Hiroshima Mathematical Journal

The Herglotz wave function, the Vekua transform and the enclosure method

Masaru Ikehata

Full-text: Open access

Abstract

This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal sound-hard obstacles and piecewise linear cracks from the far field pattern of the scattered field at a fixed wave number and at most two incident directions are given. The main new points of this paper are: a combination of the enclosure method and the Herglotz wave function; explicit construction of the density in the Herglotz wave function by using the idea of the Vekua transform. By virtue of the construction, one can avoid any restriction on the wave number in the extraction formulae. An attempt for the case when the far field pattern is given on limited angles is also given.

Article information

Source
Hiroshima Math. J., Volume 35, Number 3 (2005), 485-506.

Dates
First available in Project Euclid: 22 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1150998324

Digital Object Identifier
doi:10.32917/hmj/1150998324

Mathematical Reviews number (MathSciNet)
MR2210721

Zentralblatt MATH identifier
1106.35134

Subjects
Primary: 35R30: Inverse problems
Secondary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]

Citation

Ikehata, Masaru. The Herglotz wave function, the Vekua transform and the enclosure method. Hiroshima Math. J. 35 (2005), no. 3, 485--506. doi:10.32917/hmj/1150998324. https://projecteuclid.org/euclid.hmj/1150998324


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