  The paper considers the transmission problems for Helmholtz equation with
bodies that have negative material parameters. Such material parameters are
used to model metals on optical frequencies and so-called metamaterials. As the
absorption of the materials in the model tends to zero the fields may blow up.
When the speed of the blow up is suitable, this is called the Anomalous
Localized Reconance (ALR). In this paper we study this phenomenon and formulate
a new condition, the weak Anomalous Localized Reconance (w-ALR), where the
speed of the blow up of fields may be slower. Using this concept, we can study
the blow up of fields in the presence of negative material parameters without
the commonly used quasi-static approximation. We give simple geometric
conditions under which w-ALR or ALR may, or may not appear. In particular, we
show that in a case of a curved layer of negative material with a strictly
convex boundary neither ALR nor w-ALR appears with non-zero frequencies (i.e.
in the dynamic range) in dimensions $d\ge 3$. In the case when the boundary of
the negative material contains a flat subset we show that the w-ALR always
happens with some point sources in dimensions $d\ge 2$. These results, together
with the earlier results of Milton et al. ( [22, 23]) and Ammari et al. ([2])
show that for strictly convex bodies ALR may appear only for bodies so small
that the quasi-static approximation is realistic. This gives limits for size of
the objects for which invisibility cloaking methods based on ALR may be used.
