A high-frequency approximation of sound propagation in a stratified moving atmosphere above a porous ground surface

Abstract

This paper examines the sound propagation in a moving stratified atmosphere above a porous ground surface. A generalized expression is derived for the sound field above a porous half-space and the existence of an asymptotic behavior is demonstrated. The leading term of the solution is not only consistent with ray theory but mathematically rigorous. The analysis starts from the fundamental hydrodynamic equations of pressure and particle velocity, etc., and the assumption that the atmosphere is vertically stratified. These hydrodynamic equations are reduced to a one-dimensional Helmholtz equation by applying the Fourier transform method. The solution for the simplified Helmholtz equation is approximated in the high-frequency limit and then the sound pressure is represented by a Fourier integral. The method of stationary phase is used to evaluate the Fourier integral in order to give an asymptotic expression for the sound pressure. It should be emphasized that the Weyl–Van der Pol formula has been generalized to give the field due to a point source in a moving stratified medium above an extended reaction boundary. The total sound field is derived as the sum of three components: a direct contribution from the source, a geometrically reflected component, and a ground wave term, respectively. The ground wave gives a significant contribution to the total sound field especially at low frequencies and at the ground effect dip.