Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval

This paper considers an inverse problem for the classical wave equation in an exterior domain. It is a mathematical interpretation of an inverse obstacle problem which employs the dynamical scattering data of acoustic wave over a finite time interval. It is assumed that the wave satisfies a Robin type boundary condition with an unknown variable coefficient. The wave is generated by the initial data localized outside the obstacle and observed over a finite time interval at the same place as the support of the initial data. It is already known that, using the enclosure method, one can extract the maximum sphere whose exterior encloses the obstacle, from the data. In this paper, it is shown that the enclosure method enables us to extract also: (i) a quantity which indicates the deviation of the geometry between the maximum sphere and the boundary of the obstacle at the first reflection points of the wave; (ii) the value of the coefficient of the boundary condition at an arbitrary first reflection point of the wave provided, for example, the surface of the obstacle is known in a neighbourhood of the point. Another new obtained knowledge is that: the enclosure method can cover the case when the data are taken over a sphere whose centre coincides with that of the support of an initial data and yields corresponding results to (i) and (ii).