Wavelength
From Wikipedia, the free encyclopedia
For other uses, see Wavelength (disambiguation).

Wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests, or troughs, or corresponding zero crossings as shown.

In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.[1] It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns.[2][3] Wavelength is commonly designated by the Greek letter lambda (λ). The concept can also be applied to periodic waves of non-sinusoidal shape.[1][4] The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.[5]

Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.[6]

Examples of wave-like phenomena are sound waves, light, and water waves. A sound wave is a periodic variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are periodic variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary periodically in both lattice position and time.

Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength.[7]Contents [hide]
1 Sinusoidal waves
1.1 Standing waves
1.2 Mathematical representation
1.3 General media
1.3.1 Nonuniform media
1.3.2 Crystals
2 More general waveforms
2.1 Envelope waves
2.2 Wave packets
3 Interference and diffraction
3.1 Double-slit interference
3.2 Single-slit diffraction
3.3 Diffraction-limited resolution
4 Subwavelength
5 See also
6 References
7 External links

[edit]
Sinusoidal waves

In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components.

The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by:[8]


Refraction: when a plane wave encounters a medium in which it has a slower speed, the wavelength decreases, and the direction adjusts accordingly.

where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency.

In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s. For sound waves in air, the speed of sound is 343 m/s (1238 km/h) (at room temperature and atmospheric pressure). As an example, the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×108 m/s divided by 108 Hz = 3 metres.

Visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm (430–750 THz) (for other examples, see electromagnetic spectrum). The wavelengths of sound frequencies audible to the human ear (20 Hz–20 kHz) are between approximately 17 m and 17 mm, respectively, assuming a typical speed of sound of about 343 m/s; the wavelengths in audible sound are much longer than those in visible light.

Frequency and wavelength can change independently, but only when the speed of the wave changes. For example, when light enters another medium, its speed and wavelength change while its frequency does not; this change of wavelength causes refraction, or a change in propagation direction of waves that encounter the interface between media at an angle.

Sinusoidal standing waves in a box that constrains the end points to be nodes will have an integer number of half wavelengths fitting in the box.
[edit]
Standing waves

A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).

A standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, and the wavelength is twice the distance between nodes. The wavelength, period, and wave velocity are related as before, if the stationary wave is viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities.[9]
[edit]
Mathematical representation

Traveling sinusoidal waves are often represented mathematically in terms of their velocity v (in the x direction), frequency f and wavelength λ as:


where y is the value of the wave at any position x and time t, and A is the amplitude of the wave. They are also commonly expressed in terms of (radian) wavenumber k (2π times the reciprocal of wavelength) and angular frequency ω (2π times the frequency) as:


in which wavelength and wavenumber are related to velocity and frequency as:


or


Dispersion causes separation of colors when light is refracted by a prism.

The relationship between ω and λ (or k) is called a dispersion relation. This is not generally a simple inverse relation because the wave velocity itself typically varies with frequency.[10]

Wavelength is decreased in a medium with higher refractive index.

In the second form given above, the phase (kx − ωt) is often generalized to (k•r − ωt), by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r. In that case, the wavenumber k, the magnitude of k, is still in the same relationship with wavelength as shown above, with v being interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction.

Generalizations to sinusoids of other phases, and to complex exponentials, are also common; see plane wave. The typical convention of using the cosine phase instead of the sine phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave

[edit]
General media

The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in most media is lower than in vacuum, which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum. The wavelength in the medium is


Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore.[11]

where λ0 is the wavelength in vacuum, and n(λ0) is the refractive index of the medium, which varies with wavelength. This variation, called dispersion, causes different colors of light to be separated when light is refracted by a prism.

When wavelengths of electromagnetic radiation are quoted, the vacuum wavelength is usually intended unless the wavelength is specifically identified as the wavelength in some other medium. In acoustics, where a medium is essential for the waves to exist, the wavelength value is given for a specified medium.
[edit]
Nonuniform media

A sinusoidal wave in a nonuniform medium, with loss. As the wave slows down, the wavelength gets shorter and the amplitude increases; after a place of maximum response, the short wavelength is associated with a high loss and the wave dies out.

Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.[11]

Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (an inhomogeneous medium) may propagate at a velocity that varies with position, and as a result may not be sinusoidal in space. The analysis of differential equations of such systems is often done approximately, using the WKB method (also known as the Liouville–Green method). The method integrates phase through space using a local wavenumber, which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space.[12][13] This method treats the system locally as if it were uniform with the local properties; in particular, the local wave velocity associated with a frequency is the only thing needed to estimate the corresponding local wavenumber or wavelength. In addition, the method computes a slowly changing amplitude to satisfy other constraints of the equations or of the physical system, such as for conservation of energy in the wave.
[edit]
Crystals

A wave on a line of atoms can be interpreted according to a variety of wavelengths.

Waves in crystalline solids are not continuous, because they are composed of vibrations of discrete particles arranged in a regular lattice. This produces aliasing because the same vibration can be considered to have a variety of different wavelengths, as shown in the figure.[14] Descriptions using more than one of these wavelengths are redundant; it is conventional to choose the longest wavelength that fits the phenomenon. The range of wavelengths sufficient to provide a description of all possible waves in a crystalline medium corresponds to the wave vectors confined to the Brillouin zone.[15]

This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such as energy bands and lattice vibrations. It is mathematically equivalent to the aliasing of a signal that is sampled at discrete intervals.
[edit]
More general waveforms

A wave moving in space is called a traveling wave. If the shape repeats itself, it is also a periodic wave.[16] In the special case of uniform and dispersionless media (see Dispersion relation), at a fixed moment in time, a snapshot of the wave shows a repeating form in space, with characteristics such as peaks and troughs repeating at equal intervals. To an observer at a fixed location the amplitude appears to vary in time, and repeats itself with a certain period, for example T. If the spatial period of this wave is referred to as its wavelength, then during every period, one wavelength of the wave passes the observer. In dispersion and uniform media, the wave propagates with unchanging shape and the velocity in the medium is uniform, so this period implies the wavelength is:


Near-periodic waves over shallow water have sharper crests and flatter troughs than those of a sinusoid.

This duality of space and time is expressed mathematically by the fact that, in such special media, the wave's behavior does not depend independently on position x and time t, but rather on the combination of position and time x − vt. The wave's amplitude u is then expressed as u(x − vt) and in the case of a periodic function u with period λ, that is, u(x + λ − vt) = u(x − vt), the periodicity of u in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ. In a similar fashion, this periodicity of u implies a periodicity in time as well: u(x − v(t + T)) = u(x − vt) using the relation vT = λ described above, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.[16]

Traveling waves with non-sinusoidal wave shapes can occur in linear dispersionless media such as free space, but also may arise in nonlinear media under certain circumstances. For example, large-amplitude ocean waves with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.[17] An example is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobi elliptic function of m-th order, usually denoted as cn (x; m).[18]
[edit]
Envelope waves

The term wavelength is also sometimes applied to the envelopes of waves, such as the traveling sinusoidal envelope patterns that result from the interference of two sinusoidal waves close in frequency; such envelope characterizations are used in illustrating the derivation of group velocity, the speed at which slow envelope variations propagate.[19]
[edit]
Wave packets

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.[20]
Main article: Wave packet

Localized wave packets, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics; the notion of a wavelength also may be applied to these wave packets.[21] The wave packet has an envelope that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called a local wavelength.[22][23] Using Fourier analysis, wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different wavenumbers or wavelengths.[24]

Louis de Broglie postulated that all particles with a specific value of momentum have a wavelength


where h is Planck's constant, and p is the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a De Broglie wavelength of about 10−13 m. To prevent the wave function for such a particle being spread over all space, De Broglie proposed using wave packets to represent particles that are localized in space.[25] The spread of wavenumbers of sinusoids that add up to such a wave packet corresponds to an uncertainty in the particle's momentum, one aspect of the Heisenberg uncertainty principle.[24]
[edit]
Interference and diffraction
[edit]
Double-slit interference
Main article: Interference (wave propagation)

Pattern of light intensity on a screen for light passing through two slits. The labels on the right refer to the difference of the path lengths from the two slits, which are idealized here as point sources.

When sinusoidal waveforms add, they may reinforce each other (constructive interference) or cancel each other (destructive interference) depending upon their relative phase. This phenomenon is used in the interferometer. A simple example is an experiment due to Young where light is passed through two slits.[26] As shown in the figure, light is passed through two slits and shines on a screen. The path of the light to a position on the screen is different for the two slits, and depends upon the angle θ the path makes with the screen. If we suppose the screen is far enough from the slits (that is, s is large compared to the slit separation d) then the paths are nearly parallel, and the path difference is simply d sin θ. Accordingly the condition for constructive interference is:[27]


where m is an integer, and for destructive interference is:


Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern or fringes, and vice versa.

It should be noted that the effect of interference is to redistribute the light, so the energy contained in the light is not altered, just where it shows up.[28]
[edit]
Single-slit diffraction
Main articles: Diffraction and Diffraction formalism

The notion of path difference and constructive or destructive interference used above for the double-slit experiment applies as well to the display of a single slit of light intercepted on a screen. The main result of this interference is to spread out the light from the narrow slit into a broader image on the screen. This distribution of wave energy is called diffraction.

Two types of diffraction are distinguished, depending upon the separation between the source and the screen: Fraunhofer diffraction or far-field diffraction at large separations and Fresnel diffraction or near-field diffraction at close separations.

In the analysis of the single slit, the non-zero width of the slit is taken into account, and each point in the aperture is taken as the source of one contribution to the beam of light (Huygen's wavelets). On the screen, the light arriving from each position within the slit has a different path length, albeit possibly a very small difference. Consequently, interference occurs.

In the Fraunhofer diffraction pattern sufficiently far from a single slit, within a small-angle approximation, the intensity spread S is related to position x via a squared sinc function:[29]
 with 

where L is the slit width, R is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The function S has zeros where u is a non-zero integer, where are at x values at a separation proportion to wavelength.
[edit]
Diffraction-limited resolution
Main articles: Angular resolution and Diffraction-limited system

Diffraction is the fundamental limitation on the resolving power of optical instruments, such as telescopes (including radiotelescopes) and microscopes.[30] For a circular aperture, the diffraction-limited image spot is known as an Airy disk; the distance x in the single-slit diffraction formula is replaced by radial distance r and the sine is replaced by 2J1, where J1 is a first order Bessel function.[31]

The resolvable spatial size of objects viewed through a microscope is limited according to the Rayleigh criterion, the radius to the first null of the Airy disk, to a size proportional to the wavelength of the light used, and depending on the numerical aperture:[32]


where the numerical aperture is defined as for θ being the half-angle of the cone of rays accepted by the microscope objective.

The angular size of the central bright portion (radius to first null of the Airy disk) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:[33]


where λ is the wavelength of the waves that are focused for imaging, D the entrance pupil diameter of the imaging system, in the same units, and the angular resolution δ is in radians.

As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.
[edit]
Subwavelength

The term subwavelength is used to describe an object having one or more dimensions smaller than the length of the wave with which the object interacts. For example, the term subwavelength-diameter optical fibre means an optical fibre whose diameter is less than the wavelength of light propagating through it.

A subwavelength particle is a particle smaller than the wavelength of light with which it interacts (see Rayleigh scattering). Subwavelength apertures are holes smaller than the wavelength of light propagating through them. Such structures have applications in extraordinary optical transmission, and zero-mode waveguides, among other areas of photonics.

