Nonlinear wave equations for low-frequency acoustic gravity waves

It is shown that low-frequency acoustic gravity waves propagating parallel to the Earth's surface satisfy the Korteweg-De Vries equation or the Kadomtsev-Petviashvili equation and have a discrete spectrum of group velocities. The atmosphere is considered to be incompressible, homogeneous in composition, and isothermal; and the gravitational acceleration depends upon the height. The hydrostatic approximation is used and adapted in such a way that dispersion is not neglected. The nonlinear wave equations are obtained by using the reductive perturbation technique. Changes for a compressible atmosphere are discussed.