1. Introduction

The canonical problem of plane wave scattering from a wedge with anisotropic impedance is of great importance in many electromagnetic issues, such as radar cross section (RCS) predictions, electromagnetic wave propagation, and antennas design. The anisotropic impedance boundary conditions (IBC’s) are the most effective way to describe the EM scattering problems of surfaces coated with a thin anisotropic layer. Previous solutions to this canonical problem were based either on numerical approaches [1–3] or on approximate analytical techniques based on Maliuzhinets method in the framework of the UTD [4–14]. In the case of normal to the edge incidence, the approximate analytical solutions have been studied and proposed in several literatures [5,6]. While in the case of oblique incidence, the problem becomes much more complicated due to the difficulty of the solution for the coupled equations generated by the boundary conditions. By resorting to perturbation approach [8,9], the UTD solutions in the case of oblique incidence were derived for the case of small deviation from normal to the edge incidence (normal incidence) or grazing to the edge incidence (grazing incidence). According to [8,9], the UTD solution is valid only in the angle region deviating at most ${40}^o$ from normal incidence $({\beta }_0\ge {40}^o)$ and ${15}^o$from the grazing incidence $({\beta }_0\le {15}^o)$. Here, ${\beta }_0$is a measure of the incident skewness with respect to the edge of the wedge. Therefore, the perturbation approach has the limits of applicability and could not provide satisfactory solutions for the case of large skewness.

The perturbation approach employs the asymptotic iterative method to solve the coupled difference equations and the spectral function can be approximated by the first two leading terms. For the case of large deviations, an accurate solution could hardly be obtained by one or two iteration steps. On the other hand, more iterations are extremely difficult to be implemented with asymptotic iterative method. Therefore, the total fields calculated with the perturbation approach could not satisfy the boundary conditions on the wedge faces strictly when the incident wave deviates a lot from the normal incidence or grazing incidence. In our previous work, a hybrid MM-PO technique in [15] has provided numerical solutions for this canonical problem at an arbitrary skew incidence. However, it is still worthwhile to obtain the valid UTD solutions at arbitrary skew incidences over the whole angle space$(0^o<{\beta }_0\le {90}^o).$ To overcome the limitation of the perturbation technique, the restriction of the boundary conditions on the wedge faces need to be enhanced during the solution of the coupled equations. Different to the previous Maliuzhinets method, we start from the coupled integral equations before they are converted into the coupled difference equations as the classic Maliuzhinets method. The key of the solution is to figure out the approximate expression of the spectral functions in the coupled integral equations.

The spectral functions are then numerically derived by resorting to a proposed numerical matching method (NMM). The approximate expression of the spectral function in the Sommerfeld integral representation of the longitudinal components of the EM field is introduced by a suitable expansion including a series of the spectrum and the skew incident angle. The expansion of the spectral function takes the expression in perturbation technique as a reference. Then, the unknown expanded coefficients of the spectral functions are determined numerically by solving a system of algebraic equations constructed from the coupled integral equations, after choosing the finite numerical matching regions on the wedge faces and setting a proper Sommerfeld numerical integration path. For efficiency consideration, the AWE technique is employed to construct the spectral function with the data set of the sampled angles by for an arbitrary skew incidence in the whole angle space. Once obtaining the spectral function, the integral representations of the total fields could be asymptotically evaluated by residue theorem and steepest descent procedure (SDP) to obtain the UTD solution [8,9]. Finally the satisfactory UTD solutions are validated through numerical results by comparing with available examples and methods. And it is worth pointing out that the solutions proposed yield a uniform behavior of the total fields at the shadow boundaries.

The paper is organized as follows. Section 2 provides the outline of the solution. Then the spectral function for one arbitrary given incidence is worked out with the numerical matching method in Section 3. Section 4 gives the fast and efficient construction of the spectral functions over the whole angle space by employing AWE techniques. Section 5 validates the UTD solutions through numerical results. Finally, a conclusion is presented in Section 6.

