Transition operators in acoustic-wave diffraction theory. I. General theory.

Abstract

The objective of this paper is the establishment of a formal theory of the scattering of time-harmonic acoustic scalar waves from impenetrable, immobile obstacles; the time-independent formal scattering theory of nonrelativistic quantum mechanics, in particular the theory of the complete Green's function and the transition (T) operator, provides the model. The quantum-mechanical approach is modified to allow the treatment of acoustic-wave scattering with imposed boundary conditions of impedance type on the surface \ensuremath{\partial}\ensuremath{\Omega} of an impenetrable obstacle. With ${\mathit{k}}_{0}$ as the free-space wave number of the signal, a simplified expression is obtained for the ${\mathit{k}}_{0}$-dependent T operator for a general case of homogeneous impedance boundary conditions for the acoustic wave on \ensuremath{\partial}\ensuremath{\Omega}. All the nonelementary operators that enter the expression for the T operator are formally simple, rational algebraic functions of a certain invertible linear operator Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}$, which is called the radiation impedance operator, and which maps any sufficiently well-behaved, complex-valued function on \ensuremath{\partial}\ensuremath{\Omega} into another such function on \ensuremath{\partial}\ensuremath{\Omega}. The nonlocal operators Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}$ and Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}^{\mathrm{\ensuremath{-}}1}$ are defined only implicitly, in that Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}^{\mathrm{\ensuremath{-}}1}$ is the operator that maps the limiting-value function on \ensuremath{\partial}\ensuremath{\Omega} of an outgoing-wave solution to the scalar Helmholtz equation into the uniquely corresponding limiting-normal-derivative function, and Z${\mathrm{\ifmmode \check{}\else \v{}\fi{}}}_{\mathit{k}0}$ does the inverse operation. Previous appearances in the literature of these operators, or their analogs for other elliptic, linear partial-differential equation systems, are cited. An analytical study of the dominant singularities of the operators (considered as two-point kernels on a smooth \ensuremath{\partial}\ensuremath{\Omega}), and of their behavior in the geometrical acoustics limit, is the subject of a second paper [G. E. Hahne, following paper, Phys. Rev. A 43, 1990 (1991)].