An oscillating system responds especially strongly when it is driven near one of its natural frequencies. This phenomenon is resonance.

Natural frequency

A system capable of oscillation has one or more characteristic frequencies determined by its physical parameters and boundary conditions.

A mass-spring oscillator, for example, has angular frequency

$$\omega_0=\sqrt{\frac{k}{m}}.$$

A string or air column can have several normal-mode frequencies.

Driven oscillation

If an external periodic force drives the system, the steady oscillation amplitude depends on the driving frequency. Near a natural frequency, energy is transferred efficiently from the driver to the oscillator and the response can become large.

Damping

Real systems lose energy. Damping limits the resonant amplitude and broadens the range of frequencies over which the response is strong.

Weak damping produces a sharper resonance; stronger damping produces a broader and smaller peak.

Resonance is not unlimited growth in real systems

Ideal undamped models can predict amplitudes that grow without bound at exact resonance, but real systems always contain losses, nonlinearities or physical limits.

Resonance appears in mechanical structures, musical instruments, electrical circuits, antennas and many other wave and oscillation systems.