Abstract
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MATHEMATICAL PHYSICS
P. E. KRASNUSHKIN
SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF RADIO-WAVE PROPAGATION AROUND THE EARTH WITH ACCOUNT OF THE PRINCIPAL GEOPHYSICAL FACTORS
(Presented by Academician N. N. Bogolyubov on 2 January 1961)
The problem considered here is a special case of the solution \({}^{(1)}\) for three spherical layers: the 0th layer, \(0 \le r \le a\), is the Earth of radius \(a\), \(\varepsilon_{rr}=\varepsilon_{\theta\theta}=\varepsilon_{\varphi\varphi}=\varepsilon_0+i\cdot 4\pi\sigma_0/\omega\), where \(\varepsilon_0\) and the conductivity \(\sigma_0\) do not depend on \(r\), and \(\varepsilon_{\theta\varphi}=0\); the 1st layer, \(a \le r \le c\), is the ideal atmosphere, \(\varepsilon_{rr}=\varepsilon_{\theta\theta}=\varepsilon_{\varphi\varphi}=1,\ \varepsilon_{\theta\varphi}=0\); the 2nd layer, \(c \le r < \infty\), is the ionosphere, regarded as an electron-ion plasma placed in the vertical magnetic field of the Earth \(H_0\), characterized by the electron concentration \(N_e(r)\) and the collision frequency \(\nu_{\mathrm{eff}}(r)\) of electrons with other particles.
Taking into account only the motion of the electrons and neglecting the compressibility of the plasma, we write the components of the dielectric-constant tensor as
\[ \varepsilon_{rr}=1-\frac{v}{1+is}; \qquad v=\frac{\omega_0^2}{\omega^2}; \qquad \omega_0^2=\frac{4\pi N_e e^2}{m}; \]
\[ \varepsilon_{\theta\theta}=\varepsilon_{\varphi\varphi} =1-\frac{v(1+is)}{(1+is)^2-u^2}; \qquad u=\frac{\omega_H}{\omega}; \qquad \omega_H=-\frac{eH_0}{mc\omega}; \tag{1} \]
\[ \varepsilon_{\theta\varphi}=-\varepsilon_{\varphi\theta} =\frac{ivu}{(1+is)^2-u^2}; \qquad s=\frac{\nu_{\mathrm{eff}}}{\omega}, \]
where \(\omega\) is the frequency of oscillations of the currents exciting the field; \(c\) is the speed of light; \(e\) and \(m\) are the charge and mass of the electron; \(N_e\) and \(\nu_{\mathrm{eff}}\) are continuous and differentiable functions of \(r\) in the interval \((c,\infty)\), and in \((c,d)\) they are arbitrary, while for \((r \ge d)\) (the exosphere and outer space) they vary so slowly that, for sufficiently large \(r_0 \ge d\), in \((r \ge r_0)\) the approximation of geometrical optics is valid for the wave equations.
As follows from \({}^{(1)}\), the problem consists in finding a stationary solution of Maxwell’s equations under the corresponding conditions at \(r=0\) and \(r\to\infty\), and the conditions at the discontinuities of the medium for \(r=a\) and \(r=c\). Wishing to exclude the continuous spectrum \(\chi\) from consideration, we replace the boundedness condition at \(r=0\) by the condition of absence of waves traveling away from the center of the Earth. The general solution for a Hertz dipole is described by formulas \((14')\) of \({}^{(1)}\). We shall now construct a concrete solution for a three-layer medium.
