This work is a generalization of Lighthill's acoustic analogy in which it is proved that the jet noise problem can be modelled exactly by equivalent sources near a vortex sheet. Mani's work has shown that this type of scheme can agree well with experiment. This theory justifies Mani's general procedure but gives in addition the equivalent sources needed for an exact analogy. Each moving fluid particle supports a quadrupole whose strength per unit mass is given by Lighthill's stress tensor and the sound radiates as if it were adjacent to a laminar instability free vortex sheet. Though we show that the sound is determined in terms of the turbulence stress tensor, sound is also generated by the flow's instability waves as they grow into turbulence, and this sound appears as an exponentially growing precursor of the main field. Some well known features of the mean flow acoustic interaction issue are an immediate consequence of the theory. We examine the case of a round jet in some detail and concentrate on an aspect that we think is new. When the mean jet density is much lower than that of its environment then the mean flow-acoustic interaction results in a considerable amplification of the quadrupole field, and the intensity of its sound can scale on an unusually low power of jet speed. We show that a fourth power law is possible and even a second power law when the density difference is large enough. This may be part of the 'excess noise' problem in which the sound of engine-produced hot jets is often insensitive to changes in jet speed at low exhaust power.