Subwavelength may also refer to a phenomenon involving subwavelength objects; for example, subwavelength imaging.
[edit]
See also
Emission spectrum
Fraunhofer lines – dark lines in the solar spectrum, traditionally used as standard optical wavelength references
Spectral line
Spectrum
Spectrum analysis
[edit]
References
^ a b Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. pp. 15–16. ISBN 0-201-11609-X.
^ Raymond A. Serway, John W. Jewett. Principles of physics (4th ed.). Cengage Learning. pp. 404, 440. ISBN 053449143X.
^ A. A. Sonin (1995). The surface physics of liquid crystals. Taylor & Francis. p. 17. ISBN 2881249957.
^ Brian Hilton Flowers (2000). "§21.2 Periodic functions". An introduction to numerical methods in C++ (2nd ed.). Cambridge University Press. p. 473. ISBN 0198506937.
^ Keqian Zhang and Dejie Li (2007). Electromagnetic Theory for Microwaves and Optoelectronics. Springer,. p. 533. ISBN 9783540742951.
^ Theo Koupelis and Karl F. Kuhn (2007). In Quest of the Universe. Jones & Bartlett Publishers. ISBN 0763743879.
^ Paul R Pinet (2008). Invitation to Oceanography (5th ed.). Jones & Bartlett Publishers. p. 237. ISBN 0763759937.
^ David C. Cassidy, Gerald James Holton, Floyd James Rutherford (2002). Understanding physics. Birkhäuser. pp. 339 ff. ISBN 0387987568.
^ John Avison (1999). The World of Physics. Nelson Thornes. p. 460. ISBN 9780174387336.
^ John A. Adam (2003). Mathematics in nature. Princeton University Press. p. 148. ISBN 0691114293. "The relation between the frequency of a wave and its wavelength λ ... is referred to as a dispersion relation."
^ a b Paul R Pinet. op. cit.. p. 242. ISBN 0763759937.
^ Bishwanath Chakraborty. Principles of Plasma Mechanics. New Age International. p. 454. ISBN 9788122414462.
^ Jeffrey A. Hogan and Joseph D. Lakey (2005). Time-frequency and time-scale methods: adaptive decompositions, uncertainty principles, and sampling. Birkhäuser. p. 348. ISBN 9780817642761.
^ See Figure 4.20 in A. Putnis (1992). Introduction to mineral sciences. Cambridge University Press. p. 97. ISBN 0521429471. and Figure 2.3 in Martin T. Dove (1993). Introduction to lattice dynamics (4th ed.). Cambridge University Press. p. 22. ISBN 0521392934.
^ Manijeh Razeghi (2006). Fundamentals of solid state engineering (2nd ed.). Birkhäuser. pp. 165 ff. ISBN 0387281525.
^ a b Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0470185902.
^ Ken'ichi Okamoto (2001). Global environment remote sensing. IOS Press. p. 263. ISBN 9781586031015.
^ Roger Grimshaw (2007). "Solitary waves propagating over variable topography". In Anjan Kundu. Tsunami and Nonlinear Waves. Springer. pp. 52 ff. ISBN 3540712550.
^ Mark W. Denny (1995). Air and Water: The Biology and Physics of Life's Media. Princeton University Press. p. 289. ISBN 9780691025186.
^ A. T. Fromhold (1991). "Wave packet solutions". Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff. ISBN 0486667413. "(p. 61) ... the individual waves move more slowly than the packet and therefore pass back through the packet as it advances"
^ Paul A. LaViolette (2003). Subquantum Kinetics: A Systems Approach to Physics and Cosmology. Starlane Publications. p. 80. ISBN 9780964202559.
^ Peter R. Holland (1995). The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press. p. 160. ISBN 9780521485432.
^ Jeffery Cooper (1998). Introduction to partial differential equations with MATLAB. Springer. p. 272. ISBN 0817639675. "The local wavelength λ of a dispersing wave is twice the distance between two successive zeros. ... the local wave length and the local wave number k are related by k = 2π / λ."
^ a b See, for example, Figs. 2.8–2.10 in Joy Manners (2000). "Heisenberg's uncertainty principle". Quantum Physics: An Introduction. CRC Press. pp. 53–56. ISBN 9780750307208.
^ Ming Chiang Li (1980). "Electron Interference". In L. Marton and Claire Marton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 0120146533.
^ Greenfield Sluder and David E. Wolf (2007). "IV. Young's Experiment: Two-Slit Interference". Digital microscopy (3rd ed.). Academic Press. p. 15. ISBN 0123740258.
^ Halliday, Resnick, Walker (2008). "§35-4 Young's interference experiment". Fundamentals of Physics (Extended 8th ed.). Wiley-India. p. 965. ISBN 8126514426.
^ Douglas B. Murphy (2002). Fundamentals of light microscopy and electronic imaging. Wiley/IEEE. p. 64. ISBN 047123429X.
^ John C. Stover (1995). Optical scattering: measurement and analysis (2nd ed.). SPIE Press. p. 64. ISBN 9780819419347.
^ Graham Saxby (2002). "Diffraction limitation". The science of imaging. CRC Press. p. 57. ISBN 075030734X.
^ Grant R. Fowles (1989). Introduction to Modern Optics. Courier Dover Publications. pp. 117–120. ISBN 9780486659572.
^ James B. Pawley (1995). Handbook of biological confocal microscopy (2nd ed.). Springer. p. 112. ISBN 9780306448263.
^ Ray N. Wilson (2004). Reflecting Telescope Optics I: Basic Design Theory and Its Historical Development. Springer. p. 302. ISBN 9783540401063.
[edit]
External links
Conversion: Wavelength to Frequency and vice versa – Sound waves and radio waves
Teaching resource for 14–16 years on sound including wavelength
The visible electromagnetic spectrum displayed in web colors with according wavelengths
Categories: Waves | Fundamental physics concepts | Length
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Phonon
From Wikipedia, the free encyclopedia
(Redirected from Lattice vibration)
For KDE Software Compilation 4's multimedia framework, see Phonon (KDE). This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (February 2010)


Normal modes of vibration progression through a crystal. The amplitude of the motion has been exaggerated for ease of viewing; in an actual crystal, it is typically much smaller than the lattice spacing.

A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids. Often referred to as a quasiparticle,[1] it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.

Phonons play a major role in many of the physical properties of solids, including a material's thermal and electrical conductivities. Hence the study of phonons is an important part of solid state physics.

A phonon is a quantum mechanical description of a special type of vibrational motion, in which a lattice uniformly oscillates at the same frequency. In classical mechanics this is known as normal mode. Normal mode is important because any arbitrary lattice vibration can be considered as a superposition of these elementary vibrations (cf. Fourier analysis). While normal modes are wave-like phenomena in classical mechanics, they have particle-like properties in the wave–particle duality of quantum mechanics.

The name phonon comes from the Greek word φωνή (phonē), which translates as sound or voice because long-wavelength phonons give rise to sound.

The concept of phonons was introduced by Russian physicist Igor Tamm.Contents [hide]
1 Lattice dynamics
1.1 Lattice waves
1.2 Phonon dispersion
1.3 Dispersion relation
2 Acoustic and optical phonons
3 Crystal momentum
4 Thermodynamics
5 Operator formalism
6 See also
7 Notes
8 References
9 External links

[edit]
Lattice dynamics

The equations in this section either do not use axioms of quantum mechanics or use relations for which there exists a direct correspondence in classical mechanics.

For example, consider a rigid regular, crystalline, i.e. not amorphous, lattice composed of N particles. We will refer to these particles as atoms, although in a real solid these may be molecules. N is some large number, say around 1023 (on the order of Avogadro's number) for a typical sample of solid. If the lattice is rigid, the atoms must be exerting forces on one another to keep each atom near its equilibrium position. These forces may be Van der Waals forces, covalent bonds, electrostatic attractions, and others, all of which are ultimately due to the electric field force. Magnetic and gravitational forces are generally negligible. The forces between each pair of atoms may be characterized by a potential energy function V that depends on the distance of separation of the atoms. The potential energy of the entire lattice is the sum of all pairwise potential energies:[2]


where is the position of the th atom, and is the potential energy between two atoms.

It is difficult to solve this many-body problem in full generality, in either classical or quantum mechanics. In order to simplify the task, we introduce two important approximations. First, we perform the sum over neighboring atoms only. Although the electric forces in real solids extend to infinity, this approximation is nevertheless valid because the fields produced by distant atoms are screened. Secondly, we treat the potentials as harmonic potentials: this is permissible as long as the atoms remain close to their equilibrium positions. (Formally, this is done by Taylor expanding about its equilibrium value to quadratic order, giving proportional to the displacement and the elastic force simply proportional to . The error in ignoring higher order terms remains small if remains close to the equilibrium position).

The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modelling phonons. Other common lattices may be found in the article on crystal structure.


The potential energy of the lattice may now be written as


Here, is the natural frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. is the position coordinate of the th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted as "(nn)".
[edit]
Lattice waves

Phonon propagating through a square lattice (atom displacements greatly exaggerated)

Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration waves propagating through the lattice. One such wave is shown in the figure to the right. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength is marked.

There is a minimum possible wavelength, given by twice the equilibrium separation a between atoms. As we shall see in the following sections, any wavelength shorter than this can be mapped onto a wavelength longer than 2a, due to the periodicity of the lattice.

Not every possible lattice vibration has a well-defined wavelength and frequency. However, the normal modes do possess well-defined wavelengths and frequencies.
[edit]
Phonon dispersion

Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions. The Hamiltonian for this system is


where is the mass of each atom, and and are the position and momentum operators for the th atom. A discussion of similar Hamiltonians may be found in the article on the quantum harmonic oscillator.

We introduce a set of "normal coordinates" , defined as the discrete Fourier transforms of the 's and "conjugate momenta" defined as the Fourier transforms of the 's:


The quantity will turn out to be the wave number of the phonon, i.e. divided by the wavelength. It takes on quantized values, because the number of atoms is finite. The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is


The upper bound to comes from the minimum wavelength, which is twice the lattice spacing , as discussed above.

By inverting the discrete Fourier transforms to express the 's in terms of the 's and the 's in terms of the 's, and using the canonical commutation relations between the 's and 's, we can show that


In other words, the normal coordinates and their conjugate momenta obey the same commutation relations as position and momentum operators! Writing the Hamiltonian in terms of these quantities,


where


Notice that the couplings between the position variables have been transformed away; if the 's and 's were Hermitian (which they are not), the transformed Hamiltonian would describe uncoupled harmonic oscillators.

This may be generalized to a three-dimensional lattice. The wave number k is replaced by a three-dimensional wave vector k. Furthermore, each k is now associated with three normal coordinates.

The new indices s = 1, 2, 3 label the polarization of the phonons. In the one dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to longitudinal waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like transverse waves. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.
[edit]
Dispersion relation

Dispersion curve

In the above discussion, we have obtained an equation that relates the frequency of a phonon, , to its wave number :


This is known as a dispersion relation.

The speed of propagation of a phonon, which is also the speed of sound in the lattice, is given by the slope of the dispersion relation, (see group velocity.) At low values of (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately , independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of , i.e. short wavelengths, due to the microscopic details of the lattice.

For a crystal that has at least two atoms in a unit cell (which may or may not be different), the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper and lower sets of curves in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the wave-vector. The boundaries at -km and km are those of the first Brillouin zone. The blue, violet, and brown curves are those of longitudinal acoustic, transverse acoustic 1, and transverse acoustic 2 modes, respectively.

In some crystals the two transverse acoustic modes have exactly the same dispersion curve. It is also interesting that for a crystal with N ( > 2) different atoms in a primitive cell, there are always three acoustic modes. The number of optical modes is 3N - 3. Many phonon dispersion curves have been measured by neutron scattering.

