A Wave Diffraction Problem with Higher Order Impedance Boundary Conditions

A M Simoes

doi:10.46300/9101.2022.16.12

Abstract

In this paper, we consider an impedance boundary transmission problem for the Helmholtz equation originated by a problem of wave diffraction by an infinite strip with higher order imperfect boundary conditions. Operator theoretical methods and relations between operators are built to deal with the problem and, as consequence, a transparent interpretation of the problem in an operator theory framework are associated to the problem. In particular, different types of operator relations are exhibited for different types of operators acting between Lebesgue and Sobolev spaces on a finite interval and the positive half-line. All this has consequences in the understanding of the structure of this type of problems. At the end, we describe when the operators associated with the problem enjoy the Fredholm property in terms of the initial space order parameters.

Highlights

By using methods from operator theory, in this paper, inspired by the work [14], we will consider a boundarytransmission problem for the Helmholtz equation which arises within the context of wave diffraction theory [3]–[5], [7]–[19], [20], [21] and [24]–[28] on a finite strip [9], [10] and [15] with impedance boundary conditions [7] and [9]
One of the main goals of the present work is the use of an operator theoretical machinery that will translate the problem into the study of properties of certain known types of operators associated to the problem
In the present paper we were able to characterize the Fredholm property of particular operators associated with an impedance boundary problem which are a generalization of the results presented in [14]

Summary

A Wave Diffraction Problem with Higher Order Impedance Boundary Conditions

Operator theoretical methods and relations between operators are built to deal with the problem and, as consequence, a transparent interpretation of the problem in an operator theory framework are associated to the problem. Different types of operator relations are exhibited for different types of operators acting between Lebesgue and Sobolev spaces on a finite interval and the positive half-line. All this has consequences in the understanding of the structure of this type of problems. We describe when the operators associated with the problem enjoy the Fredholm property in terms of the initial space order parameters

INTRODUCTION

PRELIMINARIES AND FORMULATION OF THE PROBLEM

REDUCTION OF THE PROBLEM TO A SYSTEM OF CONVOLUTION TYPE OPERATORS

EXTENSION METHODS AND RELATIONS BETWEEN OPERATORS

FREDHOLM ANALYSIS

CONCLUSION