An explicit factorization of the Green's function for an acoustic half-space problem with impedance boundary conditions into an oscillatory exponential and a slowly varying function

TL;DR

This paper introduces a new explicit factorization of the Green's function for acoustic half-space problems with impedance boundary conditions, separating it into an oscillatory exponential and a slowly varying function for improved analysis and computation.

Contribution

The paper presents a novel explicit factorization of the Green's function for impedance boundary conditions, enabling efficient approximation and uniform representation across parameters.

Findings

Green's function is factored into an exponential and a slowly varying function.

The new representation allows for efficient polynomial approximation.

The formula is uniform for all parameters using a product of two analytic functions.

Abstract

In this paper, new representations of the Green's function for an acoustic d-dimensional half-space problem with impedance boundary conditions are presented. The main features of the new representation are: a) in addition to additive terms that appear also in the case of Dirichlet or Neumann boundary conditions, the remaining part of the Green's function is factored into an oscillatory complex exponential function (with the product of the wavenumber and the eikonal as argument) and a remaining function which is slowly varying and hence allows for efficient polynomial approximation; b) the representation is given uniformly for all parameters by a single formula which consists of the product of two analytic functions.