• The phase velocity is given in terms of the wavelength λ (lambda) and period T as 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.

• Plane waves can be specified by a vector of unit length 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 () and time ().

• 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: In almost all cases, a wave is mainly a movement of energy through a medium.

• Mathematical description Single waves [edit] See also: Solitary wave A wave can be described just like a field, namely as a function where is a position and is a time.

• Sound pressure standing wave in a half-open pipe playing the 7th harmonic of the fundamental (n = 4) For example, the sound pressure inside a recorder that is playing a “pure”
  note is typically a standing wave, that can be written as The parameter defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); is the speed of sound; is the length
  of the bore; and is a positive integer (1,2,3,…) that specifies the number of nodes in the standing wave.

• Waves are often described by a wave equation (standing wave field of two opposite waves) or a one-way wave equation for single wave propagation in a defined direction.

• Then the temperatures at later times can be expressed by a function that depends on the function (that is, a functional operator), so that the temperature at a later time
  is Differential wave equations [edit] Another way to describe and study a family of waves is to give a mathematical equation that, instead of explicitly giving the value of , only constrains how those values can change with time.

• [clarification needed] Wave velocity [edit] Further information: Phase velocity, Group velocity, and Signal velocity Seismic wave propagation in 2D modelled using FDTD method
  in the presence of a landmine 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 angular frequency ω cannot be chosen independently from the wavenumber k, but both are related through the dispersion relationship: 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.

• Since the wave profile only depends on the position in the combination , any displacement in directions perpendicular to cannot affect the value of the field.

• Mathematically, the simplest wave is a sinusoidal plane wave in which at any point the field experiences simple harmonic motion at one frequency.

• For example, if represents the vibrations inside an elastic solid, the value of is usually a vector that gives the current displacement from of the material particles that
  would be at the point in the absence of vibration.

• In general, the velocities are not the same, so the wave form will change over time and space.

• Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.

• For example, if is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation where
  is the heat that is being generated per unit of volume and time in the neighborhood of at time (for example, by chemical reactions happening there); are the Cartesian coordinates of the point ; is the (first) derivative of with respect to
  ; and is the second derivative of relative to .

• Wave spectrum [edit] See also: Wind wave § Spectrum, Electromagnetic spectrum, and Spectrum (physical sciences) Wave families [edit] Sometimes one is interested in a single
  specific wave.

• [9] In the case of a periodic function F with period λ, that is, 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).

• [8] Beside the second order wave equations that are describing a standing wave field, the one-way wave equation describes the propagation of single wave in a defined direction.

• 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.

• Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave.

• 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 phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency.

• [4][5] • with constant waveform, or shape This wave can then be described by the two-dimensional functions • (waveform traveling to the right) • (waveform traveling to the
  left) or, more generally, by d’Alembert’s formula:[6]representing two component waveforms and traveling through the medium in opposite directions.

• 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.

• Superposition [edit] Main article: Superposition principle Waves of the same type are often superposed and encountered simultaneously at a given point in space and time.

• A plane wave is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific direction of travel.

• Plane waves [edit] Main article: Plane wave A plane wave is a kind of wave whose value varies only in one spatial direction.

• This differential equation is called “the” wave equation in mathematics, even though it describes only one very special kind of waves.

• In some of those situations, one may describe such a family of waves by a function that depends on certain parameters , besides and .

• 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).

• In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero.

• Plane waves are often used to model electromagnetic waves far from a source.

• For another example, we can describe all possible sounds echoing within a container of gas by a function that gives the pressure at a point and time within that container.

• Then the vibration for all possible strikes can be described by a function .

• For example, when describing the motion of a drum skin, one can consider to be a disk (circle) on the plane with center at the origin , and let be the vertical displacement
  of the skin at the point of and at time .

• (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.)

• When any two sine waves of the same frequency (but arbitrary phase) are linearly combined, the result is another sine wave of the same frequency; this property is unique among
  periodic waves.

• 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).

• Wave in elastic medium Consider a traveling transverse wave (which may be a pulse) on a string (the medium).

• [14][15] Phase velocity and group velocity [edit] Main articles: Phase velocity and Group velocity See also: Envelope (waves) § Phase and group velocity The red square moves
  with the phase velocity, while the green circles propagate with the group velocity.

