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Wave

From Wikipedia, the free encyclopedia
(Redirected fromWave propagation)
Repeated oscillation around equilibrium
This article is about waves in the scientific sense. For waves on seas and lakes, seeWind wave. For the human hand gesture, seeWaving. For other uses, seeWave (disambiguation) andWave motion (disambiguation).
Surface waves in water showing water ripples

Inphysics,mathematics,engineering, and related fields, awave is a propagating dynamic disturbance (change fromequilibrium) of one or morequantities.Periodic waves oscillate repeatedly about an equilibrium (resting) value at somefrequency. When the entirewaveform moves in one direction, it is said to be atravelling wave; by contrast, a pair ofsuperimposed periodic waves traveling in opposite directions makes astanding wave. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero.

There are two types of waves that are most commonly studied inclassical physics:mechanical waves andelectromagnetic waves. In a mechanical wave,stress andstrain fields oscillate about a mechanical equilibrium. A mechanical wave is a localdeformation (strain) in some physical medium that propagates from particle to particle by creating localstresses that cause strain in neighboring particles too. For example,sound waves are variations of the localpressure andparticle motion that propagate through the medium. Other examples of mechanical waves areseismic waves,gravity waves,surface waves andstring vibrations. In an electromagnetic wave (such as light), coupling between the electric and magnetic fields sustains propagation of waves involving these fields according toMaxwell's equations. Electromagnetic waves can travel through avacuum and through somedielectric media (at wavelengths where they are consideredtransparent). Electromagnetic waves, as determined by their frequencies (orwavelengths), have more specific designations includingradio waves,infrared radiation,terahertz waves,visible light,ultraviolet radiation,X-rays andgamma rays.

Other types of waves includegravitational waves, which are disturbances inspacetime that propagate according togeneral relativity;heat diffusion waves;plasma waves that combine mechanical deformations and electromagnetic fields;reaction–diffusion waves, such as in theBelousov–Zhabotinsky reaction; and many more. Mechanical and electromagnetic waves transferenergy,[1]momentum, andinformation, but they do not transfer particles in the medium. In mathematics andelectronics waves are studied assignals.[2] On the other hand, some waves haveenvelopes which do not move at all such asstanding waves (which are fundamental to music) andhydraulic jumps.

Example of biological waves expanding over the brain cortex, an example ofspreading depolarizations.[3]

A physical wavefield is almost always confined to some finite region of space, called itsdomain. For example, the seismic waves generated byearthquakes 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.

Aplane wave is an important mathematical idealization where the disturbance is identical along any (infinite) planenormal to a specific direction of travel. Mathematically, the simplest wave is asinusoidal plane wave in which at any point the field experiencessimple harmonic motion at one frequency. In linear media, complicated waves can generally be decomposed as the sum of many sinusoidal plane waves havingdifferent directions of propagation and/ordifferent frequencies. A plane wave is classified as atransverse wave if the field disturbance at each point is described by a vector perpendicular to the direction of propagation (also the direction of energy transfer); orlongitudinal wave if those vectors are aligned with the propagation direction. Mechanical waves include both transverse and longitudinal waves; on the other hand electromagnetic plane waves are strictly transverse while sound waves in fluids (such as air) can only be longitudinal. That physical direction of an oscillating field relative to the propagation direction is also referred to as the wave'spolarization, which can be an important attribute.

Modern physics
H^|ψn(t)⟩=iℏddt|ψn(t)⟩{\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {d}{dt}}|\psi _{n}(t)\rangle }
Gμν+Λgμν=κTμν{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
Categories

Mathematical description

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Single waves

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See also:Solitary wave

A wave can be described just like a field, namely as afunctionF(x,t){\displaystyle F(x,t)} wherex{\displaystyle x} is a position andt{\displaystyle t} is a time.

The value ofx{\displaystyle x} is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually avector in theCartesian three-dimensional spaceR3{\displaystyle \mathbb {R} ^{3}}. However, in many cases one can ignore one dimension, and letx{\displaystyle x} be a point of the Cartesian planeR2{\displaystyle \mathbb {R} ^{2}}. This is the case, for example, when studying vibrations of a drum skin. One may even restrictx{\displaystyle x} to a point of the Cartesian lineR{\displaystyle \mathbb {R} } – that is, the set ofreal numbers. This is the case, for example, when studying vibrations in aviolin string orrecorder. The timet{\displaystyle t}, on the other hand, is always assumed to be ascalar; that is, a real number.

