(double) tensor contraction operator. The above equation is the celebrated Lighthill equation of aeroacoustics. It is a wave equation with a source term on the right-hand side, i.e. an inhomogeneous wave equation. The argument of the "double-divergence operator" on the right-hand side of last equation, i.e. , is the so-called Lighthill turbulence stress tensor for the acoustic field, and it is commonly denoted by . Using Einstein notation, Lighthill’s equation can be written as where and is the Kronecker delta. Each of the acoustic source terms, i.e. terms in , may play a significant role in the generation of noise depending upon flow conditions considered. describes unsteady convection of flow (or Reynolds' Stress, developed by Osborne Reynolds), describes sound generated by viscosity, and describes non-linear acoustic generation processes. In practice, it is customary to neglect the effects of viscosity on the fluid, i.e. one takes , because it is generally accepted that the effects of the latter on noise generation, in most situations, are orders of magnitude smaller than those due to the other terms. Lighthill provides an in-depth discussion of this matter. In aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation in order to make statements regarding the relevant aerodynamic noise generation mechanisms present. Finally, it is important to realize that Lighthill's equation is exact in the sense that no approximations of any kind have been made in its derivation. Related model equations In their classical text on fluid mechanics, Landau and Lifshitz derive an aeroacoustic equation analogous to Lighthill's (i.e., an equation for sound generated by "turbulent" fluid motion), but for the incompressible flow of an inviscid fluid. The inhomogeneous wave equation that they obtain is for the pressure rather than for the density of the fluid. Furthermore, unlike Lighthill's equation, Landau and Lifshitz's equation is not exact; it is an approximation. If one is to allow for approximations to be made, a simpler way (without necessarily assuming the fluid is incompressible) to obtain an approximation to Lighthill's equation is to assume that , where and are the (characteristic) density and pressure of the fluid in its equilibrium state. Then, upon substitution the assumed relation between pressure and density into we obtain the equation (for an inviscid fluid, σ = 0) And for the case when the fluid is indeed incompressible, i.e. (for some positive constant )