![2006-01-14 Surface waves](https://upload.wikimedia.org/wikipedia/commons/4/43/2006-01-14_Surface_waves.jpg)
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
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
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. When the entire
waveform
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electronic ...
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
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
'' (standing wave field of two opposite waves) or a
one-way wave equation
A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resu ...
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
In physics, a mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a medium. While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefor ...
'',
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
and
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
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
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
waves are variations of the local
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
and
particle motion that propagate through the medium. Other examples of mechanical waves are
seismic wave
A seismic wave is a wave of acoustic energy that travels through the Earth. It can result from an earthquake, volcanic eruption, magma movement, a large landslide, and a large man-made explosion that produces low-frequency acoustic energy. S ...
s,
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 fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
. In an ''
electromagnetic wave
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, (visib ...
'' (such as light), coupling between the electric and magnetic fields which sustains propagation of a wave involving these fields according to
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
. Electromagnetic waves can travel through a
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
and through some
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
media (at wavelengths where they are considered
transparent
Transparency, transparence or transparent most often refer to:
* Transparency (optics), the physical property of allowing the transmission of light through a material
They may also refer to:
Literal uses
* Transparency (photography), a still, ...
). Electromagnetic waves, according to their frequencies (or
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
s) 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
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
,
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
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
,
ultraviolet radiation
Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30 PHz) to 400 nm (750 THz), shorter than that of visible light, but longer than X-rays. UV radiation i ...
,
X-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
s and
gamma ray
A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically ...
s.
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
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
that propagate according to
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
;
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
A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in ...
; and many more. Mechanical and electromagnetic waves transfer
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
,
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
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
, but they do not transfer particles in the medium. In mathematics and
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
waves are studied as
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
s. 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 jump
A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in th ...
s. Some, like the
probability wave
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
s of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
'' is an important mathematical idealization where the disturbance is identical along any (infinite) plane
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
to a specific direction of travel. Mathematically, the simplest wave is a ''
sinusoidal plane wave In physics, a sinusoidal (or monochromatic) plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane.
For any position \vec x in space and any time t, the val ...
'' in which at any point the field experiences
simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
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
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
'' 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
Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
'' 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'', which can be an important attribute.
Mathematical description
Single waves
A wave can be described just like a field, namely as a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. 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
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, or
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
. 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
A drumhead or drum skin is a membrane stretched over one or both of the open ends of a drum. The drumhead is struck with sticks, mallets, or hands, so that it vibrates and the sound resonates through the drum.
Additionally outside of percus ...
, 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
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
echos one could get from an
airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, propeller, or rocket engine. Airplanes come in a variety of sizes, shapes, and wing configurations. The broad spe ...
that may be approaching an
airport
An airport is an aerodrome with extended facilities, mostly for commercial air transport. Airports usually consists of a landing area, which comprises an aerially accessible open space including at least one operationally active surface ...
.
In some of those situations, one may describe such a family of waves by a function
that depends on certain
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s
, 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
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
of the emitted note, and
is its
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
.) 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
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Solutio ...
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
A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or " ...
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
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
in mathematics, even though it describes only one very special kind of waves.
Wave in elastic medium
Consider a traveling
transverse wave
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
(which may be a
pulse
In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the nec ...
) 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
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
* with constant velocity
, where
is
** independent of
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
(no
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
)
** independent of amplitude (
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
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
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electronic ...
, 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
In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation u_(x,t) = c^2 u_(x,t) (where subscript indices indicate partial differentiation, using the d' ...
:
:
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
A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resu ...
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
In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation u_(x,t) = c^2 u_(x,t) (where subscript indices indicate partial differentiation, using the d' ...
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
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
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
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
'' 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
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
(lambda) and period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
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
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
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
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 ...
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 extr ...
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
The violin, sometimes known as a ''fiddle'', is a wooden chordophone (string instrument) in the violin family. Most violins have a hollow wooden body. It is the smallest and thus highest-pitched instrument (soprano) in the family in regular ...
string is displaced, transverse waves propagate out to where the string is held in place at the bridge
A bridge is a structure built to span a physical obstacle (such as a body of water, valley, road, or rail) without blocking the way underneath. It is constructed for the purpose of providing passage over the obstacle, which is usually somethi ...
