MATHEMATICAL PHYSICS

P. E. KRASNUSHKIN

THE BOUNDARY-VALUE PROBLEM FOR THE PROPAGATION OF ELECTROMAGNETIC WAVES IN A SPHERICALLY LAYERED ANISOTROPIC DISSIPATIVE MEDIUM

(Presented by Academician N. N. Bogolyubov, January 9, 1961)

In paper (¹) a method of normal waves was published. Its formulation was based on the spectral theory of self-adjoint linear operators, as a result of which it was suitable for solving a comparatively narrow class of boundary-value problems on wave propagation in layered media without losses. The spectral theory of linear non-self-adjoint operators developed in recent decades makes it possible to extend the application of the method of normal waves (¹) to layered media with losses and, naturally, to take radiation into account. Here a sufficiently general case of such media is considered, having a direct relation to the spherical semiconducting Earth surrounded by a magneto-anisotropic ionosphere. It includes Watson’s problems (²) and their analogues from acoustics and seismology.

1. We seek a stationary electromagnetic field of frequency $\omega$ in a spherical coordinate system $r,\theta,\varphi$, produced by current densities

$$ I_r(r,\theta)e^{-i\omega t}; \qquad I_\theta = 0; \qquad I_\varphi = 0. \tag{1} $$

The medium is determined by the tensor of dielectric constant

$$ \left\| \varepsilon^k \right\| = \left\| \begin{matrix} \varepsilon_{rr}^k & 0 & 0 \\ 0 & \varepsilon_{\theta\theta}^k & \varepsilon_{\theta\varphi}^k \\ 0 & -\varepsilon_{\theta\varphi}^k & \varepsilon_{\varphi\varphi}^k \end{matrix} \right\|, \qquad k = 0,1,2,\ldots,N. \tag{2} $$

Its components are complex functions of $r$, continuous on the intervals $r_{k-1} \leqslant r \leqslant r_k$ and differentiable. As $r \to \infty$, $\varepsilon_{rr}$, $\varepsilon_{\theta\theta}$, and $\varepsilon_{\varphi\varphi}$ tend adiabatically slowly to $1+i\Delta$, where $\Delta$ is a small but finite positive quantity, while $\varepsilon_{\theta\varphi}\to 0$. The magnetic permeability is $\mu=1$.

The field components satisfy Maxwell’s equations:

$$ \frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta H_\varphi^k\right) = -\frac{i\omega}{c}\varepsilon_{rr}^k E_r^k + \frac{4\pi}{c}I_r; \qquad \frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta E_\varphi^k\right) = \frac{i\omega}{c}H_r^k, $$

$$ \frac{1}{r}\frac{\partial}{\partial r}\left(rH_\varphi^k\right) = \frac{i\omega}{c}\left[\varepsilon_{\theta\theta}^k E_\theta^k+\varepsilon_{\theta\varphi}^k E_\varphi^k\right]; \qquad -\frac{1}{r}\frac{\partial}{\partial r}\left(rE_\varphi^k\right) = \frac{i\omega}{c}H_\theta^k, \tag{3} $$

$$ \frac{1}{r} \left[ \frac{\partial}{\partial r}\left(rH_\theta^k\right) - \frac{\partial H_r^k}{\partial\theta} \right] = \frac{i\omega}{c} \left[\varepsilon_{\theta\varphi}^k E_\theta-\varepsilon_{\varphi\varphi}^k E_\varphi^k\right]; $$

$$ \frac{1}{r} \left[ \frac{\partial}{\partial r}\left(rE_\theta^k\right) - \frac{\partial E_r^k}{\partial\theta} \right] = \frac{i\omega}{c}H_\varphi^k. $$

At the boundaries of the layers $r=r_k$ the continuity conditions hold for the tangential components $E_\theta$, $E_\varphi$, $H_\theta$, and $H_\varphi$; at the point $r_{-1}=0$ the boundedness conditions hold: $\operatorname{mod} E < M$, $\operatorname{mod} H < M$, while as $r\to\infty$, $\operatorname{mod} E$ and $\operatorname{mod} H \to 0$.

