  Using the framework of transformation optics, this paper presents a detailed
analysis of a non-singular square cloak for acoustic, out-of-plane shear
elastic, and electromagnetic waves. The generating map is examined in detail
and linked to the material properties of the cloak. Analysis of wave
propagation through the cloak is presented and accompanied by numerical
illustrations. The efficacy of the regularised cloak is demonstrated and an
objective numerical measure of the quality of the cloaking effect is provided.
It is demonstrated that the cloaking effect persists over a wide range of
frequencies. As a demonstration of the effectiveness of the regularised cloak,
a Young's double slit experiment is presented. The stability of the
interference pattern is examined when a cloaked and uncloaked obstacle are
successively placed in front of one of the apertures. This novel link with a
well-known quantum mechanical experiment provides an additional method through
which the quality of cloaks may be examined. In the second half of the paper,
it is shown that an approximate cloak may be constructed using a discrete
lattice structure. The geometry and material properties of the lattice are
derived from the continuum cloak. The efficiency of the approximate lattice
cloak is analysed and a series of illustrative simulations presented. It is
demonstrated that effective cloaking may be obtained by using a relatively
simple lattice structure, particularly in the low frequency regime.
