3. Mathematical Formulation of the Problem

[10]  We chose the spherical coordinate system Q, j, r, the Q=0 axis passing through the vertical dipole. The source excites the falling field independent of j, so in the chosen model of a one-dimensional irregular waveguide we have an axis-symmetric problem. We write the Maxwell equation in the matrix form [Felsen and Marcuvitz, 1973; Lutchenko and Bulakh, 1986], taking the following dependence on time exp(-iwt):

(1)

(2)

I is a unit matrix, k₀ is the wave number in the vacuum k₀=w(e₀ m₀)^1/2, e is the dimensionless tensor of the dielectric permittivity of the magnetoactive medium depending on r and Q, j^m is the density of the external magnetic current, j^e is the density of the external electric current (A m^-2 ) and in the case of a vertical point-like electric dipole located over the Earth surface at r=b

J is the current at the antenna input, l_p is the antenna virtual height, and l_r is a unit vector. The problem solution F has to satisfy the impedance boundary conditions at r=a, the conditions of a field decrease at Im k₀ > 0 and r, and its boundedness at Q=0 and Q=p. In the accepted model Im k₀ = 0, however for choosing the solution, the commonly accepted [Makarov et al., 1993] principle of the limiting amplitude in which a presence of losses in the medium leading to Im k₀ > 0 is used. After construction of unambiguous solution, we come back to the model Im k₀ = 0 and e = 0.

[11]  The solution of system (1) is constructed by the cross-section method [Katsenelenbaum, 1961], presenting the solution in the form of the expansion in terms of orthogonal system of functions Y_m

(3)

As Y_m, the eigenfunctions of the lateral operator K for the regular waveguides of comparison are chosen. The waveguides of comparison are spherical waveguides with the polar axis coinciding with the axis of the initial waveguide and the dielectric permittivity e(r) which depends only on the coordinate r and coincides to the dielectric permittivity of the initial waveguide in the Q cross section. To construct the eigenfunctions in the regular waveguides of comparison, we consider homogeneous Maxwell equations and take approximately that ( dA_m/dQ)=in_mA_m, where m is the number of the mode. Then we neglect cot Q in the operator _t assuming that cot Q n_m , ( n_m k₀a).

(4)

with the boundary conditions

q is the incidence angle of the plain wave on the Earth surface; e is the relative dielectric permittivity of the Earth surface, s is its conductivity in S m^-1, and by the condition Y_m(r) 0 at Im k₀ > 0 and r, G have the form (2). In real conditions (except the Antarctics, Greenland and the permafrost regions) e sin²q, so d_e with a high accuracy does not depend on the spectral parameter. Equation (4) is an equation for the eigenfunctions. In order to obtain the orthogonality relation for the eigenfunctions a scalar product is introduced and the conjugated operator K⁺ is determined in the following way

(5)

The vector standing at the first place is taken with the complex conjugation and the parenthesis mean a scalar product

(5')

e^T is the transposed tensor e with the boundary conditions

(6)

the eigenvalues of equation (4) at the same indices coincide with the complex conjugated value of equation (6). We multiply the left-hand side of (4) to rY⁺_n, and the right-hand side of (6) to rY_m and subtract the latter from the former. Then

or

We integrate both sides of the latter equality forming a scalar product

where

(7)

E_m= ( E_m/n_m) and H_m=( H_m/n_m) are derivatives with respect to the spectral parameter and N_m is the normalizing multiplier. Expression (3) for the solution we substitute into the initial equation (1) outside the sources region and multiply its left-hand side scalarly to Y⁺_n. Then we obtain

(8)

The solution of the homogeneous equation (8) has an exponential dependence on the Q coordinate. At the same time, it is known that in a regular waveguide, the angular dependence of normal waves is described by the Legendre function. Only while using asymptotic presentations of these functions in the wave zone relative to the source and its antipode, the Legendre function becomes approximately exponential. Therefore we obtain the limits of applicability of the solution (3) cot Q   n_m  or Q (1/k₀a) and p - Q (1/k₀a), because n_m k₀a.

[12]  Assuming the eigenfunctions to be normalized, i.e., N_n=1, rewrite equation (8) with respect to a new function L_n(Q) outside the source area for a path segment, within which the waveguide characteristics are supposed constant. Let's denote Q₀ the initial coordinate of such a segment. Assuming the eigenfunctions to be normalized, i.e., N_n=1, we rewrite equation (8) with respect to a new function L_n(Q) outside the sources for that part of the path within which the characteristics of the waveguide are considered to be constant. Let us designate the initial coordinate of such part as Q₀.

(9)

(9')

The scalar product (1/2)(Y⁺_n, GY_m/ Q)=S_mn describes the differential matrix of transformation of normal waves [Lutchenko et al., 1986]. In the real conditions, the electric properties of the ionosphere vary smoothly, so the S_mn elements only due to the ionosphere would be continuous functions of the Q argument. However variations in the conductivity of the lower boundary (in terms of the scales of field changes) in the accepted model occurs in a jump-like way.

[13]  Let a jump-like change of the medium properties occurs in the cross section with a coordinate Q₀. We designate F⁽¹⁾ field in the cross section Q₀ - e and F⁽²⁾ in the cross section Q₀ + e. The tangential components of the vectors GF⁽¹⁾ = GF⁽²⁾ should be continuous or

(10)

(11)

[14]  The scalar product

(12)

may be interpreted as an element of the matrix P of transformation of normal waves and the aggregate L⁽²⁾_n may be considered as the L⁽²⁾ vector which is determined by the product of the P matrix to the A⁽¹⁾ vector in the cross section Q₀.

[15]  The solution of system of differential equations ( 9' ) presents a rather tiresome problem, so we will approximately take that the ionosphere is changing in a jump-like way. A sequence of cross sections is chosen along the path. In each cross section, characteristics of normal waves are determined and to the next cross section the waveguide is considered as homogeneous. At the joints of the cross sections, the matrix P is calculated neglecting the reflected waves. Thus in the source vicinity, normal waves with the amplitudes L⁽⁰⁾_m are excited. At the next cross section located at the angular distance of DQ⁽⁰⁾ the waves will have amplitudes A⁽⁰⁾_m=L⁽⁰⁾_m e^in(0)_mDQ(0). In the next cross section the amplitudes will be

(13)

(14)

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