1. Formulation of the problem in the language of operators. Introduce the vector function

$$ \left| \begin{matrix} B(r,\theta)\\ A(r,\theta) \end{matrix} \right|, \qquad \text{where} \qquad E_\varphi=\frac{1}{r}\frac{\partial A}{\partial\theta}; \qquad H_\varphi=\frac{1}{r}\frac{\partial B}{\partial\theta}. \tag{4} $$

Then equations (3) are written in operator-matrix form:

\[ l_r^{(k)} \begin{vmatrix} B_k\\ A_k \end{vmatrix} + l_\theta^{(k)} \begin{vmatrix} B_k\\ A_k \end{vmatrix} = \frac{4\pi}{c}r^2 \begin{vmatrix} I_r\\ 0 \end{vmatrix}, \qquad k=0,1,2,\ldots,N, \]

where

\[ l_r^{(k)}= \begin{vmatrix} \varepsilon_{rr}^k r^2 \dfrac{\partial}{\partial r} \left[\dfrac{1}{\varepsilon_{\theta\theta}^k}\,\cdot\right] +k_0^2\varepsilon_{rr}^k r^2\,\cdot; & -ik_0\varepsilon_{rr}^k r^2\dfrac{\partial}{\partial r} \left[\dfrac{\varepsilon_{\theta\varphi}^k}{\varepsilon_{\theta\theta}^k}\,\cdot\right] \\[1.2em] \dfrac{ik_0\varepsilon_{\theta\varphi}^k r^2}{\varepsilon_{\theta\theta}^k} \dfrac{\partial}{\partial r}\,\cdot; & r^2\dfrac{\partial^2}{\partial r^2}\,\cdot +k_0^2r^2 \left[ \varepsilon_{\varphi\varphi}^k+ \dfrac{(\varepsilon_{\theta\varphi}^k)^2}{\varepsilon_{\theta\theta}^k} \right]\cdot \end{vmatrix}, \tag{5} \]

\[ l_\theta^{(k)} = \begin{vmatrix} \mathcal L; & 0\\ 0; & \mathcal L \end{vmatrix}, \qquad \text{where } \mathcal L= \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial}{\partial\theta}\,\cdot\right). \tag{6} \]

The conditions at \(r=r_k\), \(r=0\), and \(r\to\infty\) take the form

\[ \Delta^{(k)} \begin{vmatrix} B_k\\ A_k \end{vmatrix}_{r=r_k} = \Delta^{(k+1)} \begin{vmatrix} B_{k+1}\\ A_{k+1} \end{vmatrix}_{r=r_k}; \qquad \begin{vmatrix} B_k\\ A_k \end{vmatrix}_{r=r_k} = \begin{vmatrix} B_{k+1}\\ A_{k+1} \end{vmatrix}_{r=r_k}, \tag{7} \]

where

\[ \Delta^{(k)}= \begin{vmatrix} \dfrac{1}{\varepsilon_{\theta\theta}^k}\dfrac{\partial}{\partial r}\,\cdot & \dfrac{-ik_0\varepsilon_{\theta\varphi}^k}{\varepsilon_{\theta\theta}^k}\,\cdot \\[1.2em] 0 & \dfrac{\partial}{\partial r}\,\cdot \end{vmatrix}, \qquad k_0=\omega/c \text{ is the wave number;} \]

\[ \bmod \begin{vmatrix} B\\ A \end{vmatrix}_{r\to 0} < \begin{vmatrix} M'\\ M' \end{vmatrix}; \qquad \bmod \begin{vmatrix} B\\ A \end{vmatrix}_{r\to\infty} \to 0. \tag{8} \]

We write (4) in the form

\[ \begin{vmatrix} Y^{(k)}(r)\Psi(\theta)\\ Z^{(k)}(r)\Psi(\theta) \end{vmatrix}. \]