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 Co ...
, 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
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two or more curves, lines, ...
. Halfway between two nodes there is an antinode
A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effect ...
, 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
Fundamental may refer to:
* Foundation of reality
* Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental"
* Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
mode and the first 5 overtone
An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s.
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
A transmission medium is a system or substance that can mediate the propagation of signals for the purposes of telecommunication. Signals are typically imposed on a wave of some kind suitable for the chosen medium. For example, data can modulate ...
''. 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
Loss may refer to:
Arts, entertainment, and media Music
* ''Loss'' (Bass Communion album) (2006)
* ''Loss'' (Mull Historical Society album) (2001)
*"Loss", a song by God Is an Astronaut from their self-titled album (2008)
* Losses "(Lil Tjay son ...
. 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
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
which is complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
. 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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
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
Medium may refer to:
Science and technology
Aviation
*Medium bomber, a class of war plane
* Tecma Medium, a French hang glider design
Communication
* Media (communication), tools used to store and deliver information or data
* Medium of ...
into another. The amount by which a wave is refracted by a material is given by the refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
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
In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
.
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 wave
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
s 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
In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term ''linear polarizati ...
. 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
A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well ...
.
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
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
, the result of which is to produce the spectrum of colors of the rainbow. Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
performed experiments with light and prisms, presenting his findings in the ''Opticks
''Opticks: or, A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light'' is a book by English natural philosopher Isaac Newton that was published in English in 1704 (a scholarly Latin translation appeared in 1706). (''Optick ...
'' (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
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 ...
(''v'') is directly proportional to the square root of the tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
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
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
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
Ripple may refer to:
Science and technology
* Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid
** Ripple, more generally a disturbance, for example of spacetime in gravitational waves
* Ripple (electri ...
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
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
– 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
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
;
* Ocean surface 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 ...
s, 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
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of a wave in relation to an observer
An observer is one who engages in observation or in watching an experiment.
Observer may also refer to:
Computer science and information theory
* In information theory, any system which receives information from an object
* State observer in con ...
who is moving relative to the wave source. It is named after the Austria
Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
n physicist Christian Doppler
Christian Andreas Doppler ( (); 29 November 1803 – 17 March 1853) was an Austrian mathematician and physicist. He is celebrated for his principle – known as the Doppler effect – that the observed frequency of a wave depends on the relative ...
, 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
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
and density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of the medium.
Other
* Waves of traffic
Traffic comprises pedestrians, vehicles, ridden or herded animals, trains, and other conveyances that use public ways (roads) for travel and transportation.
Traffic laws govern and regulate traffic, while rules of the road include 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
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
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
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
showed that, in vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
, the electric and magnetic fields satisfy the wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
both with speed equal to that of the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. From this emerged the idea that light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves
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 ...
, microwaves
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency rang ...
, infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
, visible light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
, ultraviolet
Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nanometer, nm (with a corresponding frequency around 30 Hertz, PHz) to 400 nm (750 Hertz, THz), shorter than that of visible light, but longer than ...
, X-rays
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
, and Gamma rays
A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically ...
.
Quantum mechanical waves
Schrödinger equation
The Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
describes the wave-like behavior of particles in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Solutions of this equation are wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
s which can be used to describe the probability density of a particle.
Dirac equation
The Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac part ...
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
Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to quantum theory. In his 1924 PhD thesis, he postulated the wave na ...
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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Nowadays, this wavelength is called the de Broglie wavelength
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
. For example, the electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s in a CRT 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
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
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
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diffe ...
,[
] a waveform often used in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
to describe the wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
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
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diffe ...
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
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
''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
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that Wave propagation, propagate as waves outward from their source at the speed of light. They were first proposed by Oliv ...
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
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, predicted by Einstein's theory of general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.
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, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mea ...
, in fluid dynamics
* Capillary wave
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.
Capillary waves are common in nature, and are often referred to as ripples. The w ...
, in fluid dynamics
* Cnoidal wave
In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function ''cn'', which is why they are coined ''cn''oidal waves. They are ...
, in fluid dynamics
* Edge wave
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
, a surface gravity wave fixed by refraction against a rigid boundary
* Faraday wave
Faraday waves, also known as Faraday ripples, named after Michael Faraday (1791–1867), are nonlinear standing waves that appear on liquids enclosed by a vibrating receptacle. When the vibration frequency exceeds a critical value, the flat hydros ...
, 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
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
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 physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
, in aerodynamics
* Internal wave
Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change (continuously or discontinuously) with depth/height due to changes, for example, in ...
, a wave within a fluid medium
* Tidal wave, a scientifically incorrect name for a tsunami
A tsunami ( ; from ja, 津波, lit=harbour wave, ) is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater explo ...
* Tollmien–Schlichting wave In fluid dynamics, a Tollmien–Schlichting wave (often abbreviated T-S wave) is a streamwise unstable wave which arises in a bounded shear flow (such as boundary layer and channel flow). It is one of the more common methods by which a laminar bound ...
, in fluid dynamics
In quantum mechanics
In relativity
Other specific types of waves
* Alfvén wave
In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of plasma wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines.
Definition
An Alfvén wave is a ...
, in plasma physics
* Atmospheric wave
An atmospheric wave is a periodic disturbance in the fields of atmospheric variables (like surface pressure or geopotential height, temperature, or wind velocity) which may either propagate (''traveling wave'') or not (''standing wave''). Atmosp ...
, a periodic disturbance in the fields of atmospheric variables
* Fir wave
A fir wave is a set of alternating bands of fir trees in sequential stages of development, observed in forests on exposed mountain slopes in several areas, including northeastern North America and Japan. Fir waves develop by wave-regeneration fol ...
, a forest configuration
* Lamb waves Lamb waves propagate in solid plates or spheres. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the direction perpendicular to the plate. In 1917, the English mathematician Horace L ...
, in solid materials
* Rayleigh waves
Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They can be produced in materials in many ways, such as by a localized impact or by piezo-electric transduction, and are frequently used in non-destructive ...
, surface acoustic waves that travel on solids
* Spin wave
A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as ...
, in magnetism
* Spin density wave
Spin-density wave (SDW) and charge-density wave (CDW) are names for two similar low-energy ordered states of solids. Both these states occur at low temperature in anisotropic, low-dimensional materials or in metals that have high densities of st ...
, in solid materials
* Trojan wave packet, in particle science
* Waves in plasmas In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A Plasma (physics), plasma is a Plasma (physics)#Plasma_potential, quasineutral, electrical conductivity, elec ...
, in plasma physics
Related topics
* Beat (acoustics)
In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, ''perceived'' as a periodic variation in volume whose rate is the difference of the two frequencies.
With tuning instruments that can produce su ...
* 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
An envelope detector (sometimes called a peak detector) is an electronic circuit that takes a (relatively) high-frequency amplitude modulated signal as input and provides an output, which is the demodulated ''envelope'' of the original signal.
...
* 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
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 ...
* Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...
for computing periodicity in unevenly spaced data
* List of waves named after people
This is a list of waves named after people (eponymous waves).
See also
* Eponym
* List of eponymous laws
* Waves
* Scientific phenomena named after people
References
{{Reflist
Waves
Waves most often refers to:
* Waves, oscillations accomp ...
* 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
Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the s ...
* Resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
* 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
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
* Shallow water equations
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). T ...
* Shive wave machine
John Northrup Shive (February 22, 1913 – June 1, 1984) was an American physicist and inventor. He made notable contributions in electronic engineering and solid-state physics during the early days of transistor development at Bell Labora ...
* Sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
* 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
A transmission medium is a system or substance that can mediate the propagation of signals for the purposes of telecommunication. Signals are typically imposed on a wave of some kind suitable for the chosen medium. For example, data can modulate ...
* 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
*
*
*
* .
*
* .
*
External links
The Feynman Lectures on Physics: Waves
Linear and nonlinear waves
{{Authority control
Differential equations
Articles containing video clips