text{cm}\). What pow-

ness must a resonator with copper walls have in order that the amplitude of the electric-field intensity in the cavity be 1 million V?

Answer: 1720 kW.

§ 10. Spherical resonator

The theory of natural oscillations in a hollow resonator of spherical form can be developed as a special case of a certain general method, which may have a much more general application.

Let us introduce orthogonal curvilinear coordinates \(x_1, x_2\), and \(x_3\) such that the element of arc length is expressed in these coordinates by the formula

\[ ds^2=e_1^2dx_1^2+e_2^2dx_2^2+e_3^2dx_3^2 . \tag{10.1} \]

For example, in spherical coordinates

\[ ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta\,d\varphi^2 \tag{10.2} \]

and, consequently,

\[ x_1=r;\quad x_2=\theta;\quad x_3=\varphi;\quad e_1=1;\quad e_2=r;\quad e_3=r\sin\theta . \]

In such curvilinear coordinates the Maxwell equations for \(\operatorname{rot}\mathbf E\) and \(\operatorname{rot}\mathbf H\) can be represented in the form

\[ \left. \begin{aligned} ike_2e_3E_1&=\frac{\partial}{\partial x_2}(e_3H_3)-\frac{\partial}{\partial x_3}(e_2H_2),\\ ike_3e_1E_2&=\frac{\partial}{\partial x_3}(e_1H_1)-\frac{\partial}{\partial x_1}(e_3H_3),\\ ike_1e_2E_3&=\frac{\partial}{\partial x_1}(e_2H_2)-\frac{\partial}{\partial x_2}(e_1H_1),\\ -ike_2e_3H_1&=\frac{\partial}{\partial x_2}(e_3E_3)-\frac{\partial}{\partial x_3}(e_2E_2),\\ -ike_3e_1H_2&=\frac{\partial}{\partial x_3}(e_1E_1)-\frac{\partial}{\partial x_1}(e_3E_3),\\ -ike_1e_2H_3&=\frac{\partial}{\partial x_1}(e_2E_2)-\frac{\partial}{\partial x_2}(e_1E_1). \end{aligned} \right\} \tag{10.3} \]

Suppose that the coordinate \(x_1\) is chosen so that \(e_1=1\), as in the case of ordinary spherical coordinates, and, in addition, that in the chosen coordinate system the ratio \(e_2/e_3\) does not depend on \(x_1\).

In this case, as will now be proved, the natural oscillations split into two independent classes, each class of oscillations being described by its own wave function.

These two classes of oscillations we shall call oscillations of \(E\)-type and \(H\)-type, in accordance with the classification introduced earlier.

In the case of oscillations of the \(E\)-type, \(H_1=0\). The fourth of equations (10.3) will be satisfied if we put

\[ e_2 E_2=\frac{\partial P}{\partial x_2},\qquad e_3 E_3=\frac{\partial P}{\partial x_3}. \]

If we introduce a scalar function \(U\) such that \(P=\partial U/\partial x_1\), the second and third equations give

\[ H_2=\frac{ik}{e_3}\frac{\partial U}{\partial x_3},\qquad H_3=-\frac{ik}{e_2}\frac{\partial U}{\partial x_2}. \]

The first and fifth of equations (10.3) give two different expressions for \(E_1\) in terms of \(U\). The compatibility condition for both expressions leads to the equation for \(U\):

\[ \frac{\partial^2 U}{\partial x_1^2} +\frac{1}{e_2 e_3}\left( \frac{\partial}{\partial x_2}\frac{e_3}{e_2}\frac{\partial U}{\partial x_2} + \frac{\partial}{\partial x_3}\frac{e_2}{e_3}\frac{\partial U}{\partial x_3} \right) +k^2 U=0. \tag{10.4} \]

The most convenient expression of \(E_1\) in terms of \(U\) has the form

\[ E_1=k^2U+\frac{\partial^2 U}{\partial x_1^2}. \]

With this choice of the relation between \(U\) and \(E_1\), the sixth equation (10.3) will be satisfied identically.

