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
resonant frequencies (or
resonance 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 resonant frequency is approximately equal to the natural
frequency of the system, which is the frequency of free vibrations.
Resonance 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 (e.g. musical instruments), or pick out
specific frequencies from a complex vibration containing many
frequencies.
Resonance was discovered by
Galileo
Galilei with his investigations of
pendulums and
musical
strings 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
Electrical
resonance
Optical resonance
Orbital resonance in
astronomy
Atomic, particle, and molecular resonance
Theory
The exact response of a resonance, especially for frequencies far
from the resonant frequency, depends on the details of the physical
system, and is usually not exactly symmetric about the resonant
frequency, as illustrated for the
simple harmonic oscillator
above.For a lightly
damped linear oscillator
with a resonant frequency Ω, the
intensity of oscillations
I when the system is driven with a driving frequency ω is
typically approximated by a formula that is symmetric about the
resonant frequency:
- I(\omega) \propto \frac{\frac{\Gamma}{2}}{(\omega - \Omega)^2 +
\left( \frac{\Gamma}{2} \right)^2 }.
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 resonant frequency. The linewidth is
inversely proportional to the
Q factor, which is a measure of the
sharpness of the resonance.
In
electrical engineering,
this approximate symmetric response is known as the
universal
resonance curve, a concept introduced by
Frederick E. Terman in 1932 to simplify the
approximate analysis of radio circuits with a range of center
frequencies and Q values.
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 d\,, the length of a round trip is 2d\,. 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, 2d\,, be equal to an integer
number of wavelengths \lambda\, of the wave:
- 2d = N\lambda,\qquad\qquad N \in \{1,2,3...\}
If the velocity of a wave is v\,, the frequency is f = v /
\lambda\, so the resonance frequencies are:
- f = \frac{Nv}{2d}\qquad\qquad N \in \{1,2,3...\}
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
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
Electrical resonance occurs in an
electric circuit at a particular
resonance frequency when the
impedance between the input and
output of the circuit is at a minimum (or when the
transfer function is at a maximum). Often
this happens when the impedance between the input and output of the
circuit is almost zero and when the transfer function is close to
one.
Optical resonance
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
reflects 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.
Ring resonators and whispering
galleries are example of optical resonators which do not form
standing waves.
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.
Additional optical resonances are
Guided-mode resonances and surface
plasmon resonance, which result in anomalus 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-metalic interface. These modes are ususally excited by a
subwavelength grating.
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
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,
crystal 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 ( ) 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
: 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 \Gamma is the
decay rate and \Omega
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 M+i\Gamma. The formula is further related to the
particle's
decay rate by
the
optical theorem.
Failure of the original Tacoma Narrows Bridge
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.
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
References
- Olson,
Harry F. Music, Physics and Engineering. Dover
Publications, 1967, pp. 248–249. "Under very favorable conditions
most individuals can obtain tonal characteristics as low as 12
cycles."
-
http://www.physics.ucla.edu/demoweb/demomanual/acoustics/effects_of_sound/breaking_glass_with_sound.html
- http://www.rp-photonics.com/q_factor.html
- K. Billah and R. Scanlan (1991), Resonance, Tacoma Narrows
Bridge Failure, and Undergraduate Physics Textbooks,
American Journal of Physics,
59(2), 118--124 (PDF)
- Continued Shaking on space station rattles NASA.
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