# American Institute of Mathematical Sciences

December  2017, 11(6): 1091-1105. doi: 10.3934/ipi.2017050

## Near-field imaging of sound-soft obstacles in periodic waveguides

 1 School of Mechanical Engineering, University of Shanghai for Science and Technology, 200093 Shanghai, China 2 Center for Industrial Mathematics, University of Bremen, 28359 Bremen, Germany

* Corresponding author: Ruming Zhang.

Received  January 2017 Revised  May 2017 Published  September 2017

Fund Project: The first author was supported by the Program for Fostering of Young Teachers in the Higher Education Institutions of Shanghai, China, No. ZZslg16032.
The second author was supported by the University of Bremen and the European Union FP7.

In this paper, we introduce a direct method for the inverse scattering problems in a periodic waveguide from near-field scattered data. The direct scattering problem is to simulate the point sources scattered by a sound-soft obstacle embedded in the periodic waveguide, and the aim of the inverse problem is to reconstruct the obstacle from the near-field data measured on line segments outside the obstacle. Firstly, we will approximate the scattered field by some solutions of a series of Dirichlet exterior problems, and then the shape of the obstacle can be deduced directly from the Dirichlet boundary condition. We will also show that the approximation procedure is reasonable as the solutions of the Dirichlet exterior problems are dense in the set of scattered fields. Finally, we will give several examples to show that this method works well for different periodic waveguides.

Citation: Ming Li, Ruming Zhang. Near-field imaging of sound-soft obstacles in periodic waveguides. Inverse Problems & Imaging, 2017, 11 (6) : 1091-1105. doi: 10.3934/ipi.2017050
##### References:

show all references

##### References:
The scattering problem in the periodic waveguide
Direct method for inverse scattering problems.
A periodic half guide.
Waveguide 1
Waveguide 2
Four scatterers
(a)-(b): numerical result for scatter 1 with waveguides
(a)-(b): numerical result for scatter 2 with waveguides
(a)-(b): numerical result for scatter 3 with waveguides
(a)-(b): numerical result for scatter 4 with waveguides
(a)-(b): numerical result for scatter 1 with waveguides
(a)-(b): numerical result for scatter 2 with waveguides
(a)-(b): numerical result for scatter 3 with waveguides
(a)-(b): numerical result for scatter 4 with waveguides
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