Wavenumbers are identified with spatial structures in the same fashion that frequency is the transform of time. The wavenumber- frequency spectrum is used to decompose array data when details of the spatial structure of a homogeneous field are important. The acoustic far field of a noise source is one example.  If the field is not homogeneous then a more general decomposition known as the Karhunen-Loeve expansion is necessary.

The wavenumber frequency spectrum is constructed from the cross spectral density (CSD) by projecting a spatial harmonic waveform onto the data. The quantity inside the square brackets is the projection vector. It operates on the kernel, Fjl , to produce the result b(w,k).

b(w,k)