Diffraction of a Plane Electromagnetic Wave by Two Pairs of Parallel Coaxial Cylinders

E. A. Ivanov, S. F. Il’yukevich

A rigorous solution is given for the problem of diffraction of a linearly polarized plane electromagnetic wave by a system consisting of two pairs of parallel, infinitely long coaxial cylinders, when the direction of propagation is normal to the axes of the cylinders and forms an arbitrary angle with the line of centers. The vector problem of finding the resulting field is reduced, as usual, to two scalar problems: for the transverse-magnetic wave (TM) and for the transverse-electric wave (TE), each of which is solved in cylindrical coordinates by the method of separation of variables. In both cases the expansion coefficients are found from infinite systems, solvable by truncation. Examples of a numerical solution of the problem are presented. For the case in which the distance between the centers of the pairs is much greater than the wavelength of the excitation, approximate formulas for the coefficients are obtained in “closed” form.

The problem of diffraction of a linearly polarized plane wave by two pairs of parallel, infinitely long coaxial cylinders of circular cross section is solved under the assumption that the radii of both the inner and outer cylinders, as well as the physical properties of each of the media (I), (II), (III) (Fig. 1), do not vary along the longitudinal axes of the cylinders, and that the wave propagates in the direction of the unit vector $\vec n$, forming an arbitrary angle $\alpha$ with the line of centers and an angle $\pi/2$ with the axes of the cylinders. The coordinate system $Oxyz$ is introduced so that its origin $O$ is the midpoint of the segment of length $l$ joining the centers of the cross sections of the cylinders by the normal plane coinciding with the plane $Oxy$; the axis $Oz$ is parallel to the axes of the cylinders, and the axis $Ox$ is directed along the line of centers. In addition, local coordinate systems $O_s x_s y_s z_s$, $s=\pm 1$, are introduced, together with the associated formulas
\[ x_s=\rho_s\cos\varphi_s,\quad y_s=\rho_s\sin\varphi_s,\quad z_s=z,\quad s=\pm 1, \]
and local cylindrical coordinates $\rho,\varphi,z$, referred to the corresponding pairs of cylinders (the radii of the cross sections of the inner cylinders

Fig. 1. Normal section of the cylinders.
Physical constants:

\[ I-\varepsilon_1,\mu_1,\sigma_1;\quad II-\varepsilon_2,\mu_2,\sigma_2;\quad III-\varepsilon_3,\mu_3,\sigma_3 \]

are assumed equal to \(\rho_s=b\), and the outer ones to \(\rho_s=a,\ s=\pm 1\). By virtue of the linearity of Maxwell’s equations with respect to \(\vec E,\vec H\), the problem of finding the secondary fields arising in media (I)—(III) as a result of their excitation by the incident wave can, as is known, be reduced to the solution of two particular problems: a) the problem of diffraction of a wave of TM type, for which

\[ \vec E^{0}=\{0,\,0,\,E_z^{0}\},\quad \vec H^{0}=\{H_{\rho}^{0},\,H_{\varphi}^{0},\,0\},\quad E_z^{0}=E_0 e^{ik\vec n\vec R-i\omega t}, \tag{1} \]

and b) the problem of diffraction of a wave of TE type, for which

\[ \vec E^{0}=\{E_{\rho}^{0},\,E_{\varphi}^{0},\,0\},\quad \vec H^{0}=\{0,\,0,\,H_z^{0}\},\quad H_z^{0}=H_0 e^{ik\vec n\vec R-i\omega t}. \tag{2} \]

In mathematical formulation, problem a) consists in finding a unique solution of the two-dimensional Helmholtz equation (the components \(\vec E\) and \(\vec H\) in our case do not depend on \(z\))

\[ \Delta E_z^{(j)}+k_j^2 E_z^{(j)}=0,\quad j=1,\,2,\,3, \tag{3} \]

in each of the regions (I)—(III), satisfying the boundary conditions

\[ \left. \begin{aligned} E_z^{(1)}&=E_z^{(0)}+E_{z,b}^{(1)}=E_{z,s}^{(2)},\\ \mu_2\frac{\partial E_z^{(1)}}{\partial \rho_s} &=\mu_1\frac{\partial E_{z,s}^{(2)}}{\partial \rho_s} \end{aligned} \right\},\quad \rho_s=a,\ s=\pm 1, \tag{4} \]

\[ \left. \begin{aligned} E_{z,s}^{(2)}&=E_{z,s}^{(3)},\\ \mu_3\frac{\partial E_{z,s}^{(2)}}{\partial \rho_s} &=\mu_2\frac{\partial E_{z,s}^{(3)}}{\partial \rho_s} \end{aligned} \right\},\quad \rho_s=b,\ s=\pm 1, \tag{5} \]

where the function \(E_{z,b}^{(1)}\), in addition, must satisfy the radiation condition at infinity. The solution of problem b) likewise consists in finding a unique solution of equation (3) for the function \(H_z^{(j)}\) in (I)—(III), satisfying the conditions

\[ H_z^{(1)}=H_z^{(0)}+H_{z,b}^{(1)}=H_{z,s}^{(2)}, \]

\[ \varepsilon_2(4\pi\sigma_2-i\omega)\frac{\partial H_z^{(1)}}{\partial \rho_s} = \varepsilon_1(4\pi\sigma_1-i\omega)\frac{\partial H_{z,s}^{(2)}}{\partial \rho_s}, \quad \rho_s=a,\ s=\pm 1, \tag{6} \]

\[ H_{z,s}^{(2)}=H_{z,s}^{(3)}, \]

\[ \varepsilon_3(4\pi\sigma_3-i\omega)\frac{\partial H_{z,s}^{(2)}}{\partial \rho_s} = \varepsilon_2(4\pi\sigma_2-i\omega)\frac{\partial H_{z,s}^{(3)}}{\partial \rho_s}, \quad \rho_s=b,\ s=\pm 1, \tag{7} \]

where the function \(H_{z,b}^{(1)}\) must satisfy the radiation condition at infinity. In (3) it is assumed that

\[ k_1^2=\frac{\varepsilon_1\mu_1\omega^2}{c^2} \quad \text{for medium (I),} \]

\[ k_j^2=\frac{\varepsilon_j\mu_j\omega^2+4\pi\mu_j\sigma_j\omega i}{c^2},\qquad j=2,3,\ \text{for media (II), (III).} \]