The value ofF(x,t){\displaystyle F(x,t)} can be any physical quantity of interest assigned to the pointx{\displaystyle x} that may vary with time. For example, ifF{\displaystyle F} represents the vibrations inside an elastic solid, the value ofF(x,t){\displaystyle F(x,t)} is usually a vector that gives the current displacement fromx{\displaystyle x} of the material particles that would be at the pointx{\displaystyle x} in the absence of vibration. For an electromagnetic wave, the value ofF{\displaystyle F} can be theelectric field vectorE{\displaystyle E}, or themagnetic field vectorH{\displaystyle H}, or any related quantity, such as thePoynting vectorE×H{\displaystyle E\times H}. Influid dynamics, the value ofF(x,t){\displaystyle F(x,t)} could be the velocity vector of the fluid at the pointx{\displaystyle x}, or any scalar property likepressure,temperature, ordensity. In a chemical reaction,F(x,t){\displaystyle F(x,t)} could be the concentration of some substance in the neighborhood of pointx{\displaystyle x} of the reaction medium.

For any dimensiond{\displaystyle d} (1, 2, or 3), the wave's domain is then asubsetD{\displaystyle D} ofRd{\displaystyle \mathbb {R} ^{d}}, such that the function valueF(x,t){\displaystyle F(x,t)} is defined for any pointx{\displaystyle x} inD{\displaystyle D}. For example, when describing the motion of adrum skin, one can considerD{\displaystyle D} to be adisk (circle) on the planeR2{\displaystyle \mathbb {R} ^{2}} with center at the origin(0,0){\displaystyle (0,0)}, and letF(x,t){\displaystyle F(x,t)} be the vertical displacement of the skin at the pointx{\displaystyle x} ofD{\displaystyle D} and at timet{\displaystyle t}.

Superposition

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Main article:Superposition principle

Waves of the same type are often superposed and encountered simultaneously at a given point in space and time. The properties at that point are the sum of the properties of each component wave at that point. In general, the velocities are not the same, so the wave form will change over time and space.

Wave spectrum

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See also:Wind wave § Spectrum,Electromagnetic spectrum, andSpectrum (physical sciences)
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This sectionneeds expansion with: concept summary. You can help byadding to it.(May 2023)

Wave families

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Sometimes one is interested in a single specific wave. 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 adrum stick, or all the possibleradar echoes one could get from anairplane that may be approaching anairport.

In some of those situations, one may describe such a family of waves by a functionF(A,B,…;x,t){\displaystyle F(A,B,\ldots ;x,t)} that depends on certainparametersA,B,…{\displaystyle A,B,\ldots }, besidesx{\displaystyle x} andt{\displaystyle t}. Then one can obtain different waves – that is, different functions ofx{\displaystyle x} andt{\displaystyle t} – by choosing different values for those parameters.

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

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

F(A,L,n,c;x,t)=A(cos⁡2πx2n−14L)(cos⁡2πct2n−14L){\displaystyle F(A,L,n,c;x,t)=A\left(\cos 2\pi x{\frac {2n-1}{4L}}\right)\left(\cos 2\pi ct{\frac {2n-1}{4L}}\right)}

The parameterA{\displaystyle A} defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note);c{\displaystyle c} is the speed of sound;L{\displaystyle L} is the length of the bore; andn{\displaystyle n} is a positive integer (1,2,3,...) that specifies the number ofnodes in the standing wave. (The positionx{\displaystyle x} should be measured from themouthpiece, and the timet{\displaystyle t} from any moment at which the pressure at the mouthpiece is maximum. The quantityλ=4L/(2n−1){\displaystyle \lambda =4L/(2n-1)} is thewavelength of the emitted note, andf=c/λ{\displaystyle f=c/\lambda } is itsfrequency.) 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 distancer{\displaystyle r} from the center of the skin to the strike point, and on the strengths{\displaystyle s} of the strike. Then the vibration for all possible strikes can be described by a functionF(r,s;x,t){\displaystyle F(r,s;x,t)}.

