Wave Number

The wavenumber (also known as the wave number or repetency) is the spatial frequency of a wave in the physical sciences, expressed in cycles per unit distance (ordinary wavenumber) or radians per unit distance (radian wavenumber) (angular wavenumber). It is comparable to temporal frequency, which is defined as the number of wave cycles per unit time (ordinary frequency) or radians per unit time (radians per unit time) (angular frequency).

The wavenumber is the magnitude of the wave vector in multidimensional systems. The space of wave vectors is known as reciprocal space. In optics and the physics of wave scattering, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics, wave numbers and wave vectors are critical. The canonical momentum of quantum mechanical waves is defined as the wavenumber multiplied by the decreased Planck’s constant.

Other than spatial frequency, the wavenumber can be used to express other parameters. It is commonly used in optical spectroscopy as a unit of temporal frequency.

Formula for Wavenumbers

The wavenumber equation is defined mathematically as the number of complete cycles of a wave multiplied by its wavelength, denoted by –

k = 1/𝜆

Where,

• where k denotes the wavenumber
• 𝜆 is the wave’s wavelength

Calculate in rad/m

Equation of Wavenumbers

By and large, we presume that wave number is a property of a wave and that it is constant for a wave. It fluctuates according to the wave. However, there are certain exceptional circumstances in which the value can be dynamic.

In the field of spectroscopy:

The angular wavenumber k is calculated as follows:

k = 2𝜋/𝜆 = 2𝜋v/vp = ⍵/vp

Where,

• vp denotes a wave’s phase velocity.
• ⍵ = 2𝜋𝜈 denotes the angle frequency.
• The dispersion relation expresses this relationship between frequency and wavenumber.

In terms of matter-wave:

For instance, the non-relativistic approximation for an electron wave is:

k = 2𝜋/𝜆 = p/ħ = √(2mE)/ħ

Where,

• E denotes the particle’s kinetic energy.
• m denotes the particle’s mass
• p denotes the particle’s momentum

ħ is Planck’s constant in its reduced form.

In the particular situation of electromagnetic wave propagation in vacuum, the wavenumber is defined as:

k = f/c

Where,

• f denotes the wave’s frequency.

• c is the speed of light

Quantum Physics Of Wave Numbers

The canonical momentum of quantum mechanical waves is defined as the wavenumber multiplied by the decreased Planck’s constant. Other than spatial frequency, the wavenumber can be used to express other parameters. Assuming a constant light speed, it is widely employed as a temporal frequency unit.

Significance of Wavenumber

In essence, wavenumber is to wavelength what frequency is to the time period. The frequency of a wave is the number of oscillations that occur at a point in unit time. The wave-number specifies the number of complete waves present at a given moment in a unit length. Their product is equal to the reciprocal of the wave speed.

Wavenumber Applications

• The spatial frequency is calculated using a wavenumber.
• In addition to spatial frequency, wavenumber may be used to describe other concepts in physics, including optics and wave scattering.
• In X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics, wavenumbers and wave vectors are critical.
• In quantum mechanical waves, the canonical momentum is equal to the wavenumber multiplied by the decreased Planck’s constant.
• A wavenumber can be used to determine the group velocity.

Problems on Wave Number

In this part, we will learn how to calculate the wavenumber utilising the various formulae mentioned thus far. Due to the fact that there are several formulas for calculating the wavenumber based on the various linked variables. Let us solve some questions to have a better understanding of wavenumber ideas.

Conclusion

Therefore it is concluded that it is utilised specifically to determine the number of cycles per unit distance. The magnitude of a wavenumber is directionless. A wave number is defined as “the spatial frequency of a wave, expressed in cycles per unit distance or radians per unit distance.” In multidimensional systems, the Wavenumber denotes the magnitude of the wave vector. The wavenumber is a scalar value. In spectroscopy and chemistry, the wavenumber is defined as the number of wavelengths per unit distance. The angular wavenumber is the number of radians per unit distance.