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Keywords:

  • diffraction;
  • Wiener-Hopf;
  • quarter plane

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References

[1] In this paper, we demonstrate that Radlow's solution to diffraction by the soft quarter plane is incorrect. We derive an explicit expression for the Wiener-Hopf factorizing function and verify that a nonvanishing term, which was not accounted for by Radlow, arises from a correct evaluation of the field on the quarter plane. Consequently, the field function proposed by Radlow does not satisfy the boundary conditions, and therefore it is not the correct solution to diffraction by a quarter plane.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References

[2] The problem of calculating the diffraction of an electromagnetic or acoustic wave by a quarter plane (or more generally by a plane angular sector) is still a challenge. As a matter of fact, the 2004 URSI Commission B international electromagnetics prize dealt with a readily computable solution for the scattering by a quarter plane. Indeed, existing solutions consider the quarter plane as a degenerated elliptic cone and express the field as a spherical wave multipole series [Kraus and Levine, 1961; Satterwhite, 1974; Sahalos and Thiele, 1983; Bowman et al., 1987; Hansen, 1990] (see [Blume, 1996] for a recent review). However, such a series exhibits a poor convergence rate when both the source and the observation point move far from the quarter-plane tip. In the plane wave incidence–far field observation regime, which is of great interest in high-frequency modeling of scattering, the series computation becomes critical [Blume and Krebs, 1998]. Furthermore, each term of the series requires the numerical solution of the transverse (on the sphere) eigenvalue problem with the calculation of eigenvalues and eigenfunctions (Lamè functions [Whittaker and Watson, 1990]). For this reason many works proposed alternative formulations based on heuristic models [Sikta et al., 1983; Pathak, 1988; Hill and Pathak, 1991; Maci et al., 1994; Capolino and Maci, 1996] or hybrid analytical-numerical approaches [Hansen, 1991; Babich et al., 1996]. In 1961 Radlow published a paper [Radlow, 1961], followed by the improved version [Radlow, 1965], where the problem is attacked with a Wiener-Hopf [Noble, 1958] (convolution integral) technique, somehow inspired by Sommerfeld's half plane solution [Sommerfeld, 1896; Born and Wolf, 1964], but rephrased in two complex variables. The papers aroused a discussion because the order of the field singularity at the tip of Radlow's solution differs from the correct value [Van Bladel, 1991], revealing a suspicion of incorrectness. A researcher who gave him credit was Albertsen who extended the solution to the electromagnetic case [Albertsen, 1997], also providing a closed form vertex diffraction coefficient by means of an explicit calculation of the Wiener-Hopf factorization involved in Radlow's solution. From the same canonical problem, Albertsen [2000] also derived a uniform high-frequency description for the doubly diffracted rays, i.e., rays which experience two successive diffractions at the two edges. Indeed, the plane wave spectral representation which is obtained by Radlow is very appealing to derive high-frequency description of the scattering via asymptotic ray optics approximation, however, as we will demonstrate in this paper, it is not correct.

[3] In what follows we will adopt the notation of Radlow [1965] to which we address for all the details. For reader convenience we briefly abstract the statement of the problem and Radlow's solution in section 2. In section 3 we conduct a detailed analysis of the various functions introduced by Radlow [1961, 1965], providing a closed form expression for the Wiener-Hopf factorization and a description of the singularity topology of the factorized function. This discussion is fundamental for the understanding of the following sections. Next, in section 4 we briefly summarize why Radlow's demonstration that his solution satisfies boundary conditions is incorrect. Finally, in section 5 we shows that Radlow's solution does not respect required boundary conditions and therefore it is not the exact solution. By means of a direct check of the boundary conditions, an extra term, not accounted for by Radlow, is shown to be present, thus preventing the fulfillment of boundary condition on the quarter plane. Some conclusions about Radlow's solution end the paper.

2. Statement of the Problem and Radlow's Solution

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References

[4] Radlow [1965] presents a Wiener-Hopf solution for the soft quarter plane problem. A Cartesian (x1, x2, x3) reference system is chosen in which the quarter plane is described by x1 ≥ 0, x2 ≥ 0, x3 = 0 (see Figure 1). The quarter plane is illuminated by an impinging plane wave

  • equation image

in which a1 = ik sin α0 cos β0, a2 = ik cos α0, a3 = ik sin α0 sin β0, with k = piq denoting the ambient wave number (with small losses q > 0), and α0, β0 the incidence aspect angles. A eiωt harmonic dependence is intended and suppressed. The scattered wave ϕ(x1, x2, x3) solution of the problem must satisfies the mixed boundary problem

  • equation image
  • equation image
  • equation image

and the radiation condition at infinity and “edge and corner condition”, i.e., exhibits physically acceptable singularities at edges and at the corner.

image

Figure 1. Geometry of the quarter-plane problem.

