An asymptotic analysis has been carried out to study the propagation of sound at long ranges in an arbitrary downward refracting sound‐speed profile. It has been shown that the solution can be expressed in a form of the Weyl–Van der Pol formula and the present theory offers an extension for the established theory to the case of a vertically stratified medium. The analysis starts from the full wave equation and the derivation assumes that the ambient properties only depend on the height above an impedance ground and that there is no wind or turbulence in the medium. In addition, the theory also assumes that the sound speed of the atmosphere increases monotonically with height. For such an atmospheric condition, the medium forms a waveguide that enhances the sound levels at long ranges along the ground surface. It is demonstrated that the acoustical path length derived by the asymptotic method is identical to that derived by the analytical ray trace model. Furthermore, the asymptotic method also allows a proper inclusion of the surface wave term rigorously.