Wave Number - Definition, Formula, Symbol, FAQs

What is Wave Number?

Wavenumber is defined as the number of complete wave cycles present in a unit distance. It is an important concept in wave mechanics, optics, and physics.

The wavenumber equation is given by:

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

• $k$ is the wavenumber (also called angular wavenumber)
• $\lambda$ is the wavelength of the wave

Unit: The SI unit of wavenumber is rad/m (radian per meter).

Another commonly used form is spectroscopic wavenumber:

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

which is measured in $\mathbf{m}^{\mathbf{- 1}}$ or $\mathbf{c m}^{\mathbf{- 1}}$.

Also Read:

• NCERT solutions for Class 12 Physics Chapter 8 Electromagnetic Waves
• NCERT Exemplar Class 12 Physics Solutions Chapter 8 Electromagnetic Waves
• NCERT notes Class 12 Physics Chapter 8 Electromagnetic Waves

Wavenumber Definition

Wavenumber is a fundamental quantity used in physics, chemistry, and spectroscopy to describe wave properties in terms of spatial variation.

1. In Theoretical Physics

Wavenumber is defined as the number of radians per unit distance. It represents how rapidly the phase of a wave changes in space.

2. In Chemistry and Spectroscopy

Wavenumber is defined as the number of waves per unit distance. It is commonly used in spectroscopy and is expressed as:

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

Unit: $\mathbf{c m}^{\boldsymbol{-} \mathbf{1}}$ (commonly used in spectroscopy)

3. Complex Wavenumber

In a medium, the wavenumber can be written as:

$
k=k_0 \sqrt{\varepsilon \mu}=k_0 n
$

Where:

• $k_0=$ free-space wavenumber
• $\varepsilon=$ relative permittivity
• $\mu=$ relative permeability
• $n=$ refractive index

The imaginary part of wavenumber represents attenuation, which is important in studying exponentially decaying (evanescent) waves.

Wavenumber Equation
The angular wavenumber is given by:

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

Where:

• $v_p=$ phase velocity
• $\omega=2 \pi \nu=$ angular frequency

Also read -

• NCERT Solutions for Class 11 Physics
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for All Subjects

Dimensional Formula of Wavenumber (k)

Wavenumber represents the number of waves per unit length. Since it is the reciprocal of wavelength, its dimensional formula is:

$
[k]=\left[M^0 L^{-1} T^0\right]
$

• No dependence on mass or time
• Depends only on length (inverse of length)

Applications of Wavenumber

Wavenumber is widely used in physics and chemistry to describe wave properties in a simple form.

• Spectroscopy: Used to represent spectral lines in $\mathbf{c m}^{-\mathbf{1}}$ for easy analysis of molecular and atomic transitions.
• Energy Relation: Directly related to energy of radiation $(E \propto \bar{\nu})$, useful in quantum studies.
• Wave Mechanics: Helps in expressing wave equations and phase variation using $k=\frac{2 \pi}{\lambda}$.
• Optics: Used to study refraction and dispersion as wavenumber changes with medium ( $k=k_0 n$ ).
• Quantum Mechanics: Represents particle momentum ( $p=\hbar k$ ).

Also check-

• NCERT Exemplar Class 11th Physics Solutions
• NCERT Exemplar Class 12th Physics Solutions
• NCERT Exemplar Solutions for All Subjects

NCERT Physics Notes:

• NCERT Notes Class 11th Physics
• NCERT Notes Class 12th Physics
• NCERT Notes For All Subjects