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 functionh{\displaystyle h} such thath(x){\displaystyle h(x)} is the initial temperature at each pointx{\displaystyle x} of the bar. Then the temperatures at later times can be expressed by a functionF{\displaystyle F} that depends on the functionh{\displaystyle h} (that is, afunctional operator), so that the temperature at a later time isF(h;x,t){\displaystyle F(h;x,t)}

Differential wave equations

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Another way to describe and study a family of waves is to give a mathematical equation that, instead of explicitly giving the value ofF(x,t){\displaystyle F(x,t)}, only constrains how those values can change with time. Then the family of waves in question consists of all functionsF{\displaystyle F} that satisfy those constraints – that is, allsolutions 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, ifF(x,t){\displaystyle F(x,t)} is the temperature inside a block of somehomogeneous andisotropic solid material, its evolution is constrained by thepartial differential equation

∂F∂t(x,t)=α(∂2F∂x12(x,t)+∂2F∂x22(x,t)+∂2F∂x32(x,t))+βQ(x,t){\displaystyle {\frac {\partial F}{\partial t}}(x,t)=\alpha \left({\frac {\partial ^{2}F}{\partial x_{1}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{2}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{3}^{2}}}(x,t)\right)+\beta Q(x,t)}

whereQ(p,f){\displaystyle Q(p,f)} is the heat that is being generated per unit of volume and time in the neighborhood ofx{\displaystyle x} at timet{\displaystyle t} (for example, by chemical reactions happening there);x1,x2,x3{\displaystyle x_{1},x_{2},x_{3}} are the Cartesian coordinates of the pointx{\displaystyle x};∂F/∂t{\displaystyle \partial F/\partial t} is the (first) derivative ofF{\displaystyle F} with respect tot{\displaystyle t}; and∂2F/∂xi2{\displaystyle \partial ^{2}F/\partial x_{i}^{2}} is the second derivative ofF{\displaystyle F} relative toxi{\displaystyle x_{i}}. (The symbol "∂{\displaystyle \partial }" 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 thediffusion of heat in solid media. For that reason, it is called theheat 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 functionF(x,t){\displaystyle F(x,t)} that gives the pressure at a pointx{\displaystyle x} and timet{\displaystyle t} within that container. If the gas was initially at uniform temperature and composition, the evolution ofF{\displaystyle F} is constrained by the formula

∂2F∂t2(x,t)=α(∂2F∂x12(x,t)+∂2F∂x22(x,t)+∂2F∂x32(x,t))+βP(x,t){\displaystyle {\frac {\partial ^{2}F}{\partial t^{2}}}(x,t)=\alpha \left({\frac {\partial ^{2}F}{\partial x_{1}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{2}^{2}}}(x,t)+{\frac {\partial ^{2}F}{\partial x_{3}^{2}}}(x,t)\right)+\beta P(x,t)}

HereP(x,t){\displaystyle P(x,t)} is some extra compression force that is being applied to the gas nearx{\displaystyle x} by some external process, such as aloudspeaker orpiston right next top{\displaystyle p}.

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∂2F/∂t2{\displaystyle \partial ^{2}F/\partial t^{2}}, the second derivative ofF{\displaystyle F} with respect to time, rather than the first derivative∂F/∂t{\displaystyle \partial F/\partial t}. Yet this small change makes a huge difference on the set of solutionsF{\displaystyle F}. 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

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Main articles:Wave equation andd'Alembert's formula

Consider a travelingtransverse wave (which may be apulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling

Wavelengthλ 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 thatf(x,t) +g(x,t) =u(x,t).

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

or, more generally, byd'Alembert's formula:[6]u(x,t)=F(x−vt)+G(x+vt).{\displaystyle u(x,t)=F(x-vt)+G(x+vt).}representing two component waveformsF{\displaystyle F} andG{\displaystyle G} traveling through the medium in opposite directions. A generalized representation of this wave can be obtained[7] as thepartial differential equation1v2∂2u∂t2=∂2u∂x2.{\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}.}

General solutions are based uponDuhamel's principle.[8]

Wave forms

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Main article:Waveform
Sine,square,triangle andsawtooth waveforms

The form or shape ofF ind'Alembert's formula involves the argumentx −vt. Constant values of this argument correspond to constant values ofF, and these constant values occur ifx increases at the same rate thatvt increases. That is, the wave shaped like the functionF will move in the positivex-direction at velocityv (andG will propagate at the same speed in the negativex-direction).[9]

In the case of a periodic functionF with periodλ, that is,F(x +λ −vt) =F(x −vt), the periodicity ofF in space means that a snapshot of the wave at a given timet finds the wave varying periodically in space with periodλ (thewavelength of the wave). In a similar fashion, this periodicity ofF implies a periodicity in time as well:F(x −v(t +T)) =F(x −vt) providedvT =λ, so an observation of the wave at a fixed locationx finds the wave undulating periodically in time with periodT =λ/v.[10]

Amplitude and modulation

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Main article:Amplitude modulation
See also:Frequency modulation andPhase modulation
Amplitude modulation can be achieved throughf(x,t) = 1.00×sin(2π/0.10×(x−1.00×t)) andg(x,t) = 1.00×sin(2π/0.11×(x−1.00×t)) only the resultant is visible to improve clarity of waveform.
Illustration of theenvelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is thecarrier wave, which is being modulated.