2. Outline of the solution

The three-dimensional (3-D) geometry of wedge scattering problem is shown in Fig. 1 . An arbitrary polarization harmonic plane wave impinges on the edge from the direction determined by the two angles: ${\beta }_0$ and ${\varphi }_0$. The longitudinal components of the incident field can be expressed as

 

Fig. 1 Scattering of a plane wave by a wedge at skew incidence.

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An appropriate representation of the total field longitudinal components solution is expressed as the form of Sommerfeld integrals [8,9]:

 

Fig. 2 Integration paths in the $\alpha$-complex plane.

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The key of the UTD solution is to figure out the approximate expression of the spectral function at an arbitrary skew incidence ${\beta }_0$ $(0^o<{\beta }_0\le {90}^o).$ A NMM method is proposed to derive the spectral functions numerically. As shown in Fig. 3 , firstly, we start from the boundary conditions on the wedge faces to obtain the coupled integral equations, which could be based on normal to the edge incidence case or the grazing to the edge incidence case. Then the spectral function is expanded properly by a product of Maliuzhinets special function, meromorphic function and the chosen series about spectrum and the skew incident angle with constant coefficients. The series expansion of the spectral function refers to the expression in the perturbation technique. The expanded spectral function is then substituted into the coupled integral equation to construct a system of algebraic equations with the constant coefficients of the spectral function as unknowns. For the purpose of numerically solving the algebraic equation, the numerical matching procedure follows by choosing the finite numerical matching regions on the wedge faces and setting a proper Sommerfeld numerical integration path. By solving the algebraic equations, the constant coefficients of the spectral functions (viz. the unknowns of the algebraic equations) are determined. Therefore, given one arbitrary incidence, the spectral function could be obtained with the NMM method. Once obtaining the spectral function, the UTD solutions could be worked out by the same procedure as in the previous literatures [8,9].

 

Fig. 3 Outline of the solution.

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For efficiency consideration, the asymptotic waveform evaluation (AWE) technique is employed for the interpolation of the spectral function at any other skew incidence in the whole angle space$0^o<{\beta }_0\le {90}^o.$ When ${25}^o\le {\beta }_0\le {90}^o,$the coupled integral equations are based on the normal incidence case. When $0^o<{\beta }_0\le {30}^o,$ the coupled integral equations are based on the grazing incidence case. In the overlapped angle range${25}^o\le {\beta }_0\le {30}^o,$ solutions provided based on two different coupled integral equations will be examined for the uniformity.

Finally, the UTD solution is examined and validated. The numerical results in this paper are compared with available examples in the UTD literatures. For the cases when a valid UTD solution is not found in the previous publishes, the UTD solution presented in this paper is verified through the numerical results solved by the code of the MM-PO hybrid method in our previous research [15].

3. Spectral function for one arbitrary given incidence with NMM

3.1 Based on normal to the edge incidence case

Given an arbitrary incidence of ${\beta }_0$in the range ${25}^o\le {\beta }_0\le {90}^o,$ the spectral functions based on normal incidence case are determined with the application of the numerical matching method in the following. We consider the impedance tensors of the wedge surface have their principal anisotropy axes along directions parallel or perpendicular to the edge, and the two surface impedances are represented by the tensors as${\overline{\overline{Z}}}_{0,n}=\left[\begin{array}{cc}{Z}_{\rho \rho }^{0,n}& 0\\ 0& Z_{zz}^{0,n}\end{array}\right],$with $\mathrm{Re}[Z_{\rho \rho }^{0,n},Z_{zz}^{0,n}]\ge 0$ to meet the restriction of passive anisotropic impedance. Accordingly, the IBC’s can be expressed in the cylindrical coordinate as

We substitute the Sommerfeld integral expressions of $E_z$and ${\eta }_0{H}_z$ in Eq. (2) into the boundary condition (4), then two coupled integral equations are obtained.