The normal-wave operator \(L_r\), according to (5) and (9) of \({}^{(1)}\), will be
\[ r^2\varepsilon_{rr}^k\frac{d}{dr} \left[ \frac{1}{\varepsilon_{\theta\theta}^k}\frac{dY^k}{dr} \right] +r^2k^2\varepsilon_{rr}^kY^k -ir^2k\varepsilon_{rr}^k\frac{d}{dr} \left[ \frac{\varepsilon_{\theta\varphi}^k}{\varepsilon_{\theta\theta}^k}Z^k \right] =\chi Y^k, \tag{2} \]
\[ r^2\frac{d^2Z^k}{dr^2} +r^2k^2\widetilde{\varepsilon}^{\,k} +ir^2k\frac{\varepsilon_{\theta\varphi}^k}{\varepsilon_{\theta\theta}^k} \frac{dY^k}{dr} =\chi Z^k,\qquad k=0,1,2, \tag{3} \]
with the conditions at the boundaries of the layers
\[ Y^k\big|_{r_k}=Y^{k+1}\big|_{r_k},\quad Z^k\big|_{r_k}=Z^{k+1}\big|_{r_k},\quad \frac{\widetilde{dY^k}}{dr}\bigg|_{r_k} = \frac{\widetilde{dY^{k+1}}}{dr}\bigg|_{r_k},\quad \frac{dZ^k}{dr}\bigg|_{r_k} = \frac{dZ^{k+1}}{dr}\bigg|_{r_k}, \]
\[ k=k_1=\frac{\omega}{c},\qquad \widetilde{\varepsilon}=\varepsilon_{\theta\theta} +\frac{\varepsilon_{\theta\varphi}^2}{\varepsilon_{\theta\theta}},\qquad \frac{\widetilde{dY}}{dr}\bigg|_{r_k} = \left[ \frac{1}{\varepsilon_{\theta\theta}}\frac{dY}{dr} \right]_{r_k} -ik \left[ \frac{\varepsilon_{\theta\varphi}}{\varepsilon_{\theta\theta}}Z \right]_{r_k}. \tag{4} \]
Solutions of equations (2), (3) in layers 0 and 1 are expressed through linear combinations of the spherical Bessel functions \(h_\nu^{(1)}(\rho)=\sqrt{\dfrac{\pi\rho}{2}}H_\nu^{(1)}(\rho)\) and \(h_\nu^{(2)}(\rho)=\sqrt{\dfrac{\pi\rho}{2}}H_\nu^{(2)}(\rho)\); therefore, introducing the two-argument functions
\[ D_\nu(x,y)=\frac{i}{2} \begin{vmatrix} h_\nu^{(1)}(kx) & h_\nu^{(2)}(kx)\\ h_\nu^{(1)}(ky) & h_\nu^{(2)}(ky) \end{vmatrix}; \qquad D_\nu(x,y')=\frac{i}{2} \begin{vmatrix} h_\nu^{(1)}(kx) & h_\nu^{(2)}(kx)\\ \dot h_\nu^{(1)}(ky) & \dot h_\nu^{(2)}(ky) \end{vmatrix}; \tag{5} \]
\[ D_\nu(x',y)=\frac{i}{2} \begin{vmatrix} \dot h_\nu^{(1)}(kx) & \dot h_\nu^{(2)}(kx)\\ h_\nu^{(1)}(ky) & h_\nu^{(2)}(ky) \end{vmatrix}; \qquad D_\nu(x',y')=\frac{i}{2} \begin{vmatrix} \dot h_\nu^{(1)}(kx) & \dot h_\nu^{(2)}(kx)\\ \dot h_\nu^{(1)}(ky) & \dot h_\nu^{(2)}(ky) \end{vmatrix}, \]
we write the solution of (2), (3) in the atmospheric layer, satisfying condition (4) at \(r_0=a\) and the radiation condition at \(r_{-1}=0\), in the form
\[ Y^{(1)}=C_1\{D_\nu(r,a')-Z_yD_\nu(r,a)\}=C_1\overline{D_\nu(r,a')}; \tag{6} \]
\[ Z^{(1)}=C_2\left\{\frac{1}{Z_z}D_\nu(r,a')-D_\nu(r,a)\right\}=C_2\overline{D_\nu(r,a)}, \tag{7} \]
where \(Z_y=\dfrac{\dot h_\nu^{(2)}(k\sqrt{\varepsilon_0}a)}{\sqrt{\varepsilon_0}\,h_\nu^{(2)}(k\sqrt{\varepsilon_0}a)}\) and \(Z_z=\dfrac{\sqrt{\varepsilon_0}\dot h_\nu^{(2)}(k\sqrt{\varepsilon_0}a)}{h_\nu^{(2)}(k\sqrt{\varepsilon_0}a)}\) are the impedances of the Earth for spherical waves of vertical (\(Z_y\)) and horizontal (\(Z_z\)) polarizations. As \(\sigma_0\to\infty\), \(Z_y\to 0\), \(Z_z\to\infty\), and \(Y,\ Z\) pass into the corresponding expressions without the overbar. Dots denote derivatives of the functions \(h_\nu\) with respect to the argument. Equations (2), (3) in the second layer have four independent solutions. In the upper part of the layer (\(r>d\)) they can be represented by traveling waves in the directions \(r\to0\) and \(r\to\infty\) of two types: ordinary and extraordinary. Their analytic continuations into the region \((c\leq r\leq d)\) determine the four independent solutions of (2), (3) of interest to us. Only two solutions satisfy the condition at \(r\to\infty\), namely those passing into waves escaping to infinity. Denoting them by the symbols \(o\) (ordinary) and \(e\) (extraordinary), we write the solution of equations (2), (3) in \((r_1=c\leq r<\infty)\):
\[ Y^{(2)}(r,\chi)=C_3Y_e(r,\chi)+C_4Y_o(r,\chi); \tag{8} \]
\[ Z^{(2)}(r,\chi)=C_3Z_e(r,\chi)+C_4Z_o(r,\chi). \tag{9} \]
The specific form of (8), (9) for arbitrary functions \(N_e^2(r)\) and \(\nu(r)\) in the region where interaction occurs, i.e., conversion of \(o\)-waves into \(e\)-waves and conversely, as well as their reflection, was obtained by the author by the method of numerical integration on a high-speed electronic computer.
