# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 8 Radiation through a Small-Sized Cavity Aperture  
The solution of the problem of wave penetration through a small-sized hole allows the scattered field characteristics to be obtained in analytical form. The well-known results in this area are presented in [47; 48]. The general theory for the solution to this problem is developed in [49]. It is worth noting that the main problem this study engages is correct accounting for the field singularity at the aperture edge. A technique for the solution of the problem of wave radiation from a small-sized hole of a sphere-conical cavity, which rigorously accounted for the field singularities at the edge, was considered in [31]. Here, we develop this technique for the solution of the problem of wave radiation from a cavity with a small-sized hole into a biconical region. For this purpose, we take into account the small dimension of the hole (\(|sa_{1}/2|<<1\)). Thus, we apply the asymptotic expressions for the modified Bessel and Macdonald functions [44] to estimate the known terms of the Equations (73), (74) and, taking into account the resonant properties of the cavity, we derive the approximate equations as  
\[x_{k}+\sum_{q=1}^{\infty}\tau_{kq}^{(1)}\kappa_{\xi_{q}}(sc_{1})\bar{x}_{q}= \sum_{p=1}^{\infty}\,\tau_{kp}^{(1)}f_{p}, \tag{88}\]\[y_{k}+\sum_{q=1}^{\infty}\eta_{kq}\kappa_{\xi_{q}}(sc_{1})y_{q}=\sum_{n=1}^{ \infty}\tau_{kn}^{(2)}\bar{\varphi}_{n}. \tag{89}\]  
Here,  
\[\bar{x}_{q}=\sum_{n=1}^{\infty}\frac{x_{n}}{\xi_{q}+z_{n}}, \tag{90}\]  
\[\eta_{kq}=\sum_{n=1}^{\infty}\frac{\tau_{kn}^{(2)}}{\xi_{q}+z_{n}}=\frac{1}{ \left\{\left[M_{-}(\xi_{k})\right]^{-1}\right\}^{{}^{\prime}}M_{+}(\xi_{q})( \xi_{q}+\xi_{k})}, \tag{91}\]  
\[\kappa_{\xi_{q}}(sc_{1})=\left\{ \begin{array}{l}\frac{2(sa_{1}/2)^{2\xi_{q} }}{\Gamma(\xi_{q})\Gamma(\xi_{q}+1)}\left[\frac{K_{\xi_{q}}(sc_{1})+2 sc_{1}K_{ \xi_{q}}^{\prime}(sc_{1})}{I_{\xi_{q}}(sc_{1})+2 sc_{1}I_{\xi_{q}}^{\prime}(sc_ {1})}\right]\\ \\ 0\,\ \xi_{q}\in\{v_{k}\}_{k=1}^{\infty},\end{array} \right. \tag{92}\]  
\[f_{p}=-\sum_{n=1}^{\infty}A_{n}q(z_{n},\gamma_{2})\frac{(r_{0}/a_{1})^{z_{n}} }{2z_{n}}\left\{ \begin{array}{l}\frac{1}{\xi_{p}+z_{n}}+\frac{\kappa_{\xi_{p }}}{\xi_{p}-z_{n}}\,\ \xi_{p}\in\{\mu_{p}\}_{p=1}^{\infty},\\ \\ \frac{1}{\xi_{p}+z_{n}}\,\ \xi_{p}\in\{v_{k}\}_{k=1}^{\infty}, \end{array} \right. \tag{93a}\]  
\[\bar{\varphi}_{n}=\frac{A_{n}(r_{0}/a_{1})^{z_{n}}}{\bar{x}(z_{n},\gamma_{1}, \gamma_{2})}. \tag{93b}\]  
Let us consider the resonant excitation of a cavity with the small dimension of the hole. Taking into account the time factor \(e^{-i\omega t}\), the real and imaginary parts of the complex resonance frequency of an open cavity \(\omega_{res}=\text{Re}(\omega_{res})+i\text{Im}(\omega_{res})\) are determined as \(\text{Re}(\omega_{res})>0\) and \(\text{Im}(\omega_{res})\leq 0\). Under these conditions, the resonance frequency has a physical sense: the wave moves from the scatterer to infinity, the corresponding vibration grows in volume and decreases with time. Let us introduce \(\tilde{\rho}_{pj}=(sc_{1})_{pj}=-i\omega_{\mu_{pj}}c_{1}\sqrt{\epsilon\mu}\)\((p,j=1,2,3,\ldots)\), where \(\omega_{\mu_{pj}}\) is the real resonant frequency \((\text{Re}(\omega_{\mu_{pj}})>0,\text{Im}(\omega_{\mu_{pj}})=0)\) of a closed sphere-conical resonator, which corresponds to the resonant \(TM_{\mu_{p}0j}\) mode; \(\omega_{\mu_{pj}}\) is determined from the solution of the transcendental equation as  
\[I_{\mu_{p}}(\bar{\rho}_{pj})+2\bar{\rho}_{pj}I_{\mu_{p}}^{\prime}(\bar{\rho}_{ pj})=0. \tag{94}\]  
Let \(\rho_{pj}=\bar{\rho}_{pj}+\Delta\rho_{pj}\), \((\Delta\rho_{pj}\equiv-i\Delta\omega_{\mu_{pj}}c_{1}\sqrt{\epsilon\mu}\), \(\Delta\omega_{\mu_{pj}}=\text{Re}(\Delta\omega_{\mu_{pj}})+i\text{Im}(\Delta \omega_{\mu_{pj}}))\), and \(|\Delta\rho_{pj}|<<1\).