The amplitude of a wave may be constant (in which case the wave is ac.w. orcontinuous wave), or may bemodulated so as to vary with time and/or position. The outline of the variation in amplitude is called theenvelope of the wave. Mathematically, themodulated wave can be written in the form:[11][12][13]u(x,t)=A(x,t)sin⁡(kx−ωt+ϕ),{\displaystyle u(x,t)=A(x,t)\sin \left(kx-\omega t+\phi \right),}whereA(x, t){\displaystyle A(x,\ t)} is the amplitude envelope of the wave,k{\displaystyle k} is thewavenumber andϕ{\displaystyle \phi } is thephase. If thegroup velocityvg{\displaystyle v_{g}} (see below) is wavelength-independent, this equation can be simplified as:[14]u(x,t)=A(x−vgt)sin⁡(kx−ωt+ϕ),{\displaystyle u(x,t)=A(x-v_{g}t)\sin \left(kx-\omega t+\phi \right),}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 anenvelope equation.[14][15]

Phase velocity and group velocity

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Main articles:Phase velocity andGroup velocity
See also:Envelope (waves) § Phase and group velocity
The red square moves with thephase velocity, while the green circles propagate with thegroup velocity.

There are two velocities that are associated with waves, thephase velocity and thegroup velocity.

Phase velocity is the rate at which thephase of the wavepropagates in space: any given phase of the wave (for example, thecrest) will appear to travel at the phase velocity. The phase velocity is given in terms of thewavelengthλ (lambda) andperiodT asvp=λT.{\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}

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.

Special waves

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Sine waves

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This section is an excerpt fromSine wave.[edit]
Tracing the y component of acircle while going around the circle results in a sine wave (red). Tracing the x component results in acosine wave (blue). Both waves are sinusoids of the same frequency but different phases.

Asine wave, sinusoidal wave, or sinusoid (symbol: ∿) is aperiodic wave whosewaveform (shape) is thetrigonometricsine function. Inmechanics, as a linearmotion over time, this issimple harmonic motion; asrotation, it corresponds touniform circular motion. Sine waves occur often inphysics, includingwind waves,sound waves, andlight waves, such asmonochromatic radiation. Inengineering,signal processing, andmathematics,Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.

When any two sine waves of the samefrequency (but arbitraryphase) arelinearly combined, the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, thesine andcosinecomponents, respectively.

Plane waves

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Main article:Plane wave

Aplane wave is a kind of wave whose value varies only in one spatial direction. That is, its value is constant on a plane that is perpendicular to that direction. Plane waves can be specified by a vector of unit lengthn^{\displaystyle {\hat {n}}} indicating the direction that the wave varies in, and a wave profile describing how the wave varies as a function of the displacement along that direction (n^⋅x→{\displaystyle {\hat {n}}\cdot {\vec {x}}}) and time (t{\displaystyle t}). Since the wave profile only depends on the positionx→{\displaystyle {\vec {x}}} in the combinationn^⋅x→{\displaystyle {\hat {n}}\cdot {\vec {x}}}, any displacement in directions perpendicular ton^{\displaystyle {\hat {n}}} cannot affect the value of the field.

Plane waves are often used to modelelectromagnetic waves far from a source. For electromagnetic plane waves, the electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other.

Standing waves

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Main articles:Standing wave,Acoustic resonance,Helmholtz resonance, andOrgan pipe
Standing wave. The red dots represent the wavenodes.

A standing wave, also known as astationary wave, is a wave whoseenvelope remains in a constant position. This phenomenon arises as a result ofinterference between two waves traveling in opposite directions.

Thesum of two counter-propagating waves (of equal amplitude and frequency) creates astanding 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 aviolin string is displaced, transverse waves propagate out to where the string is held in place at thebridge and thenut, where the waves are reflected back. At the bridge and nut, the two opposed waves are inantiphase and cancel each other, producing anode. Halfway between two nodes there is anantinode, where the two counter-propagating wavesenhance each other maximally. There is no netpropagation of energy over time.

Solitary waves

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Main article:Soliton
Solitary wave in a laboratorywave channel

Asoliton orsolitary wave is a self-reinforcingwave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation ofnonlinear anddispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersivepartial differential equations describing physical systems.

Physical properties

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Propagation

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Wave propagation is any of the ways in which waves travel. With respect to the direction of theoscillation relative to the propagation direction, we can distinguish betweenlongitudinal wave andtransverse waves.

