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In [[mathematics]] and [[science]], a '''wave''' is a disturbance that travels through [[space]] and [[time]], usually by the transfer of [[energy]]. Waves are described by a wave equation that can take on many forms depending on the type of wave. A mechanical wave is a wave that propagates through a [[medium (optics)|medium]] owing to [[restoring force]]s 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 actually traveling with the wave.
In [[mathematics]] and [[science]], a '''wave''' is a disturbance that travels through [[space]] and [[time]], usually by the transfer of [[energy]]. Waves are described by a wave equation that can take on many forms depending on the type of wave. A mechanical wave is a wave that propagates through a [[medium (optics)|medium]] owing to [[restoring force]]s 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 actually traveling with the wave.


Waves travel and transfer [[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 [[oscillation]]s or vibrations around almost fixed locations. For example, a cork on rippling water will bob up and down while staying in about the same place while the wave itself moves onwards. Waves carry energy but not mass because even as a wave travels outward from the center (carrying energy of motion), the medium itself does not flow with it.

There are also waves capable of traveling through a [[vacuum]], e.g. [[electromagnetic radiation]] (including visible light, ultraviolet radiation, infrared radiation, gamma rays, X-rays, and radio waves). They consist of period oscillations in electrical and magnetic properties that grow, reach a peak, and diminish to zero in a periodic fashion.

Researchers believe that gravitational waves travel through space, although gravitational waves have never been directly detected. (See [[gravitational radiation]].)

== 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 disturbances in space, these disturbances 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 {{Harv|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 (physics)|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 (chemistry)|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.<ref name=Ostrovsky>
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.<ref name=Ostrovsky>


{{cite book |title=Modulated waves: theory and application |url=http://www.amazon.com/gp/product/0801873258 |author= Lev A. Ostrovsky & Alexander I. Potapov |publisher=Johns Hopkins University Press |isbn=0801873258 |year=2002 }}
{{cite book |title=Modulated waves: theory and application |url=http://www.amazon.com/gp/product/0801873258 |author= Lev A. Ostrovsky & Alexander I. Potapov |publisher=Johns Hopkins University Press |isbn=0801873258 |year=2002 }}

</ref> 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 (waves)|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 (scientist)|Thomas Young]]'s principle of interference (Young, 1802, in {{Harvnb|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.

== Mathematical description ==
=== Wave equation ===
{{Main|Wave equation|D'Alembert's formula}}

For a wave moving in one dimension traveling along the x-axis whose shape stays the same, whether or not it be a [[pulse (physics)|pulse]], the wave function takes one of the forms,

: <math>u(x, \ t) = F(x - v \ t)</math> (traveling to the right)

: <math>u(x, \ t) = G(x + v \ t)</math> (traveling to the left)

It can be shown that both these wave functions satisfy the wave equation,

:<math>
\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}. \,
</math>

The [[wave equation]] is a [[partial differential equation]] that describes the evolution of a wave over time in a medium where the wave propagates at the same speed independent of wavelength (no [[dispersion relation|dispersion]]), and independent of amplitude ([[linear]] media, not [[nonlinear]]).<ref name= Helbig>

{{cite book |title=Seismic waves and rays in elastic media |url=http://books.google.com/?id=s7bp6ezoRhcC&pg=PA134 |pages=131 ''ff'' |author=Michael A. Slawinski, Klause Helbig |chapter=Wave equations |isbn=0080439306 |year=2003 |publisher=Elsevier}}

</ref> General solutions are based upon [[Duhamel's principle]].<ref name=Struwe>

{{cite book |title=Geometric wave equations |author=Jalal M. Ihsan Shatah, Michael Struwe |url=http://books.google.com/?id=zsasG2axbSoC&pg=PA37 |chapter=The linear wave equation |pages=37 ''ff'' |isbn=0821827499 |year=2000 |publisher=American Mathematical Society Bookstore}}

</ref>

In particular, consider the wave equation in one dimension, for example, as applied to a string. Suppose a one-dimensional wave is traveling along the ''x'' axis with velocity <math>v</math> and amplitude <math>u</math> (which generally depends on both ''x'' and ''t''), the wave equation is

The velocity ''v'' will depend on the medium through which the wave is moving.

