Resonance

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Increase of amplitude as damping decreases and frequency approaches resonance frequency.

In physics, resonance is the tendency of a system to oscillate at larger amplitude at some frequencies than at others. These are known as the system's resonance frequencies (or resonant frequencies). At these frequencies, even small periodic driving forces can produce large amplitude vibrations, because the system stores vibrational energy. When damping is small, the resonance frequency is approximately equal to the natural frequency of the system, which is the frequency of free vibrations. Resonant phenomena occur with all types of vibrations or waves: there is mechanical 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, or pick out specific frequencies from a complex vibration containing many frequencies.

Resonance was discovered by Galileo Galilei with his investigations of pendulums beginning in 1602.

Examples

One familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonance frequency) will make the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo will result in smaller arcs. This is because the energy the swing absorbs is maximized when the pushes are 'in phase' with the swing's oscillations, while some of the swing's energy is actually extracted by the opposing force of the pushes when they are not.

Resonance occurs widely in nature, and is exploited in many man-made 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 are:

Mechanical and acoustic resonance

• the timekeeping mechanisms of all modern clocks and watches: the balance wheel in a mechanical watch and the quartz crystal in a quartz watch
• the tidal resonance of the Bay of Fundy
• acoustic resonances of musical instruments and human vocal cords
• the resonance of the basilar membrane in the cochlea of the ear, which enables people to distinguish different frequencies or tones in the sounds they hear.
• the shattering of a crystal wineglass when exposed to a musical tone of the right pitch (its resonance frequency).

Electrical resonance

• electrical resonance of tuned circuits in radios and TVs that allow individual stations to be picked up

Optical resonance

• creation of coherent light by optical resonance in a laser cavity

Orbital resonance in astronomy

• orbital resonance as exemplified by some moons of the solar system's gas giants

Atomic, particle, and molecular resonance

• material resonances in atomic scale are the basis of several spectroscopic techniques that are used in condensed matter physics.
  • Nuclear Magnetic Resonance
  • Mössbauer effect
  • Electron Spin Resonance.

Theory

For a linear oscillator with a resonance frequency Ω, the intensity of oscillations I when the system is driven with a driving frequency ω is given by:

<math>I(\omega) \propto \frac{\frac{\Gamma}{2}}{(\omega - \Omega)^2 + \left( \frac{\Gamma}{2} \right)^2 }.</math>

The intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, and this response is found in many physical situations involving resonant systems. Γ is a parameter dependent on the damping of the oscillator, and is known as the linewidth of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonance frequency. The linewidth is inversely proportional to the Q factor, which is a measure of the sharpness of the resonance.

Resonators

A physical system can have as many resonance frequencies as it has degrees of freedom; each degree of freedom can vibrate as a harmonic oscillator. Systems with one degree of freedom, such as a mass on a spring, pendulums, balance wheels, and LC tuned circuits have one resonance frequency. Systems with two degrees of freedom, such as coupled pendulums and resonant transformers can have two resonance frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer energy from one to the next becomes significant. The vibrations in them begin to travel through the coupled harmonic oscillators in waves, from one oscillator to the next.

Extended objects that experience resonance due to vibrations inside them are called resonators, such as organ pipes, vibrating strings, quartz crystals, microwave cavities, and laser rods. Since these can be viewed as being made of millions of coupled moving parts (such as atoms), they can have millions of resonance frequencies. The vibrations inside them travel as waves, at an approximately constant velocity, bouncing back and forth between the sides of the resonator. If the distance between the sides is <math>d\,</math>, the length of a round trip is <math>2d\,</math>. In order to cause resonance, the phase of a sinusoidal wave after a round trip has to be equal to the initial phase, so the waves will reinforce. So the condition for resonance in a resonator is that the round trip distance, <math>2d\,</math>, be equal to an integral number of wavelengths <math>\lambda\,</math> of the wave:

<math>2d = N\lambda,\qquad\qquad N \in \{1,2,3...\}</math>

If the velocity of a wave is <math>v\,</math>, the frequency is <math>f = v / \lambda\,</math> so the resonance frequencies are:

<math>f = \frac{Nv}{2d}\qquad\qquad N \in \{1,2,3...\}</math>

So the resonance frequencies of resonators, called normal modes, are equally spaced multiples of a lowest frequency called the fundamental frequency. The multiples are often called overtones. There may be several such series of resonance frequencies, corresponding to different modes of vibration.

Mechanical and acoustic resonance

Main articles: Mechanical resonance, Acoustic resonance, and String resonance

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, and airplanes. Engineers when designing objects must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other oscillating parts a phenomenon known as resonance disaster.

Avoiding resonance disasters is a major concern in every building, tower and bridge construction project. As a countermeasure, shock mounts can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 730-ton pendulum — a tuned mass damper — to cancel resonance. Furthermore, the structure is designed to resonate at a frequency which does not typically occur. Buildings in seismic zones are often constructed to take into account the oscillating frequencies of expected ground motion. In addition, Engineers designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts.

