Wave
In physics, a wave is an oscillation accompanied by a transfer of energy. Frequency refers to the addition of time. Wave motion transfers energy from one point to another, which displace particles of the transmission medium–that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations (of a physical quantity), around almost fixed locations.
A wave is a disturbance that transfers energy through matter or space. There are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed. Restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When the molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave.
The second main type, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields originally generated by charged particles, and can therefore travel through a vacuum. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, Xrays and gamma rays.
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. Further, the behavior of particles in quantum mechanics are described by waves. In addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields.
A wave can be transverse, where a disturbance creates oscillations that are perpendicular to the propagation of energy transfer, or longitudinal: the oscillations are parallel to the direction of energy propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse in free space.
Contents
 1 General features
 2 Mathematical description of onedimensional waves
 3 Sinusoidal waves
 4 Plane waves
 5 Standing waves
 6 Physical properties
 7 Mechanical waves
 8 Electromagnetic waves
 9 Quantum mechanical waves
 10 Gravity waves
 11 Gravitational waves
 12 WKB method
 13 See also
 14 References
 15 Sources
 16 External links
General features
A single, allencompassing definition for the term wave is not straightforward. A vibration can be defined as a backandforth motion around a reference value. However, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon as a wave results in a blurred line.
The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a 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 motion is problematic for a standing wave (for example, a wave on a string), where energy is moving in both directions equally, or for electromagnetic (e.g., light) waves in a vacuum, where the concept of medium does not apply and interaction with a target is the key to wave detection and practical applications. There are water waves on the ocean surface; gamma waves and light waves emitted by the Sun; microwaves used in microwave ovens and in radar equipment; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsets and living creatures (as voices), to mention only a few wave phenomena.
It may appear that the description of waves is closely related to their physical origin for each 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 transfer caused by vibration. 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; and so on....
Other properties, however, although usually described in terms of origin, 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 of their physical origin.^{[1]} For example, based on the mechanical origin of acoustic waves, a moving disturbance in space–time can exist 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 repeatedly .
Mathematical description of onedimensional waves
Wave equation
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
 in the direction in space. E.g., let the positive direction be to the right, and the negative direction be to the left.
 with constant amplitude
 with constant velocity , where 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 twodimensional functions
 (waveform traveling to the right)
 (waveform traveling to the left)
or, more generally, by d'Alembert's formula:^{[3]}
representing two component waveforms and traveling through the medium in opposite directions. A generalized representation of this wave can be obtained^{[4]} as the partial differential equation
General solutions are based upon Duhamel's principle.^{[5]}
Wave forms
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 xdirection at velocity v (and G will propagate at the same speed in the negative xdirection).^{[6]}
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.^{[7]}
Amplitude and 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:^{[8]}^{[9]}^{[10]}
where is the amplitude envelope of the wave, is the wavenumber and is the phase. If the group velocity (see below) is wavelengthindependent, this equation can be simplified as:^{[11]}
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.^{[11]}^{[12]}
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.
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 xdirection with a constant phase velocity .^{[13]}
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(k_{1}) (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 k_{1} = k. In exponential form:
with A_{o} the magnitude of A. For example, a common choice for A_{o} is a Gaussian wave packet:^{[14]}
where σ determines the spread of k_{1}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_{1}), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.^{[13]} 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.^{[15]}) The condition for φ to vary slowly is that its rate of change with k_{1} be small; this rate of variation is:^{[13]}
where the evaluation is made at k_{1} = k because A(k_{1}) 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.^{[13]} Because the group velocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized.^{[16]} 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.
Sinusoidal waves
This section duplicates the scope of other sections, specifically, Sinusoidal wave and Frequency. (July 2015)

Mathematically, the most basic wave is the (spatially) onedimensional sine wave (or harmonic wave or sinusoid) with an amplitude described by the equation:
where
 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.
 is the space coordinate
 is the time coordinate
 is the wavenumber
 is the angular frequency
 is the phase constant.