Electromagnetic waves propagate invacuum as well as in material media. Propagation of other wave types such as sound may occur only in atransmission medium.

Reflection of plane waves in a half-space

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Further information:Reflection coefficient

The propagation and reflection of plane waves—e.g. Pressure waves (P wave) orShear waves (SH or SV-waves) are phenomena that were first characterized within the field of classical seismology, and are now considered fundamental concepts in modernseismic tomography. The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding theHelmholtz decomposition of the displacement field, which is then substituted into thewave equation. From here, theplane wave eigenmodes can be calculated.[citation needed][clarification needed]

SV wave propagation

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The propagation of SV-wave in a homogeneous half-space (the horizontal displacement field)
The propagation of SV-wave in a homogeneous half-space (The vertical displacement field)[clarification needed]

The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of the reflected SV wave is identical to the incidence wave, while the angle of the reflected P wave is greater than the SV wave. For the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture.[16]

P wave propagation

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Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different.[clarification needed]

Wave velocity

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Further information:Phase velocity,Group velocity, andSignal velocity
Seismic wave propagation in 2D modelled usingFDTD method in the presence of a landmine

Wave velocity is a general concept, of various kinds of wave velocities, for a wave'sphase andspeed concerning energy (and information) propagation. Thephase velocity is given as:vp=ωk,{\displaystyle v_{\rm {p}}={\frac {\omega }{k}},}where:

The phase speed gives you the speed at which a point of constantphase of the wave will travel for a discrete frequency. The angular frequencyω cannot be chosen independently from the wavenumberk, but both are related through thedispersion relationship:ω=Ω(k).{\displaystyle \omega =\Omega (k).}

In the special caseΩ(k) =ck, withc a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speedc. For instanceelectromagnetic waves invacuum are non-dispersive. In case of other forms of the dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instanceelectromagnetic,sound orwater waves).

The speed at which a resultantwave packet from a narrow range of frequencies will travel is called thegroup velocity and is determined from thegradient of thedispersion relation:vg=∂ω∂k{\displaystyle v_{\rm {g}}={\frac {\partial \omega }{\partial k}}}

In almost all cases, a wave is mainly a movement of energy through a medium. Most often, the group velocity is the velocity at which the energy moves through this medium.

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

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Main articles:Rectilinear propagation,Transmittance, andTransmission medium

Waves normally move in a straight line (that is, rectilinearly) through atransmission medium. Such media can be classified into one or more of the following categories:

  • Abounded medium if it is finite in extent, otherwise anunbounded medium
  • Alinear medium if the amplitudes of different waves at any particular point in the medium can be added
  • Auniform medium orhomogeneous medium if its physical properties are unchanged at different locations in space
  • Ananisotropic medium if one or more of its physical properties differ in one or more directions
  • Anisotropic medium if its physical properties are thesame in all directions

Absorption

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Main articles:Absorption (acoustics) andAbsorption (electromagnetic radiation)

Waves are usually defined in media which allow most or all of a wave's energy to propagate withoutloss. However materials may be characterized as "lossy" if they remove energy from a wave, usually converting it into heat. This is termed "absorption." A material which absorbs a wave's energy, either in transmission or reflection, is characterized by arefractive index which iscomplex. The amount of absorption will generally depend on the frequency (wavelength) of the wave, which, for instance, explains why objects may appear colored.

Reflection

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Main article:Reflection (physics)

When a wave strikes a reflective surface, it changes direction, such that the angle made by theincident wave and linenormal to the surface equals the angle made by the reflected wave and the same normal line.

Refraction

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Main article: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 thephase velocity changes. Typically, refraction occurs when a wave passes from onemedium into another. The amount by which a wave is refracted by a material is given by therefractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials bySnell's law.

Diffraction

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Main article: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.

Interference

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Main article:Wave interference
Identical waves from two sources undergoinginterference. Observed at the bottom one sees 5 positions where the waves add in phase, but in between which they are out of phase and cancel.

When waves in a linear medium (the usual case) cross each other in a region of space, they do not actually interact with each other, but continue on as if the other one were not present. However at any pointin that region thefield quantities describing those waves add according to thesuperposition principle. If the waves are of the same frequency in a fixedphase relationship, then there will generally be positions at which the two waves arein phase and their amplitudesadd, and other positions where they areout of phase and their amplitudes (partially or fully)cancel. This is called aninterference pattern.

Polarization

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Main article:Polarization (waves)

The phenomenon of polarization arises when wave motion can occur simultaneously in twoorthogonal 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 oflinear 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 apolarizing filter.

Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.

Dispersion

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Schematic of light being dispersed by a prism. Click to see animation.
Main articles:Dispersion relation,Dispersion (optics), andDispersion (water waves)

Dispersion is the frequency dependence of therefractive index, a consequence of the atomic nature of materials.[17]: 67 A wave undergoes dispersion when either thephase velocity or thegroup velocity depends on the wave frequency. Dispersion is seen by letting white light pass through aprism, the result of which is to produce the spectrum of colors of the rainbow.Isaac Newton was the first to recognize that this meant that white light was a mixture of light of different colors.[17]: 190 

Doppler effect

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Main article:Doppler effect

The Doppler effect or Doppler shift is the change infrequency of a wave in relation to an observer who is moving relative to the wave source.[18] It is named after theAustrian physicistChristian Doppler, who described the phenomenon in 1842.

Mechanical waves

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Main article:Mechanical wave

A mechanical wave is an oscillation ofmatter, and therefore transfers energy through amedium.[19] While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefore, the oscillating material does not move far from its initial position. Mechanical waves can be produced only in media which possesselasticity andinertia. There are three types of mechanical waves:transverse waves,longitudinal waves, andsurface waves.

Waves on strings

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Main article:String vibration

The transverse vibration of a string is a function of tension and inertia, and is constrained by the length of the string as the ends are fixed. This constraint limits the steady state modes that are possible, and thereby the frequencies.The speed of a transverse wave traveling along avibrating string (v) is directly proportional to the square root of thetension of the string (T) over thelinear mass density (μ):

v=Tμ,{\displaystyle v={\sqrt {\frac {T}{\mu }}},}

where the linear densityμ is the mass per unit length of the string.

Acoustic waves

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Main article:Acoustic wave

Acoustic orsound waves are compression waves which travel as body waves at the speed given by:

v=Bρ0,{\displaystyle v={\sqrt {\frac {B}{\rho _{0}}}},}

or the square root of the adiabatic bulk modulus divided by the ambient density of the medium (seespeed of sound).

Water waves

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Main article:Water waves
  • Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
  • Sound, a mechanical wave that propagates through gases, liquids, solids and plasmas.
  • Inertial waves, which occur in rotating fluids and are restored by theCoriolis effect.
  • Ocean surface waves, which are perturbations that propagate through water.

Body waves

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Main article:Body wave (seismology)

Body waves travel through the interior of the medium along paths controlled by the material properties in terms of density and modulus (stiffness). The density and modulus, in turn, vary according to temperature, composition, and material phase. This effect resembles the refraction of light waves. Two types of particle motion result in two types of body waves: Primary and Secondary waves.

Seismic waves

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Main article:Seismic wave

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. They include body waves—the primary (P waves) and secondary waves (S waves)—and surface waves, such asRayleigh waves,Love waves, andStoneley waves.

Shock waves

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Formation of a shock wave by a plane
Main article:Shock wave

A shock wave is a type of propagating disturbance. When a wave moves faster than the localspeed of sound in afluid, 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 inpressure,temperature anddensity of the medium.[20]

See also:Sonic boom andCherenkov radiation

Shear waves

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Main article:Shear wave

Shear waves are body waves due to shear rigidity and inertia. They can only be transmitted through solids and to a lesser extent through liquids with a sufficiently high viscosity.

Other

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  • Waves oftraffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves[21][22]
  • Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.

Electromagnetic waves

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Main article:Electromagnetic wave
Further information:Electromagnetic spectrum

An electromagnetic wave consists of two waves that are oscillations of theelectric andmagnetic 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, invacuum, the electric and magnetic fields satisfy thewave equation both with speed equal to that of thespeed of light. From this emerged the idea thatlight is an electromagnetic wave. The unification of light and electromagnetic waves was experimentally confirmed byHertz in the end of the 1880s. Electromagnetic waves can have different frequencies (and thus wavelengths), and are classified accordingly in wavebands, such asradio waves,microwaves,infrared,visible light,ultraviolet,X-rays, andgamma rays. The range of frequencies in each of these bands is continuous, and the limits of each band are mostly arbitrary, with the exception of visible light, which must be visible to the normal human eye.

Quantum mechanical waves

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Main article:Schrödinger equation
See also:Wave function

Schrödinger equation

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TheSchrödinger equation describes the wave-like behavior ofparticles inquantum mechanics. Solutions of this equation arewave functions which can be used to describe the probability density of a particle.

Dirac equation

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TheDirac 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-1⁄2 particles.

A propagating wave packet; in general, theenvelope of the wave packet moves at a different speed than the constituent waves.[23]

de Broglie waves

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Main articles:Wave packet andMatter wave

Louis de Broglie postulated that all particles withmomentum have a wavelength

λ=hp,{\displaystyle \lambda ={\frac {h}{p}},}

whereh is thePlanck constant, andp is the magnitude of themomentum of the particle. This hypothesis was at the basis ofquantum mechanics. Nowadays, this wavelength is called thede Broglie wavelength. For example, theelectrons in aCRT display have a de Broglie wavelength of about 10−13 m.

A wave representing such a particle traveling in thek-direction is expressed by the wave function as follows:

ψ(r,t=0)=Aeik⋅r,{\displaystyle \psi (\mathbf {r} ,\,t=0)=Ae^{i\mathbf {k\cdot r} },}

where the wavelength is determined by thewave vectork as:

λ=2πk,{\displaystyle \lambda ={\frac {2\pi }{k}},}

and the momentum by:

p=ℏk.{\displaystyle \mathbf {p} =\hbar \mathbf {k} .}

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 awave packet,[24] a waveform often used inquantum mechanics to describe thewave 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, thewave packet is often taken to have aGaussian shape and is called aGaussian wave packet.[25][26][27] Gaussian wave packets also are used to analyze water waves.[28]

For example, a Gaussian wavefunctionψ might take the form:[29]

ψ(x,t=0)=Aexp⁡(−x22σ2+ik0x),{\displaystyle \psi (x,\,t=0)=A\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}+ik_{0}x\right),}

at some initial timet = 0, where the central wavelength is related to the central wave vectork0 asλ0 = 2π /k0. It is well known from the theory ofFourier analysis,[30] or from theHeisenberg 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. TheFourier transform of a Gaussian is itself a Gaussian.[31] Given the Gaussian:

f(x)=e−x2/(2σ2),{\displaystyle f(x)=e^{-x^{2}/\left(2\sigma ^{2}\right)},}

the Fourier transform is:

f~(k)=σe−σ2k2/2.{\displaystyle {\tilde {f}}(k)=\sigma e^{-\sigma ^{2}k^{2}/2}.}

The Gaussian in space therefore is made up of waves:

f(x)=12π∫−∞∞ f~(k)eikx dk;{\displaystyle f(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\ {\tilde {f}}(k)e^{ikx}\ dk;}

that is, a number of waves of wavelengthsλ such thatkλ = 2 π.

The parameter σ decides the spatial spread of the Gaussian along thex-axis, while the Fourier transform shows a spread inwave vectork determined by 1/σ. That is, the smaller the extent in space, the larger the extent ink, and hence inλ = 2π/k.

Animation showing the effect of a cross-polarized gravitational wave on a ring oftest particles

Gravity waves

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Main article:Gravity wave

Gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy works to restore equilibrium. Surface waves on water are the most familiar example.

Gravitational waves

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Main article:Gravitational wave

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 ofspacetime, predicted by Einstein's theory ofgeneral relativity.

See also

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Waves in general

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Parameters

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Waveforms

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Electromagnetic waves

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In fluids

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In quantum mechanics

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In relativity

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Other specific types of waves

