In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

The waves most commonly studied in physics are mechanical and electromagnetic. A mechanical wave is a local deformation (strain) in some physical medium that propagates from particle to particle by creating local stresses that cause strain in neighboring particles too. For example, sound waves in air are variations of the local pressure that propagate by collisions between gas molecules. Other examples of mechanical waves are seismic waves, gravity waves, vortices, and shock waves. An electromagnetic wave consists of a combination of variable electric and magnetic fields, that propagates through space according to Maxwell's equations. Electromagnetic waves can travel through suitable dielectric media or through vacuum; examples include radio waves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.

Other types of waves include gravitational waves, which are disturbances in a gravitational field that propagate according to general relativity; heat diffusion waves; plasma waves, that combine mechanical deformations and electromagnetic fields; reaction-diffusion waves, such as in the Belousov–Zhabotinsky reaction; and many more.

Mechanical and electromagnetic waves may often seem to travel through space; but, while they can carry energy,[1] momentum, and information through matter or empty space, they may do that without transferring any mass. In mathematics and electronics waves are studied as signals.[2] On the other hand, some waves do not appear to move at all, like standing waves (which are fundamental to music) and hydraulic jumps. Some, like the probability waves of quantum mechanics, may be completely static.

A physical wave is almost always confined to some finite region of space, called its domain. For example, the seismic waves generated by earthquakes are significant only in the interior and surface of the planet, so they can be ignored outside it. However, waves with infinite domain, that extend over the whole space, are commonly studied in mathematics, and are very valuable tools for understanding physical waves in finite domains.

A plane wave seems to travel in a definite direction, and has constant value over any plane perpendicular to that direction. Mathematically, the simplest waves are the sinusoidal ones. Complicated waves can often be described as the sum of many sinusoidal plane waves. A plane wave can be transverse, if its effect at each point is described by a vector that is perpendicular to the direction of propagation or energy transfer; or longitudinal, if the describing vectors are parallel to the direction of energy propagation. While mechanical waves can be both transverse and longitudinal, electromagnetic waves are transverse in free space.

2006-01-14 Surface waves
Surface waves in water showing water ripples
Rectified waves
Different types of wave with varying rectifications

Mathematical Description

Single waves

A wave can be described just like a field, namely as a function where is a position and is a time.

The value of is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space . However, in many cases one can ignore one or two dimensions, and let be a point of the Cartesian plane . This is the case, for example, when studying vibrations of a drum skin). One may even restrict to a point of the Cartesian line — that is, the set of real numbers. This is the case, for example, when studying vibrations in a violin string or recorder. The time , on the other hand, is always assumed to be a scalar; that is, a real number.

The value of can be any physical quantity of interest assigned to the point that may vary with time. For example, if represents the vibrations inside an elastic solid, the value of is usually a vector that gives the current displacement from of the material particles that would be at the point in the absence of vibration. For an electromagnetic wave, the value of can be the electric field vector , or the magnetic field vector , or any related quantity, such as the Poynting vector . In fluid dynamics, the value of could be the velocity vector of the fluid at the point , or any scalar property like pressure, temperature, or density. In a chemical reaction, could be the concentration of some substance in the neighborhood of point of the reaction medium.

For any dimension (1, 2, or 3), the wave's domain is then a subset of , such that the function value is defined for any point in . For example, when describing the motion of a drum skin, one can consider to be a disk (circle) on the plane with center at the origin , and let be the vertical displacement of the skin at the point of and at time .

Wave families

Sometimes one is interested in a single specific wave, like how the Earth vibrated after the 1929 Murchison earthquake. More often, however, one needs to understand large set of possible waves; like all the ways that a drum skin can vibrate after being struck once with a drum stick, or all the possible radar echos one could get from an airplane that may be approaching an airport.

In some of those situations, one may describe such a family of waves by a function that depends on certain parameters , besides and . Then one can obtain different waves — that is, different functions of and — by choosing different values for those parameters.

Half-open pipe wave
Sound pressure wave in an half-open pipe playing the 7th harmonic of the fundamental (n = 4).

For example, the sound pressure inside a recorder that is playing a "pure" note is typically a standing wave, that can be written as

The parameter defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); is the speed of sound; is the length of the bore; and is a positive integer (1,2,3,...) that specifies the number of nodes in the standing wave. (The position should be masured from the mouthpiece, and the time from any moment at which the pressure at the mouthpiece is maximum. The quantity is the wavelength of the emitted note, and is its frequency.) Many general properties of these waves can be inferred from this general equation, without choosing specific values for the parameters.

