In their classical text on fluid mechanics, Landau and Lifshitz[4] 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 p rather than for the density \rho 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 p-p_0=c_0^2(\rho-\rho_0), where \rho_0 and p_0 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