Reduced equations for models of laminated materials in thin domains. II

Abstract

In a companion paper, we considered a scalar wave equation on a thin, laminated, three‐dimensional plate. We showed that if the plate was sufficiently thin, then there exists a hierarchy of two‐dimensional equations whose dynamics model the dynamics of the full plate, each of which successively lengthens the time interval over which the approximation is valid. In certain cases, these approximating equations may formally be ill‐posed. In this paper, we consider modifications of the approximating equations which are themselves well‐posed, and which qualitatively afford the same approximation. We also present an algorithm to compute the coefficients in the approximating equations in closed form, and show numerical evidence that the estimates on the efficacy of the approximating equations are sharp.