## Table of Contents

# Wavenumber

Wavenumber, also called wave number, a unit of frequency, often used in atomic, molecular, and nuclear spectroscopy, equal to the true frequency divided by the speed of the wave and thus equal to the number of waves in a unit distance.

Physicists and pharmacists often use two different types of wavenumber;

The **Spatial Wavenumber** or also called as spatial frequency is referred to as the number of wavelengths per unit distance. Angular wavenumber is generally used in physics and geophysics. Fundamentally, the equations for both angular and spatial wavenumber are the same except the fact that the angular wavenumber uses 2π in its numerator as this is the number of radians in a complete circle (that is 180 x 2 = 360°).

The **Angular Wavenumber** is also termed as circular wavenumber gives us the number of radians (a measure of angle) per unit distance. Spatial wavenumber is generally used in chemistry.

Waves can define sound, light, or the wavefunction of given particles, but each wave has a wavenumber.

Theoretically, the wavenumber is also termed as propagation number or angular wavenumber is referred to as the number of the complete cycle of a wave over its wavelength. It is denoted as a scalar quantity and is represented by the symbol k and the mathematical depiction is as follows:

k=1/λ

Where,

• k represents the wavenumber

• λ represents the wavelength

## Wavenumber Formula

Using the equation mentioned above to calculate the spatial wavenumber (ν)

ν = 1 / 𝜆

= f / v

Where,

𝜆 represents wavelength

f represents frequency

v represents the speed of the wave.

k = 2π / 𝜆

where

𝜆 = v/f

Thus

k = 2πf / v

This equation is used to calculate angular wavenumber (k).

### Wavenumber Formula in Spectroscopy

Wavenumber is a term that is used in spectroscopy to describe a frequency that has been separated by the speed of light in a vacuum. The formula of Wave number in spectroscopy and chemistry fields is given as follows;

¯v¯ = 1/ λ = ω/ 2πc = v/ c

Where,

¯v¯represents the spectroscopy wavenumber.

Λ represents the wavelength often called as spectroscopic wavenumber

Thus,

ω= 2 πv is the angular frequency

Spectroscopic wavenumber can also be converted into energy per photon by using Planck’s relation as given below;

E = hcv¯

Where

E represents the energy per photon

h represents the reduced Planck’s constant = 6.62607004 × 10^{-34} m2 kg / s

c represents the speed of light

v¯ represents the Spectroscopic Wavenumber

Spectroscopic wavenumber can also be converted to the wavelength of light as mentioned below;

λ (1n/v¯)

Where,

λ represents the wavelength

n represents the refractive index of the medium

v¯represents the Spectroscopic Wavenumber

The SI unit of measurement of Spectroscopic Wavenumber is often expressed as the reciprocal of meter that is written as m^{-1}.

• The CGS unit of measurement of Spectroscopic Wavenumber is often expressed as the reciprocal of a centimeter that is written as cm^{-1}.

#### Wavenumber Formula for Wave Equations

The formula for wave number in theoretical physics is given by

k = 2π/λ = ω/vp

Where;

k represents the angular wavenumber

λ represents the wavelength

ω = 2πv represents the angular frequency

When an electromagnetic wave propagate at the speed of light or c in a vacuum, then the wave equation k is represented as follows;

k =E/hc

Where,

k represents the angular wave number

E represents the energy of the wave

h represents the Planck’s constant which is equal to 6.62607004 × 10^{-34} m2 kg / s

c represents the speed of light

##### Applications of Wavenumber

• A wavenumber helps to calculate the spatial frequency.

• Apart from spatial frequency, wavenumber is also used to explain other quantities for example optics and wave scatterings in physics.

• Wavenumbers and wave vectors are often used to explain in X-ray diffraction and neutron diffraction, electron diffraction, and also in elementary particles in physics.

• Group velocity can also be explained with the help of a wavenumber.

##### Wavenumber Examples

**Example 1:** Calculate the Angular Wavenumber if the W wvelength of the Light wave is given as 500 Nanometers.

The formula for angular wavenumber is as follows;

k = 2π/λ

Where,

λ represents the wavelength of the light wave and is given as 500 nanometers which is further equal to 500 × 10-9 m.

[we know that 1nm =10−9m]

Now substituting the values in the formula to get the angular wavenumber as follows:

k = 2π/500×10−9

Thus,

k = 12.56 x 106 m-1

**Example 2:** Calculate the wavelength, frequency, and wavenumber of a light wave whose time period is given as 5.0×10^{−10} s.

The frequency or represented by symbol v of the light wave is given by;

1/ period = 1 / 5.0×10^{−10} seconds = 2×10^{9} hertz or Hz

The wavelength of the light wave is given by;

c/v = 3×10^{8} m / 2×10^{9}/s = 15×10^{−2} m. ( c is the speed of light)

The wavenumber of the light wave is given by;

νˉ= 1/λ= 1/ 15.0×10^{−2} =6.6 /m

**Example 3:** Calculate the frequency and wavenumber of radiation with wavelength 380 nm.

Given that; wavelength or

λ = 380nm = 380×10−9m [ we know that 1nm=10−9m]

Speed of light or c = 3×108 m/sec

Thus the Frequency (v) is given as,

v = c/ λ = 3×108 ms-1 / 380×10−9m

= 7.89 × 1014 hertz or Hz.

##### Wavenumber Citations

- Linear-in-wavenumber actively-mode-locked wavelength-swept laser. Opt Lett . 2020 Oct 1;45(19):5327-5330.
- Frequency-wavenumber domain analysis of guided wavefields. Ultrasonics . 2011 May;51(4):452-66.
- Wavenumber selection method to determine the concentration of cocaine and adulterants in cocaine samples. J Pharm Biomed Anal . 2018 Apr 15;152:120-127.

Share