Oscillations of the \(H\)-type are obtained if one puts \(E_1=0\). This leads to an analogous system of equations that makes it possible to express the field components through a scalar function \(U\) satisfying the wave equation (10.4). As a result we obtain the following relations connecting the field components with the scalar function \(U\):

\(E\)-type:

\[ \begin{aligned} E_1&=k^2U+\frac{\partial^2U}{\partial x_1^2}, &\qquad H_1&=0,\\ E_2&=\frac{1}{e_2}\frac{\partial^2U}{\partial x_1\partial x_2}, &\qquad H_2&=\frac{ik}{e_3}\frac{\partial U}{\partial x_3},\\ E_3&=\frac{1}{e_3}\frac{\partial^2U}{\partial x_1\partial x_3}, &\qquad H_3&=-\frac{ik}{e_2}\frac{\partial U}{\partial x_2}. \end{aligned} \tag{10.5} \]

\(H\)-type:

\[ \begin{aligned} E_1&=0, &\qquad H_1&=k^2U+\frac{\partial^2U}{\partial x_1^2},\\ E_2&=-\frac{ik}{e_3}\frac{\partial U}{\partial x_3}, &\qquad H_2&=\frac{1}{e_2}\frac{\partial^2U}{\partial x_1\partial x_2},\\ E_3&=\frac{ik}{e_2}\frac{\partial U}{\partial x_2}, &\qquad H_3&=\frac{1}{e_3}\frac{\partial^2U}{\partial x_1\partial x_3}. \end{aligned} \tag{10.6} \]

Let us now specialize our general solution to the case of the sphere. For this purpose we choose as the coordinates \(x_1,x_2\), and \(x_3\) the spherical coordinates \(r,\theta,\varphi\). Then the equation for \(U\) assumes the form

\[ \frac{\partial^2U}{\partial r^2} +\frac{1}{r^2}\left( \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\,\frac{\partial U}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2U}{\partial\varphi^2} \right) +k^2U=0. \tag{10.7} \]

We seek its general solution in the form

\[ U=R(r)\Theta(\theta,\varphi), \]

where the function \(\Theta(\theta,\varphi)\) satisfies the equation

\[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\,\frac{\partial\Theta}{\partial\theta}\right) +\frac{1}{\sin^{2}\theta}\frac{\partial^{2}\Theta}{\partial\varphi^{2}} +l(l+1)\Theta=0, \tag{10.8} \]

and the function \(R(r)\) satisfies the equation

\[ \frac{d^{2}R}{dr^{2}}+\left(k^{2}-\frac{l(l+1)}{r^{2}}\right)R=0. \tag{10.9} \]

In order that a finite and single-valued solution of equation (10.8) exist, it is necessary that the parameter \(l\) take only a series of integral values.

The functions satisfying equation (10.8) are called harmonic functions. The theory of harmonic functions has been fully developed, and its content is presented in a large number of mathematical manuals.

The solutions of equation (10.9) are called cylindrical functions:

\[ R(r)=(kr)z_l(kr), \tag{10.10} \]

where

\[ z_l(x)=\left(\frac{\pi}{2x}\right)^{1/2} z_{l+\frac12}(x). \]

Bessel functions satisfy the recurrence relations

\[ \left. \begin{aligned} z_{n-1}+z_{n+1}&=\frac{2n+1}{x}z_n,\\ \frac{dz_n}{dx}(x)&=\bigl[nz_{n-1}-(n-1)z_{n+1}\bigr]\frac{1}{2n+1},\\ \frac{d x^{n+1}z_n}{dx}&=x^{n+1}z_{n-1},\\ \frac{d x^{-n}z_n}{dx}&=-x^{-n}z_{n+1}. \end{aligned} \right\} \tag{10.11} \]

We need to choose those of the Bessel functions which remain finite as \(z=0\). We shall denote them by \(j_l(x)\). The first functions \(j_l(x)\) are given in Table 5.

