On the Effect of Atmospheric Temperature Gradients and Ground Impedance on Sound

Abstract

The acoustic wave equation is solved exactly in a polytropic atmosphere over a flat impedance ground. The effects of atmospheric stratification on sound are presented as a correction to the amplitude and phase of a plane wave compared to a homogeneous medium corresponding to sea level conditions. It is shown that: (i) the direct effect of atmospheric stratification on sound amplitude is very small over the entire audible range 20Hz-20kHz, even for altitude ranges in excess of 1 Km: (ii) the atmospheric stratification causes significant phase changes relative to an homogeneous medium for frequencies above 500Hz even over altitude ranges of less than 100m; (iv) the phase changes due to atmospheric stratification can affect significantly interference between acoustic waves, for example in the case of multiple sound sources or multiple reflections by nearby obstacles. The exact solution of the wave equation for the acoustic potential specifies also the velocity and pressure perturbation that appear in the impedance boundary condition over flat ground. This specifies the amplitude and phase of the ground reflection coefficient and also the amplitude and phase of the total obsorption factor of the atmosphere plus ground. The amplitude and phase of the reflection coefficient are plotted as a function of the angle of incidence for a fixed altitude of propagation of the wave for a source above ground, such as an aircraft near an airport and several values of the ground impedance. The theory applies to wavelengths smaller, comparable or larger than the lenghthscales of variation of atmospheric properties, since it is based on an exact solution of the wave equation, without ray, parabolic equation, or any other approximations.