frequencywavenumber

Frequency-wavenumber analysis refers to the joint representation of a wavefield in the frequency domain (temporal) and the wavenumber domain (spatial). In practice, one applies a Fourier transform in time and in space to a signal u(x, t), producing a spectrum U(k, ω) or S(k, ω). Here ω is the angular frequency (2πf) and k is the spatial wavenumber vector (m^-1), with magnitude |k| = 2π/λ in one dimension or components in higher dimensions.

The frequency-wavenumber spectrum reveals how energy is distributed across temporal and spatial scales. Dispersive waves satisfy a dispersion relation ω = ω(k), so peaks or ridges in the k-ω diagram trace the phase or group velocity v_p = ω/k and v_g = dω/dk. In homogeneous media, distinct wave modes appear as lines with characteristic slopes; in complex media, energy can spread across k and ω due to scattering.

Common applications include seismology, acoustics, and optics, where k-ω analysis helps identify wave modes, estimate velocity models, separate source and noise, and create images of subsurface structures. The method underpins dispersion analysis, seismic tomography, and array processing.

Computationally, a 2D Fourier transform (in time and space) yields U(k, ω). In nonstationary data, short-time or windowed transforms, 2D spectrograms, or wavelet methods are used. Limitations include the uncertainty principle between time and frequency and spatial resolution constraints, finite data length, and noise.