Reciprocity and Representation Theorems for Flux- and Field-Normalised Decomposed Wave Fields

Kees Wapenaar

doi:10.1155/2020/9540135

Abstract

We consider wave propagation problems in which there is a preferred direction of propagation. To account for propagation in preferred directions, the wave equation is decomposed into a set of coupled equations for waves that propagate in opposite directions along the preferred axis. This decomposition is not unique. We discuss flux-normalised and field-normalised decomposition in a systematic way, analyse the symmetry properties of the decomposition operators, and use these symmetry properties to derive reciprocity theorems for the decomposed wave fields, for both types of normalisation. Based on the field-normalised reciprocity theorems, we derive representation theorems for decomposed wave fields. In particular, we derive double- and single-sided Kirchhoff-Helmholtz integrals for forward and backward propagation of decomposed wave fields. The single-sided Kirchhoff-Helmholtz integrals for backward propagation of field-normalised decomposed wave fields find applications in reflection imaging, accounting for multiple scattering.

Highlights

In many wave propagation problems, it is possible to define a preferred direction of propagation
To account for propagation in preferred directions, the wave equation for the full wave field can be decomposed into a set of coupled equations for waves that propagate in opposite directions along the preferred axis
These equations form the basis for reciprocity theorems for the decomposed field and source vectors p and s in the two sections

Summary

Introduction

In many wave propagation problems, it is possible to define a preferred direction of propagation. Exploiting the symmetry of the flux-normalised decomposition operators, the author derived reciprocity and representation theorems for flux-normalised one-way wave fields [16, 17]. The first aim of this paper is to discuss flux-normalised versus field-normalised decomposition in a systematic way. It will be shown that reciprocity theorems for field-normalised one-way wave fields can be derived in a similar way as those for flux-normalised one-way wave fields, even though the operators for field-normalised decomposition exhibit less symmetry. The second aim is to discuss representation theorems for field-normalised one-way wave fields in a systematic way. In. Section 4, we extensively discuss representation theorems for field-normalised one-way wave fields and indicate applications.

Unified Wave Equation and Its Symmetry Properties

Decomposed Wave Equation and Its Symmetry Properties

Field-Normalised Decomposition and Reciprocity

Field-Normalised Representation Theorems

Conclusions