The ducting of acoustic gravity waves by the thermosphere: An analytic solution

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Abstract
An equation describing the propagation of acoustic‐gravity waves (AGW) is solved analytically for an atmosphere whose sound speed squared, c2(z), varies with altitude, z, as: with co 2, Δc 2, l and z o all constant. Such a profile represents the actual atmosphere above 85 kilometers, the thermosphere, fairly well. Solutions are obtained for the AGW equation with this c2(z) profile in terms of Heun functions, which are rather general special functions. The effects of the ground and lower atmosphere are neglected by applying boundary conditions to these solutions which require the finiteness of the altitude integral of AGW energy density when integrated from z = ‐ 8 to z = + 8. Ducted or trapped propagation modes are obtained by this procedure. The dispersion relations for these analytically obtained ducted modes are found to correspond at high phase speeds and long periods with the dispersion relations for internal gravity modes obtained numerically by simulating the atmosphere with a large number of isothermal layers. The propagation parameters (periods 7–20 minutes, wavelengths 100–300 km and velocities, 120–280 meters/second) of the modes are in general agreement with the observed values for a class of travelling ionospheric disturbances. The role of the thermosphere in the ducting of AGW is thus elucidated and these analytic solutions may have value in determining AGW spectra generated by extended sources at thermospheric altitudes.

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