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[5] In order to adopt a Wiener-Hopf technique, the solution is represented using the plane wave spectrum [Radlow, 1965, equations. (2.5)–(2.5)]

  • equation image

for which the wave equation (2) is automatically fulfilled, and the radiation condition is ensured by choosing the principal branch of the square root, i.e., equation imageeequation image ≥ 0; the same branch choice is adopted for square roots throughout the paper. In (5)

  • equation image

is the spectral free-space Green's function [Radlow, 1965, equation (2.2)]. Accordingly to Radlow's notation, for the sake of compactness, the complex variable pair (w1, w2) ∈ equation image2 is somewhere alternatively denoted by w. Also, real and imaginary parts of complex variables are denoted by wj = uj + ivj, j = 1, 2. Following the Wiener-Hopf strategy conditions (3) and (4) has to be imposed directly on the field spectrum. To this end Bochner's theorem [Bochner and Martin, 1948] is invoked.

[6] When applying standard (one dimensional) Wiener-Hopf technique, bilateral functions are split into the sum of two monolateral functions, one vanishing on the negative semi axis and the other vanishing on the positive semiaxis. Consequently, the function spectrum, i.e., its Laplace transform, is in turn split into the sum of two terms, one analytic on the left complex half plane (positive real part) and the other analytic on the right complex half plane (negative real part). Such a procedure is generalized in several variables for multiple Laplace transform by Bochner's theorem [Bochner and Martin, 1948]. In our two dimensional case a spatial function h(x1, x2) can be written as the sum of four functions hn(x1, x2) each vanishing outside the nth quadrant Qn of the (x1, x2) plane. Consequently, its double Laplace transform H(w) is written as the sum of four functions Hn(w) each analytic on the respective tube T(qn), with qn denoting the nth quadrant in the (u1, u2) plane. We remind to the reader that a tube T(D) of basis D is defined [Bochner and Martin, 1948] as the point set in the w = (w1, w2) double complex plane for which (u1, u2) ∈ D for any −∞ < v1 < +∞ and −∞ < v2 < +∞. In other words T(qn) is the point set (u1, u2) ∈ qn. In particular, a function h1(x1, x2) vanishing outside the quarter plane x1 ≥ 0 ∩ x2 ≥ 0, i.e., outside the first quadrant Q1, presents a double Laplace transform H1(w) analytic on T(q1), i.e., for w1 and w2 lying in the respective right complex half planes u1 > 0 and u2 > 0.

[7] Using Bochner's theorem, boundary conditions (3) and (4) are translated directly on the field spectrum, thus becoming condition I and II of theorem 4.1 of Radlow [1965], respectively. Namely they state that: (I) A(w) is analytic in the tube T(a), i.e., for u1 > −ℜe{a1} and u2 > −ℜe{a2}; (II) the first term of Bochner's decomposition of A(w)B(w) is

  • equation image

[8] Although Wiener-Hopf approach is usually a constructive algorithm, Radlow does not derive the solution via usual Wiener-Hopf schemes, but he introduces his solution [Radlow, 1965, equations. (7.1)–(7.2.2)]

  • equation image

without derivation. Radlow's solution (8) is expressed in terms of the functions Mn(w) resulting from the product factorization of the Green's function spectrum (6), for which he gives a formal definition. Radlow [1965] then discussed the proposed solution to prove the fulfillment of the two conditions I and II. The demonstration that condition I is met is quite obvious; on the contrary, the proof for condition II is more involved and it is based on the determination of the analyticity domain of M13(w) = M1(w) M3(w). To better understand the behavior of Radlow's solution and to allow its numerical calculation a closed form expression was derived for the Mn(w) functions, which is presented in the next section.

