In mathematics and physical science, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Periodic waves oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a traveling wave; by contrast, a pair of identical superimposed periodic waves traveling in opposite directions makes a standing 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 in classical physics: mechanical waves and electromagnetic waves. In a mechanical wave, stress and strain fields oscillate about a mechanical equilibrium. A mechanical wave is a local deformation (strain) in some physical medium that propagates from particle to particle by creating local stresses that cause strain in neighboring particles too. For example, sound waves are variations of the local pressure and particle motion that propagate through the medium. Other examples of mechanical waves are seismic waves, gravity waves, surface waves and string vibrations. In an electromagnetic wave (such as light), coupling between the electric and magnetic fields sustains propagation of waves involving these fields according to Maxwell's equations. Electromagnetic waves can travel through a vacuum and through some dielectric media (at wavelengths where they are considered transparent). Electromagnetic waves, as determined by their frequencies (or wavelengths), have more specific designations including radio waves, infrared radiation, terahertz waves, visible light, ultraviolet radiation, X-rays and gamma rays. Other types of waves include gravitational waves, which are disturbances in spacetime that propagate according to general relativity; heat diffusion waves; plasma waves that combine mechanical deformations and electromagnetic fields; reaction–diffusion waves, such as in the Belousov–Zhabotinsky reaction; and many more. Mechanical and electromagnetic waves transfer energy, momentum and information. In the case of waves in a liquid medium, matter can also be transferred (see Stokes drift). In mathematics and electronics waves are studied as signals. On the other hand, some waves have envelopes which do not move at all such as standing waves (which are fundamental to music) and hydraulic jumps.
A physical wave field is almost always confined to some finite region of space, called its domain. For example, the seismic waves generated by earthquakes are significant only in the interior and surface of the planet, so they can be ignored outside it. However, waves with infinite domain, that extend over the whole space, are commonly studied in mathematics, and are very valuable tools for understanding physical waves in finite domains. A plane wave is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific direction of travel. Mathematically, the simplest wave is a sinusoidal plane wave in which at any point the field experiences simple harmonic motion at one frequency. In linear media, complicated waves can generally be decomposed as the sum of many sinusoidal plane waves having different directions of propagation and/or different frequencies. A plane wave is classified as a transverse 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); or longitudinal 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's polarization, which can be an important attribute.
Definition While waves are ubiquitous features of physical systems, no single definition adequately describes the topic. Examples and descriptions of common characteristics are used as an alternative to a single definition. In abstract, waves are the dynamic manifestation of time-dependent field theory analogous to ballistics, the dynamic manifestation of particle mechanics, but this point of view misses the many visually appealing examples of waves like vibrating stringed instruments and fluid ripples. Viewed microscopically, waves are changes in the value of physical property at a point in space that results from a delayed response to changes in adjacent regions. Examples of properties include pressure, temperature, height, or gravitational force. A physical medium, like vacuum, air, water, or solid rock may exhibit waves in different properties which may or may not be related. Properties like the heights spectators at a sporting event or the level of anxiety among political protestors may also be considered waves when the property depends upon delayed responds to physically adjacent events. In physics, a physical quantity that has a value at points in space is called a field, so a wave is disturbance in a field resulting from delayed response to adjacent disturbances. Viewed macroscopically, a wave is the dynamic response of a field caused by effects that can only propagate at a finite speed. Rotation of an electric dipole produce electromagnetic waves and the mutual rotation of binary stars produce gravitational waves, each of which propagate at the speed of light. Periodic or sinusoidal waves are useful idealized examples, but the essential neighbor-to-neighbor interaction with delay means waves may propagate in non-linear, granular, or noisy media which do not produce these ideal results. Temperature waves or chemical reaction waves demonstrate that waves need not be a material displacement.