To solve the approximate expression of the spectral function from Eq. (5), it is assumed that the unknown spectral function $S_{e,h}(\alpha )$ could be expanded as the product of Maliuzhinets special function, meromorphic function and the chosen series of $\tan (\alpha /8)\cos {\beta }_0$with constant coefficient $a_{e,h}^m$,

Here, during the solution of expanded coefficients in Eq. (7), the infinite series need to be truncated with finite terms. So the spectral function $S_{e,h}(\alpha )$ is approximated by the major orders of the series. The first five terms are taken into account and the convergence of the finite series has been verified by including an increasing number of terms. It is found that the effect of the left terms of higher orders is negligible and it could be demonstrated in the following study.

According to the framework of the UTD, the expansion of the spectral function with the first five terms is rewritten as follows,

By letting $a_{e,h}^0=1/{\Psi }_{e,h}(n\pi /2-{\varphi }_0),$ the first leading terms of the series are taken as the spectral function coefficient at normal incidence, which are the same as in the perturbation method [8]. Thus, eight unknown coefficients of the spectral functions, viz. $a_{e,h}^m$,$m=1,2,3,4$, remain to be determined.

To solve the eight constant coefficients numerically, eight algebraic equations is going be constructed from the coupled integral equations with the application of the numerical matching method. Firstly, the numerical procedure needs to choose the finite numerical matching regions properly to take into account the diffraction effect as much as possible. In our investigation, two numerical matching regions on each side of the wedge faces are chosen and illustrated in Fig. 4 as $(0,{\rho }_1)$ and $({\rho }_1,{\rho }_2)$. Performing an integral over the two numerical matching regions, the boundary conditions are satisfied on each region in an average sense. The surface field on the wedge faces around the edge contributes greatly to the diffraction field, so the numerical matching regions are chosen close to the edge of the wedge. In the following, we choose $k_t{\rho }_1=0.\text{1}$and $k_t{\rho }_2=0.2$. The choice has been tested through many numerical simulations and the convergence of the numerical results is guaranteed.

 

Fig. 4 Numerical matching regions.

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Secondly, the numerical matching procedure includes setting a Sommerfeld numerical integration path where the infinite integration can be easily truncated and computed. As shown in Fig. 5 , A SIP ${\gamma }_0$ is defined to evaluate the integral numerically. The integration path is set that when $\mathrm{Re}(\alpha )$ is changing, $|\mathrm{Im}(\alpha )|$ remains the same, and when $|\mathrm{Im}(\alpha )|$ is changing, $\mathrm{Re}(\alpha )$ keeps unchanging. According to the previous analysis, the real part of $\alpha$ satisfies the condition$\left|\mathrm{Re}(\alpha )\right|\le n\pi /2$, so the key to implement the infinite integral numerically is to identify the upper limit and lower limit of $|\mathrm{Im}(\alpha )|$ properly for a convergent result. The detailed choice of the truncation points on the integration path will be demonstrated in the following computation.

 

Fig. 5 Numerical integration path ${\gamma }_0$ (dashed lines).

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Since the Sommerfeld numerical integral path and the numerical matching regions are properly chosen, the eight algebraic equations could be written in the form as follows,