The general solution for the wave equation in one dimension was given by [[d'Alembert]]; it is known as [[d'Alembert's formula]]:<ref name=Graaf >
{{cite book
|title=Wave motion in elastic solids
|author =Karl F Graaf
|edition=Reprint of Oxford 1975
|publisher=Dover
|year=1991
|url=http://books.google.com/?id=5cZFRwLuhdQC&printsec=frontcover
|pages=13–14
|isbn=9780486667454
}}</ref>

:<math>
u(x,t)=F(x-vt)+G(x+vt). \,
</math>

This formula represents two shapes traveling through the medium in opposite directions; ''F'' in the positive ''x'' direction, and ''G'' in the negative ''x'' direction, of arbitrary functional shapes ''F'' and ''G''.

[[File:Nonsinusoidal wavelength.JPG|thumb|right|200 px|Wavelength ''λ'', can be measured between any two corresponding points.]]

=== Wave forms ===

[[File:Waveforms.svg|thumb|right|280 px|Sine, [[square wave|square]], triangle and sawtooth waveforms.]]

The form of the forward propagating wave ''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).<ref name=Lyons>

{{cite book |url=http://books.google.com/?id=WdPGzHG3DN0C&pg=PA128 |pages=128 ''ff'' |title=All you wanted to know about mathematics but were afraid to ask |author=Louis Lyons |isbn=052143601X |publisher=Cambridge University Press |year=1998 }}</ref>

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'' (sometimes called 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''.<ref name="McPherson0">{{cite book|title=Introduction to Macromolecular Crystallography
|author=Alexander McPherson |url=http://books.google.com/?id=o7sXm2GSr9IC&pg=PA77
|page=77
|chapter=Waves and their properties
|isbn=0470185902
|year=2009
|edition=2
|publisher=Wiley
}}
</ref>

===Phase velocity and group velocity===


[[Image:Wave group.gif|thumb|frame|right|[[Dispersion (water waves)|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.]]
[[Image:Wave group.gif|thumb|frame|right|[[Dispersion (water waves)|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.]]
Line 139: Line 51:
where σ determines the spread of ''k''<sub>1</sub>-values about ''k'', and ''N'' is the amplitude of the wave.
where σ determines the spread of ''k''<sub>1</sub>-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(''k''<sub>1</sub>), and where it varies rapidly, the exponentials cancel each other out, [[Interference (wave propagation)|interfere]] destructively, contributing little to ψ.<ref name=Messiah/> 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 approximation|stationary phase]] for evaluation of such integrals.<ref name=Orland>

{{cite book |title=Quantum many-particle systems |author=John W. Negele, Henri Orland |url=http://books.google.com/?id=mx5CfeeEkm0C&pg=PA121 |page=121 |isbn=0738200522 |year=1998 |publisher=Westview Press |edition=Reprint in Advanced Book Classics}}

</ref>) The condition for φ to vary slowly is that its rate of change with ''k''<sub>1</sub> be small; this rate of variation is:<ref name=Messiah/>

:<math>\left . \frac{d \varphi }{d k_1} \right | _{k_1 = k } = x - t \left . \frac{d \omega}{dk_1}\right | _{k_1 = k } +\left . \frac{d \alpha}{d k_1}\right | _{k_1 = k } \ ,</math>

where the evaluation is made at ''k''<sub>1</sub> = ''k'' because ''A(k''<sub>1</sub>'')'' 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'':

:<math>v_g = \frac{d \omega}{dk} \ . </math>

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 = (ħ''k'')<sup>2</sup>/(2''m''). Consequently, for that wave situation, the group velocity is

:<math> v_g = \frac {\hbar k}{m} \ , </math>

showing that the velocity of a localized particle in quantum mechanics is its group velocity.<ref name=Messiah/> Because the group velocity varies with ''k'', the shape of the wave packet broadens with time, and the particle becomes less localized.<ref name=Fitt>

{{cite book |title=Principles of quantum mechanics: as applied to chemistry and chemical physics |author=Donald D. Fitts |url=http://books.google.com/?id=8t4DiXKIvRgC&pg=PA15 |pages=15 ''ff'' |isbn=0521658411 |year=1999 |publisher=Cambridge University Press }}

</ref> 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.

=== Modulated waves ===
[[File:Wave packet.svg|right|thumb|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|Amplitude modulation}}

{{See also|Frequency modulation|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 [[Amplitude modulation|modulated wave]] can be written in the form:<ref name=Jirauschek>

{{cite book |url= http://books.google.com/?id=6kOoT_AX2CwC&pg=PA9 |page=9 |title=FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection |author=Christian Jirauschek |isbn=3865374190 |year=2005 |publisher=Cuvillier Verlag}}

</ref><ref name=Kneubühl>

{{cite book |title=Oscillations and waves |author=Fritz Kurt Kneubühl |url=http://books.google.com/?id=geYKPFoLgoMC&pg=PA365 |page=365 |year=1997 |isbn=354062001X |publisher=Springer}}

</ref><ref name=Lundstrom>

{{cite book |url=http://books.google.com/?id=FTdDMtpkSkIC&pg=PA33 |page=33 |author=Mark Lundstrom |isbn=0521631343 |year=2000 |title=Fundamentals of carrier transport |publisher=Cambridge University Press}}

</ref>

:<math>u(x, \ t) = A(x, \ t)\sin (kx - \omega t + \phi) \ , </math>

where <math>A(x,\ t)</math> is the amplitude envelope of the wave, <math>k</math> is the ''wave number'' and <math>\phi</math> is the ''[[phase (waves)|phase]]''. If the [[group velocity]] (see below) is wavelength independent, this equation can be simplified as:<ref name=Chen>

{{cite book |url=http://books.google.com/?id=LxzWPskhns0C&pg=PA363 |author=Chin-Lin Chen |title=Foundations for guided-wave optics |page=363 |chapter=§13.7.3 Pulse envelope in nondispersive media |isbn=0471756873 |year=2006 |publisher=Wiley}}

</ref>

:<math>u(x, \ t) = A(x - v_g \ t)\sin (kx - \omega t + \phi) \ , </math>

where ''v''<sub>g</sub> is the group velocity, showing that the envelope moves with velocity ''v''<sub>g</sub> 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''.<ref name=Chen/><ref name=Recami>

{{cite book |title=Localized Waves |chapter=Localization and Wannier wave packets in photonic crystals |author=Stefano Longhi, Davide Janner |editor=Hugo E. Hernández-Figueroa, Michel Zamboni-Rached, Erasmo Recami |url=http://books.google.com/?id=xxbXgL967PwC&pg=PA329 |page=329 |isbn=0470108851 |year=2008 |publisher=Wiley-Interscience}}

</ref>

== Sinusoidal waves ==

Mathematically, the most basic wave is the [[sine wave]] (or harmonic wave or sinusoid), with an amplitude ''u'' described by the equation:

:<math>u(x, \ t)= A \sin (kx - \omega t + \phi) \ , </math>

[[File:Onda 2.jpg|thumb|300px|Graphic of the <math>y=\sin[2\pi(\frac{x}{2\pi}-\frac{t}{2\pi})]</math> progressive sinusoidal wave.]]

where ''A'' is the [[amplitude]] of the wave – the 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]] (spatial frequency), ''ω'' is the temporal frequency, and ''φ'' is a phase offset.

The units of the amplitude depend on the type of wave — waves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals) and electromagnetic waves as the amplitude of the [[electric field]] (volts/meter).

The [[wavelength]] (denoted as ''λ'') is the distance between two sequential crests (or troughs), and generally is measured in meters.

A [[wavenumber]] ''k'', the spatial frequency of the wave in [[radian]]s per unit distance (typically per meter), can be associated with the wavelength by the relation

:<math>
k = \frac{2 \pi}{\lambda}. \,
</math>

[[File:Simple harmonic motion animation.gif|thumb|right|Sine waves correspond to [[simple harmonic motion]].]]

The [[period (physics)|period]] ''T'' is the time for one complete cycle of an oscillation of a wave. The [[frequency]] ''f'' (also frequently denoted as ''ν'' ) is the number of periods per unit time (per second) and is measured in [[hertz]]. These are related by:

:<math>
f=\frac{1}{T}. \,
</math>

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 by

:<math>
\omega = 2 \pi f = \frac{2 \pi}{T}. \,
</math>

The wavelength ''λ'' of a sinusoidal waveform traveling at constant speed ''v'' is given by:<ref name= Cassidy>

