On the causality of the Rayleigh wave

Abstract

An explicit hyperbolic-elliptic formulation for surface Rayleigh waves is analysed with an emphasis on the causality of obtained results. As an example, a 3D moving load problem for a distributed vertical load is considered. A simple approximate solution is derived for a near-resonant regime, and the related point load solution is recast as a limiting case. It is shown that causality is characteristic only for the longitudinal wave potential along the surface, where it is governed by a hyperbolic equation modelling the small dilatation disturbances propagating at the Rayleigh wave speed.

Asymptotic Theory for Rayleigh and Rayleigh-Type Waves

“…assuming absence of surface disturbances in front of a travelling load, we obtain, see Erbaş & Şahin (2016) for further detail,…”

“…The interpretation of the formulae in this subsection, written in terms of |ξ| and |η|, relies on the implementation of the causality principle, see also Erbaş & Şahin (2016). In absence of a coating, when h = 0, the equation ( Similarly to the sub-Rayleigh case, we analyse the longitudinal and transverse cross-sections of the scaled horizontal displacement 7 Parabolic-elliptic model for a bending edge wave on a thin plate…”

“…Indeed, hyperbolicity stands for propagation along the surface, whereas ellipticity may be associated with decay into the interior. At the same time, it should be noted that hyperbolicity is characteristic only for one of the Lamé potentials along the surface, see Erbaş & Şahin (2016) for more detail.…”

Surface Displacement Field of a Coated Elastic Half-Space Under the Influence of a Moving Distributional Load

“…We will therefore investigate the far-field asymptotic behaviour of the integrals as | | ≫ 1 and * ∼ 1. It may be shown that while the first integral in (17) does not have any stationary points in the integration interval, i.e., any points for which the derivative vanishes, the second integral has one, which is obtained by taking the derivative of the exponent with respect to , given by * = 2(1 − 3 * 2 + √1 + 3 * 2 ) 9 *…”

“…It is clearly seen in these graphs that the greater values of the parameter reduces the magnitude of the displacement resulting in a more uniform distributed profile, which is an expected result (cf. [17,18]). It should be noted that the values of the parameter either corresponds to the amplitude of the profile of the load for fixed values of the material parameter and the coating thickness ℎ; or, to the coating thickness ℎ for fixed values of , and .…”