On Rayleigh wave field induced by surface stresses under the effect of gravity

Abstract

The paper is concerned with development of the asymptotic formulation for surface wave field induced by vertical surface stress under the effect of gravity in the short-wave region. The approach relies on the methodology of hyperbolic-elliptic models for the Rayleigh wave and results in a regularly perturbed hyperbolic equation on the surface acting as a boundary condition for the elliptic equation governing decay over the interior. A special value of the Poisson’s ratio v = 0.25 is pointed out, at which the effect of gravity disappears at leading order.

Surface Wave Propagation in a Rotating Doubly Coated Nonhomogeneous Half Space with Application

“…Thus, we have shown in what follows the comparison between the exact and approximate dispersion relations with a loaded end surface condition determined in Equations (54) and (32), respectively, in Figure 2, while the comparison between the exact and approximate dispersion curves with an unloaded end surface condition determined in Equations ( 55) and (28), respectively, is depicted in Figure 3. Importantly, these figures showed perfect agreement between the exact and approximate fundamental modes in both cases.…”

“…We must mention here the recent study on the influence of magnetic and gravitational fields on the propagation of surface waves on a coated viscoelastic half-space in a rotating reference [31]. We also recall the recent work on the propagation of Rayleigh waves on a mechanically loaded compressible half space [32]. The effects of surface stresses were examined, in addition to the consequences of the presence of gravity.…”

Surface Waves on a Coated Homogeneous Half-Space under the Effects of External Forces

“…Deploying the expressions expressed in Equations (13) and (14) into the boundary and continuity conditions prescribed in (17) and (19), we obtain a homogeneous system of order six with the non-zero components. The dispersion relation, or rather, the secular equation, is then obtained as follows:…”

“…We also recall their significance in material science, glazing and flooring, and above all, its applicability in the design and construction of the multi-layered media [3][4][5][6][7]. Moreover, the propagation of surface waves in coated elastic solids have inspired many research questions in recent times, having frequently arisen, for instance, in modeling seismic protection, improving highways and rail transportation quality to mention a few, see, for example, [8][9][10][11][12][13][14] and the references provided therein for more readings. The Rayleigh wave propagation, being a form of a surface wave on a homogeneous elastic half-space, is widely known to occur only for stress-free surfaces [15,16].…”

“…However, in this regard, the idea for the construction of the associated effective boundary conditions, which is also analogous to the interfacial conditions for hyperbolic-elliptic formulation to study the Rayleigh-type waves which firstly appeared in the works of Kaplunov and Kossovich [37], with further advances in [33]. The same method was also used for various propagation problems in elastic media, including coated half-spaces [8][9][10][11][12][13][14], where a pseudo-differential operator is realized, and the crack propagation scenario related to the mixed boundary value problems [22]. More so, the method provides the asymptotic formulas for the corrections to the Rayleigh wave speed in favor of the thin light coating that follows from the pseudo-differential equation along the half-space surface.…”