Wavenumbers

Wavenumber is a measure of the spatial frequency of a wave. For a one-dimensional sinusoidal wave with wavelength λ, the wavenumber is defined as k = 2π/λ. The wavenumber can be treated as a vector k⃗, whose magnitude is |k⃗| = 2π/λ and whose direction indicates the wave’s propagation. In a plane wave, the phase is φ = k⃗·x − ωt, and dispersion relations express how the temporal frequency ω depends on k⃗. In non-dispersive media, ω = c|k⃗|, where c is the wave speed.

In mathematics and signal analysis, the wavenumber is the spatial frequency variable in Fourier transforms. A spatial function can be decomposed into plane waves with different wavenumbers, and k⃗ plays the role conjugate to position. In multiple dimensions, k⃗ has components along each axis, forming a spectrum of spatial frequencies.

In spectroscopy, the term wavenumber ν̃ refers to the reciprocal of wavelength, commonly expressed in cm⁻¹. Since ν̃ = 1/λ (with λ measured in centimeters), photon energy is E = h c ν̃. Wavenumbers are convenient in infrared and vibrational spectroscopy because they scale with energy. For example, a wavelength near 500 nm corresponds to a wavenumber ν̃ ≈ 2.0 × 10⁴ cm⁻¹, while the corresponding spatial wavenumber magnitude is |k⃗| ≈ 1.26 × 10⁵ cm⁻¹.

Applications extend to solid-state and optical physics, where k determines electron and phonon dispersion, as well as to the study of wave phenomena in fluids and acoustics. Boundary conditions and crystal symmetry set allowed k-values, leading to band structures and Brillouin zones.