Local spectra and wave distribution functions

Wave spectra observed by satellites in space are local, in the sense that they depend on the time and place of observation. Fourier transforms and power spectra are on the other hand global quantities, defined by integrals over infinite intervals. This discrepancy between theory and practice often causes difficulties, which can be overcome if a new definition of a local power spectrum is adopted. The proposed definition is closely related to experimental practice, yet very convenient in analytical calculations. A distribution function, and an associated continuity equation, for the wave energy density in a plasma is derived.