wavenumber

Wavenumber is a quantity that describes the spatial frequency of a wave along a given direction. In the standard form of a monochromatic plane wave, ψ(r,t) = A e^{i(k⃗ · r − ωt)}, the wavevector k⃗ has magnitude k = |k⃗|, called the wavenumber. The wavenumber has units of inverse length (typically m⁻¹) and is related to the wavelength λ by k = 2π/λ. Thus longer wavelengths correspond to smaller wavenumbers.

There are two related, but not always identical, uses of the term wavenumber. In physics and wave mechanics, k is the angular wavenumber—the magnitude of the wavevector. In some spectroscopic and data-analysis contexts, wavenumber ν̃ is defined as the reciprocal of wavelength (ν̃ = 1/λ) and is commonly expressed in cm⁻¹; in that convention, k = 2π ν̃, so ν̃ = k/(2π). In quantum mechanics, the wavenumber is tied to momentum via p = ħk, where ħ is the reduced Planck constant.

The wavenumber is a vector quantity in three dimensions: k⃗, whose magnitude is k and whose direction indicates the propagation direction of the wave. Related quantities include the phase velocity v_p = ω/k and the group velocity v_g = dω/dk; in nondispersive media, ω = v k, where v is the phase speed.

Applications span many fields. In optics and acoustics, wavenumber characterizes wave propagation and dispersion. In Fourier analysis, spatial frequency corresponds to k/2π. In crystallography and solid-state physics, the wavevector is central to describing wave-like behavior in periodic media.