{{cite book |title=Understanding physics |author= David C. Cassidy, Gerald James Holton, Floyd James Rutherford |url=http://books.google.com/?id=rpQo7f9F1xUC&pg=PA340 |pages=339 ''ff'' |isbn=0387987568 |year=2002 |publisher=Birkhäuser}}

</ref>
:<math>\lambda = \frac{v}{f},</math>

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 function|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.<ref name=Pinet2>
{{cite book |title=op. cit.
|author = Paul R Pinet
|url = http://books.google.com/?id=6TCm8Xy-sLUC&pg=PA242
|page = 242
|isbn = 0763759937 |year=2009
}}</ref>

Although arbitrary wave shapes will propagate unchanged in lossless [[linear time-invariant system]]s, 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.<ref>
{{cite book
| title = Communication Systems and Techniques
| author = Mischa Schwartz, William R. Bennett, and Seymour Stein
| publisher = John Wiley and Sons
| year = 1995
| isbn = 9780780347151
| page = 208
| url = http://books.google.com/?id=oRSHWmaiZwUC&pg=PA208&dq=sine+wave+medium++linear+time-invariant
}}</ref> Due to the [[Kramers–Kronig relation]]s, 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.<ref name=Tielens>See Eq. 5.10 and discussion in {{cite book |author= A. G. G. M. Tielens |title=The physics and chemistry of the interstellar medium |url=http://books.google.com/?id=wMnvg681JXMC&pg=PA119 |pages=119 ''ff'' |isbn=0521826349 |year=2005 |publisher=Cambridge University Press}}; Eq. 6.36 and associated discussion in {{cite book |title=Introduction to solid-state theory |author=Otfried Madelung |url=http://books.google.com/?id=yK_J-3_p8_oC&pg=PA261 |pages =261 ''ff'' |isbn=354060443X |year=1996 |edition=3rd |publisher=Springer}}; and Eq. 3.5 in {{cite book |author=F Mainardi |chapter=Transient waves in linear viscoelastic media |editor=Ardéshir Guran, A. Bostrom, Herbert Überall, O. Leroy |title=Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering |url=http://books.google.com/?id=UfSk45nCVKMC&pg=PA134 |page=134 |isbn=9810242719 |year=1996 |publisher=World Scientific}}</ref>
}}</ref> Due to the [[Kramers–Kronig relation]]s, 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.<ref name=Tielens>See Eq. 5.10 and discussion in {{cite book |author= A. G. G. M. Tielens |title=The physics and chemistry of the interstellar medium |url=http://books.google.com/?id=wMnvg681JXMC&pg=PA119 |pages=119 ''ff'' |isbn=0521826349 |year=2005 |publisher=Cambridge University Press}}; Eq. 6.36 and associated discussion in {{cite book |title=Introduction to solid-state theory |author=Otfried Madelung |url=http://books.google.com/?id=yK_J-3_p8_oC&pg=PA261 |pages =261 ''ff'' |isbn=354060443X |year=1996 |edition=3rd |publisher=Springer}}; and Eq. 3.5 in {{cite book |author=F Mainardi |chapter=Transient waves in linear viscoelastic media |editor=Ardéshir Guran, A. Bostrom, Herbert Überall, O. Leroy |title=Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering |url=http://books.google.com/?id=UfSk45nCVKMC&pg=PA134 |page=134 |isbn=9810242719 |year=1996 |publisher=World Scientific}}</ref>
The [[sine function]] is periodic, so the [[sine wave]] or sinusoid has a [[wavelength]] in space and a period in time.<ref name=Filippov>
The [[sine function]] is periodic.
{{cite book
|url=http://books.google.com/?id=TC4MCYBSJJcC&pg=PA106
|page=106
|author=Aleksandr Tikhonovich Filippov
|title=The versatile soliton
|year=2000
|publisher=Springer
|isbn=0817636358
}}</ref><ref name=Stein1>
{{cite book
|title=An introduction to seismology, earthquakes, and earth structure
|author=Seth Stein, [[Michael E. Wysession]]
|page=31
|url=http://books.google.com/?id=Kf8fyvRd280C&pg=PA31
|isbn=0865420785
|year=2003
|publisher=Wiley-Blackwell
}}</ref>

