In their classical text on fluid mechanics, Landau and Lifshitz derive an aeroacoustic equation analogous to Lighthills (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 Lighthills 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 ) everywhere, then we obtain exactly the equation given in Landau and Lifshitz, namely
A similar approximation [in the context of equation ], namely , is suggested by Lighthill [see Eq. (7) in the latter paper].
Of course, one might wonder whether we are justified in assuming that . The answer is affirmative, if the flow satisfies certain basic assumptions. In particular, if  and , then the assumed relation follows directly from the linear theory of sound waves (see, e.g., the linearized Euler equations and the acoustic wave equation). In fact, the approximate relation between  and  that we assumed is just a linear approximation to the generic barotropic equation of state of the fluid.
However, even after the above deliberations, it is still not clear whether one is justified in using an inherently linear relation to simplify a nonlinear wave equation. Nevertheless, it is a very common practice in nonlinear acoustics as the textbooks on the subject show: e.g., Naugolnykh and Ostrovsky and Hamilton and Morfey.