Wavenumber

In the

, which is defined as the number of wave cycles per unit time (ordinary frequency) or radians per unit time (angular frequency).

In

.

Wavenumber can be used to specify quantities other than spatial frequency. For example, in

.

Definition

Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically centimeters (cm⁻¹):

${\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}$

where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber".^[1] It equals the spatial frequency.

For example, a wavenumber in inverse centimeters can be converted to a frequency in gigahertz by multiplying by 29.9792458 cm/ns (the speed of light, in centimeters per nanosecond);^[5] conversely, 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:^[6]

${\displaystyle k\;=\;{\frac {2\pi }{\lambda }}}$

When wavenumber is represented by the symbol ν, a frequency is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationship ${\textstyle {\frac {\nu _{s}}{c}}\;=\;{\frac {1}{\lambda }}\;\equiv \;{\tilde {\nu }}}$, where ν_s is a frequency in hertz. This is done for convenience as frequencies tend to be very large.^[7]

Wavenumber has

.

For

in spectroscopy.

Complex

A complex-valued wavenumber can be defined for a medium with complex-valued relative permittivity ${\displaystyle \varepsilon _{r}}$, relative permeability ${\displaystyle \mu _{r}}$ and

${\displaystyle k=k_{0}{\sqrt {\varepsilon _{r}\mu _{r}}}=k_{0}n}$

where k₀ 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 fields.

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^[10]: 51

${\displaystyle P=e^{-jkx}}$

where

• ${\displaystyle k=k'-jk''={\sqrt {-\left(\omega \mu ''+j\omega \mu '\right)\left(\sigma +\omega \varepsilon ''+j\omega \varepsilon '\right)}}\;}$
• ${\displaystyle k'=}$
  phase constant in the units of radians
  /meter
• ${\displaystyle k''=}$
  attenuation constant in the units of nepers
  /meter
• ${\displaystyle \omega =}$ frequency in the units of radians/second
• ${\displaystyle x=}$ distance traveled in the x direction
• ${\displaystyle \sigma =}$ conductivity in Siemens/meter
• ${\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''=}$ complex permittivity
• ${\displaystyle \mu =\mu '-j\mu ''=}$ complex permeability
• ${\displaystyle j={\sqrt {-1}}}$

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, phase velocity, and skin depth have simple relationships to the components of the wavenumber:

${\displaystyle \lambda ={\frac {2\pi }{k'}}\qquad v_{p}={\frac {\omega }{k'}}\qquad \delta ={\frac {1}{k''}}}$

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

for discussion of the case when these quantities are not constant.

In general, the angular wavenumber k (i.e. the magnitude of the wave vector) is given by

${\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi \nu }{v_{\mathrm {p} }}}={\frac {\omega }{v_{\mathrm {p} }}}}$

where ν is the frequency of the wave, λ is the wavelength, ω = 2πν is the angular frequency of the wave, and v_p is the phase velocity 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.

For the special case of an

in a vacuum, in which the wave propagates at the speed of light, k is given by:

${\displaystyle k={\frac {E}{\hbar c}}={\frac {\omega }{c}}}$

where E is the

in a vacuum.

For the special case of a matter wave, 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):

${\displaystyle k\equiv {\frac {2\pi }{\lambda }}={\frac {p}{\hbar }}={\frac {\sqrt {2mE}}{\hbar }}}$

Here p is the

.

Wavenumber is also used to define the group velocity.

In spectroscopy

In spectroscopy, "wavenumber" ${\displaystyle {\tilde {\nu }}}$ (in

(usually in centimeters per second, cm⋅s⁻¹):

${\displaystyle {\tilde {\nu }}={\frac {\nu }{c}}={\frac {\omega }{2\pi c}}.}$

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

 : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum:

${\displaystyle \lambda _{\rm {vac}}={\frac {1}{\tilde {\nu }}},}$

which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from

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:

${\displaystyle {\tilde {\nu }}=R\left({\frac {1}{{n_{\text{f}}}^{2}}}-{\frac {1}{{n_{\text{i}}}^{2}}}\right),}$

where R is the Rydberg constant, and n_i and n_f are the principal quantum numbers of the initial and final levels respectively (n_i is greater than n_f for emission).

A spectroscopic wavenumber can be converted into

:

${\displaystyle E=hc{\tilde {\nu }}.}$

It can also be converted into wavelength of light:

${\displaystyle \lambda ={\frac {1}{n{\tilde {\nu }}}},}$

where n is the refractive index of the medium. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.

Often spatial frequencies are stated by some authors "in wavenumbers",^[11] incorrectly transferring the name of the quantity to the CGS unit cm⁻¹ itself.^[12]

See also

• Angular wavelength
• Spatial frequency
• Refractive index
• Zonal wavenumber

References

External links

• Media related to Wavenumber at Wikimedia Commons