For the special case of two parallel infinitely long metallic cylinders, problem a) was studied theoretically and experimentally by P. Row [1]. In a rigorous formulation, this problem was reduced in [1] to a system of integral equations, the solution of which in turn leads to the solution of an infinite system of linear equations for the coefficients of the expansion into a complex Fourier series of the functions of the surface currents excited on each of the cylinders. An analogous method for solving diffraction problems on two cylinders was subsequently used by other authors as well, for example in [2–3]. For particular values of the system parameters, the equations obtained in [1] were solved by truncation, without a theoretical justification of the applicability of this method in the general case. To solve problems a), b) we apply a method which is, in essence, a generalization of the known method for the rigorous solution of the problem of diffraction of a plane wave by one pair of coaxial cylinders (see, for example, [4]–[5]) to the case of two pairs of coaxial cylinders. The infinite systems of linear equations for the unknown expansion coefficients that arise in the process of solving problems a), b) are more transparent than in [1]. Their study has shown that they are always solvable by truncation, provided that the outer cylinders of the system do not touch. Section 1 is devoted to the solution of problems a), b). Section 2 gives examples of a numerical solution of problem a) for particular parameter values, and Section 3 considers certain asymptotic cases in the solution of problems a), b).

1. Solution of problems a) and b).

We seek the solution of problem a) in each of the media in the form of series

\[ E_z^{(1)}=E_z^{(0)}+E_{z,b}^{(1)},\qquad E_{z,b}^{(1)}=\sum_{s=\pm1}\sum_{n=-\infty}^{\infty} x_n^{(s)}H_n^{(1)}(k_1\rho_s)e^{in\varphi_s} \quad \text{medium (I),} \tag{8} \]

\[ E_{z,s}^{(2)}=\sum_{n=-\infty}^{\infty} y_n^{(s)}J_n(k_2\rho_s)e^{in\varphi_s} + \sum_{n=-\infty}^{\infty} z_n^{(s)}H_n^{(1)}(k_2\rho_s)e^{in\varphi_s} \quad \text{medium (II),} \tag{9} \]

\[ E_{z,s}^{(3)}=\sum_{n=-\infty}^{\infty} u_n^{(s)}J_n(k_3\rho_s)e^{in\varphi_s} \quad \text{medium (III),} \tag{10} \]

\(s=\pm1\), in terms of the eigen wave functions of a circular cylinder, written in local polar coordinates, where in the coordinates of the \(s\)-th pair of cylinders

\[ E_z^{(0)}=E_0e^{isk_1l_0\cos\alpha} \sum_{n=-\infty}^{\infty} i^nJ_n(k_1\rho_s)e^{in(\varphi_s-\alpha)},\qquad l_0=\frac{l}{2}. \tag{11} \]

The unknown expansion coefficients in (8)—(10) are found from the boundary conditions (4), (5) with the aid of the addition theorem, well known in the theory of cylindrical functions,

\[ H_n^{(1)}(k\rho_s)e^{in\varphi_s} = \sum_{m=-\infty}^{\infty} J_m(k\rho_{-s})H_{n-m}^{(1)}(k\rho_{s,-s}) e^{im\varphi_{-s}-i(n-m)\varphi_{s,-s}}, \tag{12} \]

where \(\rho_{s,-s} > \rho_s\), and \(\rho_{s,-s}\) and \(\varphi_{s,-s}\) are the coordinates of the origin of the \(-s\)-th coordinate system in the coordinates of the \(s\)-th system. In our case \(\varphi_{-1,+1}=0\), \(\varphi_{+1,-1}=\pi\), \(\rho_{s,-s}=l\). On the basis of (4), (5), and (8)—(12), after some transformations, for \(x_n^{(s)}\) one obtains the infinite systems of linear equations

\[ x_n^{(s)}+\sum_{m=-\infty}^{\infty} \alpha_{nm}^{(-s,s)} x_m^{(-s)} = f_n^{(s)}, \quad n=0,\ \pm 1,\ldots,\quad s=\pm 1 . \tag{13} \]

The remaining coefficients of the expansions (9), (10) are found in terms of \(x_n^{(s)}\) from the formulas:

\[ y_n^{(s)}=A_n x_n^{(s)}, \quad z_n^{(s)}=B_n x_n^{(s)}, \quad u_n^{(s)}=C_n x_n^{(s)} . \tag{14} \]

Here

\[ \alpha_{nm}^{(-s,s)} H_{m-n}^{(1)}(k_1 l) W_n e^{-i(m-n)\varphi_{-s,s}}, \tag{15} \]

\[ f_n^{(s)}=-E_0 e^{i s k_1 l_0 \cos\alpha+i\left(\frac{\pi}{2}-\alpha\right)n} W_n, \quad W_n=\frac{\alpha_n+\beta_n K_n}{\gamma_n+\tau_n K_n}, \tag{16} \]

\[ A_n=\frac{-k_1\sigma_n}{\alpha_n+\beta_n K_n}, \quad B_n=A_nK_n, \]

\[ C_n=\frac{J_n(k_2 b)+H_n^{(1)}(k_2 b)K_n}{J_n(k_3 b)}\,A_n, \tag{17} \]

where, in turn,

\[ \alpha_n= \left| \begin{matrix} J_n(k_2 a) & J_n(k_1 a)\\ \dfrac{\mu_1 k_2}{\mu_2} J_n'(k_2 a) & k_1 J_n'(k_1 a) \end{matrix} \right|, \quad \beta_n= \left| \begin{matrix} H_n^{(1)}(k_2 a) & J_n(k_1 a)\\ \dfrac{\mu_1 k_2}{\mu_2} H_n^{(1)'}(k_2 a) & k_1 J_n'(k_1 a) \end{matrix} \right|, \]

