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Keywords:

  • wedge diffraction;
  • anisotropic impedance;
  • skew incidence

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information

[1] This paper presents, as an extension of the authors' recent work, an exact solution to diffraction of a skew incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces. Applying the Sommerfeld-Malyuzhinets technique to the boundary-value problem yields a coupled system of difference equations for the spectra; on elimination, a functional difference (FD) equation of higher order for one spectrum arises; after simplification in terms of a generalized Malyuzhinets function and accounting for the Meixner's edge condition as well as the poles and residues of the spectrum in the basic strip of the complex plane, the FD equation is converted, via the so-called S integrals, to an integral equivalent; for points on the imaginary axis which belong to the basic strip the integral equivalent becomes a Fredholm equation of the second kind with a nonsingular, wave number–free and exponentially decreasing kernel; solving this integral equation by the quadrature method the spectrum can be determined by integral extrapolation and by analytical continuation; a first-order uniform asymptotic solution follows from evaluating the Sommerfeld integrals with the saddle-point method. Comparison with available exact solutions in several special cases shows that this approach leads to a fast and accurate solution of the problem under study.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information

[2] Diffraction of electromagnetic waves in wedge-shaped regions remains the subject matter of many works since the seminal paper of Sommerfeld [1896], and is analyzed often with the Sommerfeld-Malyuzhinets method [cf. Senior and Volakis, 1995; Buldyrev and Lyalinov, 2001; Babich et al., 2004]. The key step of the Sommerfeld-Malyuzhinets technique lies in solving the resulting functional difference equations of higher order. Recently, it has been shown that a generalized Malyuzhinets function χΦ [Bobrovnikov and Fisanov, 1988; Avdeev, 1994] and the so-called S integral [Tuzhlin, 1973] can be combined to convert a functional difference (FD) equation of the second order to a Fredholm integral equation of the second kind with the integral equation solved numerically by quadrature method [Lyalinov and Zhu, 2003b]. Furthermore, this approach has been extended to solve an FD equation of higher order resulting from diffraction of a skew incident plane electromagnetic wave by a wedge with scalar impedance faces [Lyalinov and Zhu, 2006].

[3] In the present paper, this approach is applied, with due modification, to a more general diffraction problem, namely, diffraction of a skew incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces. The extensions are direct and within the framework of the general procedure for the isotropic impedance faces.

[4] It is worth mentioning that recently, this problem has been tackled on use of two other ways: the probabilistic random-walk method [Budaev and Bogy, 2006] and a generalized Wiener-Hopf technique [Daniele and Lombardi, 2006]. These two approaches manage to deal with more general anisotropic impedance faces than the way which is followed in the present paper. This notwithstanding, the authors do believe that for problems solvable using all three approaches, the present one is at least as efficient as the other two in terms of accuracy and computational speed (see section 3), thanks to the nonsingular, wave number–free and rapidly decreasing kernel of the resulting Fredholm integral equation obtained on use of, among other things, the special function χΦ. Furthermore, just like the Wiener-Hopf technique, which was regarded for many decades as not being suitable for wedge problems but now has outperformed the Sommerfeld-Malyuzhinets technique in certain aspects, the Sommerfeld-Malyuzhinets technique could one day get the upper hand again. In addition, a closed-form explicit solution to the problem under study ought to be found yet (for such solutions in special cases see, for instance, Lyalinov and Zhu [1999, 2003a] and Antipov and Silvestrov [2006]). Hence this solution procedure deserves to be presented to the electromagnetics community.

