TheInfoList

In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, 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 Phase or phases may refer to: Science * State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter) In the physical sciences, a phase is a region of space (a thermodynamic system A thermodynamic system is a ...
on the wave, such as two adjacent crests, troughs, or
zero crossing A zero-crossing is a point where the sign of a mathematical function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ar ...

s, and is a characteristic of both traveling waves and
standing wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

s, as well as other spatial wave patterns. The
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...

of the wavelength is called the
spatial frequency In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Wavelength is commonly designated by the
Greek letter The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European languages, Indo-European family of languages, nat ...
''
lambda Lambda (; uppercase , lowercase ; el, λάμ(β)δα, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the sound Dental, alveolar and postalveolar lateral approximants, /l/. In the system of Greek numerals, lambda has a v ...

'' (λ). The term ''wavelength'' is also sometimes applied to
modulated In electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. It uses active devices to control electron flow by amplifier, amplifi ...

waves, and to the sinusoidal
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 (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one of ...
of modulated waves or waves formed by
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 extran ...
of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to
frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. Wavelength depends on the medium (for example, vacuum, air, or water) that a wave travels through. Examples of waves are
sound wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...

s,
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 ...

,
water wave In fluid dynamics, a wind wave, or wind-generated wave, is a water surface wave that occurs on the free surface of Body of water, bodies of water. Wind waves result from the wind blowing over a fluid surface, where the contact distance in the d ...
s and periodic electrical signals in a
conductor Conductor or conduction may refer to: Music * Conductor (music), a person who leads a musical ensemble like, for example, an orchestra. * Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas * Conduction, a type of ...
. A
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 b ...

wave is a variation in air
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...
, while in
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 ...

and other
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

the strength of the
electric Electricity is the set of physics, physical Phenomenon, 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 electromagnet ...

and the
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

vary. Water waves are variations in the height of a body of water. In a crystal
lattice vibration In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
, atomic positions vary. The range of wavelengths or frequencies for wave phenomena is called a
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a Continuum (theory), continuum. The word was first used scientifically in optics to describe the ...

. The name originated with the
visible light spectrum File:Light Amplification by Stimulated Emission of Radiation.jpg, Laser beams with visible spectrum The visible spectrum is the portion of the electromagnetic spectrum that is visual perception, visible to the human eye. Electromagnetic radiation ...
but now can be applied to the entire
electromagnetic spectrum The electromagnetic spectrum is the range of frequencies Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existe ...

as well as to a
sound spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without steps, across a Continuum (theory), continuum. The word was first used scientifically in optics Optics is th ...
or
vibration spectrumA molecular vibration is a periodic motion of the atoms An atom is the smallest unit of ordinary matter that forms a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical e ...
.

# Sinusoidal waves

In
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength ''λ'' of a sinusoidal waveform traveling at constant speed ''v'' is given by :$\lambda = \frac\,\,,$ where ''v'' is called the phase speed (magnitude of the
phase velocity in groups of gravity wave , Croatia in July 2009. Image:wave clouds.jpg, Wave clouds over Theresa, Wisconsin, United States in August 2005. In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface (matter), inter ...

) of the wave and ''f'' is the wave's
frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

. In a
dispersive mediumA dispersive medium is a medium in which waves of different frequencies travel at different velocities. With electromagnetic radiation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of ...
, the phase speed itself depends upon the frequency of the wave, making the nonlinear. In the case of
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

—such as light—in
free space A vacuum is space devoid of matter. The word stems from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicist ...
, the phase speed is the
speed of light The speed of light in vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure m ...
, about 3×108 m/s. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep
red Red is the color at the long wavelength end of the visible spectrum of light, next to orange and opposite violet. It has a dominant wavelength Image:dominant wavelength.png, frame, Dominant/complementary wavelength example on the CIE color ...

, roughly 700 nm, to
violet Violet may refer to: Common meanings * Violet (color), a spectral color with wavelengths shorter than blue * One of a list of plants known as violet, particularly: ** Viola (plant), ''Viola'' (plant), a genus of flowering plants Places United ...
, roughly 400 nm (for other examples, see
electromagnetic spectrum The electromagnetic spectrum is the range of frequencies Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existe ...

). For
sound wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...

s in air, the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elasticity (solid mechanics), elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends s ...
is 343 m/s (at room temperature and atmospheric pressure). The wavelengths of sound frequencies audible to the human ear (20 –20 kHz) are thus between approximately 17  m and 17  mm, respectively. Somewhat higher frequencies are used by
bat Bats are mammal Mammals (from Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the po ...

s so they can resolve targets smaller than 17 mm. Wavelengths in audible sound are much longer than those in visible light.

## Standing waves

A
standing wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called
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 *Node (autonomous system), behaviour fo ...
, and the wavelength is twice the distance between nodes. The upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box (an example of
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the di ...
) determining which wavelengths are allowed. For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall. The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Consequently, wavelength, period, and wave velocity are related just as for a traveling wave. For example, the
speed of light The speed of light in vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure m ...
can be determined from observation of standing waves in a metal box containing an ideal vacuum.

## Mathematical representation

Traveling sinusoidal waves are often represented mathematically in terms of their velocity ''v'' (in the x direction), frequency ''f'' and wavelength ''λ'' as: :$y \left(x, \ t\right) = A \cos \left\left( 2 \pi \left\left( \frac - ft \right \right) \right \right) = A \cos \left\left( \frac \left(x - vt\right) \right \right)$ where ''y'' is the value of the wave at any position ''x'' and time ''t'', and ''A'' is the
amplitude The amplitude of a 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 * Period, a descriptor for ...

of the wave. They are also commonly expressed in terms of
wavenumber In the physical science Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical scienc ...
''k'' (2π times the reciprocal of wavelength) and
angular frequency In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
''ω'' (2π times the frequency) as: :$y \left(x, \ t\right) = A \cos \left\left( kx - \omega t \right\right) = A \cos \left\left(k\left(x - v t\right) \right\right)$ in which wavelength and wavenumber are related to velocity and frequency as: :$k = \frac = \frac = \frac,$ or :$\lambda = \frac = \frac = \frac.$ In the second form given above, the phase is often generalized to , by replacing the wavenumber ''k'' with a
wave vector In , a wave vector (also spelled wavevector) is a which helps describe a . Like any vector, it has a , both of which are important. Its magnitude is either the or of the wave (inversely proportional to the ), and its direction is ordinarily the ...
that specifies the direction and wavenumber of a
plane wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

in
3-space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., Point (mathematics), point). This is the info ...
, parameterized by position vector r. In that case, the wavenumber ''k'', the magnitude of k, is still in the same relationship with wavelength as shown above, with ''v'' being interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction. Generalizations to sinusoids of other phases, and to complex exponentials, are also common; see
plane wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

. The typical convention of using the
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave :$A e^.$

## General media

The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in a medium is less than in
vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gree ...
, which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum, as shown in the figure at right. This change in speed upon entering a medium causes
refraction In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

, or a change in direction of waves that encounter the interface between media at an angle. To aid imagination, this bending of the wave often is compared to the analogy of a column of marching soldiers crossing from solid ground into mud. See, for example, For
electromagnetic waves In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

, this change in the angle of propagation is governed by
Snell's law of light at the interface between two media of different refractive index, refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of in ...

. The wave velocity in one medium not only may differ from that in another, but the velocity typically varies with wavelength. As a result, the change in direction upon entering a different medium changes with the wavelength of the wave. For electromagnetic waves the speed in a medium is governed by its ''
refractive index In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes how fast EM radiation, light travels through the material. It is defined as :n = \frac ...

'' according to :$v = \frac,$ where ''c'' is the
speed of light The speed of light in vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure m ...
in vacuum and ''n''(λ0) is the refractive index of the medium at wavelength λ0, where the latter is measured in vacuum rather than in the medium. The corresponding wavelength in the medium is :$\lambda = \frac.$ When wavelengths of electromagnetic radiation are quoted, the wavelength in vacuum usually is intended unless the wavelength is specifically identified as the wavelength in some other medium. In acoustics, where a medium is essential for the waves to exist, the wavelength value is given for a specified medium. The variation in speed of light with wavelength is known as
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 variation ...
, and is also responsible for the familiar phenomenon in which light is separated into component colors by a
prism A prism An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical ...

. Separation occurs when the refractive index inside the prism varies with wavelength, so different wavelengths propagate at different speeds inside the prism, causing them to
refract In physics, refraction is the change in direction of a wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that ...
at different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as a
dispersion relation causes different colors to refract at different angles, splitting white light into a rainbow of colors. In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a ...

.

### Nonuniform media

Wavelength can be a useful concept even if the wave is not in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying ''local'' wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth. Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (an ''inhomogeneous'' medium) may propagate at a velocity that varies with position, and as a result may not be sinusoidal in space. The figure at right shows an example. As the wave slows down, the wavelength gets shorter and the amplitude increases; after a place of maximum response, the short wavelength is associated with a high loss and the wave dies out. The analysis of
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

s of such systems is often done approximately, using the '' WKB method'' (also known as the ''Liouville–Green method''). The method integrates phase through space using a local
wavenumber In the physical science Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical scienc ...
, which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space. This method treats the system locally as if it were uniform with the local properties; in particular, the local wave velocity associated with a frequency is the only thing needed to estimate the corresponding local wavenumber or wavelength. In addition, the method computes a slowly changing amplitude to satisfy other constraints of the equations or of the physical system, such as for
conservation of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
in the wave.

### Crystals

Waves in crystalline solids are not continuous, because they are composed of vibrations of discrete particles arranged in a regular lattice. This produces
aliasing In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...

because the same vibration can be considered to have a variety of different wavelengths, as shown in the figure.See Figure 4.20 in and Figure 2.3 in Descriptions using more than one of these wavelengths are redundant; it is conventional to choose the longest wavelength that fits the phenomenon. The range of wavelengths sufficient to provide a description of all possible waves in a crystalline medium corresponds to the wave vectors confined to the
Brillouin zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is ...

. This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such as energy bands and lattice vibrations. It is mathematically equivalent to the
aliasing In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...

of a signal that is
sampled Sample or samples may refer to: Base meaning * Sample (statistics) In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sci ...
at discrete intervals.

# More general waveforms

The concept of wavelength is most often applied to sinusoidal, or nearly sinusoidal, waves, because in a linear system the sinusoid is the unique shape that propagates with no shape change – just a phase change and potentially an amplitude change. See The wavelength (or alternatively
wavenumber In the physical science Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical scienc ...
or
wave vector In , a wave vector (also spelled wavevector) is a which helps describe a . Like any vector, it has a , both of which are important. Its magnitude is either the or of the wave (inversely proportional to the ), and its direction is ordinarily the ...
) is a characterization of the wave in space, that is functionally related to its frequency, as constrained by the physics of the system. Sinusoids are the simplest
traveling wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities, sometimes as described by a wave equation. In physical waves, at least two field (physics), field quant ...
solutions, and more complex solutions can be built up by superposition. In the special case of dispersion-free and uniform media, waves other than sinusoids propagate with unchanging shape and constant velocity. In certain circumstances, waves of unchanging shape also can occur in nonlinear media; for example, the figure shows ocean waves in shallow water that have sharper crests and flatter troughs than those of a sinusoid, typical of a
cnoidal wave 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 ...
, a traveling wave so named because it is described by the Jacobi elliptic function of ''m''-th order, usually denoted as . Large-amplitude
ocean wave In fluid dynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entitie ...
s with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium. If a traveling wave has a fixed shape that repeats in space or in time, it is a ''periodic wave''. Such waves are sometimes regarded as having a wavelength even though they are not sinusoidal. As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform.

## Wave packets

Localized
wave packet In physics, a wave packet (or wave train) is a short "burst" or "Wave envelope, 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 wa ...

s, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics. A wave packet has an ''envelope'' that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called a ''local wavelength''. An example is shown in the figure. In general, the ''envelope'' of the wave packet moves at a speed different from the constituent waves. Using
Fourier analysis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
, wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different
wavenumber In the physical science Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical scienc ...
s or wavelengths.See, for example, Figs. 2.8–2.10 in
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 nat ...

postulated that all particles with a specific value of
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

''p'' have a wavelength ''λ = h/p'', where ''h'' is . This hypothesis was at the basis of
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
. Nowadays, this wavelength is called the
de Broglie wavelength Matter waves are a central part of the theory of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic ...
. For example, the
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

s in a
CRT CRT may refer to: Science and technology Medicine * Calreticulin, a protein *Capillary refill time, the rate at which blood refills empty capillaries *Cardiac resynchronization therapy, a treatment for heart failure, and CRT defibrillator (CRT-D ...

display have a De Broglie wavelength of about 10−13 m. To prevent 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 number, complex-valued probability amplitude, and the probabilities for the possible results of ...

for such a particle being spread over all space, de Broglie proposed using wave packets to represent particles that are localized in space. The spatial spread of the wave packet, and the spread of the
wavenumber In the physical science Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical scienc ...
s of sinusoids that make up the packet, correspond to the uncertainties in the particle's position and momentum, the product of which is bounded by
Heisenberg uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of Inequality (mathematics), mathematical inequalities asserting a fundamental limit to the accuracy with which the values for ...
.

# Interference and diffraction

## Double-slit interference

When sinusoidal waveforms add, they may reinforce each other (constructive interference) or cancel each other (destructive interference) depending upon their relative phase. This phenomenon is used in the
interferometer . The two light rays with a common source combine at the half-silvered mirror to reach the detector. They may either interfere constructively (strengthening in intensity) if their light waves arrive in phase, or interfere destructively (weakening i ...
. A simple example is an experiment due to
Young Young may refer to: * Offspring, the product of reproduction of a new organism produced by one or more parents * Youth, the time of life when one is young, often meaning the time between childhood and adulthood Music * The Young, an American rock ...

where light is passed through two slits. As shown in the figure, light is passed through two slits and shines on a screen. The path of the light to a position on the screen is different for the two slits, and depends upon the angle θ the path makes with the screen. If we suppose the screen is far enough from the slits (that is, ''s'' is large compared to the slit separation ''d'') then the paths are nearly parallel, and the path difference is simply ''d'' sin θ. Accordingly, the condition for constructive interference is: :$d \sin \theta = m \lambda \ ,$ where ''m'' is an integer, and for destructive interference is: :$d \sin \theta = \left(m + 1/2 \right)\lambda \ .$ Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern or ''fringes'', and ''vice versa''. For multiple slits, the pattern is :$I_q = I_1 \sin^2 \left\left( \frac \right\right) / \sin^2 \left\left( \frac\right\right) \ ,$ where ''q'' is the number of slits, and ''g'' is the grating constant. The first factor, ''I''1, is the single-slit result, which modulates the more rapidly varying second factor that depends upon the number of slits and their spacing. In the figure ''I''1 has been set to unity, a very rough approximation. The effect of interference is to ''redistribute'' the light, so the energy contained in the light is not altered, just where it shows up.

## Single-slit diffraction

The notion of path difference and constructive or destructive interference used above for the double-slit experiment applies as well to the display of a single slit of light intercepted on a screen. The main result of this interference is to spread out the light from the narrow slit into a broader image on the screen. This distribution of wave energy is called
diffraction Diffraction refers to various phenomena that occur when a wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that ...

. Two types of diffraction are distinguished, depending upon the separation between the source and the screen:
Fraunhofer diffraction In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object (in the far-field region), and also when it is viewed at the focal plan ...
or far-field diffraction at large separations and
Fresnel diffraction Augustin-Jean Fresnel ( or ; ; 10 May 1788 – 14 July 1827) was a French civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engineering, professional engi ...

or near-field diffraction at close separations. In the analysis of the single slit, the non-zero width of the slit is taken into account, and each point in the aperture is taken as the source of one contribution to the beam of light (''Huygens' wavelets''). On the screen, the light arriving from each position within the slit has a different path length, albeit possibly a very small difference. Consequently, interference occurs. In the Fraunhofer diffraction pattern sufficiently far from a single slit, within a
small-angle approximation The small-angle approximations can be used to approximate the values of the main trigonometric functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical ...
, the intensity spread ''S'' is related to position ''x'' via a squared
sinc function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

: :$S\left(u\right) = \mathrm^2\left(u\right) = \left\left( \frac \right\right) ^2 \ ;$  with  $u = \frac \ ,$ where ''L'' is the slit width, ''R'' is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The function ''S'' has zeros where ''u'' is a non-zero integer, where are at ''x'' values at a separation proportion to wavelength.

## Diffraction-limited resolution

Diffraction is the fundamental limitation on the of optical instruments, such as
telescope A telescope is an optical instrument An optical instrument (or "optic" for short) is a device that processes light waves (or photons), either to enhance an image for viewing or to analyze and determine their characteristic properties. Common ...

s (including
s) and
microscopes A microscope (from the grc, μικρός, ''mikrós'', "small" and , ''skopeîn'', "to look" or "see") is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science Scie ...
. For a circular aperture, the diffraction-limited image spot is known as an
Airy disk A computer-generated Airy disk from diffracted white light ( D65 spectrum). Note that the red component is diffracted more than the blue, so that the center appears slightly bluish. In optics Optics is the branch of physics Physics (f ...
; the distance ''x'' in the single-slit diffraction formula is replaced by radial distance ''r'' and the sine is replaced by 2''J''1, where ''J''1 is a first order
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematici ...
. The resolvable ''spatial'' size of objects viewed through a microscope is limited according to the
Rayleigh criterion Angular resolution describes the ability of any image-forming device such as an optical or radio telescope A radio telescope is a specialized antenna and radio receiver radio in the 1940s. During the golden age of radio, 1925–1955, fa ...
, the radius to the first null of the Airy disk, to a size proportional to the wavelength of the light used, and depending on the
numerical aperture of light goes through a flat plane of glass, its half-angle changes to . Due to Snell's law, the numerical aperture remains the same:\text = n_1 \sin \theta_1 = n_2 \sin\theta_2. In optics Optics is the branch of physics Physics (from ...

: :$r_ = 1.22 \frac \ ,$ where the numerical aperture is defined as $\mathrm = n \sin \theta\;$ for θ being the half-angle of the cone of rays accepted by the microscope objective. The ''angular'' size of the central bright portion (radius to first null of the
Airy disk A computer-generated Airy disk from diffracted white light ( D65 spectrum). Note that the red component is diffracted more than the blue, so that the center appears slightly bluish. In optics Optics is the branch of physics Physics (f ...
) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is: :$\delta = 1.22 \frac \ ,$ where λ is the wavelength of the waves that are focused for imaging, ''D'' the entrance pupil diameter of the imaging system, in the same units, and the angular resolution δ is in radians. As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.

# Subwavelength

The term ''subwavelength'' is used to describe an object having one or more dimensions smaller than the length of the wave with which the object interacts. For example, the term ''subwavelength-diameter optical fibre'' means an optical fiber, optical fibre whose diameter is less than the wavelength of light propagating through it. A subwavelength particle is a particle smaller than the wavelength of light with which it interacts (see Rayleigh scattering). Subwavelength apertures are holes smaller than the wavelength of light propagating through them. Such structures have applications in extraordinary optical transmission, and zero-mode waveguides, among other areas of photonics. ''Subwavelength'' may also refer to a phenomenon involving subwavelength objects; for example, subwavelength imaging.

# Angular wavelength

A quantity related to the wavelength is the angular wavelength (also known as reduced wavelength), usually symbolized by ''ƛ'' (lambda-bar). It is equal to the "regular" wavelength "reduced" by a factor of 2π (''ƛ'' = ''λ''/2π). It is usually encountered in quantum mechanics, where it is used in combination with the reduced Planck constant (symbol ''ħ'', h-bar) and the
angular frequency In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
(symbol ''ω'') or angular wavenumber (symbol ''k'').