\[ \gamma_n= \left| \begin{matrix} H_n^{(1)}(k_1 a) & k_1 H_n^{(1)'}(k_1 a)\\ J_n(k_2 a) & \dfrac{\mu_1 k_2}{\mu_2} J_n'(k_2 a) \end{matrix} \right|, \quad \tau_n= \left| \begin{matrix} H_n^{(1)}(k_1 a) & k_1 H_n^{(1)'}(k_1 a)\\ H_n^{(1)}(k_2 a) & \dfrac{\mu_1 k_2}{\mu_2} H_n^{(1)'}(k_2 a) \end{matrix} \right|, \]

\[ \sigma_n= \left| \begin{matrix} H_n^{(1)}(k_1 a) & J_n(k_1 a)\\ H_n^{(1)'}(k_1 a) & J_n'(k_1 a) \end{matrix} \right|, \quad K_n=\frac{\delta_n}{\lambda_n}, \]

\[ \lambda_n= \left| \begin{matrix} k_2 H_n^{(1)'}(k_2 b) & J_n'(k_3 b)\\ \dfrac{\mu_2 k_3}{\mu_3} H_n^{(1)}(k_2 b) & J_n(k_3 b) \end{matrix} \right|, \quad \delta_n= \left| \begin{matrix} \dfrac{\mu_2 k_3}{\mu_3} J_n(k_3 b) & k_2 J_n'(k_2 b)\\ J_n(k_3 b) & J_n(k_2 b) \end{matrix} \right| . \]

It is not difficult to verify that to the system (13) one cannot justifiably apply any of the known methods for the direct solution of infinite systems of linear equations, since, in particular, here for each \(n\), \(\left|\alpha_{nm}^{(-s,s)}\right|\to\infty\) as \(|m|\to\infty\). Consequently, the theory of the solution of infinite systems of linear equations set forth, for example, in [6] is not applicable to (13). However, if in (13) and (14) the old coefficients are replaced by new ones according to the formulas

\[ x_n^{(s)}=J_n(k_1 a)X_n^{(s)}, \quad y_n^{(s)}=H_n^{(1)}(k_2 a)Y_n^{(s)}, \]

\[ z_n^{(s)}=J_n(k_2 b)Z_n^{(s)}, \quad u_n^{(s)}=H_n^{(1)}(k_3 b)U_n^{(s)}, \tag{18} \]

then the infinite system for \(X_n^{(s)}\) obtained from (13),

\[ X_n^{(s)}+\sum_{m=-\infty}^{\infty}\sigma_{nm}^{(-s,s)}X_m^{(-s)} =F_n^{(s)},\quad n=0\pm1,\ldots;\ s=\pm1 \tag{19} \]

will be solvable by the method of reduction for arbitrary values of the parameters, provided that the outer cylinders of the system do not touch. Here now

\[ \sigma_{nm}^{(-s,s)}=\frac{J_m(k_1a)}{J_n(k_1a)}\,\alpha_{nm}^{(-s,s)},\quad F_n^{(s)}=\frac{f_n^{(s)}}{J_n(k_1a)}, \tag{20} \]

while \(Y_n^{(s)}, Z_n^{(s)}, U_n^{(s)}\) are expressed in terms of \(X_n^{(s)}\) by the formulas

\[ Y_n^{(s)}=\overline{A}_nX_n^{(s)},\quad Z_n^{(s)}=\overline{B}_nX_n^{(s)},\quad U_n^{(s)}=\overline{C}_nX_n^{(s)}, \]

where

\[ \overline{A}_n=-\frac{J_n(k_1a)}{H_n^{(1)}(k_2a)}\,A_n;\quad \overline{B}_n=\frac{J_n(k_1a)}{J_n(k_2b)}\,B_n \quad\text{and}\quad \overline{C}_n=\frac{J_n(k_1a)}{H_n^{(1)}(k_3b)}\,C_n. \]

One can verify what has just been stated regarding system (19) if one estimates, in modulus, its matrix elements \(\sigma_{nm}^{(-s,s)}\). Thus, for example, representing \(\sigma_{nm}^{(-s,s)}\) in the form

\[ \sigma_{nm}^{(-s,s)} =\frac{J_m(k_1a)}{H_n^{(1)}(k_1a)} H_{m-n}^{(1)}(k_1l)R_ne^{-i(m-n)\varphi_{-s,s}}, \tag{21} \]

where \(|R_n|<\mathrm{const}\) for all \(n\), and repeating for (21) all the arguments analogous to those given in [7] in estimating there the terms of sequence (38), we find that for all \(n\) and \(m\)

\[ \left|\sigma_{nm}^{(-s,s)}\right| <\mathrm{const}_1 \frac{(|n|+|m|)!}{|n|!\,|m|!} \left(\frac{a}{l}\right)^{|n|+|m|}, \quad s=\pm1. \tag{22} \]

For the right-hand sides of system (19) we obtain that

\[ \left|F_n^{(s)}\right|<\mathrm{const}_2 q^{|n|},\quad q<1. \tag{23} \]

Infinite systems with matrix elements and right-hand sides of the form (22), (23), on the basis of the corresponding theorems from [6], will possess a completely continuous form, be quasiregular, and be solvable by the method of reduction. Consequently, the majorized system (19) is then solvable by reduction. Its solutions, as solutions of a system with a completely continuous form and with free terms satisfying the condition

\[ \sum_{n=-\infty}^{\infty}|F_n^{(s)}|^2<\infty, \]

will belong to the Hilbert space \(l^2\) [6], whence

\[ \sum_{n=-\infty}^{\infty}|X_n^{(s)}|^2<\infty,\quad s=\pm1. \]

Then also

\[ \sum_{n=-\infty}^{\infty}|Y_n^{(s)}|^2<\infty,\quad \sum_{n=-\infty}^{\infty}|Z_n^{(s)}|^2<\infty,\quad \sum_{n=-\infty}^{\infty}|U_n^{(s)}|^2<\infty. \]

The latter circumstance makes it possible, without computing the coefficients \(X_n^{(s)}, Y_n^{(s)}, Z_n^{(s)}, U_n^{(s)}\), to show

absolute and uniform convergence of the series (8)—(10) at each point of the corresponding medium.

