New equations for nonlinear acoustics in a low Mach number and weakly heterogeneous atmosphere

Various scalar equations are proposed, modeling the pressure field in the linear and nonlinear acoustical regimes. They are derived by assuming a flow with a small Mach number and a smaller medium heterogeneity. Such assumptions are well satisfied in the atmospheric boundary layer. Further simplifications can be obtained when less intense turbulent fluctuations are superimposed to a sheared mean flow. In the linear regime, a hierarchy of equations with increasing orders of precision is established. A new equation is found where all terms quadratic with respect to the ambient flow are retained, either related to sound convection by the flow, or to the flow inhomogeneity. Numerical solutions indicate that it is more precise than the equations in the literature for small Mach numbers, but less robust for larger negative Mach numbers. Two generalizations of Lilley’s equation incorporate the effects of turbulent fluctuations. Nonlinear terms are of different origins, either thermodynamical, inertial, or related to the flow shear. For a locally plane wave, they simplify into a single term which appears as the classical Westervelt quadratic nonlinearity convected by the flow. Consequently, all linear equations can easily be generalized to nonlinear ones, such as a new Lilley’s equation augmented with acoustical nonlinearities and turbulent flow fluctuations.