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.<ref name=Stein2>
{{cite book
|title=''op. cit.''
|author=Seth Stein, [[Michael E. Wysession]]
|page=32
|url=http://books.google.com/?id=Kf8fyvRd280C&pg=PA32
|isbn=0865420785
|year=2003
}}</ref><ref name=Schwinger>
{{cite book
|title=Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators
|author=Kimball A. Milton, Julian Seymour Schwinger
|url=http://books.google.com/?id=x_h2rai2pYwC&pg=PA16
|page=16
|isbn=3540293043
|publisher=Springer
|year=2006
|quote=Thus, an arbitrary function ''f''('''''r''''', ''t'') can be synthesized by a proper superposition of the functions ''exp''[i ('''''k·r'''''−ω''t'')]…
}}</ref> 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.<ref name=Jewett>
{{cite book
|url=http://books.google.com/?id=1DZz341Pp50C&pg=PA433
|page=433 |title=Principles of physics
|author=Raymond A. Serway and John W. Jewett
|chapter=§14.1 The Principle of Superposition
|isbn=053449143X |year=2005
|edition=4th
|publisher=Cengage Learning
}}</ref> When a medium is [[nonlinear]], the response to complex waves cannot be determined from a sine-wave decomposition.

== Plane waves ==
{{Main|Plane wave}}

== Standing waves ==
{{Main|Standing wave}}

[[File:Standing wave.gif|frame|right|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|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 (wave propagation)|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, longitudinal waves propagate out to where the string is held in place at the [[Bridge (instrument)|bridge]] and the "[[Nut (string instrument)|nut]]", whereupon the waves are reflected back. At the bridge and nut, the two opposed waves are in [[antiphase]] and cancel each other, producing a [[node (physics)|node]]. Halfway between two nodes there is an [[antinode]], where the two counter-propagating waves ''enhance'' each other maximally. There is on [[average]] no net propagation of energy.

Also see: [[Acoustic resonance]], [[Helmholtz resonator]], and [[organ pipe]]

== Physical properties ==
[[File:Dispersive Prism Illustration by Spigget.jpg|thumb|right|280 px|Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism]]

Periodic waves are characterized by ''[[crest (physics)|crests]]'' (highs) and ''[[trough (physics)|troughs]]'' (lows), and may usually be categorized as either longitudinal or transverse. [[Transverse wave]]s are those with vibrations perpendicular to the direction of the propagation of the wave; examples include waves on a string, and electromagnetic waves. [[Longitudinal wave]]s are those with vibrations parallel to the direction of the propagation of the wave; examples include most sound waves.

When an object bobs up and down on a ripple in a pond, it experiences an orbital trajectory because ripples are not simple transverse sinusoidal waves.

All waves have common behavior under a number of standard situations. All waves can undergo the following:
* [[Reflection (physics)|Reflection]] — change in wave direction after it strikes a reflective surface, causing the angle the wave makes with the reflective surface in relation to a normal line to the surface to equal the angle the reflected wave makes with the same normal line

* [[Rectilinear propagation]] — the movement of waves in a straight line, in the absence of any obstacles or change in media

=== Interference ===
{{Main|Interference (wave propagation)}}
Waves that encounter each other combine through [[superposition principle|superposition]] to create a new wave called an [[Interference (wave propagation)|interference pattern]]. Important interference patterns occur for waves that are in phase

===Reflection, absorption and transmission===
{{Main|Reflection|Absorption (acoustics)|Absorption (electromagnetic radiation)|Transmission|Transmission medium}}

The medium that carries a wave is called a ''transmission medium''. It 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 ''isotropic medium'' if its physical properties are the ''same'' in different directions

=== Refraction ===
{{Main|Refraction}}
[[File:Wave refraction.gif|thumb|right|200 px|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 it's speed. Mathematically, this means that the size of the [[phase velocity]] changes. Typically, refraction occurs when a wave passes from one [[Transmission medium|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]].

=== Diffraction ===
{{Main|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.

=== Polarization ===
{{Main|Polarization (waves)}}
[[File:Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg|thumb|left]]
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.