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Related topics

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References

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  1. ^(Hall 1980, p. 8)
  2. ^Pragnan Chakravorty, "What Is a Signal? [Lecture Notes]", IEEESignal Processing Magazine, vol. 35, no. 5, pp. 175–177, Sept. 2018. doi:10.1109/MSP.2018.2832195
  3. ^Santos, Edgar; Schöll, Michael; Sánchez-Porras, Renán; Dahlem, Markus A.; Silos, Humberto; Unterberg, Andreas; Dickhaus, Hartmut; Sakowitz, Oliver W. (2014-10-01). "Radial, spiral and reverberating waves of spreading depolarization occur in the gyrencephalic brain".NeuroImage.99:244–255.doi:10.1016/j.neuroimage.2014.05.021.ISSN 1095-9572.PMID 24852458.S2CID 1347927.
  4. ^Michael A. Slawinski (2003)."Wave equations".Seismic waves and rays in elastic media. Elsevier. pp. 131ff.ISBN 978-0-08-043930-3.
  5. ^Lev A. Ostrovsky & Alexander I. Potapov (2001).Modulated waves: theory and application. Johns Hopkins University Press.ISBN 978-0-8018-7325-6.
  6. ^Graaf, Karl F (1991).Wave motion in elastic solids (Reprint of Oxford 1975 ed.). Dover. pp. 13–14.ISBN 978-0-486-66745-4.
  7. ^For an example derivation, see the steps leading up to eq. (17) inRedfern, Francis."Kinematic Derivation of the Wave Equation".Physics Journal. Archived fromthe original on 2013-07-24. Retrieved2012-12-11.
  8. ^Jalal M. Ihsan Shatah; Michael Struwe (2000)."The linear wave equation".Geometric wave equations. American Mathematical Society Bookstore. pp. 37ff.ISBN 978-0-8218-2749-9.
  9. ^Louis Lyons (1998).All you wanted to know about mathematics but were afraid to ask. Cambridge University Press. pp. 128ff.ISBN 978-0-521-43601-4.
  10. ^McPherson, Alexander (2009)."Waves and their properties".Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77.ISBN 978-0-470-18590-2.
  11. ^Christian Jirauschek (2005).FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection. Cuvillier Verlag. p. 9.ISBN 978-3-86537-419-6.
  12. ^Fritz Kurt Kneubühl (1997).Oscillations and waves. Springer. p. 365.ISBN 978-3-540-62001-3.
  13. ^Mark Lundstrom (2000).Fundamentals of carrier transport. Cambridge University Press. p. 33.ISBN 978-0-521-63134-1.
  14. ^abChin-Lin Chen (2006)."§13.7.3 Pulse envelope in nondispersive media".Foundations for guided-wave optics. Wiley. p. 363.ISBN 978-0-471-75687-3.
  15. ^Longhi, Stefano; Janner, Davide (2008)."Localization and Wannier wave packets in photonic crystals". In Hugo E. Hernández-Figueroa; Michel Zamboni-Rached; Erasmo Recami (eds.).Localized Waves. Wiley-Interscience. p. 329.ISBN 978-0-470-10885-7.
  16. ^The animations are taken fromPoursartip, Babak (2015)."Topographic amplification of seismic waves". UT Austin. Archived fromthe original on 2017-01-09. Retrieved2023-02-24.
  17. ^abHecht, Eugene (1998).Optics (3 ed.). Reading, Mass. Harlow: Addison-Wesley.ISBN 978-0-201-83887-9.
  18. ^Giordano, Nicholas (2009).College Physics: Reasoning and Relationships. Cengage Learning. pp. 421–424.ISBN 978-0534424718.
  19. ^Giancoli, D. C. (2009) Physics for scientists & engineers with modern physics (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
  20. ^Anderson, John D. Jr. (January 2001) [1984],Fundamentals of Aerodynamics (3rd ed.),McGraw-Hill Science/Engineering/Math,ISBN 978-0-07-237335-6
  21. ^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 10.1.1.205.4573.doi:10.1098/rspa.1955.0088.S2CID 18301080.
  22. ^P.I. Richards (1956). "Shockwaves on the highway".Operations Research.4 (1):42–51.doi:10.1287/opre.4.1.42.
  23. ^A.T. Fromhold (1991)."Wave packet solutions".Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59ff.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
  24. ^Ming Chiang Li (1980)."Electron Interference". In L. Marton; Claire Marton (eds.).Advances in Electronics and Electron Physics. Vol. 53. Academic Press. p. 271.ISBN 978-0-12-014653-6.
  25. ^Walter Greiner; D. Allan Bromley (2007).Quantum Mechanics (2 ed.). Springer. p. 60.ISBN 978-3-540-67458-0.
  26. ^John Joseph Gilman (2003).Electronic basis of the strength of materials. Cambridge University Press. p. 57.ISBN 978-0-521-62005-5.
  27. ^Donald D. Fitts (1999).Principles of quantum mechanics. Cambridge University Press. p. 17.ISBN 978-0-521-65841-6.
  28. ^Chiang C. Mei (1989).The applied dynamics of ocean surface waves (2nd ed.). World Scientific. p. 47.ISBN 978-9971-5-0789-3.
  29. ^Greiner, Walter; Bromley, D. Allan (2007).Quantum Mechanics (2nd ed.). Springer. p. 60.ISBN 978-3-540-67458-0.
  30. ^Siegmund Brandt; Hans Dieter Dahmen (2001).The picture book of quantum mechanics (3rd ed.). Springer. p. 23.ISBN 978-0-387-95141-6.
  31. ^Cyrus D. Cantrell (2000).Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 677.ISBN 978-0-521-59827-9.
  32. ^"Gravitational waves detected for 1st time, 'opens a brand new window on the universe'". Canadian Broadcasting Corporation. 11 February 2016.

Sources

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External links

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