As another example, it may be that the vibrations of a drum skin after a single strike depend only on the distance from the center of the skin to the strike point, and on the strength of the strike. Then the vibration for all possible strikes can be described by a function .

Sometimes the family of waves of interest has infinitely many parameters. For example, one may want to describe what happens to the temperature in a metal bar when it is initially heated at various temperatures at different points along its length, and then allowed to cool by itself in vacuum. In that case, instead of a scalar or vector, the parameter would have to be a function such that is the initial temperature at each point of the bar. Then the temperatures at later times can be expressed by a function that depends on the function (that is, a functional operator), so that the temperature at a later time is

Differential wave equations

Another way to describe and study a family of waves is to give a mathematical equation that, instead of explicitly giving the value of , only constrains how those values can change with time. Then the family of waves in question consists of all functions that satisfy those constraints — that is, all solutions of the equation.

This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve. For example, if is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation

where is the heat that is being generated per unit of volume and time in the neighborhood of at time (for example, by chemical reactions happening there); are the Cartesian coordinates of the point ; is the (first) derivative of with respect to ; and is the second derivative of relative to . (The simbol "" is meant to signify that, in the derivative with respect to some variable, all other variables must be considered fixed.)

This equation can be derived from the laws of physics that govern the diffusion of heat in solid media. For that reason, it is called the heat equation in mathematics, even though it applies to many other physical quantities besides temperatures.

For another example, we can describe all possible sounds echoing within a container of gas by a function that gives the pressure at a point and time within that container. If the gas was initially at uniform temperature and composition, the evolution of is constrained by the formula

Here is some extra compression force that is being applied to the gas near by some external process, such as a loudspeaker or piston] right next to .

This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in a homogeneous isotropic non-conducting solid. Note that this equation differs from that of heat flow only in that the left-hand side is , the second derivative of with respect to time, rather than the first derivative . Yet this small change makes a huge difference on the set of solutions . This differential equation is called "the" wave equation in mathematics, even though it describes only one very special kind of waves.

Wave in elastic medium

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

Nonsinusoidal wavelength
Wavelength λ, can be measured between any two corresponding points on a waveform
Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t)
  • in the direction in space. For example, let the positive direction be to the right, and the negative direction be to the left.
  • with constant amplitude
  • with constant velocity , where is
  • with constant waveform, or shape

This wave can then be described by the two-dimensional functions

(waveform traveling to the right)
(waveform traveling to the left)

or, more generally, by d'Alembert's formula:[5]

representing two component waveforms and traveling through the medium in opposite directions. A generalized representation of this wave can be obtained[6] as the partial differential equation

General solutions are based upon Duhamel's principle.[7]

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).[8]

In the case of a periodic function F with period λ, that is, F(x + λvt) = 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(xv(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.[9]

Amplitude and modulation

Amplitude modulation can be achieved through f(x,t) = 1.00*sin(2*pi/0.10*(x-1.00*t)) and g(x,t) = 1.00*sin(2*pi/0.11*(x-1.00*t))only the resultant is visible to improve clarity of waveform.
Wave packet
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.

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:[10][11][12]

where is the amplitude envelope of the wave, is the wavenumber and is the phase. If the group velocity (see below) is wavelength-independent, this equation can be simplified as:[13]

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.[13][14]

Phase velocity and group velocity

Wave group
The red square moves with the phase velocity, while the green circles propagate with the group velocity

There are two velocities that are associated with waves, the phase velocity and the group velocity.

Phase velocity is the rate at which the phase of the wave propagates in space: any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as

Wave opposite-group-phase-velocity
A wave with the group and phase velocities going in different directions

Group velocity is a property of waves that have a defined envelope, measuring propagation through space (that is, phase velocity) of the overall shape of the waves' amplitudes – modulation or envelope of the wave.

Sine waves

Simple harmonic motion animation
Sinusoidal waves correspond to simple harmonic motion.

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


  • 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 (for example, a wave on a string) have an amplitude expressed as a distance (for example, meters), longitudinal mechanical waves (for example, sound waves) use units of pressure (for example, pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (for example, 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:[15]

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.[16]

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.[17] 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.[18] The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.[19][20]

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. 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.[21][22] 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.[23] When a medium is nonlinear, then the response to complex waves cannot be determined from a sine-wave decomposition.

Standing waves

Standing wave
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.

Standing waves on a string

One-dimensional standing waves; the fundamental mode and the first 5 overtones.

Drum vibration mode01

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

Drum vibration mode21

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

Physical properties

Light dispersion of a mercury-vapor lamp with a flint glass prism IPNr°0125
Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism

Waves exhibit common behaviors under a number of standard situations, for example:

Transmission and media

Waves normally move in a straight line (that is, 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 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.


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.


Wave 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.


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.


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.


Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View

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.


Light dispersion conceptual waves
Schematic of light being dispersed by a prism. Click to see animation.

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.[24]

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

Shallow water wave
  • 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

Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy.

Shock waves

Formation of a shock wave by a plane.

A shock wave is a type of propagating disturbance. When a wave moves faster than the local speed of sound in a fluid, it is a shock wave. Like an ordinary wave, a shock wave carries energy and can propagate through a medium; however, it is characterized by an abrupt, nearly discontinuous change in pressure, temperature and density of the medium.[25]


  • 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.

Electromagnetic waves

Onde electromagnétique

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, X-rays, and Gamma rays.

Quantum mechanical waves

Schrödinger equation

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.

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 spin-½ particles.

Wave packet (dispersion)
A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.[27]

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 k-direction 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,[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.

GravitationalWave CrossPolarization
Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

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.[34] Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

See also

Waves in general



Electromagnetic waves

In fluids

In quantum mechanics

In relativity

Other specific types of waves

Related topics


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  26. ^ 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 (1178): 281–345. Bibcode:1955RSPSA.229..281L. CiteSeerX 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.
  27. ^ 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 978-0-486-66741-6. (p. 61) ...the individual waves move more slowly than the packet and therefore pass back through the packet as it advances
  28. ^ Ming Chiang Li (1980). "Electron Interference". In L. Marton; Claire Marton (eds.). Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 978-0-12-014653-6.
  29. ^ See for example Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics (2 ed.). Springer. p. 60. ISBN 978-3-540-67458-0. and John Joseph Gilman (2003). Electronic basis of the strength of materials. Cambridge University Press. p. 57. ISBN 978-0-521-62005-5.,Donald D. Fitts (1999). Principles of quantum mechanics. Cambridge University Press. p. 17. ISBN 978-0-521-65841-6..
  30. ^ Chiang C. Mei (1989). The applied dynamics of ocean surface waves (2nd ed.). World Scientific. p. 47. ISBN 978-9971-5-0789-3.
  31. ^ Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics (2nd ed.). Springer. p. 60. ISBN 978-3-540-67458-0.
  32. ^ Siegmund Brandt; Hans Dieter Dahmen (2001). The picture book of quantum mechanics (3rd ed.). Springer. p. 23. ISBN 978-0-387-95141-6.
  33. ^ Cyrus D. Cantrell (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 677. ISBN 978-0-521-59827-9.
  34. ^ "Gravitational waves detected for 1st time, 'opens a brand new window on the universe'". CBC. 11 February 2016.


  • Fleisch, D.; Kinnaman, L. (2015). A student's guide to waves. Cambridge: Cambridge University Press. Bibcode:2015sgw..book.....F. ISBN 978-1107643260.
  • Campbell, Murray; Greated, Clive (2001). The musician's guide to acoustics (Repr. ed.). Oxford: Oxford University Press. ISBN 978-0198165057.
  • French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes. ISBN 978-0-393-09936-2. OCLC 163810889.
  • Hall, D.E. (1980). Musical Acoustics: An Introduction. Belmont, CA: Wadsworth Publishing Company. ISBN 978-0-534-00758-4..
  • Hunt, Frederick Vinton (1978). Origins in acoustics. Woodbury, NY: Published for the Acoustical Society of America through the American Institute of Physics. ISBN 978-0300022209.
  • Ostrovsky, L.A.; Potapov, A.S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns Hopkins University Press. ISBN 978-0-8018-5870-3..
  • 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

Doppler effect

The Doppler effect (or the Doppler shift) is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.

A common example of Doppler shift is the change of pitch heard when a vehicle sounding a horn approaches and recedes from an observer. Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing by, and lower during the recession.The reason for the Doppler effect is that when the source of the waves is moving towards the observer, each successive wave crest is emitted from a position closer to the observer than the crest of the previous wave. Therefore, each wave takes slightly less time to reach the observer than the previous wave. Hence, the time between the arrivals of successive wave crests at the observer is reduced, causing an increase in the frequency. While they are traveling, the distance between successive wave fronts is reduced, so the waves "bunch together". Conversely, if the source of waves is moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the frequency. The distance between successive wave fronts is then increased, so the waves "spread out".

