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71-595: A wave is a disturbance that transfers energy through matter or space. Wave or waves may also refer to: Wave 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

142-401: 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. 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

213-470: 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

284-428: A transmission medium . 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

355-634: 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

426-413: 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

497-618: 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,

568-402: 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

639-537: 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

710-442: 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. 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

781-517: A longitudinal component in the electric and/or magnetic fields when traversing birefringent materials, or inhomogeneous materials especially at interfaces (surface waves for instance) such as Zenneck waves . In the development of modern physics, Alexandru Proca (1897–1955) was known for developing relativistic quantum field equations bearing his name (Proca's equations) which apply to the massive vector spin-1 mesons. In recent decades some other theorists, such as Jean-Pierre Vigier and Bo Lehnert of

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852-410: A sum of sine waves of various frequencies, relative phases, and magnitudes. 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 {n}}} indicating the direction that

923-401: 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. Waves exhibit common behaviors under a number of standard situations, for example: Longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel to the direction the wave travels and displacement of

994-635: 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)}

1065-456: A wave propagating through a poly-crystal crosses a grain boundary, a scattering event occurs causing scattering based attenuation of the wave. Additionally it has been shown that the ratio rule for viscoelastic materials, applies equally successfully to polycrystalline materials. A current prediction for modeling attenuation of waves in polycrystalline materials with elongated grains is the second-order approximation (SOA) model which accounts

1136-435: A wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as: v p = ω k , {\displaystyle v_{\rm {p}}={\frac {\omega }{k}},} where: 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

1207-563: Is a good visualization. Real-world examples include sound waves ( vibrations in pressure, a particle of displacement, and particle velocity propagated in an elastic medium) and seismic P-waves (created by earthquakes and explosions). The other main type of wave is the transverse wave , in which the displacements of the medium are at right angles to the direction of propagation. Transverse waves, for instance, describe some bulk sound waves in solid materials (but not in fluids ); these are also called " shear waves" to differentiate them from

1278-582: 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

1349-440: 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

1420-494: 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

1491-538: 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

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1562-448: 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

1633-458: 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

1704-520: 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 } "

1775-648: 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

1846-417: 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

1917-549: 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}

1988-536: 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)}

2059-495: The Belousov–Zhabotinsky reaction ; and many more. Mechanical and electromagnetic waves transfer energy , momentum , and information , but they do not transfer particles in the medium. 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

2130-463: The Helmholtz decomposition of the displacement field, which is then substituted into the wave equation . From here, the plane wave eigenmodes can be calculated. 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

2201-454: The electric field vector E {\displaystyle E} , or the magnetic field vector H {\displaystyle H} , or any related quantity, such as the Poynting vector E × H {\displaystyle E\times H} . In fluid dynamics , the value of F ( x , t ) {\displaystyle F(x,t)} could be

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2272-442: The frequency and wavelength can be described by the formula where: The quantity     x   c   {\displaystyle \ {\frac {\ x\ }{c}}\ } is the time that the wave takes to travel the distance   x   . {\displaystyle \ x~.} The ordinary frequency (   f   {\displaystyle \ f\ } ) of

2343-586: The (longitudinal) pressure waves that these materials also support. "Longitudinal waves" and "transverse waves" have been abbreviated by some authors as "L-waves" and "T-waves", respectively, for their own convenience. While these two abbreviations have specific meanings in seismology (L-wave for Love wave or long wave ) and electrocardiography (see T wave ), some authors chose to use "ℓ-waves" (lowercase 'L') and "t-waves" instead, although they are not commonly found in physics writings except for some popular science books. For longitudinal harmonic sound waves,

2414-472: 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. 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 is a general concept, of various kinds of wave velocities, for

2485-468: 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 ),

2556-400: 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)} Another way to describe and study

2627-430: 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

2698-435: The combination n ^ ⋅ x → {\displaystyle {\hat {n}}\cdot {\vec {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,

2769-611: 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 instance electromagnetic , sound or water waves). The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation : v g = ∂ ω ∂ k {\displaystyle v_{\rm {g}}={\frac {\partial \omega }{\partial k}}} In almost all cases,

2840-435: The electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other. 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

2911-446: The electromagnetic field. After Heaviside 's attempts to generalize Maxwell's equations , Heaviside concluded that electromagnetic waves were not to be found as longitudinal waves in " free space " or homogeneous media. Maxwell's equations, as we now understand them, retain that conclusion: in free-space or other uniform isotropic dielectrics, electro-magnetic waves are strictly transverse. However electromagnetic waves can display

