Equation (3) is tractable in the usual way if the right-hand side is known independently of any acoustic consideration. The following approximations are generally retained; they can be partially released if the needed information is provided by a CFD code, for instance.
First the Reynolds number of the flow is assumed high and the fluctuating Mach number is assumed low. This leads to Lighthill’s approximation Tj ~ p0 UiUj in which po is the mean density and U stands for the aerodynamic velocity, cleaned of the acoustic motion. The second source term represents all contact forces, say P, applied from the surfaces onto the fluid, and can be written V – P. It involves both aerodynamic forces and acoustic pressure forces which represent sound scattering effects: P = Paero + p n. In many applications of interest, typically dealing with rotating-blade noise technology, the diffraction effects are ignored. This is accepted for convenience but only valid as long as the surfaces are acoustically compact. Any surface in a disturbed flow is both a source of sound and a scattering screen. Substantial errors can be generated if the surfaces are not compact anymore. The third term does not require any approximation, since it is
Lighthill’s tensor is now defined by the relative velocity V’ = V — V0, as well as the thickness-noise term involving the surface velocity vector Vs. Furthermore time derivatives are accounting for convection by the mean flow : D/Dt = d/dt + V0 – V.