2. Analysis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information

2.1. Statement of the Problem and Decoupling of the Functional Difference Equations

[5] A wedge is placed in a circular cylindrical coordinate system (r, ϕ, z) in such a way that its edge coincides with the z axis and its faces are the half-planes ϕ = ±Φ with π > Φ > 0 (Figure 1). On the wedge faces the following boundary conditions must be met by the tangential components of the electric (Ez, Er) and the magnetic (Hz, Hr) fields:

  • equation image

Z0 being the intrinsic impedance of the surrounding medium, a12± and a21± the normalized axially anisotropic surface impedances. The passivity requires that Re a12± ≥ 0 and Re a21± ≥ 0 hold good. Obviously, a scalar impedance wedge with a12± = a21± studied by Lyalinov and Zhu [2006] represents a special case of this work.

image

Figure 1. A wedge illuminated obliquely by a plane electromagnetic wave.

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[6] Let a plane electromagnetic wave fall on this wedge. The z components of the incident field are given by (a time-dependence exp(−iωt) has been assumed, but is dropped in the remaining part of this work)

  • equation image

Here k0 stands for the wave number in the surrounding medium, the angles ϑ0 and ϕ0 characterize the incident direction of the plane wave, U10/Z0 and U20 denote the amplitude of the magnetic and electric components along the edge, respectively.

[7] Therefore the z components of the total field take the form

  • equation image

and equation image(r, ϕ) meet on the wedge faces following conditions [see Lyalinov and Zhu, 1999, 2003a, equations (3) and (4)]:

  • equation image
  • equation image

As r [RIGHTWARDS ARROW] 0, equation image(r, ϕ) remain finite (Meixner's edge condition).

[8] Inserting the Sommerfeld integral expressions for equation image(r, equation image) [Sommerfeld, 1896; Senior and Volakis, 1995; Buldyrev and Lyalinov, 2001; Babich et al., 2004],

  • equation image

into the boundary condition (4) and making use of the inversion formulae leads to a matrix equation for the spectra equation image(α),

  • equation image

The radiation condition demands that equation image(α) − equation image0/(α − ϕ0) be regular in Π(−Φ, Φ).

[9] Equation (7) can be rewritten as

  • equation image

where

  • equation image
  • equation image
  • equation image
  • equation image
  • equation image

Auxiliary angles ϑ± and χ± have been introduced above according to

  • equation image
  • equation image

[10] Eliminating f2(−α) from (8) one obtains a wholly decoupled FD equation for f1(α),

  • equation image

After having obtained a solution for f1(α) in the strip Π(−2Φ, 2Φ), f2(α) in the same strip is given by, on use of (8),

  • equation image

The spectra equation image(α) can be analytically extended to α outside of the basic strip Π(−2Φ, 2Φ) by virtue of (8).

2.2. Simplified Difference Equation

[11] It is worth noting that there exists only one nonconstant coefficient at the left-hand side of (16), namely,

  • equation image

and this coefficient can be written in an equivalent form,

  • equation image

with the auxiliary function Ψ0(α) given by

  • equation image

[12] The defining equation for the generalized Malyuzhinets function χΦ(α),

  • equation image

has been used above [cf. Bobrovnikov and Fisanov, 1988; Avdeev, 1994; Babich et al., 2004].

[13] In case of Φ > π/2, the two zeros of Ψ0(α) inside the basic strip can be eliminated,

  • equation image

This new function F0(α) is free of both zeros and poles in the basic strip.

[14] Introducing a new spectrum equation image1(α) according to f1(α) = F0(α)equation image1(α), it is not difficult to see that equation image1(α) obeys a difference equation with constant coefficients at the left-hand side,

  • equation image

2.3. Fredholm Integral Equation of the Second Kind

[15] Expressing both the spectrum equation image1(α) and the right-hand side of (23) in their even and odd parts, equation image1(α) = equation image1e(α) + equation image1o(α), H1(α) = Q1(α)equation image1(−α) = H1e(α) + H1o(α), it is evident that (23) is identical to

  • equation image
  • equation image

To get the particular solution to the functional difference equation (25), which is equivalent to a system of equations,

  • equation image
  • equation image

we resort to the so-called S integral [Tuzhlin, 1973; Buldyrev and Lyalinov, 2001; Babich et al., 2004] [see also Lyalinov and Zhu, 2003a, 2006].

