# Oscillations

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## Oscillations

motions (changes in state) that have some degree of repetition. During the oscillation of a pendulum (Figure l,a) it deviates repeatedly to either side of the vertical position. During the oscillation of a spring pendulum—a bob suspended on a spring (Figure 1, b)—there are recurring deviations upward and downward from some middle position During oscillations in an electric circuit with capacitance *C* and inductance *L* (Figure 2), the magnitude and sign of the charge *q* are repeatedly changed on each plate of the capacitor.

The oscillations of a pendulum take place because the force of gravity returns the pendulum to the state of equilibrium, and when the pendulum returns to the state of equilibrium, the velocity it gained during the excursion makes it continue to move

(by inertia) and again deflects it from the state of equilibrium to the side opposite that from which it came. The oscillations of the bob (Figure 1, b) take place because the elastic force of a compressed or stretched spring returns the bob from the upward or downward position to the state of equilibrium, and after returning to the state of equilibrium, it has velocity and by inertia “jumps” through the center position. This results in extension or compression of the spring. The oscillations in an electric circuit take place because the potential difference between the plates of a charged capacitor induces a current *i* in the coil, and the current does not stop when the capacitor is fully discharged: as a result of the coil’s inductance, the current continues to flow, recharging the capacitor.

Physics and technology deal with oscillations of extremely varied physical nature, character, and recurrence; rapidity of the succession of states; and “mechanism” of occurrence. In particular, the following types of oscillations can be distinguished according to their physical nature: (1) mechanical oscillations, such as the oscillations of a pendulum, a bridge, a ship on a wave, or a string, and oscillations of the density and pressure of air during the propagation of elastic (acoustic) waves, particularly audible sound waves; (2) electromagnetic oscillations, such as the oscillations in an oscillatory circuit (Figure 2), cavity resonator, or wave guide, and the oscillations of the intensity of electric and magnetic fields in radio waves, visible light waves, and any other electromagnetic waves; (3) electromechanical oscillations (the oscillations of the diaphragm of a telephone or of a quartz-crystal or magnetostriction ultrasonic radiator); (4) chemical oscillations (oscillations of the concentration of reagents during periodic chemical reactions); and (5) thermodynamic oscillations (such as the so-called singing flame) and other thermal self-oscillations that are encountered in acoustics, as well as in some types of jet engines. Fluctuations in the brightness of cepheids are of great interest in astrophysics. Thus, oscillations encompass a broad range of physical phenomena and industrial processes. In particular, they are of primary importance in shipbuilding, aircraft construction, electrical engineering, and automatic control technology. Radio engineering and technical acoustics are based entirely on the use of oscillations. They are also encountered in meteorology, chemistry, physiology (for example, the beating of the heart), and a number of other natural sciences.

Certain characteristic principles that are identical for oscillations of different physical natures are inherent in oscillations. As a result, a branch of physics (oscillation theory) that studies the general principles of oscillations has arisen. Differential equations are the main mathematical apparatus of oscillation theory. There are groups of oscallations of various physical natures—for example, the oscillation of a pendulum, a bob on a spring, and an electric circuit *(see above);* the oscillation of clocks and tube oscillators; and the oscillations of an elastic rod or electric cable —and analogous differential equations correspond to them. The analogous nature of these equations reflects the community of certain objective principles inherent in the oscillations of this group. However, the analogies between oscillations of different physical natures, like any analogies, are limited to a well-defined framework; them encompass far from all of the significant features of oscillations.

The study of the oscillations of a pendulum that was undertaken at the start of the 17th century by the Italian scientist Galileo and later by the Dutch scientist C. Huygens played an extremely important role in the rise of classical mechanics. The study of electromagnetic oscillations in the late 19th century by the Scottish physicist W. Thomson (Lord Kelvin) was of great importance in understanding electromagnetic phenomena. Much important information and many important results in the theory of oscillations can be found in the works of the English physicist Lord Rayleigh.