Table 5

Values for \(j_l(x)\)

\(l\)	\(j_l(x)\)
0	\(\sin x/x\)
1	\(\sin x/x^2-\cos x/x\)
2	\((3/x^3-1/x)\sin x-(3/x^2)\cos x\)
3	\((15/x^4-6/x^2)\sin x-(15/x^3-1/x)\cos x\)

For spherical functions we introduce the following notation

\[ \theta(\vartheta,\varphi)=\Theta(l,m)e^{im\varphi}, \tag{10.12} \]

where

\[ m\geq 0 \begin{cases} \Theta(l,m)= (-1)^m\left[(2l+1)\dfrac{(l-m)!}{(l+m)!}\right]^{1/2} \sin^m\vartheta\, \dfrac{d^m}{d(\cos\vartheta)^m}\times \\[6pt] \qquad\qquad\qquad\qquad\qquad\qquad {}\times P_l(\cos\vartheta),\\[6pt] \Theta(l,-m)= \end{cases} \]

Here \(P_l(\cos\vartheta)\) is the Legendre polynomial of order \(l\).
The first spherical functions have the form

\[ \Theta(0,0)=1, \]

\[ \Theta(1,0)=\sqrt{3}\cos\vartheta, \]

\[ \Theta(2,0)=\frac{\sqrt{5}}{2}(3\cos^2\vartheta-1), \]

\[ \Theta(3,-0)=\frac{\sqrt{7}}{2}(2\cos^3\vartheta-3\cos\vartheta\sin^2\vartheta). \]

The coefficients in \(\Theta(l,m)\) are chosen so that the normalization condition is satisfied

\[ \iint |\Theta|^2\sin\vartheta\,d\vartheta\,d\varphi=4\pi, \]

the integration being carried out over all angles \(\vartheta\varphi\).

In what follows we shall need the following formulas from the theory of spherical functions

\[ \begin{aligned} \frac{\partial}{\partial\vartheta}\Theta(l,m) &=\frac{1}{2}[(l-m)(l+m+1)]^{1/2}\Theta(l,m+1) \\ &\quad -\frac{1}{2}[(l+m)(l-m+1)]^{1/2}\Theta(l,m-1), \\[6pt] \cos\vartheta\,\Theta(l,m) &=\Theta(l+1,m) \left[ \frac{(l+1-m)(l+1+m)}{(2l+1)(2l+3)} \right]^{1/2} \\ &\quad+\Theta(l-1,m) \left[ \frac{(l-m)(l+m)}{(2l-1)(2l+1)} \right]^{1/2}, \\[6pt] \sin\vartheta\,\Theta(l,m) &=-\Theta(l+1,m+1) \left[ \frac{(l+m+1)(l+m+2)}{(2l+1)(2l+3)} \right]^{1/2} \\ &\quad+\Theta(l-1,m+1) \left[ \frac{(l-m)(l-m-1)}{(2l-1)(2l+1)} \right]^{1/2} \\ &=\Theta(l+1,m-1)-\Theta(l-1,m-1). \end{aligned} \tag{10.13} \]

Finally, the function \(U\) can be written in the form

\[ U=krj_l(kr)\Theta(l,m)e^{im\varphi}, \tag{10.14} \]

where \(|m|\leq l\) and \(l=0,1,2,3,\ldots\).

