Evolution of the Concept of Resonance *)

N. D. Papaleksi

Before proceeding to my report, allow me to express my sincere gratitude for the great honor accorded to me by the invitation to deliver a report before such esteemed scientific and technical institutions of a country friendly to us. I see in this not so much a recognition of my modest scientific merits as an expression of the sincere desire of your freedom-loving people to strengthen cultural ties with our great country.

After long reflection I have allowed myself to choose as the topic of my report “The Evolution of the Concept of Resonance.” This choice may, at first glance, seem somewhat strange. The concept of resonance belongs among the most fundamental concepts, well known to everyone, and undoubtedly the thought may occur to many: what new can be said about such an old, well-worn thing as resonance, and what new interest can this hold for physics and technology? After all, everyone knows well that the word “resonance” itself comes from “to resonate,” which means to echo, to increase the duration or intensity of a sound, as, for example, Larousse says. Everyone also knows well that resonance phenomena occur not only in the domain of sound, but also in mechanics, optics, and electricity; that resonance may, for example, be the cause of the destruction of a bridge under the action of a periodic load, the breaking of shafts at critical numbers of revolutions, the rolling of ships, the breakdown of an electric cable (the Ferranti phenomenon). Almost every schoolboy now knows that radio engineering is based on the broad use of resonance. What, then, can still be said that is new about resonance?

Precisely because the role of resonance in science and technology is extraordinarily great, because in life at every step we encounter one or another manifestation of resonance—whether harmful, the destructive consequences of which we strive to avoid, or useful, which

*) A report prepared at the suggestion of the Presidium of the Academy of Sciences for the forthcoming spring of 1946 trip to Romania at the invitation of the Romanian Academy of Sciences, the Iași Polytechnic Institute, and the Romanian Telephone Society. The report was not delivered because the trip did not take place.

we try to use as fully as possible—is a very important, deep and clear understanding of what we mean by resonance. If we first became acquainted with the phenomena of resonance in the field of acoustic and mechanical oscillations, and then encountered analogous phenomena in the field of electrical oscillations and in optics, as a result of which the concept of “classical” resonance crystallized, then subsequently new forms of manifestation of resonant phenomena were discovered, such as “generalized” resonance, “parametric” resonance, and various “nonlinear resonances.” In connection with this, the very concept of “classical” resonance was refined. A special role in the evolution of the concept of resonance was undoubtedly played by the development of radio, which posed new problems and, thanks to the electron tube, made it possible to create oscillatory systems with new properties, different from those of earlier systems. New kinds of resonance have already acquired noticeable practical significance, and not only in the field of radio, and there is reason to believe that their importance will increase still more in the future.

Since, on the one hand, these questions, important for practical applications, have not yet become widely disseminated, and, on the other hand, it may be that the most essential results were obtained precisely in the scientific institutions of our country, chiefly by the school of scholars associated with the name of the recently deceased Academician L. I. Mandelstam, I have permitted myself to choose as the topic of my report “The Evolution of the Concept of Resonance.”

Let us turn first of all to the concept of ordinary classical resonance. When one speaks of resonance, what is usually meant is a remarkable property of an oscillatory system—be it a string, a pendulum, or an electrical circuit—to enter into especially intense oscillations under the action upon it of an external alternating force of a certain kind. Thus the concept of resonance is connected with the reaction of oscillatory systems of a certain type to the action of a certain external alternating force. What, then, are these systems, and how can one characterize the external force that produces ordinary resonance? Here we must enlist the aid of mathematics, that incomparable instrument of formulation in brevity, accuracy, and definiteness. Since the behavior of the systems in which the phenomena of resonance were first studied is described by linear differential equations: ordinary ones in the case, for example, of small oscillations of a pendulum or of an electrical oscillatory circuit, and partial differential equations for a string or a radio antenna, such systems are called, as is known, linear. How, then, is one to characterize an external alternating force that produces resonance in a linear system with constant parameters, say, for simplicity, in an electrical oscillatory circuit—the so-called linear electrical resonator? It is well known that under the action of an external harmonic force

A linear resonator enters into especially strong oscillations if the period of its natural oscillations coincides with the period of the external force. The resonant oscillations excited in the resonator are likewise sinusoidal and have the same period as the external force. Further, the amplitude of the resonant oscillations is proportional to the amplitude of the acting force, and it is the greater (the sharper the resonance), the smaller the damping of the oscillatory system. These properties of “classical” or “linear” resonance (perhaps it would be appropriate to call it “harmonic” resonance) characterize the ability of a linear harmonic resonator to extract from a complex oscillation the harmonic component of the same frequency as its own natural oscillations. On these properties is based the evaluation of the action of a variable force on any linear system, namely: the linear resonator makes it possible to decompose a variable force into a sum of harmonic components, to find, as they say by analogy with optics, its spectrum. Then, determining the action of each component separately on the given system and summing all these actions on the basis of the applicability of the superposition principle to linear systems, we thus obtain the total action of the entire force on the given system.

