# Scalar wave diffraction by an open-ended sphere-conical cavity: the Abel integral transform in the Dirichlet and Neumann problems

## TL;DR

This paper studies wave diffraction in a cavity shaped like a truncated cone with a spherical cap, using mathematical techniques to analyze resonance and scattering.

## Contribution

The paper introduces a novel regularization method for solving wave diffraction problems using Abel integral transforms and mode matching.

## Key findings

- Analytical solutions for cavity resonance frequency perturbations are derived for small aperture sizes.
- New approximate formulas are developed for determining resonance behavior based on cavity geometry.
- Scattering characteristics of various structures are analyzed for different surface properties.

## Abstract

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’.