# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 3 Fields Representation  
Let us split the area surrounding the biconical scatterer \(Q\) into three canonical subregions as (see Figure 1a)  
\[D_{1}:\{r\in(0,a_{1}),\theta\in(\gamma_{1},\pi)\},\ D_{2}:\{r\in(a_{1},c_{1}), \theta\in(\gamma_{2},\pi]\},\ D_{3}:\{r\in(a_{1},\infty),\theta\in[\gamma_{1}, \gamma_{2})\}. \tag{8}\]  
Here, for further convenience, the coordinate \(\varphi\in[0,2\pi)\) is omitted in the notation. Since the unknown \(H_{\varphi}\) component satisfies Equation (3), we apply for the solution the method of separation variables and represent it using eigenfunctions in the appropriate domains \(D_{l}(l=1,2,3)\), determined in (8) as \(H_{\varphi}^{(l)}=\sum\left(H_{\varphi}^{(l)}\right)_{n}\), where \(\left(H_{\varphi}^{(l)}\right)_{n}\) is a normal magnetic TM-mode in a spherical coordinate system [43];  
\[\left(H_{\varphi}^{(l)}\right)_{n}=\frac{i\omega\varepsilon}{\sqrt{sr}}\Big{[} g_{n}^{(l)}I_{\eta_{n}}(sr)+t_{n}^{(l)}K_{\eta_{n}}(sr)\Big{]}\Phi_{\eta_{n}-1/2} ^{(l)}(\cos\theta). \tag{9}\]  
Here, \(I_{\eta_{n}}(sr)\) and \(K_{\eta_{n}}(sr)\) are modified Bessel and Macdonald functions; the indices \(\eta_{n}\) and unknown coefficients \(g_{n}^{(l)}\) and \(t_{n}^{(l)}\) (\(n=1,\,2,\,3,\ldots\)) are to be determined from the problem solution. In order to satisfy the energy limitation and the radiation conditions, we accept that \(t_{n}^{(l)}\equiv 0\) for \(l=1\) and \(g_{n}^{(l)}\equiv 0\) for \(l=3\). The eigenfunctions \(\Phi_{\eta_{n}-1/2}^{(l)}(\cos\theta)\) (\(n=1,\,2,\,3,\ldots\)) are determined in the regions \(D_{l}\) from the solution of Sturm-Liouville problems for the differential equation  
\[\widehat{\Delta}_{\theta}\,\Phi_{\eta_{n}-1/2}^{(l)}(\cos\theta)=-\ (\eta_{n}^{2}-1/4)\Phi_{\eta_{n}-1/2}^{(l)}(\cos\theta), \tag{10}\]  
where  
\[\widehat{\Delta}_{\theta}=(\sin\theta)^{-1}\partial_{\theta}(\sin\theta\ \partial_{\theta})-(\sin\theta)^{-2}\]  
is the Beltrami operator. We consider here the limited solution of Equation (10), which satisfies the boundary condition (4) that is taken in the form  
\[(\sin\gamma)^{-1}\partial_{\theta}\Big{[}\sin\gamma\,\Phi_{\eta_{n}-1/2}^{(l) }(\cos\gamma)\Big{]}=0 \tag{11}\]  
with \(\gamma=\gamma_{1}\) if \(l=1\), and \(\gamma=\gamma_{2}\) if \(l=2\), \(3\). Let us represent them as  
\[ \begin{split}\Phi_{z_{n}-1/2}^{(1)}(\cos\theta)&= \partial_{\theta}P_{z_{n}-1/2}(-\cos\theta),\\ &\gamma_{1}\leq\varrho\leq\pi\\ \Phi_{\mu_{n}-1/2}^{(2)}(\cos\theta)&=\partial_{\theta}P_{ \mu_{n}-1/2}(-\cos\theta),\\ \Phi_{\
u_{n}-1/2}^{(3)}(\cos\theta)&=\Psi_{\
u_{n}-1/2}( \cos\theta).\\ &\gamma_{1}\leq\varrho\leq\gamma_{2}\end{split}  \tag{12}\]Here, \(\eta_{n}=\mu_{n}(v_{n},\,z_{n})\); \(P_{\eta-1/2}(\pm\cos\theta)\) and \(\partial_{\theta}P_{\eta-1/2}(\pm\cos\theta)=\pm\,P_{\eta-1/2}^{1}(\pm\cos\theta)\) are the Legendre and the associated Legendre functions of the first order, respectively [44];  
\[\Psi_{v_{n}-1/2}(\cos\theta)=\left\{ \begin{array}{l}(\sin\theta)^{-1},\,\,n=1,\\ \\ \partial_{\theta}[R_{v_{n}-1/2}(\cos\theta)],\,\,n>1,\end{array} \right. \tag{13}\]  
where  
\[R_{v_{n}-1/2}(\cos\theta)=P_{v_{n}-1/2}(\cos\theta)P_{v_{n}-1/2}(-\cos\gamma_ {1})-P_{v_{n}-1/2}(-\cos\theta)P_{v_{n}-1/2}(\cos\gamma_{1})\,\,\,. \tag{14}\]  
Taking into account the relationship  
\[(\sin\theta)^{-1}\,\partial_{\theta}\Big{[}\sin\theta\,\partial_{\theta}P_{ \eta-1/2}(\pm\cos\theta)\Big{]}=-\,(\eta^{2}-1/4)P_{\eta-1/2}(\pm\cos\theta) \tag{15}\]  
we find that the functions (12) satisfy the boundary condition (11) if their indices are determined from the solution of transcendental equations as  
\[P_{z_{n}-1/2}(-\cos\gamma_{1})=0,\]  
\[P_{\mu_{n}-1/2}(-\cos\gamma_{2})=0, \tag{16}\]  
\[R_{v_{n}-1/2}(\cos\gamma_{2})=0.\]  
The roots of these equations are real and in view that \(P_{\eta-1/2}(\pm\cos\theta)=P_{-\eta-1/2}(\pm\cos\theta)\) are arranged symmetrically above zero on the number axis [45].