Harmonic Oscillations

Harmonic Oscillations


oscillations during which a physical quantity varies with time according to a sine or cosine law. Harmonic oscillations are represented graphically by a sine or cosine curve (see Figure 1); they can be written in the form x = A sin(ωt + φ) or x = A cos(ω + φ), where x is the value of the oscillating quantity at time t (for mechanical harmonic oscillations, for example, the displacement or velocity; for electrical harmonic oscillations, voltage or current strength), A is the amplitude of the oscillation, ω is the angular frequency, (ωt + φ) is the phase of the oscillation, and φ is the initial phase of the oscillation.

Figure 1. Harmonic oscillations

Harmonic oscillations occupy an important place among the various types of oscillations; this is determined by two circumstances. First, oscillatory processes whose form is close to harmonic oscillations are often encountered in nature and technology. Second, a very broad class of systems whose properties can be considered invariable (for example, electrical circuits in which the inductance, capacitance, and resistance do not depend on the voltage or current strength) behave in a special way in relation to harmonic oscillations: when acted upon by harmonic oscillations, the induced oscillations they perform are also harmonic oscillations. (When the form of the external action is not harmonic oscillation, the system’s induced oscillations always differ from the form of the external action.) In other words, in most cases harmonic oscillations are the only type of oscillation whose form is not distorted upon reproduction; this fact determines the special value of harmonic oscillations, as well as the possibility of representing nonharmonic oscillations in the form of a harmonic spectrum of oscillations.


Elementarnyi uchebnik fiziki, 3rd ed., vol. 3. Edited by G. S. Landsberg. Moscow, 1962.
Khaikin, S. E. Fizicheskie osnovy mekhaniki. Moscow, 1963.
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