The study of oscillations is greatly indebted to the work of Russian scientists. The invention of radio by A. S. Popov (1895) was a very important technical application of electromagnetic oscillations. P. N. Lebedev devoted a great deal of outstanding research to the generation of very high frequency electromagnetic oscillations, to ultrasonic oscillations, and to the behavior of matter upon exposure to rapidly alternating electric fields. A. N. Krylov conducted fundamental research in the theory of the pitching of ships. The works of such Soviet scientists as L. I. Mandel’shtam, N. D. Papaleksi, N. M. Krylov, N. N. Bogoliubov, and A. A. Andronov were of great importance in the study of oscillations, particularly nonlinear oscillations. The works of A. N. Kolmogorov and A. la. Khinchin contain the mathematical foundation of the theory of random processes in oscillating systems, which has been of great practical importance.

** Kinematics** From the standpoint of kinematics it is possible to distinguish some of the most important types of oscillations (Figure 3), in which the oscillating quantity

*s*may be of any physical nature (such as the mechanical displacement of a solid, the compression of a gas, or current strength). A general case of a periodic oscillation is shown in Figure 3,a; here every value of

*s*is repeated an unlimited number of times at identical time intervals

*t = T:*

*s(t + T) = s(t)**(-∞ < t < ∞)*

The quantity *T* is called the period. The number of oscillations per unit time, *v* = 1/T, is called the oscillation frequency.

Rectangular oscillations (Figure 3,b), sawtooth oscillations (Figure 3,c) and sinusoidal (or harmonic) oscillations (Figure 3,d) are particular cases of periodic oscillations. In the last case,

*s = A* cos *(ωt - φ)*

where *A,* ω, and φ are constants. The quantity *A* (the maximum value of *s)* is called the amplitude. Since the values of cos (ωt — φ) repeat when the argument increases by 2π, *ωT = 2π,* and consequently

*ω = 2π/T* = 2πν

The quantity ω is called the angular, or cyclic, frequency and is equal to the number of oscillations per *2π* units of time. The time function *ωt* — φ is called the oscillation phase, and the constant φ is called the initial phase shift (often called simply the phase). Figure 3,e depicts a damped oscillation,

*s = Ae ^{-δt}* cos

*(ωt — φ)*

where *A,* δ, ω, and φ are constants *(A* is called the initial amplitude, *Ae ^{-δt}* is the instantaneous value of the amplitude, δ is the damping factor, and π = 1/δ is the time constant). Here the quantity δ is positive. When the sign of δ is negative the oscillation is rising (Figure 3,f). The quantities

*ωt —*φ, ω and φ have the same names as in the case of a sinusoidal oscillation. Although a damped oscillation is not precisely periodic, the quantity

*T = 2π/ω*is also called the period.

Modulated oscillations, that is, oscillations of the type

*s = A(t)* cos *[ωt — φ(t)]*

are of great importance in physics and radio engineering, and the functions *A(t*) and *φ(t*) change slowly in comparison with cos *ωt* (where ω is a constant). If φ*(t)* = const., then the oscillation is said to be amplitude-modulated (Figure 3,g), and if *A(t) =* const. (Figure 3,h) they are said to be phase-modulated (or frequency-modulated). In the general case (Figure 3,i) the oscillations are modulated with respect to both amplitude and phase. Figures 3,g, h, and i correspond to periodic amplitude and phase modulation: *A(t)* and *φ(t*) are periodic functions. The case in which *Aφt*) or φ(t) or both simultaneously are random functions (Figure 3,j) is of great importance in technology (radio telephony and television) and physics. Random oscillations (Figure 3,k), such as white light and acoustic and electrical “white” noise, are frequently encountered in nature and technology.