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 is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber , 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 is the time for one complete cycle of an oscillation of a wave. The frequency is the number of periods per unit time (per second) and is typically measured in hertz denoted as Hz. These are related by:
In other words, the frequency and period of a wave are reciprocals.
The angular frequency represents the frequency in radians per second. It is related to the frequency or period by
The wavelength of a sinusoidal waveform traveling at constant speed is given by:^{[17]}
where is called the phase speed (magnitude of the phase velocity) of the wave and 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.^{[18]}
Although arbitrary wave shapes will propagate unchanged in lossless linear timeinvariant 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.^{[19]} 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.^{[20]} The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.^{[21]}^{[22]}
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.^{[23]}^{[24]} 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.^{[25]} When a medium is nonlinear, the response to complex waves cannot be determined from a sinewave decomposition.
Plane waves
This section should include a summary of Plane wave. See Wikipedia:Summary style for information on how to incorporate it into this article's main text. (July 2015)

Standing waves
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 counterpropagating 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 counterpropagating 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 counterpropagating waves enhance each other maximally. There is no net propagation of energy over time.

Onedimensional standing waves; the fundamental mode and the first 5 overtones.

A twodimensional 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.
Physical properties
Waves exhibit common behaviors under a number of standard situations, e. g.
Transmission and media
Waves normally move 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
Absorption
Absorption of waves means, if a kind of wave strikes a matter, it will be absorbed by the matter. When a wave with that same natural frequency impinges upon an atom, then the electrons of that atom will be set into vibrational motion. If a wave of a given frequency strikes a material with electrons having the same vibrational frequencies, then those electrons will absorb the energy of the wave and transform it into vibrational motion.
Reflection
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.
Interference
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.
Refraction
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.
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
The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter.
Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.
Dispersion
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.^{[26]}
Mechanical waves
Waves on strings
The speed of a transverse 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.
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).
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.
Seismic waves
Shock waves
Other
 Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves^{[27]}
 Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.
Electromagnetic waves
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, Xrays, and Gamma rays.
Quantum mechanical waves
Schrödinger equation
The Schrödinger equation describes the wavelike 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.
Dirac equation
The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin½ particles.
de Broglie waves
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 kdirection is expressed by the wave function as follows:
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,^{[29]} 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.^{[30]} Gaussian wave packets also are used to analyze water waves.^{[31]}
For example, a Gaussian wavefunction ψ might take the form:^{[32]}
at some initial time t = 0, where the central wavelength is related to the central wave vector k_{0} as λ_{0} = 2π / k_{0}. It is well known from the theory of Fourier analysis,^{[33]} 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.^{[34]} 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 xaxis, 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.
Gravity waves
Gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. A ripple on a pond is one example.
Gravitational waves
Gravitational waves also travel through space. The first observation of gravitational waves was announced on 11 February 2016.^{[35]} Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.
WKB method
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.