[16] To find the particular solution to (24) by using the same procedure, it is advantageous to introduce a new odd function equation image1o(α) via equation image1e(α) = cot(να)equation image1o(α). The respective difference equation follows from (24),

  • equation image

and can be dealt with in the same way as (25). Hence the particular solution is given by

  • equation image

The sought-for equivalent integral representation of (23) in the basic strip Π(−2Φ, 2Φ) reads

  • equation image

The nonintegral terms at the right-hand side recover the geometrical-optics poles and the asymptotic behavior of equation image1(α) at infinity.

[17] The constants A1± are determined by the radiation condition according to which the second spectrum f2(α) given in the basic strip by (17) must be free of nonphysical poles. This leads to two additional conditions,

  • equation image

where

  • equation image
  • equation image

[18] As indicated clearly by (30), especially by the last term at its right-hand side (the particular solution), equation image1(α) in the basic strip depends upon its value along the imaginary axis of the complex plane. In line with (31), the same is true for the constants A1±. Therefore, for points on the imaginary axis of the complex α plane, (30), together with (31), amounts to a Fredholm integral equation of the second kind.

[19] The kernel of the integral equation (30), Q1(−t)/cos[ν(α + t)], is free from singularity and does not contain the wave number k0 which may be a large parameter, as can be verified by recalling the definition of Q1(t) and the properties of F0(t) and q1(t). Furthermore, this kernel decreases exponentially with ∣Im t∣, or more precisely,

  • equation image

as t [RIGHTWARDS ARROW] ±i∞. It is precisely these properties that render the above described solution procedure a very efficient, that is, fast and accurate one; see section 3.

[20] From the asymptotic behavior of Q1(α) and equation image1(α) it can be inferred that the kernel of (30) is square integrable. Hence its solvability follows from the uniqueness of solution to the present problem, which has been shown elsewhere [cf. Lyalinov and Zhu, 2003b, 2006].

2.4. Uniform Asymptotic Solution

[21] Together with (31), (30) is solved by quadrature method, with the asymptotic behavior of equation image1 accounted for explicitly [see Lyalinov and Zhu, 2003b, 2006]. Evaluating the Sommerfeld integrals (6) by the saddle-point method, a uniform asymptotic solution for the problem under study has been derived,

  • equation image

where the superscripts “go”, “sw” and “d” stand for the geometrical-optics part, the surface waves and diffracted waves, respectively. The formulae for these wave ingredients are given explicitly by Lyalinov and Zhu [2006], where now in the residues of the surface waves the following multiplication factors have to be used instead:

  • equation image

with

  • equation image
  • equation image
  • equation image
  • equation image

[22] The explicit expression for equation imageequation image results from interchanging equation image± and χ± with each other in the above formula for equation imageequation image.

[23] On setting a12± = a21± = η±, the above formulae reduce precisely to the ones for diffraction of a plane electromagnetic wave by a wedge with scalar impedance faces [Lyalinov and Zhu, 2006].

3. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information

[24] As mentioned briefly, the integral equation (30) has been solved with the quadrature method, that is, evaluating approximately the integral at the left-hand side of (30) with a Gauß-Legendre rule of N abscissae and then enforcing the fulfillment of the equation (30) precisely at these N abscissae.

[25] The above described solution procedure has been coded in Fortran and compiled using the Visual Fortran Professional 5.0.A. To verify the numerical results based on the present exact solution we take recourse to the closed-form exact solution given by Lyalinov and Zhu [1999, 2003a].

