  Transformation optics constructions have allowed the design of cloaking
devices that steer electromagnetic, acoustic and quantum waves around a region
without penetrating it, so that this region is hidden from external
observations. The material pa- rameters used to describe these devices are
anisotropic, and singular at the interface between the cloaked and uncloaked
regions, making physical realization a challenge. These singular material
parameters correspond to singular coefficient functions in the partial
differential equations modeling these constructions and the presence of these
singularities causes various mathematical problems and physical effects on the
interface surface. In this paper, we analyze the two dimensional cloaking for
Helmholtz equation when there are sources or sinks present inside the cloaked
region. In particular, we consider nonsingular approximate invisibility cloaks
based on the truncation of the singular transformations. Using such truncation
we analyze the limit when the approximate cloaking approaches the ideal
cloaking. We show that, surprisingly, a non-local boundary condition appears on
the inner cloak interface. This effect in the two dimensional (or cylindrical)
invisibility cloaks, which seems to be caused by the infinite phase velocity
near the interface between the cloaked and uncloaked regions, is very different
from the earlier studied behavior of the solutions in the three dimensional
cloaks.
