# Wave Number

Wavelength and wavenumber are two very important concepts discussed in physics and various other fields. Wavelength is the distance between two consecutive points which are in the same phase. Wavenumber is the number of wavelengths in a given distance along the propagation of the wave. In this article, we will learn about the wave number in detail.

## What Is Wavenumber?

In physics, the wavenumber is also known as propagation number, or angular wavenumber is defined as the number of wavelengths per unit distance.It is equal to the inverse of wavelength. It is a scalar quantity represented by k and the mathematical representation is given as follows:

$$\begin{array}{l}k=\frac{1}{\lambda }\end{array}$$
Where,

• k is the wavenumber
• Ξ» 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 –

$$\begin{array}{l}k=\frac{2\pi }{\lambda }\end{array}$$

Where,

• k is the wavenumber
• π is the wavelength of the wave

## Wavenumber Definition

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

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

$$\begin{array}{l}\bar{\nu }=\frac{1}{\lambda }\end{array}$$

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

$$\begin{array}{l}k=k_{0}\sqrt{\varepsilon \mu }=k_{0}n\end{array}$$

Where,

• k0 is the free-space wavenumber
• π is the relative permittivity
• π is the 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

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 wavenumber k is given by-

$$\begin{array}{l}k=\frac{2\pi }{\lambda }=\frac{2\pi \nu }{\vartheta _{p}}=\frac{\omega}{\vartheta _{p}}\end{array}$$

Where,

• vp is the phase velocity of a wave
• β΅ = 2ππ is the 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

$$\begin{array}{l}k=\frac{2\pi }{\lambda }=\frac{p }{\hbar}=\frac{\sqrt{2mE}}{\hbar}\end{array}$$

Where,

• E is the kinetic energy of the particle
• m is the mass of the particle
• p is the momentum of the particle
• Δ§ is the reduced Planckβs constant
• In the special case of propagation of an electromagnetic wave in vacuum, the wavenumber is given by-
$$\begin{array}{l}k=\frac{E}{\hbar}\end{array}$$
Where,
• E is the energy of the wave
• Δ§ is the reduced Planckβs constant
• c is the velocity of light
• In some cases, the wavenumber also defines group velocity.

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