![2006-01-14 Surface waves](https://upload.wikimedia.org/wikipedia/commons/4/43/2006-01-14_Surface_waves.jpg)
In
physics,
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and related fields, a wave is a propagating dynamic disturbance (change from
equilibrium) of one or more
quantities
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
. Waves can be
periodic, in which case those quantities 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
superimposed
Superimposition is the placement of one thing over another, typically so that both are still evident.
Graphics
In graphics, superimposition is the placement of an image or video on top of an already-existing image or video, usually to add to ...
periodic waves traveling in opposite directions makes a ''
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
''. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. 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.
Two types of waves are most commonly studied in
classical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
. 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 wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
s,
surface wave
In physics, a surface wave is a mechanical wave that propagates along the Interface (chemistry), interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occu ...
s,
string vibration
A vibration in a strings (music), string is a wave. Acoustic resonance#Resonance of a string, Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch (music), pitch. If the length or tension of the strin ...
s, and
vortices. In an ''
electromagnetic wave'' (such as light), coupling between the electric and magnetic fields which sustains propagation of a wave 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, according to their frequencies (or
wavelengths) have more specific designations including
radio wave
Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz (GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (short ...
s,
infrared radiation,
terahertz waves
Terahertz radiation – also known as submillimeter radiation, terahertz waves, tremendously high frequency
(THF), T-rays, T-waves, T-light, T-lux or THz – consists of electromagnetic waves within the ITU-designated band of freq ...
,
visible light,
ultraviolet radiation,
X-rays and
gamma rays.
Other types of waves include
gravitational wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s, which are disturbances in
spacetime that propagate according to
general relativity;
heat diffusion waves;
plasma wave In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electron ...
s 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
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
, 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
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...
which do not move at all such as
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s (which are fundamental to music) and
hydraulic jumps. Some, like the
probability waves of
quantum mechanics, may be completely static.
![Santos E et al Neuroimage 2014](https://upload.wikimedia.org/wikipedia/commons/b/b3/Santos_E_et_al_Neuroimage_2014_.gif)
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
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
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
Polarization or polarisation may refer to:
Mathematics
*Polarization of an Abelian variety, in the mathematics of complex manifolds
*Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
'', which can be an important attribute.
Mathematical description
Single waves
A wave can be described just like a field, namely as a
function where
is a position and
is a time.
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
Recorder or The Recorder may refer to:
Newspapers
* ''Indianapolis Recorder'', a weekly newspaper
* ''The Recorder'' (Massachusetts newspaper), a daily newspaper published in Greenfield, Massachusetts, US
* ''The Recorder'' (Port Pirie), a news ...
. The time
, on the other hand, is always assumed to be a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
; 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
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
vector
, or the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
vector
, or any related quantity, such as the
Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
. In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, 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
(1, 2, or 3), the wave's domain is then a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
(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
.
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
A drum stick (or drumstick) is a type of percussion mallet used particularly for playing snare drum, drum kit, and some other percussion instruments, and particularly for playing unpitched percussion.
Specialized beaters used on some other pe ...
, or all the possible
radar echos 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.
![Half-open pipe wave](https://upload.wikimedia.org/wikipedia/commons/3/33/Half-open_pipe_wave.gif)
For example, the sound pressure inside a
recorder
Recorder or The Recorder may refer to:
Newspapers
* ''Indianapolis Recorder'', a weekly newspaper
* ''The Recorder'' (Massachusetts newspaper), a daily newspaper published in Greenfield, Massachusetts, US
* ''The Recorder'' (Port Pirie), a news ...
that is playing a "pure" note is typically a
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
, 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 general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two ...
in the standing wave. (The position
should be measured from the
mouthpiece
Mouthpiece may refer to:
* The part of an object which comes near or in contact with one's mouth or nose during use
** Mouthpiece (smoking pipe) or cigarette holder
** Mouthpiece (telephone handset)
** Mouthpiece (woodwind), a component of a woodw ...
, and the time
from any moment at which the pressure at the mouthpiece is maximum. The quantity
is the
wavelength of the emitted note, and
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
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
(that is, a
functional operator), so that the temperature at a later time is
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
, only constrains how those values can change with time. Then the family of waves in question consists of all functions
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
is the temperature inside a block of some
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
and
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
solid material, its evolution is constrained by the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
:
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
. (The symbol "
" 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 and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
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
that gives the pressure at a point
and time
within that container. 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
A piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tig ...
right next to
.
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
, the second derivative of
with respect to time, rather than the first derivative
. Yet this small change makes a huge difference on the set of solutions
. 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
![Superpositionprinciple](https://upload.wikimedia.org/wikipedia/commons/c/c8/Superpositionprinciple.gif)
* in the
direction in space. For example, let the positive
direction be to the right, and the negative
direction be to the left.