For waves that propagate in a medium, such as sound waves, the velocity of the observer and of the source are relative to the medium in which the waves are transmitted. The total Doppler effect may therefore result from motion of the source, motion of the observer, or motion of the medium. Each of these effects is analyzed separately. For waves which do not require a medium, such as light or gravity in general relativity, only the relative difference in velocity between the observer and the source needs to be considered.

Electromagnetic radiation

In physics, electromagnetic radiation (EM radiation or EMR) refers to the waves (or their quanta, photons) of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.Classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric and magnetic fields that propagate at the speed of light, which, in a vacuum, is commonly denoted c. In homogeneous, isotropic media, the oscillations of the two fields are perpendicular to each other and perpendicular to the direction of energy and wave propagation, forming a transverse wave. The wavefront of electromagnetic waves emitted from a point source (such as a light bulb) is a sphere. The position of an electromagnetic wave within the electromagnetic spectrum can be characterized by either its frequency of oscillation or its wavelength. Electromagnetic waves of different frequency are called by different names since they have different sources and effects on matter. In order of increasing frequency and decreasing wavelength these are: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.Electromagnetic waves are emitted by electrically charged particles undergoing acceleration, and these waves can subsequently interact with other charged particles, exerting force on them. EM waves carry energy, momentum and angular momentum away from their source particle and can impart those quantities to matter with which they interact. Electromagnetic radiation is associated with those EM waves that are free to propagate themselves ("radiate") without the continuing influence of the moving charges that produced them, because they have achieved sufficient distance from those charges. Thus, EMR is sometimes referred to as the far field. In this language, the near field refers to EM fields near the charges and current that directly produced them specifically, electromagnetic induction and electrostatic induction phenomena.

In quantum mechanics, an alternate way of viewing EMR is that it consists of photons, uncharged elementary particles with zero rest mass which are the quanta of the electromagnetic force, responsible for all electromagnetic interactions. Quantum electrodynamics is the theory of how EMR interacts with matter on an atomic level. Quantum effects provide additional sources of EMR, such as the transition of electrons to lower energy levels in an atom and black-body radiation. The energy of an individual photon is quantized and is greater for photons of higher frequency. This relationship is given by Planck's equation E = hν, where E is the energy per photon, ν is the frequency of the photon, and h is Planck's constant. A single gamma ray photon, for example, might carry ~100,000 times the energy of a single photon of visible light.

The effects of EMR upon chemical compounds and biological organisms depend both upon the radiation's power and its frequency. EMR of visible or lower frequencies (i.e., visible light, infrared, microwaves, and radio waves) is called non-ionizing radiation, because its photons do not individually have enough energy to ionize atoms or molecules or break chemical bonds. The effects of these radiations on chemical systems and living tissue are caused primarily by heating effects from the combined energy transfer of many photons. In contrast, high frequency ultraviolet, X-rays and gamma rays are called ionizing radiation, since individual photons of such high frequency have enough energy to ionize molecules or break chemical bonds. These radiations have the ability to cause chemical reactions and damage living cells beyond that resulting from simple heating, and can be a health hazard.


Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second (60 seconds divided by 120 beats). Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.


Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to visible light, which is the visible spectrum that is visible to the human eye and is responsible for the sense of sight. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), or 4.00 × 10−7 to 7.00 × 10−7 m, between the infrared (with longer wavelengths) and the ultraviolet (with shorter wavelengths). This wavelength means a frequency range of roughly 430–750 terahertz (THz).

The main source of light on Earth is the Sun. Sunlight provides the energy that green plants use to create sugars mostly in the form of starches, which release energy into the living things that digest them. This process of photosynthesis provides virtually all the energy used by living things. Historically, another important source of light for humans has been fire, from ancient campfires to modern kerosene lamps. With the development of electric lights and power systems, electric lighting has effectively replaced firelight. Some species of animals generate their own light, a process called bioluminescence. For example, fireflies use light to locate mates, and vampire squids use it to hide themselves from prey.

