Single oscillator

• The harmonic oscillator, when excited by a small, sudden external impulse, oscillates at a well-defined frequency that depends on the properties of the physical system, in particular on its "elasticity" and its "inertia", but not on the nature of the impulse. This frequency is called the natural frequency of the system.
• What makes this property of the harmonic oscillator interesting is that every vibrating system made of a single oscillating element behaves like a harmonic oscillator when its oscillation has a small enough amplitude.

Example

a small push on a swing makes it oscillate at a very precise frequency that depends on the length of the ropes (as well as on the strength of gravity). Although a swing is not a point mass, like the ideal harmonic oscillator, but a real extended system, it has only one degree of freedom. That is, a single coordinate is enough to describe its motion: the angle of inclination. Of course reality is always more complex than our models, but this model describes it quite well, at least as long as we limit ourselves to small-amplitude oscillations.

• For the calculation of the natural frequency of the harmonic oscillator see the section dynamics of harmonic motion.
• "Elasticity" and "inertia" are written in quotation marks above because the specific names of the equivalent properties can change depending on the nature of the medium and on the type of excitations and waves passing through it. On this point see the page on analogies in physics, and the pages on the calculation of natural frequencies in different physical systems.

Systems of many oscillators

• When we excite a real vibrating system by applying an external force, we generally obtain very complex oscillations. This generally depends on two factors:

1. the system can no longer be described as a single harmonic oscillator.
2. the external forces of interest may have a complicated form

Examples:

• The vibrations of a musical instrument played by a human performer
• The behaviour of a power line, or of a complex circuit exposed to the electromagnetic waves in the environment.
• The vibrations in a bridge, a building or a structure subjected to an earthquake, or simply to passing traffic

The good news is that the two difficulties can be separated, at least in linear systems (see the discussion of the superposition principle).

Here we deal essentially only with the first problem, examining the free response of the system, that is, its response to a small external impulse. It can be shown, in fact, that once the impulse response of a system is known, its entire frequency response can be reconstructed.

What is the frequency of a compound oscillator?

Let us therefore focus on the first problem: how do we describe the free response of a system that cannot be described as a single oscillator?

A musical example:

• when the string of a musical instrument is plucked, the force applied by the finger or the plectrum only serves to impose the shape the string takes at the start of the oscillation, after which the string oscillates completely freely (that is, subject only to the constraint reactions) at a very precise frequency (see natural frequencies of a string)
• even though we know that the string has sinusoidal normal modes (see normal modes of the string and natural frequencies of a string), the plucked string vibrates in a much more complex way, and the sound it produces is "richer" than the single pure sound corresponding to the fundamental frequency. This "richness" (which musicians call timbre) indicates that the string cannot be considered a single oscillator, but a compound one.
• The vibration (and therefore the sound) becomes even more complex when the string is part of an instrument such as the guitar. In this case the vibration of the string sets many other parts of the instrument, and the air it encloses, vibrating. So the musical instrument too, although to our ear it produces a single "note", must be considered, from the physical point of view, a compound oscillator.

The answer comes from the analysis of the normal modes of the system: a system made of n coupled oscillators has n independent normal modes of oscillation, each with its own natural frequency. To excite a particular normal mode, the system must be put in resonance with a source oscillating at the natural frequency of that mode. However, with an impulsive excitation followed by free motion of the system, all the normal modes are excited, though to different extents.

The answer, then, is that a compound system generally oscillates with a motion that contains all the frequencies of its normal modes. It can be thought of as the superposition of simple oscillations, each at a different frequency.

A powerful mathematical-physics technique for expanding a compound oscillation into (a particular set of) harmonic oscillations is spectral analysis, or Fourier analysis.

It remains to explain, in the string example, why our ear perceives a single "note". This, however, is a problem of sound perception, and in particular of the perception of pitch and the perception of timbre, for which we refer to the relevant sections.

Further reading and links

• Examples of calculating natural frequencies in various mechanical systems:
  • Natural frequencies of a bar
  • Natural frequencies of pipes
  • Natural frequencies of a string
  • Natural frequencies of a circular membrane
  • Natural frequencies of a rectangular membrane
  • Natural frequencies of circular plates
• Definition and characteristics of normal modes: normal modes
• Examples of normal modes in mechanical systems: normal modes of a string, normal modes of a circular membrane, normal modes of a rectangular membrane
• General remarks on resonance: resonance
• Examples of resonance in various systems, natural and otherwise: examples of resonance phenomena
• On spectral analysis see Fourier's theorem for the theory, and the guided tours of the signal spectral analysis app

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