For correlated signals has information regarding the phase difference between and . For example, if and are harmonic with phase difference , is also harmonic with a starting value that depends on . It is thus a convenient measure of the phase difference itself. If and are random signals will be a function of the frequency variable . In such problems we work with the cross-spectral function defined as

In wave propagation problems, has information regarding and hence the wave number distribution and the wave speeds as a function of frequency. In nonlinear dynamics this is further interpreted in terms of appearance of coherent structures.

The following results can be easily derived.

If and , and is a delta function centred at . One can calculate from . In a travelling wave problem, and may be two signals obtained from probes separated by a distance . The wave number is then given as . If is white noise and , i.e. is a time shifted form of , one can show that where is the power spectrum of (and a constant if is white noise). Subsequently it is easy to show that and .