The primary properties of visible light are intensity, propagation direction, frequency or wavelength spectrum, and polarization, while its speed in a vacuum, 299,792,458 metres per second, is one of the fundamental constants of nature. Visible light, as with all types of electromagnetic radiation (EMR), is experimentally found to always move at this speed in a vacuum.In physics, the term light sometimes refers to electromagnetic radiation of any wavelength, whether visible or not. In this sense, gamma rays, X-rays, microwaves and radio waves are also light. Like all types of EM radiation, visible light propagates as waves. However, the energy imparted by the waves is absorbed at single locations the way particles are absorbed. The absorbed energy of the EM waves is called a photon, and represents the quanta of light. When a wave of light is transformed and absorbed as a photon, the energy of the wave instantly collapses to a single location, and this location is where the photon "arrives." This is what is called the wave function collapse. This dual wave-like and particle-like nature of light is known as the wave–particle duality. The study of light, known as optics, is an important research area in modern physics.

New wave music

New wave is a genre of pop-oriented rock music popular in the late 1970s and the 1980s with ties to mid-1970s punk rock. New wave moved away from blues and rock and roll sounds to create rock music (early new wave) or pop music (later) that incorporated disco, mod, and electronic music. Initially new wave was similar to punk rock, before becoming a distinct genre. It subsequently engendered subgenres and fusions, including synth-pop.New wave differs from other movements with ties to first-wave punk as it displays characteristics common to pop music, rather than the more "artsy" post-punk. Although it incorporates much of the original punk rock sound and ethos, new wave exhibits greater complexity in both music and lyrics. Common characteristics of new wave music include the use of synthesizers and electronic productions, and a distinctive visual style featured in music videos and fashion.New wave has been called one of the definitive genres of the 1980s, after it was promoted heavily by MTV (the Buggles' "Video Killed the Radio Star" music video was broadcast as the first music video to promote the channel's launch). The popularity of several new wave artists is often attributed to their exposure on the channel. In the mid-1980s, differences between new wave and other music genres began to blur. New wave has enjoyed resurgences since the 1990s, after a rising "nostalgia" for several new wave-influenced artists. Subsequently, the genre influenced other genres. During the 2000s, a number of acts, such as the Strokes, Interpol, Franz Ferdinand and The Killers explored new wave and post-punk influences. These acts were sometimes labeled "new wave of new wave".


The photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light, and the force carrier for the electromagnetic force (even when static via virtual particles). Invariant mass of the photon is zero; it always moves at the speed of light within a vacuum.

Like all elementary particles, photons are currently best explained by quantum mechanics and exhibit wave–particle duality, exhibiting properties of both waves and particles. For example, a single photon may be refracted by a lens and exhibit wave interference with itself, and it can behave as a particle with definite and finite measurable position or momentum, though not both at the same time as per Heisenberg's uncertainty principle. The photon's wave and quantum qualities are two observable aspects of a single phenomenon—they cannot be described by any mechanical model; a representation of this dual property of light that assumes certain points on the wavefront to be the seat of the energy is not possible. The quanta in a light wave are not spatially localized.

The modern concept of the photon was developed gradually by Albert Einstein in the early 20th century to explain experimental observations that did not fit the classical wave model of light. The benefit of the photon model is that it accounts for the frequency dependence of light's energy, and explains the ability of matter and electromagnetic radiation to be in thermal equilibrium. The photon model accounts for anomalous observations, including the properties of black-body radiation, that others (notably Max Planck) had tried to explain using semiclassical models. In that model, light is described by Maxwell's equations, but material objects emit and absorb light in quantized amounts (i.e., they change energy only by certain particular discrete amounts). Although these semiclassical models contributed to the development of quantum mechanics, many further experiments beginning with the phenomenon of Compton scattering of single photons by electrons, validated Einstein's hypothesis that light itself is quantized. In December 1926, American physical chemist Gilbert N. Lewis coined the widely-adopted name "photon" for these particles in a letter to Nature. After Arthur H. Compton won the Nobel Prize in 1927 for his scattering studies, most scientists accepted that light quanta have an independent existence, and the term "photon" was accepted.

In the Standard Model of particle physics, photons and other elementary particles are described as a necessary consequence of physical laws having a certain symmetry at every point in spacetime. The intrinsic properties of particles, such as charge, mass, and spin, are determined by this gauge symmetry. The photon concept has led to momentous advances in experimental and theoretical physics, including lasers, Bose–Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics. It has been applied to photochemistry, high-resolution microscopy, and measurements of molecular distances. Recently, photons have been studied as elements of quantum computers, and for applications in optical imaging and optical communication such as quantum cryptography.

Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.Classical physics, the physics existing before quantum mechanics, describes nature at ordinary (macroscopic) scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.

Quantum mechanics differs from classical physics in that energy, momentum, angular momentum and other quantities of a bound system are restricted to discrete values (quantization); objects have characteristics of both particles and waves (wave-particle duality); and there are limits to the precision with which quantities can be measured (uncertainty principle).Quantum mechanics gradually arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle.

Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.

Radio wave

Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. Radio waves have frequencies as high as 300 gigahertz (GHz) to as low as 30 hertz (Hz). At 300 GHz, the corresponding wavelength is 1 mm, and at 30 Hz is 10,000 km. Like all other electromagnetic waves, radio waves travel at the speed of light. They are generated by electric charges undergoing acceleration, such as time varying electric currents. Naturally occurring radio waves are emitted by lightning and astronomical objects.

Radio waves are generated artificially by transmitters and received by radio receivers, using antennas. Radio waves are very widely used in modern technology for fixed and mobile radio communication, broadcasting, radar and other navigation systems, communications satellites, wireless computer networks and many other applications. Different frequencies of radio waves have different propagation characteristics in the Earth's atmosphere; long waves can diffract around obstacles like mountains and follow the contour of the earth (ground waves), shorter waves can reflect off the ionosphere and return to earth beyond the horizon (skywaves), while much shorter wavelengths bend or diffract very little and travel on a line of sight, so their propagation distances are limited to the visual horizon.

To prevent interference between different users, the artificial generation and use of radio waves is strictly regulated by law, coordinated by an international body called the International Telecommunications Union (ITU), which defines radio waves as "electromagnetic waves of frequencies arbitrarily lower than 3 000 GHz, propagated in space without artificial guide". The radio spectrum is divided into a number of radio bands on the basis of frequency, allocated to different uses.


In physics refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

For light, refraction follows Snell's law, which states that, for a given pair of media, the ratio of the sines of the angle of incidence θ1 and angle of refraction θ2 is equal to the ratio of phase velocities (v1 / v2) in the two media, or equivalently, to the indices of refraction (n2 / n1) of the two media.

Optical prisms and lenses use refraction to redirect light, as does the human eye. The refractive index of materials varies with the wavelength of light, and thus the angle of the refraction also varies correspondingly. This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral colors.

Schrödinger equation

The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who derived the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In classical mechanics, Newton's second law (F = ma) is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force on the system. Those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation.

The concept of a wave function is a fundamental postulate of quantum mechanics; the wave function defines the state of the system at each spatial position, and time. Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian. This derivation is explained below.

In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe. Schrödinger's equation is central to all applications of quantum mechanics including quantum field theory which combines special relativity with quantum mechanics. Theories of quantum gravity, such as string theory, also do not modify Schrödinger's equation.[citation needed]

The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. Paul Dirac incorporated matrix mechanics and the Schrödinger equation into a single formulation.

Second-wave feminism

Second-wave feminism is a period of feminist activity and thought that began in the United States in the early 1960s and lasted roughly two decades. It quickly spread across the Western world, with an aim to increase equality for women by gaining more than just enfranchisement.

Whereas first-wave feminism focused mainly on suffrage and overturning legal obstacles to gender equality (e.g., voting rights and property rights), second-wave feminism broadened the debate to include a wider range of issues: sexuality, family, the workplace, reproductive rights, de facto inequalities, and official legal inequalities. Second-wave feminism also drew attention to the issues of domestic violence and marital rape, engendered rape-crisis centers and women's shelters, and brought about changes in custody laws and divorce law. Feminist-owned bookstores, credit unions, and restaurants were among the key meeting spaces and economic engines of the movement.Many historians view the second-wave feminist era in America as ending in the early 1980s with the intra-feminism disputes of the feminist sex wars over issues such as sexuality and pornography, which ushered in the era of third-wave feminism in the early 1990s.


Ska (; Jamaican: [skjæ]) is a music genre that originated in Jamaica in the late 1950s and was the precursor to rocksteady and reggae. It combined elements of Caribbean mento and calypso with American jazz and rhythm and blues. Ska is characterized by a walking bass line accented with rhythms on the off beat. It was developed in Jamaica in the 1960s when Prince Buster, Clement "Coxsone" Dodd, and Duke Reid formed sound systems to play American rhythm and blues and then began recording their own songs. In the early 1960s, ska was the dominant music genre of Jamaica and was popular with British mods. Later it became popular with many skinheads.Music historians typically divide the history of ska into three periods: the original Jamaican scene of the 1960s; the 2 Tone ska revival of the late 1970s in Britain, which fused Jamaican ska rhythms and melodies with the faster tempos and harder edge of punk rock; and third wave ska, which involved bands from the UK, other European countries (notably Germany), Australia, Japan, South America and the United States, beginning in the 1980s and peaking in the 1990s.


In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

In human physiology and psychology, sound is the reception of such waves and their perception by the brain. Humans can only hear sound waves as distinct pitches when the frequency lies between about 20 Hz and 20 kHz. Sound waves above 20 kHz are known as ultrasound and is not perceptible by humans. Sound waves below 20 Hz are known as infrasound. Different animal species have varying hearing ranges.


Surfing is a surface water sport in which the wave rider, referred to as a surfer, rides on the forward or face of a moving wave, which usually carries the surfer towards the shore. Waves suitable for surfing are primarily found in the ocean, but can also be found in lakes or rivers in the form of a standing wave or tidal bore. However, surfers can also utilize artificial waves such as those from boat wakes and the waves created in artificial wave pools.

The term surfing refers to the act of riding a wave, regardless of whether the wave is ridden with a board or without a board, and regardless of the stance used. The native peoples of the Pacific, for instance, surfed waves on alaia, paipo, and other such craft, and did so on their belly and knees. The modern-day definition of surfing, however, most often refers to a surfer riding a wave standing up on a surfboard; this is also referred to as stand-up surfing.

Another prominent form of surfing is body boarding, when a surfer rides a wave on a bodyboard, either lying on their belly, drop knee, or sometimes even standing up on a body board. Other types of surfing include knee boarding, surf matting (riding inflatable mats), and using foils. Body surfing, where the wave is surfed without a board, using the surfer's own body to catch and ride the wave, is very common and is considered by some to be the purest form of surfing.

Three major subdivisions within stand-up surfing are stand-up paddling, long boarding and short boarding with several major differences including the board design and length, the riding style, and the kind of wave that is ridden.

In tow-in surfing (most often, but not exclusively, associated with big wave surfing), a motorized water vehicle, such as a personal watercraft, tows the surfer into the wave front, helping the surfer match a large wave's speed, which is generally a higher speed than a self-propelled surfer can produce. Surfing-related sports such as paddle boarding and sea kayaking do not require waves, and other derivative sports such as kite surfing and windsurfing rely primarily on wind for power, yet all of these platforms may also be used to ride waves. Recently with the use of V-drive boats, Wakesurfing, in which one surfs on the wake of a boat, has emerged. The Guinness Book of World Records recognized a 23.8 m (78 ft) wave ride by Garrett McNamara at Nazaré, Portugal as the largest wave ever surfed.

Third-wave feminism

Third-wave feminism is an iteration of the feminist movement that began in the early 1990s United States and continued until the rise of the fourth wave in the 2010s. Born in the 1960s and 1970s as members of Generation X, and grounded in the civil-rights advances of the second wave, third-wave feminists embraced individualism and diversity and sought to redefine what it meant to be a feminist. According to feminist scholar Elizabeth Evans, the "confusion surrounding what constitutes third-wave feminism is in some respects its defining feature."The third wave is traced to the emergence of the riot grrrl feminist punk subculture in Olympia, Washington, in the early 1990s, and to Anita Hill's televised testimony in 1991—to an all-male, all-white Senate Judiciary Committee—that Clarence Thomas, nominated for the Supreme Court of the United States, had sexually harassed her. The term third wave is credited to Rebecca Walker, who responded to Thomas's appointment to the Supreme Court with an article in Ms. magazine, "Becoming the Third Wave" (1992). She wrote:

So I write this as a plea to all women, especially women of my generation: Let Thomas' confirmation serve to remind you, as it did me, that the fight is far from over. Let this dismissal of a woman's experience move you to anger. Turn that outrage into political power. Do not vote for them unless they work for us. Do not have sex with them, do not break bread with them, do not nurture them if they don't prioritize our freedom to control our bodies and our lives. I am not a post-feminism feminist. I am the Third Wave.

