Wave Number

What is wavenumber?

In physics, the wave number is also known as propagation number or angular wave number is defined as the number of wavelengths per unit distance the spacial wave frequency and is known as spatial frequency. It is a scalar quantity represented by k and the mathematical representation is given as follows:
$k=\frac{1}{\lambda }$ Where,

• k is the wavenumber
• λ is the wavelength

Wavenumber formula

Wavenumber equation is mathematically expressed as the number of the complete cycle of a wave over its wavelength, given by –

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

Where,

• k is the wavenumber
• 𝜆 is the wavelength of the wave

Wave number definition

In theoratical physics: It is the number of radians present in unit distance.

In chemistry and spectroscopy: It is the number of waves per unit distance. Typically measure using cm-1. Is given by-

$\bar{\nu }=\frac{1}{\lambda }$

In complex form: The complex values wave number for a medium can be expressed as –

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

Where,

• k0 is the free-space wave number
• 𝜀 is the relative permittivity
• 𝜇 is relative permeability
• n is the refractive index
• The imaginary/complex part of the above equation represents the attenuation per unit distance. It is often used to study the exponentially decaying evanescent fields.

Wavenumber Equation

In general, we assume wave number is a characteristic of a wave and is constant for a wave. It varies from one wave to another wave. But there are some special cases where the value can be dynamic.

In spectroscopy:

The angular wave number k is given by-

$k=\frac{2\pi }{\lambda }=\frac{2\pi \vartheta }{\vartheta _{p}}=\frac{\omega}{\vartheta _{p}}$

Where,

• vp is the phase velocity of a wave
• ⍵ = 2𝜋𝜈 is angular frequency
• This dependency of frequency on wavenumber is expressed as the dispersion relation.

In matter-wave:

For example, a non- relativistic approximation for an electron wave is given by

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

Where,

• E is the kinetic energy of the particle
• m is the mass of the particle
• p is the momentum of the particle
• ħ is the reduced Planck’s constant
• In the special case of propagation of an electromagnetic wave in vacuum, the wavenumber is given by-

$k=\frac{E}{\hbar}$

Where,

• E is the energy of the wave
• ħ is the reduced Planck’s constant
• c is the velocity of light
• In some case, the wavenumber also defines group velocity.

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