Dynamical homogenization of a transmission grating

Abstract

A periodic assembly of acoustically-rigid blocks (termed 'grating'), situated between two half spaces occupied by fluid-like media, lends itself to a rigorous theoretical analysis of its response to an acoustic homogeneous plane wave. This theory gives rise to two sets of linear equations, the first for the amplitudes of the waves in the space between successive blocks, and the second for the amplitudes of the waves in the two half spaces. The first set is solved numerically to furnish reference solutions. The second set is submitted to low-frequency approximation procedure whereby the pressure fields are found to be those for a configuration in which the grating becomes a homogeneous layer of the same thickness as the height of the blocks in the grating. A simple formula is derived for the constitutive properties of this layer in terms of those of the fluid-like medium in between the blocks. The homogeneous layer model scattering amplitude transfer functions and spectral reflectance, transmittance and absorptance reproduce quite well the corresponding rigorous numerical functions of the grating over a non-negligible range of low frequencies. Due to its simplicity, the homogeneous layer model enables theoretical predictions of many of the key features of the acoustic response of the grating.

Effective medium description of the resonant elastic-wave response at the periodically-uneven boundary of a half-space

“…Homogenization of empirical nature is a very old and even contemporary practice in geophysical problems. At present, homogenization has developed into a full branch of applied mathematics [5,15,17,24,26,31] which has attracted the attention of physicists (mostly solid-state) interested in developing materials with unusual properties [12,15,34,38,43,48]. These so-called 'metamaterials' are usually composed of periodic, or nearly-periodic assemblies of resonating elements similar to the blocks or buildings (thought of as single-degree-of-freedom oscillators) in [9].…”