Mathematical description
Single waves Mathematically, a wave is described by a function F ( x , t ) {\displaystyle F(x,t)} that maps a point in space and time onto a field. For a scalar field its value is a number; for a vector field it is a vector; in general a tensor field has a tensor value. The value of x {\displaystyle x} is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . However, in many cases one can ignore one dimension, and let x {\displaystyle x} be a point of the Cartesian plane R 2 {\displaystyle \mathbb {R} ^{2}} . This is the case, for example, when studying vibrations of a drum skin. One may even restrict x {\displaystyle x} to a point of the Cartesian line R {\displaystyle \mathbb {R} } – that is, the set of real numbers. This is the case, for example, when studying vibrations in a violin string or recorder. The time t {\displaystyle t} , on the other hand, is always assumed to be a scalar; that is, a real number. The value of F ( x , t ) {\displaystyle F(x,t)} can be any physical quantity of interest assigned to the point x {\displaystyle x} that may vary with time. For example, if F {\displaystyle F} represents the vibrations inside an elastic solid, the value of F ( x , t ) {\displaystyle F(x,t)} is usually a vector that gives the current displacement from x {\displaystyle x} of the material particles that would be at the point x {\displaystyle x} in the absence of vibration. For an electromagnetic wave, the value of F {\displaystyle F} can be the electric field vector E {\displaystyle \mathbf {E} } , or the magnetic field vector H {\displaystyle \mathbf {H} } , or any related quantity, such as the Poynting vector E × H {\displaystyle \mathbf {E} \times \mathbf {H} } . In fluid dynamics, the value of F ( x , t ) {\displaystyle F(x,t)} could be the velocity vector of the fluid at the point x {\displaystyle x} , or any scalar property like pressure, temperature, or density. In a chemical reaction, F ( x , t ) {\displaystyle F(x,t)} could be the concentration of some substance in the neighborhood of point x {\displaystyle x} of the reaction medium. For any dimension d {\displaystyle d} (1, 2, or 3), the wave's domain is then a subset D {\displaystyle D} of R d {\displaystyle \mathbb {R} ^{d}} , such that the function value F ( x , t ) {\displaystyle F(x,t)} is defined for any point x {\displaystyle x} in D {\displaystyle D} . For example, when describing the motion of a drum skin, one can consider D {\displaystyle D} to be a disk (circle) on the plane R 2 {\displaystyle \mathbb {R} ^{2}} with center at the origin ( 0 , 0 ) {\displaystyle (0,0)} , and let F ( x , t ) {\displaystyle F(x,t)} be the vertical displacement of the skin at the point x {\displaystyle x} of D {\displaystyle D} and at time t {\displaystyle t} .
Superposition
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
A wave spectrum is a representation that describes how the energy of a sea surface (or wave field) is distributed across different wave frequencies (or wave numbers) and directions. Real ocean surfaces consist of many overlapping waves of different wavelengths, periods, amplitudes, and directions. A wave spectrum offers a statistical (spectral) description rather than tracking individual waves. The observed sea-surface elevation at a point can be thought of as the sum (superposition) of many sinusoidal wave components; the wave spectrum quantifies how much energy is associated with each component.
Wave families
Sometimes one is interested in a single specific wave. More often, however, one needs to understand a large set of possible waves; like all the ways that a drum skin can vibrate after being struck once with a drum stick, or all the possible radar echoes one could get from an airplane that may be approaching an airport.
In some of those situations, one may describe such a family of waves by a function F ( A , B , … ; x , t ) {\displaystyle F(A,B,\ldots ;x,t)} that depends on certain parameters A , B , … {\displaystyle A,B,\ldots } , besides x {\displaystyle x} and t {\displaystyle t} . Then one can obtain different waves – that is, different functions of x {\displaystyle x} and t {\displaystyle t} – by choosing different values for those parameters.For example, the sound pressure inside a recorder that is playing a "pure" note is typically a standing wave, that can be written as
F ( A , L , n , c ; x , t ) = A ( cos 2 π x 2 n − 1 4 L ) ( cos 2 π c t 2 n − 1 4 L ) {\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 parameter A {\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; and n {\displaystyle n} is a positive integer (1,2,3,...) that specifies the number of nodes in the standing wave. (The position x {\displaystyle x} should be measured from the mouthpiece, and the time t {\displaystyle t} from any moment at which the pressure at the mouthpiece is maximum. The quantity λ = 4 L / ( 2 n − 1 ) {\displaystyle \lambda =4L/(2n-1)} is the wavelength of the emitted note, and f = c / λ {\displaystyle f=c/\lambda } is its frequency.) Many general properties of these waves can be inferred from this general equation, without choosing specific values for the parameters. As another example, it may be that the vibrations of a drum skin after a single strike depend only on the distance r {\displaystyle r} from the center of the skin to the strike point, and on the strength s {\displaystyle s} of the strike. Then the vibration for all possible strikes can be described by a function F ( 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 function h {\displaystyle h} such that h ( x ) {\displaystyle h(x)} is the initial temperature at each point x {\displaystyle x} of the bar. Then the temperatures at later times can be expressed by a function F {\displaystyle F} that depends on the function h {\displaystyle h} (that is, a functional operator), so that the temperature at a later time is F ( h ; x , t ) {\displaystyle F(h;x,t)}
Differential wave equations Another way to describe and study a family of waves is to give a mathematical equation that, instead of explicitly giving the value of F ( 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 functions F {\displaystyle F} that satisfy those constraints – that is, all solutions of the equation. This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve. For example, if F ( x , t ) {\displaystyle F(x,t)} is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation
∂ F ∂ t ( x , t ) = α ( ∂ 2 F ∂ x 1 2 ( x , t ) + ∂ 2 F ∂ x 2 2 ( x , t ) + ∂ 2 F ∂ x 3 2 ( x , t ) ) + β Q ( 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)}
