# Anharmonicity

In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using perturbation theory. If the anharmonicity is large, then other numerical techniques have to be used. In reality all oscillating systems are anharmonic, but most approximate the harmonic oscillator the smaller the amplitude of the oscillation is. Potential energy of a diatomic molecule as a function of atomic spacing. When the molecules are too close or too far away, they experience a restoring force back towards u0. (Imagine a marble rolling back and forth in the depression.) The blue curve is close in shape to the molecule's actual potential well, while the red parabola is a good approximation for small oscillations. The red approximation treats the molecule as a harmonic oscillator, because the restoring force, -V'(u), is linear with respect to the displacement u.

As a result, oscillations with frequencies $2\omega$ and $3\omega$ etc., where $\omega$ is the fundamental frequency of the oscillator, appear. Furthermore, the frequency $\omega$ deviates from the frequency $\omega _{0}$ of the harmonic oscillations. See also intermodulation and combination tones. As a first approximation, the frequency shift $\Delta \omega =\omega -\omega _{0}$ is proportional to the square of the oscillation amplitude $A$ :

$\Delta \omega \propto A^{2}$ In a system of oscillators with natural frequencies $\omega _{\alpha }$ , $\omega _{\beta }$ , ... anharmonicity results in additional oscillations with frequencies $\omega _{\alpha }\pm \omega _{\beta }$ .

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