Diffraction Theory of Propagation of High-Frequency Radio Waves in a Spherically Layered Ionospheric Radio Channel

Abstract

A special analytical method is developed to describe the propagation of high-frequency electromagnetic waves emitted by a vertical point dipole from the surface of perfectly conducting Earth in a regular spherically layered background ionosphere. The basic representation of the wave field employs an integral over the set of wave components, each of which is related to a specific ray trajectory. Such an approach makes it possible to analytically take into account the effect of medium-scale 3D inclusions in the background ionosphere as an additional phase shift of the wave component with allowance for the distortion of its ray trajectory and to numerically calculate the effect of wave-field focusing.

APPENDIX

CONSTRUCTION OF RADIAL FUNCTIONS ABOVE THE PERFECTLY CONDUCTIVE EARTH SURFACE

Determination of radial functions R_n(r) for a point source placed on the surface of perfectly conductive Earth surface requires the use of special mathematical methods considered in this Appendix. As was mentioned in Section 1, the problem under consideration is particularly difficult due to the fact that boundary condition (4) is set at $r = {{R}_{{\text{e}}}}$ (i.e., at the singularity of Eq. (3)). To separate these two problems, we must, first, write and solve Eq. (9) with the field source located at arbitrary point r = b > R_e above the Earth surface:

$$~\frac{{{{d}^{2}}{{R}_{n}}}}{{d{{x}^{2}}}} + \left( {1 - \frac{{n\left( {n + 1} \right)}}{{{{x}^{2}}}}} \right){{R}_{n}} = - \frac{1}{{b\xi }}\delta \left( {x - \xi } \right),$$

(A1)

where auxiliary independent variable $x = kr\sqrt {{{\varepsilon }_{m}}\left( r \right)}$ is introduced, and parameter $\xi = kb\sqrt {{{\varepsilon }_{m}}\left( b \right)}$ corresponds to the value of this variable at the source point. After finding a solution to equation (A1), one should pass to the limit at $b \to {{R}_{{\text{e}}}}$.

It is easy to show that homogeneous equation (A1) has the fundamental system of solutions

$$R_{n}^{{\left( 0 \right)}}\left( x \right) = \sqrt x {{Z}_{{n + \frac{1}{2}}}}\left( x \right),$$

(A2)

where Z_ν(x) are the cylindrical functions of order ν. Assuming time dependence of the wave field in the form ${\text{exp}}\left( { - i\omega t} \right)$, we must choose the Hankel functions of the first kind in the region above the source x > ξ, since they describe the upward propagating waves. Since both upward and downward propagation takes place in the layer between the Earth surface and the source at R_e < x < ξ, the system is correctly described using the Bessel functions. Thus, a particular solution to Eq. (A1) is represented as

$$\begin{gathered} R_{n}^{{\left( 1 \right)}}\left( x \right) = \frac{{i\pi }}{{2b\xi }}\left[ {\sqrt x H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( x \right)\sqrt \xi {{J}_{{n + \frac{1}{2}}}}\left( \xi \right)\Theta \left( {x - \xi } \right)} \right. \\ \left. { + \,\,\sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)\sqrt x {{J}_{{n + \frac{1}{2}}}}\left( x \right)\Theta \left( {\xi - x} \right)} \right] \\ \end{gathered} $$

(A3)

using Heaviside function Θ(x).

The boundary condition formulated as

$$R_{n}^{'}\left( x \right) \to 0\,\,\,\,{\text{at}}\,\,\,\,x \to \xi = k{{R}_{{\text{e}}}}$$

(A4)

can be satisfied by adding function (A2) with an appropriate coefficient. It can be easily shown that the expression

$$\begin{gathered} R_{n}^{{{\text{total}}}}\left( x \right) = \frac{{i\pi }}{{2b\xi }} \\ \times \,\,\left[ {\sqrt x H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( x \right){\kern 1pt} \sqrt \xi {{J}_{{n + \frac{1}{2}}}}{\kern 1pt} \left( \xi \right){\kern 1pt} \Theta {\kern 1pt} \left( {x - \xi } \right) + \sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)\sqrt x {{J}_{{n + \frac{1}{2}}}}{\kern 1pt} \left( x \right){\kern 1pt} \Theta \left( {\xi - x} \right) - \sqrt x H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}{\kern 1pt} \left( x \right){\kern 1pt} \sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}{\kern 1pt} \left( \xi \right)\frac{{{{{\left( {\sqrt \xi {{J}_{{n + \frac{1}{2}}}}\left( \xi \right)} \right)}}^{'}}}}{{{{{\left( {\sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)} \right)}}^{'}}}}} \right] \\ \end{gathered} $$

(A5)

satisfies both Eq. (A1) and boundary condition (A4). In this work, we study the space above the Earth surface and, hence, select the corresponding part R_n(x). After substituting b = R_e and ξ = kR_e, this function is written as

$$\begin{gathered} {{R}_{n}}\left( x \right) = \frac{{i\pi }}{{2{{R}_{{\text{e}}}}\xi }}\left[ {\sqrt {x~} H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( x \right)\sqrt \xi {{J}_{{n + \frac{1}{2}}}}\left( \xi \right) - \sqrt {x~} H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( x \right)\sqrt \xi \frac{{{{{\left( {\sqrt \xi {{J}_{{n + \frac{1}{2}}}}\left( \xi \right)} \right)}}^{'}}}}{{{{{\left( {\sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)} \right)}}^{'}}}}} \right] \\ = \frac{{i\pi }}{{2{{R}_{{\text{e}}}}\xi }}\frac{{\sqrt x H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( x \right)}}{{{{{\left( {\sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)} \right)}}^{'}}}}\mathcal{W}\left[ {\sqrt \xi {{J}_{{n + \frac{1}{2}}}}\left( \xi \right),\sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)} \right] = - \frac{1}{{{{R}_{{\text{e}}}}\xi }}\frac{{\sqrt x H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( x \right)}}{{{{{\left( {\sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)} \right)}}^{'}}}}, \\ \end{gathered} $$

(A6)

where the known value of the Wronskian is used:

$$\mathcal{W}\left[ {\sqrt \xi {{J}_{{n + \frac{1}{2}}}}\left( \xi \right),\sqrt \xi H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)} \right] = \frac{{2i}}{\pi }.$$

The expression for the derivative in the denominator in expression (A6) can be simplified, and the result is

$$\begin{gathered} {{R}_{n}}\left( x \right) = - \frac{2}{{{{R}_{{\text{e}}}}}}\sqrt {\frac{x}{\xi }} \\ \times \,\,\frac{{H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( x \right)}}{{\xi \left( {H_{{n - \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right) - H_{{n + \frac{3}{2}}}^{{\left( 1 \right)}}\left( \xi \right)} \right) + H_{{n + \frac{1}{2}}}^{{\left( 1 \right)}}\left( \xi \right)}}. \\ \end{gathered} $$

(A7)

Radial coordinate r is substituted in formula (A7), and, in this form, it is used in subsequent analytical transformations. Note that radial function (A7) fully corresponds to the solution of the problem of the field of a point source above a spherical perfectly conductive Earth with surface impedance $\delta = 0$ [1].