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Time-Dependent Polarization Tensors: Derivation of Asymptotic Expansions for the Transient Wave Equation

by L. Baldassari

(Report number 2019-16)

Abstract
This paper aims at introducing the concept of time-dependent polarization tensors for the Helmholtz equation. One considers solutions to the frequency-domain Helmholtz equation in two and three dimensions. Based on layer potential techniques one provides for such solutions a rigorous systematic derivation of complete asymptotic expansions of perturbations resulting from the presence of diametrically small targets with constitutive parameters different from those of the background and size less than the operating wavelength. Such asymptotic expansions are based on careful and precise estimates of the dependence with respect to the frequency of the remainders. By truncating the high frequencies of the Fourier transform of these asymptotic expansions, one recovers the time-domain formulas. The threshold of the truncation is determined by the size of the target. The time-dependent asymptotic expansions are written in terms of the new concept of time-dependent polarization tensors. It is expected that our results will find important applications for developing time-domain algorithms for target classification.

Keywords: Helmholtz equation, transient imaging, small inclusion, full-asymptotic expansions, polarization tensors, boundary integral equation, Fourier transform

@Techreport{B19_820,
  author = {L. Baldassari},
  title = {Time-Dependent Polarization Tensors: Derivation of Asymptotic Expansions for the Transient Wave Equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-16.pdf },
  year = {2019}
}

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