AOA

Coupled oscillations

The harmonic oscillator and the systems it models have a single degree of freedom. More complicated systems have more degrees of freedom, for example two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of the others. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise. This phenomenon was first observed by Christiaan Huygens in 1665.[1] The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into normal modes.

More special cases are the coupled oscillators where the energy alternates between two forms of oscillation. Well-known is the Wilberforce pendulum, where the oscillation alternates between an elongation of a vertical spring and the rotation of an object at the end of that spring.
[edit] Continuous systems – waves

As the number of degrees of freedom becomes arbitrarily large, a system approaches continuity; examples include a string or the surface of a body of water. Such systems have (in the classical limit) an infinite number of normal modes and their oscillations occur in the form of waves that can characteristically prop

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