A theoretical discussion is presented of the influence of gravity on sound propagation from a small source in an isothermal atmosphere where ambient pressure and density decrease exponentially with height. A solution for the free-space case is derived that indicates that waves with angular frequency ω between (γ − 1)12g/c and (γ/2)g/c will not be propagated, while those with ω between 0 and (γ − 1)12(g/c) cosθ will not be propagated in a direction making an angle of θ with the vertical axis. A formal solution incorporating appropriate boundary conditions at the ground is derived and discussed. The field along the vertical line passing through the source is found explicitly. A consideration of the energy intensity shows that no energy is propagated within a cone above and below the source if ω < (γ − 1)12g/c, A calculation of the intensity for the case when (γ − 1)12g/c < ω < (γ/2)g/c indicates that the energy flowing from the source tends to concentrate in the lowest layers of the atmosphere. The field for large horizontal distances appears as a sum of a direct wave, a reflected wave, and a surface wave. Reflection coefficients are derived, and the criteria for the surface wave to be dominant are discussed.
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Allan D. Pierce (1963) studied this question.