Wave
In mathematics and physical science, a wave is a propagating dynamic disturbance of one or more quantities. Periodic waves oscillate repeatedly about an equilibrium value at some frequency. When the entire waveform moves in one direction, it is said to be a travelling wave; by contrast, a pair of 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 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, 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. Electromagnetic waves, as determined by their frequencies, 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, 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 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 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 ; 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 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 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 wave may propagate in non-linear, granular, or noisy medium 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 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 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. However, in many cases one can ignore one dimension, and let be a point of the Cartesian plane. This is the case, for example, when studying vibrations of a drum skin. One may even restrict to a point of the Cartesian line – that is, the set of real numbers. This is the case, for example, when studying vibrations in a violin string or recorder. The time, on the other hand, is always assumed to be a scalar; that is, a real number.
The value of can be any physical quantity of interest assigned to the point that may vary with time. 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. For an electromagnetic wave, the value of can be the electric field vector, or the magnetic field vector, or any related quantity, such as the Poynting vector. In fluid dynamics, the value of could be the velocity vector of the fluid at the point, or any scalar property like pressure, temperature, or density. In a chemical reaction, could be the concentration of some substance in the neighborhood of point of the reaction medium.
For any dimension , the wave's domain is then a subset of, such that the function value is defined for any point in. For example, when describing the motion of a drum skin, one can consider to be a disk on the plane with center at the origin, and let be the vertical displacement of the skin at the point of and at time.
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 is distributed across different wave frequencies and directions.Real ocean surfaces consist of many overlapping waves of different wavelengths, periods, amplitudes, and directions. A wave spectrum offers a statistical description rather than tracking individual waves. The observed sea-surface elevation at a point can be thought of as the sum 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 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 that depends on certain parameters, besides and. Then one can obtain different waves – that is, different functions of and – 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
The parameter defines the amplitude of the wave ; is the speed of sound; is the length of the bore; and is a positive integer that specifies the number of nodes in the standing wave. 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 from the center of the skin to the strike point, and on the strength of the strike. Then the vibration for all possible strikes can be described by a function.
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 such that is the initial temperature at each point of the bar. Then the temperatures at later times can be expressed by a function that depends on the function , so that the temperature at a later time is