Most people are familiar with wavelengths, but a “wavenumber” is a little more puzzling. If you’re trying to make sense of this term and work out what to do with it, learning to convert the wavenumber to a wavelength helps you understand what a wavenumber is and extract some more usable information about the wave it describes. The conversion is simple as soon as you learn the definition of a wavenumber.
TL;DR (Too Long; Didn't Read)
Convert from wavenumber to wavelength by dividing 1 by the wavenumber. If the wavenumber is expressed in 1/m, you will get a result in m. If the wavenumber is expressed in 1/cm, you will get a result in cm. You can convert the result into the required unit in the usual way.
What Is a Wavenumber?
A wavenumber is the reciprocal of the wavelength of the wave. This tells you how many wavelengths fit into a unit of distance. It is analogous to frequency, which tells you how often a wave completes a cycle per unit of time (for a traveling wave, this is how many complete wavelengths pass a given point per second).
The standard scientific (SI) unit of distance is the meter (m), but in many cases wavelengths may be expressed in centimeters (cm) or other units. The wavelength is given the symbol λ, and the wavenumber is given the symbol k. It is defined by:
k = 1 ÷ λ
The wavenumber has units of 1/distance, or distance−1. For wavelengths expressed in meters, this is m−1, and if the wavelength is expressed in cm, the units of the wavenumber are cm−1.
Converting From Wavenumber to Wavelength
The definition of a wavenumber is quite simple, and it only depends on the wavelength. Converting from a wavenumber to a wavelength is a straightforward process. You can convert wavenumbers to wavelength using the formula:
λ = 1 ÷ k
So if you have a wavenumber (k), divide 1 by this number to get the wavelength. Using a wavenumber of 100 m−1 as an example, the wavelength is:
λ = 1 ÷ 100 m−1 = 0.01 m
The wavelength of this is 1 cm. If this wavelength represents electromagnetic radiation, it would be a microwave, just beyond the infrared spectral region.
Getting the Right Units
Wavenumbers may be expressed in different units, notably cm−1. If you have a wavenumber in a different unit, you can convert it to a wavelength in the same way as in the previous section. The only difference is that the wavelength you’ll end up with will be in a different unit. If the wavenumber was expressed in cm−1, the resulting wavelength will be in cm. If the wavenumber was expressed in nm−1 (nanometers−1), then the wavelength will be in nm.
If you need your answer in a specific unit, convert your resulting wavelength into the required unit. In general, to change to a smaller unit of measurement, you multiply by the conversion factor (the number of smaller units per bigger unit). To change to a bigger unit of measurement, divide by the conversion factor.
For example, if you get a result in meters and you need it in nanometers, multiply the result in meters by 1,000,000,000 (or 109). To convert from nanometers to meters, you divide the result by 1,000,000,000. If you get a result in centimeters, but need it in meters, divide your result by 100. To convert from meters to centimeters, multiply your result by 100. You can use a conversion chart or an online converter to do this if you’re unsure.
In some areas of physics (such as geophysics), you might encounter an “angular wavenumber.” This is very similar to wavenumber, except it’s multiplied by 2π, so it describes rotations or oscillations. The unit of an angular wavenumber is radians per meter. To convert an angular wavenumber to a wavelength, divide 2π by the angular wavenumber.