=== Dispersion ===
[[File:Light dispersion conceptual.gif|thumb|right|270 px]]
{{Main|Dispersion (optics)|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.<ref name=Newton>{{cite book
|last = Newton
|first = Isaac
|year=1704
|authorlink= Isaac Newton
|title= Opticks: Or, A treatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of Curvilinear Figures
|page= 118
|location=London
|chapter=Prop VII Theor V
|quote= All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, or compounded of these...
|volume= 1
|url= http://gallica.bnf.fr/ark:/12148/bpt6k3362k.image.f128.pagination

}}
</ref>

== Mechanical waves ==
{{Main|Mechanical wave}}

=== Waves on strings ===

{{Main|Vibrating string}}

The speed of a wave traveling along a [[vibrating string]] ('' v '') is directly proportional to the square root of the [[Tension (mechanics)|tension]] of the string ('' T '') over the [[linear mass density]] ('' μ ''):

:<math>
v=\sqrt{\frac{T}{\mu}}, \,
</math>

where the linear density ''μ'' is the mass per unit length of the string.

=== Acoustic waves ===
Acoustic or [[sound]] waves travel at speed given by

:<math>
v=\sqrt{\frac{B}{\rho_0}}, \,
</math>

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

=== Water waves ===
[[File:Shallow water wave.gif|thumb|]]
{{Main|Water waves}}
* [[ripple tank|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 wave]]s, which are perturbations that propagate through water

=== Seismic waves ===
{{Main|Seismic waves}}
[[Seismic wave]]s in [[earthquake]]s, of which there are three types, called S, P, and L

=== Shock waves ===
[[File:Transonico-en.svg|thimb|right|300 px|Formation of a shock wave by a plane.]]
{{Main|Shock wave}}
{{See also|Sonic boom|Cerenkov radiation}}

=== Other ===
* Waves of [[Traffic wave|traffic]], that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves<ref name=Lighthill>{{cite journal | author1=M. J. Lighthill | author1-link=James Lighthill | author2=G. B. Whitham | author2-link=Gerald B. Whitham | year=1955 | title=On kinematic waves. II. A theory of traffic flow on long crowded roads | journal=Proceedings of the Royal Society of London. Series A | volume=229 | pages=281–345 | ref=harv | postscript=. }} And: {{cite journal | doi=10.1287/opre.4.1.42 | author=P. I. Richards | year=1956 | title=Shockwaves on the highway | journal=Operations Research | volume=4 | pages=42–51 | ref=harv | postscript=. }}</ref>

== Electromagnetic waves ==
[[File:Onde electromagnétique.png|thumb|right|200 px]]
{{Main|Electromagnetic radiation|Electromagnetic spectrum}}
(infrared, micro, radio, visible, uv)

An electromagnetic wave consists of two waves that are oscillations of the [[electric field|electric]] and [[magnetic field|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 [[visible light|light]] is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as, [[infrared]], [[radio waves]], visible light, [[ultraviolet]], [[microwaves]] and [[X-rays]].

== Quantum mechanical waves ==
{{Main|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 function]]s 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.

[[File:Wave packet (dispersion).gif|thumb|A propagating wave packet; in general, the ''envelope'' of the wave packet moves at a different speed than the constituent waves.<ref name= Fromhold>{{cite book |title=Quantum Mechanics for Applied Physics and Engineering
|author=A. T. Fromhold |chapter=Wave packet solutions
|pages=59 ''ff''
|quote=(p. 61) …the individual waves move more slowly than the packet and therefore pass back through the packet as it advances
|url=http://books.google.com/?id=3SOwc6npkIwC&pg=PA59
|isbn=0486667413 |publisher=Courier Dover Publications
|year=1991
|edition=Reprint of Academic Press 1981
}}</ref>]]

=== de Broglie waves ===
{{Main|Wave packet|Matter wave}}

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

:<math>\lambda = \frac{h}{p},</math>

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 [[electron]]s in a [[cathode ray tube|CRT]] display have a de Broglie wavelength of about 10<sup>−13</sup> m.'''

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

:<math>\psi (\mathbf{r}, \ t=0) =A\ e^{i\mathbf{k \cdot r}} \ , </math>

where the wavelength is determined by the [[wave vector]] '''k''' as:

:<math> \lambda = \frac {2 \pi}{k} \ , </math>

and the momentum by:

:<math> \mathbf p = \hbar \mathbf{k} \ . </math>

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]],<ref name=Marton>

{{cite book |title=Advances in Electronics and Electron Physics |page=271 |url=http://books.google.com/?id=g5q6tZRwUu4C&pg=PA271 |isbn=0120146533 |year=1980 |publisher=Academic Press |volume=53 |editor=L. Marton & Claire Marton |author=Ming Chiang Li |chapter=Electron Interference}}