Strictly periodic (particularly strictly harmonic) oscillations are never encountered, either in nature or in technology. Nevertheless, harmonic oscillations are extremely important for two reasons. First, oscillations that differ little from harmonic oscillations for a fairly long time occur frequently in nature and technical devices. Second, many physical systems that belong to the class of spectral instruments in the broad sense of the word or to the class of harmonic analyzers convert arbitrary oscillations into a set of oscillations that are close to harmonic. When we speak of harmonic oscillations, we always have in mind oscillations that are merely close to harmonic. Even harmonic oscillations of identical physical nature (such as the fluctuations in air pressure or in the intensity of an electric field) but different frequency may have markedly different properties (in addition to similar properties); they may affect in a totally different manner various physical systems and living organisms and, in particular, the sense organs of humans and animals.

*Initiation.* Here we will examine the initiation of oscillations in a system that does not receive oscillations from without but rather is a source of oscillations. If the system begins to oscillate upon application of oscillations from without we speak not of the generation of oscillations but of the effect of oscillations on the system and their transformation by the system. In passive systems, which contain no energy sources, this effect induces forced oscillations.

There are three main types of oscillations in systems that are oscillation sources: free oscillations, fluctuation oscillations, and self-oscillations.

Free (or natural) oscillations occur when the system is left to itself after disruption of equilibrium by external intervention, such as the oscillations of a spring pendulum (Figure l,b) and the current oscillations in an electric circuit (Figure 2). The free oscillations of spring pendulums and oscillation circuits are a particular type of free oscillations in linear oscillatory systems that is, systems whose parameters are virtually constant and can be described with sufficient accuracy by linear differential equations) with one degree of freedom. In linear systems with *N* degrees of freedom *(N* > 1) the free oscillations at each point are superpositions of *N* oscillations. In linear distributed systems (ignoring the atomistic structure of matter), such as a string, rod, pipe, electric cable, or cavity resonator, the free oscillations at each point are superpositions of an infinite number of oscillations. If the restoring force—that is, the force that restores the system to a state of equilibrium is not proportional to the deviation from equilibrium, the free oscillations may be described by a nonlinear differential equation (for example, in the case of a pendulum, when the amplitude cannot be considered very small). Such systems are said to be nonlinear. Here, in contrast to linear systems, free oscillations (even if damping is not taken into account) are not sinusoidal, and in addition, their period depends on the initial conditions—for example, in a pendulum the greater the amplitude the longer the period of the free oscillations. Only at the limit, when the amplitude tends toward zero, does the system become linear and its oscillations isochronous: the period is independent of amplitude.

Fluctuation oscillations take place as a result of the thermal motion of matter. Since a pendulum, bob, or circuit takes part in the thermal motion of matter, it performs unending fluctuations—one type of Brownian motion. These oscillations are particularly easy to detect and observe in the case of an oscillatory circuit, in which fluctuations of voltage and current take place, using an amplifier with a high gain and an oscilloscope. Fluctuations in oscillation circuits and antennas are the most important factor limiting the sensitivity of receivers.

Self-oscillations are sustained oscillations that can exist in the absence of a variable external influence; the amplitude and period of the oscillations are determined only by the properties of the system itself and, within certain limits, are independent of the initial conditions. Examples include the oscillations of a pendulum or the balance wheel of a watch, which are maintained by the lowering of a weight or the unwinding of a helical spring; the sounding of wind and string musical instruments; and oscillations of all types of electronic tube oscillators that are used in radio engineering.

** Propagation**. An oscillating pendulum (Figure 1) sets in motion the frame on which it is suspended; the frames sets the table in motion, and so on. Thus, oscillations do not remain localized but rather propagate, encompassing all surrounding bodies. The phenomenon of the propagation of oscillations is much more pronounced in the case of faster mechanical (sound) oscillations —strings, bells, and air in the pipes of wind instruments. Here the propagation of oscillations takes place primarily through the air. Variable electric and magnetic fields that propagate from point to point into the distance through dielectrics (including vacuum) arise around the sources of electrical oscillations. The processes of propagation of oscillations (and also of all perturbations) are called waves.

** General character of oscillatory influences**. The greater the load, the greater will be the bending of a beam under the constant load; the greater the electromotive force, the greater the strength of the current that arises under its action. In the case of an oscillating load, variable electromotive force and other oscillatory influences, the problem is much more complex: here forced oscillations take place. In this case the result of an effect depends not only on its intensity but also largely on its rate (on the way in which it changes over time). This is one of the main and characteristic features of oscillations.