For the purpose of numerical evaluation, the integration path ${\gamma }_0$ in Fig. 5 has to be truncated and the convergence of the truncated integrations should be examined. Considering the incident angle of ${\beta }_0={30}^o$, the integration of $A{1}_h^0(1)$ is taken as an example for the test of convergence in integration evaluation and the other parameters in the double integrals are arbitrarily set as $Z_{zz}^{0,n}/{\eta }_0=1,Z_{\rho \rho }^{0,n}/{\eta }_0=2$ and ${\varphi }_0={45}^{\circ }.$ In the numerical integration, the integration steps along SIP and the wedge faces in Eq. (11) are both chosen as 0.001. Table 1 shows the numerical results of $A{1}_h^0(1)$when the truncation point on the numerical integration path ${\gamma }_0$ is different. First the lower limit of $|\mathrm{Im}(\alpha )|$is fixed and the upper limit changes slowly to test the convergence. Then, the upper limit is fixed and the lower limit changes. It is observed that the integration is convergent when the upper limit of $|\mathrm{Im}(\alpha )|$ is set as 10 and the lower limit of $|\mathrm{Im}(\alpha )|$ is set as 0.03. The upper and lower limit is applicable to the other double integrals, which are calculated with the same condition as $A{1}_h^0(1)$.The convergence of the numerical integration in Table 1 shows that the choice of the lower limit has already included the contribution of the most evanescent modes. All the 80 double integrals in Eq. (10) have been calculated in the same way.

Table 1. Numerical integration of $A{1}_h^0(1)$

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It should be pointed out that as the incident angle ${\varphi }_0$ or the impedance tensors ${\overline{\overline{Z}}}_{0,n}$ changes, the 80 double integrals in Eq. (10) vary slowly. Meanwhile, the variation trends of all the 80 integrals in Eq. (10) keeps the same when the incident angle ${\varphi }_0$ or the impedance tensors ${\overline{\overline{Z}}}_{0,n}$changes, which means the solution of Eq. (10) is independent of the incident angle ${\varphi }_0$ and the impedance tensors. The detail study of this independence has been performed in our research.

Based on the above analysis, the unknown expanded coefficients of the spectral functions $a_{e,h}^m$ could be obtained for the arbitrary given incident angle ${\beta }_0$ by solving the algebraic equations. For the purpose of interpolations in the next section, the data set of the sampled incident angles ${\beta }_k$ is prepared. It is found that the coefficients of the spectral functions vary slowly when the skew incident angles ${\beta }_0$ deviate from the normal incidence not too much. However, the coefficients change rapidly when the incidence is approaching the boundary incidence of the valid angle range based on normal to the edge incidence case. So it is logical to set large sampling steps in the angle range where the coefficients varies smoothly but small sampling steps in the angle range where the coefficients varies quickly. The sampled incident angels are chosen ${\beta }_0={70}^o,{50}^o,{40}^o,{25}^o.$The coefficients of the spectral functions at the sampled incidence ${\beta }_k$ are illustrated in Table 2 . It could be noted in the table that the first two terms contributes mostly while the value of the high order terms is very small, as expected.

Table 2. Coefficients of spectral functions based on normal to the edge incidence case

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3.2 Based on grazing to the edge incidence case

Given an arbitrary incidence in $0^o<{\beta }_0\le {30}^o$, the spectral functions based on grazing to the edge incidence case are determined with the application of the numerical matching method.

Introducing two new components $P$and $T$, with $P={\eta }_0{H}_z-j{E}_z,T=E_z-j{\eta }_0{H}_z.$Since $P$and$T$are the linear combination of $E_z$ and ${\eta }_0{H}_z$, they also satisfy all the constraints which $E_z$ and ${\eta }_0{H}_z$do. So the solutions of $P$ and $T$should be in the form of the Sommerfeld integral as

To solve the approximate expression of the spectral function from Eq. (13), it is assumed that the unknown spectral function $S_{P,T}(\alpha )$ could be expanded as the series of $\tan (\alpha /8)$$\sin ({\beta }_0)$with constant coefficients$a_{P,T}^m$,

Similar to the case based on the normal to the edge incidence, the spectral function $S_{P,T}(\alpha )$ can be worked out with the NMM. The coefficients of the spectral functions at the sampled incidence ${\beta }_k$ are illustrated in Table 3 .