The determination of the coefficients \(X_n^{(s)}, Y_n^{(s)}, Z_n^{(s)}, U_n^{(s)}\) completely solves problem a). Problem b) is solved in the same way. If its solution is sought in the form of the same series (8)—(20) for the functions \(H_z\), then, after the corresponding procedures and the replacement of the coefficients according to formulas (18), for \(X_n^{(s)}\) we obtain a system of the form (19), where, as in problem a),

\[ \sigma_{nm}^{(-s,s)}= \frac{J_m(k_1a)}{J_n(k_1a)}\,a_{nm}^{(-s,s)},\qquad F_n^{(s)}=\frac{f_n^{(s)}}{J_n(k_1a)}, \]

\[ a_{nm}^{(-s,s)}= H_{m-n}^{(1)}(k_1l)\,W_n e^{-i(m-n)\varphi_{-s,s}},\qquad f_n^{(s)}= \tag{24} \]

\[ =H_0 e^{i k_1 l_0\cos \alpha+i\left(\frac{\pi}{2}-\alpha\right)n}\,W_n,\qquad W_n=\frac{\alpha_n+\beta_nK_n}{\gamma_n+\tau_nK_n}, \]

and

\[ \alpha_n= \left| \begin{array}{cc} J_n(k_2a) & J_n(k_1a)\\[2mm] \dfrac{\varepsilon_1(4\pi\sigma_1-i\omega)k_2}{\varepsilon_2(4\pi\sigma_2-i\omega)}\,J_n'(k_2a) & k_1J_n'(k_1a) \end{array} \right|, \]

\[ \beta_n= \left| \begin{array}{cc} H_n^{(1)}(k_2a) & J_n(k_1a)\\[2mm] \dfrac{\varepsilon_1(4\pi\sigma_1-i\omega)k_2}{\varepsilon_2(4\pi\sigma_2-i\omega)}\,H_n^{(1)\prime}(k_2a) & k_1J_n'(k_1a) \end{array} \right|, \]

\[ \gamma_n= \left| \begin{array}{cc} H_n^{(1)}(k_1a) & k_1H_n^{(1)\prime}(k_1a)\\[2mm] J_n(k_2a) & \dfrac{\varepsilon_1(4\pi\sigma_1-i\omega)k_2}{\varepsilon_2(4\pi\sigma_2-i\omega)}\,J_n'(k_2a) \end{array} \right|, \]

\[ \tau_n= \left| \begin{array}{cc} H_n^{(1)}(k_1a) & k_1H_n^{(1)\prime}(k_1a)\\[2mm] H_n^{(1)}(k_2a) & \dfrac{\varepsilon_1(4\pi\sigma_1-i\omega)k_2}{\varepsilon_2(4\pi\sigma_2-i\omega)}\,H_n^{(1)\prime}(k_2a) \end{array} \right|, \tag{25} \]

\[ \sigma_n= \left| \begin{array}{cc} H_n^{(1)}(k_1a) & J_n(k_1a)\\[2mm] H_n^{(1)\prime}(k_1a) & J_n'(k_1a) \end{array} \right|, \qquad K_n=\frac{\delta_n}{\lambda_n}, \]

\[ \lambda_n= \left| \begin{array}{cc} k_2H_n^{(1)\prime}(k_2b) & J_n'(k_3b)\\[2mm] \dfrac{\varepsilon_2(4\pi\sigma_2-i\omega)k_3}{\varepsilon_3(4\pi\sigma_3-i\omega)}\,H_n^{(1)}(k_2b) & J_n(k_3b) \end{array} \right|, \]

\[ \delta_n= \left| \begin{array}{cc} \dfrac{\varepsilon_2(4\pi\sigma_2-i\omega)k_3}{\varepsilon_3(4\pi\sigma_3-i\omega)}\,J_n'(k_3b) & k_2J_n'(k_2b)\\[2mm] J_n(k_3b) & J_n(k_2b) \end{array} \right|. \]

The coefficients \(Y_n^{(s)}, Z_n^{(s)}, U_n^{(s)}\) are determined through \(X_n^{(s)}\) by formulas of the previous form.

2. Examples of a numerical solution of problem a)

As examples illustrating the method, we give the results of a numerical solution of problem a) for the case of metallic inner cylinders, when \(\sigma_3=\infty\). Then \(\vec E^{(3)}=\vec H^{(3)}\equiv 0\), and in system (19)

\[ \sigma_{nm}^{(-s,s)} = \frac{J_m(k_1a)}{J_n(k_1a)} H_{m-n}^{(1)}(k_1l)\, W_n^{(0)} e^{-i(m-n)\varphi_{-s,s}}, \qquad F_n^{(s)} = \]

\[ = -\,E_0 e^{\,isk_1l_0\cos\alpha-i\left(\alpha-\frac{\pi}{2}\right)n} \frac{W_n^{(0)}}{J_n(k_1a)}, \]

where

\[ W_n^{(0)} = \frac{ \left| \begin{array}{cc} \dfrac{\mu_1 k_2}{\mu_2} W_n^{(1)} & W_n^{(2)}\\[4pt] k_1J_n'(k_1a) & J_n(k_1a) \end{array} \right| }{ \left| \begin{array}{cc} \dfrac{\mu_1 k_2}{\mu_2} W_n^{(1)} & W_n^{(2)}\\[4pt] k_1H_n^{(1)\prime}(k_1a) & H_n^{(1)}(k_1a) \end{array} \right| }, \]

\[ W_n^{(1)} = \left| \begin{array}{cc} J_n'(k_2a) & J_n(k_2b)\\ H_n^{(1)\prime}(k_2a) & H_n^{(1)}(k_2b) \end{array} \right|, \]

\[ W_n^{(2)} = \left| \begin{array}{cc} H_n^{(1)}(k_2b) & H_n^{(1)}(k_2a)\\ J_n(k_2b) & J_n(k_2a) \end{array} \right|, \]

and

\[ \overline{B}_n = - \frac{J_n(k_1a)}{H_n^{(1)}(k_2b)} \overline{A}_n, \]

\[ \overline{A}_n = \frac{ -\,k_1H_n^{(1)}(k_2b)\,J_n(k_1a)\,W_n^{(1)} }{ H_n^{(1)}(k_2a) \left| \begin{array}{cc} \dfrac{\mu_1 k_2}{\mu_2} W_n^{(1)} & W_n^{(2)}\\[4pt] k_1J_n'(k_1a) & J_n(k_1a) \end{array} \right| }. \]

