## 4 Asymptotic Estimation of the Field in the Angular Region
### Classical Solution of the GWHEs of the Problem in Different Complex Planes
Equations (99)-(100) respectively become the following system of FIEs:  
\[G(\eta)H_{oz}(\eta)+\frac{1}{2\pi j}\int\limits_{-\infty}^{\infty}\frac{(G(t)- G(\eta))H_{az}(t)}{t-\eta}dt=\frac{1}{(2\pi j)^{2}}\int\limits_{-\infty}^{ \infty}M_{e}(\eta,\eta^{\prime})H_{az}(\eta^{\prime})d\eta^{\prime}+s_{1}(\eta) \tag{108}\]  
and  
\[G(\eta)H_{az}(\eta)+\frac{1}{2\pi j}\int\limits_{-\infty}^{\infty}\frac{(G(t )-G(\eta))H_{az}(t)}{t-\eta}dt=\frac{1}{(2\pi j)^{2}}\int\limits_{-\infty}^{ \infty}M_{e}(\eta,\eta^{\prime})H_{oz}(\eta^{\prime})d\eta^{\prime}+s_{2}(\eta) \tag{109}\]  
with  
\[M_{e}(\eta,\eta^{\prime})=\int_{\Gamma_{1}}\Delta(\frac{1}{(t-\eta)(\eta^{ \prime}+m(t)})dt, \tag{102}\]0  
Let us focus our attention on the source term (110); for simplicity, we assume that only \(F_{+}(\eta)=H_{oz}(\eta)\) is nonstandard:  
\[M_{e}(\eta,\eta^{\prime})=\int_{\Gamma_{1}}\Delta(\frac{1}{(t-\eta)(\eta^{ \prime}+m(t)})dt, \tag{102}\]1  
where \(\eta_{o}=-k\cos(\varphi_{o})\), \(0<\varphi_{o}<\pi/2\) and \(k\) has small losses (\(k=k_{r}-jk_{i}\), \(k_{i}<<k_{r}\)). From (111), according to the properties of \(-m(\eta)\) (see also Figure 4), \(H_{oz}^{ns}(-m(\eta))\) in the proper lower half of the complex \(\eta\) plane shows poles originating from the zeros of \(m(\eta)+\eta_{o}\) (in the \(m\) plane, we have the pole \(m_{o}=-\eta_{o}\)). The poles can be related to GO waves, i.e., connected to the last couple of reflections from faces \(a\) and \(o\) that create shadow boundaries (for instance, see [40]). As an example, if \(\varphi_{o}<\pi-\gamma\), then we have one reflection from face \(a\) and one reflection from face \(o\) reflected again by face \(a\). In fact, from a mathematical point of view, in this case we have pole \(m_{o}\) related to poles \(\eta_{ra}=-k\cos(\gamma-\varphi_{o})\) (reflection from face \(a\)) and \(\eta_{raro}=-k\cos(\gamma+\varphi_{o})\) (reflection from \(a\) after \(o\)), which are associated with incoming azimuthal directions \(\gamma\mp\varphi_{o}\) with respect to the reference face \(a\), i.e., the incoming directions \(2\gamma\mp\varphi_{o}\) with respect to face \(o\). However, we also need to note that the residues of the poles in the selected test problems are always related only to the incident field. This means that the primary spectra of \(H_{oz}^{ns}(-m(\eta))\) in (110) are more similar to a replica of the incident spectrum for \(\eta_{ra},\eta_{raro}\), similar to what was described in the \(w\) plane in [13]. Indeed, the integrand of the source term in (110) also exhibits a branch cut of \(-m(\eta)\). Thus, we estimate (110) by warping \(\gamma_{tt}\) into \(\lambda_{1}\), as follows:  
\[M_{e}(\eta,\eta^{\prime})=\int_{\Gamma_{1}}\Delta(\frac{1}{(t-\eta)(\eta^{ \prime}+m(t)})dt, \tag{102}\]2  
where \(R_{a}\) and \(R_{o}\) are the residues of \(H_{oz}(-m(\eta))\) in \(\eta_{ra}\) and \(\eta_{raro}\), respectively. \[M_{e}(\eta,\eta^{\prime})=\int_{\Gamma_{1}}\Delta(\frac{1}{(t-\eta)(\eta^{ \prime}+m(t)})dt, \tag{102}\]3  
\[M_{e}(\eta,\eta^{\prime})=\int_{\Gamma_{1}}\Delta(\frac{1}{(t-\eta)(\eta^{ \prime}+m(t)})dt, \tag{102}\]4  
Using the same mathematical procedure that yields (102)-(104) from (101), from (112) we obtain  
\[M_{e}(\eta,\eta^{\prime})=\int_{\Gamma_{1}}\Delta(\frac{1}{(t-\eta)(\eta^{ \prime}+m(t)})dt, \tag{102}\]5  
Thus,  
\[M_{e}(\eta,\eta^{\prime})=\int_{\Gamma_{1}}\Delta(\frac{1}{(t-\eta)(\eta^{ \prime}+m(t)})dt, \tag{102}\]6  
The final set of FIEs for \(H_{z}\) polarization when the structure is illuminated by a plane wave with \(0<\varphi_{o}<\pi/2\) is then (108)-(109) (a specialization of (99)-(100)), with sources \(s_{1,2}(\eta)\) being defined and calculated in (110), (111), and (116). Note that \(s_{1}(\eta)\) and \(s_{2}(\eta)\) are spectral components defined in the \(\eta\) plane of \(H_{oz}(\eta)\) and \(H_{az}(\eta)\), respectively, i.e., with the reference coordinate system of face \(o\) and face \(a\). We now examine the convergence properties of the FIEs in (108)-(109) to obtain accurate numerical results [41].