Table 3. Coefficient of spectral functions based on grazing to the edge incidence case

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4. Spectral functions for arbitrary skew incidence with AWE technique

In this part, the AWE technique is applied to construct the spectral functions for an arbitrary incidence in the whole angle space. The asymptotic waveform evaluation technique has already been successfully used in various electromagnetic problems [16]. On the basis of AWE technique, the spectral function at an arbitrary angle ${\beta }_0$ is expanded by a product of Maliuzhinets special function, meromorphic function and a rational function achieved via Pade approximation. In the representation of the spectral functions, the rational function as a factor of the spectral function is much more general compared with the series expression in Eqs. (7) or (14). For the case based on normal to the edge incidence when ${25}^o\le {\beta }_0\le {90}^o,$ the spectral function with Pade rational function representation could be written as follows,

According to the AWE, the coefficients $c_{e,h}^i(i=0,1,2)$ and $d_{e,h}^j(j=1,2)$ in the Pade representation are determined by the following matrix equation,

By solving the matrix equation, the expanded coefficients is determined and illustrated in Table 5.

Then the spectral function could be written as,

While in the case of based on grazing to the edge incidence, the expansion of the spectral function could be written as:

Table 4. Coefficients of the spectral function with AWE

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Then the spectral functions can be written as,

5. UTD solution and numerical results

Once obtaining the spectral functions, by applying the residue theorem, the Sommerfeld integration can be deformed into the collective contribution of: (1) the residues of the poles of $S_{e,h}(\alpha )$, which account for the geometrical optical (GO) field and the surface wave fields. The latter will not be discussed here because of its minor contribution when talking about the fields in the far zone. (2) the two steepest descent path (SDP) integrals through the saddle points ($\pm \pi$) (Fig. 2), which account for the diffraction field by the edge. The diffraction coefficients could be obtained by the same procedure as in the previous literatures [9]. In the following examples, we choose a right-angled wedge (n = 3/2), due to the fact that in this configuration, the Maliuzhinets function is known in a closed form. Meanwhile, in all the numerical examples below, the independent factor $\exp (-jkz\cos {\beta }_0)$ is suppressed, and all the fields are evaluated at a normalized distance$k_t\rho =10.$

To validate the proposed method, the numerical results are illustrated and compared with other methods. For the cases that based on normal incidence, some available examples in the UTD literatures could be found for comparison in 5.1. However, for the cases when a valid UTD solution is difficult to found, the numerical results are verified through the MM-PO hybrid method in our previous research [15] in 5.2. In the overlapped angle range${25}^o\le {\beta }_0\le {30}^o,$ solutions provided based on two different coupled integral equations are examined for the uniformity in 5.3. Finally, for the cases that the spectral functions are interpolated through AWE technique, the UTD solutions are validated by the MM-PO hybrid method for arbitrary skew incidences in 5.4.

5.1 Cases based on normal incidence ${25}^o\le {\beta }_0\le {90}^o$

In this part, we take ${\beta }_0$as ${30}^o,{50}^o,{70}^o,{90}^o$ respectively for example. The results have been compared with those obtained by reference solutions presented in the literatures, including the probabilistic random walk method [3], the parabolic equation method [2], and the Maliuzhinets method [9,10].

As shown in Fig. 6(a) and 6(b), the amplitude of the copular and cross-polar components of the total fields with respect to the observation points is plotted respectively. When ${\beta }_0={50}^o,{70}^o,$ the data calculated by the proposed solution (solid lines) are compared with the parabolic equation method in [2] (marked by * and o) respectively. When ${\beta }_0={30}^o,$ the data is compared with MM-PO method (marked by + ).

 

Fig. 6 Validation based on normal incidence: copolar (a) and cross-polar (b). The impedance tensors are $Z_{zz}^{0,n}/{\eta }_0=1,$$Z_{\rho \rho }^{0,n}/{\eta }_0=2.$ The incident plane wave is TM polarized $E_{{\beta }_0}^i=1,E_{{\varphi }_0}^i=0,$ and the geometrical parameters: ${\varphi }_0={45}^{\circ }$ and ${\beta }_0={30}^o,{50}^o,{70}^o,{90}^o.$ The proposed solution: copolar and cross-polar components, respectively (solid lines); parabolic equation method in [2]: ${\beta }_0={70}^o$ (marked by *),${\beta }_0={50}^o$ (marked by o); the MM-PO method when${\beta }_0={30}^o$ (marked by + ).

