Comment on "Characterization of an acoustic spherical cloak", Inverse Problems 31(3):035001, 2015

TL;DR

This paper critiques and extends previous work on acoustic spherical cloaks by providing explicit solutions and compatibility conditions for radial mass density profiles, enhancing the understanding of inverse problems in transformation acoustics.

Contribution

It offers explicit solutions for radial mass density in acoustic cloaks and derives a valid compatibility condition, addressing gaps in prior inverse problem approaches.

Findings

Explicit solutions for radial mass density profiles are provided.

A valid compatibility condition for the inverse problem is derived.

The approach clarifies the spectral properties involved in acoustic cloaking.

Abstract

The paper [1] considers a spherical cloak described by three radially varying acoustical quantities. For a given radial mass density in the cloak the question posed is whether the remaining two parameters, tangential density and compressibility, are uniquely determined. A method is proposed in [1] to solve this inverse question based upon the solution of a Riemann-Hilbert problem involving spectral properties of a one-dimensional inhomogeneous Schr\"odinger equation. However, no constructive examples of the solution procedure are given. This comment provides explicit solutions for any radial mass density that conforms with the requirements of transformation acoustics [2]. A valid form of the compatibility condition [1, eq. (22)] for $n=0$ is derived.