3. Analysis of the Solution and Closed Form Expression for M(w)

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References

[9] In this section a novel closed form expression for the Mn(w) functions is reported. The derivation is similar to that of Albertsen [1997], but here rephrased in original Radlow's notation, and with special care to singularities topology. Radlow [1961, 1965] showed that the spectral Green's function (6) is analytic in T(b), with b any compact subset of u12 + u22 < q2, and can be factorized [Radlow, 1965, equation (5.1)]

  • equation image

as the product of four functions Mn(w), each analytic and nonzero in the respective tube T(b, n) = Tbqn〉, given by [Radlow, 1965, equation (5.2)]

  • equation image

Wn functions are explicitly calculated by Radlow [1961, 1965] as [Radlow, 1965, equations (5.4) and (5.4.1)]

  • equation image

and

  • equation image

[10] In (11), principal branches are chosen for square roots and logarithms; that is, cuts are traced for real negative argument, accordingly to (6). Note that W1(w; k) exhibits branch point singularities in each variable at w1 = −ik and w2 = −ik, whereas it is regular at w1 = ik and w2 = ik and along the respective apparent cuts, where square root and logarithm jump compensate each other (Figure 2).

image

Figure 2. Singularities of W1(w; k) in the w1 complex plane. The same figure applies to the w2 complex plane by interchanging w1 and w2.

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[11] Furthermore, W1(w; k) exhibits an interlaced pole singularity in the left half plane u1 ≤ 0 of the w1 principal Riemann sheet at w1 = −equation image when u2 ≤ 0, or, equivalently, in the left half plane equation imagee{w2} ≤ 0 of the w2 principal Riemann sheet at w2 = −equation image when u1 ≤ 0. Indeed, when u2 > 0 (u1 > 0) the function is bounded in the principal Riemann sheet at w1 = −equation image (w2 = −equation image) because the pole occurs at the same point but in a different Riemann sheet. Conversely, unlike B(w), W1(w; k) is regular in the w1 and w2 principal Riemann sheets at w1 = equation image and w2 = equation image, respectively, where the numerator vanishes and compensates for the denominator pole. Thereby, all singularities have a negative real part in both variables and are outside the q1 quadrant, as expected.

[12] Then, introducing the quantities

  • equation image

for which wj = ik cos equation imagej and equation image = k sin equation imagej, and by using equation image, from which equation image and some trigonometric manipulations, equation (11) can be rearranged as

  • equation image

thus the function M1(w) can be explicitly calculated from (10), except for an integration constant α0, as

  • equation image

where the function

  • equation image

(see Figure 3) is simply related to the first-order Debye function [Abramowitz and Stegun, 1970] Debye1 (x) = equation image by

  • equation image
image

Figure 3. Real (solid curves) and imaginary (dashed curves) part of m(x) in the complex x plane.

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[13] The Debye function is in turn related to the Dilogarithm function (Spence's integral for n = 2) [Abramowitz and Stegun, 1970] Dilog(x) = −equation imagedt by

  • equation image

leading to

  • equation image

[14] Equation (18), and consequently (19), strictly hold for −π < equation imagem{x} < π, however, it can be extended on the entire x plane mapping each strip (2n − 1)π < equation imagem{x} < (2n + 1)π onto the pertinent Riemann sheet of the Dilog function. This is not straightforward, and therefore, unlike Albertsen [1997], our solution is not expressed in term of the Dilogarithm function, to better control the topology of branch singularities. For numerical calculation of m(x), either in the small argument Taylor expansion

  • equation image

with Bn denoting Bernoulli's numbers, and in the large argument asymptotic expansions

  • equation image

18 terms were found sufficient to reach double precision accuracy in the strip −π < ℜe{x} < π. An arbitrary argument outside that strip can be reduced to the previous case via the translation relation

  • equation image

with n and n′ denoting the nearest integer and the nearest integer toward zero to ℜe{equation image}, respectively.

[15] The limit for k [RIGHTWARDS ARROW] ∞ determines the value of α0. Since equation imageB(w) = equation image becomes regular in both the variables as the singularities approach infinity, then

  • equation image

must hold. Considering that equation imageequation imagej = equation image and m(equation image) + m(equation image) = 2πlog 2, equation (23) implies α0 = equation imagelog 2.

