Wave

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

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

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

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

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

Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave.

General features

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 . However, this notion is problematic for a standing wave (for example, a wave on a string), where energy is moving in both directions equally, or for electromagnetic / light waves in a vacuum, where the concept of medium does not apply. There are water waves on the ocean surface; light waves emitted by the Sun; microwaves used in microwave ovens; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsets and living creatures (as voices).

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 etc.; and so on.

Other properties, however, although they are usually described in an origin-specific manner, may be generalized to all waves. For such reasons, wave theory represents a particular branch of physics that is concerned with the properties of wave processes independently from their physical origin. 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. This is impossible because it would violate general relativity. On the other hand, if all the parts were independent, then there would not be any transmission of the vibration and again, no wave motion. Although the above statements are meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times.

Similarly, wave processes revealed from the study of waves other than sound waves can be significant to the understanding of sound phenomena. A relevant example is Thomas Young's principle of interference (Young, 1802, in ). This principle was first introduced in Young's study of light and, within some specific contexts (for example, scattering of sound by sound), is still a researched area in the study of sound.

Mathematical description of one-dimensional 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

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

: $u(x,\; \backslash \; t)\; =\; F(x\; -\; v\; \backslash \; t)$ (waveform $F$ traveling to the right) : $u(x,\; \backslash \; t)\; =\; G(x\; +\; v\; \backslash \; t)$ (waveform $G$ traveling to the left)

or, more generally, by d'Alembert's formula:

:$u(x,t)=F(x-vt)+G(x+vt).\; \backslash ,$

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

:$\backslash frac\{1\}\{v^2\}\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; t^2\}=\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; x^2\}.\; \backslash ,$

General solutions are based upon Duhamel's principle.

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 x-direction at velocity v (and G will propagate at the same speed in the negative x-direction).

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.

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:

:$u(x,\; \backslash \; t)\; =\; A(x,\; \backslash \; t)\backslash sin\; (kx\; -\; \backslash omega\; t\; +\; \backslash phi)\; \backslash \; ,$

where $A(x,\backslash \; t)$ is the amplitude envelope of the wave, $k$ is the wavenumber and $\backslash phi$ is the phase. If the group velocity $v\_g$ (see below) is wavelength-independent, this equation can be simplified as:

:$u(x,\; \backslash \; t)\; =\; A(x\; -\; v\_g\; \backslash \; t)\backslash sin\; (kx\; -\; \backslash omega\; t\; +\; \backslash phi)\; \backslash \; ,$

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.

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:

:$\backslash psi\; (x,\; \backslash \; t)\; =\; A\; e^\{i\; \backslash left(\; kx\; -\; \backslash omega\; t\; \backslash right)\}\; \backslash \; ,$

which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument, $kx-\backslash omega\; t\; =\; (\backslash frac\{2\backslash pi\}\{\backslash lambda\})(x\; -\; vt)$, makes clear that this expression describes a vibration of wavelength $\backslash lambda\; =\; \backslash frac\{2\backslash pi\}\{k\}$ traveling in the x-direction with a constant phase velocity $v\_p\; =\; \backslash frac\{\backslash omega\}\{k\}\backslash ,$.

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:

:$\backslash psi\; (x,\; \backslash \; t)\; =\; \backslash int\_\{-\backslash infty\}\; ^\{\backslash infty\}\backslash \; dk\_1\; \backslash \; A(k\_1)\backslash \; e^\{i\backslash left(k\_1x\; -\; \backslash omega\; t\; \backslash right)\}\; \backslash \; ,$

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

:$A\; =\; A\_o\; (k\_1)\; e^\; \{i\; \backslash alpha\; (k\_1)\}\; \backslash \; ,$

with A_o the magnitude of A. For example, a common choice for A_o is a Gaussian wave packet:

:$A\_o\; (k\_1)\; =\; N\backslash \; e^\{-\backslash sigma^2\; (k\_1-k)^2\; /\; 2\}\; \backslash \; ,$

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

The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(k₁), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.) The condition for φ to vary slowly is that its rate of change with k₁ be small; this rate of variation is: 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

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

:$u(x,\; \backslash \; t)=\; A\; \backslash sin\; (kx\; -\; \backslash omega\; t\; +\; \backslash phi)\; \backslash \; ,$

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

:$k\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash lambda\}.\; \backslash ,$

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

:$f=\backslash frac\{1\}\{T\}.\; \backslash ,$

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

The angular frequency $\backslash omega$ represents the frequency in radians per second. It is related to the frequency or period by

:$\backslash omega\; =\; 2\; \backslash pi\; f\; =\; \backslash frac\{2\; \backslash pi\}\{T\}.\; \backslash ,$

The wavelength $\backslash lambda$ of a sinusoidal waveform traveling at constant speed $v$ is given by: :$\backslash lambda\; =\; \backslash frac\{v\}\{f\},$

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

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

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. 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. The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.

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. 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. When a medium is nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition.

Plane waves

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 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.

Physical properties

Waves exhibit common behaviors under a number of standard situations, e.g.,

Transmission and media

Waves normally move in in a straight line (i.e. rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories:

Absorption

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

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

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

Dispersion

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.

Mechanical waves

Waves on strings

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

:$v=\backslash sqrt\{\backslash frac\{T\}\{\backslash mu\}\},\; \backslash ,$

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

Acoustic waves

:$v=\backslash sqrt\{\backslash frac\{B\}\{\backslash rho\_0\}\},\; \backslash ,$

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

Water waves

Seismic waves

Shock waves

Other

Electromagnetic waves

(radio, micro, infrared, visible, uv)

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

Quantum mechanical waves

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

de Broglie waves

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

:$\backslash lambda\; =\; \backslash frac\{h\}\{p\},$

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:

:$\backslash psi\; (\backslash mathbf\{r\},\; \backslash \; t=0)\; =A\backslash \; e^\{i\backslash mathbf\{k\; \backslash cdot\; r\}\}\; \backslash \; ,$

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

:$\backslash lambda\; =\; \backslash frac\; \{2\; \backslash pi\}\{k\}\; \backslash \; ,$

and the momentum by:

:$\backslash mathbf\; p\; =\; \backslash hbar\; \backslash mathbf\{k\}\; \backslash \; .$

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, 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. Gaussian wave packets also are used to analyze water waves.

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

:$\backslash psi(x,\backslash \; t=0)\; =\; A\backslash \; \backslash exp\; \backslash left(\; -\backslash frac\{x^2\}\{2\backslash sigma^2\}\; +\; i\; k\_0\; x\; \backslash right)\; \backslash \; ,$

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, 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. Given the Gaussian:

:$f(x)\; =\; e^\{-x^2\; /\; (2\backslash sigma^2)\}\; \backslash \; ,$

the Fourier transform is:

:$\backslash tilde\{\; f\}\; (k)\; =\; \backslash sigma\; e^\{-\backslash sigma^2\; k^2\; /\; 2\}\; \backslash \; .$

The Gaussian in space therefore is made up of waves:

:$f(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash \; \backslash tilde\{f\}\; (k)\; e^\{ikx\}\; \backslash \; dk\; \backslash \; ;$

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.

Gravitational waves

Researchers believe that gravitational waves also travel through space, although gravitational waves have never been directly detected. Not to be confused with gravity waves, gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

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.

References

See also

Sources

External links

Category:Fundamental physics concepts Category:Partial differential equations