The circumstance that in linear systems with constant parameters—whether simple or complex systems—a harmonic oscillation passes without distortion through all the links of the chain undoubtedly explains the gradually entrenched view of the harmonic oscillation as the simplest one. This same circumstance also determined the broad, almost exclusive significance that the representation of a variable force as a sum of harmonic components has deservedly acquired in physics and technology, especially in radio, for the consideration of periodic and nearly periodic processes. On this representation is based the development and dissemination of very economical symbolic (complex) methods for solving oscillatory problems (Heaviside, Carson, and others). The exceptional fruitfulness of treating a variable force as consisting of a spectrum of harmonic components, or briefly of a spectrum of frequencies, was undoubtedly the reason why a spectral approach to the solution of oscillatory problems was developed and became established.

The property of a linear harmonic resonator to extract from a complex oscillation, containing an entire spectrum of frequencies, only one harmonic oscillation whose frequency coincides with its natural oscillations was, as is known, especially widely used in the field of communications (wire and radio), both for freeing from interference created by other sources of oscillations and for carrying out multiple telegraphy and telephony over a single wire or on a single carrier wave. It should be noted that, for solving this problem, highly effective harmonic resonators of a new type were created, namely electromechanical ones—with a very small damping coefficient, piezoelectric ones, magnetostrictive ones—

and others. The remarkable practical successes achieved in this field of communication, and the above-mentioned spectral approach to questions of oscillations, gradually strengthened the conviction, which became almost universal among specialists, that for the best use of wave communication the oscillations employed must be as close as possible to harmonic ones, and that selection can best be carried out only by means of a linear resonant system with constant parameters and with the least possible damping. However, as is known, the following difficulty arises here. As is well known, a signal cannot be transmitted by a single harmonic: to carry out transmission it is necessary to change the form of the oscillation (to modulate it), or, in other words, to transmit an entire spectrum of frequencies; moreover, this spectrum will be the more complex and broader the greater the rate of signal transmission (an especially broad frequency spectrum occurs in television transmission—a million or more impulses per second). Thus a harmonic resonator cannot completely solve the problem of freeing radio reception from interference, since if one takes a highly selective resonator (with small damping), it will not be able to receive any sufficiently rapid transmission, even telegraphic; whereas if the resonator is made weakly selective (with large damping), then, besides the signal, it will also pass extraneous interfering oscillations. Such an antagonism between the rate of transmission and the sharpness of linear selection, in a certain sense analogous to the well-known uncertainty principle in quantum physics, naturally raised the question of whether other methods of selection are possible, not based on harmonic resonance.

First of all the question arises: are there other oscillatory systems, besides linear ones with constant coefficients, for which the principle of superposition is applicable? The answer here is very simple: yes, there are—these are systems with variable parameters depending only on time. Examples of such systems may be: a pendulum with periodically varying length, or an electric oscillatory circuit with periodically varying capacitance (rotating plates of a capacitor), a flexible (elastic) rotating rod of rectangular cross-section carrying a load at one end, a two-pole rotor of a turbogenerator, etc. Such systems are described by linear differential equations with periodic coefficients, and to them, as to linear systems, the principle of superposition is applicable. Thus, for example, in the case of the simplest electric oscillatory circuit with a capacitance varying periodically according to the law

\[ \frac{1}{C}=\frac{1}{C_0}(1+m\cos\omega t), \tag{1} \]

we obtain the equation

\[ \ddot{x}+2\delta\dot{x}+\omega_0^2(1+m\cos\omega t)x=0. \tag{2} \]

What will happen if we subject such a system to the action of an external

variable force \(f(t)\), when we shall, in other words, have the equation

\[ \ddot{x}+2\delta\dot{x}+\omega_0^2(1+m\cos\omega t)x=f(t)? \tag{3} \]