* with constant
amplitude
* with constant velocity
, where
is
** independent of
wavelength (no
dispersion)
** independent of amplitude (
linear media, not
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
).
* 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:
:
representing two component waveforms
and
traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
:
General solutions are based upon
Duhamel's principle In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to homogeneous differential equation, inhomogeneous linear evolution equations like the heat equation, wave equa ...
.
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.
Wave forms
![Waveforms](https://upload.wikimedia.org/wikipedia/commons/7/77/Waveforms.svg)
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
![Amplitudemodulation](https://upload.wikimedia.org/wikipedia/commons/9/97/Amplitudemodulation.gif)
The amplitude of a wave may be constant (in which case the wave is a ''c.w.'' or ''
continuous wave
A continuous wave or continuous waveform (CW) is an electromagnetic wave of constant amplitude and frequency, typically a sine wave, that for mathematical analysis is considered to be of infinite duration. It may refer to e.g. a laser or particle ...
''), 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:
[
]
:
where
is the amplitude envelope of the wave,
is the ''
wavenumber'' and
is the ''
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
''. If the
group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
(see below) is wavelength-independent, this equation can be simplified as:
:
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
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
and the group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
.
Phase velocity is the rate at which the phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
of the wave propagates in space: any given phase of the wave (for example, the crest
Crest or CREST may refer to:
Buildings
*The Crest (Huntington, New York), a historic house in Suffolk County, New York
*"The Crest", an alternate name for 63 Wall Street, in Manhattan, New York
*Crest Castle (Château Du Crest), Jussy, Switzerla ...
) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength (lambda) and period as
:
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 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 (). 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.
Plane waves are often used to model electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
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
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extra ...
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
Nut often refers to:
* Nut (fruit), fruit composed of a hard shell and a seed, or a collective noun for dry and edible fruits or seeds
* Nut (hardware), fastener used with a bolt
Nut or Nuts may also refer to:
Arts, entertainment, and media Com ...
, 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.
Image:Standing waves on a string.gif, One-dimensional standing waves; the fundamental mode and the first 5 overtones.
Image:Drum vibration mode01.gif, A two-dimensional standing wave on a disk; this is the fundamental mode.
Image:Drum vibration mode21.gif, A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.
Physical properties
Waves exhibit common behaviors under a number of standard situations, for example:
Transmission and media
Waves normally move in a straight line (that is, rectilinearly) through a '' transmission medium''. 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
Waves are usually defined in media which allow most or all of a wave's energy to propagate without loss. However materials may be characterized as "lossy" if they remove energy from a wave, usually converting it into heat. This is termed "absorption." 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.
Reflection
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.
Refraction
Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
.
Diffraction
A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.
Interference
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 weren't present. However at any point ''in'' that region the ''field quantities'' describing those waves add according to the superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
. If the waves are of the same frequency in a fixed phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
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''. This is called an interference pattern.
Polarization
The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. 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. Electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter.
Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.
Dispersion
A wave undergoes dispersion when either the phase velocity
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
or the group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
depends on the wave frequency.
Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce the spectrum of colors of the rainbow. Isaac Newton performed experiments with light and prisms, presenting his findings in the '' Opticks'' (1704) that white light consists of several colors and that these colors cannot be decomposed any further.[
]
Mechanical waves
Waves on strings
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
Linear density is the measure of a quantity of any characteristic value per unit of length. Linear mass density (titer in textile engineering, the amount of mass per unit length) and linear charge density (the amount of electric charge per unit ...
(''μ''):
:
where the linear density ''μ'' is the mass per unit length of the string.
Acoustic waves
Acoustic or sound waves travel at speed given by
:
or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
).
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.
* Sound – a mechanical wave that propagates through gases, liquids, solids and plasmas;
* Inertial waves
Inertial waves, also known as inertial oscillations, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow through the interior of the fluid, not ...
, which occur in rotating fluids and are restored by the Coriolis effect;
* Ocean surface waves, which are perturbations that propagate through water.
Seismic waves
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.
Doppler effect
The Doppler effect (or the Doppler shift) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.
Shock waves
A shock wave is a type of propagating disturbance. When a wave moves faster than the local speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
in a fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
, it is a shock wave. 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.
Other
* Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves[ And: ]
* Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.
Electromagnetic waves
An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. 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. From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and Gamma rays.
Quantum mechanical waves
Schrödinger equation
The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle.
Dirac equation
The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. 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. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin- particles.
de Broglie waves
Louis de Broglie postulated that all particles with momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
have a wavelength
:
where ''h'' is Planck's constant, and ''p'' is the magnitude of the momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT
CRT or Crt may refer to:
Science, technology, and mathematics Medicine and biology
* Calreticulin, a protein
*Capillary refill time, for blood to refill capillaries
*Cardiac resynchronization therapy and CRT defibrillator (CRT-D)
* Catheter-re ...
display have a de Broglie wavelength of about 10−13 m.
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. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet,[
] a waveform often used in quantum mechanics to describe the wave function of a particle. 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.
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''.[
See for example and ,.
] Gaussian wave packets also are used to analyze water waves.[
]
For example, a Gaussian wavefunction ''ψ'' might take the form:[
]
:
at some initial time ''t'' = 0, where the central wavelength is related to the central wave vector ''k''0 as λ0 = 2π / ''k''0. It is well known from the theory of Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
,[
] or from the Heisenberg uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
(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. The Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of a Gaussian is itself a Gaussian.[
] 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 ''k''λ = 2 π.
The parameter σ decides the spatial spread of the Gaussian along the ''x''-axis, while the Fourier transform shows a spread in wave vector ''k'' determined by 1/''σ''. That is, the smaller the extent in space, the larger the extent in ''k'', and hence in λ = 2π/''k''.
Gravity waves
Gravity waves
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. A ripple on a pond is one example.
Gravitational waves
Gravitational waves also travel through space. The first observation of gravitational waves was announced on 11 February 2016.
Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.
See also
* Index of wave articles
This is a list of Wave topics.
0–9
*21 cm line
A
*Abbe prism
*Absorption spectroscopy
*Absorption spectrum
*Absorption wavemeter
*Acoustic wave
*Acoustic wave equation
*Acoustics
*Acousto-optic effect
*Acousto-optic modulator
*Acousto- ...
Waves in general
Parameters
Waveforms
Electromagnetic waves
In fluids
* Airy wave theory, in fluid dynamics
* Capillary wave, in fluid dynamics
* Cnoidal wave, in fluid dynamics
* Edge wave, a surface gravity wave fixed by refraction against a rigid boundary
* Faraday wave, a type of wave in liquids
* Gravity wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
, in fluid dynamics
* Sound wave, a wave of sound through a medium such as air or water
* Sea wave spectrum
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...
* Shock wave, in aerodynamics
* Internal wave, a wave within a fluid medium
* Tidal wave, a scientifically incorrect name for a tsunami
* Tollmien–Schlichting wave, in fluid dynamics
In quantum mechanics
In relativity
Other specific types of waves
* Alfvén wave, in plasma physics
* Atmospheric wave, a periodic disturbance in the fields of atmospheric variables
* Fir wave, a forest configuration
* Lamb waves, in solid materials
* Rayleigh waves, surface acoustic waves that travel on solids
* Spin wave, in magnetism
* Spin density wave, in solid materials
* Trojan wave packet
A trojan wave packet is a wave packet that is nonstationary and nonspreading. It is part of an artificially created system that consists of a nucleus and one or more electron wave packets, and that is highly excited under a continuous electromagn ...
, in particle science
* Waves in plasmas, in plasma physics
Related topics
* Beat (acoustics)
* Cymatics
Cymatics (from grc, κῦμα, translit=kyma, translation=wave) is a subset of modal vibrational phenomena. The term was coined by Hans Jenny (1904-1972), a Swiss follower of the philosophical school known as anthroposophy. Typically the surf ...
* Doppler effect
The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...
* Envelope detector
* Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
for computing periodicity in evenly spaced data
* Group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
* Harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
* Index of wave articles
This is a list of Wave topics.
0–9
*21 cm line
A
*Abbe prism
*Absorption spectroscopy
*Absorption spectrum
*Absorption wavemeter
*Acoustic wave
*Acoustic wave equation
*Acoustics
*Acousto-optic effect
*Acousto-optic modulator
*Acousto- ...
* Inertial wave
* Least-squares spectral analysis for computing periodicity in unevenly spaced data
* List of waves named after people
* Phase velocity
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
* Reaction–diffusion system
* Resonance
* Ripple tank
In physics, a ripple tank is a shallow glass tank of water used to demonstrate the basic properties of waves. It is a specialized form of a wave tank. The ripple tank is usually illuminated from above, so that the light shines through the water. ...
* Rogue wave
* Shallow water equations
* Shive wave machine
* Sound
* Standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
* Transmission medium
* Wave turbulence
In continuum mechanics, wave turbulence is a set of nonlinear waves deviated far from thermal equilibrium. Such a state is usually accompanied by dissipation. It is either decaying turbulence or requires an external source of energy to sustain it. ...
* Wind wave
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...
References
Sources
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External links
The Feynman Lectures on Physics: Waves
Linear and nonlinear waves
{{Authority control
Differential equations
Articles containing video clips