Asymptotic Theory for Rayleigh and Rayleigh-Type Waves

Abstract

Abstract:Explicit asymptotic formulations are derived for Rayleigh and Rayleigh-type interfacial and edge waves. The hyperbolic-elliptic duality of surface and interfacial waves is established, along with the parabolic-elliptic duality of the dispersive edge wave on a Kirchhoff plate. The effects of anisotropy, piezoelectricity, thin elastic coatings, and mixed boundary conditions are taken into consideration. The advantages of the developed approach are illustrated by steady-state and transient problems for a moving load on an elastic half-space.

Explicit formulation for the Rayleigh wave field induced by surface stresses in an orthorhombic half-plane

“…Examples of such occurrence include the near-resonance regime of a moving load or a far-field approximation, for more details see [14]. For the sake of definiteness, we consider waves propagating in the positive direction according to the moving co-ordinate ξ = x 1 − c R t, c R > 0, associated with the Rayleigh wavefront.…”

“…where j = 1, 2. It should be emphasized that the presence of the leading terms of order −1 is due to the near-resonant nature of the excitation, given that a homogeneous problem is expected at leading order [14].…”

“…For this setup, we let for the loading f 1 = 0 and f 2 = f (x 1 − ct). Following the original approach in [2], we first derive the problem general solution in the subsonic regime c < c R and then draw a comparison with the results of the asymptotic formulation letting the speed of the load approach the resonant surface wave speed (see also [14]). In going through the derivations, it will become apparent that the asymptotic model is basically obtained by perturbation of this steady-state moving load problem solution around Rayleigh's critical speed.…”

“…In parallel, parabolic-elliptic models were constructed for bending edge Rayleigh-type waves [16]. Some more results on the asymptotic theories for Rayleigh and Rayleigh-type waves may be found in [13] and [14].…”

Composite wave models for elastic plates

“…Consider antisymmetric deformation of an elastic layer of thickness 2h (−∞ x 1 , x 2 ∞, −h x 3 h) subject to prescribed normal stresses ± 1 2 P (x 1 , x 2 , t) at faces x 3 = ±h, see Figure 1, starting from shortened forms of 3D dynamic equations in linear elasticity exposed, in particular, in the paper [7], the book [9], and in the annual volume chapter [21]; in this section we use the results of these publications without further reference to them. where L and T are typical wavelength and time scale, respectively, E is Young's modulus, and ρ is mass density.…”