Then \(l_r^{(k)}\) acts only on

\[ \begin{vmatrix} Y\\ Z \end{vmatrix}, \]

and \(l_\theta\) on \(\Psi\); therefore in them, and in (7), (8), we replace partial derivatives by total derivatives and introduce the operator \(L_r\), generated by the differential expressions \(l_r^{(k)}\), the conditions at the discontinuities

\[ \Delta^{(k)} \begin{vmatrix} Y^{(k)}\\ Z^{(k)} \end{vmatrix}_{r_k} = \Delta^{(k+1)} \begin{vmatrix} Y^{(k+1)}\\ Z^{(k+1)} \end{vmatrix}_{r_k}; \qquad \begin{vmatrix} Y^{(k)}\\ Z^{(k)} \end{vmatrix}_{r_k} = \begin{vmatrix} Y^{(k+1)}\\ Z^{(k+1)} \end{vmatrix}_{r_k} \tag{9} \]

and the boundary conditions

\[ \bmod \begin{vmatrix} Y\\ Z \end{vmatrix}_{r\to 0} < \begin{vmatrix} M''\\ M'' \end{vmatrix}; \qquad \bmod \begin{vmatrix} Y\\ Z \end{vmatrix}_{r\to\infty} \to 0. \tag{9'} \]

We also introduce the operator \(L_\theta\), generated by the differential expression \(l_\theta\) and the boundedness conditions at \(0\) and \(\pi\). Then the boundary-value problem formulated above can be written as the inhomogeneous operator equation

\[ L_r \begin{vmatrix} B\\ A \end{vmatrix} + L_\theta \begin{vmatrix} B\\ A \end{vmatrix} = \frac{4\pi}{c}r^2 \begin{vmatrix} I_r\\ \end{vmatrix}. \tag{10} \]

3. Method of solution.

The operator on the right-hand side of (10) is separated in the coordinates \(r\) and \(\theta\), and therefore we apply the method of normal waves \({}^{(1)}\), which consists in expanding spectrally

\[ \begin{vmatrix} B\\ A \end{vmatrix} \]

with respect to one of the coordinates \(r\) or \(\theta\) and representing it in source form with respect to the other. We shall expand into the spectrum of \(L_r\) and into source form with respect to \(L_\theta\). The resulting expansion converges considerably faster than an expansion in the spectrum of \(L_\theta\). The detour followed by Watson \({}^{(2)}\) was caused by the absence of a theory of non-self-adjoint operators \({}^{(3-5)}\). The operator \(L_r\) is non-self-adjoint and singular. To find its spectrum one may use methods \({}^{(5)}\); however, we introduce an auxiliary operator \(L_r'\) with the same \(l_r^{(k)}\), but with the boundary conditions at the singular points \(r=0\) and \(r=\infty\) replaced by zero conditions on spheres of small radius \(r=\rho\) for \(r=0\) and of large radius

\(r=R\) for \(r\to\infty\). The operator \(L'_r\) is regular and acts in the space of piecewise-continuously differentiable functions. Its spectrum consists only of discrete complex eigenvalues \(\{\chi_j\}\), determined by the equation

\[ L'_r \begin{vmatrix} Y\\ Z \end{vmatrix} = \chi \begin{vmatrix} Y\\ Z \end{vmatrix}. \]

Except for special cases, the eigenvalues \(\chi_j\) are simple. The orthogonality condition has the form:

\[ \int_{0}^{\infty} Y_j(r)\,U_p(r)\,dr + \int_{0}^{\infty} Z_j(r)\,V_p(r)\,dr = \begin{cases} N_j, & j=p,\\ 0, & j\ne p, \end{cases} \tag{11} \]

where

\[ \begin{vmatrix} U_p\\ V_p \end{vmatrix} \]

are the eigenfunctions of the adjoint operator \(L_r^{\prime *}\), determined from Lagrange’s identity \((4)\). Since the eigenvalues \(\mu_j\) of the operator \(L_r^{\prime *}\) are equal to \(\bar{\chi}_j\) (the bar denotes complex conjugation), it is not difficult to show that (11) reduces to the form

\[ \int_{0}^{\infty}\frac{Y_jY_p}{\varepsilon_{rr}r^2}\,dr + \int_{0}^{\infty}\frac{Z_jZ_p}{r^2}\,dr = \begin{cases} N_j, & j=p,\\ 0, & j\ne p; \end{cases} \tag{11'} \]

the quantity \(N_j\) will be called the normalizing factor.

