Wave

In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities.

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

Electromagnetic waves can travel through a vacuum and through some dielectric media (at wavelengths where they are considered transparent).

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.

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 temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation where

This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in a homogeneous isotropic non-conducting solid.

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

In engineering, signal processing, and mathematics, Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.

For electromagnetic plane waves, the electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other.

Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

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.

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:

However materials may be characterized as "lossy" if they remove energy from a wave, usually converting it into heat.

A material which absorbs a wave's energy, either in transmission or reflection, is characterized by a refractive index which is complex.

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

Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.

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

The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions.

Dispersion is the frequency dependence of the refractive index, a consequence of the atomic nature of materials.

Dispersion is seen by letting white light pass through a prism, the result of which is to produce the spectrum of colors of the rainbow.

Body waves travel through the interior of the medium along paths controlled by the material properties in terms of density and modulus (stiffness).

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.

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.

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.

[32] Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

Surface waves in water showing water ripples
Example of biological waves expanding over the brain cortex, an example of spreading depolarizations . [ 3 ]
Sound pressure standing wave in a half-open pipe playing the 7th harmonic of the fundamental ( n = 4)
Wavelength λ 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 that f ( x , t ) + g ( x , t ) = u ( x , t ) .
Sine , square , triangle and sawtooth waveforms
Amplitude modulation can be achieved through f ( x , t ) = 1.00×sin(2π/0.10×( x −1.00× t )) and g ( x , t ) = 1.00×sin(2π/0.11×( x −1.00× t )) only the resultant is visible to improve clarity of waveform.
Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.
The red square moves with the phase velocity , while the green circles propagate with the group velocity .
A wave with the group and phase velocities going in different directions
Tracing the y component of a circle while going around the circle results in a sine wave (red). Tracing the x component results in a cosine wave (blue). Both waves are sinusoids of the same frequency but different phases.
Standing wave. The red dots represent the wave nodes .
Solitary wave in a laboratory wave channel
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 ]
Seismic wave propagation in 2D modelled using FDTD method in the presence of a landmine
Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism
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
Identical waves from two sources undergoing interference . 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.
Schematic of light being dispersed by a prism. Click to see animation.
Formation of a shock wave by a plane
A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves. [ 23 ]
Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles