  Wave interaction with graded metamaterials exhibits the phenomenon of rainbow
reflection, in which broadband wave signals slow down and separate into their
frequency components before being reflected. This phenomenon has been
qualitatively understood by describing the wave field in the metamaterial using
the local Bloch wave approximation (LBWA), which locally represents the wave
field as a superposition of propagating wave solutions in the cognate infinite
periodic media (so-called Bloch waves). We evaluate the performance of the LBWA
quantitatively in the context of two-dimensional linear water-wave scattering
by graded arrays of surface-piercing vertical barriers. To do this, we
implement the LBWA numerically so that the Bloch waves in one region of the
graded array are coupled to Bloch waves in adjacent regions. This coupling is
computed by solving the scattering of Bloch waves across the interface between
two semi-infinite arrays of vertical barriers, where the barriers in each
semi-infinite array can have different submergence depths. Our results suggest
that the LBWA accurately predicts the free surface amplitude across a wide
range of frequencies, except those just above the cutoff frequencies associated
with each of the vertical barriers in the array. This highlights the importance
of decaying Bloch modes above the cutoff in rainbow reflection.
