Question

What is the $SI$ unit of wave number?
(A) $s$
(B) ${s^{ - 1}}$
(C) $m$
(D) ${m^{ - 1}}$

Answer 1

Hint:The wavenumber is the scalar quantity that can be measured in the spatial frequency or the temporal frequency of the wave. The spatial frequency is measured in cycles per unit distance or the radians per unit distance. The temporal frequency is measured from the wavelength.

Useful formula:
The formula of the wave number is given as

$w = \dfrac{1}{\lambda }$

Where $w$ is the wave number and $\lambda $ is the wavelength.

Complete step by step solution:
The $SI$ unit of the wavenumber provides the temporal frequency by finding the relationship between the wavelength and the wavenumber of the wave. The wavelength is defined as the length of one wave which includes both the crest and the trough part of the wave or it may also be said as the distance between one crest and the other crest or the distance between the one trough and the other trough. The unit of the wavelength is specified by metre, centimeter or millimeters.
The wavenumber is defined as the reciprocal of the wavelength of the considered wave. This measures the total number of waves that passes per unit length.

$w = \dfrac{1}{\lambda }$

This wave number is constant for one wave but it may vary from one wave to the other. The standard system of unit of the wavelength is metre and hence the unit of the wavenumber must be inverse of the metre (metre inverse). Hence the $SI$ unit of the wave number is ${m^{ - 1}}$ .

Thus the option (D) is correct.

Note:The wave number is also known as angular wavenumber or the propagation number. The wave number may be of complex value, if its relative permittivity, relative permeability and the reflective index of the medium at which the wave is propagating is complex.