Anharmonicity

As a result, oscillations with frequencies ${\displaystyle 2\omega }$ and ${\displaystyle 3\omega }$ etc., where ${\displaystyle \omega }$ is the fundamental frequency of the oscillator, appear. Furthermore, the frequency ${\displaystyle \omega }$ deviates from the frequency ${\displaystyle {\omega }_0}$ of the harmonic oscillations. See also intermodulation and combination tones. As a first approximation, the frequency shift ${\displaystyle \mathrm{\Delta }\omega =\omega -{\omega }_0}$ is proportional to the square of the oscillation amplitude ${\displaystyle A}$:

${\displaystyle \mathrm{\Delta }\omega \propto A^2}$

In a system of oscillators with natural frequencies ${\displaystyle {\omega }_{\alpha }}$, ${\displaystyle {\omega }_{\beta }}$, ... anharmonicity results in additional oscillations with frequencies ${\displaystyle {\omega }_{\alpha }\pm {\omega }_{\beta }}$.

Anharmonicity also modifies the energy profile of the resonance curve, leading to interesting phenomena such as the foldover effect and superharmonic resonance.

General principle

An oscillator is a physical system characterized by periodic motion, such as a pendulum, tuning fork, or vibrating diatomic molecule. Mathematically speaking, the essential feature of an oscillator is that for some coordinate x of the system, a force whose magnitude depends on x will push x away from extreme values and back toward some central value x₀, causing x to oscillate between extremes. For example, x may represent the displacement of a pendulum from its resting position x=0. As the absolute value of x increases, so does the restoring force acting on the pendulums weight that pushes it back towards its resting position.

In harmonic oscillators, the restoring force is proportional in magnitude (and opposite in direction) to the displacement of x from its natural position x₀. The resulting differential equation implies that x must oscillate sinusoidally over time, with a period of oscillation that is inherent to the system. x may oscillate with any amplitude, but will always have the same period.

Anharmonic oscillators, however, are characterized by the nonlinear dependence of the restorative force on the displacement x. Consequently, the anharmonic oscillator's period of oscillation may depend on its amplitude of oscillation.

As a result of the nonlinearity of anharmonic oscillators, the vibration frequency can change, depending upon the system's displacement. These changes in the vibration frequency result in energy being coupled from the fundamental vibration frequency to other frequencies through a process known as parametric coupling.

Treating the nonlinear restorative force as a function F(x − x₀) of the displacement of x from its natural position, we may replace F by its linear approximation F₁ = F′(0) ⋅ (x−x₀) at zero displacement. The approximating function F₁ is linear, so it will describe simple harmonic motion. Further, this function F₁ is accurate when x − x₀ is small. For this reason, anharmonic motion can be approximated as harmonic motion as long as the oscillations are small.