</ref> 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 function|Gaussian shape]] and is called a ''Gaussian wave packet''.<ref name=wavepacket>
See for example {{cite book |url=http://books.google.com/?id=7qCMUfwoQcAC&pg=PA60 |title=Quantum Mechanics |author=Walter Greiner, D. Allan Bromley |page=60 |isbn=3540674586 |edition=2 |year=2007 |publisher=Springer}} and {{cite book |title=Electronic basis of the strength of materials |author=John Joseph Gilman |url=http://books.google.com/?id=YWd7zHU0U7UC&pg=PA57 |page=57 |year=2003 |isbn=0521620058 |publisher=Cambridge University Press}},{{cite book |title=Principles of quantum mechanics |author= Donald D. Fitts |url=http://books.google.com/?id=8t4DiXKIvRgC&pg=PA17 |page =17 |isbn=0521658411 |publisher=Cambridge University Press |year=1999}}.
</ref> Gaussian wave packets also are used to analyze water waves.<ref name=Mei>

{{cite book |url=http://books.google.com/?id=WHMNEL-9lqkC&pg=PA47 |page=47 |author=Chiang C. Mei |title=The applied dynamics of ocean surface waves |isbn=9971507897 |year=1989 |edition=2nd |publisher=World Scientific}}

</ref>

For example, a Gaussian wavefunction ψ might take the form:<ref name= Bromley>

{{cite book |title=Quantum Mechanics |author= Walter Greiner, D. Allan Bromley |page=60 |url=http://books.google.com/?id=7qCMUfwoQcAC&pg=PA60 |edition=2nd |year=2007 |publisher=Springer |isbn=3540674586}}

</ref>

:<math> \psi(x,\ t=0) = A\ \exp \left( -\frac{x^2}{2\sigma^2} + i k_0 x \right) \ , </math>

at some initial time ''t'' = 0, where the central wavelength is related to the central wave vector ''k''<sub>0</sub> as λ<sub>0</sub> = 2π / ''k''<sub>0</sub>. It is well known from the theory of [[Fourier analysis]],<ref name= Brandt>

{{cite book |page=23 |url=http://books.google.com/?id=VM4GFlzHg34C&pg=PA23 |title=The picture book of quantum mechanics |author=Siegmund Brandt, Hans Dieter Dahmen |isbn=0387951415 |year=2001 |edition =3rd |publisher=Springer}}

</ref> 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.<ref name=Gaussian>

{{cite book |title=Modern mathematical methods for physicists and engineers |author= Cyrus D. Cantrell |page=677 |url=http://books.google.com/?id=QKsiFdOvcwsC&pg=PA677 |isbn=0521598273 |publisher=Cambridge University Press |year=2000}}

</ref> Given the Gaussian:

:<math>f(x) = e^{-x^2 / (2\sigma^2)} \ , </math>

the Fourier transform is:

:<math>\tilde{ f} (k) = \sigma e^{-\sigma^2 k^2 / 2} \ . </math>

The Gaussian in space therefore is made up of waves:

:<math>f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \ \tilde{f} (k) e^{ikx} \ dk \ ; </math>

that is, a number of waves of wavelengths λ such that ''k''λ = 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''.

== Gravitational waves ==
{{Main|Gravitational wave}}
{{See also|Gravitational wave detector}}

[[File:GravitationalWave CrossPolarization.gif|thumb|right|200 px|Animation showing the effect of a cross-polarized gravitational wave on a ring of [[test particles]]]]

Not to be confused with [[gravity waves]], gravitational waves are disturbances in the curvature of [[spacetime]], predicted by Einstein's theory of [[General Relativity]].

==WKB method==
{{Main|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.

