In "physics, a wave is a disturbance that transfers "energy through "matter or space, with little or no associated "mass transport. Waves consist, instead, of "oscillations or "vibrations of a physical "medium or a "field, around relatively fixed locations.
There are two main types of waves: "mechanical and "electromagnetic. Mechanical waves propagate through a physical matter, whose substance is being 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. 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.
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 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 1982, 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 .
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
This wave can then be described by the twodimensional functions
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]}
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]}
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]}
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
Group velocity is a property of waves that have a defined envelope, measuring propagation through space (i.e. phase velocity) of the overall shape of the waves' amplitudes—modulation or envelope of the wave.
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
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:^{[13]}
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.^{[14]}
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.^{[15]} 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.^{[16]} The "sine function is periodic, so the "sine wave or sinusoid has a "wavelength in space and a period in time.^{[17]}^{[18]}
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.^{[19]}^{[20]} 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.^{[21]} When a medium is "nonlinear, the response to complex waves cannot be determined from a sinewave decomposition.
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)

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.
Waves exhibit common behaviors under a number of standard situations, e. g.
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:
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.
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 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.
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.
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.^{[22]}
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 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).
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 manmade explosions that give out lowfrequency acoustic energy.
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.^{[23]}
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.
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.
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.
"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,^{[26]} 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.^{[27]} Gaussian wave packets also are used to analyze water waves.^{[28]}
For example, a Gaussian wavefunction ψ might take the form:^{[29]}
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,^{[30]} 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.^{[31]} 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 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 also travel through space. The first observation of gravitational waves was announced on 11 February 2016.^{[32]} Gravitational waves are disturbances in the curvature of "spacetime, predicted by Einstein's theory of "general relativity.
Thus, an arbitrary function f(r, t) can be synthesized by a proper superposition of the functions exp[i (k·r−ωt)]...
All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, or compounded of these...
(p. 61) ...the individual waves move more slowly than the packet and therefore pass back through the packet as it advances