[26] Of prime importance is the efficiency of the above solution procedure, that is, the accuracy of the obtained results and required CPU time. As a typical example, Table 1 lists the performance of the computer code for the wedge considered in Figure 2 with ϑ0 = 30° as a function of the number of abscissae. The amplitude of the incident wave is equal to 1 V/m, and the code has been run on a notebook (Pentium III Mobile CPU 1066 MHz, 504 MB RAM). For each abscissa, all field ingredients contained in equation (35) have been computed for 1010 different angles. Then the results have been compared with the ones based on the exact closed-form solution reported by Lyalinov and Zhu [1999, 2003a] at each of the 1010 angles with the maximum deviation shown in Table 1. The required total CPU time for 1010 angles has been estimated using the Fortran 95 intrinsic procedure CPU_TIME.

image

Figure 2. Diffracted fields U2d of (left) copolarization and (right) cross polarization as a function of ϑ0 and ϕ (Φ = 167°).

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Table 1. Performance as a Function of the Number of Abscissaea
Number of AbscissaeMaximum Deviation, V/mTotal CPU Time, s
  • a

    Read 3.18E-04 as 3.18 × 10−4.

103.18E-041.27
201.39E-051.33
401.50E-071.35

[27] Obviously, 40 abscissae lead already to a highly accurate and very fast solution of the problem under study. Therefore the results shown in Figures 2 and 3 have been calculated using 40 abscissae.

image

Figure 3. Total fields of (left) copolarization and (right) cross polarization as a function of the anisotropic impedance faces (Case 1, a12+ = 1/a21+ = 0.3 + i0.4, a12 = 1/a21 = 0.1 − i0.2; Case 2, a12+ = 0.3 + i0.2, a21+ = 1.2 − i0.8, a12 = 0.1 − i0.1, a21 = 2 + i2; Case 3, a12+ = 0.3, a21+ = 1.2, a12 = 0.1, a21 = 2) and ϕ (Φ = 160°).

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[28] The achieved efficiency rewards the analytical efforts described above, namely, to decouple the original functional difference equations (7) or (8) and to simplify the resultant equation (16) to (23) in terms of the generalized Malyuzhinets function χΦ(α), to transform (23) into its integral equivalent (30) by virtue of the S integrals and at last to obtain a Fredholm integral equation of the second kind for points on the imaginary axis of the α plane.

[29] An example of such a comparison is also depicted in Figure 2. For the sake of clarity, only the diffracted fields are shown. Evidently, the numerical results obtained in this work (squares) agree completely with the ones based on the procedure described by Lyalinov and Zhu [1999, 2003a].

[30] It is also of interest to observe the variation of the diffracted fields as a function of the skewness cos ϑ0: for cross-polarized diffracted fields, the influence of the skewness is more pronounced than for the diffracted fields of copolarization. As made clear by Table 2, though the required total CPU time for 1010 angles does depend on the skewness, but only weakly.

Table 2. Dependence of Total CPU Time Upon ϑ0a
ϑ0Total CPU Time, s
10°1.37
30°1.35
50°1.40
70°1.41
90°1.37

[31] That the total fields are everywhere continuous and smooth can be seen for instance in Figure 3, where the skewness is fixed at ϑ0 = 30°, but the anisotropic surface impedances vary, with the amplitude of the imaginary parts decreasing from Case 1 via Case 2 to Case 3. For copolarization, this change in the surface impedance is visible only in regions with reflected waves (in Figure 3, ϕ ≥ 63°), it is discernible almost everywhere for cross polarization. Except in Case 1 with a12± = 1/a21±, the amplitudes of the cross-polarized fields are not polarization-reciprocal, that is

  • equation image

[32] The required total CPU times (for 1010 angles) for the cases reported in Figure 3 are summarized in Table 3, which shows a slight dependence of the computational speed on the values of the anisotropic impedances.

Table 3. Dependence of Total CPU Time Upon Face Impedancesa
CaseTotal CPU Time, s
11.36
21.35
31.36

4. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information

[33] This paper presents an exact solution to diffraction of a skew incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces, on use and with due modification of a procedure proposed recently by Lyalinov and Zhu [2003b, 2006]. The key steps consist in simplifying and then converting the resultant functional difference equation of higher order to a Fredholm integral equation of the second order, and solving the integral equation numerically. Owing to the achieved accuracy and computational speed reported above, the problem under study can be regarded in a “generalized sense” as solved.

