Antiresonance

anti-resonanceanti-resonantantiresonatorsantiresonant
In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large shift in its oscillation phase.wikipedia
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RLC circuit

LRC circuitRLCRLC circuits
The simplest system in which antiresonance arises is a system of coupled harmonic oscillators, for example pendula or RLC circuits.
For this reason they are often described as antiresonators, it is still usual, however, to name the frequency at which this occurs as the resonance frequency.

Resonance

resonantresonant frequencyresonance frequency
In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large shift in its oscillation phase.

Oscillation

oscillatorvibrationoscillators
In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large shift in its oscillation phase.

Normal mode

modesnormal modesmode
: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.

Physics

physicistphysicalphysicists
In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large shift in its oscillation phase.

Amplitude

peak-to-peakintensityvolume
In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large shift in its oscillation phase.

Frequency

frequenciesperiodperiodic
In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large shift in its oscillation phase.

Phase (waves)

phasephase shiftout of phase
In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large shift in its oscillation phase.

Physical system

systemphysical systemsphysical
Such frequencies are known as the system's antiresonant frequencies, and at these frequencies the oscillation amplitude can drop to almost zero.

Wave interference

interferenceconstructive interferencedestructive interference
Antiresonances are caused by destructive interference, for example between an external driving force and interaction with another oscillator.

Mechanics

mechanicaltheoretical mechanicsmechanician
Antiresonances can occur in all types of coupled oscillator systems, including mechanical, acoustic, electromagnetic and quantum systems.

Acoustics

acousticacousticianacoustical
Antiresonances can occur in all types of coupled oscillator systems, including mechanical, acoustic, electromagnetic and quantum systems.

Electromagnetism

electromagneticelectrodynamicselectromagnetic force
Antiresonances can occur in all types of coupled oscillator systems, including mechanical, acoustic, electromagnetic and quantum systems.

Quantum mechanics

quantum physicsquantum mechanicalquantum theory
Antiresonances can occur in all types of coupled oscillator systems, including mechanical, acoustic, electromagnetic and quantum systems.

Harmonic oscillator

harmonic oscillatorsharmonic oscillationdamped harmonic oscillator
The simplest system in which antiresonance arises is a system of coupled harmonic oscillators, for example pendula or RLC circuits.

Pendulum

pendulumssimple pendulumpendula
The simplest system in which antiresonance arises is a system of coupled harmonic oscillators, for example pendula or RLC circuits.

Ordinary differential equation

ordinary differential equationsordinaryODE
The situation is described by the coupled ordinary differential equations

Damping ratio

dampingdampedoverdamped
:where the \omega_i represent the resonance frequencies of the two oscillators and the \gamma_i their damping rates.

Complex number

complexreal partimaginary part
Changing variables to the complex parameters, allows us to write these as first-order equations:

Rotating wave approximation

rotating-wave approximationrapidly rotating'' in the frame of the atom
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).

Angular frequency

angular rateangular speedangular frequencies
:Without damping, driving or coupling, the solutions to these equations are, which represent a rotation in the complex \alpha plane with angular frequency \Delta.

Steady state

steady-stateequilibriumsteady
The steady-state solution can be found by setting, which gives:

Spectral density

frequency spectrumpower spectrumspectrum
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.