[16] Rearranging (15), we eventually obtain

  • equation image

[17] The topology of the singularities of M1(w) is similar to that of W1(w; k) but poles are converted into branches by the k integration (Figure 4). Although (13) introduces branches at wj = ±ik, since (24) is an even function of equation imagej, the branch at wj = ik becomes fictitious. Therefore M1(w) exhibits branch point singularities in each variable only at w1 = −ik and w2 = −ik. A further interlaced branch singularity occurs in the principal Riemann sheet at w1 = −equation image only when ℜe{w2} ≤ 0, or, equivalently, at w2 = −equation image only when ℜe{w1} ≤ 0. Again, all singularities have a negative real part in both variables and are outside the q1 quadrant, as expected. It is simple to verify the symmetry M1(w1, w2) = M1(w2, w1) and the algebraic behavior M1(w1, w2) = O[∣w12 + w22equation image] for large values of ∣w12 + w22∣. Finally, note that M1(w) is bounded at the branch points w1,2 = −ik, whereas it is singular at w1,2 = −equation image (for ℜe{w2,1} ≤ 0), where it behaves like

  • equation image
image

Figure 4. Singularities of M1(w) in the w1 complex plane. The interlaced branch w1 = −equation image in the left half plane equation imagee{w1} < 0 occurs in the principal Riemann sheet only if equation imagee{w2} < 0. The same figure applies to the w2 complex plane by interchanging w1 and w2.

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[18] Other Mn(w) functions for n = 2, 3, 4 are readily obtained using (12) in (10), whence

  • equation image

4. Incorrectness of Radlow's Demonstration

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References

[19] The author went deeply through papers [Radlow, 1961, 1965] and found the demonstration contained in section 7 “The result for A(w)” of Radlow [1965], that Radlow's solution A(w) meets condition II of theorem 4.1 of [Radlow, 1965] (i.e., satisfies boundary condition), is not correct. However, since this demonstration is very long and complicated, also to show where it fails is not straightforward, and probably only specialist readers would be interested in it. For the sake of brevity, we only mention here that there is an error in section 6 “Factorization, II” of Radlow [1965], where Radlow analyzes the analyticity domain of M13(w) = M1(w) M3(w). Such an error renders invalid the use of Cauchy's residue theorem in section 7 “The result for A(w)” of Radlow [1965], for the calculation of equation (7.10) and the consequent verification of (3). Indeed the presence of a branch at w1 = −ik in the left complex half plane introduces an additional contribution to the field that Radlow did not account for. Therefore Radlow's proof that his solution meets condition II, is not correct. In principle, this does not necessarily imply that Radlow's solution does not satisfies the prescribed boundary conditions, which is what we are definitely interested in verifying. To this end, in the next section we present a direct detailed check of (3).

5. Test on Boundary Conditions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References

[20] Now we are interested in understanding whether condition II is met or not, i.e., if the field satisfies the required soft boundary condition on the quarter plane. To this end we consider the field on the quarter plane [Radlow, 1965, equations (2.5)–(2.6)],

  • equation image

[21] The topology of the singularities of the field spectrum (8), i.e., the integrand of (27), is inferred from the topology of the singularities of M functions. Namely, when observed in the w1 complex plane, (8) exhibit a simple pole at w1 = −a1 that is located slightly to the left of the imaginary axis in the “visible region” −kv1k, depending on the incidence aspect of the illuminating plane wave. Also in

  • equation image

the denominator is regular in the left half plane and introduces a branch point at w1 = equation image in the right half plane, regardless the sign of u2. On the other hand, M4(w1, w2) introduces a branch in the principal Riemann sheet at w1 = −ik and, only if u2 > 0, another branch at w1 = −equation image. Since by definition equation imagee{a2} > 0, M4(w1, a2) only introduces the branch at w1 = −ik, whereas the branch w1 = −equation image is off the principal Riemann sheet. Finally, M2(−a1, w2) M3(−a1, −a2) is a constant when observed in the w1 complex plane. A sketch of the singularity topology of the integrand A(w) B(w) in the w1 complex plane is shown in Figure 5. An analogous singularity analysis and sketch, by symmetry, applies to the w2 complex plane.

image

Figure 5. Singularity of the plane wave spectrum of the solution A(w)B(w) in the w1 complex plane. The same picture applies to w2, interchanging 1 and 2.

