The relationship between upward refraction above a complex impedance plane and the spherical wave evaluation for a homogeneous atmosphere

Abstract

An investigation has been carried out on the relationship between the residue series method for the prediction of sound propagation in an upward refracting atmosphere and the spherical wave analysis of sound propagation in a homogeneous atmosphere. It is shown that, in the limit of small sound velocity gradients, the Airy function solution developed by Pierce for sound propagation over the earth’s surface does approach the Sommerfeld integral for spherical wave reflection in a homogeneous atmosphere for a limited range of values of wave number. For complex impedance with phases greater than π/3, it is found that the surface wave pole identified for sound propagation in a homogeneous atmosphere is present in the upward refracting atmosphere. This pole arises from the residue series. Methods suggested in the literature for searching for the poles will fail for impedances with phase angles greater than π/3.