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Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration that matches its natural frequency. When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger amplitude. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases.

All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; this frequency is known as a resonant frequency or resonance frequency. When an oscillating force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher amplitude (with more force) than when the same force is applied at other, non-resonant frequencies.[3]

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Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, orbital resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions. Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters).

The term resonance (from Latin resonantia, 'echo', from resonare, 'resound') originated from the field of acoustics, particularly the sympathetic resonance observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck.

Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all sinusoidal waves and vibrations are generated. Many sounds we hear, such as when hard objects of metal, glass, or wood are struck, are caused by brief resonant vibrations in the object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms. Other examples of resonance:

Resonance manifests itself in many linear and nonlinear systems as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large.

Since many linear and nonlinear systems that oscillate are modeled as harmonic oscillators near their equilibria, this section begins with a derivation of the resonant frequency for a driven, damped harmonic oscillator. The section then uses an RLC circuit to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, the section then generalizes these relationships for higher-order linear systems with multiple inputs and outputs.

The general solution of Equation (2) is the sum of a transient solution that depends on initial conditions and a steady state solution that is independent of initial conditions and depends only on the driving amplitude F0, driving frequency tag_hash_120, undamped angular frequency tag_hash_1210, and the damping ratio tag_hash_122. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution.

tag_hash_123r is the resonant frequency for this system. Again, the resonant frequency does not equal the undamped angular frequency tag_hash_1250 of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including tag_hash_1260, but the maximum response is at the resonant frequency.

Here, the resonance corresponds physically to having a relatively large amplitude for the steady state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies.

Some systems exhibit antiresonance that can be analyzed in the same way as resonance. For antiresonance, the amplitude of the response of the system at certain frequencies is disproportionately small rather than being disproportionately large. In the RLC circuit example, this phenomenon can be observed by analyzing both the inductor and the capacitor combined.

Rather than look for resonance, i.e., peaks of the gain, notice that the gain goes to zero at tag_hash_128 = tag_hash_1290, which complements our analysis of the resistor's voltage. This is called antiresonance, which has the opposite effect of resonance. Rather than result in outputs that are disproportionately large at this frequency, this circuit with this choice of output has no response at all at this frequency. The frequency that is filtered out corresponds exactly to the zeroes of the transfer function, which were shown in Equation (7) and were on the imaginary axis.

In the RLC circuit example, the first generalization relating poles to resonance is observed in Equation (5). The second generalization relating zeroes to antiresonance is observed in Equation (7). In the examples of the harmonic oscillator, the RLC circuit capacitor voltage, and the RLC circuit inductor voltage, "poles near the imaginary axis" corresponds to the significantly underdamped condition < 1/ 2 {\displaystyle {\sqrt {2}}} .

Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, engineers must ensure the mechanical resonance frequencies of the component parts do not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster.

Acoustic resonance is a branch of mechanical resonance that is concerned with the mechanical vibrations across the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between about 20 Hz and 20,000 Hz (20 kHz),[12] Many objects and materials act as resonators with resonant frequencies within this range, and when struck vibrate mechanically, pushing on the surrounding air to create sound waves. This is the source of many percussive sounds we hear.

Acoustic resonance is an important consideration for instrument builders, as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in a flute, and the shape of, and tension on, a drum membrane.

Like mechanical resonance, acoustic resonance can result in catastrophic failure of the object at resonance. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass, although this is difficult in practice.[13]

Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedance of the circuit is at a minimum in a series circuit or at maximum in a parallel circuit (usually when the transfer function peaks in absolute value). Resonance in circuits are used for both transmitting and receiving wireless communications such as television, cell phones and radio.

Additional optical resonances are guided-mode resonances and surface plasmon resonance, which result in anomalous reflection and high evanescent fields at resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are usually excited by a subwavelength grating.

In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa, and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting nearly everything else around them; this effect is used in the current definition of a planet.

Nuclear magnetic resonance (NMR) is the name given to a physical resonance phenomenon involving the observation of specific quantum mechanical magnetic properties of an atomic nucleus in the presence of an applied, external magnetic field. Many scientific techniques exploit NMR phenomena to study molecular physics, crystals, and non-crystalline materials through NMR spectroscopy. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI).

Electron paramagnetic resonance, otherwise known as electron spin resonance (ESR), is a spectroscopic technique similar to NMR, but uses unpaired electrons instead. Materials for which this can be applied are much more limited since the material needs to both have an unpaired spin and be paramagnetic. 006ab0faaa