### Abstract

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

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

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

[4] *Radlow* [1965] presents a Wiener-Hopf solution for the soft quarter plane problem. A Cartesian (*x*_{1}, *x*_{2}, *x*_{3}) reference system is chosen in which the quarter plane is described by *x*_{1} ≥ 0, *x*_{2} ≥ 0, *x*_{3} = 0 (see Figure 1). The quarter plane is illuminated by an impinging plane wave

in which *a*_{1} = *ik* sin *α*_{0} cos *β*_{0}, *a*_{2} = *ik* cos *α*_{0}, *a*_{3} = *ik* sin *α*_{0} sin *β*_{0}, with *k* = *p* − *iq* denoting the ambient wave number (with small losses *q* > 0), and *α*_{0}, *β*_{0} the incidence aspect angles. A *e*^{iωt} harmonic dependence is intended and suppressed. The scattered wave ϕ(*x*_{1}, *x*_{2}, *x*_{3}) solution of the problem must satisfies the mixed boundary problem

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

[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)]

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., *e* ≥ 0; the same branch choice is adopted for square roots throughout the paper. In (5)

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 (*w*_{1}, *w*_{2}) ∈ ^{2} is somewhere alternatively denoted by *w*. Also, real and imaginary parts of complex variables are denoted by *w*_{j} = *u*_{j} + *iv*_{j}, *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*(*x*_{1}, *x*_{2}) can be written as the sum of four functions *h*_{n}(*x*_{1}, *x*_{2}) each vanishing outside the *n*th quadrant *Q*_{n} of the (*x*_{1}, *x*_{2}) plane. Consequently, its double Laplace transform *H*(*w*) is written as the sum of four functions *H*_{n}(*w*) each analytic on the respective tube *T*(*q*_{n}), with *q*_{n} denoting the *n*th quadrant in the (*u*_{1}, *u*_{2}) 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* = (*w*_{1}, *w*_{2}) double complex plane for which (*u*_{1}, *u*_{2}) ∈ *D* for any −∞ < *v*_{1} < +∞ and −∞ < *v*_{2} < +∞. In other words *T*(*q*_{n}) is the point set (*u*_{1}, *u*_{2}) ∈ *q*_{n}. In particular, a function *h*_{1}(*x*_{1}, *x*_{2}) vanishing outside the quarter plane *x*_{1} ≥ 0 ∩ *x*_{2} ≥ 0, i.e., outside the first quadrant *Q*_{1}, presents a double Laplace transform *H*_{1}(*w*) analytic on *T*(*q*_{1}), i.e., for *w*_{1} and *w*_{2} lying in the respective right complex half planes *u*_{1} > 0 and *u*_{2} > 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 *u*_{1} > −ℜ*e*{*a*_{1}} and *u*_{2} > −ℜ*e*{*a*_{2}}; (II) the first term of Bochner's decomposition of *A*(*w*)*B*(*w*) is

[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)]

without derivation. Radlow's solution (8) is expressed in terms of the functions *M*_{n}(*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 *M*_{13}(*w*) = *M*_{1}(*w*) *M*_{3}(*w*). To better understand the behavior of Radlow's solution and to allow its numerical calculation a closed form expression was derived for the *M*_{n}(*w*) functions, which is presented in the next section.

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

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

[9] In this section a novel closed form expression for the *M*_{n}(*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 *u*_{1}^{2} + *u*_{2}^{2} < *q*^{2}, and can be factorized [*Radlow*, 1965, equation (5.1)]

as the product of four functions *M*_{n}(*w*), each analytic and nonzero in the respective tube *T*(*b*, *n*) = *T* 〈*b* ∪ *q*_{n}〉, given by [*Radlow*, 1965, equation (5.2)]

*W*_{n} functions are explicitly calculated by *Radlow* [1961, 1965] as [*Radlow*, 1965, equations (5.4) and (5.4.1)]

and

[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 *W*_{1}(*w*; *k*) exhibits branch point singularities in each variable at *w*_{1} = −*ik* and *w*_{2} = −*ik*, whereas it is regular at *w*_{1} = *ik* and *w*_{2} = *ik* and along the respective apparent cuts, where square root and logarithm jump compensate each other (Figure 2).