Will we here too observe phenomena analogous to harmonic resonance, and under what conditions? Are harmonic functions privileged functions here as well, or not? In order to obtain an answer to these questions, let us return again to the linear system with constant parameters, described by the equation

\[ \ddot{x}+2\delta\dot{x}+\omega_0^2x=f(t). \tag{4} \]

It is, obviously, a special case of equation (3) for \(m=0\). Let us try to formulate the resonance condition in a mathematically rigorous way. The solution of equation (4) is obtained, as is known, by representing \(f(t)\) in the form:

\[ f(t)=a\cos\omega_0t+b\sin\omega_0t+\text{ terms not containing } \cos\omega_0t \text{ and } \sin\omega_0t, \]

or, as mathematicians express it, orthogonal to them.

Then

\[ x=\frac{a}{\delta}\sin\omega_0t-\frac{b}{\delta}\cos\omega_0t+\text{ nonresonant terms, giving forced oscillations.} \]

If now \(\delta\to0\), then the resonant terms will tend to infinity, while the nonresonant terms will remain finite. Thus we arrive at the following criterion of classical harmonic resonance: let a force \(\delta\cdot\varphi(t)\) act on the system; then, as follows from equation (4), which will now be written in the form

\[ \ddot{x}+\omega_0^2x=\delta[\varphi(t)-2\dot{x}], \tag{5} \]

if, as \(\delta\to0\), the established forced oscillation remains finite and different from zero, then we say that resonance takes place. The solution will be one of the solutions of the equation

\[ \ddot{x}+\omega_0^2x=0, \tag{6} \]

i.e. one of the natural oscillations of the harmonic resonator. This subtle analysis of the essence of harmonic resonance, which we owe to Academician L. I. Mandelstam, also formed the basis of the theory of resonance in systems with periodic parameters, developed by L. I. Mandelstam’s pupil, G. S. Gorelik.

Let us first note the following. As we have seen, for

\[ f(t)=\delta(a\cos\omega_0t+b\sin\omega_0t) \]

a solution of equation (4) is the established oscillation

\[ x=\frac{1}{2\omega_0}(a\sin\omega_0t-b\cos\omega_0t), \]

i.e.

\[ f(t)=2\delta\dot{x}, \tag{7} \]

which also follows directly from equation (5).

We thus arrive at the following definition of linear resonance: resonance occurs when the forced oscillation caused by \(f(t)=\delta\varphi(t)\), as \(\delta\to 0\), tends to a nonzero proper oscillation of the resonator; moreover, this takes place when the force contains terms proportional to the derivative of the proper oscillations of the resonator. This definition can be applied to a linear oscillatory system with periodic parameters, or, briefly, to a parametric system.

Indeed, let us rewrite equation (3) in the form

\[ \ddot{x}+\omega_0^2(1+m\cos\omega t)x=f(t)-2\delta\dot{x}. \tag{8} \]

Let \(u\) and \(v\) be particular integrals (proper oscillations) of the equation

\[ \ddot{x}+\omega_0^2(1+m\cos\omega t)x=0. \tag{9} \]

Then, if

\[ f(t)=2\delta(a\dot{u}+b\dot{v}), \]

the solution of (7) will indeed be

\[ x=au+bv, \tag{10} \]

remaining finite as \(\delta\to 0\). It can further be rigorously shown that if \(f(t)=\delta\varphi(t)\) does not contain \(\dot{u}\) and \(\dot{v}\) in its composition (if, expressed mathematically, it is orthogonal to both \(u\) and \(v\)), i.e.

\[ \frac{1}{T}\int_{-T/2}^{+T/2} f(t)\cdot u\,dt=0 \quad\text{and}\quad \frac{1}{T}\int_{-T/2}^{+T/2} f(t)\cdot v\,dt=0, \]

then the solution of equation (8) as \(\delta\to 0\) will also tend to zero, i.e. there will be no resonance. Hence there follows the following criterion of resonance for a linear resonator with periodic parameters, or, as we shall call it, for a parametric resonator: if the variable force \(f(t)\) can be represented in the form

\[ f(t)=P\dot{u}+Q\dot{v}+g, \]

where \(g\) is orthogonal to \(u\) and \(v\) and \(P\) or \(Q\) is not equal to zero, then resonance takes place. Thus the parametric resonator selects from the composition of the variable force not a harmonic function, but the term \(P\dot{u}+Q\dot{v}\); moreover, the smaller the damping \(\delta\), the more accurately the forced oscillation at resonance coincides with one of the proper ...