Many clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal

Acoustic resonance is a branch of mechanical resonance that is concerned the mechanical vibrations in the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between about 12 Hz and 20,000 Hz (20 kHz),

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 a drum membrane. Acoustic resonance is also important for hearing. For example, resonance of a stiff structural element, called the basilar membrane within the cochlea of the inner ear allows hairs on the membrane to detect sound. (For mammals the membrane by having different resonance on either end so that high frequencies are concentrated on one end and low frequencies on the other.)

Like mechanical resonance, acoustic resonance can result in catastrophic failure of the vibrator. 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.

Electrical resonance

Main article: Electrical resonance

Optical Resonance

Main article: optical cavity

An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. Light confined in the cavity reflect multiple times producing standing waves for certain resonance frequencies. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam.

Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them. (Flat mirrors are not often used because of the difficulty of aligning them to the needed precision.) The geometry (resonator type) must be chosen so that the beam remains stable (that the size of the beam does not continually grow with multiple reflections. Resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point (and therefore intense light at that point) inside the cavity.

Optical cavities are designed to have a very large Q factor; a beam will reflect a very large number of times with little attenuation. Therefore the frequency line width of the beam is very small indeed compared to the frequency of the laser.

Orbital resonance

Main article: Orbital resonance

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.

Atomic, particle, and molecular resonance

Main articles: Nuclear magnetic resonance and Resonance (particle)

900MHz, 21.2 T NMR Magnet at HWB-NMR, Birmingham, UK being loaded with a sample

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).

All nuclei that contain odd numbers of nucleons have an intrinsic magnetic moment and angular momentum. A key feature of NMR is that the resonance frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonance frequencies of the sample's nuclei depend on where in the field they are located. Therefore, the particle can be located quite precisely from its resonance frequency.

Electron paramagnetic resonance, otherwise known as Electron Spin Resonance (ESR) is a spectroscopic technique similar to NMR used with 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.

The Mössbauer effect (Template:Lang-de) is a physical phenomenon discovered by Rudolf Mößbauer in 1957; it refers to the resonant and recoil-free emission and absorption of gamma ray photons by atoms bound in a solid form.

Resonance (particle): In quantum mechanics and quantum field theory resonances may appear in similar circumstances to classical physics. However, they can also be thought of as unstable particles, with the formula above still valid if the <math>\Gamma</math> is the decay rate and <math>\Omega</math> replaced by the particle's mass M. In that case, the formula just comes from the particle's propagator, with its mass replaced by the complex number <math>M+i\Gamma</math>. The formula is further related to the particle's decay rate by the optical theorem.

Failure of the original Tacoma Narrows Bridge

Main article: Tacoma Narrows Bridge (1940)

The dramatically visible, rhythmic twisting that resulted in the 1940 collapse of "Galloping Gertie," the original Tacoma Narrows Bridge, has sometimes been characterized in physics textbooks as a classical example of resonance; however, this description is misleading. The catastrophic vibrations that destroyed the bridge were not due to simple mechanical resonance, but to a more complicated oscillation between the bridge and the winds passing through it — a phenomenon known as aeroelastic flutter. Robert H. Scanlan, father of the field of bridge aerodynamics, wrote an article about this misunderstanding.

Further information: mechanical resonance

Resonance causing vibration on the International Space Station

The rocket engines for the International Space Station are controlled by autopilot. Ordinarily the uploaded parameters for controlling the engine control system for the Zvezda module will cause the rocket engines to boost the International Space Station to a higher orbit. The rocket engines are hinge-mounted, and ordinarily the operation is not noticed by the crew. But on January 14, 2009, the uploaded parameters caused the autopilot to swing the rocket engines in larger and larger oscillations, at a 2 second frequency. These oscillations were captured on video, and lasted for 142 seconds.

See also

• Beat (acoustics)
• Center frequency
• Driven harmonic motion
• Earthquake engineering
• Formant
• Harmonic oscillator
• Impedance
• Q factor
• Resonance disaster
• Resonator
• Vibration
• Schumann resonance
• Simple harmonic motion
• Tuned circuit
• Wave
• Sympathetic string
• cymatics

External links

• Definition of Resonance - "The increase in amplitude of oscillation of an electric or mechanical system exposed to a periodic force whose frequency is equal or very close to the natural undamped frequency of the system."
• Resonance - a chapter from an online textbook
• Greene, Brian, "Resonance in strings". The Elegant Universe, NOVA (PBS)
• Hyperphysics section on resonance concepts
• A short FAQ on quantum resonances
• Resonance versus resonant (usage of terms)
• Wood and Air Resonance in a Harpsichord
• Java applet demonstrating resonances on a string when the frequency of the driving force is varied
• Breaking glass with sound, including high-speed footage of glass breaking