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[22] In (27), we assume to keep the original integration paths in the w1 and w2 complex planes slightly on the left of the imaginary axis; i.e., with an arbitrary small negative real part. Thereby, in the w1 left half plane only the branch at w1 = −ik occurs in the principal Riemann sheet, for each w2 integration point. As shown in Figure 5, the original integration path (black line) in the w1 complex plane is then deformed, onto the contour C1 (gray line) around the branch cut and a closing path at infinity (gray dashed line), whose contribution is vanishing because of the exponential decaying of exp(w1x1) to infinity in the left half plane u1 < 0, when observing on the quarter plane (x1 > 0). In this deformation the pole at w1 = −a1 is captured, thus the pertinent residue contribution is added, whose expression is simplified by using M3(w1, w2) M4(w1, w2) = equation image ,

  • equation image

[23] Note that the second term in (29), arising from the residue at w1 = −a1, is exactly Sommerfeld's solution [Sommerfeld, 1896] for the field scattered by a soft half plane x3 = 0, x2 ≥ 0, with edge along x1. The integrand of this second term exhibits a branch at w2 = equation image in the right half plane u2 > 0 and only a pole at w2 = −a2 in the left half plane u2 < 0; indeed, when evaluating (8) at w1 = −a1, the M2 functions in the numerator and in the denominator cancel out and the corresponding branches at w2 = −ik disappear. Hence, by closing the integration path at infinity to the left, the second term is exactly calculated as the residue at the Geometrical Optics pole. In fact the field diffracted by the soft half plane vanishes on the half plane itself, since it satisfies the boundary condition; thus the scattered field on the soft half plane surface merely coincides with the reflected field; i.e., minus the incident field

  • equation image

[24] Next, we consider the first term in (29). Because of the uniform convergence of integrals given by the exponential decaying at infinity, the order of integration is permissibly interchanged. As stated above, the analysis of singularities in the w2 complex plane is analogous to that of w1; hence the integration path in the w2 complex plane is also deformed into a path C2 around the branch cut, thus capturing a residue contribution at the pole w2 = −a2. By symmetry, the expression for such a residue contribution is the same as in the LHS of (30) except for a swap between 1 and 2, and for the integration path that is now on C1. Analogously to before, it is easily recognized to be the field diffracted (because the Geometrical Optic Pole at w1 = −a1 is now excluded by the integration path) by a soft half plane x3 = 0, x1 ≥ 0, with edge along x2. Again when evaluating the function at w2 = −a2, M4 functions simplify and the integrand loses its branch in the left half plane ℜe{w1} < 0. In conclusion, the integration along C1 of the residue contribution at w2 = −a2 vanishes, as expected since the field diffracted by the soft half plane satisfies the boundary conditions

  • equation image

[25] Finally, using (30) and (31) in (29), (27) reduces to

  • equation image

where an extra integral term appears that is missing in the work of Radlow [1961, 1965] and corresponds to branch contributions in both the variables. The extra term can be evaluated asymptotically for kr [RIGHTWARDS ARROW] ∞ at the dominant spectral point w = (−ik, −ik). Via the variable transform (13), the double integral is remapped into the θ1 and θ2 complex planes (Figure 6); integration contours C1,2 are mapped onto the integration contours Γ1,2, and dw1,2 = −ik sin θ1,2dθ1,2:

  • equation image
image

Figure 6. Singularity of the plane wave spectrum of the solution A(w)B(w) in the θ1 complex plane (a) before and (b) after the deformation of the integration path onto the SDP through the saddle point at π. The same picture applies to θ2, interchanging 1 and 2.

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[26] The w1 principal Riemann sheet is mapped onto the grey region of the equation image1 complex plane shown in Figure 6, the right and left boundary of the grey region correspond to the branches at w1 = ik and w1 = −ik, respectively; other w1 Riemann sheets are mapped onto the rest of the equation image1 complex plane. In the equation image1 planes the integrand still exhibits poles and branches relevant to M functions. Namely, if α1,2 denote the mapping of a1,2 through (13), the pole singularity at w1 = −a1 is mapped into pole singularities at equation image1 = ±(πα1), and the branch point at w1 = equation image is mapped into branches at equation image1 = ±(equation imageequation image2). Furthermore, the branches at w1 = −equation image and w1 = −equation image, that are off the principal Riemann sheet but just at the border of the branch cut, are mapped into the branches at equation image1 = equation image + equation image2 and equation image1 = equation imageα2, respectively. Note that the interlaced branch at θ1 = equation image + θ2 occurs at a point on the dashed line in Figure 6a when θ2 ∈ Γ2. A similar singularity description applies to the θ2 complex plane simply interchanging 1 and 2. The integration paths Γ1,2 can be deformed onto the Steepest Descent Paths (SDP) S1,2 through the saddle point at θ1,2 = π, which is the asymptotically dominant spectral constituent, corresponding to w1,2 = −ik. Note that the integrand presents a zero in both variables at the saddle point, given by the Jacobian of the change of variables sin θ1 sin θ2, whereas the plane wave spectrum (8) is there bounded. The leading asymptotic term is then obtained via standard asymptotic expansion

