  We seek the response, in particular the spectral absorptance, of a
rigidly-backed periodically-(in one horizontal~~ direction) ~inhomogeneous
~layer ~composed ~of ~alternating rigid and macroscopically-homogeneous porous
portions, submitted to an airborne acoustic plane body wave. The rigorous
theory of this problem is given and the means by which the latter can be
numerically solved are outlined. At low frequencies, a suitable approximation
derives from one linear equation in one unknown. This approximate solution is
shown to be equivalent to that of the problem of the same wave incident on a
homogeneous, isotropic layer. The thickness $h$ of this layer is identical to
that of the inhomogeneous layer, the effective complex body wave velocity
therein is identical to that of the porous portion of the inhomogeneous layer,
but the complex effective mass density, whose expression is given in explicit
algebraic form, is that of the reference homogeneous macroscopically-porous
layer divided by the filling factor (fraction of porous material to the total
material in one grating period). This difference of density is the reason why
it is possible for the lowest-frequency absorptance peak to be higher than that
of a reference layer. Also, it is shown how to augment the height of this peak
so that it attains unity (i.e., total absorption) and how to shift it to lower
frequencies, as is required in certain applications.
