The plane wave scattering problem for a rigid spherical shell with two coaxial circular holes has been solved rigorously by the Method of Analytical Regularization. An original approach to solving the triple series equations with Legendre polynomials as kernels that arise in the formulation of the problem led to its solution, which is uniformly valid with respect to aperture size. The solution presents two independent infinite systems of linear algebraic equations of the second kind for the even and odd Fourier coefficients respectively. The fast convergence of the systems when truncated is the key to accurate and comprehensive studies of scattering characteristics (total and sonar cross-sections, mechanical force on the walls of doubly-connected sphere) in a wide frequency range. The solution is valid for investigation of the Rayleigh scattering region (where the sphere radius a is much smaller that the wavelength $\lambda$), for the resonance region ($a\sim \lambda$), and also for the deep quasi-optics region ($a\gg \lambda )$, where the radius $a$ of the sphere may be thousands of wavelengths. The unique properties of the obtained solution have been used for calculation of the first 16 complex eigenvalues (in the rigid doubly-connected sphere) with a guaranteed accuracy of 8 significant decimal digits. The new phenomenon of a huge drop in the sonar cross-section ${\sigma }_B$ near the Helmholtz mode (${\sigma }_B/\pi a^2\sim {10}^{-10}$) was identified and studied both analytically and numerically. The resonance behaviour of the reflectivity and mechanical force factor is studied in a wide frequency range.

用解析正则法严格解决了具有两个同轴圆孔的刚性球形壳的平面波散射问题。。解决问题的方式是用勒让德多项式作为内核来求解三级数列方程的一种原始方法导致了它的求解，这在孔径大小方面是统一有效的。该解决方案针对偶数和奇数傅里叶系数分别提出了两个独立的第二种线性代数方程式无限系统。截断后，系统的快速收敛是在广泛的频率范围内准确，全面地研究散射特性（总截面和声纳截面，双连通球壁上的机械力）的关键。该解决方案对于研究瑞利散射区域（球半径a比波长小得多）有效。$\lambda$），用于谐振区域（$\mathrm{一种}〜\lambda$），也适用于深准光学区域（$\mathrm{一种}\gg \lambda ）$，半径 $\mathrm{一种}$球体的波长可能是数千个波长。所获得解决方案的独特属性已用于计算前16个复数特征值（在刚性双连通球体中），并保证8个有效十进制数字的精度。声纳横截面大幅下降的新现象${\sigma }_{乙}$ 接近亥姆霍兹模式（${\sigma }_{乙}/\pi {\mathrm{一种}}^{\mathrm{2个}}〜{10}^{-10}$）被识别并进行了分析和数值研究。在较宽的频率范围内研究了反射率和机械力因数的共振行为。