Invertibility of Helmholtz operators for nonhomogeneous medias

✍ Scribed by Vladimir Rabinovich; Manuel Quino Cerdan

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 207 KB
• Volume: 33
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1215

✦ Synopsis

Abstract

The paper is devoted to the investigation of the Helmholtz operators
describing the propagation of acoustic waves in non‐homogeneous space. We consider the operator A with a wave number k such that
where k~0~ is a positive function, k~±~ are complex constants with ℑ︁(k)__>0. The Helmholtz operator A with such wave number describes the propagation of acoustic waves in the waveguides being no homogeneous layer between two absorbing half‐space. We prove that the operator A has an inverse operator A^‐__1^ bounded in the Hilbert space L^2^(ℝ^n^). Our proof is based on the limit operators method.

We also consider the construction of the inverse operator for the Helmholtz operator A~ε~ with the density ρ = ρ(x~n~) depending on x~n~ only and wave number k~0~ = k~0~(ε__x__^′^, x~n~) depending on a small parameter ε__>__0 which characterizes the slowness of variation of the wave number in the horizontal direction. Copyright © 2009 John Wiley & Sons, Ltd.