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In Fig. 7 , the amplitude variations of the total longitudinal field and the diffraction field are shown with respect to the observation points. The numerical results calculated by the proposed solution: the total longitudinal field (solid line) and the diffraction field (dashed line) are compared with the solutions in [3] (marked by * and o), respectively. Good agreements are obtained for the comparisons.

 

Fig. 7 Validation based on normal incidence. The impedance tensors are $Z_{zz}^{0,n}/{\eta }_0=\text{4}\text{.1+3}\text{.5i},Z_{\rho \rho }^{0,n}/{\eta }_0=\text{4}\text{.1-3}\text{.5i}\text{.}$ The incident plane wave is TE polarized $E_{{\beta }_0}^i=\text{0},E_{{\varphi }_0}^i=\text{1,}$ and the geometrical parameters: ${\varphi }_0={45}^{\circ }$ and ${\beta }_0={\text{50}}^{\circ }.$ The proposed solution: the total fields (solid line) and its diffracted component (dashed line); the probabilistic random walk method solution in [3]: the total fields (marked by *) and its diffracted component (marked by o).

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In Figs. 8(a) and 8(b), the amplitude variations of the total longitudinal field (${\eta }_0{H}_z$) and ($E_z$) is shown with respect to the observation points, respectively. The numerical results calculated by the proposed solution (solid lines) for ${\beta }_0={50}^o$ and ${\beta }_0={70}^o$are compared with the UTD solutions found in the published papers [10] (marked by o and *). For the case of ${\beta }_0={30}^o$, the numerical results are compared with the MM-PO method in [15] (marked by + ). It is observed that very good agreements are obtained for the comparisons.

 

Fig. 8 Validation based on normal incidence: copolar (a) and cross-polar (b). The impedance tensors are$Z_{zz}^0/{\eta }_0=Z_{\rho \rho }^0/{\eta }_0=j/2,$ $Z_{zz}^n/{\eta }_0=Z_{\rho \rho }^n/{\eta }_0=0.$ The incident plane wave is TE polarized $E_{{\beta }_0}^i=0,E_{{\varphi }_0}^i=1$, and the geometrical parameters: ${\varphi }_0={45}^{\circ }$ and ${\beta }_0={30}^o,{50}^o,{70}^o,{90}^o.$ The proposed solution: copolar and cross-polar components, respectively (solid lines); the solution in [10]: ${\beta }_0={70}^o$ (marked by *),${\beta }_0={50}^o$ (marked by o); the MM-PO method in [15] ${\beta }_0={30}^o$ (marked by + ).

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In Fig. 9 , the numerical results calculated by the proposed solution (solid lines) for ${\beta }_0={50}^o$and ${\beta }_0={70}^o$are compared with the uniform asymptotic approximation of the exact solution in [9] (marked by o and *). For the case of ${\beta }_0={30}^o,$ the numerical results are compared with the MM-PO method (marked by +). Here, an anisotropic impedance wedge with $Z_{zz}^0/{\eta }_0=Z_{zz}^n/{\eta }_0=0,$$Z_{\rho \rho }^0/{\eta }_0=(1+j)/2,$$Z_{\rho \rho }^n/{\eta }_0=(1-j)/2$ is considered and the cross-polar field of the total field longitudinal components ($E_z$) does not exist. So only the amplitude of the longitudinal components of the total field (${\eta }_0{H}_z$) solved by the numerical matching method are validated by the reference solutions.