The physics of sound in fluids differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids can't support shear stresses. (but see viscoelastic fluids, which only apply to high frequencies, though).

In fact, the above-derived Hamiltonian looks like the classical Hamiltonian function, but if it is interpreted as an operator, then it describes a quantum field theory of non-interacting bosons. This leads to new physics.

The energy spectrum of this Hamiltonian is easily obtained by the method of ladder operators, similar to the quantum harmonic oscillator problem. We introduce a set of ladder operators defined by


The ladder operators satisfy the following identities:




As with the quantum harmonic oscillator, we can then show that and respectively create and destroy one excitation of energy . These excitations are phonons.

We can immediately deduce two important properties of phonons. Firstly, phonons are bosons, since any number of identical excitations can be created by repeated application of the creation operator . Secondly, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the ladder operators contain sums over the position and momentum operators of every atom.

It is not a priori obvious that these excitations generated by the operators are literally waves of lattice displacement, but one may convince oneself of this by calculating the position-position correlation function. Let denote a state with a single quantum of mode excited, i.e.


One can show that, for any two atoms and ,


which is exactly what we would expect for a lattice wave with frequency and wave number .

In three dimensions the Hamiltonian has the form

[edit]
Acoustic and optical phonons

Solids with more than one type of atom, with either different masses or bonding strengths, in the smallest unit cell exhibit two types of phonons: acoustic phonons and optical phonons.

In terms of the dispersion relations for phonons, acoustic phonons exhibit linear dispersion, i.e., a linear relationship between frequency and phonon wavevector, in the long-wavelength limit. The frequencies of acoustic phonons, which are the phonons described above, tend to zero with longer wavelength, and correspond to sound waves in the lattice. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively.

Optical phonons, however, have a non-zero frequency at the Brillouin zone center and show no dispersion near that long wavelength limit. They are called optical because in ionic crystals, such as sodium chloride, they are excited by infrared radiation. This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying electrical dipole moment. Optical phonons that interact in this way with light are called infrared active. Optical phonons that are Raman active can also interact indirectly with light, through Raman scattering. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse modes respectively.
[edit]
Crystal momentum
Main article: Crystal momentum

k-Vectors exceeding the first Brillouin zone (red) do not carry more information than their counterparts (black) in the first Brillouin zone.

It is tempting to treat a phonon with wave vector as though it has a momentum , by analogy to photons and matter waves. This is not entirely correct, for is not actually a physical momentum; it is called the crystal momentum or pseudomomentum. This is because is only determined up to multiples of constant vectors, known as reciprocal lattice vectors. For example, in our one-dimensional model, the normal coordinates and are defined so that


where


for any integer . A phonon with wave number is thus equivalent to an infinite "family" of phonons with wave numbers , , and so forth. Physically, the reciprocal lattice vectors act as additional "chunks" of momentum which the lattice can impart to the phonon. Bloch electrons obey a similar set of restrictions.

Brillouin zones, a) in a square lattice, and b) in a hexagonal lattice

It is usually convenient to consider phonon wave vectors which have the smallest magnitude in their "family". The set of all such wave vectors defines the first Brillouin zone. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.

It is interesting that similar consideration is needed in analog-to-digital conversion where aliasing may occur under certain conditions.
[edit]
Thermodynamics

The thermodynamic properties of a solid are directly related to its phonon structure. The entire set of all possible phonons that are described by the above phonon dispersion relations combine in what is known as the phonon density of states which determines the heat capacity of a crystal.

At absolute zero temperature, a crystal lattice lies in its ground state, and contains no phonons. A lattice at a non-zero temperature has an energy that is not constant, but fluctuates randomly about some mean value. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. (The random motion of the atoms in the lattice is what we usually think of as heat.) Because these phonons are generated by the temperature of the lattice, they are sometimes referred to as thermal phonons.

Unlike the atoms which make up an ordinary gas, thermal phonons can be created and destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero. This behavior is an extension of the harmonic potential, mentioned earlier, into the anharmonic regime. The behavior of thermal phonons is similar to the photon gas produced by an electromagnetic cavity, wherein photons may be emitted or absorbed by the cavity walls. This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators; see Black-body radiation. Both gases obey the Bose-Einstein statistics: in thermal equilibrium and within the harmonic regime, the probability of finding phonons (or photons) in a given state with a given angular frequency is:


where is the frequency of the phonons (or photons) in the state, is Boltzmann's constant, and is the temperature.
[edit]
Operator formalism

The phonon Hamiltonian is given by


In terms of the operators, these are given by


Here, in expressing the Hamiltonian (quantum mechanics) in operator formalism, we have not taken into account the term, since if we take an infinite lattice or, for that matter a continuum, the terms will add up giving an infinity. Hence, it is "renormalized" by putting the factor of to 0 arguing that the difference in energy is what we measure and not the absolute value of it. Hence, the factor is absent in the operator formalised expression for the Hamiltonian.
The ground state also called the "vacuum state" is the state composed of no phonons. Hence, the energy of the ground state is 0. When, a system is in state , we say there are nα phonons of type α. The nα are called the occupation number of the phonons. Energy of a single phonon of type α being , the total energy of a general phonon system is given by . In other words, the phonons are non-interacting. The action of creation and annihilation operators are given by


and,


i.e. creates a phonon of type α while aα annihilates. Hence, they are respectively the creation and annihilation operator for phonons. Analogous to the Quantum harmonic oscillator case, we can define particle number operator as . The number operator commutes with a string of products of the creation and annihilation operators if, the number of a's are equal to number of 's.
Phonons are bosons since, i.e. they are symmetric under exchange.[3]
[edit]
See also Physics portal

Boson
Fracton
Rayleigh wave
Brillouin scattering
Linear elasticity
Phononic crystal
Relativistic heat conduction
Surface acoustic wave
Rigid Unit Modes
Surface phonon
SASER
Thermal conductivity
[edit]
Notes
[edit]
References
^ F. Schwabel, Advanced Quantum Mechanics, 4th Ed., Springer (2008), p. 253
^ Krauth, Werner (2006-04). Statistical mechanics: algorithms and computations. International publishing locations: Oxford University Press. pp. 231–232. ISBN 9780198515364.
^ Feynman, Richard P. (1982). Statistical Mechanics, A Set of Lectures. Reading, Massachusetts: The Benjamin/Cummings Publishing Company, Inc.. p. 159. ISBN Clothbound: 0-8053-2508-5, Paperbound: 0-8053-2509-3.
[edit]
External links
Optical and acoustic modes
Phonons in a One Dimensional Microfluidic Crystal [1] and [2] with movies in [3].[hide]
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Vibration
From Wikipedia, the free encyclopedia
For the soul music group, see The Vibrations. For the machining context, see Machining vibrations.Classical mechanics

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Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.

Vibration is occasionally "desirable". For example the motion of a tuning fork, the reed in a woodwind instrument or harmonica, or the cone of a loudspeaker is desirable vibration, necessary for the correct functioning of the various devices.

More often, vibration is undesirable, wasting energy and creating unwanted sound – noise. For example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations can be caused by imbalances in the rotating parts, uneven friction, the meshing of gear teeth, etc. Careful designs usually minimize unwanted vibrations.

The study of sound and vibration are closely related. Sound, or "pressure waves", are generated by vibrating structures (e.g. vocal cords); these pressure waves can also induce the vibration of structures (e.g. ear drum). Hence, when trying to reduce noise it is often a problem in trying to reduce vibration.

One of the possible modes of vibration of a circular drum (see other modes).

One of the possible modes of vibration of a cantilevered I-beam.Contents [hide]
1 Types of vibration
2 Vibration testing
3 Vibration analysis
3.1 Free vibration without damping
3.1.1 What causes the system to vibrate: from conservation of energy point of view
3.2 Free vibration with damping
3.2.1 Damped and undamped natural frequencies
3.3 Forced vibration with damping
3.3.1 What causes resonance?
3.3.2 Applying "complex" forces to the mass–spring–damper model
3.3.3 Frequency response model
4 Multiple degrees of freedom systems and mode shapes
4.1 Eigenvalue problem
4.2 Illustration of a multiple DOF problem
4.3 Multiple DOF problem converted to a single DOF problem
5 See also
6 References
7 Further reading
8 External links

[edit]
Types of vibration

Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring. The mechanical system will then vibrate at one or more of its "natural frequency" and damp down to zero.

Forced vibration is when an alternating force or motion is applied to a mechanical system. Examples of this type of vibration include a shaking washing machine due to an imbalance, transportation vibration (caused by truck engine, springs, road, etc.), or the vibration of a building during an earthquake. In forced vibration the frequency of the vibration is the frequency of the force or motion applied, with order of magnitude being dependent on the actual mechanical system.
[edit]
Vibration testing

Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT (device under test) is attached to the "table" of a shaker. For relatively low frequency forcing, servohydraulic (electrohydraulic) shakers are used. For higher frequencies, electrodynamic shakers are used. Generally, one or more "input" or "control" points located on the DUT-side of a fixture is kept at a specified acceleration.[1] Other "response" points experience maximum vibration level (resonance) or minimum vibration level (anti-resonance).

Two typical types of vibration tests performed are random- and sine test. Sine (one-frequency-at-a-time) tests are performed to survey the structural response of the device under test (DUT). A random (all frequencies at once) test is generally considered to more closely replicate a real world environment, such as road inputs to a moving automobile.

Most vibration testing is conducted in a single DUT axis at a time, even though most real-world vibration occurs in various axes simultaneously. MIL-STD-810G, released in late 2008, Test Method 527, calls for multiple exciter testing.
[edit]
Vibration analysis

The fundamentals of vibration analysis can be understood by studying the simple mass–spring–damper model. Indeed, even a complex structure such as an automobile body can be modeled as a "summation" of simple mass–spring–damper models. The mass–spring–damper model is an example of a simple harmonic oscillator. The mathematics used to describe its behavior is identical to other simple harmonic oscillators such as the RLC circuit.

Note: In this article the step by step mathematical derivations will not be included, but will focus on the major equations and concepts in vibration analysis. Please refer to the references at the end of the article for detailed derivations.
[edit]
Free vibration without damping


To start the investigation of the mass–spring–damper we will assume the damping is negligible and that there is no external force applied to the mass (i.e. free vibration).

The force applied to the mass by the spring is proportional to the amount the spring is stretched "x" (we will assume the spring is already compressed due to the weight of the mass). The proportionality constant, k, is the stiffness of the spring and has units of force/distance (e.g. lbf/in or N/m)


The force generated by the mass is proportional to the acceleration of the mass as given by Newton’s second law of motion.


The sum of the forces on the mass then generates this ordinary differential equation:


Simple harmonic motion of the mass–spring system

If we assume that we start the system to vibrate by stretching the spring by the distance of A and letting go, the solution to the above equation that describes the motion of mass is:


This solution says that it will oscillate with simple harmonic motion that has an amplitude of A and a frequency of fn. The number fn is one of the most important quantities in vibration analysis and is called the undamped natural frequency. For the simple mass–spring system, fn is defined as:


Note: Angular frequency ω (ω = 2πf) with the units of radians per second is often used in equations because it simplifies the equations, but is normally converted to “standard” frequency (units of Hz or equivalently cycles per second) when stating the frequency of a system.