... of the resonator’s oscillations. What, then, are the proper oscillations of a parametric resonator? As is seen from equation (9), in the simplest case these are solutions of Mathieu’s equation, whose theory was developed in connection with problems of celestial mechanics and which has also acquired great significance for various fields of physics and technology. In the general case we are dealing with solutions of Hill’s equation.

In the case of equation (9), \(u\) may be represented in the form

\[ u=C_1 e^{\mu t}F(t)+C_2 e^{-\mu t}F(t), \]

where

\[ \mu \simeq \frac{\omega_0^2}{2\omega}\, m \sin 2\sigma, \]

\[ F(t)=\sin\left(\frac{\omega t}{2}-\sigma\right)+ \sum_{p=1}^{\infty}\left\{ a_{2p+1}\cos\left[\left(p+\frac12\right)\omega t-\sigma\right]+ \right. \]

\[ \left. +\,b_{2p+1}\sin\left[\left(p+\frac12\right)\omega t-\sigma\right]\right\} \]

and

\[ \frac{4\omega_0^2-\omega^2}{\omega^2} \simeq \frac{2\omega_0^2}{\omega^2}\,m\cos 2\sigma+ \left(-1+\frac12\cos 4\sigma\right)\frac{\omega_0^4}{\omega^4}\,m^4. \]

The question may arise: what will result if a parametric resonator is acted upon by a harmonic force, for example of the form \(\cos \nu t\)? It turns out that the parametric resonator will single out from the composition of such a harmonic force the component \(Q_\nu\). The same component will also be singled out by the parametric resonator from the oscillations \(\cos(\nu+2\omega)t\), \(\cos(\nu+4\omega)t\), and so on. This phenomenon of multiple resonance, which was first discovered experimentally by G. S. Gorelik and Gans in the study of a superregenerative receiver and was explained by them, shows clearly that for linear systems with periodic parameters a harmonic force is by no means simple.

The analysis of resonance phenomena in linear systems with periodic parameters not only made it possible to clarify the concept of classical, or harmonic, resonance and, in a certain sense, to generalize the concept of linear resonance, but at the same time it brought to the fore the question of the “proper” oscillations of systems with periodic parameters. Since, however, such systems are not autonomous, it may be more correct to speak of what oscillations they perform under periodic action on their parameters. What could one expect here from the mathematical point of view? The mathematical theory of equations of the Mathieu type shows that, depending on the relation between the quantities \(\omega_0/\omega\), \(m\), and \(\delta\), solutions of two kinds are possible: stable ones, i.e. such that the oscillations that arise initially gradually die away, and un-

stable, for which any oscillation that arises increases. These regions of unstable solutions can be graphically represented in the plane \((m,\omega_0/\omega)\) (Fig. 1). Here, for a definite value of \(\delta\), these regions (shaded) are plotted. As is seen from Fig. 1, these regions are located near the values \(\omega_0/\omega=p/2\) \((p=1,2,3,\ldots)\), the first region being reached at smaller values of \(m\) (the depth of modulation of the parameter) than the others. What, then, is the physical meaning of such unstable solutions? Their physical meaning consists in the fact that if, in a real oscillatory system, one of its parameters is varied periodically—for example, the length of a pendulum or the capacitance of the condenser of an electric circuit—then, when the frequency of the natural oscillations of the system is tuned to a frequency equal to one half of, equal to, or a multiple of the frequency of variation of the parameter, ever-increasing oscillations must arise in it under any initial disturbance. In other words, the system responds to the external action, and in it there arises a peculiar resonance, which may be called “parametric” resonance.

Fig. 1. Regions of instability of oscillations of systems with periodic parameters.

Are such resonance phenomena actually observed? In essence, we ourselves, even in childhood, repeatedly and unconsciously produced such a resonance while swinging on swings, since swinging on swings is nothing other than a periodic change of the moment of inertia of the oscillating system—the swing—in time with the swing. As a physical phenomenon, parametric resonance was apparently first realized in 1859 by Melde in his well-known experiment on the excitation of transverse oscillations of a string by periodically changing its tension with the aid of a tuning-fork prong attached to its free end. The possibility of realizing such phenomena in electrical oscillatory systems was pointed out as early as 1883 by Lord Rayleigh, who also first gave the correct theoretical explanation of Melde’s experiment.

