Kuryliak Dozyslav, Lysechko Victor
Department of the Theory of Wave Processes and Optical Systems of Diagnostics, Karpenko Physico-Mechanical Institute of the NAS of Ukraine, Lviv, Ukraine.
Philos Trans A Math Phys Eng Sci. 2025 Aug 14;383(2303):20240338. doi: 10.1098/rsta.2024.0338.
Two scalar wave diffraction problems for an open-ended sphere-conical cavity formed by a semi-infinite truncated cone with an internal termination in the form of the spherical cap in one of the conical regions are considered in the case of an axial excitation by a plane wave. The problems are formulated in terms of mixed boundary value ones with respect to the scalar potentials for the Helmholtz equation with Dirichlet or Neumann boundary conditions. Our technique is based on the mode matching, which is applied to reduce the problems to the infinite system of linear algebraic equations (ISLAEs) of the second kind by the method of analytical regularization. This includes the Abel integral transformation of the Legendre series equation to the Dirichlet one to justify the unique transition to ISLAE, separating the singular operators from them and deriving their inverse ones. To extend the applicability of our technique, two types of the regularization procedures are applied for the solutions, and the general scheme for designing the family of regularizing operators is proposed. The analytical solutions of the problems are obtained for small size of the cavity aperture. Based on this, the new approximate formulas are obtained to determine the cavity resonance frequency perturbations. Depending on the geometry parameters and the physical interpretation of the potentials, the scattering characteristics of probes, reflectors, resonators and subsurface defects are analysed numerically for two limiting cases of the physical properties of the scatterer's surfaces.This article is part of the theme issue 'Analytically grounded full-wave methods for advances in computational electromagnetics'.
研究了平面波轴向激励下，由半无限截锥在其中一个锥形区域内以球冠形式进行内部终端形成的开放式球锥腔的两个标量波衍射问题。这些问题是根据具有狄利克雷或诺伊曼边界条件的亥姆霍兹方程标量势的混合边值问题来表述的。我们的技术基于模式匹配，通过解析正则化方法将问题简化为第二类线性代数方程组（ISLAEs）的无穷系统。这包括将勒让德级数方程进行阿贝尔积分变换为狄利克雷方程，以证明向ISLAE的唯一过渡，从其中分离奇异算子并推导其逆算子。为了扩展我们技术适用范围，对解应用了两种正则化程序，并提出了设计正则化算子族的一般方案。对于腔体孔径较小的情况，得到了问题的解析解。在此基础上，得到了确定腔体共振频率扰动的新近似公式。根据几何参数和势的物理解释，针对散射体表面物理性质的两种极限情况，对探头、反射器、谐振器和地下缺陷的散射特性进行了数值分析。本文是主题为“计算电磁学进展的解析基础全波方法”的一部分。
