  In this paper, we investigate on the direct and inverse scattering problem by
an unbounded penetrable rough surface in a lossless medium. The cases that the
transmission coefficient $\mu\neq1$ and $\mu=1$, which creates certain
difficulties in the direct and inverse problem, respectively, are both
considered. We first estalish the well-posedness of the direct problem using
the integral equation method through an elaborate analysis. Then we carefully
consider the singularity of the solutions to the problem with incident point
source or hypersingular point source, where a simple and novel perspective is
given for the derivation of the singularity. Finally, a global uniqueness
result is proven for the inverse problem on the unique determination of the
unbounded rough surface, the transmission coefficient and the wave number in
the lower half plane from the measurements of the near field only on a line
segment above the interface at a fixed frequency.