In order for the solutions (6), (7) and (8), (9) to turn into eigenfunctions of the operator \(L_r\), they must satisfy the last condition (4) at the point \(r_1=c\). It gives four homogeneous algebraic equations for \(C_1, C_2, C_3\), and \(C_4\). The compatibility condition of this system of equations can be written in the form
\[ \left[kD_{\nu_j}(a',c')-I_y^e\overline{D_{\nu_j}(a',c)}\right] \left[k\overline{D_{\nu_j}(a,c')}-I_z^o\overline{D_{\nu_j}(a,c)}\right] - \frac{\chi^e}{\chi^o} \left[k\overline{D_{\nu_j}(a',c)}-I_y^o\overline{D_{\nu_j}(a',c)}\right] \left[k\overline{D_{\nu_j}(a,c')}-I_z^e\overline{D_{\nu_j}(a,c)}\right]=0, \tag{10} \]
where the following notation has been introduced:
\[ I_y^e=(\dot Y_e/Y_e)_{r=c},\qquad I_z^e=(\dot Z_e/Z_e)_{r=c},\qquad \chi^e=(Z_e/Y_e)_{r=c}, \]
\[ I_y^o=(\dot Y_o/Y_o)_{r=c},\qquad I_z^o=(\dot Z_o/Z_o)_{r=c},\qquad \chi^o=(Z_o/Y_o)_{r=c}. \tag{11} \]
We shall call \(I_y^e, I_z^e, I_y^o\), and \(I_z^o\) the impedances of ordinary and extraordinary waves of vertical and horizontal polarization, and \(\chi^e\) and \(\chi^o\) the polarization coefficients of the corresponding waves at the boundary \(r=c\). These 6 quantities, which are functions of \(\chi\) (or of the angle of incidence \(\theta\) of the wave on the 2nd layer, equal to \(\arc\sin \dfrac{\sqrt{\chi}}{kc}\)), completely determine the reflection properties of the 2nd layer. Together with the impedances of the Earth’s layer \(Z_y\) and \(Z_z\), they determine all parameters of the normal waves \(\nu_j, N_j\), and \(\chi_j=(C_2/C_1)_j\), called the polarization coefficient. The wave numbers \(\nu_j\) are the roots of the transcendental equation (10). The coefficients \(\chi_j\) are determined from the solution (4) for \(r=c\):
\[ \chi_j = k\chi^e \frac{I_z^e-I_z^o}{I_y^e-I_y^o} \frac{k\overline{D}_{\nu_j}(c',a')-I_y^o \overline{D}_{\nu_j}(c,a')} {k\overline{D}_{\nu_j}(c',a)-I_z^o \overline{D}_{\nu_j}(c,a)} . \tag{12} \]
The normalizing factor \(N_j\) in (11′) from (1) may be represented as
\[ N_j=C_1^2 \frac{\overline{D}_{\nu_j}(a',c)}{2\nu_j+1} \left[\frac{\partial f(\nu)}{\partial \nu}\right]_{\nu=\nu_j}, \tag{13} \]
where \(f(\nu)\) is the left-hand side of equation (10), represented in the form \(k\overline{D}_{\nu}(a',c')-a\overline{D}(a',c)\). Then the expression for the potentials \(B\) and \(A\), given in (14′) of (1), for the atmospheric layer,
\[ B=-\frac{\pi P}{cb^2} \sum_{j=0}^{\infty} \frac{\overline{D}_{\nu_j}(r,a')\overline{D}_{\nu_j}(b,a')(2\nu_j+1)} {\sin \nu_j\pi \cdot \overline{D}_{\nu_j}(c,a')[\partial f/\partial \nu]_{\nu_j}} P_{\nu_j}[\cos(\pi-\theta)]; \tag{14} \]
\[ A=-\frac{\pi P}{cb^2} \sum_{j=0}^{\infty} \frac{\chi_j D_{\nu_j}(r,a)D_{\nu_j}(b,a')(2\nu_j+1)} {\sin \nu_j\pi \cdot D_{\nu_j}(c,a')[\partial f/\partial \nu]_{\nu_j}} P_{\nu_j}[\cos(\pi-\theta)], \tag{15} \]
where \(P\) is the electric moment of the Hertz dipole.