Using (10.5), one can find the following expressions for the field components in the case of \(E\)-type oscillations:

\[ \left. \begin{aligned} E_r&=\frac{k^2}{r^2}l(l+1)U,\\ E_\theta&=\frac{k}{r}\frac{\partial}{\partial(kr)}[krj_l(kr)]\frac{\partial}{\partial\theta}\Theta(l,m)e^{im\varphi},\\ E_\varphi&=\frac{ikm}{r}\frac{1}{\sin\theta}\frac{\partial}{\partial(kr)}[krj_l(kr)]\Theta(l,m)e^{im\varphi},\\ H_r&=0,\\ H_\theta&=-\frac{km}{r\sin\theta}U,\\ H_\varphi&=-\frac{ik}{r}[krj_l(kr)]\frac{\partial\Theta}{\partial\theta}e^{im\varphi}. \end{aligned} \right\} \tag{10.15} \]

The boundary conditions require that at the boundary, for \(r=R\), where \(R\) is the radius of the sphere, the components \(E_\theta\) and \(E_\varphi\) vanish. Consequently, for \(E\)-type oscillations the wave number \(k\) must satisfy the relation

\[ kR=S_{nl}, \tag{10.16} \]

where \(S_{nl}\) are the roots of the equation

\[ \frac{d}{dx}xj_l(x)=0. \]

Similarly, for \(H\)-type oscillations we have the following expressions for the field components:

\[ \left. \begin{aligned} E_r&=0,\qquad H_r=\frac{k^2}{r^2}l(l+1)U,\\ E_\theta&=\frac{km}{r\sin\theta}U,\qquad H_\theta=-\frac{k}{r}\frac{\partial}{\partial(kr)}[krj_l(kr)]\times\frac{\partial\Theta(l,m)}{\partial\theta}e^{im\varphi},\\ E_\varphi&=\frac{ik}{r}krj_l(kr)\frac{\partial\Theta(l,m)}{\partial\theta}e^{im\varphi},\qquad H_\varphi=\frac{ikm}{r\sin\theta}\frac{\partial}{\partial(kr)}\times\\ &\hspace{4.2cm}\times[krj_l(kr)]\Theta(l,m)e^{im\varphi}. \end{aligned} \right\} \tag{10.17} \]

For oscillations of the magnetic type the boundary conditions lead to the equality

\[ kR=T_{nl}, \tag{10.18} \]

where \(T_{nl}\) are the roots of the equation \(j_l(x)=0\).

We shall denote the two types of oscillations by the symbols \(E(n,l,m)\) and \(H(n,l,m)\). Analysis of equations (10.5) and (10.17) shows that there are no solutions corresponding to the values \(l=0\) and \(m=0\). The smallest value of \(l\) for which a solution exists is \(l=1\).

Since the roots \(S_{nl}\) and \(T_{nl}\) do not depend on the number \(m\), for each

for given values of \(n\) and \(l\) there exist \(2l+1\) different oscillations of the \(E\)-type and \(H\)-type, differing from one another in the value of \(m\), but having one and the same frequency. This \((2l+1)\)-fold degeneracy arises because of the spherical symmetry of the resonator.

The values of the roots of equations (10.16) and (10.18), corresponding to the fundamental frequencies of the oscillations, are

\[ S_{11}=2.74,\qquad T_{11}=4.49. \tag{10.19} \]

Consequently, the resonant wavelength for the oscillation \(E(11)\) is equal to \(2.29R\). The spherical functions determined by equation (10.12) remain finite over the entire sphere, including also the singular points \(\theta=0\) and \(\theta=\pi\) of equation (10.8), and therefore constitute a solution of the problem on the whole sphere.

Spherical coordinates can also be used in considering a conical resonator bounded by the surfaces \(r=R\) and \(\theta=\theta_0\), and also for a resonator in the form of a sphere with two conical cutouts, i.e. the region \(0<r<R,\ \theta_0<\theta<\theta_1\). In these cases, however, it is necessary to use the somewhat more general solution (10.5), which has singularities at the poles \(\theta=0,\ \theta=\pi\), which are excluded from the region.