By virtue of the theorem of M. V. Keldysh \((3)\), we expand the solution (10) in the form

\[ \begin{vmatrix} B\\ A \end{vmatrix} = \sum_{j=0}^{\infty} \begin{vmatrix} Y_j(r)\\ Z_j(r) \end{vmatrix} \Phi_j(\theta). \tag{12} \]

For the Fourier coefficients of this expansion, by virtue of (10), (11), we obtain the equation

\[ \frac{1}{\sin\theta}\frac{d}{d\theta} \left( \sin\theta\,\frac{d\Phi_j}{d\theta} \right) + \chi_j\Phi_j = \frac{4\pi}{cN_j} \int_{0}^{\infty} \frac{I_rY_j}{\varepsilon_{rr}}\,dr. \tag{13} \]

Representing its solution, in the usual way, with the aid of Green’s function, we obtain the general solution in the form

\[ \begin{vmatrix} B\\ A \end{vmatrix} = -\frac{2\pi^2}{c} \sum_{j=0}^{\infty} \begin{vmatrix} Y_j\\ Z_j \end{vmatrix} \frac{1}{\sin(\nu_j\pi)N_j} \left\{ P_{\nu_j}[\cos(\pi-\theta)] \int_{0}^{\theta} I_jP_{\nu_j}(\cos\theta')\sin\theta'\,d\theta' + \right. \]

\[ \left. + P_{\nu_j}[\cos\theta] \int_{0}^{\theta} I_jP_{\nu_j}\cos(\pi-\theta')\sin\theta'\,d\theta' \right\}, \tag{14} \]

where \(\chi_j=\nu_j(\nu_j+1)\), and for the case when the field is excited by a Hertz dipole at the point \(\theta=0,\ r=b\):

\[ \begin{vmatrix} B\\ A \end{vmatrix} = -\frac{\pi P}{cb^2} \sum_{j=0}^{\infty} \begin{vmatrix} Y_j(r)\\ Z_j(r) \end{vmatrix} \frac{Y_j(b)}{N_j\sin\nu_j\pi} P_{\nu_j}[\cos(\pi-\theta)], \tag{14'} \]

where \(P\) is the electric moment of the Hertz dipole. Expanding \(P_{\nu_j}\) into the sum

\[ \frac{1}{\pi i} \left\{ L_{\nu_j}^{(1)}[\cos(\pi-\theta)] - L_{\nu_j}^{(2)}[\cos(\pi-\theta)] \right\} \]

and using the asymptotic representation

\[ L_{\nu_j}^{(1,2)} = Q_{\nu_j} \pm i\frac{\pi}{2}P_{\nu_j} \sim \sqrt{\frac{\pi}{2\nu_j\sin\theta}}\, e^{\pm i[\nu_j^*\theta+\pi/4]}, \qquad \nu^*=\nu+\frac{1}{2}, \]

we obtain for \((14')\) the approximate expression

\[ \left| \begin{matrix} B\\ A \end{matrix} \right| = \frac{2P}{cb^{2}}\sqrt{\frac{\pi}{2\sin\theta}} \sum_{j=0}^{\infty} \left| \begin{matrix} \dot Y_j(r)\\ Z_j(r) \end{matrix} \right| \frac{Y_j(b)\nu_j^{1/2}}{N_j} \left\{ \sum_{n=0}^{\infty} e^{\,i(n+1)2\pi\nu_j-i(\nu_j^{*}\theta+\pi/4)} + e^{\,i2\pi n\nu_j+i(\nu_j^{*}\theta+\pi/4)} \right\}, \tag{15} \]

where \(1/\sin \nu_j\pi\) has been expanded in a series—a geometric progression.

Each term of (15) is a normal wave running along the layers \(\|\varepsilon\|=\mathrm{const}\), characterized by the wave number \(\alpha_j\) and the attenuation coefficient \(\beta_j\), which are determined by the complex eigenvalue \(\chi_j\) according to the formula \(\nu_j^{*}=\alpha_j+i\beta_j=\sqrt{\chi_j+1/4}\), and also by the distribution of amplitudes over the wave front,
\[ \left| \begin{matrix} Y_j(r)\\ Z_j(r) \end{matrix} \right|; \]
\(n\) indicates the number of circuits of the wave around the sphere; waves with \(n\ne0\) are called round-the-world echoes. Because of attenuation, the waves with \(n=0\), arriving at the observation point \(\theta\) along the shortest arc, stand out. For these waves the electromagnetic field is expressed by Dirichlet series:

