Zonal harmonics in low frequency terrestrial radio wave propagation.

Johler, J. R.

doi:10.6028/nbs.tn.335

Cited by 9 publications

(4 citation statements)

References 4 publications

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“…The ELF or VLF frequencies are not, however, the upper limit of usefulness for spherical wave functions in terrestrial radio wave problems. It has been shown [Johler, 1964[Johler, , 1966] that this spherical wave solution can be reformulated as a geometric series to improve the slow convergence of the series of zonal harmonics.…”

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Spherical Wave Theory for MF, LF, and VLF Propagation

Johler¹

1970

Radio Science

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If (n) is the set of positive integers, spherical waves like ζn(1, 2)(kr)Pn(cos θ) in r, θ, ϕ spherical coordinates are used to develop a new approach to terrestrial radio wave field calculations. The spherical wave functions of integer order are used as building blocks for the solution of Maxwell's equations and boundary conditions. Instead of a direct summation of the zonal harmonics or n series, a double summation of a modified zonal harmonics series and the geometric series is used to obtain rapid convergence of the former. The most important disadvantage of an n‐series summation is the construction of the solution at the ground interface, which generally causes the n series to converge slowly. This difficulty is overcome by the geometric series approach that permits removal of the ground wave from the n series. The ground wave is then calculated by using classical methods. The remaining ionospheric waves are then calculated with the integer‐order spherical wave functions. The formulation presented is applicable to VLF, LF, and MF; anisotropic reflection coefficients based on electron and ion density profiles of the ionosphere are introduced into the analysis. Results obtained with a computer simulation of the formulas using integer‐order spherical wave functions have been found to be in close agreement with another computer simulation of formulas using complex‐order spherical wave functions at VLF. An interesting standing wave as a function of distance along the ionosphere and the ground has been found by using the integer‐order techniques.

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Spherical Wave Theory for MF, LF, and VLF Propagation

Johler¹

1970

Radio Science

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“…The waveguide-mode representations normally employed to describe VLF propagation around the earth below an azimuthally symmetric ionosphere (Watson, 1919;Bremmer, 1949;Budden, 1951Budden, , 1952Budden, , 1961Wait, 1960Wait, , 1962Wait, , 1963Wait, , 1964aJohler andBerry, 1962, 1964;Galejs, 1964), involve basically only two coordinates, the vertical and the outward (parallel to the earth's surface) directions relative to the source. These correspond to the z and the p coordinates here.…”

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Propagation of Electromagnetic Waves into Anisotropic Media From an External Point‐Dipole Source

Price

1967

Radio Science

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“…The source is assumed to be a vertical electric dipole (current moment = Iol amp-m = 1) located in the wave guide. The wave number kj, j = 1, 2, 3, is With the source at the earth's surface, b --a, the vertical electric field E•, for example, can be written [Johler, 1966] At this point it would be instructive to depart from the present development and consider a very crude model. Let us seek a quasi-static approximation to the boundary problem of a dipole source at the earth's surface oscillating at ELF.…”

Section: Theory Of I:)ropagationmentioning

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Extra low-frequency terrestrial radio-wave field calculations with the zonal harmonics series

Johler¹,

Lewis²

1969

J. Geophys. Res.

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Use of the zonal harmonics series for calculating the terrestrial wave guide fields directly is described. The analysis is extended to include radio waves propagating into sea water or below the earth's surface. A sample calculation of ELF radio waves is analyzed into a direct wave and a wave that has traveled the circumference of the earth. The location of the ripples in the standing wave pattern is compared with the locations predicted by less exact but simpler mathematical models. It has been found that approximate formulas check within a few percent of the exact analysis when used to estimate positions of maxima and minima near the antipode. It is concluded that the exact analysis can be used to check various approximate formulas. Where possible the extension of the analysis to more complex models should be accomplished with the exact solution.

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Zonal harmonics in low frequency terrestrial radio wave propagation.

Cited by 9 publications

References 4 publications

Spherical Wave Theory for MF, LF, and VLF Propagation

Spherical Wave Theory for MF, LF, and VLF Propagation

Propagation of Electromagnetic Waves into Anisotropic Media From an External Point‐Dipole Source

Extra low-frequency terrestrial radio-wave field calculations with the zonal harmonics series

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