Fig. 2. Graphs of the function \(|E_{z,b}^{(1)}(\varphi)|^2\) for the parameters
\(k_1l=6.25,\ k_2=1.6k_1,\ \sigma_1=\sigma_2=0,\ \sigma_3=\infty,\ \mu_1=\mu_2=1,\ \alpha=\pi/2;\)
\(1\) — \(k_2b=3.2,\ k_2a=4;\quad 2\) — \(k_2b=3.2,\ k_2a=3.2;\quad 3\) — \(k_2b=3.2,\ k_2a=4.8\).

These results are presented in Figs. 2–3, which show radiation patterns constructed for the function \(|E_{z,b}^{(1)}|^2\) in the far zone, where it is considered by us as a function of the angle \(\varphi\). The values of the parameters for which the calculations were performed are indicated in the figures. The coefficients \(X_n^{(s)}\) were found by truncating system (19), preliminarily rewritten in the form

\[ X_n^{(s)}-\sum_{k=-N}^{N}\tau_{nk}^{(s)}X_k^{(s)}=\varphi_n^{(s)}, \]

where

\[ \tau_{nk}^{(s)}=\sum_{m=-N}^{N}\sigma_{nm}^{(-s,s)}\sigma_{mk}^{(s,-s)}; \]

\[ \varphi_n^{(s)}=F_n^{(s)}-\sum_{m=-N}^{N}\sigma_{nm}^{(-s,s)}F_m^{(-s)},\quad n=0,\pm1,\ldots,\pm N,\quad s=\pm1 \]

and where, depending on the values of the parameters, \(N\) was taken from 4 to 10. The truncated systems, as well as \(|E_{z,b}^{(1)}|^2\), were computed with the aid of the Minsk-2 electronic computer, although in general, for comparatively small values of the parameters \(ka\) and \(kb\) (of the order of 1–2 and less), the calculation of the matrix elements, the free terms, and the systems themselves can be performed using only tables [8–10] and the simplest calculating devices.

Fig. 4 illustrates the rate of convergence of the computational process: here the function \(|E_{z,b}^{(1)}|^2\) is plotted for different orders of truncation, beginning with \(N=0\), and for comparatively large values of the parameters \(ka\), \(kb\) (in comparison with others taken by us).

Let us note here that in all the cases considered above it turned out that, already starting with \(N=4; 5\), all the remaining truncation orders give practically coincident results.

Let us also note that in the wave zone the function \(E_{z,b}^{(1)}\) (and, analogously, the function \(H_{z,b}^{(1)}\)) is given by the expression

\[ E_{z,b}^{(1)}=\sqrt{\frac{2}{\pi k_1\rho}}\,e^{i\left(k_1\rho-\frac{\pi}{4}\right)}\times \]

\[ \times\sum_{n=-\infty}^{\infty}\left(\sum_{s=\pm1}X_n^{(s)}e^{-isk_1l_0\cos\varphi}\right)(-i)^nJ_n(k_1a)e^{in\varphi}, \]

since at points of the far zone

\[ H_n^{(1)}(k_1\rho_s)=(-i)^n\sqrt{\frac{\pi}{k_1\rho}}\,e^{i\left(k_1\rho-\frac{\pi}{4}\right)-isk_1l_0\cos\varphi}, \]

and \(\varphi_s=\varphi,\ s=\pm1\). Here, if \(\alpha=\dfrac{\pi}{2}\), \(X_n^{(s)}=(-1)^nX_{-n}^{(-s)},\ s=\pm1\).

3. Asymptotic cases of the solution of problems a), b)

Expressions (15)–(17) and (20) for system (19) in problem a) and the corresponding expressions in problem b) can be simplified if, for example:

1) the outer (inner) cylinders are highly conducting, when

\[ \frac{\varepsilon_2}{\varepsilon_1}\to\infty\left(\frac{\varepsilon_3}{\varepsilon_2}\to\infty\right); \]

2) the outer (inner) cylinders have a high magnetic permeability, when

\[ \frac{\mu_2}{\mu_1}\to\infty\left(\frac{\mu_3}{\mu_2}\to\infty\right); \]

3) the cylinders have a large density, when the magnetic or dielectric permeability remains finite, but such that \((|k_i a| \gg 1), (|k_i b| \gg 1)\), and it is possible to apply asymptotic formulas, with respect to the argument, for Bessel functions;

4) the cylinders have small radii, when \(|k_i a| \ll 1\) \((|k_i b| \ll 1)\), etc. We shall not study all these cases here, but shall dwell only on the case when \(k_1 l \gg 1\) (the wavelength of the excitation is much smaller than the distance between the centers of the cylinder cross sections) and \(k_1 l\) is such that the asymptotic formula
\[ H_n^{(1)}(k_1 l)=(-i)^n \sqrt{\frac{2}{\pi k_1 l}}\, e^{i\left(k_1 l-\frac{\pi}{4}\right)} \]
holds. Then the matrix elements of system (19) in problem a) can be written in the form
\[ \sigma_{nm}^{(-s,s)} = L\left(a_m e^{-im\varphi_{-s,s}}\right) \left(b_n e^{in\varphi_{-s,s}}\right), \tag{26} \]
where
\[ L=\sqrt{\frac{2}{\pi k_1 l}}\, e^{i\left(k_1 l-\frac{\pi}{4}\right)}, \]
\[ a_n=(-i)^n J_n(k_1 a), \]
\[ b_n=i^n\,\frac{W_n}{J_n(k_1 a)} . \tag{27} \]

