Discussion: “Boundary Element Analysis of Multiple Scattering Waves in High Performance Concretes” (Sato, Hirotaka, Kitahara, Michihiro, and Shoji, Tetsuo, 2005, ASME J. Appl. Mech., 72, pp. 165–171)

Abstract

This discussion intends to make comments on the recent article of Sato et al. (1) regarding their descriptions of self-consistency conditions for the effective medium in the multiple scattering theory. Equations (13) and (14) in Ref. 1 were obtained originally by Yang and Mal (2) by extending the static generalized self-consistent method (GSCM) with recourse to Waterman and Truell’s theory (3). Recently, the dynamic GSCM was recast (4) based on a simple consideration of wave energy in the model applying the energy theorem for the scattering in absorbing media (5), and also the equivalence between Eqs. (13) and (14) andA1*(Kp,e1)=B1*(Ksv,e1)=0was shown. In the dynamic GSCM, the self-consistency condition of the effective medium (or the dispersion relation) reduces to vanishing of the wave energy extinction in the model. It should be noted that this can be further generalized to different self-consistent-type formulations of the multiple scattering problem. A simple but fairly general way to show this might be to express the Ewald formula (3) in the self-consistent formK=K+(2πnoK)F(e1)or thereforeF(e1)=0where no is the scatter number density and F(e1) denotes the forward scattering amplitude defined in the effective medium with consideration of multiple scattering effect. Note that subscript denoting the wave mode is suppressed for brevity.