  This paper concerns the analysis of a passive, broadband approximate cloaking
scheme for the Helmholtz equation in ${\mathbb R}^d$ for $d=2$ or $d=3$. Using
ideas from transformation optics, we construct an approximate cloak by
``blowing up" a small ball of radius $\epsilon>0$ to one of radius $1$. In the
anisotropic cloaking layer resulting from the ``blow-up" change of variables,
we incorporate a Drude-Lorentz-type model for the index of refraction, and we
assume that the cloaked object is a soft (perfectly conducting) obstacle. We
first show that (for any fixed $\epsilon$) there are no real transmission
eigenvalues associated with the inhomogeneity representing the cloak, which
implies that the cloaking devices we have created will not yield perfect
cloaking at any frequency, even for a single incident time harmonic wave.
Secondly, we establish estimates on the scattered field due to an arbitrary
time harmonic incident wave. These estimates show that, as $\epsilon$
approaches $0$, the $L^2$-norm of the scattered field outside the cloak, and
its far field pattern, approach $0$ uniformly over any bounded band of
frequencies. In other words: our scheme leads to broadband approximate cloaking
for arbitrary incident time harmonic waves.