Although phenomena of excitation of oscillations in electrical oscillatory systems by means of a periodic change of self-inductance had long been observed in electrical engineering (cases of self-excitation of electrical machines in circuits containing capacitance), only in recent years has such excitation of electrical oscillations been consciously realized in the laboratories of Academician L. I. Mandelstam and the author of the present report; its theory has been given and its resonant character investigated. It proved possible to excite strong resonant oscillations in an electrical oscillatory system in the absence of any-

explicit electric or magnetic fields, by means only of a mechanical periodic change either of its self-inductance (1931) or of its capacitance (1933). Figure 2 shows the implementation of variable self-inductance. Since in our first experiments the depth of modulation of the parameter was not very large (0.2–0.4), it was possible to excite parametric resonance only in the first region of instability, i.e., the first or fundamental parametric resonance. Recently (1945), in connection with the attainment of large \(m\) (greater than 0.5), we obtained and studied also the second parametric resonance for \(p = 2\), i.e., for the ratio of the frequencies of the change of the parameter and of the natural oscillations of the system \(1 : 1\).

Parametric resonance, in its properties, differs sharply from classical (harmonic) resonance. First of all, the frequency of excitation of parametric oscillations only in the second parametric resonance is equal to the frequency of the action; in the case of the first or principal parametric resonance, which is easiest to excite, the frequency of the excited oscillations is equal to one half of the frequency of the action. In addition, the region of excitation of parametric resonance, in contrast to harmonic resonance, is sharply limited. Further, parametric resonance occurs only when the value \(m\), i.e., the magnitude of the action, reaches a certain value; in other words, for parametric excitation there exists a threshold value of the magnitude of the action. Finally, the established parametric oscillations are not harmonic, but contain clearly expressed harmonics.

Fig. 2. Construction providing a periodic change of self-inductance.

The implementation of parametric resonance in electrical systems served as the basis for creating alternating-current electric generators of a new type, in which, in the absence of special magnetic or electric excitation fields, the conversion of the mechanical energy expended on the periodic change of self-inductance (or capacitance) into electric current is carried out. These so-called parametric alternating-current generators, developed in our laboratories on the basis of the ideas of Academician L. I. Mandelstam and of the present speaker, thus differ from ordinary alternators by the absence of excitation windings or permanent magnets and by the presence of capacitance in their circuit. The simplicity of construction, the substantial saving of active materials, especially copper, and also the specific features of the external operating characteristics make these machines especially suitable in a number of cases where it is more convenient to use alternating current of increased frequency (500 cycles and higher) (Fig. 3).

In parametric machines, as in dynamo machines with self-excitation, the steady-state regime is determined by the saturation of the iron. This circumstance introduces much that is essentially new into the behavior of the oscillatory system, making its inductance dependent not only on time but also on the magnitude of the current. As a result, the differential equation describing the behavior of the system ceases to be linear: it becomes “nonlinear,” and for this reason the system itself, as is known, is customarily called “nonlinear.” Electrical engineers have long encountered peculiar resonance phenomena in nonlinear oscillatory systems containing iron, and these phenomena, as is known, have received the name “ferroresonance.” Such systems, to which, as nonlinear ones, the principle of superposition is not applicable, are characterized by the fact that in them there is no proportionality between the amplitude of the forced oscillations and the amplitude of the excitation. In addition, the period of the natural oscillations of such systems, as in the case of large oscillations of a pendulum, also depends on the amplitude of the oscillations, and the oscillations themselves here are strongly anharmonic. Therefore such systems are also called “anharmonic” or “pseudoharmonic.” As can be seen from Fig. 4, the form of the ferroresonance curve not only differs substantially from the ordinary resonance curve, but at large amplitudes there occur phenomena of oscillation breakdown and a peculiar oscillatory hysteresis.

Fig. 3. Characteristics of an inductive parametric generator.