If you know the mass and stiffness of the system you can determine the frequency at which the system will vibrate once it is set in motion by an initial disturbance using the above stated formula. Every vibrating system has one or more natural frequencies that it will vibrate at once it is disturbed. This simple relation can be used to understand in general what will happen to a more complex system once we add mass or stiffness. For example, the above formula explains why when a car or truck is fully loaded the suspension will feel “softer” than unloaded because the mass has increased and therefore reduced the natural frequency of the system.
[edit]
What causes the system to vibrate: from conservation of energy point of view

Vibrational motion could be understood in terms of conservation of energy. In the above example we have extended the spring by a value of x and therefore have stored some potential energy () in the spring. Once we let go of the spring, the spring tries to return to its un-stretched state (which is the minimum potential energy state) and in the process accelerates the mass. At the point where the spring has reached its un-stretched state all the potential energy that we supplied by stretching it has been transformed into kinetic energy (). The mass then begins to decelerate because it is now compressing the spring and in the process transferring the kinetic energy back to its potential. Thus oscillation of the spring amounts to the transferring back and forth of the kinetic energy into potential energy.

In our simple model the mass will continue to oscillate forever at the same magnitude, but in a real system there is always something called damping that dissipates the energy and therefore the system eventually bringing it to rest.
[edit]
Free vibration with damping

Mass Spring Damper Model

We now add a "viscous" damper to the model that outputs a force that is proportional to the velocity of the mass. The damping is called viscous because it models the effects of an object within a fluid. The proportionality constant c is called the damping coefficient and has units of Force over velocity (lbf s/ in or N s/m).


By summing the forces on the mass we get the following ordinary differential equation:


The solution to this equation depends on the amount of damping. If the damping is small enough the system will still vibrate, but eventually, over time, will stop vibrating. This case is called underdamping – this case is of most interest in vibration analysis. If we increase the damping just to the point where the system no longer oscillates we reach the point of critical damping (if the damping is increased past critical damping the system is called overdamped). The value that the damping coefficient needs to reach for critical damping in the mass spring damper model is:


To characterize the amount of damping in a system a ratio called the damping ratio (also known as damping factor and % critical damping) is used. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. The formula for the damping ratio (ζ) of the mass spring damper model is:


For example, metal structures (e.g. airplane fuselage, engine crankshaft) will have damping factors less than 0.05 while automotive suspensions in the range of 0.2–0.3.

The solution to the underdamped system for the mass spring damper model is the following:



The value of X, the initial magnitude, and φ, the phase shift, are determined by the amount the spring is stretched. The formulas for these values can be found in the references.
[edit]
Damped and undamped natural frequencies

The major points to note from the solution are the exponential term and the cosine function. The exponential term defines how quickly the system “damps” down – the larger the damping ratio, the quicker it damps to zero. The cosine function is the oscillating portion of the solution, but the frequency of the oscillations is different from the undamped case.

The frequency in this case is called the "damped natural frequency", fd, and is related to the undamped natural frequency by the following formula:


The damped natural frequency is less than the undamped natural frequency, but for many practical cases the damping ratio is relatively small and hence the difference is negligible. Therefore the damped and undamped description are often dropped when stating the natural frequency (e.g. with 0.1 damping ratio, the damped natural frequency is only 1% less than the undamped).

The plots to the side present how 0.1 and 0.3 damping ratios effect how the system will “ring” down over time. What is often done in practice is to experimentally measure the free vibration after an impact (for example by a hammer) and then determine the natural frequency of the system by measuring the rate of oscillation as well as the damping ratio by measuring the rate of decay. The natural frequency and damping ratio are not only important in free vibration, but also characterize how a system will behave under forced vibration.
[edit]
Forced vibration with damping

In this section we will see the behavior of the spring mass damper model when we add a harmonic force in the form below. A force of this type could, for example, be generated by a rotating imbalance.


If we again sum the forces on the mass we get the following ordinary differential equation:


The steady state solution of this problem can be written as:


The result states that the mass will oscillate at the same frequency, f, of the applied force, but with a phase shift φ.

The amplitude of the vibration “X” is defined by the following formula.


Where “r” is defined as the ratio of the harmonic force frequency over the undamped natural frequency of the mass–spring–damper model.


The phase shift , φ, is defined by the following formula.




The plot of these functions, called "the frequency response of the system", presents one of the most important features in forced vibration. In a lightly damped system when the forcing frequency nears the natural frequency () the amplitude of the vibration can get extremely high. This phenomenon is called resonance (subsequently the natural frequency of a system is often referred to as the resonant frequency). In rotor bearing systems any rotational speed that excites a resonant frequency is referred to as a critical speed.

If resonance occurs in a mechanical system it can be very harmful – leading to eventual failure of the system. Consequently, one of the major reasons for vibration analysis is to predict when this type of resonance may occur and then to determine what steps to take to prevent it from occurring. As the amplitude plot shows, adding damping can significantly reduce the magnitude of the vibration. Also, the magnitude can be reduced if the natural frequency can be shifted away from the forcing frequency by changing the stiffness or mass of the system. If the system cannot be changed, perhaps the forcing frequency can be shifted (for example, changing the speed of the machine generating the force).

The following are some other points in regards to the forced vibration shown in the frequency response plots.
At a given frequency ratio, the amplitude of the vibration, X, is directly proportional to the amplitude of the force F0 (e.g. if you double the force, the vibration doubles)
With little or no damping, the vibration is in phase with the forcing frequency when the frequency ratio r < 1 and 180 degrees out of phase when the frequency ratio r > 1
When r ≪ 1 the amplitude is just the deflection of the spring under the static force F0. This deflection is called the static deflection δst. Hence, when r ≪ 1 the effects of the damper and the mass are minimal.
When r ≫ 1 the amplitude of the vibration is actually less than the static deflection δst. In this region the force generated by the mass (F = ma) is dominating because the acceleration seen by the mass increases with the frequency. Since the deflection seen in the spring, X, is reduced in this region, the force transmitted by the spring (F = kx) to the base is reduced. Therefore the mass–spring–damper system is isolating the harmonic force from the mounting base – referred to as vibration isolation. Interestingly, more damping actually reduces the effects of vibration isolation when r ≫ 1 because the damping force (F = cv) is also transmitted to the base.
[edit]
What causes resonance?

Resonance is simple to understand if you view the spring and mass as energy storage elements – with the mass storing kinetic energy and the spring storing potential energy. As discussed earlier, when the mass and spring have no force acting on them they transfer energy back and forth at a rate equal to the natural frequency. In other words, if energy is to be efficiently pumped into both the mass and spring the energy source needs to feed the energy in at a rate equal to the natural frequency. Applying a force to the mass and spring is similar to pushing a child on swing, you need to push at the correct moment if you want the swing to get higher and higher. As in the case of the swing, the force applied does not necessarily have to be high to get large motions; the pushes just need to keep adding energy into the system.

The damper, instead of storing energy, dissipates energy. Since the damping force is proportional to the velocity, the more the motion, the more the damper dissipates the energy. Therefore a point will come when the energy dissipated by the damper will equal the energy being fed in by the force. At this point, the system has reached its maximum amplitude and will continue to vibrate at this level as long as the force applied stays the same. If no damping exists, there is nothing to dissipate the energy and therefore theoretically the motion will continue to grow on into infinity.
[edit]
Applying "complex" forces to the mass–spring–damper model

In a previous section only a simple harmonic force was applied to the model, but this can be extended considerably using two powerful mathematical tools. The first is the Fourier transform that takes a signal as a function of time (time domain) and breaks it down into its harmonic components as a function of frequency (frequency domain). For example, let us apply a force to the mass–spring–damper model that repeats the following cycle – a force equal to 1 newton for 0.5 second and then no force for 0.5 second. This type of force has the shape of a 1 Hz square wave.

How a 1 Hz square wave can be represented as a summation of sine waves(harmonics) and the corresponding frequency spectrum. Click and go to full resolution for an animation

The Fourier transform of the square wave generates a frequency spectrum that presents the magnitude of the harmonics that make up the square wave (the phase is also generated, but is typically of less concern and therefore is often not plotted). The Fourier transform can also be used to analyze non-periodic functions such as transients (e.g. impulses) and random functions. With the advent of the modern computer the Fourier transform is almost always computed using the Fast Fourier Transform (FFT) computer algorithm in combination with a window function.

In the case of our square wave force, the first component is actually a constant force of 0.5 newton and is represented by a value at "0" Hz in the frequency spectrum. The next component is a 1 Hz sine wave with an amplitude of 0.64. This is shown by the line at 1 Hz. The remaining components are at odd frequencies and it takes an infinite amount of sine waves to generate the perfect square wave. Hence, the Fourier transform allows you to interpret the force as a sum of sinusoidal forces being applied instead of a more "complex" force (e.g. a square wave).

In the previous section, the vibration solution was given for a single harmonic force, but the Fourier transform will in general give multiple harmonic forces. The second mathematical tool, "the principle of superposition", allows you to sum the solutions from multiple forces if the system is linear. In the case of the spring–mass–damper model, the system is linear if the spring force is proportional to the displacement and the damping is proportional to the velocity over the range of motion of interest. Hence, the solution to the problem with a square wave is summing the predicted vibration from each one of the harmonic forces found in the frequency spectrum of the square wave.
[edit]
Frequency response model

We can view the solution of a vibration problem as an input/output relation – where the force is the input and the output is the vibration. If we represent the force and vibration in the frequency domain (magnitude and phase) we can write the following relation:


H(ω) is called the frequency response function (also referred to as the transfer function, but not technically as accurate) and has both a magnitude and phase component (if represented as a complex number, a real and imaginary component). The magnitude of the frequency response function (FRF) was presented earlier for the mass–spring–damper system.
where

The phase of the FRF was also presented earlier as:


For example, let us calculate the FRF for a mass–spring–damper system with a mass of 1 kg, spring stiffness of 1.93 N/mm and a damping ratio of 0.1. The values of the spring and mass give a natural frequency of 7 Hz for this specific system. If we apply the 1 Hz square wave from earlier we can calculate the predicted vibration of the mass. The figure illustrates the resulting vibration. It happens in this example that the fourth harmonic of the square wave falls at 7 Hz. The frequency response of the mass–spring–damper therefore outputs a high 7 Hz vibration even though the input force had a relatively low 7 Hz harmonic. This example highlights that the resulting vibration is dependent on both the forcing function and the system that the force is applied to.

Frequency response model

The figure also shows the time domain representation of the resulting vibration. This is done by performing an inverse Fourier Transform that converts frequency domain data to time domain. In practice, this is rarely done because the frequency spectrum provides all the necessary information.

The frequency response function (FRF) does not necessarily have to be calculated from the knowledge of the mass, damping, and stiffness of the system, but can be measured experimentally. For example, if you apply a known force and sweep the frequency and then measure the resulting vibration you can calculate the frequency response function and then characterize the system. This technique is used in the field of experimental modal analysis to determine the vibration characteristics of a structure.
[edit]
Multiple degrees of freedom systems and mode shapes

The simple mass–spring damper model is the foundation of vibration analysis, but what about more complex systems? The mass–spring–damper model described above is called a single degree of freedom (SDOF) model since we have assumed the mass only moves up and down. In the case of more complex systems we need to discretize the system into more masses and allow them to move in more than one direction – adding degrees of freedom. The major concepts of multiple degrees of freedom (MDOF) can be understood by looking at just a 2 degree of freedom model as shown in the figure.

2 degree of freedom model

The equations of motion of the 2DOF system are found to be:



We can rewrite this in matrix format:


A more compact form of this matrix equation can be written as:


where and are symmetric matrices referred respectively as the mass, damping, and stiffness matrices. The matrices are NxN square matrices where N is the number of degrees of freedom of the system.