Let a number of periodic brief impulses act upward on the bob of a spring pendulum. By virtue of the linearity of the system, the superposition principle is valid: the actions of the individual impulses are added. In general, the action of each successive impulse will both amplify and weaken the action of all preceding impulses with identical probability, and the amplitude of the oscillations will alternately increase and decrease, remaining comparatively low. However, if the period of the impulses is equal to or a multiple of the period of the natural oscillations, every impulse, acting “in step” with the oscillations, will amplify the action of the preceding impulses, and the spring pendulum will swing to a very great amplitude. The increase in amplitude will end only because the attenuation of the oscillations in the interval between two impulses acquires considerable magnitude when there is a large buildup. The buildup of a linear oscillatory system under the influence of periodical impulses, when limited solely by attenuation, is the phenomenon of resonance. Another important case of resonance occurs when such a system is acted on by a continuous force that varies sinusoidally if its frequency of change coincides with the frequency ω_{0} of the system’s free oscillations.

When there is a periodic change in a parameter of an oscillatory system, such as the length of the pendulum filament or the capacitance of an oscillatory circuit, in general the pendulum will not swing, or electrical oscillations will not appear in the circuit. Even in this case, however, with an appropriate rate of influence (or, best of all, if the parameter changes with a frequency equal to 2ω), oscillations may occur. Fluctuations, which have a continuous spectrum with harmonic components having all possible phase shifts, always exist in any oscillatory system as a result of the influence of various random factors. Therefore, periodic changes in a parameter of a system will always coincide in phase with one of the harmonic components, and its amplitude will increase in such a way that the pendulum will begin to swing about the vertical, or intensifying electromagnetic oscillations will appear in the circuit.

**Frequencies of Some of The Most Important Oscillations**. Rotation is the superposition of two mutually perpendicular harmonic oscillations. The revolution of the planets around the sun occurs with frequencies ranging from 1.28 X 10^{-9} hertz (Hz) in the case of Pluto, with a period of 250 years, to 1.32 X 10^{-7} Hz in the case of Mercury, with a period of 88 days. A day—the period of the earth’s rotation about its axis—corresponds to a frequency of about 1.16 X 10^{-5} Hz. The tides occur with a frequency of the same order. Sea waves, which arise under the action of the wind, have a frequency of approximately 10^{-1} Hz. The vibrations of structures and vibration speed of rotation of machines have frequencies ranging from fractions of a hertz to about 10^{4} Hz. Mechanical oscillations, which are perceived by the normal human ear as sound, occur with frequencies ranging from 20 Hz to about 2 X 10^{4} Hz. Faster, inaudible elastic oscillations with a frequency of up to 10^{9} Hz are called ultrasonic, and those with frequencies of up to 10^{12}-10^{13} Hz are called hypersonic. Frequencies of the order of 10^{13} Hz are inherent in atomic oscillations, which constitute thermal motions in solids and fluids.

In the USSR and most other countries the alternating current generated by electric power plants have a standard frequency of 50 Hz. Radio engineering uses electromagnetic oscillations and waves with frequencies ranging from 10^{5} Hz (long waves) to 10^{11} Hz (millimeter waves). Optics deals with electromagnetic waves, in which the oscillation of the intensity of electric and magnetic fields takes place with a frequency ranging from 10^{12} to 10^{17} Hz. Visible light (red, 0.4 X 10^{14} Hz; violet, 0.75 X 10^{14} Hz) falls in this interval. The interval running from 10^{12} to 10^{14} Hz corresponds to infrared radiation, and the interval from 10^{15} to 10^{17} Hz corresponds to ultraviolet radiation. Then, in ascending order of frequency, come X radiation (10^{18}-10^{19} Hz), gamma radiation (10^{20} Hz), and the electromagnetic radiation found in cosmic rays (up to 10^{22} Hz or more).

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