[34] Of particular interest to practicing radio scientists should be a solution to scattering of an electromagnetic spherical wave by a wedge with axially anisotropic impedance faces. The authors hope to be able to present an efficient solution based on the procedure described in the present paper in the near future.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information

[35] The authors thank the Guest Editor G. Uslenghi for the opportunity, to report their work in this special issue on “Analytical scattering and diffraction.” They are also grateful to the reviewers for their insightful comments. This paper is partly based on materials presented at the 28th General Assembly of URSI, October 2005, in New Delhi, India.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information
  • Antipov, Y. A., and V. V. Silvestrov (2006), Electromagnetic scattering from an anisotropic impedance half-plane at oblique incidence: the exact solution, Q. J. Mech. Appl. Math., 59(2), 211251.
  • Avdeev, A. D. (1994), On the special function of the problem of diffraction by a wedge in an anisotropic plasma, J. Commun. Technol. Electr., 39(10), 7078.
  • Babich, V. M., M. A. Lyalinov, and V. E. Grikurov (2004), Sommerfeld-Malyuzhinets Method in Diffraction Theory, St. Petersburg Univ. Press, St. Petersburg, Russia.
  • Bobrovnikov, M. S., and V. V. Fisanov (1988), Diffraction of Waves in Angular Regions, Tomsk Univ. Press, Tomsk, Russia.
  • Budaev, B. V., and D. B. Bogy (2006), Diffraction of a plane skew electromagnetic wave by a wedge with general anisotropic impedance boundary conditions, IEEE Trans. Antennas Propag., 54(5), 15591567.
  • Buldyrev, V. S., and M. A. Lyalinov (2001), Mathematical Methods in Modern Electromagnetic Diffraction Theory, Int. Ser. Monogr. Adv. Electromagn., vol. 1, Science House, Tokyo.
  • Daniele, V. G., and G. Lombardi (2006), Wiener-Hopf solution for impenetrable wedges at skew incidence, IEEE Trans. Antennas Propag., 54(9), 24722485.
  • Lyalinov, M. A., and N. Y. Zhu (1999), Diffraction of a skewly incident plane wave by an anisotropic impedance wedge—A class of exactly solvable cases, Wave Motion, 30(3), 275288.
  • Lyalinov, M. A., and N. Y. Zhu (2003a), Exact solution to diffraction problem by wedges with a class of anisotropic impedance faces: Oblique incidence of a plane electromagnetic wave, IEEE Trans. Antennas Propag., 51(6), 12161220.
  • Lyalinov, M. A., and N. Y. Zhu (2003b), A solution procedure for second-order difference equations and its application to electromagnetic-wave diffraction in a wedge-shaped region, Proc. R. Soc., Ser. A., 459(2040), 31593180.
  • Lyalinov, M. A., and N. Y. Zhu (2006), Diffraction of a skew incident plane electromagnetic wave by an impedance wedge, Wave Motion, 44(1), 2143.
  • Senior, T. B. A., and J. L. Volakis (1995), Approximate Boundary Conditions in Electromagnetics, Electromagn. Waves Ser., vol. 41, Inst. of Electr. Eng., London.
  • Sommerfeld, A. (1896), Mathematische Theorie der Diffraction, Math. Ann., 47, 317374.
  • Tuzhlin, A. A. (1973), The theory of Malyuzhinets' inhomogeneous functional equations, Diff. Uravn., 9, 18751888.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis
  5. 3. Numerical Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References
  9. Supporting Information
FilenameFormatSizeDescription
rds5450-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5450-sup-0002-t02.txtplain text document0KTab-delimited Table 2.
rds5450-sup-0003-t03.txtplain text document0KTab-delimited Table 3.

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