[4] The Earth surface is modeled by a sphere of the radius a, at which the impedance conditions are taken. In this case we exclude out of the regime the waves propagating mainly under the Earth surface because they are insignificant at distances R exceeding the thickness of the skin layer in the Earth. For determination of the impedance in any observational point, a model of the Earth homogeneous by the depth described by Morgan [Morris and Lha, 1974] is used.
[5] Over the Earth surface, we have an ionized medium located within a
constant magnetic field. The Earth magnetic field is presented in
the form
BE= u, where the potential
u is
approximated by the expression
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where Qg and jg are the geographic latitude and longitude, respectively, a is the Earth radius, Pn0( sinQg) = Pn( sinQg) is the Legendre polynomial, and Pnm( sinQg) is the associated Legendre function. The coefficients anm and bnm different from zero are presented below expressed in nT:
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The concentration and collision frequency of electrons are functions of the height over the Earth surface and are taken according to any known model.
[6] In undisturbed conditions, the plasma frequency
wpl at
altitudes up to 50 km over the Earth surface is considerably less
than the electron collision frequency with neutral particles, so
in the near-Earth region the relative dielectric permittivity of
the medium
e does not differ from its value in the
vacuum. So in the used model the lower boundary of the ionosphere
is determined as the level where
e(rl)-1 eik/
r) |-1,
where
eik is the tensor of relative dielectric
permittivity of the ionosphere)
is much larger
than the characteristic scale of the field changes in
r, so
calculating this scale
1/h one may take the medium as
homogeneous (Booker roots)
[Budden, 1961a]
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[7] We chose the level ru in such a way that h/ k0=p2 were large enough. Actually, the accuracy of calculation of eigenvalues of the normal waves is determined by the choice of rl and ru (the boundaries of the ionospheric layer called the region important for propagation). In particular, the choice of the p1=10-4 and p2=30 parameters provides the relative error of calculation of the eigennumber not more than 10-4.
[8] The problem of calculation of the fields within the waveguide channel Earth-ionosphere excited by the antenna located at some height over the Earth surface is formulated. The antenna orientation is given by the angle Qp relative to the vertical axis and by the azimuthal angle jp, counted clockwise from the direction to the receiver. Jlp is the current moment of the transmitting antenna. In the observational point, the field is received by the antenna located at any level relative to the Earth surface. The orientation of the receiving antenna is given by the angle Qa relative the vertical and by the azimuth angle ja, counted clockwise from the transmitter-receiver direction. The induced electromotive force at the short receiving antenna is calculated by the formula
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where la is the virtual height of the short receiving antenna, and Er, EQ, and Ej are the corresponding components of the electric field intensity in the place of the receiving antenna location.
[9] Mathematically we formulate the problem for a vertical dipole and after its solution we will form the field for an arbitrary oriented short antenna applying the generalized reciprocity theorem.
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