A modified Zabolotskaya–Khokhlov equation for systems having small quadratic nonlinearity

We examine three-dimensional, weakly diffracting, weakly nonlinear waves propagating in a uniform medium. A method of multiple scales is used to derive an extension of the Zabolotskaya–Khokhlov equation which is valid for systems having a quadratic nonlinearity parameter which is either zero or small. The effects of relaxation, dissipation, and dispersion are also considered resulting in general forms of the modified Kadomtsev–Petviashvili equation and the modified Khokhlov–Zabolotskaya–Kuznetsov equation. Further results include the delineation of the simplifications required when the system is such that waves propagate isotropically, i.e., where the shock and characteristic speeds are independent of the direction of propagation. We also illustrate our results with physical examples corresponding to the nonlinear acoustics of pressurized gases with a uniform wind and obliquely propagating Alfvén waves.