In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of an oscillator at a particular frequency, accompanied by a large, abrupt shift in its oscillation phase. Such frequencies are known as the system's antiresonant frequencies, and at these frequencies the oscillation amplitude can drop to almost zero. Antiresonances are caused by destructive interference, for example between an external driving force and interaction with another oscillator.
Antiresonances can occur in all types of coupled oscillator systems, including mechanical, acoustical, electromagnetic, and quantum systems. They have important applications in the characterization of complicated coupled systems.
The term antiresonance is used in electrical engineering for a form of resonance in a single oscillator with similar effects.
Antiresonance in electrical engineering
In electrical engineering, antiresonance is the condition for which the reactance vanishes and the impedance of an electrical circuit is very high, approaching infinity.
In an electric circuit consisting of a capacitor and an inductor in parallel, antiresonance occurs when the alternating current line voltage and the resultant current are in phase. Under these conditions the line current is very small because of the high electrical impedance of the parallel circuit at antiresonance. The branch currents are almost equal in magnitude and opposite in phase.
Antiresonance in coupled oscillators
The simplest system in which antiresonance arises is a system of coupled harmonic oscillators, for example pendula or RLC circuits.
Consider two harmonic oscillators coupled together with strength and with one oscillator driven by an oscillating external force . The situation is described by the coupled ordinary differential equations
where the represent the resonance frequencies of the two oscillators and the their damping rates. Changing variables to the complex parameters:
allows us to write these as first-order equations:
We transform to a frame rotating at the driving frequency
yielding
where we have introduced the detunings between the drive and the oscillators' resonance frequencies. Finally, we make a rotating wave approximation, neglecting the fast counter-rotating terms proportional to , which average to zero over the timescales we are interested in (this approximation assumes that , which is reasonable for small frequency ranges around the resonances). Thus we obtain:
Without damping, driving or coupling, the solutions to these equations are:
which represent a rotation in the complex plane with angular frequency .
The steady-state solution can be found by setting , which gives:
Examining these steady state solutions as a function of driving frequency, it is evident that both oscillators display resonances (peaks in amplitude accompanied by positive phase shifts) at the two normal mode frequencies. In addition, the driven oscillator displays a pronounced dip in amplitude between the normal modes which is accompanied by a negative phase shift. This is the antiresonance. Note that there is no antiresonance in the undriven oscillator's spectrum; although its amplitude has a minimum between the normal modes, there is no pronounced dip or negative phase shift.
Interpretation as destructive interference
The reduced oscillation amplitude at an antiresonance can be regarded as due to destructive interference or cancellation of forces acting on the oscillator.
In the above example, at the antiresonance frequency the external driving force acting on oscillator 1 cancels the force acting via the coupling to oscillator 2, causing oscillator 1 to remain almost stationary.
Complicated coupled systems
The frequency response function (FRF) of any linear dynamic system composed of many coupled components will in general display distinctive resonance-antiresonance behavior when driven.
As a rule of thumb, it can be stated that as the distance between the driven component and the measured component increases, the number of antiresonances in the FRF decreases. For example, in the two-oscillator situation above, the FRF of the undriven oscillator displayed no antiresonance. Resonances and antiresonances only alternate continuously in the FRF of the driven component itself.