  We propose to homogenize a periodic (along one direction) structure, first in
order to verify the quasi-static prediction of its response to an acoustic wave
arising from mixing theory, then to address the question of what becomes of
this prediction at higher frequencies. This homogenization is treated as an
inverse (parameter retrieval) problem, i.e., by which we: (1) generate
far-field (i.e., specular reflection and transmission coefficients) response
data for the given periodic structure, (2) replace (initially by thought) this
(inhomgoeneous) structure by a homogeneous (surrogate) layer, (3) compute the
response of the surrogate layer response for various trial constitutive
properties, (4) search for the global minimum of the discrepancy between the
response data of the given structure and the various trial parameter responses
(5) attribute the homogenized properties of the surrogate layer for which the
minimum of the discrepancy is attained. The result is that: (i) at low
frequencies and/or large filling factors, the effective constitutive properties
are close to their static equivalents, i.e., the effective mass density is the
product of a factor related to the given structure filling factor with the mass
density of a generic substructure of the given structure and the effective
velocity is equal to the velocity in the said generic substructure, 2) at
higher frequencies and/or smaller city filling factors, the effective
constitutive properties are dispersive and do not take on a simple mathematical
form, with this dispersion compensating for the discordance between the ways
the inhomogeneous given structure and the homogeneous surrogate layer respond
to the acoustic wave.