Applying then to (19) the method of successive approximations, taking as the zeroth approximations the values \(X_{n,0}^{(s)}=0\), \(s=\pm 1\), and using some ideas of V. Tverskoi from [11], for the \(k\)-th approximation, on the basis of (26), we obtain the expression
\[ X_{n,k}^{(s)}=F_n^{(s)}+\sum_{t=1}^{k-1}(-1)^t L^t \times \]
\[ \times \sum_{n',\,n'',\,\ldots,\,n^{(t)}} \left(b_n e^{in\varphi_{-s,s}}\right) \left(a_{n'} b_{n'} \times\right. \]
\[ \left.\times (-1)^{n'}\right)\cdots \left(((-1)^{n^{(t-1)}}a_{n^{(t-1)}}b_{n^{(t-1)}})\right) \left(a_{n^{(t)}} e^{-in^{(t)}\varphi_{(-1)^t s,\,(-1)^{t-1}s}} F_{n^{(t)}}^{(-1)^t s}\right), \]
where
\[ \sum_{n',\,n'',\,\ldots,\,n^{(t)}}= \sum_{n'}\sum_{n''}\cdots\sum_{n^{(t)}} . \]

Fig. 3. Graphs of the function \(|E_{z,b}^{(1)}(\varphi)|^2\) for the parameters \(k_1l=8.2,\ k_2=1.6k_1,\ \sigma_1=\sigma_2=0,\ \sigma_3=\infty,\ \mu_1=\mu_2=1,\ \alpha=\pi/2\).

\(1\) — \(k_1b=4.8,\ k_2a=5.6\); \(2\) — \(k_2b=4.8,\ k_2a=4.8\); \(3\) — \(k_2b=4.8,\ k_2a=6.4\).

After introducing the notation
\[ Q=\sum_n(-1)^n a_n b_n,\qquad P^{(s)}=\sum_n a_n F_n^{(s)}e^{-in\varphi_{s,-s}}= \]
\[ =-E_0 p^{(s)} e^{isk_1 l_0\cos\alpha}, \tag{28} \]

\[ p^{(s)}=\sum_n e^{-in\varphi_{-s,s}}W_n,\qquad s=\pm1 \]

we obtain, under the assumption that \(X_n^{(s)}=\lim\limits_{k\to\infty}X_{nk}^{(s)}\), that

\[ \begin{aligned} X_n^{(s)}={}&F_n^{(s)} -\left(b_ne^{in\varphi_{-s,s}}\right) \sum_{t=1,3\ldots}L^t \prod_{\mu=1}^{t-1} \left\{\sum_{n(\mu)}(-1)^{n(\mu)}a_n^{(\mu)}b_n^{(\mu)}\right\}P^{(-s)} \\ &+\left(b_ne^{in\varphi_{-s,s}}\right) \sum_{t=2,4\ldots}L^t \prod_{\mu=1}^{t-1} \left\{\sum_{n(\mu)}(-1)^{n(\mu)}a_n^{(\mu)}b_n^{(\mu)}\right\}P^{(s)}, \end{aligned} \]

whence

\[ X_n^{(s)}=F_n^{(s)}-\frac{b_nP^{(-s)}Le^{in\varphi_{-s,s}}}{1-(LQ)^2} +\frac{b_nP^{(s)}L^2Qe^{in\varphi_{-s,s}}}{1-(LQ)^2}, \tag{29} \]

if \(|LQ|<1\) (because of the convergence of the series for \(Q\), this condition can always be satisfied by a corresponding choice of the value of the parameter \(k_1l\)). Then, on the basis of (8), (28), (18), (27), (29), \(E_{z,b}^{(1)}=E_1+E_2\),

where

\[ E_1=-E_0\sum_{s=\pm1}\sum_{n=-\infty}^{\infty} W_nH_n^{(1)}(k_1\rho_s) e^{isk_1l_0\cos\alpha+in\left(\frac{\pi}{2}-\alpha+\varphi_s\right)}, \tag{30} \]

\[ E_2=\frac{L}{1-(LQ)^2}\sum_{s=\pm1}\sum_{n=-\infty}^{\infty} \left[P^{(-s)}-LQP^{(s)}\right]i^nW_nH_n^{(1)}(k_1\rho_s) e^{in(\varphi_{-s,s}+\varphi_s)}. \tag{31} \]

The functions \(E_1\) and \(E_2\) have here an obvious meaning; namely, the function \(E_1\) determines, at each point of region (I), the secondary field \(E_{z,b}^{(1)}\) created there as a result of excitation of the cylinders of the system by only the incident plane wave, without taking into account the mutual diffraction of the scattered waves, while the function \(E_2\) determines, at the same points, the result of all mutual successive diffractions of the scattered waves on the cylinders. In the wave zone

\[ E_1=-2E_0\sqrt{\frac{2}{\pi k_1\rho}}\, e^{i\left(k_1\rho-\frac{\pi}{4}\right)} \cos(k_1l_0\delta)\sum_{n=-\infty}^{\infty}W_ne^{in(\varphi-\alpha)}, \tag{32} \]

\[ \begin{aligned} E_2={}& \frac{\sqrt{2}\,E_0Le^{i\left(k_1\rho-\frac{\pi}{4}\right)}}{\sqrt{\pi k_1\rho}\left(1-(LQ)^2\right)} \sum_{s=\pm1}\sum_{n=-\infty}^{\infty} \left[p^{(-s)}e^{-isk_1l_0\delta_1}-LQp^{(s)}e^{isk_1l_0\delta}\right] \\ &\times W_ne^{in(\varphi_{-s,s}+\varphi)}, \end{aligned} \tag{33} \]

where

\[ \delta=\cos\alpha-\cos\varphi,\qquad \delta_1=\cos\alpha+\cos\varphi. \]

On the basis of (9), (10), and (18), the relations between \(X_n^{(s)}\) and \(Y_n^{(s)}\), \(U_n^{(s)}\), and (28), the fields \(E_{z,s}^{(2)}\), \(E_{z,s}^{(3)}\), \(s=\pm1\), in regions (II), (III) can also be represented as the sum of two functions of analogous meaning. Turning now to a system of the form (19) in problem b) with matrix elements and right-hand sides from (24), (25), for \(k_1l\gg1\) we similarly for \(X_n^{(s)}\) ...

in the same way we obtain expressions of the form (29), and for the function \(H_{z,b}^{(1)}\) itself, expansions of the form (30)—(33).