Let us consider several special cases.
I. \(H_0=0\). Then \(\varepsilon_{\theta\varphi}=0\), and equation (10) for \(\nu_j\) splits into two:
\[ k\overline{D}_{\nu_j}(a',c')-I_y^e D_{\nu_j}(a',c)=0; \tag{16} \]
\[ k\overline{D}_{\nu_j}(a,c')-I_z^o D_{\nu_j}(a,c)=0. \tag{17} \]
The first gives rise to a spectrum of normal waves of type \(TH_j\), the second of type \(TE_j\). We shall retain this classification for \(H_0\ne0\), calling the waves quasi-\(TH\) or \(TE\), depending on what they become when \(H_0=0\).
II. Watson’s waveguide case. \(H_0=0\), \(N_e\) and \(\nu_{\mathrm{eff}}\) are constant in \((c,\infty)\). Then \(\|\varepsilon_2\|=\varepsilon_2'+i\varepsilon_2''\), and the solutions (8), (9) will be \(h_{\nu}^{(1)}(k\sqrt{\varepsilon_2}r)\). Equation (16) is simplified to
\[ \overline{D}_{\nu_j}(a',c')- \frac{1}{\sqrt{\varepsilon_2}} \left\{ \frac{\dot h_{\nu}^{(1)}(k\sqrt{\varepsilon_2}c)} {h_{\nu}^{(1)}(k\sqrt{\varepsilon_2}c)} \right\} \overline{D}_{\nu_j}(a',c)=0. \tag{16′} \]
Upon replacing \(h_{\nu}^{(1)}(k\sqrt{\varepsilon_0}a)\) by \(J_\nu(k\sqrt{\varepsilon_0}a)\), it coincides with Watson’s equation \(\bigl((2^a),\) p. 561\(\bigr)\). The inessential difference in \(Z_y\) for \(4\pi\sigma_0/\omega \gg 1\) is due to the fact that Watson took into account a continuous spectrum associated with \(r=0\). Equation (16′) has three branches \(\nu_n\): \(\nu_i(0,a)\), \(\nu_j(a,c)\), and \(\nu_k(c,\infty)\), associated respectively with layers 0, 1, and 2. Only \(\nu_j(a,c)\), which pass, as \(\varepsilon_2\to\infty\) and \(\varepsilon_0\to\infty\), into ideal normal waves of a waveguide with metallic walls \(r=a\) and \(r=c\), have small attenuation.