See also
Waves in general
 Wave equation, general
 Wave propagation, any of the ways in which waves travel
 Interference (wave propagation), a phenomenon in which two waves superpose to form a resultant wave
 Mechanical wave, in media transmission
 Wave Motion (journal), a scientific journal
 Wavefront, an advancing surface of wave propagation
Parameters
 Phase (waves), offset or angle of a sinusoidal wave function at its origin
 Standing wave ratio, in telecommunications
 Wavelength
 Wavenumber
 Wave period
Waveforms
 Creeping wave, a wave diffracted around a sphere
 Evanescent wave
 Longitudinal wave
 Periodic travelling wave
 Sine wave
 Square wave
 Standing wave
 Transverse wave
Electromagnetic waves
 Electromagnetic wave
 Electromagnetic wave equation, describes electromagnetic wave propagation
 EarthIonosphere waveguide, in radio transmission
 Microwave, a form of electromagnetic radiation
In fluids
 Airy wave theory, in fluid dynamics
 Capillary wave, in fluid dynamics
 Cnoidal wave, in fluid dynamics
 Edge wave, a surface gravity wave fixed by refraction against a rigid boundary
 Faraday wave, a type of wave in liquids
 Gravity wave, in fluid dynamics
 Sound wave, a wave of sound through a medium such as air or water
 Shock wave, in aerodynamics
 Internal wave, a wave within a fluid medium
 Tidal wave, a scientifically incorrect name for a tsunami
 Tollmien–Schlichting wave, in fluid dynamics
In quantum mechanics
 Bloch wave
 Matter wave
 Pilot wave, in Bohmian mechanics
 Wave function
 Wave packet
 Waveparticle duality
In relativity
 Gravitational wave, in relativity theory
 Relativistic wave equations, wave equations that consider special relativity
 ppwave spacetime, a set of exact solutions to Einstein's field equation
Other specific types of waves
 Alfvén wave, in particle science
 Atmospheric wave, a periodic disturbance in the fields of atmospheric variables
 Fir wave, a forest configuration
 Lamb waves, in solid materials
 Rayleigh waves, surface acoustic waves that travel on solids
 Spin wave, in magnetism
 Spindensity wave, in solid materials
 Trojan wave packet, in particle science
 Waves in plasmas, in particle science
Related topics
 Beat (acoustics)
 Cymatics
 Doppler effect
 Envelope detector
 Group velocity
 Harmonic
 Index of wave articles
 Inertial wave
 List of waves named after people
 Multiplicative calculus
 Phase velocity
 Reaction–diffusion system
 Resonance
 Ripple tank
 Rogue wave
 Shallow water equations
 Shive wave machine
 Sound
 Standing wave
 Transmission medium
 Wave turbulence
 Wind wave
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.
 ^ For an example derivation, see the steps leading up to eq. (17) in Francis Redfern. "Kinematic Derivation of the Wave Equation". Physics Journal.
 ^ 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). FEWcycle Laser Dynamics and Carrierenvelope 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} ChinLin Chen (2006). "§13.7.3 Pulse envelope in nondispersive media". Foundations for guidedwave optics. Wiley. p. 363. ISBN 0471756873.
 ^ Stefano Longhi; Davide Janner (2008). "Localization and Wannier wave packets in photonic crystals". In Hugo E. HernándezFigueroa; Michel ZamboniRached; Erasmo Recami. Localized Waves. WileyInterscience. p. 329. ISBN 0470108851.
 ^ ^{a} ^{b} ^{c} ^{d} Albert Messiah (1999). Quantum Mechanics (Reprint of twovolume 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 manyparticle 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 & 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 solidstate 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. WileyBlackwell. 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 & 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. Bibcode:1955RSPSA.229..281L. doi:10.1098/rspa.1955.0088. 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.
 ^ "Gravitational waves detected for 1st time, 'opens a brand new window on the universe'". CBC. 11 February 2016.
Sources
 Fleisch, D.; Kinnaman, L. (2015). A student's guide to waves. Cambridge, UK: Cambridge University Press. ISBN 9781107643260.
 Campbell, Murray; Greated, Clive (2001). The musician's guide to acoustics (Repr. ed.). Oxford: Oxford University Press. ISBN 9780198165057.
 French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes. ISBN 0393099369. OCLC 163810889.
 Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont, California: Wadsworth Publishing Company. ISBN 0534007589..
 Hunt, Frederick Vinton (1978). Origins in acoustics. Woodbury, NY: Published for the Acoustical Society of America through the American Institute of Physics. ISBN 9780300022209.
 Ostrovsky, L. A.; Potapov, A. S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns Hopkins University Press. ISBN 0801858704..
 Griffiths, G.; Schiesser, W. E. (2010). Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple. Academic Press. ISBN 9780123846532.
External links
 Interactive Visual Representation of Waves
 Linear and nonlinear waves
 Science Aid: Wave properties—Concise guide aimed at teens
 Easy JavaScript Simulation Model of One Dimensional Wave Interference
 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 ALevel learning resource for sound and waves
 an online textbook, ch. 1920
 Simulation of waves on a string