How To Calculate A Wavenumber

Waves can describe sound, light or even the wavefunction of particles, but every wave has a wavenumber. This describes how it varies through space, and this depends crucially on the wavelength of the wave or its speed and frequency. For physics or chemistry students, learning to calculate a wavenumber forms a vital part of mastering the subject. The good news is that there is a simple formula for the wavenumber, and you need only very basic information about the wave to calculate it.

Physicists and chemists use two different types of wavenumber – either the spatial wavenumber (often called spatial frequency) or the angular wavenumber (sometimes called the circular wavenumber). The spatial wavenumber tells you the number of wavelengths per unit distance, whereas the angular wavenumber tells you the number of radians (a measure of angle) per unit distance. Generally speaking, angular wavenumber is used in physics and geophysics, whereas spatial wavenumber is used in chemistry. Essentially, the equations are the same except the angular wavenumber uses 2π as the numerator, because this is the number of radians in a whole circle (equivalent to 360°).

1. Find the Information You Need About the Wave

Find the wavelength of the wave before calculating the angular or spatial wavenumber. Both quantities depend only on the wavelength, denoted by the symbol ​λ​, and you can even read this directly from a visual representation of the wave as the distance between successive "peaks" or "troughs" of the wave.

If you don't have the wavelength, you can use the relationship:

\(\lambda = \frac{v}{f}\)

Where ​v​ stands for the speed of the wave and ​f​ stands for its frequency. This means you can calculate the wavenumber with a frequency and a speed, noting that for light waves, the speed is always ​v​ = ​c​ = 2.998 × 10⁸ meters per second.

2. Choose the Right Form of the Equation

Use the following relationship to calculate the spatial wavenumber (represented here by ​n​, although other symbols are sometimes used):

\(n=\frac{1}{\lambda}=\frac{f}{v}\)

Where the first definition simply represents the reciprocal of the wavelength, and the second expresses this as the frequency divided by the speed of the wave. Wavenumbers have units of length⁻¹, e.g., for meters (m), this would be m⁻¹.

For the angular wavenumber (denoted by ​k​), the formula is:

\(k=\frac{2\pi}{\lambda}=\frac{2\pi f}{v}\)

Where again the first uses wavelength and the second translates this into a frequency and a speed.

3. Calculate the Wavenumber

Calculate the wavenumber using the appropriate equation. For a light wave with a wavelength of 700 nanometers or 700 × 10⁻⁹ m, representing red light, the calculation of angular wavenumber is:

\(k=\frac{2\pi}{\lambda}=\frac{2\pi}{700\times 10^{-9}}=8.98\times 10^6\text{ m}^{-1}\)

For a sound wave, with a frequency of 200 Hz and a speed of 343 meters per second (m s⁻¹), the calculation of spatial wavenumber gives:

\(n=\frac{f}{v}=\frac{200}{343}=0.583\text{ m}^{-1}\)