Abstract
We consider the problem of fixed frequency acoustic scattering from a sound-soft flat screen. More precisely, the obstacle is restricted to a two-dimensional plane and interacting with an arbitrary incident wave, it scatters acoustic waves to three-dimensional space. The model is particularly relevant in the study and design of reflecting sonars and antennas, cases where one cannot assume that the incident wave is a plane wave. Our main result is that given the plane where the screen is located, the far-field pattern produced by any single arbitrary incident wave determines the exact shape of the screen, as long as it is not antisymmetric with respect to the plane. This holds even for screens whose shape is an arbitrary simply connected smooth domain. This is in contrast to earlier work where the incident wave had to be a plane wave, or more recent work where only polygonal scatterers are determined.
Highlights
The motivation for the study of wave scattering from thin and large objects lies in the antenna theory
The mathematical question of finding a screen that scatters a given incident wave into a particular far-field has applications like the following, for example: how to reduce echo in an office space? How to direct acoustic vibrations or reduce them? it answers the probing question: can we determine the shape and location of a passive sonar by how it reflects sound? These are complex questions, only one part of which we are going to solve, namely that a single input–output pair of sound waves uniquely determines the shape of a flat acoustic screen
Our work in this paper shows that given the far-field caused by any single given incident wave scattering off a smooth flat screen, the latter’s shape is determined uniquely
Summary
The motivation for the study of wave scattering from thin and large objects lies in the antenna theory. The competition was won in 1882 by young Heinrich Hertz, in favor of Maxwell’s theory He did this by constructing a dipole antenna radiating EM waves which he could measure. A major step in such a design strategy is to solve the inverse scattering problem: given an input–output pair of waves, which antenna shape produces it?. The mathematical question of finding a screen that scatters a given incident wave into a particular far-field has applications like the following, for example: how to reduce echo in an office space? These are complex questions, only one part of which we are going to solve, namely that a single input–output pair of sound waves uniquely determines the shape of a flat acoustic screen The mathematical question of finding a screen that scatters a given incident wave into a particular far-field has applications like the following, for example: how to reduce echo in an office space? How to direct acoustic vibrations or reduce them? it answers the probing question: can we determine the shape and location of a passive sonar by how it reflects sound? These are complex questions, only one part of which we are going to solve, namely that a single input–output pair of sound waves uniquely determines the shape of a flat acoustic screen
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