== Notes ==

{{Reflist|2}}

== Bibliography ==

{{Expand|date=March 2009}}

== See also ==

<div style="-moz-column-count:3; column-count:3;">
* [[Audience wave]]
* [[Beat wave]]s
* [[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]]

</div>

== Sources ==

* Campbell, M. and Greated, C. (1987). ''The Musician’s Guide to Acoustics''. New York: Schirmer Books.
* {{cite book | first = A.P. | last = French | title = Vibrations and Waves (M.I.T. Introductory physics series) | publisher = Nelson Thornes | year = 1971 | isbn = 0-393-09936-9 | oclc = 163810889 }}
* {{Cite book | last=Hall | first=D. E. | year=1980 | title=Musical Acoustics: An Introduction | location=Belmont, California | publisher=Wadsworth Publishing Company | isbn=0534007589 | ref=harv | postscript=. }}.
* {{Cite document | last=Hunt | first=F. V. | origyear=1966 | title=Origins in Acoustics | location=New York | publisher=Acoustical Society of America Press | year=1992 | url=http://asa.aip.org/publications.html#pub17 | format={{Dead link|date=May 2010}} | ref=harv | postscript=. }}.
* {{Cite book | last1=Ostrovsky | first1=L. A. | last2=Potapov | first2=A. S. | year=1999 | title=Modulated Waves, Theory and Applications | location=Baltimore | publisher=The Johns Hopkins University Press | isbn=0801858704 | ref=harv | postscript=.}}.
* [http://www.acousticslab.org/papers/diss.htm Vassilakis, P.N. (2001)]. ''Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance''. Doctoral Dissertation. University of California, Los Angeles.

== External links ==

{{commons|Wave|Wave}}
{{Wiktionary}}
* [http://resonanceswavesandfields.blogspot.com/2007/08/true-waves.html Interactive Visual Representation of Waves]
* [http://www.scienceaid.co.uk/physics/waves/properties.html Science Aid: Wave properties — Concise guide aimed at teens]
* [http://www.phy.hk/wiki/englishhtm/Diffraction.htm Simulation of diffraction of water wave passing through a gap]
* [http://www.phy.hk/wiki/englishhtm/Interference.htm Simulation of interference of water waves]
* [http://www.phy.hk/wiki/englishhtm/Lwave.htm Simulation of longitudinal traveling wave]
* [http://www.phy.hk/wiki/englishhtm/StatWave.htm Simulation of stationary wave on a string]
* [http://www.phy.hk/wiki/englishhtm/TwaveA.htm Simulation of transverse traveling wave]
* [http://www.acoustics.salford.ac.uk/feschools/ Sounds Amazing — AS and A-Level learning resource for sound and waves]
* [http://www.lightandmatter.com/html_books/3vw/ch03/ch03.html Vibrations and Waves — an online textbook]
* [http://www.physics-lab.net/applets/mechanical-waves Simulation of waves on a string]
* [http://www.cbu.edu/~jvarrian/applets/waves1/lontra_g.htm-simulation of longitudinal and transverse mechanical wave]

{{Velocities of Waves}}

[[Category:Fundamental physics concepts]]
[[Category:Partial differential equations]]
[[Category:Waves| ]]


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[[ar:موجة]]
[[arc:ܓܠܠܐ]]
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[[bn:তরঙ্গ]]
[[be:Хваля]]
[[be-x-old:Хваля]]
[[bg:Вълна]]
[[ca:Ona]]
[[cs:Vlnění]]
[[cy:Ton]]
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[[et:Laine]]
[[el:Κύμα]]
[[es:Onda (física)]]
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[[fr:Onde]]
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[[it:Onda (fisica)]]
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[[ka:ტალღა]]
[[ht:Vag]]
[[ku:Pêl]]
[[lv:Vilnis]]
[[lt:Banga]]
[[jbo:boxna]]
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[[mr:तरंग]]
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[[new:अलै (सन् २००३या संकिपा)]]
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[[ru:Волна]]
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Revision as of 12:19, 28 September 2010

This article is about waves in the most general scientific sense. For waves in the ocean, see Wind wave. For other uses of wave or waves, see Wave (disambiguation).
Amplitude modulation and frequency modulation of a "Carrier wave" by a signal

In mathematics and science, a wave is a disturbance that travels through space and time, usually by the transfer of energy. Waves are described by a wave equation that can take on many forms depending on the type of wave. A mechanical wave is a wave that propagates through a medium 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 actually traveling with the wave.


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.Cite error: A <ref> tag is missing the closing </ref> (see the help page).

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:[1]

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

}}</ref>  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.[2]

The sine function is periodic.

The rest was wrong

  1. ^ 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.
  2. ^ 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 (ed.). Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering. World Scientific. p. 134. ISBN 9810242719.{{cite book}}: CS1 maint: multiple names: editors list (link)
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