In the following analysis we will consider the case where there is no damping and no applied forces (i.e. free vibration). The solution of a viscously damped system is somewhat more complicated.[2]


This differential equation can be solved by assuming the following type of solution:


Note: Using the exponential solution of is a mathematical trick used to solve linear differential equations. If we use Euler's formula and take only the real part of the solution it is the same cosine solution for the 1 DOF system. The exponential solution is only used because it easier to manipulate mathematically.

The equation then becomes:


Since eiωt cannot equal zero the equation reduces to the following.

[edit]
Eigenvalue problem

This is referred to an eigenvalue problem in mathematics and can be put in the standard format by pre-multiplying the equation by


and if we let and


The solution to the problem results in N eigenvalues (i.e. ), where N corresponds to the number of degrees of freedom. The eigenvalues provide the natural frequencies of the system. When these eigenvalues are substituted back into the original set of equations, the values of that correspond to each eigenvalue are called the eigenvectors. These eigenvectors represent the mode shapes of the system. The solution of an eigenvalue problem can be quite cumbersome (especially for problems with many degrees of freedom), but fortunately most math analysis programs have eigenvalue routines.

The eigenvalues and eigenvectors are often written in the following matrix format and describe the modal model of the system:
and

A simple example using our 2 DOF model can help illustrate the concepts. Let both masses have a mass of 1 kg and the stiffness of all three springs equal 1000 N/m. The mass and stiffness matrix for this problem are then:
and

Then

The eigenvalues for this problem given by an eigenvalue routine will be:


The natural frequencies in the units of hertz are then (remembering ) and .

The two mode shapes for the respective natural frequencies are given as:


Since the system is a 2 DOF system, there are two modes with their respective natural frequencies and shapes. The mode shape vectors are not the absolute motion, but just describe relative motion of the degrees of freedom. In our case the first mode shape vector is saying that the masses are moving together in phase since they have the same value and sign. In the case of the second mode shape vector, each mass is moving in opposite direction at the same rate.
[edit]
Illustration of a multiple DOF problem

When there are many degrees of freedom, the best method of visualizing the mode shapes is by animating them. An example of animated mode shapes is shown in the figure below for a cantilevered I-beam. In this case, a finite element model was used to generate the mass and stiffness matrices and solve the eigenvalue problem. Even this relatively simple model has over 100 degrees of freedom and hence as many natural frequencies and mode shapes. In general only the first few modes are important.The mode shapes of a cantilevered I-beam

1st lateral bending
1st torsional
1st vertical bending

2nd lateral bending
2nd torsional
2nd vertical bending

[edit]
Multiple DOF problem converted to a single DOF problem

The eigenvectors have very important properties called orthogonality properties. These properties can be used to greatly simplify the solution of multi-degree of freedom models. It can be shown that the eigenvectors have the following properties:



and are diagonal matrices that contain the modal mass and stiffness values for each one of the modes. (Note: Since the eigenvectors (mode shapes) can be arbitrarily scaled, the orthogonality properties are often used to scale the eigenvectors so the modal mass value for each mode is equal to 1. The modal mass matrix is therefore an identity matrix)

These properties can be used to greatly simplify the solution of multi-degree of freedom models by making the following coordinate transformation.


If we use this coordinate transformation in our original free vibration differential equation we get the following equation.


We can take advantage of the orthogonality properties by premultiplying this equation by


The orthogonality properties then simplify this equation to:


This equation is the foundation of vibration analysis for multiple degree of freedom systems. A similar type of result can be derived for damped systems.[2] The key is that the modal and stiffness matrices are diagonal matrices and therefore we have "decoupled" the equations. In other words, we have transformed our problem from a large unwieldy multiple degree of freedom problem into many single degree of freedom problems that can be solved using the same methods outlined above.

Instead of solving for x we are instead solving for q, referred to as the modal coordinates or modal participation factors.

It may be clearer to understand if we write as:


Written in this form we can see that the vibration at each of the degrees of freedom is just a linear sum of the mode shapes. Furthermore, how much each mode "participates" in the final vibration is defined by q, its modal participation factor.
[edit]
See also
Balancing machine
Base isolation
Cushioning
Critical speed
Damping
Dunkerley's Method
Earthquake engineering
Fast Fourier transform
Journal of Sound and Vibration
Mechanical resonance
Modal analysis
Mode shape
Noise and vibration on maritime vessels
Noise, Vibration, and Harshness
Quantum vibration
Random vibration
Ride quality
Shock
Simple harmonic oscillator
Sound
Structural acoustics
Structural dynamics
Tire balance
Torsional vibration
Vibration control
Vibration isolation
Vibration of rotating structures
Wave
Whole body vibration
[edit]
References
^ Tustin, Wayne. Where to place the control accelerometer: one of the most critical decisions in developing random vibration tests also is the most neglected, EE-Evaluation Engineering, 2006
^ a b Maia, Silva. Theoretical And Experimental Modal Analysis, Research Studies Press Ltd., 1997, ISBN 0471970670
[edit]
Further reading
Tongue, Benson, Principles of Vibration, Oxford University Pres, 2001, ISBN 0-195-142462
Inman, Daniel J., Engineering Vibration, Prentice Hall, 2001, ISBN 013726142X
Rao, Singiresu, Mechanical Vibrations, Addison Wesley, 1990, ISBN 0-201-50156-2
Thompson, W.T., Theory of Vibrations, Nelson Thornes Ltd, 1996, ISBN 0-412-783908
Hartog, Den, Mechanical Vibrations, Dover Publications, 1985, ISBN 0-486-647854
[edit]
External links Look up vibration in Wiktionary, the free dictionary.

Hyperphysics Educational Website, Oscillation/Vibration Concepts
Nelson Publishing, Evaluation Engineering Magazine
Structural Dynamics and Vibration Laboratory of McGill University
Normal vibration modes of a circular membrane
Categories: Mechanical vibrations | Mechanical engineering
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Wave
From Wikipedia, the free encyclopedia
This article is about waves in the scientific sense. For other uses of wave or waves, see Wave (disambiguation).

Surface waves in water

In mathematics and science, a wave is a disturbance that travels through space and time, usually accompanied by the transfer of energy.

Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist instead of oscillations or vibrations around almost fixed locations. For example, a cork on rippling water will bob up and down, staying in about the same place while the wave itself moves onwards.

One type of wave is a mechanical wave, which propagates through a medium in which the substance of this medium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For example, sound waves propagate via air molecules bumping into their neighbors. This transfers some energy to these neighbors, which will cause a cascade of collisions between neighbouring molecules. When air molecules collide with their neighbors, they also bounce away from them (restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

Another type of wave can travel through a vacuum, e.g. electromagnetic radiation (including visible light, ultraviolet radiation, infrared radiation, gamma rays, X-rays, and radio waves). This type of wave consists of periodic oscillations in electrical and magnetic fields.

A main distinction is between transverse waves, in which the disturbance occurs in a direction perpendicular (at right angles) to the motion of the wave, and longitudinal waves, in which the disturbance is in the same direction as the wave.

Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave.Contents [hide]
1 General features
2 Mathematical description of one-dimensional waves
2.1 Wave equation
2.2 Wave forms
2.3 Amplitude and modulation
2.4 Phase velocity and group velocity
3 Sinusoidal waves
4 Plane waves
5 Standing waves
6 Physical properties
6.1 Transmission and media
6.2 Absorption
6.3 Reflection
6.4 Interference
6.5 Refraction
6.6 Diffraction
6.7 Polarization
6.8 Dispersion
7 Mechanical waves
7.1 Waves on strings
7.2 Acoustic waves
7.3 Water waves
7.4 Seismic waves
7.5 Shock waves
7.6 Other
8 Electromagnetic waves
9 Quantum mechanical waves
9.1 de Broglie waves
10 Gravitational waves
11 WKB method
12 References
13 See also
14 Sources
15 External links

[edit]
General features

A single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value. However, a vibration is not necessarily a wave. Defining the necessary and sufficient characteristics that qualify a phenomenon to be called a wave is flexible.

The term wave is often understood intuitively as the transport of spatial disturbances that are generally not associated with motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium (Hall 1980, p. 8). However, this notion is problematic for a standing wave (for example, a wave on a string), where energy is moving in both directions equally, or for electromagnetic / light waves in a vacuum, where the concept of medium does not apply. There are water waves in the ocean; light waves from the sun; microwaves inside the microwave oven; radio waves transmitted to the radio; and sound waves from the radio, telephone, and person.

It may be seen that the description of waves is accompanied by a heavy reliance on physical origin when describing any specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic wave-like transfer / transformation of vibratory energy. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as distinct from optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved (for example, in the case of air: vortices, radiation pressure, shock waves, etc., in the case of solids: Rayleigh waves, dispersion, etc., and so on).

Other properties, however, although they are usually described in an origin-specific manner, may be generalized to all waves. For such reasons, wave theory represents a particular branch of physics that is concerned with the properties of wave processes independently from their physical origin.[1] For example, based on the mechanical origin of acoustic waves there can be a moving disturbance in space–time if and only if the medium involved is neither infinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly bound, then they would all vibrate as one, with no delay in the transmission of the vibration and therefore no wave motion. On the other hand, if all the parts were independent, then there would not be any transmission of the vibration and again, no wave motion. Although the above statements are meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times.

Similarly, wave processes revealed from the study of waves other than sound waves can be significant to the understanding of sound phenomena. A relevant example is Thomas Young's principle of interference (Young, 1802, in Hunt 1992, p. 132). This principle was first introduced in Young's study of light and, within some specific contexts (for example, scattering of sound by sound), is still a researched area in the study of sound.
[edit]
Mathematical description of one-dimensional waves
[edit]
Wave equation
Main articles: Wave equation and D'Alembert's formula

Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling

Wavelength λ, can be measured between any two corresponding points on a waveform
in the x direction in space. E.g., let the positive x direction be to the right, and the negative x direction be to the left.
with constant amplitude u
with constant velocity v, where v is
independent of wavelength (no dispersion)
independent of amplitude (linear media, not nonlinear).[2]
with constant waveform, or shape

This wave can then be described by the two-dimensional functions
(waveform F traveling to the right)
(waveform G traveling to the left)

or, more generally, by d'Alembert's formula:[3]


representing two component waveforms F and G traveling through the medium in opposite directions. This wave can also be represented by the partial differential equation


General solutions are based upon Duhamel's principle.[4]
[edit]
Wave forms

Sine, square, triangle and sawtooth waveforms.

The form or shape of F in d'Alembert's formula involves the argument x − vt. Constant values of this argument correspond to constant values of F, and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x-direction at velocity v (and G will propagate at the same speed in the negative x-direction).[5]

In the case of a periodic function F with period λ, that is, F(x + λ − vt) = F(x − vt), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F(x − v(t + T)) = F(x − vt) provided vT = λ, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.[6]
[edit]
Amplitude and modulation

Illustration of the envelope (the slowly varying red curve) of an amplitude modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.
Main article: Amplitude modulation
See also: Frequency modulation and Phase modulation

The amplitude of a wave may be constant (in which case the wave is a c.w. or continuous wave), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form:[7][8][9]


where is the amplitude envelope of the wave, k is the wavenumber and φ is the phase. If the group velocity vg (see below) is wavelength-independent, this equation can be simplified as:[10]


showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.[10][11]
[edit]
Phase velocity and group velocity

Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity.
Main articles: Phase velocity and Group velocity

There are two velocities that are associated with waves, the phase velocity and the group velocity. To understand them, one must consider several types of waveform. For simplification, examination is restricted to one dimension.

This shows a wave with the Group velocity and Phase velocity going in different directions.

