simplification, all sources are present. Some of the sources are then identified as turbulent or laminar noise. The far-field sound pressure is then given in terms of a volume integral over the domain containing the sound source. The source term always includes physical sources and such sources, which describe the propagation in an inhomogeneous medium. The wave operator of Lighthill's analogy is limited to constant flow conditions outside the source zone. No variation of density, speed of sound and Mach number is allowed. Different mean flow conditions are identified as strong sources with opposite sign by the analogy, once an acoustic wave passes it. Part of the acoustic wave is removed by one source and a new wave is radiated to fix the different wave speed. This often leads very large volumes with strong sources. Several modifications to Lighthill's original theory have been proposed to account for the sound-flow interaction or other effects. To improve Lighthill's analogy different quantities inside the wave operator as well as different wave operators are considered by following analogies. All of them obtain modified source terms, which sometimes allow a more clear sight on the "real" sources. The acoustic analogies of Lilley, Pierce, Howe and Möhring are only some examples for aeroacoustic analogies based on Lighthill's ideas. All acoustic analogies require a volume integration over a source term. The major difficulty with the acoustic analogy, however, is that the sound source is not compact in supersonic flow. Errors could be encountered in calculating the sound field, unless the computational domain could be extended in the downstream direction beyond the location where the sound source has completely decayed. Furthermore, an accurate account of the retarded time-effect requires keeping a long record of the time-history of the converged solutions of the sound source, which again represents a storage problem. For realistic problems, the required storage can reach the order of 1 terabyte of data. Kirchhoff integral Kirchhoff and Helmholtz showed, that the radiation of sound from a limited source region can be described by enclosing this source region by a control surface - the so-called Kirchhoff surface. Then the sound field inside or outside the surface, where no sources are allowed and the wave operator on the left hand side applies, can be produced as a superposition of monopoles and dipoles on the surface. The theory follows directly from the wave equation. The source strength of monopoles