Suppose that in problem a), in addition to the condition there that \(k_1 l \gg 1\), the parameters \(k_j b, k_j a\) \((j=1,2,3)\) are such that \(|k_j b|\ll 1\) and \(|k_j a|\ll 1\). Then, using the asymptotic formulas for Bessel functions \((|x|\ll 1)\)
\[ J_n(x)=\frac{x^n}{2^n n!}, \]
\[ J_0'(x)=-\frac{x}{2},\qquad H_0^{(1)'}(x)=\frac{2i}{\pi x}, \]
\[ H_n^{(1)}(x)=\frac{-i(n-1)!}{\pi x^n}\,2^n,\qquad H_0^{(1)}(x)=-\frac{2i}{\pi}\ln\frac{2}{\gamma x} \]
\((\gamma=1.781)\), and retaining in all series in \(n\) only the terms with \(n=0\) (the case of isotropic scattering of a plane wave), after some transformations we obtain
\[ E_{z,b}^{(1)}=E_1+E_2=-2E^0 \sqrt{\frac{2}{\pi k_1\rho}}\, e^{i\left(k_1\rho-\frac{\pi}{4}\right)} W_0\cos(k_1l_0\hat{\delta})\times \tag{34} \]
\[ \times \frac{1-LW_0\cos(k_1l_0\hat{\delta}_1)/\cos(k_1l_0\hat{\delta})} {1-(LW_0)^2}, \]
since in this case
\[ p^{(-s)}=p^{(s)}=Q=W_0,\qquad s=\pm1, \]
where
\[ W_0=iR, \]
\[ R=\frac{\pi}{2}\left\{(k_2a)^2\left(\frac{\mu_1}{\mu_2}-\frac{k_1^2}{k_2^2}\right) -\left(\frac{\mu_1}{\mu_2}\right)(k_2b)^2 \left(1-\frac{\mu_2}{\mu_3}\frac{k_3^2}{k_2^2}\right)\right\}\times \]
\[ \times\left\{ 2-\left[ \frac{\mu_1}{\mu_2}(k_2a)^2 -(k_2b)^2\left(1-\frac{\mu_2}{\mu_3}\frac{k_3^2}{k_2^2}\right) \left(\frac{\mu_1}{\mu_2}\right) \right]\ln\frac{2}{\gamma k_1a} -\frac{\mu_2}{\mu_3}(k_3b)^2 \ln\frac{2}{\gamma k_2b} \right. \]
\[ \left. {}-(k_2b)^2\left(1-\frac{\mu_2}{\mu_3}\frac{k_3^2}{k_2^2}\right) \ln\frac{2}{\gamma k_2a} \right\}^{-1}, \tag{35} \]
if in the expression obtained for \(W_0\) one neglects the terms containing, as factors, the product \((k_j a)^2(k_j b)^2\). For the case when \(\sigma_3=\infty\), in (34) \(W_0\) will be given by the expression \(W_0=iR\),
\[ R=\frac{\pi}{2\left(\ln\frac{2}{\gamma k_1a}+\frac{\mu_2}{\mu_1}\ln\frac{a}{b}\right)}, \tag{36} \]
if here one also neglects the terms containing, as factors, \((k_j b)^2\) or \((k_j a)^2\). Putting \(LW_0=\omega e^{i\theta}\), where
\[ \omega=\sqrt{\frac{2}{\pi k_1 l}}\,R, \]

![Figure 4 graph from the source page]

Fig. 4. Graphs of the function \(|E_{z,b}^{(1)}(\varphi)|^2\) for different orders of truncation, when \(k_1l=6.25,\ k_2b=4,\ k_2a=4.8,\ k_2=1.6\,k_1,\ \sigma_1=\sigma_2=0;\ \sigma_3=\infty,\ \mu_1=\mu_2=1,\ \alpha=\pi/2\).

\(1\)—\(N=0\); \(2\)—\(N=1\); \(3\)—\(N=2\); \(4\)—\(N=3\)—\(10\).

and \(\theta=k_1l+\dfrac{\pi}{4}\), we can write (34) in the form

\[ E_{z,b}^{(1)}=-E_0\sqrt{\frac{2}{\pi k_1\rho}}\, e^{i\left(k_1\rho-\frac{\pi}{4}\right)}W_0 \times \frac{ e^{ik_1l_0\delta}\left(1+\omega e^{i(\theta-2\sigma)}\right) + e^{-ik_1l_0\delta}\left(1+\omega e^{i(\theta+2\sigma)}\right) }{ 1-\omega^2 e^{2i\theta} }, \tag{37} \]

where \(\sigma=-k_1l_0\cos\varphi\). From this one can obtain certain results analogous to those that hold in the problem of diffraction of a plane wave by two metallic circular cylinders [11] and in the problem of diffraction of a plane wave by two elliptic cylinders [12]. In particular, if the parameters \(\lambda\) (the wavelength of the excitation), \(l\) (the distance between the centers of the cylinder cross sections), and \(\varphi\) (the observation angle) are such that the equalities

\[ \theta \mp 2\sigma=2k_1l_0(1\pm \cos\varphi)+\frac{\pi}{4}=(2n^{\pm}+1)\pi, \]

\[ k_1l_0\delta=k_1l_0(\cos\alpha-\cos\varphi)=\pi m \]

are simultaneously satisfied, or the equalities \(\theta\mp2\sigma=2n^{\pm}\pi,\ k_1l_0\delta=\pi m\), where \(n^{\pm}, m\) are integers, then the maximum of the function \(E_1\) at the observation point will be most strongly enhanced or most strongly weakened by the values there of the function \(E_2\). If, however,