  • equation image

in which

  • equation image

is a nonvanishing dimensionless constant. Although the same asymptotic approximation may also be directly calculated using standard formulas for integration around a branch point in the w1,2 complex plane, the analysis in the θ1,2 plane permits to clearly isolate the dominant asymptotic term. As a matter of fact, after the deformation onto SDP S1,2, the interlaced branch point at θ1 = equation image + θ22 = equation image + θ1), where the integrand is singular, occurs at a point on the dashed line in Figure 6b, thus becoming asymptotically well isolated from the saddle point at θ1,2 = π. The same is not always true for the branch at θ1 = equation imageα22 = equation imageα1). Depending on the direction of incidence of the illuminating plane wave, the branch θ1 = equation imageα2 may occur to the left of the saddle point, i.e., if ℜe{α2} > equation image. In such a case, in the deformation of Γ1 onto SDP S1 a further asymptotic term arises at this branch point (see Figure 7), however its asymptotic order is the same of the previous because the spectrum here vanishes. The details are omitted for the sake of brevity. Analogously, a further term in the θ2 integration may arise from the branch point θ2 = equation imageα1 if ℜe{α1} > equation image.

image

Figure 7. Deformation of the integration path Γ1 onto the SDP in the θ1 complex plane. Depending on the plane wave incidence aspect, a further contribution may arise from the integration contour B1 around the cut, which exhibits the same asymptotic order of the saddle point. The same picture applies to θ2, interchanging 1 and 2.

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[27] In conclusion it is demonstrated that Radlow's solution does not satisfy boundary conditions on the quarter plane; therefore expression (8) proposed by Radlow is not the correct quarter plane solution spectrum. As a matter of fact assuming that expression, the total field on the quarter plane does not vanish but a residual field exists. Such a residual field on the quarter-plane surface rapidly decays far from the tip as (kr)−3, but dramatically affect the solution at the tip and at the edges of the quarter plane, i.e., for x1, x2 [RIGHTWARDS ARROW] 0. This explains why Radlow's solution does not experience the prescribed singularity order at the tip of the quarter plane [Van Bladel, 1991]. Furthermore, it is worth noting that the spurious contribution (34) is a slow wave that runs on the quarter-plane surface with a wave number greater than the ambient wave number k. Therefore it decays exponentially when observed for increasing distance x3 ≠ 0 from the quarter plane surface. In other words (34) is a surface wave, confined on the quarter plane, and it does not significantly affect the field in the space around the quarter plane. This may explain why numerical tests on the asymptotic corner diffraction coefficient [Albertsen, 1997] presented by Tew and Mittra [1980] gave good results. Nevertheless, Radlow's solution is not exact.

6. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References

[28] In this paper we show that Radlow's solution [Radlow, 1961, 1965] does not respect boundary conditions. The total field evaluated on the quarter plane is asymptotically (at large distance from the tip) very weak but not exactly zero. Hence Radlow's solution is not correct, however it may be regarded as a heuristic vertex diffraction solution. As a matter of fact it embeds the correct half plane Sommerfeld's solution relevant to the two edges of the quarter plane, that ensure the correctness of the solution to the asymptotic order (kr)equation image. Furthermore, as presented by Albertsen [2000], it also embeds doubly diffracted rays, despite their expression does not blend into the form prescribed by the Geometrical Theory of Diffraction (i.e., the cascading of two edge diffractions) far from the transition with the vertex ray, i.e., for diffraction points far from the vertex, as occurs in the work of Maci et al. [1994] or Capolino and Maci [1996]. Hence the challenge of deriving a readily computable vertex diffraction coefficient from an exact solution is still open.

[29] Similar argumentations apply to the solution presented by Albertsen [1997] for the hard quarter plane, which, together with Radlow's soft case, allows to construct the electromagnetic solution for the perfectly conducting quarter plane.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Statement of the Problem and Radlow's Solution
  5. 3. Analysis of the Solution and Closed Form Expression for M(w)
  6. 4. Incorrectness of Radlow's Demonstration
  7. 5. Test on Boundary Conditions
  8. 6. Conclusion
  9. References
  • Abramowitz, M., and I. A. Stegun (Eds.) (1970), Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, Natl. Bur. Stand. Appl. Math. Ser., vol. 55, U.S. Gov. Print. Off., Washington, D. C.
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