Abstract:

Acoustic wave propagation in a medium with microstructure is considered. By microstructure, it is meant that the medium properties change on a length scale that is short compared to an acoustic wavelength. An effective wave equation is derived for the ``long'' scale acoustic field using the method of multiple scales (i.e., homogenization). It is shown that the standard approximation breaks down in a few wavelengths. The effective equation is renormalized in an attempt to obtain a theory, which is valid for long propagation distances. The resulting equation is related to that derived by McDevitt et al. (McDevitt et al., ``Dispersive Elastodynamic Models of Periodic Structures Using a High-Order Homogenization Approach,'' Proceedings of Fourth U.S. National Congress on Computational Mechanics, San Francisco, August 6--8, 1997). Asymptotic validity of the equation for finite propagation distances is demonstrated. Over larger distances, solutions of this effective equation exhibits dispersion and attenuation. Strong scattering terms are shown to dominate these effects, however, when the medium is not statistically homogeneous. This asymptotic theory is validated by comparison to reference numerical calculations. [Work supported by NSF.]