The Green`s function for an acoustic, half-space impedance problem Part II: Analysis of the slowly varying and the plane wave component

TL;DR

This paper analyzes the acoustic Green's function for a half-space impedance problem, decomposing it into exponential and slowly varying components, and introduces a new theoretical framework for understanding these functions.

Contribution

It provides a novel theorem characterizing the Green's function as a sum of exponential and slowly varying functions, with a formal proof.

Findings

Decomposition of Green's function into exponential and slowly varying parts

Introduction of families of slowly varying functions

Theoretical framework applicable in arbitrary spatial dimensions

Abstract

We show that the acoustic Green`s function for a half-space impedance problem in arbitrary spatial dimension d can be written as a sum of two terms, each of which is the product of an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.