## 2 Formulation of the Problem
 \tag{29}\]\[\Phi^{r}_{m}(x,\alpha)=\frac{1}{b}\sum\limits_{n=0}^{\infty}\
u_{n}\frac{C_{mn}( \alpha)}{a^{2}+\Gamma_{mn}^{2}}\cos\frac{n\pi}{2b}(x+b),m=1,2,3,4, \tag{30}\]  
where  
\[\gamma_{0}=-ik,\ \ \gamma_{n}=\Big{[}(n\pi/2b)^{2}-k^{2}\Big{]}^{1/2}, \tag{31}\]  
\[\Gamma_{mn}=\Big{[}(n\pi/2b)^{2}-k_{mn}^{2}\Big{]}^{1/2},\ m=1,2,3,4,5, \tag{32}\]  
\[C_{5n}(\alpha)=e^{-iaL_{5}}C_{5n}^{+}(\alpha), \tag{33}\]  
\[C_{mn}(\alpha)=e^{-iaL_{m+1}}C_{(m+1)n}^{-}(\alpha)-e^{-iaL_{m}}C_{mn}^{+}( \alpha)\ \text{for}\ m=1,2,3,4 \tag{34}\]  
with  
\[C_{mn}^{-}(\alpha)=f_{mn}-iag_{mn},\ m=1,2,3,4,5 \tag{35}\]  
\[C_{mn}^{+}(\alpha)=(\varepsilon_{rm+1}/\varepsilon_{rm})f_{mn}-iag_{mn},\ m=2,3,4,5 \tag{36}\]  
Here the Fourier coefficients \(f_{mn}\) and \(g_{mn}\) for \(m=1,2,3,4,5\) are defined by (A5) and (A6) in Appendix A. The representation of the scattered field for the region \(|x|<b\) is obtained by substituting (16) and (19)-(23) into (11) and utilizing (9). Based on the above results, the desired scattered field representation in the Fourier transform domain is found to be  
\[ \begin{array}{rl}\Phi(x,\alpha)&=\mp\gamma^{-1}\Psi^{\prime}_{(+)}(\pm b, \alpha)e^{\mp\gamma(x\mp b)}\ \text{for}\ x\gtrless\pm b,\\ &=\ \Psi^{\prime}_{(+)}(b,\alpha)\frac{\cosh\gamma(x+b)}{\gamma\sinh 2\gamma b}- \Psi^{\prime}_{(+)}(-b,\alpha)\frac{\cosh\gamma(x-b)}{\gamma\sinh 2\gamma b}\\ &-\frac{1}{b}\sum\limits_{n=0}^{\infty}\
u_{n}\frac{C_{5n}(\alpha)}{a^{2}+ \gamma z}\cos\frac{n\pi}{2b}(x+b)\\ &-\frac{1}{b}\sum\limits_{n=0}^{\infty}\
u_{n}\frac{C_{5n}(\alpha)}{a^{2}+ \Gamma_{mn}^{2}}\cos\frac{n\pi}{2b}(x+b)\\ &-\frac{1}{b}\sum\limits_{m=1}^{\infty}\sum\limits_{n=0}^{\infty}\
u_{n}\frac{ C_{5m}(\alpha)}{a^{2}+\Gamma_{mn}^{2}}\cos\frac{n\pi}{2b}(x+b)\ \text{for}\ |x|<b.\end{array}  \tag{37}\]  
Equation (37) holds in the strip \(-k_{2}<\tau<k_{2}\cos\theta_{0}\) of the complex \(\alpha-\) plane. Differentiating (37) with respect to \(x\), setting \(x=\pm b\pm 0,\ \pm b\mp 0\) in the results, after manipulating the equations with the help of the boundary conditions, we can deduce that  
\[ \begin{array}{rl}J_{-}^{d}(\alpha)&=-\frac{U_{(+)}(\alpha)}{M(\alpha)}- \frac{2}{b}\sum\limits_{n=1,\text{odd}}^{\infty}\frac{e^{-iaL_{5}}C_{5n}^{+}( \alpha)}{a^{2}+\gamma z}\\ &+\frac{2}{b}\sum\limits_{n=1,\text{odd}}^{\infty}\frac{e^{-iaL_{1}}C_{1n}^{+ }(\alpha)}{a^{2}+\Gamma_{1n}^{2}}\\ &+\frac{2}{b}\sum\limits_{n=1,\text{odd}}^{\infty}\frac{e^{-iaL_{m}}C_{mn}^{-} (\alpha)-e^{-iaL_{m}-1}C_{(m-1)n}^{+}(\alpha)}{a^{2}+\Gamma_{mn}^{2}},\end{array}  \tag{38}\]  
\[ \begin{array}{rl}J_{-}^{s}(\alpha)=&-\frac{V_{(+)}(\alpha)}{N(\alpha)}+ \frac{2}{b}\sum\limits_{n=0,\text{even}}^{\infty}\
u_{n}\frac{e^{-iaL_{5}}C_{5n }^{+}(\alpha)}{a^{2}+\gamma z}\\ &-\frac{2}{b}\sum\limits_{n=0,\text{even}}^{\infty}\
u_{n}\frac{e^{-iaL_{1}}C_{ 5n}^{+}(\alpha)}{a^{2}+\Gamma_{mn}^{2}}\\ &-\frac{2}{b}\sum\limits_{m=2}^{\infty}\sum\limits_{n=0,\text{even}}^{\infty} \
u_{n}\frac{e^{-iaL_{m}}C_{mn}^{+}(\alpha)-e^{-iaL_{m}-1}C_{(m-1)n}^{+}( \alpha)}{a^{2}+\Gamma_{mn}^{2}},\end{array}  \tag{39}\]  
where  
\[U_{(+)}(\alpha)=\Psi^{\prime}_{(+)}(b,\alpha)+\Psi^{\prime}_{(+)}(-b,\alpha), \tag{40}\]  
\[V_{(+)}(\alpha)=\Psi^{\prime}_{(+)}(b,\alpha)-\Psi^{\prime}_{(+)}(-b,\alpha), \tag{41}\]  
\[\int_{-}^{d}(\alpha)=J_{-}(b,\alpha)-J_{-}(-b,\alpha), \tag{42}\]  
\[J_{-}^{s}(\alpha)=J_{-}(b,\alpha)+J_{-}(-b,\alpha), \tag{43}\]\[M(\alpha)=\gamma e^{-\gamma b}\cosh\gamma b,\ N(\alpha)=\gamma e^{-\gamma b}\text{ sinh}\gamma b, \tag{44}\]  
\[J_{-}(\pm b,\alpha)=\Phi^{\prime}{}_{-}(\pm b\pm 0,\alpha)-\Phi^{\prime}{}_{1}(\pm b \mp 0,\alpha). \tag{45}\]  
The unknown spectral functions fulfill the Wiener-Hopf equations expressed in Equations (38) and (39).