On unique determination of partially coated polyhedral scatterers with far field measurements

This work is a continuation of our early study in Liu and Zou (2006 Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers Inverse Problems 22 515-24; 2006 Uniqueness in determining multiple polygonal or polyhedral scatterers of mixed type Technical Report 2006-03(337) The Chinese University of Hong Kong) and addresses the unique determination of partially coated polyhedral scatterers in ℝ^N (N ≥ 2) along with their surface impedance from far field data. Two global uniqueness results are established for this inverse problem with a scatterer consisting of multiple solid polyhedra: the first one is to determine such a scatterer of mixed sound-soft and impedance type by a single incident plane wave and the other is to determine such a scatterer of mixed sound-soft, sound-hard and impedance type by N different incident waves in the N-dimensional case with N ≥ 3 and by only one incident wave for the two-dimensional case. Then we present some examples to show that as long as a scatterer admits the presence of (sound-hard) crack-type obstacles, then one cannot determine the scatterer uniquely by any less than N different incident plane waves. These examples also reveal that the uniqueness results achieved earlier in [15, 16] for polyhedral scatterers are optimal. Finally, the uniqueness results that have been solved or are still unsolved for the polyhedral-type scatterers with both solid and crack components are summarized in the conclusion. © 2007 IOP Publishing Ltd.