# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 9 Numerical Analysis
Comparing these curves, we can conclude that the current density on the conical surface \(\theta=\gamma_{1}\) is almost completely induced by the TEM mode. Figure 4b shows the behaviour of the radiating power \(W\) through the biconical area as a function of the parameter \(kc_{1}\), where  
\[W=\frac{\pi^{2}}{2Z}\frac{|x_{1}^{(3)}|^{2}}{|K_{\
u_{1}}(ka_{1})|^{2}}\ln \Bigl{(}\text{c}\frac{\gamma_{1}}{2}\text{t}\frac{\gamma_{2}}{2}\Bigr{)}\]  
\[+\frac{\pi^{2}\sin^{2}\gamma_{2}}{2Z}\sum_{n=2}^{\infty}\frac{|x_{n}^{(3)}|^{2 }}{|K_{\
u_{n}}(ka_{1})|^{2}}\frac{\
u_{n}^{2}-1/4}{2\
u_{n}}\frac{\partial}{ \partial\
u_{n}}\bigl{[}R_{\
u_{n}-1/2}(\cos\gamma_{2})\bigr{]}\Psi_{\
u_{n}- 1/2}(\cos\gamma_{2}).\]  
We see that the shapes of the far field in Figure 4a and the radiated power of the oscillations in Figure 4b are similar, with the period approximately equal to \(\lambda/2\) (\(\lambda\) is the dimensionless wavelength). Therefore, we can suppose that the local maxima of the radiation that we observe from these curves are caused by the resonant excitation of an open cavity. We observe their sharp jumps if \(kc_{1}\) is close to the dimensionless resonant radii \(k_{\mu_{\mu}j}c_{1}\) for \(TM_{\mu_{\mu}0j}\) modes of the closed sphere-conical resonator with the azimuthal index \(n=2\) and the radial ones \(j=1,\ldots,5\). Parameter \(k_{\mu_{\mu}j}c_{1}\) is real and determined from the solution of Equation (94). Numerical examples of this parameter are given in Table 1.  
Figure 5 represents the near-total-field distribution at the virtual spherical surface \(\{r<a_{1},\gamma_{1}\leq\theta\leq\pi\}\) for the different positions of the cavity termination. Curves 1 and 2 in this figure are plotted if \(kc_{1}\) corresponds to the maximum with \(kc_{1}=5.66\) and minimum with \(kc_{1}=8.28\) of curve 1 in Figure 4. Comparing these curves, we observe their essential difference in the angle sector \(\gamma_{1}<\theta<\gamma_{2}\). As follows from the behaviour of curve 1 in Figure 5, the near field exceeds the maximum at the plane. Therefore, for this regime of excitation, we can amplify the far density current on the plane surface owing to the intensification of the TEM wave radiation and to the increase in the near field in the conical sector \((r<a_{1},\gamma_{1}<\theta<\gamma_{2})\) outside of the bicone area. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mu_{n}/k_{\mu_{\mu}j}c_{1}\) & \(j=1\) & \(j=2\) & \(j=3\) & \(j=4\) & \(j=5\) \\ \hline \(\mu_{1}=2.27729\) & 3.6219 & 7.1526 & 10.4066 & 13.6051 & 16.7814 \\ \hline \(\mu_{2}=5.26278\) & 6.8853 & 10.9012 & 14.3663 & 17.6942 & 20.9595 \\ \hline \(\mu_{3}=8.25826\) & 10.0795 & 14.4705 & 18.128 & 21.5862 & 24.9479 \\ \hline \end{tabular}
\end{table} 
Table 1: Resonant radii of the axially symmetric \(TM_{\mu_{\mu}0j}\) mode oscillation for the closed sphere-conical resonator with \(\pi-\gamma_{2}=60^{\circ}\). Figure 4: Dependencies of the far-field characteristics on \(kc_{1}\) for a bicone with \(\gamma_{1}=89^{\circ}\), \(\gamma_{2}=120^{\circ}\), \(ka_{1}=4.5\), and \(kr_{0}=1\); (**a**) (1) modulus of the total magnetic field at the conical surface \(\theta=\gamma_{1}\); (2) modulus of the TEM mode; (**b**) radiation power. Let us consider a bicone with a semi-spherical cavity (\(\pi-\gamma_{2}=91^{\circ}\)) and a narrow conical probe (\(\gamma_{1}=30^{\circ}\)), (see Figure 1c). This geometry can be considered as a model for detection of cavity-type defects under a plane surface using a semi-infinite conical probe. For this structure in Figures 6 and 7, we represent the same scattering field characteristics as those we have studied in the previous case. In Figure 6, the sharp jumps are also observed if \(kc_{1}\) is located close to \(TM_{\mu_{0}j}\) resonant oscillations of the corresponding closed cavity. Unlike in the previous case, we observe in this figure the excitation of the two types of the azimuthal resonant vibration with the indices \((n=2;j=2,\ldots,5)\) and \((n=3;j=1)\).