Walker sought to establish that third-wave feminism was not just a reaction, but a movement in itself, because the feminist cause had more work ahead. The term intersectionality—to describe the idea that women experience "layers of oppression" caused, for example, by gender, race and class—had been introduced by Kimberlé Williams Crenshaw in 1989, and it was during the third wave that the concept flourished. As feminists came online in the late 1990s and early 2000s and reached a global audience with blogs and e-zines, they broadened their goals, focusing on abolishing gender-role stereotypes and expanding feminism to include women with diverse racial and cultural identities.The third wave saw the emergence of new feminist currents and theories, such as intersectionality, sex positivity, vegetarian ecofeminism, transfeminism, and postmodern feminism.


A tsunami (from Japanese: 津波, "harbour wave";

English pronunciation: soo-NAH-mee or ) or tidal wave,, also known as a seismic sea wave, is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater explosions (including detonations, landslides, glacier calvings, meteorite impacts and other disturbances) above or below water all have the potential to generate a tsunami. Unlike normal ocean waves, which are generated by wind, or tides, which are generated by the gravitational pull of the Moon and the Sun, a tsunami is generated by the displacement of water.

Tsunami waves do not resemble normal undersea currents or sea waves because their wavelength is far longer. Rather than appearing as a breaking wave, a tsunami may instead initially resemble a rapidly rising tide. For this reason, it is often referred to as a "tidal wave", although this usage is not favoured by the scientific community because it might give the false impression of a causal relationship between tides and tsunamis. Tsunamis generally consist of a series of waves, with periods ranging from minutes to hours, arriving in a so-called "internal wave train". Wave heights of tens of metres can be generated by large events. Although the impact of tsunamis is limited to coastal areas, their destructive power can be enormous, and they can affect entire ocean basins. The 2004 Indian Ocean tsunami was among the deadliest natural disasters in human history, with at least 230,000 people killed or missing in 14 countries bordering the Indian Ocean.

The Ancient Greek historian Thucydides suggested in his 5th century BC History of the Peloponnesian War that tsunamis were related to submarine earthquakes, but the understanding of tsunamis remained slim until the 20th century and much remains unknown. Major areas of current research include determining why some large earthquakes do not generate tsunamis while other smaller ones do; accurately forecasting the passage of tsunamis across the oceans; and forecasting how tsunami waves interact with shorelines.


In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.

It is thus the inverse of the spatial frequency. Wavelength 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.

Wavelength is commonly designated by the Greek letter lambda (λ).

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.Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.Wavelength depends on the medium (for example, vacuum, air, or water) that a wave travels through.

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

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 sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength. The range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum.

Wave–particle duality

Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be partly described in terms not only of particles, but also of waves. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behaviour of quantum-scale objects. As Albert Einstein wrote:

It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.

Through the work of Max Planck, Albert Einstein, Louis de Broglie, Arthur Compton, Niels Bohr, and many others, current scientific theory holds that all particles exhibit a wave nature and vice versa. This phenomenon has been verified not only for elementary particles, but also for compound particles like atoms and even molecules. For macroscopic particles, because of their extremely short wavelengths, wave properties usually cannot be detected.Although the use of the wave-particle duality has worked well in physics, the meaning or interpretation has not been satisfactorily resolved; see Interpretations of quantum mechanics.

Bohr regarded the "duality paradox" as a fundamental or metaphysical fact of nature. A given kind of quantum object will exhibit sometimes wave, sometimes particle, character, in respectively different physical settings. He saw such duality as one aspect of the concept of complementarity. Bohr regarded renunciation of the cause-effect relation, or complementarity, of the space-time picture, as essential to the quantum mechanical account.Werner Heisenberg considered the question further. He saw the duality as present for all quantic entities, but not quite in the usual quantum mechanical account considered by Bohr. He saw it in what is called second quantization, which generates an entirely new concept of fields that exist in ordinary space-time, causality still being visualizable. Classical field values (e.g. the electric and magnetic field strengths of Maxwell) are replaced by an entirely new kind of field value, as considered in quantum field theory. Turning the reasoning around, ordinary quantum mechanics can be deduced as a specialized consequence of quantum field theory.

Wind wave

In fluid dynamics, wind waves, or wind-generated waves, are surface waves that occur on the free surface of bodies of water (like oceans, seas, lakes, rivers, canals, puddles or ponds). They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.When directly generated and affected by local waters, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. More generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago. Wind waves in the ocean are called ocean surface waves.

Wind waves have a certain amount of randomness: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process, in combination with the physics governing their generation, growth, propagation, and decay—as well as governing the interdependence between flow quantities such as: the water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.

Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves.

Velocities of waves

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