• For example, sound waves are variations of the local pressure and particle motion that propagate through the medium.

• 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.

• 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.

• [3] A physical wave field is almost always confined to some finite region of space, called its domain.

• If the gas was initially at uniform temperature and composition, the evolution of is constrained by the formula Here is some extra compression force that is being applied
  to the gas near by some external process, such as a loudspeaker or piston right next to .

• 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.

• This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve.

• Note that this equation differs from that of heat flow only in that the left-hand side is , the second derivative of with respect to time, rather than the first derivative
  .

• Propagation of other wave types such as sound may occur only in a transmission medium.

• In that case, instead of a scalar or vector, the parameter would have to be a function such that is the initial temperature at each point of the bar.

• For that reason, it is called the heat equation in mathematics, even though it applies to many other physical quantities besides temperatures.

• This phenomenon arises as a result of interference between two waves traveling in opposite directions.

• It is well known from the theory of Fourier analysis,[30] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths
  is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths.

• To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet,[24] a waveform often used in quantum
  mechanics to describe the wave function of a particle.

• • Sound – a mechanical wave that propagates through gases, liquids, solids and plasmas; • Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;
  • Ocean surface waves, which are perturbations that propagate through water Body waves [edit] 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).

• Reflection [edit] Main article: Reflection (physics) When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line
  normal to the surface equals the angle made by the reflected wave and the same normal line.

• Like an ordinary wave, a shock wave carries energy and can propagate through a medium; however, it is characterized by an abrupt, nearly discontinuous change in pressure,
  temperature and density of the medium.

• A wave representing such a particle traveling in the k-direction is expressed by the wave function as follows: where the wavelength is determined by the wave vector k as:
  and the momentum by: However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space.

• Refraction [edit] 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.

• Gravity waves 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.

• Such media can be classified into one or more of the following categories: • A bounded medium if it is finite in extent, otherwise an unbounded medium • A linear medium if
  the amplitudes of different waves at any particular point in the medium can be added • A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space • An anisotropic medium if one or more of
  its physical properties differ in one or more directions • An isotropic medium if its physical properties are the same in all directions Absorption [edit] Main articles: Absorption (acoustics) and Absorption (electromagnetic radiation) Waves
  are usually defined in media which allow most or all of a wave’s energy to propagate without loss.

• Acoustic waves [edit] Main article: Acoustic wave Acoustic or sound waves are compression waves which travel as body waves at the speed given by: or the square root of the
  adiabatic bulk modulus divided by the ambient density of the medium (see speed of sound).

• Other [edit] • Waves of traffic, 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.

• In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet.

• The speed of a transverse wave traveling along a vibrating string (v) is directly proportional to the square root of the tension of the string (T) over the linear mass density
  (μ): where the linear density μ is the mass per unit length of the string.

• [28] For example, a Gaussian wavefunction ψ might take the form:[29] at some initial time t = 0, where the central wavelength is related to the central wave vector k0 as .

• 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.

• Electromagnetic waves can have different frequencies (and thus wavelengths), and are classified accordingly in wavebands, such as radio waves, microwaves, infrared, visible
  light, ultraviolet, X-rays, and gamma rays.

• However at any point in that region the field quantities describing those waves add according to the superposition principle.

• 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.

• [31] Given the Gaussian: the Fourier transform is: The Gaussian in space therefore is made up of waves: that is, a number of waves of wavelengths λ such that.

• The amount of absorption will generally depend on the frequency (wavelength) of the wave, which, for instance, explains why objects may appear colored.

• Interference [edit] Main article: Wave interference Identical waves from two sources undergoing interference.

• If the waves are of the same frequency in a fixed phase relationship, then there will generally be positions at which the two waves are in phase and their amplitudes add,
  and other positions where they are out of phase and their amplitudes (partially or fully) cancel.

• Polarization [edit] Main article: Polarization (waves) The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions.

• Water waves [edit] 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.

• Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way.

• In the 19th century, James Clerk Maxwell showed that, in vacuum, the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light.

• Mechanical waves A mechanical wave is an oscillation of matter, and therefore transfers energy through a medium.

• 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.

• Main articles: Dispersion relation, Dispersion (optics), and Dispersion (water waves) A wave undergoes dispersion when either the phase velocity or the group velocity depends
  on the wave frequency.

 

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