 

Fig. 9 Validation based on normal incidence. The impedance tensors are $Z_{zz}^0/{\eta }_0=Z_{zz}^n/{\eta }_0=0,$$Z_{\rho \rho }^0/{\eta }_0=(1+j)/2,$ $Z_{\rho \rho }^n/{\eta }_0=(1-j)/2.$ The incident plane wave is TE polarized $E_{{\beta }_0}^i=\text{0},E_{{\varphi }_0}^i=\text{1,}$ and the geometrical parameters: ${\varphi }_0={\text{30}}^{\circ }$ and ${\beta }_0={30}^o,{50}^o,{70}^o,{90}^o.$ The proposed solution: copolar components, respectively (solid lines); UTD solutions in the published papers [9]: ${\beta }_0={70}^o$ (marked by *),${\beta }_0={50}^o$ (marked by o); the MM-PO method in [15] ${\beta }_0={30}^o$ (marked by +).

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The numerical results have shown that the proposed UTD solution in this paper provides accurate results beyond the standard limits of applicability of perturbative methods, viz.${\beta }_0>{50}^o$.

5.2 Cases based on grazing to the edge incidence ${\text{0}}^o<{\beta }_0\le {30}^o$

For the cases based on grazing to the edge incidence, a valid UTD solution is not found in the previous publishes. The numerical results calculated by the proposed solution (solid lines) are validated by the MM-PO method (marked by + ), as shown in Fig. 10 and Fig. 11 (Two different examples are chosen).

 

Fig. 10 Validation based on grazing to the edge incidence: copolar (a) and cross-polar (b) . The impedance tensors are $Z_{zz}^{0,n}/{\eta }_0=1,$$Z_{\rho \rho }^{0,n}/{\eta }_0=2.$ The incident plane wave is TE polarized $E_{{\beta }_0}^i=\text{0},E_{{\varphi }_0}^i=\text{1,}$and the geometrical parameters: ${\varphi }_0={45}^{\circ }$ and ${\beta }_0=1^o,5^o,{15}^o,{20}^o,{25}^o,{30}^o$. The proposed solution: based on grazing to the edge incidence (solid lines); the MM-PO method (marked by +).

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Fig. 11 Validation based on based on grazing to the edge incidence: copolar (a) and cross-polar (b). The impedance tensors are $Z_{zz}^0/{\eta }_0=1,$$Z_{\rho \rho }^0/{\eta }_0=0,$$Z_{zz}^n/{\eta }_0=Z_{\rho \rho }^n/{\eta }_0=j/2.$ The incident plane wave is TM polarized $E_{{\beta }_0}^i=\text{1},E_{{\varphi }_0}^i=\text{0,}$ and the geometrical parameters: ${\varphi }_0={\text{30}}^{\circ }$ and ${\beta }_0=1^o,5^o,{15}^o,{20}^o,{25}^o,{30}^o.$This solution: based on grazing to the edge incidence (solid lines); the MM-PO method (marked by +).

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The numerical results have shown that the UTD solution proposed in this paper provides accurate results far beyond the standard limits of applicability of perturbative methods, viz.${\beta }_0\le {15}^o.$

5.3 Examination for uniformity of the overlapped angle range (${25}^o\le {\beta }_0\le {30}^o$)

In the overlapped angle range${25}^o\le {\beta }_0\le {30}^o,$ solutions could be provided based on two different coupled integral equations. In this part, the cases of ${\beta }_0={25}^o$and${\beta }_0={30}^o$ are taken as examples and the agreement based on two different integral equations is examined. Two examples are demonstrated in Fig. 12 and Fig. 13 respectively.

 

Fig. 12 Examination based on two integral equations. The impedance tensors are $Z_{zz}^{0,n}/{\eta }_0=1,$$Z_{\rho \rho }^{0,n}/{\eta }_0=2.$ The incident plane wave is TE polarized $E_{{\beta }_0}^i=0,E_{{\varphi }_0}^i=1,$and the geometrical parameters: ${\varphi }_0={\text{45}}^{\circ }$ and ${\beta }_0={25}^o,{30}^o.$ The proposed solution: based on the normal incidence case (solid lines), based on grazing to the edge incidence case (dashed lines); the MM-PO method (marked by +).