[11] Furthermore, *W*_{1}(*w*; *k*) exhibits an interlaced pole singularity in the left half plane *u*_{1} ≤ 0 of the *w*_{1} principal Riemann sheet at *w*_{1} = − when *u*_{2} ≤ 0, or, equivalently, in the left half plane *e*{*w*_{2}} ≤ 0 of the *w*_{2} principal Riemann sheet at *w*_{2} = − when *u*_{1} ≤ 0. Indeed, when *u*_{2} > 0 (*u*_{1} > 0) the function is bounded in the principal Riemann sheet at *w*_{1} = − (*w*_{2} = −) because the pole occurs at the same point but in a different Riemann sheet. Conversely, unlike *B*(*w*), *W*_{1}(*w*; *k*) is regular in the *w*_{1} and *w*_{2} principal Riemann sheets at *w*_{1} = and *w*_{2} = , 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 *q*_{1} quadrant, as expected.

[12] Then, introducing the quantities

for which *w*_{j} = *ik* cos _{j} and = *k* sin _{j}, and by using , from which and some trigonometric manipulations, equation (11) can be rearranged as

thus the function *M*_{1}(*w*) can be explicitly calculated from (10), except for an integration constant *α*_{0}, as

where the function

(see Figure 3) is simply related to the first-order Debye function [*Abramowitz and Stegun*, 1970] Debye_{1} (*x*) = by

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

leading to

[14] Equation (18), and consequently (19), strictly hold for −*π* < *m*{*x*} < *π*, however, it can be extended on the entire *x* plane mapping each strip (2*n* − 1)*π* < *m*{*x*} < (2*n* + 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

with *B*_{n} denoting Bernoulli's numbers, and in the large argument asymptotic expansions

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

with *n* and *n*′ denoting the nearest integer and the nearest integer toward zero to ℜ*e*{}, respectively.

[16] Rearranging (15), we eventually obtain

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

[18] Other *M*_{n}(*w*) functions for *n* = 2, 3, 4 are readily obtained using (12) in (10), whence

### 5. Test on Boundary Conditions

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

[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 *w*_{1} complex plane, (8) exhibit a simple pole at *w*_{1} = −*a*_{1} that is located slightly to the left of the imaginary axis in the “visible region” −*k* ≤ *v*_{1} ≤ *k*, depending on the incidence aspect of the illuminating plane wave. Also in

the denominator is regular in the left half plane and introduces a branch point at *w*_{1} = in the right half plane, regardless the sign of *u*_{2}. On the other hand, *M*_{4}(*w*_{1}, *w*_{2}) introduces a branch in the principal Riemann sheet at *w*_{1} = −*ik* and, only if *u*_{2} > 0, another branch at *w*_{1} = −. Since by definition *e*{*a*_{2}} > 0, *M*_{4}(*w*_{1}, *a*_{2}) only introduces the branch at *w*_{1} = −*ik*, whereas the branch *w*_{1} = − is off the principal Riemann sheet. Finally, *M*_{2}(−*a*_{1}, *w*_{2}) *M*_{3}(−*a*_{1}, −*a*_{2}) is a constant when observed in the *w*_{1} complex plane. A sketch of the singularity topology of the integrand *A*(*w*) *B*(*w*) in the *w*_{1} complex plane is shown in Figure 5. An analogous singularity analysis and sketch, by symmetry, applies to the *w*_{2} complex plane.

[22] In (27), we assume to keep the original integration paths in the *w*_{1} and *w*_{2} complex planes slightly on the left of the imaginary axis; i.e., with an arbitrary small negative real part. Thereby, in the *w*_{1} left half plane only the branch at *w*_{1} = −*ik* occurs in the principal Riemann sheet, for each *w*_{2} integration point. As shown in Figure 5, the original integration path (black line) in the *w*_{1} complex plane is then deformed, onto the contour *C*_{1} (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(*w*_{1}*x*_{1}) to infinity in the left half plane *u*_{1} < 0, when observing on the quarter plane (*x*_{1} > 0). In this deformation the pole at *w*_{1} = −*a*_{1} is captured, thus the pertinent residue contribution is added, whose expression is simplified by using *M*_{3}(*w*_{1}, *w*_{2}) *M*_{4}(*w*_{1}, *w*_{2}) = ,

