The diffraction of electromagnetic waves by rectangular step-like discontinuities on various ground surfaces is a critical area of research in radar cross-section analysis and numerous microwave engineering applications. An illustrative example of such step discontinuity issues can be found in Sommer and Pathak’s work [1], in which the authors analyzed the diffraction problem caused by metallic tapes on paneled compact range reflectors using the geometrical theory of diffraction. Johansen [2] addressed another step discontinuity problem in the context of surface wave illumination, which involved a reactive step created by the conjunction of two reactive half-planes. The majority of later research on this phenomenon focused on step discontinuity geometries under plane wave illumination, with significant contributions made by Buyukaksoy and Birbir [3,4], and Volakis and Ricoy [5]. Their studies explored diffraction from geometries comprising two half-planes of identical impedance types joined by steps featuring differing impedance and reactive properties. These investigations typically involved reducing the problem to the solution of a modified Wiener–Hopf equation of the second kind and then solving it numerically for various surface impedances and step heights.

However, in this context, considering line sources at finite distances from scatterers offers a more realistic approach than plane waves. Notable recent studies in this area have been conducted by Ayub et al. [6] and Ahmed [7], who examined high-frequency line source diffraction in similar geometries and solved diffraction problems related to perfect electromagnetic conductor (PEMC) steps. The aforementioned studies predominantly considered specific geometries, such as perpendicular junctions of two asymmetric parallel half-planes (single step discontinuity). However, when a scatterer includes two perpendicular joints of asymmetric halfplanes along with an intervening strip (double step discontinuity)—termed “successive step discontinuities”—the complexity increases significantly. In this context, a study conducted by Dogan et al. [8] can be cited as unique since it carried out the complete investigation of diffraction by step discontinuities, as depicted in Fig. 1, with all surfaces assumed to be perfectly conducting and illuminated by a time-harmonic infinitely long line source that is parallel to the z-axis. Furthermore, to solve the boundary-value problem, the authors of [8] applied the Fourier integral transform to the corresponding reduced wave equation. Subsequently, by accounting for the perfectly electric-conducting boundary conditions in the transform domain, the problem was addressed by employing a modified Wiener–Hopf equation of the third kind. The solution to this equation was obtained using the classical procedure—two equations were determined for the two separate functions, one regular in the upper complex half-plane and the other in the lower. These two equations yielded coupled Fredholm integral equations of the second kind, which can be solved approximately by iterations. Notably, the solutions of these integral equations involved a branch-cut integral whose evaluation could only be realized through numerical computation.

The current work investigates the diffraction problem while maintaining the same geometry depicted in Fig. 1 by employing a novel approach to reach the solution for the aforementioned modified Wiener–Hopf equation of the third kind. For this purpose, the modified Wiener–Hopf equation in question is reduced once again to obtain a pair of coupled Fredholm integral equations of the second kind. However, this time, the factorization process is conducted in such a way that the two separate functions to be solved are both regular in the upper complex half-plane. This novel approach, which may be considered unconventional compared to the classical procedure, enables the analytical evaluation of the branch-cut integral, which could only be evaluated numerically in [8]. Consequently, the pair of coupled Fredholm integral equations of the second kind is approximately solved by iterations, with the iterative solution involving five sets of infinite constants satisfying five systems of infinite linear algebraic equations. Notably, throughout this paper, the time dependence of the waves is suppressed as e^−iωt, where ω denotes the angular frequency.

The remainder of this paper is structured as follows. In Section II, the general solution to the problem is presented by accounting for the relationship between the modified Wiener–Hopf equation of the third kind and the geometry presented in Fig. 1. In Section III, the analysis of the diffracted field for y > y₀ is realized for high frequencies based on the steepest descent path method. In Section IV, the computational results for the amplitude of the diffracted field with respect to different geometric values of successive step discontinuities are presented. Finally, concluding remarks pertaining to the proposed solution for the problem are delivered in Section V.

Based on the procedure presented in Section 2 of [8], it can be established that the solution to the diffraction problem pertaining to the geometry depicted in Fig. 1 can be arrived at using the modified Wiener–Hopf equation of the third kind, expressed as follows:

with

and

Notably, in Eq. (1)G₋(α,d), G₁(α,d), and G₊(α,d) can be formulated as:

respectively. Furthermore, the total field is defined as follows:

and C(α) stands for:

Here, K(α) refers to the well-known square root function K(α)=k2-α2, which can be defined in the complex α-plane cut as shown in Fig. 2, such that K(0)=k. Furthermore, the coefficients appearing in Eq. (1) can be defined as follows:

with

and

with

for n=1,2,…, which satisfy the sets of equations presented below. Notably, the derivation of these equations is detailed in Eqs. (12)–(16) and Eqs. (19)–(23) in [8].

and

Finally, α_n’s, β_n’s, K_n’s, and γ_n’s can be expressed as:

and

for n=1,2,…

The first step in solving the modified Wiener–Hopf equation of the third kind, as expressed in Eq. (1), is factorization of the kernel functions M(α) and N(α) using the methods described in [9, 10]. However, in this paper, we follow a unique approach—we apply “two-step factorization” to reach the solution of the two different functions that are regular in the upper half-plane. Notably, this approach has not been employed in previous works since the classical approach has always been to determine the solution of a function regular in the upper half-plane and another one regular in the lower half-plane. In other words, the approach proposed in the current paper avoids the problem of numerically evaluating a crucial branch-cut integral that appears in the Wiener–Hopf procedure. By multiplying Eq. (1) by e^−iαlN₋(α) and defining:

We get:

After decomposing the terms on the right-hand side and the first term on the left-hand side of Eq. (19), and by accounting for the analytical continuation principle followed by Liouville’s theorem, the following equation was obtained:

where

and

Changes in variables τ=kcost, x₀ − l = ρ₁ cosφ₁, and y₀ − d=ρ₁ sinφ₁ allowed us to evaluate the above integral J₁(α) using the saddle-point technique, as follows:

where H(.) is the Heaviside step function, which became zero for Re{arccos(αk)}<ϕ1. Furthermore, applying a similar procedure of multiplying each term in Eq. (1) by M₋(α) yielded the following equation:

where x₀ = ρ₂ cosφ₂ and y₀ − d = ρ₂ sinφ₂. In this way, the solution to the problem was reduced to the solution of a Fredholm integral equation of the second type. This integral equation was solved using an iterative procedure that produced a Neumann series expansion of the solutions. When kl was large, the terms outside the integrals on the right side of Eq. (20) and Eq. (24) have the first-order solutions. The second-order solutions were then obtained by replacing the unknown functions appearing in the integrands with their first-order approximations. Subsequently, the next higher-order terms were obtained by following the same procedure [11]. Hence, the solutions for P(α) and Q(α) can be expressed as follows:

where the first-order solutions can be determined by the following equation:

and

Notably, in Eqs. (26) and (27), U_j and V_j(α) (j=1,2) stand for:

and

respectively. Next, the second-order solutions were determined based on the following integral equations:

and

These integrals were then evaluated asymptotically following the procedure proposed in [12, 13]. It was obvious that the solutions involved the unknown constants f_n, g_n, h_n, p_n, and q_n for n=1,2,… Therefore, to determine these unknown constants, we first substituted P^(1,2)(α) and Q^(1,2)(α) into Eq. (25), and then accounted for P(α) and Q(α) to find G₁(α,d) and G₊(α,d) using Eqs. (18) and (22). Finally, the five infinite systems of linear algebraic equations, which can easily be solved numerically, were determined by substituting G₁(α,d) and G₊(α,d) into Eqs. (13)–(15), and by accounting for Eqs. (40d)–(40e) in [8], as follows:

and

The field u₁(x,y), given by Eq. (7), can be expressed as follows:

As explained in detail in Section 2 of [8], the spectral coefficient F(α,y) appearing in Eq. (34) can be presented using the following equation:

By substituting the expressions of G₁(α,d) and G₊(α,d) into Eq. (35), while also considering the changes in variables

and

the integral in Eq. (34) was evaluated using the saddle-point technique by solving the following equation:

with

and

The numerical analysis of the diffracted field based on Eq. (38) is presented in the following section.

This section presents the numerical results, thereby providing an idea of the variations of the diffracted field, as obtained in Section III, with respect to the observation angle φ = arctan(y/x) ∈ [15°, 165°] for different parameters of the step discontinuities, such as width l, height c, and d (see Fig. 1). Notably, for all applications, the values of k=2π/λ, x₀=2λ, and y₀=λ>d were taken into account, with λ being the wavelength related to the time harmonic line source. Furthermore, the magnitude of the diffracted field was assumed to be composed of the contributions of the first-and second-order diffractions in all figures. In Fig. 3, variation of the diffracted field magnitude with respect to different values of step length l is presented, showing that the oscillation in diffraction decreases with increasing values of l, as expected. The variation of the diffracted field for different values of the d/c ratio is depicted in Fig. 4, showing that the field’s magnitude increases with an increase in this ratio, indicating a direct proportion. Finally, Fig. 5 presents the relationship between the magnitude of the diffracted field and the step height d, showing that the former decreases with a decline in the value of d. Notably, all the results presented in this section were consistent with those reported in [8]. However, by implementing the proposed approach, we obtained smoother solutions, as depicted in Fig. 6. The figure displays a jump for the results obtained in [8], along with small differences for the lower observation angles, which can be attributed to differences in the factorization process and, accordingly, getting rid of the complicated branch-cut integral, which could only have been calculated numerically with relatively lower accuracy.

In this paper, the scattering of electromagnetic waves excited by a line source, which is caused by perfectly conducting successive step discontinuities, is rigorously analyzed by employing the Wiener–Hopf technique, whose solution was reduced to that of two coupled Fredholm integral equations of the second kind. These integral equations were solved asymptotically, based on which the approximate explicit expression of the diffracted field was obtained. In the process, an unconventional novel approach was formulated based on the functions that are all regular in the upper half-plane in the Wiener–Hopf decomposition procedure. Notably, this new method offers the ability to analytically evaluate the branch-cut integral, which could only be evaluated numerically in [8]. When comparing these two approaches, it was observed that the analytical evaluation conducted in the current study achieved smoother results and avoided irrelevant discontinuities. Moreover, the effects of various parameters, such as the height and width of the step discontinuities, on the diffraction phenomena were also investigated in this study. Overall, the efficiency of the proposed approach proved to be very useful, allowing researchers to avoid similar complexities encountered in other relevant applications.