Determining a sound-soft polyhedral scatterer by a single far-field measurement

Abstract

Abstract. We prove that a sound-soft polyhedral scatterer is uniquely determined by the far-field pattern corresponding to an incident plane wave at one given wavenumber and one given incident direction.Lo duca e io per quel cammino ascoso intrammo a ritornar nel chiaro mondo; e sanza cura aver d'alcun riposo, salimmo su, el primo e io secondo, tanto ch'i' vidi de le cose belle che porta'l ciel, per un pertugio tondo; e quindi uscimmo a riveder le stelle.Dante, Inferno, C.XXXIV, 133-139.

Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers

“…Proof. For the boundedness of D u , we refer to lemma 3.1 in [1]. And the closeness of D u can be proved in a similar way to the proof of lemma 2.…”

8. Some old and some new results in inverse obstacle scattering

“…n of order n and of the first kind (see in [16,Theorem 2.16]). This implies v δ (x) = δ k i n+1 h (1) n (kR)Y n x |x| , |x| = R, and consequently, by the asymptotics of the spherical Hankel functions for larger order, it follows that…”

“…Starting in 2003 in a series of papers by Alessandrini, Cheng, Liu, Rondi and Yamamoto [1,13,34,35] it was established that one incident plane wave is sufficient to uniquely determine a sound-soft polyhedron. Assuming that there exist two polyhedral scatterers producing the same far field pattern for one incident plane wave, the main idea of their proofs is to use the reflexion principle to construct a zero field line extending to infinity.…”

Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement

“…We refer to survey papers and books [15,24,25,26,27]. We also refer to [3,1,9,10,14,20,22] for uniqueness results in recovering of obstacles in acoustic scattering problems.…”

“…By choosing ϕ 1 , ϕ 2 , ϕ 3 ∈ H 1 (∂D) such that the matrix M defined in (2.14) is invertible one can thus uniquely recover σ (1) and σ (2) .…”