The most basic wave (a form of plane wave) may be expressed in the form:


which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument, , makes clear that this expression describes a vibration of wavelength traveling in the x-direction with a constant phase velocity .[12]

The other type of wave to be considered is one with localized structure described by an envelope, which may be expressed mathematically as, for example:


where now A(k1) (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δk surrounding the point k1 = k. In exponential form:


with Ao the magnitude of A. For example, a common choice for Ao is a Gaussian wave packet:[13]


where σ determines the spread of k1-values about k, and N is the amplitude of the wave.

The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(k1), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.[12] However, an exception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basis for the method of stationary phase for evaluation of such integrals.[14]) The condition for φ to vary slowly is that its rate of change with k1 be small; this rate of variation is:[12]


where the evaluation is made at k1 = k because A(k1) is centered there. This result shows that the position x where the phase changes slowly, the position where ψ is appreciable, moves with time at a speed called the group velocity:


The group velocity therefore depends upon the dispersion relation connecting ω and k. For example, in quantum mechanics the energy of a particle represented as a wave packet is E = ħω = (ħk)2/(2m). Consequently, for that wave situation, the group velocity is


showing that the velocity of a localized particle in quantum mechanics is its group velocity.[12] Because the group velocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized.[15] In other words, the velocity of the constituent waves of the wave packet travel at a rate that varies with their wavelength, so some move faster than others, and they cannot maintain the same interference pattern as the wave propagates.
[edit]
Sinusoidal waves

Sinusoidal waves correspond to simple harmonic motion.

Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude u described by the equation:


Graphic of the progressive sinusoidal wave.

where
A is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.
x is the space coordinate
t is the time coordinate
k is the wavenumber
ω is the angular frequency
φ is the phase.

The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) have an amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (e.g., volts/meter).

The wavelength λ is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber k, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation


The period T is the time for one complete cycle of an oscillation of a wave. The frequency f is the number of periods per unit time (per second) and is typically measured in hertz. These are related by:


In other words, the frequency and period of a wave are reciprocals.

The angular frequency ω represents the frequency in radians per second. It is related to the frequency or period by


The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by:[16]


where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency.

Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.[17]

Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze.[18] Due to the Kramers–Kronig relations, a linear medium with dispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium.[19] The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.[20][21]

The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general cases.[22][23] In particular, many media are linear, or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superposition principle to find the solution for a general waveform.[24] When a medium is nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition.
[edit]
Plane waves
Main article: Plane wave
[edit]
Standing waves
Main articles: Standing wave, Acoustic resonance, Helmholtz resonator, and organ pipe

Standing wave in stationary medium. The red dots represent the wave nodes

A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.

The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time.


One-dimensional standing waves; the fundamental mode and the first 6 overtones.


A two-dimensional standing wave on a disk; this is the fundamental mode.


A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.
[edit]
Physical properties

Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism

Waves exhibit common behaviors under a number of standard situations, e.g.,
[edit]
Transmission and media
Main articles: Rectilinear propagation, Transmittance, and Transmission medium

Waves normally move in in a straight line (i.e. rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories:
A bounded medium if it is finite in extent, otherwise an unbounded medium
A linear medium if the amplitudes of different waves at any particular point in the medium can be added
A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space
An anisotropic medium if one or more of its physical properties differ in one or more directions
An isotropic medium if its physical properties are the same in all directions
[edit]
Absorption
Main articles: Absorption (acoustics) and Absorption (electromagnetic radiation)
[edit]
Reflection
Main article: Reflection (physics)

When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.
[edit]
Interference
Main article: Interference (wave propagation)

Waves that encounter each other combine through superposition to create a new wave called an interference pattern. Important interference patterns occur for waves that are in phase.
[edit]
Refraction
Main article: Refraction

Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.

Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law.
[edit]
Diffraction
Main article: Diffraction

A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.
[edit]
Polarization
Main article: Polarization (waves)


A wave is polarized if it oscillates in one direction or plane. A wave can be polarized by the use of a polarizing filter. The polarization of a transverse wave describes the direction of oscillation in the plane perpendicular to the direction of travel.

Longitudinal waves such as sound waves do not exhibit polarization. For these waves the direction of oscillation is along the direction of travel.
[edit]
Dispersion

Main articles: Dispersion (optics) and Dispersion (water waves)

A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce the spectrum of colours of the rainbow. Isaac Newton performed experiments with light and prisms, presenting his findings in the Opticks (1704) that white light consists of several colours and that these colours cannot be decomposed any further.[25]
[edit]
Mechanical waves
Main article: Mechanical wave
[edit]
Waves on strings
Main article: Vibrating string

The speed of a wave traveling along a vibrating string ( v ) is directly proportional to the square root of the tension of the string ( T ) over the linear mass density ( μ ):


where the linear density μ is the mass per unit length of the string.
[edit]
Acoustic waves

Acoustic or sound waves travel at speed given by


or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound).
[edit]
Water waves

Main article: Water waves
Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
Sound—a mechanical wave that propagates through gases, liquids, solids and plasmas;
Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;
Ocean surface waves, which are perturbations that propagate through water.
[edit]
Seismic waves
Main article: Seismic waves
[edit]
Shock waves

Main article: Shock wave
See also: Sonic boom and Cerenkov radiation
[edit]
Other
Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves[26]
Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.
[edit]
Electromagnetic waves

Main articles: Electromagnetic radiation and Electromagnetic spectrum

(radio, micro, infrared, visible, uv)

An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum, the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light. From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves, microwaves, infrared, visible light, ultraviolet and X-rays.
[edit]
Quantum mechanical waves
Main article: Schrödinger equation
See also: Wave function

The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle. Quantum mechanics also describes particle properties that other waves, such as light and sound, have on the atomic scale and below.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.[27]
[edit]
de Broglie waves
Main articles: Wave packet and Matter wave

Louis de Broglie postulated that all particles with momentum have a wavelength


where h is Planck's constant, and p is the magnitude of the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a de Broglie wavelength of about 10−13 m.

A wave representing such a particle traveling in the k-direction is expressed by the wave function:


where the wavelength is determined by the wave vector k as:


and the momentum by:


However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet,[28] a waveform often used in quantum mechanics to describe the wave function of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.

In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet.[29] Gaussian wave packets also are used to analyze water waves.[30]

For example, a Gaussian wavefunction ψ might take the form:[31]


at some initial time t = 0, where the central wavelength is related to the central wave vector k0 as λ0 = 2π / k0. It is well known from the theory of Fourier analysis,[32] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.[33] Given the Gaussian:


the Fourier transform is:


The Gaussian in space therefore is made up of waves:


that is, a number of waves of wavelengths λ such that kλ = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows a spread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, and hence in λ = 2π/k.

Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles
[edit]
Gravitational waves
Main article: Gravitational wave

Researchers believe that gravitational waves also travel through space, although gravitational waves have never been directly detected. Not to be confused with gravity waves, gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.
[edit]
WKB method
Main article: WKB method
See also: Slowly varying envelope approximation

In a nonuniform medium, in which the wavenumber k can depend on the location as well as the frequency, the phase term kx is typically replaced by the integral of k(x)dx, according to the WKB method. Such nonuniform traveling waves are common in many physical problems, including the mechanics of the cochlea and waves on hanging ropes.
[edit]
References
^ Lev A. Ostrovsky & Alexander I. Potapov (2002). Modulated waves: theory and application. Johns Hopkins University Press. ISBN 0801873258.
^ Michael A. Slawinski (2003). "Wave equations". Seismic waves and rays in elastic media. Elsevier. pp. 131 ff. ISBN 0080439306.
^ Karl F Graaf (1991). Wave motion in elastic solids (Reprint of Oxford 1975 ed.). Dover. pp. 13–14. ISBN 9780486667454.
^ Jalal M. Ihsan Shatah, Michael Struwe (2000). "The linear wave equation". Geometric wave equations. American Mathematical Society Bookstore. pp. 37 ff. ISBN 0821827499.
^ Louis Lyons (1998). All you wanted to know about mathematics but were afraid to ask. Cambridge University Press. pp. 128 ff. ISBN 052143601X.
^ Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0470185902.
^ Christian Jirauschek (2005). FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection. Cuvillier Verlag. p. 9. ISBN 3865374190.
^ Fritz Kurt Kneubühl (1997). Oscillations and waves. Springer. p. 365. ISBN 354062001X.
^ Mark Lundstrom (2000). Fundamentals of carrier transport. Cambridge University Press. p. 33. ISBN 0521631343.
^ a b Chin-Lin Chen (2006). "§13.7.3 Pulse envelope in nondispersive media". Foundations for guided-wave optics. Wiley. p. 363. ISBN 0471756873.
^ Stefano Longhi, Davide Janner (2008). "Localization and Wannier wave packets in photonic crystals". In Hugo E. Hernández-Figueroa, Michel Zamboni-Rached, Erasmo Recami. Localized Waves. Wiley-Interscience. p. 329. ISBN 0470108851.
^ a b c d Albert Messiah (1999). Quantum Mechanics (Reprint of two-volume Wiley 1958 ed.). Courier Dover. pp. 50–52. ISBN 9780486409245.
^ See, for example, Eq. 2(a) in Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics: An introduction (2nd ed.). Springer. pp. 60–61. ISBN 3540674586.
^ John W. Negele, Henri Orland (1998). Quantum many-particle systems (Reprint in Advanced Book Classics ed.). Westview Press. p. 121. ISBN 0738200522.
^ Donald D. Fitts (1999). Principles of quantum mechanics: as applied to chemistry and chemical physics. Cambridge University Press. pp. 15 ff. ISBN 0521658411.
^ David C. Cassidy, Gerald James Holton, Floyd James Rutherford (2002). Understanding physics. Birkhäuser. pp. 339 ff. ISBN 0387987568.
^ Paul R Pinet (2009). op. cit.. p. 242. ISBN 0763759937.
^ Mischa Schwartz, William R. Bennett, and Seymour Stein (1995). Communication Systems and Techniques. John Wiley and Sons. p. 208. ISBN 9780780347151.
^ See Eq. 5.10 and discussion in A. G. G. M. Tielens (2005). The physics and chemistry of the interstellar medium. Cambridge University Press. pp. 119 ff. ISBN 0521826349.; Eq. 6.36 and associated discussion in Otfried Madelung (1996). Introduction to solid-state theory (3rd ed.). Springer. pp. 261 ff. ISBN 354060443X.; and Eq. 3.5 in F Mainardi (1996). "Transient waves in linear viscoelastic media". In Ardéshir Guran, A. Bostrom, Herbert Überall, O. Leroy. Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering. World Scientific. p. 134. ISBN 9810242719.
^ Aleksandr Tikhonovich Filippov (2000). The versatile soliton. Springer. p. 106. ISBN 0817636358.
^ Seth Stein, Michael E. Wysession (2003). An introduction to seismology, earthquakes, and earth structure. Wiley-Blackwell. p. 31. ISBN 0865420785.
^ Seth Stein, Michael E. Wysession (2003). op. cit.. p. 32. ISBN 0865420785.
^ Kimball A. Milton, Julian Seymour Schwinger (2006). Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Springer. p. 16. ISBN 3540293043. "Thus, an arbitrary function f(r, t) can be synthesized by a proper superposition of the functions exp[i (k·r−ωt)]…"
^ Raymond A. Serway and John W. Jewett (2005). "§14.1 The Principle of Superposition". Principles of physics (4th ed.). Cengage Learning. p. 433. ISBN 053449143X.
^ Newton, Isaac (1704). "Prop VII Theor V". Opticks: Or, A treatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of Curvilinear Figures. 1. London. p. 118. "All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, or compounded of these..."
^ M. J. Lighthill; G. B. Whitham (1955). "On kinematic waves. II. A theory of traffic flow on long crowded roads". Proceedings of the Royal Society of London. Series A 229: 281–345. And: P. I. Richards (1956). "Shockwaves on the highway". Operations Research 4 (1): 42–51. doi:10.1287/opre.4.1.42.
^ A. T. Fromhold (1991). "Wave packet solutions". Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff. ISBN 0486667413. "(p. 61) …the individual waves move more slowly than the packet and therefore pass back through the packet as it advances"
^ Ming Chiang Li (1980). "Electron Interference". In L. Marton & Claire Marton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 0120146533.
^ See for example Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics (2 ed.). Springer. p. 60. ISBN 3540674586. and John Joseph Gilman (2003). Electronic basis of the strength of materials. Cambridge University Press. p. 57. ISBN 0521620058.,Donald D. Fitts (1999). Principles of quantum mechanics. Cambridge University Press. p. 17. ISBN 0521658411..
^ Chiang C. Mei (1989). The applied dynamics of ocean surface waves (2nd ed.). World Scientific. p. 47. ISBN 9971507897.
^ Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics (2nd ed.). Springer. p. 60. ISBN 3540674586.
^ Siegmund Brandt, Hans Dieter Dahmen (2001). The picture book of quantum mechanics (3rd ed.). Springer. p. 23. ISBN 0387951415.
^ Cyrus D. Cantrell (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 677. ISBN 0521598273.
[edit]
See also
Audience wave
Beat waves
Capillary waves
Cymatics
Doppler effect
Envelope detector
Group velocity
Harmonic
Inertial wave
List of wave topics
List of waves named after people
Ocean surface wave
Phase velocity
Reaction-diffusion equation
Resonance
Ripple tank
Rogue wave (oceanography)
Shallow water equations
Shive wave machine
Standing wave
Transmission medium
Wave turbulence
[edit]
Sources
Campbell, M. and Greated, C. (1987). The Musician’s Guide to Acoustics. New York: Schirmer Books.
French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes. ISBN 0-393-09936-9. OCLC 163810889.
Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont, California: Wadsworth Publishing Company. ISBN 0534007589..
Hunt, F. V. (1992) ([dead link]). Origins in Acoustics. New York: Acoustical Society of America Press..
Ostrovsky, L. A.; Potapov, A. S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns Hopkins University Press. ISBN 0801858704..
Vassilakis, P.N. (2001). Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance. Doctoral Dissertation. University of California, Los Angeles.
[edit]
External links Wikimedia Commons has media related to: Wave
Look up wave in Wiktionary, the free dictionary.