III. The plane waveguide is obtained by passing in the solution to \(a\to\infty\). We introduce the linear wave number \(\bar{\nu}=\nu/a\) and the linear distance \(s=a\theta\), and replace the functions \(h\nu\) by Debye approximations in region 1 according to Watson \(((^7),\) Fig. 22). Then, for \(\varepsilon_0\to\infty\), we obtain the equation
\[ \sin \int_{ka}^{kc} \sqrt{1-\left(\frac{\nu}{\rho}\right)^2}\,d\rho - i\varepsilon_2 \left\{ \frac{\sqrt{\dfrac{k^2\varepsilon_2-\nu_j^2}{k^2-\nu_j^2}}} {} \right\} \cos \int_{ka}^{kc} \sqrt{1-\left(\frac{\nu}{\rho}\right)^2}\,d\rho =0. \tag{16''} \]
Replacing, in the arguments of the trigonometric functions, the phase integral by its approximate expression \((c-a)\sqrt{k^2-\nu^2}\), we obtain the equation for \(\bar{\nu}_j(a,c)\) of paper \((^8)\), derived by the authors for a flat earth (see also \((^{11b})\)). As \(ka\to\infty\), the spectra of the branches \(\bar{\nu}_i(0,a)\) and \(\bar{\nu}_k(c,\infty)\), beginning in neighborhoods of the points \(\nu=k\sqrt{\varepsilon_0}\) and \(k\sqrt{\varepsilon_2}\), become continuous, and the corresponding sums (14) are transformed into integrals identical with contour integrals along the banks of the cuts from the branch points \(k\sqrt{\varepsilon_0}\) and \(k\sqrt{\varepsilon_2}\). As is known from papers \((^9,{}^{10})\), they describe the so-called lateral waves. They are produced by sums of normal waves localized and propagating in the waveguide layers \((0,a)\) and \((c,\infty)\), and slightly leaking into the main waveguide channel \((a,c)\). The lateral waves play no role in the far field because of their strong attenuation.
IV. Diffraction case. For \(H_0=0\) and \(\varepsilon_2=1\), equation \((16')\) is transformed into the form
\[ \dot{h}_\nu^{(1)}(ka)-Z_\nu h_\nu^{(1)}(ka)=0, \tag{16'''} \]
which coincides (with the reservation in case II) with Watson’s formula \((^{26})\), p. 98). It gives two branches \(\nu_j^D(0,a)\) and \(\nu_j^D(a,\infty)\), beginning in neighborhoods of the points \(\nu=ka\sqrt{\varepsilon_0}\) and \(\nu=ka\). The branch \(\nu_i^D(a,\infty)\), important for practice, was first investigated by B. A. Vvedenskii in a series of papers \((^3)\). He used Debye’s approximation in region 2 \(((^7),\) Fig. 22) for \(h_\nu^{(1)}(ka)\) and in region 1 for \(h_\nu(k\sqrt{\varepsilon_0}a)\). In this approximation \((16''')\) will be:
\[ \tan\left\{ \frac{\pi}{4} + \int_{ka}^{\nu_j}\sqrt{1-\left(\frac{\nu_j}{\rho}\right)^2}\,d\rho \right\} = -\frac{i}{\varepsilon_0} \left\{ \sqrt{\nu_j^2-\varepsilon_0(ka)^2}\,/\,\sqrt{\nu_j^2-(ka)^2} \right\}. \]
It is easily transformed into the phase-integral equation
\[ 2\int_{ka}^{\nu}\sqrt{1-\left(\frac{\nu_j}{\rho}\right)^2}\,d\rho + i\ln \frac{ \sqrt{\nu_j^2-(ka)^2}-\sqrt{\nu_j^2-\varepsilon_0(ka)^2} }{ \sqrt{\nu_j^2-(ka)^2}+\sqrt{\nu_j^2-\varepsilon_0(ka)^2} } = 2\pi\left(j+\frac14\right). \]
Eckersley’s idea \((^{11})\) of obtaining the phase-integral equation directly from equations (2), (3), as is done in quantum mechanics for self-adjoint operators, was not carried through, since the normal-wave operator \(L_r\) \((^1)\) is non-self-adjoint and the theory for constructing the phase-integral equation for it is still lacking.
V. Sommerfeld case \((^{12})\). We obtain it from the preceding one for \(a\to\infty\), when the branches \(\bar{\nu}_j^D(0,a)\) and \(\bar{\nu}_j^D(a,\infty)\) pass into continuous spectra, and the sums (14) into the integrals \(Q_E\) and \(Q\) of paper \((^{12})\) along the banks of the cuts from the branch points \(\nu=k\sqrt{\varepsilon_0}\) and \(\nu=k\). The exception is an isolated eigenvalue of the branch \(\bar{\nu}_j^D(a,\infty)\) with wave number \(\bar{\nu}_p=k\sqrt{\varepsilon_0}/\sqrt{1+\varepsilon_0}\), which determines the Zenneck wave.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
5 I 1961
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