\[ \begin{gathered} H_{\varphi}=Ai\sum_{j=0}^{\infty}Y_j(r)B_j;\qquad E_r=-\frac{Ai}{k_0r}\sum_{j=0}^{\infty}\nu_jY_j(r)B_j;\qquad E_\theta=\frac{A}{k_0}\sum_{j=0}^{\infty}\frac{dY_j}{dr}B_j;\\ E_{\varphi}=Ai\sum_{j=0}^{\infty}Z_j(r)B_j;\qquad H_r=\frac{Ai}{k_0r}\sum_{j=0}^{\infty}\nu_jZ_j(r)B_j;\qquad H_\theta=-\frac{A}{k_0}\sum_{j=0}^{\infty}\frac{dZ_j}{dr}B_j, \end{gathered} \tag{16} \]

where
\[ A=\frac{2P}{cb^{2}r}\sqrt{\frac{\pi}{2\sin\theta}}\,e^{-i\omega t}; \qquad B_j=\frac{Y_j(b)}{N_j}\nu_j^{1/2}e^{\,i(\nu_j^{*}\theta+\pi/4)}. \]

1. Of greatest interest is the case of such a distribution of \(\|\varepsilon\|\) in \(r\) in which intervals \((r_i,r_{i+1})\) with a slow variation of \(\|\varepsilon\|\) alternate with intervals \((r_{i+1},r_{i+2})\), \(i=0,2,4,6,\ldots\), where \(\|\varepsilon\|\) changes sharply over the length of one wavelength, which leads to a noticeable reflection of waves at the boundaries of the intervals \((r_i,r_{i+1})\). In this case the spectrum \(\chi_j\) of the operator \(L_r'\) is divided into a number of branches, each of which is associated with the corresponding interval \((r_i,r_{i+1})\). The forms of the normal waves \(|Y_j/Z_j|^{(i)}\) of the branch \(\chi_j^{(i)}\) are localized in the interval \((r_i,r_{i+1})\). A dipole placed in the interval \((r_i,r_{i+1})\) will produce a field that will propagate along \(\theta\), remaining localized within \((r_i,r_{i+1})\). Such propagation is characteristic of waveguide channels \((^{1})\). Only the first types of waves \(\chi_j^{(i)}\) have small attenuation; waves of higher numbers \(j>N^{(i)}\), for which the period of spatial modulation of \(|Y_j/Z_j|\) along the front \(\theta=\mathrm{const}\) is less than the wavelength in the given waveguide \((r_i,r_{i+1})\), have larger \(\beta_j^{(i)}\), and the series (14′) for approximate calculations of the field may be cut off at \(N^{(i)}\). The filtering of high types of waves in waveguide channels ensures the rapid convergence of the series (14′).

As \(\rho\to0\) and \(R\to\infty\), the discrete \(\chi_j^{(i)}\) of the branches \(i=1,2,\ldots\) of the operator \(L_\rho'\) continuously pass into the corresponding \(\chi_j^{(i)}\) of the operator \(L_r\). Only on one of the branches, \(i=0\), associated with the singularity at the point \(r=0\), do the \(\chi_j^{(0)}\) come closer together, turning in the limit into a continuous spectrum; and the corresponding sum (14′) becomes an integral with respect to \(d\chi\), which describes waves penetrating the central part of the medium. If the latter has losses, then such waves may be neglected.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
5 I 1961

CITED LITERATURE

1. P. E. Krasnushkin, Doctoral dissertation, Moscow State University Press, 1945; The Method of Normal Waves as Applied to the Problem of Long-Range Communications, Moscow State University Press, 1947; DAN, 56, No. 7, 687 (1947); ZhTF, 18, issue 4, 431 (1948).
2. G. N. Watson, Proc. Roy. Soc., 95a, 83 (1918); 95, 546 (1919).
3. M. V. Keldysh, DAN, 77, 11 (1951).
4. M. A. Naimark, Linear Differential Operators, Moscow, 1954; Tr. Moscow Math. Soc., 2, 3 (1953).
5. V. A. Marchenko, Matem. sbornik, 52, issue 2, 739 (1960).