Interactive Visual Representation of Waves
Science Aid: Wave properties—Concise guide aimed at teens
Simulation of diffraction of water wave passing through a gap
Simulation of interference of water waves
Simulation of longitudinal traveling wave
Simulation of stationary wave on a string
Simulation of transverse traveling wave
Sounds Amazing—AS and A-Level learning resource for sound and waves
Vibrations and Waves—an online textbook
Simulation of waves on a string
of longitudinal and transverse mechanical wave

Velocities of waves
Phase velocity • Group velocity • Front velocity • Signal velocity

Categories: Fundamental physics concepts | Partial differential equations | Waves
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Electromagnetic radiation
From Wikipedia, the free encyclopedia Electromagnetism

Electricity · Magnetism [show]
Electrostatics
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Electrodynamics
Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy current
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Electromagnetic radiation (often abbreviated E-M radiation or EMR) is a form of energy exhibiting wave-like behavior as it travels through space. EMR has both electric and magnetic field components, which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation.

Electromagnetic radiation is classified according to the frequency of its wave. In order of increasing frequency and decreasing wavelength, these are radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays (see Electromagnetic spectrum). The eyes of various organisms sense a small and somewhat variable window of frequencies called the visible spectrum. The photon is the quantum of the electromagnetic interaction and the basic "unit" of light and all other forms of electromagnetic radiation and is also the force carrier for the electromagnetic force.

EM radiation carries energy and momentum that may be imparted to matter with which it interacts.Contents [hide]
1 Physics
1.1 Theory
1.2 Properties
1.3 Wave model
1.4 Particle model
1.5 Speed of propagation
2 Thermal radiation and electromagnetic radiation as a form of heat
3 Electromagnetic spectrum
3.1 Light
3.2 Radio waves
4 Derivation
5 See also
6 References
7 External links

[edit]
Physics
[edit]
Theory

Shows the relative wavelengths of the electromagnetic waves of three different colors of light (blue, green and red) with a distance scale in micrometres along the x-axis.
Main article: Maxwell's equations

James Clerk Maxwell first formally postulated electromagnetic waves. These were subsequently confirmed by Heinrich Hertz. Maxwell derived a wave form of the electric and magnetic equations, thus uncovering the wave-like nature of electric and magnetic fields, and their symmetry. Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave.

According to Maxwell's equations, a time-varying electric field generates a time-varying magnetic field and vice versa. Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on. These oscillating fields together form a propagating electromagnetic wave.

A quantum theory of the interaction between electromagnetic radiation and matter such as electrons is described by the theory of quantum electrodynamics.
[edit]
Properties

Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarized wave propagating from right to left. The electric field is in a vertical plane and the magnetic field in a horizontal plane.

The physics of electromagnetic radiation is electrodynamics. Electromagnetism is the physical phenomenon associated with the theory of electrodynamics. Electric and magnetic fields obey the properties of superposition. Thus, a field due to any particular particle or time-varying electric or magnetic field contributes to the fields present in the same space due to other causes. Further, as they are vector fields, all magnetic and electric field vectors add together according to vector addition. For example, in optics two or more coherent lightwaves may interact and by constructive or destructive interference yield a resultant irradiance deviating from the sum of the component irradiances of the individual lightwaves.

Since light is an oscillation it is not affected by travelling through static electric or magnetic fields in a linear medium such as a vacuum. However in nonlinear media, such as some crystals, interactions can occur between light and static electric and magnetic fields — these interactions include the Faraday effect and the Kerr effect.

In refraction, a wave crossing from one medium to another of different density alters its speed and direction upon entering the new medium. The ratio of the refractive indices of the media determines the degree of refraction, and is summarized by Snell's law. Light disperses into a visible spectrum as light passes through a prism because of the wavelength dependent refractive index of the prism material (Dispersion).

EM radiation exhibits both wave properties and particle properties at the same time (see wave-particle duality). Both wave and particle characteristics have been confirmed in a large number of experiments. Wave characteristics are more apparent when EM radiation is measured over relatively large timescales and over large distances while particle characteristics are more evident when measuring small timescales and distances. For example, when electromagnetic radiation is absorbed by matter, particle-like properties will be more obvious when the average number of photons in the cube of the relevant wavelength is much smaller than 1. Upon absorption of light, it is not too difficult to experimentally observe non-uniform deposition of energy. Strictly speaking, however, this alone is not evidence of "particulate" behavior of light, rather it reflects the quantum nature of matter.[1]

There are experiments in which the wave and particle natures of electromagnetic waves appear in the same experiment, such as the self-interference of a single photon. True single-photon experiments (in a quantum optical sense) can be done today in undergraduate-level labs.[2] When a single photon is sent through an interferometer, it passes through both paths, interfering with itself, as waves do, yet is detected by a photomultiplier or other sensitive detector only once.
[edit]
Wave model

Electromagnetic radiation is a transverse wave meaning that the oscillations of the waves are perpendicular to the direction of energy transfer and travel. An important aspect of the nature of light is frequency. The frequency of a wave is its rate of oscillation and is measured in hertz, the SI unit of frequency, where one hertz is equal to one oscillation per second. Light usually has a spectrum of frequencies which sum together to form the resultant wave. Different frequencies undergo different angles of refraction.

A wave consists of successive troughs and crests, and the distance between two adjacent crests or troughs is called the wavelength. Waves of the electromagnetic spectrum vary in size, from very long radio waves the size of buildings to very short gamma rays smaller than atom nuclei. Frequency is inversely proportional to wavelength, according to the equation:


where v is the speed of the wave (c in a vacuum, or less in other media), f is the frequency and λ is the wavelength. As waves cross boundaries between different media, their speeds change but their frequencies remain constant.

Interference is the superposition of two or more waves resulting in a new wave pattern. If the fields have components in the same direction, they constructively interfere, while opposite directions cause destructive interference.

The energy in electromagnetic waves is sometimes called radiant energy.
[edit]
Particle model
See also: Quantization (physics) and Quantum optics

Because energy of an EM interaction is quantized, EM waves are emitted and absorbed as discrete packets of energy, or quanta, called photons.[3] Because photons are emitted and absorbed by charged particles, they act as transporters of energy, and are associated with waves with frequency proportional to the energy carried. The energy per photon can be related to the frequency via the Planck–Einstein equation:[4]


where E is the energy, h is Planck's constant, and f is frequency. The energy is commonly expressed in the unit of electronvolt (eV). This photon-energy expression is a particular case of the energy levels of the more general electromagnetic oscillator, whose average energy, which is used to obtain Planck's radiation law, can be shown to differ sharply from that predicted by the equipartition principle at low temperature, thereby establishes a failure of equipartition due to quantum effects at low temperature.[5]

As a photon is absorbed by an atom, it excites the atom, elevating an electron to a higher energy level. If the energy is great enough, so that the electron jumps to a high enough energy level, it may escape the positive pull of the nucleus and be liberated from the atom in a process called photoionisation. Conversely, an electron that descends to a lower energy level in an atom emits a photon of light equal to the energy difference. Since the energy levels of electrons in atoms are discrete, each element emits and absorbs its own characteristic frequencies.

Together, these effects explain the emission and absorption spectra of light. The dark bands in the absorption spectrum are due to the atoms in the intervening medium absorbing different frequencies of the light. The composition of the medium through which the light travels determines the nature of the absorption spectrum. For instance, dark bands in the light emitted by a distant star are due to the atoms in the star's atmosphere. These bands correspond to the allowed energy levels in the atoms. A similar phenomenon occurs for emission. As the electrons descend to lower energy levels, a spectrum is emitted that represents the jumps between the energy levels of the electrons. This is manifested in the emission spectrum of nebulae. Today, scientists use this phenomenon to observe what elements a certain star is composed of. It is also used in the determination of the distance of a star, using the red shift.
[edit]
Speed of propagation
Main article: Speed of light

Any electric charge which accelerates, or any changing magnetic field, produces electromagnetic radiation. Electromagnetic information about the charge travels at the speed of light. Accurate treatment thus incorporates a concept known as retarded time (as opposed to advanced time, which is not physically possible in light of causality), which adds to the expressions for the electrodynamic electric field and magnetic field. These extra terms are responsible for electromagnetic radiation. When any wire (or other conducting object such as an antenna) conducts alternating current, electromagnetic radiation is propagated at the same frequency as the electric current. At the quantum level, electromagnetic radiation is produced when the wavepacket of a charged particle oscillates or otherwise accelerates. Charged particles in a stationary state do not move, but a superposition of such states may result in oscillation, which is responsible for the phenomenon of radiative transition between quantum states of a charged particle.

Depending on the circumstances, electromagnetic radiation may behave as a wave or as particles. As a wave, it is characterized by a velocity (the speed of light), wavelength, and frequency. When considered as particles, they are known as photons, and each has an energy related to the frequency of the wave given by Planck's relation E = hν, where E is the energy of the photon, h = 6.626 × 10−34 J·s is Planck's constant, and ν is the frequency of the wave.

One rule is always obeyed regardless of the circumstances: EM radiation in a vacuum always travels at the speed of light, relative to the observer, regardless of the observer's velocity. (This observation led to Albert Einstein's development of the theory of special relativity.)

