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*orientation*and supported outside an unknown obstacle and observed on the same support over a finite time interval. It is assumed that the obstacle is a perfect conductor. Two types of analytical formulae which employ a

*single*observed wave and explicitly contain information about the geometry of the obstacle are given. In particular, an effect of the orientation of the current is catched in one of two formulae. Two corollaries concerning with the detection of the points on the surface of the obstacle nearest to the centre of the current support and curvatures at the points are also given.

An inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval is considered. It is assumed that the electric field $\boldsymbol{E}$ and magnetic field $\boldsymbol{ H}$ which are solutions of the Maxwell system are generated only by a current density at the initial time located not far a way from an unknown obstacle. The obstacle is embedded in a medium like air which has constant electric permittivity $ε$ and magnetic permeability $μ$. It is assumed that the fields on the surface of the obstacle satisfy the Leontovich boundary condition $\boldsymbol{ ν}×\boldsymbol{H}-λ\,\boldsymbol{ ν}×(\boldsymbol{ E}×\boldsymbol{ ν})=\boldsymbol{ 0}$ with admittance $λ$ an unknown positive function and $\boldsymbol{ ν}$ the unit outward normal. The observation data are given by the electric field observed at the same place as the support of the current density over a finite time interval. It is shown that an indicator function computed from the electric fields corresponding two current densities enables us to know: the distance of the center of the common spherical support of the current densities to the obstacle; whether the value of the admittance $λ$ is greater or less than the special value $\sqrt{ε/μ}$.

*single*set of the temperature and heat flux on the lateral boundary for a fixed observation time is given as an observation datum. It is shown that this datum yields the minimum length of broken paths that start at a given point outside the body, go to a point on the boundary of the unknown discontinuity and return to a point on the boundary of the body under some conditions on the input heat flux, the unknown discontinuity and the body. This is new information obtained by using enclosure method.

An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.

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