The development of radio, brought about by the appearance of the electron tube and the application of the feedback principle, brought to the fore the study of new nonlinear electrical oscillatory systems—the so-called regenerative systems, in which the energy for maintaining oscillations is supplied through feedback by a local source of electrical energy. As is well known, with a sufficiently large amount of feedback (greater than the critical value), oscillations arise in such systems and are maintained for a long time; these oscillations have received the name “self-oscillations,” or, in other words, the so-called “self-oscillatory” systems become generators of oscillations. In order to emphasize this property of regenerative systems

EVOLUTION OF THE CONCEPT OF RESONANCE

systems pass, for a definite value of the feedback, into self-oscillatory ones; they may perhaps aptly be called “potentially self-oscillatory.”

The theoretical and experimental study of the behavior of potentially self-oscillatory systems under the action of an external e.m.f., as resonant systems, has led to new and very interesting results. Since, owing to feedback, the losses in a regenerative system are compensated, with feedback close to critical it has very small damping and, consequently, very high selectivity, the degree of which, in view of the nonlinearity of the characteristic of the electron tube, depends on the amplitude of the action, and is the greater the smaller the amplitude of the action. Thanks to these properties, a regenerative system makes it possible to discriminate between the amplification of weak and strong signals. However, the distinctive features of regenerative systems are not confined to this. For a magnitude of feedback differing only slightly from the critical one, new phenomena are observed which do not fit within the framework of our conception of ordinary resonance. One of these phenomena consists in the fact that, when the amplitude of the action is greater than a certain value, intense natural oscillations are excited in the system, independently of the frequency of the external force; for this reason this type of excitation has received the name “asynchronous.” Asynchronous excitation occurs only within certain limits of the magnitude of the feedback, somewhat smaller than the critical one; in the regime of “hard” self-excitation of self-oscillations it may be compared with a “relay” action. An essentially new—and theoretically and practically more interesting—phenomenon is observed when the feedback is decreased beyond the limits of “asynchronous” excitation. Here, when the regenerative system is tuned to a frequency approximately equal to one half of the frequency of the external force, intense oscillations are excited in it with a frequency exactly equal to one half of the frequency of the action. In view of the connection between the theory of this phenomenon and Poincaré’s theory of periodic solutions of the second kind, this phenomenon has received the name resonance of the second kind. Besides the fundamental difference from classical resonance, consisting in the fact that in resonance of the second kind the frequency of the resonant oscillations is equal to one half of the frequency of the action, there are also a number of other differences between them. Thus, in resonance of the second kind the oscillations are excited—

Fig. 4. Ferroresonance curve.

are excited only within definite limits of the amplitude of the acting force, i.e., in other words, for the magnitude of the action there exists both a “threshold” and a “ceiling.” The width of the excitation region also depends on the magnitude of the action. Moreover, in contrast to ordinary resonance, in resonance of the second kind the oscillations at first increase more slowly, and then more rapidly (Fig. 5). Various applications of resonance of the second kind are based on the use of these peculiar properties, both for the transformation of frequency downward and for purposes of selection in radio reception (the autoparametric filter). Besides its practical interest, the phenomenon of resonance of the second kind is also of great fundamental interest, since in it the periodic solutions of the second kind of Poincaré found their physical embodiment. The theory, developed on the basis of Poincaré’s mathematical methods, made it possible not only to understand fully the various details of the phenomenon of resonance of the second kind, but also brought clarity into the vast and complex field of very diverse oscillatory phenomena occurring in regenerative systems (self-excited and non-self-excited), both simple and complex, under the action upon them of a variable force. Thus, for example, if, under the action of a harmonic force on a complex (say, with two degrees of freedom) linear oscillatory system with constant parameters, resonance occurs in it only when the frequency \(\omega\) of the acting harmonic coincides with the frequency \(\omega_1\) or \(\omega_2\) of one of the natural oscillations of the system, and, naturally, the frequency of the resonant oscillations is exactly equal to the frequency \(\omega\) of the action, then in the case of a complex regenerative system, under the same action and under certain conditions, there is observed the phenomenon of the so-called “combination” resonance, consisting in the fact that when \(\omega = \omega_1 + \omega_2\) or \(\omega = \omega_1 - \omega_2\), both natural oscillations \(\omega_1\) and \(\omega_2\) are excited in the system. Of very great practical and fundamental interest are the peculiar resonance phenomena that occur in self-excited systems. These include, for example, the phenomenon of “forced synchronization of frequency,” which consists in the fact that, under the action of an external harmonic e.m.f. of frequency \(\omega\), the frequency of the self-oscillations of the system becomes exactly equal to \(\omega/p\), where \(p = 1, 2, 3, \ldots\), if the frequency of the action is close to the fundamental frequency of the self-oscillations of the system or to its overtone. Such “entrainment,” or “forced synchronization,” of frequency, known in mechanics since the time of Huygens, who observed the synchronization of clocks suspended

