# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 2 Statement of the Problem  
Let us consider, in a spherical coordinate system \((r,\theta,\varphi)\), a perfectly conducting biconical scatterer conjoined with an open-ended conical cavity \(Q=Q_{1}\bigcup Q_{2}\), where \(Q_{1}\) is the semi-infinite cone and \(Q_{2}\) is the truncated semi-infinite one with an internal spherical termination; the termination forms an open-ended conical cavity:  
\[Q_{1}:\{r\in(0,\infty);\ \theta=\gamma_{1};\,\varphi\in[0,2\pi)\}, \tag{1}\]  
Here, \(\gamma_{1}\) and \(\gamma_{2}\) are the generating angles; \(a_{1}\) and \(c_{1}\) are the radial coordinates of the cavity aperture and the spherical termination, respectively, \(c_{1}>a_{1}\); \(\gamma_{2}>\gamma_{1}\) and \(\gamma_{1(2)}\
eq\pi/2\). The biconical scatterer and its particular cases are shown in Figure 1.  
Let the bicone \(Q\) be excited axially symmetric by the TM-wave, produced by an electric dipole located on the symmetry axis. The time factor \(e^{-i\omega t}\) is suppressed throughout this paper. Since the incident modes are independent of the azimuth angle \(\varphi\), then, as follows from [PERSON]'s equations, only the three nonzero field components \(E_{r},E_{\theta}\), and \(H_{\varphi}(r,\theta)\) are excited [40]. The electric field components are expressed in terms of \(H_{\varphi}\) by  
\[E_{r}=-\ (i\omega\varepsilon r\sin\theta)^{-1}\partial_{\theta}(\sin\theta H_{ \varphi}),\ \ E_{\theta}=(i\omega\varepsilon r)^{-1}\partial_{r}(rH_{\varphi})\ \, \tag{2}\]where \(\varepsilon\) is the dielectric permittivity of the medium.  
Taking into account the axial symmetry of the initial problem, let us formulate the mixed boundary value problem to determine the \(H_{\varphi}\)-field diffracted from \(Q\) as  
\[\
abla^{2}H_{\varphi}-(r\sin\theta)^{-2}H_{\varphi}+k^{2}H_{\varphi}=0, \tag{3}\]  
\[(\sin\theta)^{-1}\partial_{\theta}\Big{[}\sin\theta(H_{\varphi}+H_{\varphi}^{ i})\Big{]}=0 \tag{4}\]  
\[\text{if}\,r,\theta\in\{r\in(0,\infty),\,\theta=\tau_{1}\}\cup\{r\in(a_{1}, \infty),\,\theta=\tau_{2}-0\}\cup\{r\in(a_{1},c_{1}),\,\theta=\tau_{2}+0\}\]  
and  
\[r^{-1}\partial_{r}[r(H_{\varphi}+H_{\varphi}^{i})]=0 \tag{5}\]  
\[\text{if}\,r,\theta\in\,\{r=c_{1},\theta\in[\tau_{2},\pi]\}.\]  
Here, \(k=\omega\sqrt{\epsilon\mu}\) is the wave number; \(\mu\) is the magnetic permeability; \(\
abla^{2}\) is the axially symmetric Laplace operator in the spherical coordinate system;  
\[\
abla^{2}=\partial_{rr}^{2}+2r^{-1}\partial_{r}+(r^{2}\sin\theta)^{-1} \partial_{\theta}(\sin\theta\partial_{\theta});\]  
\(H_{\varphi}^{i}\) is the known magnetic component of the incident field, and \(H_{\varphi}+H_{\varphi}^{i}=H_{\varphi}^{t}\) is the total field.  
We search for the solution of the mixed boundary value problems (3)-(5) in the class of functions that satisfy the Silver-Muller radiation condition [41]  
Figure 1: An axially symmetric bicone with an open-ended conical cavity; (**a**) general view; (**b**) sphere-conical cavity over the plane \((r,\theta\)-plane); (**c**) conical probe over the semi-spherical cavity \((r,\theta\)-plane).  
\[\lim_{r\rightarrow\infty}r\Big{[}\begin{matrix}\stackrel{{\rightharpoonup}}{{e}} _{r}\times\breve{H}+Z^{-1}\breve{E}\end{matrix}\Big{]}=0\cdot\vec{e}_{\theta}, \tag{6}\]  
where \(\breve{H}=(0,0,\vec{e}_{\varphi}H_{\varphi})\), \(\breve{E}=(0,\vec{e}_{\theta}E_{\theta},0)\), \(Z=\sqrt{\mu/\epsilon}\) is the medium wave resistance, and the energy limitation condition is [42]  
\[\int_{V}\Big{(}\epsilon|E|^{2}+\mu|\breve{H}|^{2}\Big{)}dv<\infty. \tag{7}\]  
Here, \(V\) is any finite volume of integration, \(dv=r^{2}\sin\theta drd\theta d\varphi\). This condition, in our case, is reduced to the fulfilment of the Meixner condition at the edge of \(Q\). In order to fulfil the radiation condition and prevent the arrival of waves from infinity, we assume that \(k=k^{\prime}+ik^{\prime\prime}\), \(k^{\prime}>0\), \(k^{\prime\prime}\geq 0\). Due to these two additional conditions, the diffraction problems (3)-(5) are properly posed.