In a medium (other than vacuum), velocity factor or refractive index are considered, depending on frequency and application. Both of these are ratios of the speed in a medium to speed in a vacuum.
[edit]
Thermal radiation and electromagnetic radiation as a form of heat
Main article: Thermal radiation

The basic structure of matter involves charged particles bound together in many different ways. When electromagnetic radiation is incident on matter, it causes the charged particles to oscillate and gain energy. The ultimate fate of this energy depends on the situation. It could be immediately re-radiated and appear as scattered, reflected, or transmitted radiation. It may also get dissipated into other microscopic motions within the matter, coming to thermal equilibrium and manifesting itself as thermal energy in the material. With a few exceptions such as fluorescence, harmonic generation, photochemical reactions and the photovoltaic effect, absorbed electromagnetic radiation simply deposits its energy by heating the material. This happens both for infrared and non-infrared radiation. Intense radio waves can thermally burn living tissue and can cook food. In addition to infrared lasers, sufficiently intense visible and ultraviolet lasers can also easily set paper afire. Ionizing electromagnetic radiation can create high-speed electrons in a material and break chemical bonds, but after these electrons collide many times with other atoms in the material eventually most of the energy gets downgraded to thermal energy, this whole process happening in a tiny fraction of a second. That infrared radiation is a form of heat and other electromagnetic radiation is not, is a widespread misconception in physics. Any electromagnetic radiation can heat a material when it is absorbed.

The inverse or time-reversed process of absorption is responsible for thermal radiation. Much of the thermal energy in matter consists of random motion of charged particles, and this energy can be radiated away from the matter. The resulting radiation may subsequently be absorbed by another piece of matter, with the deposited energy heating the material. Radiation is an important mechanism of heat transfer.

The electromagnetic radiation in an opaque cavity at thermal equilibrium is effectively a form of thermal energy, having maximum radiation entropy. The thermodynamic potentials of electromagnetic radiation can be well-defined as for matter. Thermal radiation in a cavity has energy density (see Planck's Law) of


Differentiating the above with respect to temperature, we may say that the electromagnetic radiation field has an effective volumetric heat capacity given by

[edit]
Electromagnetic spectrum
Main article: Electromagnetic spectrum

Electromagnetic spectrum with light highlighted

Legend:
γ = Gamma rays
HX = Hard X-rays
SX = Soft X-Rays
EUV = Extreme ultraviolet
NUV = Near ultraviolet
Visible light
NIR = Near infrared
MIR = Moderate infrared
FIR = Far infrared

Radio waves:
EHF = Extremely high frequency (Microwaves)
SHF = Super high frequency (Microwaves)
UHF = Ultrahigh frequency
VHF = Very high frequency
HF = High frequency
MF = Medium frequency
LF = Low frequency
VLF = Very low frequency
VF = Voice frequency
ULF = Ultra low frequency
SLF = Super low frequency
ELF = Extremely low frequency

Generally, EM radiation (the designation 'radiation' excludes static electric and magnetic and near fields) is classified by wavelength into radio, microwave, infrared, the visible region we perceive as light, ultraviolet, X-rays and gamma rays. Arbitrary electromagnetic waves can always be expressed by Fourier analysis in terms of sinusoidal monochromatic waves which can be classified into these regions of the spectrum.

The behavior of EM radiation depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. When EM radiation interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. Spectroscopy can detect a much wider region of the EM spectrum than the visible range of 400 nm to 700 nm. A common laboratory spectroscope can detect wavelengths from 2 nm to 2500 nm. Detailed information about the physical properties of objects, gases, or even stars can be obtained from this type of device. It is widely used in astrophysics. For example, hydrogen atoms emit radio waves of wavelength 21.12 cm.

Soundwaves are not electromagnetic radiation. At the lower end of the electromagnetic spectrum, about 20 Hz to about 20 kHz, are frequencies that might be considered in the audio range. However, electromagnetic waves cannot be directly perceived by human ears. Sound waves are the oscillating compression of molecules. To be heard, electromagnetic radiation must be converted to air pressure waves, or if the ear is submerged, water pressure waves.
[edit]
Light
Main article: Light

EM radiation with a wavelength between approximately 400 nm and 700 nm is directly detected by the human eye and perceived as visible light. Other wavelengths, especially nearby infrared (longer than 700 nm) and ultraviolet (shorter than 400 nm) are also sometimes referred to as light, especially when visibility to humans is not relevant.

If radiation having a frequency in the visible region of the EM spectrum reflects off of an object, say, a bowl of fruit, and then strikes our eyes, this results in our visual perception of the scene. Our brain's visual system processes the multitude of reflected frequencies into different shades and hues, and through this not-entirely-understood psychophysical phenomenon, most people perceive a bowl of fruit.

At most wavelengths, however, the information carried by electromagnetic radiation is not directly detected by human senses. Natural sources produce EM radiation across the spectrum, and our technology can also manipulate a broad range of wavelengths. Optical fiber transmits light which, although not suitable for direct viewing, can carry data that can be translated into sound or an image. To be meaningful both transmitter and receiver must use some agreed-upon encoding system - especially so if the transmission is digital as opposed to the analog nature of the waves.
[edit]
Radio waves
Main article: Radio waves

Radio waves can be made to carry information by varying a combination of the amplitude, frequency and phase of the wave within a frequency band.

When EM radiation impinges upon a conductor, it couples to the conductor, travels along it, and induces an electric current on the surface of that conductor by exciting the electrons of the conducting material. This effect (the skin effect) is used in antennas. EM radiation may also cause certain molecules to absorb energy and thus to heat up; this is exploited in microwave ovens. Radio waves are not ionizing radiation, as the energy per photon is too small.
[edit]
Derivation This article's tone or style may not be appropriate for Wikipedia. Specific concerns may be found on the talk page. See Wikipedia's guide to writing better articles for suggestions. (April 2010)


Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations. Inspection of Maxwell's equations without sources (charges or currents) results in, along with the possibility of nothing happening, nontrivial solutions of changing electric and magnetic fields. Beginning with Maxwell's equations in free space:




where
is a vector differential operator (see Del).

One solution,
,

is trivial.

For a more useful solution, we utilize vector identities, which work for any vector, as follows:


To see how we can use this, take the curl of equation (2):


Evaluating the left hand side:

where we simplified the above by using equation (1).

Evaluate the right hand side:


Equations (6) and (7) are equal, so this results in a vector-valued differential equation for the electric field, namely


Applying a similar pattern results in similar differential equation for the magnetic field:.


These differential equations are equivalent to the wave equation:

where
c0 is the speed of the wave in free space and
f describes a displacement

Or more simply:

where is d'Alembertian:


Notice that in the case of the electric and magnetic fields, the speed is:


Which, as it turns out, is the speed of light in vacuum. Maxwell's equations have unified the vacuum permittivity ε0, the vacuum permeability μ0, and the speed of light itself, c0. Before this derivation it was not known that there was such a strong relationship between light and electricity and magnetism.

But these are only two equations and we started with four, so there is still more information pertaining to these waves hidden within Maxwell's equations. Let's consider a generic vector wave for the electric field.


Here is the constant amplitude, f is any second differentiable function, is a unit vector in the direction of propagation, and is a position vector. We observe that is a generic solution to the wave equation. In other words
,

for a generic wave traveling in the direction.

This form will satisfy the wave equation, but will it satisfy all of Maxwell's equations, and with what corresponding magnetic field?



The first of Maxwell's equations implies that electric field is orthogonal to the direction the wave propagates.



The second of Maxwell's equations yields the magnetic field. The remaining equations will be satisfied by this choice of .

Not only are the electric and magnetic field waves traveling at the speed of light, but they have a special restricted orientation and proportional magnitudes, E0 = c0B0, which can be seen immediately from the Poynting vector. The electric field, magnetic field, and direction of wave propagation are all orthogonal, and the wave propagates in the same direction as .

From the viewpoint of an electromagnetic wave traveling forward, the electric field might be oscillating up and down, while the magnetic field oscillates right and left; but this picture can be rotated with the electric field oscillating right and left and the magnetic field oscillating down and up. This is a different solution that is traveling in the same direction. This arbitrariness in the orientation with respect to propagation direction is known as polarization. On a quantum level, it is described as photon polarization.

More general forms of the second order wave equations given above are available, allowing for both non-vacuum propagation media and sources. A great many competing derivations exist, all with varying levels of approximation and intended applications. One very general example is a form of the electric field equation,[6] which was factorized into a pair of explicitly directional wave equations, and then efficiently reduced into a single uni-directional wave equation by means of a simple slow-evolution approximation.
[edit]
See also
Antenna (radio)
Antenna measurement
Bioelectromagnetism
Bolometer
Control of electromagnetic radiation
Electromagnetic field
Electromagnetic pulse
Electromagnetic radiation and health
Electromagnetic spectrum
Electromagnetic wave equation
Evanescent wave coupling
Finite-difference time-domain method
Helicon
Impedance of free space
Light
Maxwell's equations
Near and far field
Radiant energy
Radiation reaction
Risks and benefits of sun exposure
Sinusoidal plane-wave solutions of the electromagnetic wave equation
[edit]
References
^ [1]
^ [2]
^ Weinberg, S. (1995). The Quantum Theory of Fields. 1. Cambridge University Press. pp. 15–17. ISBN 0-521-55001-7.
^ Paul M. S. Monk (2004). Physical Chemistry. John Wiley and Sons. p. 435. ISBN 9780471491804.
^ Vu-Quoc, L., Configuration integral (statistical mechanics), 2008.
^ Kinsler, P. (2010). "Optical pulse propagation with minimal approximations" (reprint). Phys. Rev. A 81: 013819. doi:10.1103/PhysRevA.81.013819].
Hecht, Eugene (2001). Optics (4th ed.). Pearson Education. ISBN 0-8053-8566-5.
Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks Cole. ISBN 0-534-40842-7.
Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.
Reitz, John; Milford, Frederick; Christy, Robert (1992). Foundations of Electromagnetic Theory (4th ed.). Addison Wesley. ISBN 0-201-52624-7.
Jackson, John David (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN 0-471-30932-X.
Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Artech House Publishers. ISBN 1-58053-832-0.
[edit]
External links Wikisource has original text related to this article:
Pictured Electro-Magnetic Waves

Electromagnetism - a chapter from an online textbook
Electromagnetic Radiation - an introduction for electrical engineers
Electromagnetic Waves from Maxwell's Equations on Project PHYSNET.
Radiation of atoms? e-m wave, Polarisation, ...
An Introduction to The Wigner Distribution in Geometric Optics[hide]
v · d · e
Radiation (Physics & Health)

Main articles Non-ionizing radiation Ultraviolet light · Near ultraviolet · Visible light · Infrared light · Microwave · Radio waves · Acoustic Radiation

Ionizing radiation X-ray · Cosmic radiation · Gamma ray · Background radiation · Nuclear fission · Nuclear fusion · Particle accelerators · Nuclear radiation (Nuclear weapons · Nuclear reactors) · Radioactive materials (Radioactive decay)

Thermal radiation · Electromagnetic radiation · Earth's radiation balance


Radiation health effects Radiation therapy · Radiation poisoning · Radioactivity in biological research · List of civilian radiation accidents
Mobile phone radiation and health · Wireless electronic devices and health · Health physics · Laser safety · Lasers and aviation safety

Related articles Radiation hardening · Half-life · Radiobiology · Nuclear physics

See also: Category:Radiation effects · Category:Radioactivity · Category:Radiation health effects · Category:Radiobiology



Categories: Electromagnetic radiation
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