Wavenumber

Angular wavenumber, typically denoted by the letter ${\displaystyle k}$, defined as the wave phase divided by length, is a quantity with dimension of angle per length (or simply reciprocal length) and SI units of radians per metre (or m⁻¹). They are analogous to temporal frequency, respectively the ordinary frequency, defined as the number of wave cycles divided by time (in cycles per second or reciprocal seconds), and the angular frequency, defined as the phase angle divided by time (in radians per second).

In multidimensional systems, the angular wavenumber ${\displaystyle k}$ is the magnitude of the wave vector. The space of wave vectors is called reciprocal space. Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck constant is the canonical momentum.

Wavenumber can be used to specify quantities other than spatial frequency. For example, in optical spectroscopy, it is often used as a unit of temporal frequency assuming a certain speed of light.

Definition

Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance:

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

where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber". It equals the spatial frequency.

In theoretical physics, an angular wave number, defined as the number of radians per unit distance is more often used:

${\displaystyle k=\frac{2\pi }{\lambda }=2\pi \tilde{\nu }}$.

Units

The SI unit of spectroscopic wavenumber is the reciprocal m, written m⁻¹. However, it is more common, especially in spectroscopy, to give wavenumbers in cgs units i.e., reciprocal centimeters or cm⁻¹, with

${\displaystyle 1{\mathrm{c}\mathrm{m}}^{-1}=100{\mathrm{m}}^{-1}}$.

Occasionally in older references, the unit kayser (after Heinrich Kayser) is used; it is abbreviated as K or Ky, where 1 K = 1 cm⁻¹.

Angular wavenumber may be expressed in the unit radian per meter (rad⋅m⁻¹), or as above, since the radian is dimensionless.

Unit conversions

The frequency of light with wavenumber ${\displaystyle \tilde{\nu }}$ is

${\displaystyle f=\frac{c}{\lambda }=c\tilde{\nu }}$,

where ${\displaystyle c}$ is the speed of light. The conversion from spectroscopic wavenumber to frequency is therefore

${\displaystyle 1{\mathrm{c}\mathrm{m}}^{-1}\cdot c=29.9792458\mathrm{G}\mathrm{H}\mathrm{z}.}$

Wavenumber can also be used as unit of energy, since a photon of frequency ${\displaystyle f}$ has energy ${\displaystyle hf}$, where ${\displaystyle h}$ is the Planck constant. The energy of a photon with wavenumber ${\displaystyle \tilde{\nu }}$ is

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

The conversion from spectroscopic wavenumber to energy is therefore

${\displaystyle 1{\mathrm{c}\mathrm{m}}^{-1}\cdot hc=1.986446\times {10}^{-23}\mathrm{J}=1.239842\times {10}^{-4}\mathrm{e}\mathrm{V}}$

where energy is expressed either in J or eV.

Complex

A complex-valued wavenumber can be defined for a medium with complex-valued relative permittivity ${\displaystyle {\epsilon }_r}$, relative permeability ${\displaystyle {\mu }_r}$ and refraction index n as:

${\displaystyle k=k_0\sqrt{{\epsilon }_r{\mu }_r}=k_{0}n}$

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 positive x direction in a linear material is given by

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

where

• ${\displaystyle k=k^{\prime }-j{k}^{″}=\sqrt{-\left(\omega {\mu }^{″}+j\omega {\mu }^{\prime }\right)\left(\sigma +\omega {\epsilon }^{″}+j\omega {\epsilon }^{\prime }\right)}}$
• ${\displaystyle k^{\prime }=}$ phase constant in the units of radians/meter
• ${\displaystyle k^{″}=}$ attenuation constant in the units of nepers/meter
• ${\displaystyle \omega =}$ angular frequency
• ${\displaystyle x=}$ distance traveled in the x direction
• ${\displaystyle \sigma =}$ conductivity in Siemens/meter
• ${\displaystyle \epsilon ={\epsilon }^{\prime }-j{\epsilon }^{″}=}$ complex permittivity
• ${\displaystyle \mu ={\mu }^{\prime }-j{\mu }^{″}=}$ 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:

${\displaystyle \lambda =\frac{2\pi }{k^{\prime }}{v}_p=\frac{\omega }{k^{\prime }}\delta =\frac{1}{k^{″}}}$