\[ k_1l_0\delta=(2m+1)\frac{\pi}{2} \]

and simultaneously \(2k_1l_0\cos\varphi=\pi n\), then the function \(E_{z,b}^{(1)}\) at the observation point vanishes. Let, for example, \(\theta\mp2\sigma=(2n^{\pm}+1)\pi\) or \(\theta\mp2\sigma=2n^{\pm}\pi\). Then (37) can be written in the form

\[ E_{z,b}^{(1)} = -2E_0\sqrt{\frac{2}{\pi k_1\rho}}\, e^{i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{W_0\cos(k_1l_0\delta)}{1\pm\omega}, \tag{38} \]

whence it follows that, for \(|\omega|\ll1\), the result of successive mutual diffractions on the cylinders \(E_2\) will have little effect on the value of the function \(E_1\), and then approximately \(E_{z,b}^{(1)}=E_1\). Thus, in particular, if \(\varepsilon_1=\mu_1=\mu_2=1\) and \(\sigma_3=\infty\), and \(k_1b=0.05\), then from

\[ \omega=\sqrt{\frac{\lambda}{l}}\, \frac{1}{2\ln 2/\gamma\,0.05} \]

we find that \(\omega\simeq0.05\) for \(l/\lambda=10\) \((k_1l=20\pi)\); if \(k_1b=0.005\), then \(\omega\simeq0.05\) for \(l/\lambda=4\) \((k_1l=8\pi)\), etc. These results agree with the results of V. Tverskoi from [11]. They show that, for \(\mu_1=\mu_2=1\), in the case of isotropic scattering of the incident wave, when \(|k_1a|\ll1,\ |k_1b|\ll1,\ \sigma_3=\infty\), the presence of a dielectric coating on the cylinders does not affect the value of \(E_{z,b}^{(1)}\) in the far zone. The table shows

Table of the dependence of \(l/\lambda\) on \(a/b\) and \(\mu_2\) for \(k_1a=0.1\) and \(\omega=0.05\)

the dependence of the values of \(1/\lambda\) on \(a/b\) and \(\mu_2\), when \(k_1a=0.1\), while \(\omega\) remains equal to 0.05. From the expression for \(\omega\) \((\sigma_3=\infty)\) it is seen that, as \(\mu_2\) increases,

\[ \omega \to \frac{\mu_2}{\mu_1}\sqrt{\frac{\pi}{2k_1l}\ln\frac{a}{b}}. \]

Considering problem b) and the corresponding expressions for \(H_{z,b}^{(1)}\), we similarly find that

\[ H_{z,b}^{(1)}=-2H_0\sqrt{\frac{2}{\pi k_1\rho}}\,e^{i\left(k_1\rho-\frac{\pi}{4}\right)}W_0\cos(k_1l_0\delta)\times \]

\[ {}\times \frac{1-LW_0\cos k_1l_0\delta_1/\cos k_1l_0\delta} {1-(LW_0)^2}, \tag{39} \]

where \(L\), \(\delta\), and \(\delta_1\) have their former meanings, and

\[ W_0=i\frac{\pi}{2}\left\{(k_2a)\left[\frac{\varepsilon_1(4\pi\sigma_1-i\omega)} {\varepsilon_2(4\pi\sigma_2-i\omega)}-\frac{k_1^2}{k_2^2}\right] -\frac{\varepsilon_1(4\pi\sigma_1-i\omega)} {\varepsilon_2(4\pi\sigma_2-i\omega)}(k_2b)^2\left[1-\right.\right. \]

\[ \left.\left. -\frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)}\frac{k_3^2}{k_2^2}\right]\right\} \cdot \left\{2-\left[(k_2a)^2-(k_2b)^2\left(1- \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)}\frac{k_3^2}{k_2^2}\right)\right]\right\}\times \tag{40} \]

\[ {}\times \frac{\varepsilon_1(4\pi\sigma_1-i\omega)} {\varepsilon_2(4\pi\sigma_2-i\omega)} \ln\frac{2}{\gamma k_1a} -(k_2b)^2\left[1- \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)}\frac{k_3^2}{k_2^2}\right]\ln\frac{2}{\gamma k_2a} \]

\[ {}-(k_3b)^2 \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)} \ln\frac{2}{\gamma k_2b} \Bigg\}^{-1}, \]

if, as before, we neglect here the terms containing, as factors of products, \((k_ja)^2(k_jb)^2\). If \(\sigma_3=\infty\), then \(W_0\) will have the form

\[ W_0=iG,\qquad G=\frac{\pi/2} {\ln\frac{2}{\gamma k_1a}+ \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_1(4\pi\sigma_1-i\omega)} \ln\frac{a}{b}}, \tag{41} \]

if here one neglects terms with factors \((k_fb)^2\) or \((k_ja)^2\). If, moreover, it is assumed that \(\sigma_1=\sigma_2=0\), then

\[ G=\frac{\pi/2} {\ln\frac{2}{\gamma k_1a}+\frac{\varepsilon_2}{\varepsilon_1}\ln\frac{a}{b}}. \tag{42} \]

Formulas (39)—(42) make it possible to analyze the solution of problem b) for \(|k_ja|\ll 1\), \(|k_jb|\ll 1\), \(j=1,2,3\).

Retaining in the series by which the solutions of problems a) and b) are determined terms of higher orders with respect to the summation index \(n\), we obtain expressions depending on the angle of incidence \(\alpha\).

Conclusion

The method applied for solving the problem of the diffraction of a linearly polarized plane electromagnetic wave by a system of two pairs of coaxial circular cylinders is an effective method for obtaining numerical results over a sufficiently broad range of parameter values (provided that the outer cylinders of the system do not touch). It can be generalized to the case of an arbitrary finite number of pairs of coaxial cylinders.

References

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Received by the editors
September 17, 1964

Institute of Mathematics
and Computational Technology, Academy of Sciences of the BSSR