Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.

Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating of the human heart (for circulation), business cycles in economics, predator–prey population cycles in ecology, geothermal geysers in geology, vibration of strings in guitar and other string instruments, periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy. The term vibration is precisely used to describe a mechanical oscillation.

Oscillation, especially rapid oscillation, may be an undesirable phenomenon in process control and control theory (e.g. in sliding mode control), where the aim is convergence to stable state. In these cases it is called chattering or flapping, as in valve chatter, and route flapping.

Simple harmonic oscillation

The simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension. Such a system may be approximated on an air table or ice surface. The system is in an equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired momentum which keeps it moving beyond that position, establishing a new restoring force in the opposite sense. If a constant force such as gravity is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory period.

The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion. In the spring-mass system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, namely the existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium.

In the case of the spring-mass system, Hooke's law states that the restoring force of a spring is:

$${\displaystyle F=-kx}$$

By using Newton's second law, the differential equation can be derived:

$${\displaystyle {\ddot {x}}=-{\frac {k}{m}}x=-\omega ^{2}x,}$$

${\textstyle \omega ={\sqrt {k/m}}}$

The solution to this differential equation produces a sinusoidal position function:

$${\displaystyle x(t)=A\cos(\omega t-\delta )}$$

where ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and −1 infinitely, our spring-mass system would oscillate between the positive and negative amplitude forever without friction.

Two-dimensional oscillators

In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an isotropic oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions.

$${\displaystyle {\vec {F}}=-k{\vec {r}}}$$

This produces a similar solution, but now there is a different equation for every direction.

$${\displaystyle {\begin{aligned}x(t)&=A_{x}\cos(\omega t-\delta _{x}),\\y(t)&=A_{y}\cos(\omega t-\delta _{y}),\\&\;\,\vdots \end{aligned}}}$$

Anisotropic oscillators

With anisotropic oscillators, different directions have different constants of restoring forces. The solution is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is quasiperiodic. This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat.^[1]

Damped oscillations

All real-world oscillator systems are thermodynamically irreversible. This means there are dissipative processes such as friction or electrical resistance which continually convert some of the energy stored in the oscillator into heat in the environment. This is called damping. Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator.

Damped oscillators are created when a resistive force is introduced, which is dependent on the first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant b. This example assumes a linear dependence on velocity.

$${\displaystyle m{\ddot {x}}+b{\dot {x}}+kx=0}$$

This equation can be rewritten as before:

$${\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=0,}$$

${\textstyle 2\beta ={\frac {b}{m}}}$

This produces the general solution:

$${\displaystyle x(t)=e^{-\beta t}\left(C_{1}e^{\omega _{1}t}+C_{2}e^{-\omega _{1}t}\right),}$$

${\textstyle \omega _{1}={\sqrt {\beta ^{2}-\omega _{0}^{2}}}}$

The exponential term outside of the parenthesis is the decay function and β is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where β < ω₀; over-damped, where β > ω₀; and critically damped, where β = ω₀.

Driven oscillations

In addition, an oscillating system may be subject to some external force, as when an AC circuit is connected to an outside power source. In this case the oscillation is said to be driven.

The simplest example of this is a spring-mass system with a sinusoidal driving force.

$${\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=f(t),}$$

${\displaystyle f(t)=f_{0}\cos(\omega t+\delta ).}$

This gives the solution: