TheInfoList In the
physical science Physical science is a branch of natural science that studies abiotic component, 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 sciences". D ...
s, the wavenumber (also wave number or repetency) is 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 ...
of a
wave 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 su ... , measured in cycles per unit distance or radians per unit distance. Whereas temporal
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 ... can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance. In
multidimensional systems In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables. Important problems such as factorization and Stability t ...
, the wavenumber is the magnitude of the
wave vector 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 ...
. The space of wave vectors is called
reciprocal space In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fun ... . Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering, such as
X-ray diffraction X-ray crystallography (XRC) is the experimental science determining the atomic and molecular structure of a crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a ...
,
neutron diffraction The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...
,
electron diffraction #REDIRECT Electron diffraction#REDIRECT Electron diffractionElectron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons ... , and
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the fundamental fermions (quarks, leptons, antiquarks, and a ...
physics. For
quantum mechanical Quantum mechanics is a fundamental theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, ...
waves, the wavenumber multiplied by the reduced
Planck's constant is the
canonical momentum 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 ...
. Wavenumber can be used to specify quantities other than spatial frequency. In
optical spectroscopy Spectroscopy is the study of the interaction between matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately compo ...
, it is often used as a unit of temporal frequency assuming a certain
speed of light The speed of light 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 paramet ...
.

# Definition

Wavenumber, as used in
spectroscopy Spectroscopy is the study of the interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ... and most chemistry fields, is defined as the number of
wavelength 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 su ... s per unit distance, typically centimeters (cm−1): :$\tilde \;=\; \frac,$ where ''λ'' is the wavelength. It is sometimes called the "spectroscopic wavenumber". It equals 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 ...
. A wavenumber in inverse cm can be converted to a frequency in GHz by multiplying by 29.9792458 (the speed of light in centimeters per nanosecond). An electromagnetic wave at 29.9792458 GHz has a wavelength of 1 cm in free space. In theoretical physics, a wave number defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often used: :$k \;=\; \frac$ When wavenumber is represented by the symbol , a
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 ... is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationship $\frac \;=\; \frac \;\equiv\; \tilde$, where s is a frequency in
hertz The hertz (symbol: Hz) is the unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action ... . This is done for convenience as frequencies tend to be very large. Wavenumber has
dimensions thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
of
reciprocal lengthReciprocal length or inverse length is a measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other o ...
, so its
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system The metric system is a that succeeded the decimal ...
is the reciprocal of meters (m−1). In
spectroscopy Spectroscopy is the study of the interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ... it is usual to give wavenumbers in cgs unit (i.e., reciprocal centimeters; cm−1); in this context, the wavenumber was formerly called the ''kayser'', after
Heinrich Kayser Heinrich Gustav Johannes Kayser ForMemRS Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowl ... (some older scientific papers used this unit, abbreviated as ''K'', where 1K = 1cm−1). The angular wavenumber may be expressed in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ... s per meter (rad⋅m−1), or as above, since the
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ... is
dimensionless In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric curre ...
. For
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. ... in vacuum, wavenumber is directly proportional to frequency and to
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In , an elementary particle or fundamental particle is a that is not composed of other particles. Particles currently thought to be elementary include the fundamental s ... energy. Because of this, wavenumbers are used as a convenient
unit of energy Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) i ...
in spectroscopy.

## Complex

A complex-valued wavenumber can be defined for a medium with complex-valued relative
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
$\varepsilon_r$, relative permeability $\mu_r$ and
refraction 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, ...
''n'' as: :$k = k_0 \sqrt = k_0 n$ where ''k''0 is the free-space wavenumber, as above. The imaginary part of the wavenumber expresses attenuation per unit distance and is useful in the study of exponentially decaying
evanescent field In electromagnetics Electromagnetism is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matte ...
s.

## Plane waves in linear media

The propagation factor of a sinusoidal plane wave propagating in the x direction in a linear material is given by :$P = e^$ where *$k = k\text{'} - jk\text{'}\text{'} = \sqrt\;$ *$k\text{'} =$
phase constant The propagation constant of a sinusoidal electromagnetic wave Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged pa ...
in the units of
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ... s/meter *$k\text{'}\text{'} =$
attenuation constant The propagation constant of a sinusoidal electromagnetic wave Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged pa ...
in the units of
neper The neper (symbol: Np) is a logarithmic unit A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of t ... s/metre *$\omega =$ frequency in the units of
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ... s/metre *$x =$ distance traveled in the ''x'' direction *$\sigma =$ conductivity in
Siemens Siemens AG ( ) is a German multinational Multinational may refer to: * Multinational corporation, a corporate organization operating in multiple countries * Multinational force, a military body from multiple countries * Multinational state, ...
/metre *$\varepsilon = \varepsilon\text{'} - j\varepsilon\text{'}\text{'} =$
complex permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek alphabet, Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polar ...
*$\mu = \mu\text{'} - j\mu\text{'}\text{'} =$ complex permeability *$j=\sqrt$ The sign convention is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x direction.
Wavelength 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 su ... ,
phase velocity The phase velocity of a wave 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 ... , and
skin depth s use stranded coils to reduce losses due to the skin effect. Skin effect is the tendency of an alternating current, alternating electric current (AC) to become distributed within a conductor such that the current density In electromagnetism ... have simple relationships to the components of the wavenumber: :$\lambda = \frac \qquad v_p = \frac \qquad \delta = \frac 1$

# In wave equations

Here we assume that the wave is regular in the sense that the different quantities describing the wave such as the wavelength, frequency and thus the wavenumber are constants. See
wavepacket 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 ... for discussion of the case when these quantities are not constant. In general, the angular wavenumber ''k'' (i.e. the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of the
wave vector 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 ...
) is given by :$k = \frac = \frac=\frac$ where ''ν'' is the frequency of the wave, ''λ'' is the wavelength, ''ω'' = 2''πν'' is 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 ...
of the wave, and ''v''p is the
phase velocity The phase velocity of a wave 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 ... of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of #Dispersion, dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency ... . For the special case of an
electromagnetic wave 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 a vacuum, in which the wave propagates at the speed of light, ''k'' is given by: :$k = \frac$ where ''E'' is the
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 regula ... of the wave, ''ħ'' is the
reduced Planck constant The Planck constant, or Planck's constant, is the quantum of electromagnetic action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photon's energy. The Planck constant i ...
, and ''c'' is the
speed of light The speed of light 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 paramet ...
in a vacuum. For the special case of a
matter wave Matter waves are a central part of the theory 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 ...
, for example an electron wave, in the non-relativistic approximation (in the case of a free particle, that is, the particle has no potential energy): :$k \equiv \frac = \frac= \frac$ Here ''p'' is the
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 ... of the particle, ''m'' is the
mass Mass is the quantity Quantity 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 ...
of the particle, ''E'' is the
kinetic 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 ...
of the particle, and ''ħ'' is the
reduced Planck constant The Planck constant, or Planck's constant, is the quantum of electromagnetic action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photon's energy. The Planck constant i ...
. Wavenumber is also used to define the
group velocity in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the       green circles propagate with the group velocity. In this deep-water case, ''the phase velocity is twice ...
.

# In spectroscopy

In
spectroscopy Spectroscopy is the study of the interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ... , "wavenumber" $\tilde$ refers to a frequency which has been divided by the
speed of light in vacuum The speed of light in vacuum A vacuum is space devoid of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ult ...
usually in centimeters per second (cm.s−1): : :$\tilde = \frac = \frac.$ The historical reason for using this spectroscopic wavenumber rather than frequency is that it is a convenient unit when studying atomic spectra by counting fringes per cm with an
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 ... : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum: :$\lambda_ = \frac,$ which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ... s and the distance between fringes in
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 ... s, when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations of
Johannes Rydberg Johannes (Janne) Robert Rydberg (; 8 November 1854 – 28 December 1919) was a Swedish people, Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and ...
in the 1880s. The Rydberg–Ritz combination principle of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood in quantum theory as differences between energy levels, energy being proportional to wavenumber, or frequency. However, spectroscopic data kept being tabulated in terms of spectroscopic wavenumber rather than frequency or energy. For example, the spectroscopic wavenumbers of the emission spectrum of atomic hydrogen are given by the
Rydberg formula In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light Light or ...
: :$\tilde = R\left\left(\frac - \frac\right\right),$ where ''R'' is the
Rydberg constant In spectroscopy Spectroscopy is the study of the interaction between matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched ar ...
, and ''n''i and ''n''f are the
principal quantum number In 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 ...
s of the initial and final levels respectively (''n''i is greater than ''n''f for emission). A spectroscopic wavenumber can be converted into energy per photon ''E'' by Planck's relation: :$E = hc\tilde.$ It can also be converted into wavelength of light: :$\lambda = \frac,$ where ''n'' is the
refractive index In optics Optics is the branch of 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 ... of the
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 i ...
. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant. Conventionally, inverse centimeter (cm−1) units are used for $\tilde$, so often that such spatial frequencies are stated by some authors "in wavenumbers", incorrectly transferring the name of the quantity to the CGS unit cm−1 itself. 