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2982-428: 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 . 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

3053-495: 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 where Q ( p , f ) {\displaystyle Q(p,f)}

3124-459: The following ratio: where   c T   {\displaystyle \ c_{T}\ } and   c ℓ   {\displaystyle \ c_{\ell }\ } are the transverse and longitudinal wave speeds respectively. Polycrystalline materials are made up of various crystal grains which form the bulk material. Due to the difference in crystal structure and properties of these grains, when

3195-445: The formula 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 p {\displaystyle p} . This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in

3266-415: 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

3337-589: 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 . The form or shape of F in d'Alembert's formula involves

3408-444: The medium is in the same (or opposite) direction of the wave propagation . Mechanical longitudinal waves are also called compressional or compression waves , because they produce compression and rarefaction when travelling through a medium, and pressure waves , because they produce increases and decreases in pressure . A wave along the length of a stretched Slinky toy, where the distance between coils increases and decreases,

3479-624: 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. 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

3550-419: 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

3621-413: 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,

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3692-609: The overall shape of the waves' amplitudes—modulation or envelope of the wave. A sine wave , sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function . In mechanics , as a linear motion over time, this is simple harmonic motion ; as rotation , it corresponds to uniform circular motion . Sine waves occur often in physics , including wind waves , sound waves, and light waves, such as monochromatic radiation . In engineering , signal processing , and mathematics , Fourier analysis decomposes general functions into

3763-476: 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 . The amplitude of

3834-426: The prediction of electromagnetic waves in a vacuum, which are strictly transverse waves ; due to the fact that they would need particles to vibrate upon, the electric and magnetic fields of which the wave consists are perpendicular to the direction of the wave's propagation. However plasma waves are longitudinal since these are not electromagnetic waves but density waves of charged particles, but which can couple to

3905-438: The propagation direction is also referred to as the wave's polarization , which can be an important attribute. A wave can be described just like a field, namely as a function F ( x , t ) {\displaystyle F(x,t)} where x {\displaystyle x} is a position and t {\displaystyle t} is a time. The value of x {\displaystyle x}

3976-441: The same, so the wave form will change over time and space. 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 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

4047-422: The second order of inhomogeneity allowing for the consideration multiple scattering in the crystal system. This model predicts that the shape of the grains in a poly-crystal has little effect on attenuation. The equations for sound in a fluid given above also apply to acoustic waves in an elastic solid. Although solids also support transverse waves (known as S-waves in seismology ), longitudinal sound waves in

4118-420: The skin at the point x {\displaystyle x} of D {\displaystyle D} and at time t {\displaystyle t} . 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

4189-481: The solid exist with a velocity and wave impedance dependent on the material's density and its rigidity , the latter of which is described (as with sound in a gas) by the material's bulk modulus . In May 2022, NASA reported the sonification (converting astronomical data associated with pressure waves into sound ) of the black hole at the center of the Perseus galaxy cluster . Maxwell's equations lead to

4260-423: The sound pressure inside a recorder that is playing a "pure" note is typically a standing wave , that can be written as 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}

4331-437: The speed of a longitudinal wave can be described by where The attenuation of a wave in a medium describes the loss of energy a wave carries as it propagates throughout the medium. This is caused by the scattering of the wave at interfaces, the loss of energy due to the friction between molecules, or geometric divergence. The study of attenuation of elastic waves in materials has increased in recent years, particularly within

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4402-646: The study of polycrystalline materials where researchers aim to "nondestructively evaluate the degree of damage of engineering components" and to "develop improved procedures for characterizing microstructures" according to a research team led by R. Bruce Thompson in a Wave Motion publication. In viscoelastic materials, the attenuation coefficients per length   α ℓ   {\displaystyle \ \alpha _{\ell }\ } for longitudinal waves and   α T   {\displaystyle \ \alpha _{T}\ } for transverse waves must satisfy

4473-463: 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

4544-415: 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

4615-418: 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

4686-485: 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),

4757-441: 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

4828-431: The wave is given by The wavelength can be calculated as the relation between a wave's speed and ordinary frequency. For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave. Sound's propagation speed depends on the type, temperature, and composition of the medium through which it propagates. For isotropic solids and liquids,

4899-458: 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 position x → {\displaystyle {\vec {x}}} in

4970-419: 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

5041-438: 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 instance electromagnetic waves in vacuum are non-dispersive. In case of other forms of

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