[23] Note that the second term in (29), arising from the residue at *w*_{1} = −*a*_{1}, is exactly Sommerfeld's solution [*Sommerfeld*, 1896] for the field scattered by a soft half plane *x*_{3} = 0, *x*_{2} ≥ 0, with edge along *x*_{1}. The integrand of this second term exhibits a branch at *w*_{2} = in the right half plane *u*_{2} > 0 and only a pole at *w*_{2} = −*a*_{2} in the left half plane *u*_{2} < 0; indeed, when evaluating (8) at *w*_{1} = −*a*_{1}, the *M*_{2} functions in the numerator and in the denominator cancel out and the corresponding branches at *w*_{2} = −*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

[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 *w*_{2} complex plane is analogous to that of *w*_{1}; hence the integration path in the *w*_{2} complex plane is also deformed into a path *C*_{2} around the branch cut, thus capturing a residue contribution at the pole *w*_{2} = −*a*_{2}. 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 *C*_{1}. Analogously to before, it is easily recognized to be the field diffracted (because the Geometrical Optic Pole at *w*_{1} = −*a*_{1} is now excluded by the integration path) by a soft half plane *x*_{3} = 0, *x*_{1} ≥ 0, with edge along *x*_{2}. Again when evaluating the function at *w*_{2} = −*a*_{2}, *M*_{4} functions simplify and the integrand loses its branch in the left half plane ℜ*e*{*w*_{1}} < 0. In conclusion, the integration along *C*_{1} of the residue contribution at *w*_{2} = −*a*_{2} vanishes, as expected since the field diffracted by the soft half plane satisfies the boundary conditions

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

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* ∞ at the dominant spectral point *w* = (−*ik*, −*ik*). Via the variable transform (13), the double integral is remapped into the θ_{1} and θ_{2} complex planes (Figure 6); integration contours *C*_{1,2} are mapped onto the integration contours Γ_{1,2}, and *dw*_{1,2} = −*ik* sin θ_{1,2}*d*θ_{1,2}:

[26] The *w*_{1} principal Riemann sheet is mapped onto the grey region of the _{1} complex plane shown in Figure 6, the right and left boundary of the grey region correspond to the branches at *w*_{1} = *ik* and *w*_{1} = −*ik*, respectively; other *w*_{1} Riemann sheets are mapped onto the rest of the _{1} complex plane. In the _{1} planes the integrand still exhibits poles and branches relevant to *M* functions. Namely, if *α*_{1,2} denote the mapping of *a*_{1,2} through (13), the pole singularity at *w*_{1} = −*a*_{1} is mapped into pole singularities at _{1} = ±(*π* − *α*_{1}), and the branch point at *w*_{1} = is mapped into branches at _{1} = ±( − _{2}). Furthermore, the branches at *w*_{1} = − and *w*_{1} = −, that are off the principal Riemann sheet but just at the border of the branch cut, are mapped into the branches at _{1} = + _{2} and _{1} = − *α*_{2}, respectively. Note that the interlaced branch at θ_{1} = + θ_{2} occurs at a point on the dashed line in Figure 6a when θ_{2} ∈ Γ_{2}. A similar singularity description applies to the θ_{2} complex plane simply interchanging 1 and 2. The integration paths Γ_{1,2} can be deformed onto the Steepest Descent Paths (SDP) *S*_{1,2} through the saddle point at θ_{1,2} = *π*, which is the asymptotically dominant spectral constituent, corresponding to *w*_{1,2} = −*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 θ_{1} sin θ_{2}, whereas the plane wave spectrum (8) is there bounded. The leading asymptotic term is then obtained via standard asymptotic expansion

in which

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 *w*_{1,2} complex plane, the analysis in the θ_{1,2} plane permits to clearly isolate the dominant asymptotic term. As a matter of fact, after the deformation onto SDP *S*_{1,2}, the interlaced branch point at θ_{1} = + θ_{2} (θ_{2} = + θ_{1}), 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 θ_{1,2} = *π*. The same is not always true for the branch at θ_{1} = − *α*_{2} (θ_{2} = − *α*_{1}). Depending on the direction of incidence of the illuminating plane wave, the branch θ_{1} = − *α*_{2} may occur to the left of the saddle point, i.e., if ℜ*e*{*α*_{2}} > . In such a case, in the deformation of Γ_{1} onto SDP *S*_{1} 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 θ_{2} integration may arise from the branch point θ_{2} = − *α*_{1} if ℜ*e*{*α*_{1}} > .

[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 *x*_{1}, *x*_{2} 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 *x*_{3} ≠ 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.