wave number — from formula anatomyThe wave number is a measure of spatial frequency: it tells you how many complete wave cycles fit into a unit of length. In physics, it appears in the mathematical description of waves, from light and sound to quantum mechanical wavefunctions. It is the spatial analogue of temporal frequency — just as frequency counts oscillations per second, wave number counts oscillations per meter.
The concept emerged in the 19th century with the mathematical treatment of wave phenomena, though the term 'wave number' became standard in spectroscopy and quantum theory in the early 20th century.
For a wave of wavelength λ, the wave number k is defined as k = 2π/λ. This definition makes it a natural companion to angular frequency ω = 2πf, so that a traveling wave can be written compactly as A·sin(kx − ωt). In quantum mechanics, the wave number is directly proportional to momentum via the de Broglie relation p = ħk, making it a bridge between wave and particle descriptions.
Wave numbers are used in spectroscopy to label spectral lines, in crystallography to index diffraction peaks, and in oceanography to describe surface wave fields.
In spectroscopy, wave numbers are often quoted in reciprocal centimeters (cm⁻¹), so a visible red photon at 700 nm has a wave number of about 14,300 cm⁻¹.
| \psi | variable | wave function — wave function |
| x | variable | position coordinate — position metre |
| k | variable | wave number — wave number per metre |
| V(x) | variable | potential function — potential energy joule |
| \frac{d^2}{dx^2} | operator | second derivative with respect to position — second derivative |
| \tilde{\kappa} | variable | Fourier transform of convergence — convergence |
| \tilde{\kappa}_B | variable | Fourier transform of B-mode convergence — b mode convergence |
| \tilde{\gamma}_1 | variable | Fourier transform of first shear component — shear component |
| \tilde{\gamma}_2 | variable | Fourier transform of second shear component — shear component |
| k_1 | variable | First component of wave vector — wave vector component radian per metre |
| k_2 | variable | Second component of wave vector — wave vector component radian per metre |
| k | variable | Magnitude of wave vector — wave number radian per metre |
| ω | variable | angular frequency — angular frequency radian per second |
| k | variable | wave number — wave number radian per metre |
| c | variable | propagation speed — wave speed metre per second |
| | | | operator | returns the magnitude of a real number — absolute value operator |
| ω | variable | Angular frequency of the wave — angular frequency radian per second |
| k | variable | Wave number (spatial frequency) — wave number radian per metre |