# Wave Number

Wavenumber refers to the total number of complete wave cycles in an electromagnetic field that exists in one meter of linear space. In physics, wave number is also known as the propagation constant. A wavenumber is a unit of frequency in atomic, molecular physics. Wave number is sometimes defined as the reciprocal of wavelength in meters. The wavenumber for an Electromagnetic field is wave number to wavelength to the 2.A wave number of any wave is inversely proportional to its wavelength, thus a shorter wavelength will be having a larger wavenumber referring to more number complete wave cycles.

## Wave Number Definition

Wavenumber is defined as the reciprocal of wavelength in meters or sometimes it is also measured in centimeters. A wavenumber is also known as the angular wavenumber and is defined as the number of wavelengths per unit spatial frequency known as the spatial frequency. It is denoted by k. Wavenumber is a scalar quantity, it has magnitude and direction will not be required. A wavenumber is also considered as a unit of frequency in some branch of physics, where a wavenumber is equal to frequency divided by the speed of light and it will be equal to the number of waves per unit frequency.

Let us have look at the definition of wavenumber in different fields:

### In Physics:

It is considered to be the number of radians per unit length.

### In Spectroscopy:

In spectroscopy and also chemistry it is considered to be the measure of the number of waves per unit length. It is particularly considered in the CGS system i.e., it will be measured in cm-1and is inversely proportional to the wavelength. It is often confused with wavenumber to wavelengths as the same entity, it is important to keep in mind that Wavenumber and wavelength are two different quantities.

### In Medium:

In any medium the wave number is directly proportional it’s the refractive Index.

### Wave Number Formula

From the wave number definition formula of wave number can be written mathematically as,

k= $\frac{2𝜋}{λ}$……..(1)

Where,

λ-The wavelength of given waves.

Equation (1) is the most general formula for wave number in physics. Wavenumber is also expressed in terms of momentum of waves in quantum physics as,

k= $\frac{p}{ħ}$ ( we know that the momentum of wave packets is expressed by equation p=ħk.)

Where,

p-The momentum of the particles

ħ-Correction to Planck’s constant

If we consider matter waves(non-relativistic mechanics) then the wavenumber equation will be rephrased by replacing momentum with its energy equation as given below:

k= $\sqrt{\frac{2mE}{ħ}}$

Where,

m-The mass of the particle

E- The energy associated with the particle

ħ-Correction to Planck’s constant

For angular wavenumber, the mathematical expression is given by:

k=vp

Where,

-The angular frequency

vp-The phase velocity

In some special case of electromagnetic wave propagation in a vacuum, then we write angular wavenumber is considered as:

k=Eħ

Where,

E- The energy associated with the particle

ħ-Correction to Planck’s constant

In free space, the wavenumber is equal to the frequency of the wave divided by the speed of light.

k=fc

Where,

f- The frequency of the wave

C- Speed of light

The wavenumber and frequency are directly proportional to each other, greater frequency will result in a high wavenumber or more number of waves.

In any medium other than the free space, c must be multiplied by a velocity v of the particle. The velocity factor for any particular medium is the ratio of the speed of EM propagation in that medium to the speed of EM propagation in free space. As such, the velocity factor will be always greater than 0 and less than or equal to 1. So, by taking the velocity factor into account, then the equation of wavenumber is given by:

k=fcv

Where,

f- The frequency of the wave

C- Speed of light

V- The velocity of the particle

### Physical Significance of Wavenumber

• For a linear isotropic medium, wavenumber is an alternative way of expressing frequency. Though frequency is a fundamental quantity, it’s preferable to use the wavenumber for elaborating the importance of the wave vectors.

• For an absorbing medium, the wavenumber tells us how quickly the wave is absorbed. The frequency doesn't tell us this - this is a practical use for using wavenumber instead of frequency.

• The only significance of wavenumber in the absence of a frequency is the same as the significance of the speed of light in the absence of a frequency, which is standing waves.

### Example:

1. Calculate the wavenumber and frequency of yellow radiation having a wavelength of 5800A0.

Sol:

Given that we have yellow radiation of 5800A0, we are asked to determine the wavelength and frequency of the same.

Then from the formula of wavenumber we have,

k=2……..(1)

Where,

-The wavelength of given waves.

Substituting the corresponding values in equation (1) we get,

k=25800=10.833103m-1

Now, the frequency of yellow radiation is given by,

f=c

Where,

-The wavelength of given waves

C- Speed of light

Now, let us simplify the above expressions by substituting the wavelength,

f=3108580010-10=5.1721014 Hz

Therefore, the frequency and wavenumber of yellow radiation of wavelength 5800A0is 5.1721014 Hz and 10.833103m-1 respectively.