An extensive summary of classical potential solutions has been given recently by Van Nostrand and Cook (1966). This note presents a solution for the potential due to a point source of electric current placed on the earth’s surface in the vicinity of a buried spherical body of arbitrary resistivity. The analysis follows the procedure suggested by Van Nostrand and Cook and is similar to that used recently by Merkel (1969, 1971).
A method is described for calculating the resonance characteristics of the earth-ionosphere cavity. Particular attention is paid to the influence of the geomagnetic field which, for purposes of the analysis, is assumed to be purely radial. In the model used, the ionosphere is represented as a homogeneous concentric shell, which is regarded as a cold electron plasma with collisions. It is found that the geomagnetic field lowers the real resonance frequencies from those for the corresponding isotropic model.There has been some recent interest in using cavity resonance phenomena in the diagnostics of magnetoplasmas. Unfortunately, the problem is vastly more complicated than with isotropic plasma configurations because of the anisotropy introduced by the d-e magnetic field. An interesting example of the phenomenon occurs in the cavity space formed by the earth and the ionosphere. As indicated by Schumann (1952), the fundamental resonance of the system is of the order of 10 Hz. While there has been a large number of theoretical treatments of the problem, the influence of the earth's magnetic field is not fully understood. However, some progress has been made in treating limited aspects of the problem (Galejs, 1965;Madden and Thompson, 1965;Wait, 1965;Large and Wait, 1966). In this note, we discuss the concentric spherical cavity model in which the radial magnetic field is assumed to be everywhere radially oriented.The model and the spherical coordinate system (r, 8, cP) are illustrated in figure 1. The earth, of radius a, is assumed to be perfectly conducting. The ionosphere is represented by a homogeneous electron plasma layer of thickness d = r, -ro, permeated by a constant, radially directed magnetic field of strength B0• The plasma frequency is denoted CrJo and the effective mean collision frequency is denoted v. The geometry is a simplification of a more general configuration considered previously (Wait, 1963), and much of the notation of this referenee-will-be used.
A coupled multimode analysis technique is applied to a model of the earth‐ionosphere cavity at ELF in which the ionosphere is replaced by a thin, anisotropic shell. The unique feature of the analy. sis is that it is capable of illustrating the effects of a dipolar form of geomagnetic field. Numerical values of the cavity resonant frequencies and quality factors for several values of electron density and collision frequency are presented for both a homogeneous and a dipolar magnetic field configuration. The main purpose of the paper is to demonstrate that a coupled multimode analysis is applicable to such nonsymmetrical configurations.
A cylindrical model of the earth‐ionosphere cavity at ELF with an angularly dependent surface‐impedance boundary condition is analyzed. The variation of the ionospheric surface impedance is chosen to simulate the presence of a dipolar, radial, geomagnetic field. By neglecting TE‐TM mode coupling, the coupling between TM modes is analyzed. Cavity resonant frequencies and quality factors as functions of ionospheric electron concentration and collision frequency are compared between constant and dipolar field configurations. Parameters that yield resonances close to experimental values are found for both configurations, and it is shown that the constant field model with an induction equal to 2/π times the polar induction produces resonances very, close to those of the corresponding dipolar model.
The exact harmonic series solution for the fields in a cylindrically stratified plasma is utilized to examine the cavity resonant frequencies. The prolilem is restricted to two-dimensional geometry such that the d-e magnetic field is parallel to the axis of the system. Some numerical results are given for the special case of an annular air region bounded by a perfectly conducting core and an homogeneous external plasma. It is shown that the d-e magnetic field shifts the resonant frequencies from their isotropic values and causes certain "line-splitting" effects. . IntroductionIn a previous paper [Wait, 1966], an exact expression was given for the fields of a line source in the presence of an isotropic conducting cylinder which is surrounded by a concentrically stratified plasma. The d-e magnetic field was taken to be parallel to the cylindrical axis of the system. In that paper, the nature of the waveguide modes which propagate circumferentially around the cylinder was discussed in some detail. It was indicated that these modes were analogous to VLF radio waves propagating in the earth-ionosphere waveguide for directions along the magnetic equator. A related problem, which is discussed in the present paper, is to calculate the natural resonant frequencies of the system. While we restrict attention to a cylindrical model, the results are analogous to the so-called "Schumann resonances" which are actually observed in the spherical earth-ionosphere cavity [Balser and Wagner, 1960;Madden and Thompson;1965 , Galejs, 1965. General Cavity Mode EquationThe geometry of the problem is identical to that in the previous paper [Wait, 1966] which, hereafter, we shall refer to as paper L The "cavity resonances" may now be obtained by examining the condition for which the denominator of (57) in paper I vanishes. Explicitly, this condition is written (1) where the symbols have the following meaning: a0 is the~ radius of the isotropic core; a1 is the inner radius of the concentrically stratified magnetoplasma; Rm,o is the cylindrical reflection coefficient, .for the interface at p= au, as defined by (52) in I; Rm,o is the cylindrical reflection coefficient, for the interface at p =a~, as defined by (4 7) in I; HW and Hff.l are H~nkel functions of the first and second kind, respectively, with arguments as indicated; k=(Eo/LO)ll 2 w is the wave number where w is the (complex) angular frequency.The "cavity resonances" occur at the complex frequency w/2TT where (1) above is satisfied for integer values of m. Thus, it is appropriate to call m the cavity resonance mode number. This is to be contrasted with the waveguide modes which would be formally obtained from (l) by solving for the nonintegral value of m (i.e., denoted 11 in I) for a real frequency w/2TT.
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