where Q ( x , t ) {\displaystyle Q(x,t)} is the heat that is being generated per unit of volume and time in the neighborhood of x {\displaystyle x} at time t {\displaystyle t} (for example, by chemical reactions happening there); x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} are the Cartesian coordinates of the point x {\displaystyle x} ; ∂ F / ∂ t {\displaystyle \partial F/\partial t} is the (first) derivative of F {\displaystyle F} with respect to t {\displaystyle t} ; and ∂ 2 F / ∂ x i 2 {\displaystyle \partial ^{2}F/\partial x_{i}^{2}} is the second derivative of F {\displaystyle F} relative to x i {\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 the diffusion of heat in solid media. For that reason, it is called the heat equation in mathematics, even though it applies to many other physical quantities besides temperatures. For another example, we can describe all possible sounds echoing within a container of gas by a function F ( x , t ) {\displaystyle F(x,t)} that gives the pressure at a point x {\displaystyle x} and time t {\displaystyle t} within that container. If the gas was initially at uniform temperature and composition, the evolution of F {\displaystyle F} is constrained by the formula
∂ 2 F ∂ t 2 ( x , t ) = α ( ∂ 2 F ∂ x 1 2 ( x , t ) + ∂ 2 F ∂ x 2 2 ( x , t ) + ∂ 2 F ∂ x 3 2 ( x , t ) ) + β P ( 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)}
Here P ( x , t ) {\displaystyle P(x,t)} is some extra compression force that is being applied to the gas near x {\displaystyle x} by some external process, such as a loudspeaker or piston right next to it. 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 ∂ 2 F / ∂ t 2 {\displaystyle \partial ^{2}F/\partial t^{2}} , the second derivative of F {\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 solutions F {\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
Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling
in the x {\displaystyle x} direction in space. For example, let the positive x {\displaystyle x} direction be to the right, and the negative x {\displaystyle x} direction be to the left. with constant amplitude u {\displaystyle u}
with constant velocity v {\displaystyle v} , where v {\displaystyle v} is independent of wavelength (no dispersion) independent of amplitude (linear media, not nonlinear). with constant waveform, or shape This wave can then be described by the two-dimensional functions
or, more generally, by d'Alembert's formula:
u ( x , t ) = F ( x − v t ) + G ( x + v t ) . {\displaystyle u(x,t)=F(x-vt)+G(x+vt).}
representing two component waveforms F {\displaystyle F} and G {\displaystyle G} traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation
1 v 2 ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 . {\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}.}
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 , t ) = A ( x , t ) sin ( k x − ω t + ϕ ) , {\displaystyle u(x,t)=A(x,t)\sin \left(kx-\omega t+\phi \right),}
where A ( x , t ) {\displaystyle A(x,t)} is the amplitude envelope of the wave, k {\displaystyle k} is the wavenumber and ϕ {\displaystyle \phi } is the phase. If the group velocity v g {\displaystyle v_{g}} (see below) is wavelength-independent, this equation can be simplified as:
u ( x , t ) = A ( x − v g t ) sin ( k x − ω t + 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 an envelope equation.
Phase velocity and group velocity
There are two velocities that are associated with waves, the phase velocity and the group velocity. Phase velocity is the rate at which the phase of the wave propagates in space: any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as
v p = λ T . {\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.} 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
Sine waves
Plane waves A plane 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 length n ^ {\displaystyle {\hat {\mathbf {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 {\mathbf {n} }}\cdot \mathbf {x} } ) and time ( t {\displaystyle t} ). Since the wave profile only depends on the position x {\displaystyle \mathbf {x} } in the combination n ^ ⋅ x {\displaystyle {\hat {\mathbf {n} }}\cdot \mathbf {x} } , any displacement in directions perpendicular to n ^ {\displaystyle {\hat {n}}} cannot affect the value of the field. Plane waves are often used to model electromagnetic 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 A standing wave, also known as a stationary wave, is a wave whose envelope remains in a constant position. This phenomenon arises 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.
Solitary waves A soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive 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 dispersive partial differential equations describing physical systems.
Physical properties
Propagation Wave propagation is any of the ways in which waves travel. With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves. Electromagnetic waves propagate in vacuum as well as in material media. Propagation of other wave types such as sound may occur only in a transmission medium.
Reflection of plane waves in a half-space The propagation and reflection of plane waves—e.g. Pressure waves (P wave) or Shear waves (SH or SV-waves) are phenomena that were first characterized within the field of classical seismology, and are now considered fundamental concepts in modern seismic tomography. The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding the Helmholtz decomposition of the displacement field, which is then substituted into the wave equation. From here, the plane wave eigenmodes can be calculated.
SV wave propagation
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.
P wave propagation
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.
Wave velocity
Wave velocity is a general concept, of various kinds of wave velocities, for a wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as:
v p = ω k , {\displaystyle v_{\text{p}}={\frac {\omega }{k}},}
where:
vp is the phase velocity (with SI unit m/s), ω is the angular frequency (with SI unit rad/s), k is the wavenumber (with SI unit rad/m). The phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency. The angular frequency ω cannot be chosen independently from the wavenumber k, but both are related through the dispersion relationship:
ω = Ω ( k ) . {\displaystyle \omega =\Omega (k).}
In the special case Ω(k) = ck, with c a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed c. For insta