Acta Mech 207, 235–243 (2009) DOI 10.1007/s00707-008-0125-4 Stanislaw J. Matysiak · Radoslaw Mieszkowski · Dariusz M. Perkowski Surface waves in a periodic two-layered elastic half-space with the boundary normal to the layering Received: 21 February 2008 / Revised: 9 October 2008 / Published online: 27 November 2008 © Springer-Verlag 2008 Abstract The paper deals with the problem of surface waves in a vertically layered elastic half-space. The transmitting medium is composed of periodically repeated two-constituent laminae, and its boundary is assu- med to be normal to the layering. The problem is solved using an approximate model proposed by Wo´ zniak (Int J Eng Sci 25:483–499, 1987) and Matysiak and Wo´ zniak (Int J Eng Sci 25:549–559, 1987), which is referred to as the homogenized model with microlocal parameters. The velocity of the surface wave is obtained as a function of geometric and dynamic properties of the laminae. The variation of the surface wave velocity with respect to composite constituents is illustrated by numerical examples. 1 Introduction Layered composites with periodic structures represent an important type of modern materials used in various branches of technology [1]. Moreover, many rocks and soils are stratified and clearly piece-wise homogeneous [2]. These are various metamorphic rods with fabrics having parallel arrangements of flat minerals: bedded rocks like shale–sandstone or slate–sandstone, or layered soils like varved clays or flotation wastes [3]. The propagation of surface waves in elastic homogeneous bodies received much attention [4–8]. The problems of wave propagation in anisotropic bodies were considered in [9–11]. The dynamical problems of layered elastic composites are much more complicated. The behavior of structures, which are made of a large number of repeated laminae, can be described within the framework of elasticity theory by partial differential equations with discontinuous, periodic coefficients. The analytical and numerical treatments of periodically layered solids (based directly on the dynamic theory of elasticity) are rather awkward due to a large number of continuity conditions at the interfaces. It seems thus to be more appropriate to apply approximate models that describe the mechanical behavior of the periodically layered structure. Many models have been proposed using averaging procedures: classical asymptotic homogenization [12–16] (based on the mixture theory [17], the theory of thick plates [18], some physical assumptions [19] and matrix methods [20]), the method of tolerance modeling [21], and nonasymptotic treatments (based on the nonstandard analysis [22] adopted to periodically layered elastic composites [26]). The last of these models, called the homogenized model with microlocal parameters, is described by unknown macrodisplacements related with the averaged fields and additional unknown microlocal parame- ters pertinent to the layered structure of composites. The microlocal parameters are governed by a system of S. J. Matysiak (B )· R. Mieszkowski Institute of Hydrogeology and Engineering Geology, Faculty of Geology, University of Warsaw, Al. ˙ Z wirki i Wigury 93, 02-089 Warsaw, Poland E-mail: s.j.matysiak@uw.edu.pl D. M. Perkowski Faculty of Mechanical Engineering, Bialystok University of Technology, Wiejska St., 45C, 15-351 Bialystok, Poland