Resonator in the form of a sphere with conical cutouts. Let us consider, for example, the fundamental \(E\)-type oscillation for a sphere with conical cutouts\(^8\). For this case it is convenient to choose the function \(U\) in the form

\[ U=\left(\lg \operatorname{tg}\frac{\theta}{2}\right)\frac{\sin kr}{kr}. \tag{10.20} \]

In finding the solution for the whole sphere this form of \(U\) was not suitable, since \(U\) then has a logarithmic singularity at \(\theta=0\). It evidently corresponds to the case \(l=0\) excluded from the preceding solution. With the aid of (10.15) one can find that the only nonvanishing field components are

\[ \left. \begin{aligned} E_\theta&=k^2\left(-\frac{1}{\sin\theta}\right)\frac{\cos kr}{kr},\\ H_\varphi&=-ik^2\frac{1}{\sin\theta}\frac{\sin kr}{kr} \end{aligned} \right\} \tag{10.21} \]

Since \(E_r\) and \(E_\varphi\) are zero everywhere, the boundary condition on the surface \(r=R\) reads \(E_\theta=0\). This requirement will be fulfilled if

\[ kR=\left(n+\frac{1}{2}\right)\pi \]

independently of the position of the other boundaries of the region.

The oscillation with the lowest frequency corresponds to the value \(n\) equal to zero. In this case its wavelength is exactly equal to four times the radius of the sphere. In the solution written above, \(E_\theta\) tends to infinity

... infinity as \(r\) approaches zero. However, the line integral of the field-strength vector \(E\), taken along the curve \(r=\mathrm{const.}\) from one cut to the other, remains finite.

Examples. 1. Show that the value of \(Q\) for the fundamental frequency of such a resonator is equal to

\[ Q=\left(\frac{4R}{\rho\mu}\right)^{1/2} \frac{\lg \tg \frac{\theta_1}{2}-\lg \tg \frac{\theta_0}{2}} {\lg \tg \frac{\theta_1}{2}-\lg \tg \frac{\theta_0}{2}} +I(\csc\theta_1+\csc\theta_0), \]

where

\[ I=\int_0^{\pi/2}\frac{\sin^2 x}{x}\,dx=0.825. \]

Show that, if \(\theta_1=\pi-\theta_0\), \(Q\), as a function of the angle \(\theta_0\), has a maximum at \(\theta_0=34^\circ\).

1. Show that the shunt resistance of this natural oscillation, if the potential difference is measured between the vertices, is

\[ S=120\left(\frac{4R}{\rho\mu}\right)^{1/2} \frac{\lg^2\left(\tg \frac{\theta_1}{2}:\tg \frac{\theta_0}{2}\right)} {\lg\left(\tg \frac{\theta_0}{2}:\tg \frac{\theta_1}{2}\right)+I(\csc\theta_1+\csc\theta_0)}. \]

Show that, for \(\theta_1=\theta_0-\pi\), it has a maximum at \(\theta_0 \simeq 9^\circ\).

(End in the next issue.)

LITERATURE

1. Jeans, Dynamical Theory of Gases, Cambridge Univ. Press, London, 1921, 3th Edit., Ch. 16; Fowler, Statistical Mechanics, ibid. 2-nd Edit., Ch. 4.
2. E. Fermi, Rev. Mod. Phys., 4, 87, 1932; W. Heitler, Quantum Theory of Radiation, Oxford Univ. Press, London, 1936, p. 40; Heitler, Quantum Theory of Radiation, GTTI, 1940.
3. Stratton, Electromagnetic Theory, McGraw-Hill, New-York, 1941, Ch. 6; Bateman, Electrical and Optical Wave Motion, Cambridge Univ. Press, London, 1915; Borgnis, Ann. d. Physik, 35, 359, 1939.
4. Borgnis, Z. Hochfrequenztech., 56, 47, 1940.
5. W. W. Hansen, J. App. Phys., 10, 38, 1939.
6. Barrow a. Mieher, Proc. I. R. E., 28, 184, 1940.
7. Terman, Radio Engineering, McGraw-Hill, New-York, 1937, a. 37 a. Ch. 3.
8. W. W. Hansen a. R. D. Richtmyer, J. App. Phys., 10, 189, 1939.