Definition 0.1. The very sharp (or marked) increase in amplitude of oscillation of a mechanical, electrical or nuclear system O₁ exposed to a periodic force F_r(t), or another oscillator O₂ whose frequency is equal to, or very close to, the natural undamped frequency of the oscillating system O₁. The intensity of oscillation is defined as the square of the amplitude of the oscillations. More precisely, in the case of a linear oscillator with a resonance frequency ω₀, the intensity of oscillations I(ω) when the system O₁ is driven with a driving frequency ω₂ is given by:

I(ω₂) = const..

This oscillator intensity is a Lorentzian function which is characteristic of many resonant systems, with γ being known as the linewidth of the resonance, or resonance linewidth at half-heigth of the resonance peak which depends on the degree of damping of the oscillator.

Remark 0.1. Heavily damped oscillators have broad linewidths, and will respond to a wider range of driving frequencies near the resonance frequency. The resonance linewidth γ is inversely proportional to the quality factor, or Q-factor of the oscillating systems, which is a measure of the sharpness of their resonance. As an example, for NMR and rf electrical circuit probes, a high Q-factor is essential for high sensitivity detection of the signal by the rf probe.

Definition 0.2. A system is called resonant if it has the property of oscillating by itself (without external coupling) with a maximum amplitude only at a certain frequency or frequencies, known as the resonance frequencies of the system.