Hiroshima Mathematical Journal

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

Masaru Ikehata

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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

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

First available in Project Euclid: 22 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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