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Fig. 13 Examination based on two integral equations. The impedance tensors are $Z_{zz}^0/{\eta }_0=1,$ $Z_{\rho \rho }^0/{\eta }_0=0,$$Z_{zz}^n/{\eta }_0=Z_{\rho \rho }^n/{\eta }_0=j/2.$ The incident plane wave is TM polarized $E_{{\beta }_0}^i=\text{1},E_{{\varphi }_0}^i=\text{0}\text{.}$ and the geometrical parameters: ${\varphi }_0={\text{30}}^{\circ }$ and ${\beta }_0={25}^o,{30}^o.$ The proposed solution: based on the normal incidence case (solid lines), based on grazing to the edge incidence case (dashed lines); the MM-PO method (marked by +).

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It is observed that the numerical results based on the normal incidence case (solid lines) agree well with the one based on grazing to the edge incidence case (dashed lines). Besides, validation of the solutions is also presented by comparing with the MM-PO method (marked by +). It is found that the UTD solution based on the two different integral equations match well with each other for the TE and TM polarization both.

5.4 Validation of the solution derived through AWE technique

For the case of ${\beta }_0={35}^o,{65}^o,$ the UTD solution could be achieved with AWE technique based on the data of normal incidence case, as shown in Fig. 14 .

 

Fig. 14 Validation based on AWE technique. The impedance tensors are $Z_{zz}^{0,n}/{\eta }_0=1,Z_{\rho \rho }^{0,n}/{\eta }_0=2.$The incident plane wave is TE polarized $E_{{\beta }_0}^i=0,E_{{\varphi }_0}^i=1,$ and the geometrical parameters: ${\varphi }_0={\text{45}}^{\circ }$ and ${\beta }_0={\text{35}}^o,{\text{65}}^o.$ The proposed solution: by AWE (dashed lines); the MM-PO method (marked by +).

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For the case of ${\beta }_0={22}^o,{28}^o,$ the UTD solution could be achieved with AWE technique based on the data of grazing to the edge incidence case, as shown in Fig. 15 . The numerical results obtained by AWE (dashed lines) are validated by the results with MM-PO method (marked by + ) in Fig. 14 for ${\beta }_0={35}^o,{65}^o$and Fig. 15 for ${\beta }_0={22}^o,{28}^o.$

 

Fig. 15 Validation based two different interpolation techniques. The impedance tensors are $Z_{zz}^0/{\eta }_0=1,$ $Z_{\rho \rho }^0/{\eta }_0=0,$$Z_{zz}^n/{\eta }_0=Z_{\rho \rho }^n/{\eta }_0=j/2.$ The incident plane wave is TM polarized $E_{{\beta }_0}^i=\text{1},E_{{\varphi }_0}^i=\text{0,}$ and the geometrical parameters: ${\varphi }_0={\text{30}}^{\circ }$ and ${\beta }_0=2{\text{2}}^o,{\text{28}}^o.$ The proposed solution: AWE (dashed lines); the MM-PO method (marked by +).

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6. Conclusion

UTD solutions for diffraction from anisotropic impedance wedge, illuminated by plane wave at an arbitrary given oblique incidence over the whole angle space $0^o<{\beta }_0\le {90}^o,$ have been provided by resorting to a presented numerical matching method. With the numerical matching procedure, the spectral functions are numerically derived and then the rigorous spectral representations are asymptotically evaluated to yield a suitable high-frequency solution for the diffracted field in the format of the uniform geometrical theory of diffraction. The numerical results reveal good agreements compared with those obtained by reference solutions presented in the literature, including the probabilistic random walk method, the parabolic equation method, the Maliuzhinets method as well as the hybrid MM-PO method. The UTD solutions of the proposed numerical matching method show a great extension of the limits of applicability of the previous perturbation approaches.