Fig. 5. Curves of the growth of oscillations in ordinary resonance (1) and resonance of the second kind (2).

on one and the same wall, are at present widely used in radio engineering both for synchronizing transmitter frequencies and for measurement purposes. I shall not dwell on other distinctive resonance phenomena occurring in “nonlinear” systems, such as “autoparametric” or “fractional” resonance, or the various “combination” resonances. I should only like to touch upon one more group of resonance phenomena in which, alongside amplitude and frequency dependences, phase dependences also play an essential role. I have in mind peculiar phenomena which appear in an especially pronounced form in a parametric resonator when it is acted upon by an e.m.f. with a frequency exactly equal to the frequency of variation of the parameter. For the electrical parametric resonator considered above [equation (3)] we then obtain the following equation:

\[ \ddot{x}+2\delta\dot{x}+\omega_0^2(1+m\cos 2\omega t)x=a\cos(\omega t-\varphi). \]

If the quantity \(m\) is such that the parametric resonator is close to self-excitation, then, under the action of the e.m.f. \(a\cos(\omega t-\varphi)\), intense forced oscillations arise in it, their energy being supplied mainly by the work expended on changing the parameter. Thus here we have an analogy with a vacuum-tube regenerative system and may regard the parametric resonator as a parametrically regenerated system, whose degree of regeneration is the greater the closer the system is to the boundary of self-excitation. As theory and experiment show, the intensity of forced oscillations in such a system depends strongly on the phase difference \(\varphi\) between the external e.m.f. and the variation of the parameter (Fig. 6). As is seen from Fig. 6, the character of the influence of the parameter variation depends strongly on the phase: here we have both a region of positive regeneration (amplification of oscillations) and of negative regeneration (suppression of oscillations), the effect being the stronger the closer the system is to self-excitation, i.e. the smaller

\[ \frac{2\delta}{\omega}-\frac{m}{2}. \]

Fig. 6. Dependence of the amplitude of oscillations in the system on the phase difference \(\varphi\) in the regime of parametric regeneration.

Since, in this case, especially sharp effects are observed at the phase values \(\varphi_1=\pi/2\) and \(\varphi_2=3\pi/2\), one may say that here phase selection is effected and that the parametrically regenerated resonator resonates to a definite phase.

In considering various new kinds of resonance, we have made almost exclusive use of electrical oscillatory systems. This is explained by the fact that many of these resonances were first discovered or realized precisely with the aid of radio-engineering circuits, owing to their universality and flexibility. This does not mean, however, that new kinds of resonance are observed only in the domain of electrical oscillations. Thus, we encounter the undesirable and at times destructive effects of parametric resonance both in the case of a rotating flexible rod of rectangular cross-section carrying a load at one end, and in the rotation of the two-pole rotor of a turbogenerator. The phenomenon of pseudoharmonic resonance can occur in a number of mechanisms in which elasticity or flexibility changes with deformation. As A. A. Andronov and G. S. Gorelik have pointed out, in Lawrence’s cyclotron—a remarkable device for accelerating ions, which has acquired special significance in connection with questions of the use of intra-atomic energy—there occur resonance phenomena of a ferroresonant character: the relativistic change in the mass of the particle is analogous to the change in the inductance of a circuit as a function of the current strength.

In my brief survey I have tried to set forth before you, in the most general outline (a more detailed exposition would have taken too much time), how, as our knowledge in the field of oscillations has grown and as this field has expanded, the concept of resonance—at first vague and indefinite—has gradually been refined, deepened, and broadened. I wanted to show you how, as a result of this gradual deepening and refinement of our knowledge, the concept of classical, or harmonic, resonance acquired crystalline logical clarity and mathematical precision. On the other hand, the discovery of new kinds of resonance phenomena, understood as the occurrence in an oscillatory system of powerful oscillations within a narrow frequency interval definitely connected both with the frequency of the external action and with the frequency of the system’s natural oscillations, substantially broadened the concept of resonance in general, though at the cost of a loss of precision. As a result of the enrichment of our knowledge of oscillations, and in particular of resonance, the arsenal of means for solving various problems in the field of oscillations has been considerably enriched.