  An inverse obstacle problem for the wave governed by the wave equation in a
two layered medium is considered under the framework of the time domain
enclosure method. The wave is generated by an initial data supported on a
closed ball in the upper half-space, and observed on the same ball over a
finite time interval. The unknown obstacle is penetrable and embedded in the
lower half-space. It is assumed that the propagation speed of the wave in the
upper half-space is greater than that of the wave in the lower half-space,
which is excluded in the previous study: Ikehata and Kawashita (2018) to
appear, Inverse Problems and Imaging. In the present case, when the reflected
waves from the obstacle enter the upper layer, the total reflection phenomena
occur, which give singularities to the integral representation of the
fundamental solution for the reduced transmission problem in the background
medium. This fact makes the problem more complicated. However, it is shown that
these waves do not have any influence on the leading profile of the indicator
function of the time domain enclosure method.
