Fields of cylindrically curved ionosphere layers are approximated by exponential functions that consider the increase of the phase velocity and decrease of the attenuation rate with the altitude of the layers. Matrix multiplication techniques are applied to the treatment of multilayer ionosphere models. The individual ionosphere layers are thin relative to the wavelength, and the approximate field representations give in the limits of horizontal and radial static magnetic field results identical to those obtained using more accurate treatments of curvature effects. TM modes in the space between the earth and the ionosphere excite coupled TM and TE modes in the ionospheric layers, the presence of which are considered in an iterative solution of the modal equation. In the ELF range, propagation parameters computed for the east‐to‐west (EW) and west‐to‐east (WE) directions differ most for dip angles of 15‐30°. In the VLF range propagation parameters may exhibit discontinuities at nighttime for the EW direction, when the fields penetrate higher into the ionosphere. The interference distance D of the two lower wave guide modes is increased for propagation in the EW direction relative to propagation in WE direction provided that the electron concentration profiles exhibit sharp density gradients at heights in the range from 80 to 85 km. The desired propagation characteristics could not be produced in the presence of a nondipping static magnetic field or using smoothly varying electron concentration profiles.

A slot covered by a stratiiied plasma is assumed to radiate into a wide waveguide instead of free space. The slot admittance approximates the free space admittance of the slot for waveguide diameters exceeding 6 to 10h. For thick plasma layers the computed slot admittance checks with earlier admittance calculations for a laterally unbounded plasma. When approximating a plasma profile of a typical hypersonic re-entry, a multilayer plasma model in a wide waveguide appears to provide a more accurate slot admittance than a single-layer approximation in a laterally unbounded geometry.

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The surface impedance of a cylindrically stratified anisotropic ionosphere has been computed in the presence of a transverse static magnetic field without requiring explicit representations of the individual cylindrical wave functions in the lower ionospheric layers. The propagation parameters are determined from the usual transcendental modal equation which is solved starting out with an initial real solution dependent on the reactive part of the ionospheric surface impedance. The surface impedance computa~ions are continued below the ionospheric boundary height, and the impedance reflected to the ground level by the ionosphere is compared with the ground impedance. The accu· racy of the computations is shown to be comparable to a few percent uncert.iinty of the ground im· pedance for low conductivity ground.The propagation parameters of the first three waveguide modes are calQulated for a number of ionosphere models which include the recent D region models of Deeks (1966). For propagation in the east-to-west direction, the phase velocity of the second mode is shown to be discontinuous when the real part of the ionospheric reflection coefficient undergoes a sign change. This condition is characterized by an absorption peak and for ionosphere models of lower anisotropy Gower values of WH/v, where WH is the gyrofrequency and vis the effective collision frequency) the interference distance D of the two lowest modes is increased for propagation in the east-to-west direction relative tO. the WeSt·~o-east direction." but it is decrea~ed for larger WH/V ratiOS which are repJksentative of available models of stratified ionosphere. . IntroductionThe propagation of VLF waves in the guide between the curved earth and a sharply bounded homogeneous ionosphere has been discussed by Wait (1962). More refined ionosphere models have been investigated by Walters (1963, 1964), Wait (1963), and Galejs (1964b,c;1965), and extensive numerical results have been presented recently by Spies (1964, 1965). This work contains data on phase velocities and on signal attenuation rates.Recently observed sunrise and sunset fading over long VLF paths can be explained by dif. ferences between the phase velocities of the n = 1 and n = 2 modes over the nighttime portions of the propagation path. The interference pattern between the two modes changes with the movement of the sunrise or sunset line (the "terminator") along the path. The distance D of the terminator movement which corresponds to a complete interference pattern is given by (1) where Aj is the guide wavelength of the j mode, Ao is the free-space wavelength, and Re Si = c/v1 (Crombie, 1964). This distance D represents the spacing between successive minima of the interference patterns, and it provides an accurate indication of the differences in the phase velocities of the two waveguide modes. Crombie (1966) reports that Dis approximately 1900 km for west-to-east propagation (WE), 2025 km for north-to-south or south-to-north propagation (NS or SN), and 2800 km for propagation in the ...