Free Rayleigh Wave Research Articles

Elastodynamic response of an orthotropic layered elastic half-space to time-harmonic loading.

The elastodynamics of an orthotropic half-space coated by a thin orthotropic layer is theoretically investigated in this article. We newly propose explicit expressions of free Rayleigh waves in a layered half-space that are dependent on only one unknown constant representing amplitude. The main contribution is on deriving, in a simple manner, the theoretical predictions of far-field Rayleigh wave motion arising from time-harmonic loads using elastodynamic reciprocity theorems. These are the very first closed-form exact solutions found for the forced motion of Rayleigh waves in a layered half-space of orthotropic materials. To demonstrate the theoretical results, computation of Rayleigh wave motion in a jointed rock, including a layer of quartz-schist and a half-space of soil, is considered. We present the phase and group dispersion curves superimposed with the amplitude spectra that provide useful information on wave modes, frequencies, and displacement amplitudes. The inclusion of the amplitude spectra in the dispersion curves is a significant improvement over other dispersion curves currently available in the literature. The analytical predictions are compared with numerical results found by finite element analysis, and they show excellent agreement for the cases of a uniform distributed load and a varying distributed load both applied over a strip on the layer surface. The calculations obtained in the current study could generally be very useful for applications in seismology and materials characterization of coated structures.

A closed-form solution to propagation of guided waves in a layered half-space under a time-harmonic load: An application of elastodynamic reciprocity

This article is concerned with the application of reciprocity in computing guided wave motions generated by a time-harmonic load in a layer of uniform thickness joined to a half-space. Explicit expressions for free Rayleigh waves and Love waves propagating in the layered half-space are introduced. Exact solutions of Rayleigh waves and Love waves are derived from reciprocity relations between an actual state – guided waves generated by a time-harmonic line load and a virtual state – an appropriately chosen free wave traveling in the structure. Scattered amplitudes of the wave motions are thus determined. The validation of the reciprocity approach is shown through the computation of the lowest Rayleigh wave mode in the layered half-space, which approaches the calculation of the Rayleigh surface wave in the half-space once the layer thickness approaches zero in the limit.

Comment on "An Exact Solution for Effects of Topography on Free Rayleigh Waves" by William Z. Savage

The article by Savage (2004) presents analytical solutions of Rayleigh waves for topographic surfaces and discusses amplifications on the rig top and valley bottom. Analytical solutions on a topographic surface may contribute to studies of teleseismic waves, microseism measurements, and strong ground motion amplifications. We, however, find a serious problem in the analyticity of fk ( wk ) in that report. The article attempts to derive analytical solutions based on projection from a topographic surface coordinate ( u, v ) to a flat surface …

Reply to "Comment on 'An Exact Solution for Effects of Topography on Free Rayleigh Waves' by William Z. Savage," by H. Goto and S. Sawada

I appreciate Goto and Sawada’s (Goto and Sawada, 2007) interest in my article (Savage, 2004). After reading their …

Energy Spectrum Width and Nonequilibrium Energy of Rayleigh Waves Scattered on a Two-Dimensional Near-Surface Electron Layer

We obtain expressions for the energy spectrum widths of Rayleigh waves corresponding to their deformational coupling to Fermi and Boltzmann electrons in a two-dimensional layer near the surface of a semibounded solid. We evaluate the nonequilibrium energy of Rayleigh waves that depends on these widths and is caused by the same coupling to the corresponding hot electrons. We show that this energy is independent of the degeneracy degree of the electrons and is given by the mean energy of free Rayleigh waves heated up to temperature of the electrons. We find conditions under which the thermodynamics is determined by this nonequilibrium energy of Rayleigh waves in films of a certain thickness with Fermi electrons near the surface and by the equilibrium energy of bulk phonons in thicker samples. All the results are obtained using the Keldysh diagram technique applied to the case of semibounded media.

An Exact Solution for Effects of Topography on Free Rayleigh Waves

An exact solution for the effects of topography on Rayleigh wave amplification is presented. The solution is obtained by incorporating conformal mapping into complex-variable stress functions developed for free Rayleigh wave propagation in an elastic half-space with a flat upper surface. Results are presented for free Rayleigh wave propagation across isolated symmetric ridges and valleys. It is found for wavelengths that are comparable to ridge widths that horizontal Rayleigh wave amplitudes are amplified at ridge crests and that vertical amplitudes are strongly reduced near ridge crests relative to horizontal and vertical amplitudes of free Rayleigh waves in the flat case. Horizontal amplitudes are strongly deamplified at valley bottoms relative to those for the flat case for Rayleigh wavelengths comparable to valley widths. Wave amplitudes in the symmetric ridges and valleys asymptotically approach those for the flat case with increased wavelengths, increased ridge and valley widths, and with horizontal distance from and depth below the isolated ridges and valleys. Also, prograde particle motion is predicted near crests of narrow ridges and near the bottoms of narrow valleys. Finally, application of the theory at two sites known for topographic wave amplification gives a predicted surface wave amplification ratio of 3.80 at the ridge center for a frequency of 1.0 Hz at Robinwood Ridge in northern California and a predicted surface wave amplification ratio of 1.67 at the ridge center for the same frequency at the Cedar Hill Nursery site at Tarzana in southern California.

On the controversial leaky Rayleigh wave at a water/ice interface

To the author's knowledge, no experimental data have been published to date isolating and demonstrating the existence of the controversial leaky Rayleigh wave originating at a water/ice interface. In this paper, ultrasonic laboratory results are presented demonstrating the existence of a leaky Rayleigh wave at a water/ice interface not meeting Brower's (1979) existence condition. The measured phase velocity of transient leaky Rayleigh waves is about 8% higher than the free Rayleigh wave velocity from the air/ice interface. The refracted shear wave amplitude is very small compared to the leaky Rayleigh wave. The received signal is dominated by the leaky Rayleigh wave when the receiver is simultaneously close to the source and at a distance of about one Scholte wavelength from the interface. Substantial energy is associated with the leaky Rayleigh wave indicating that it may be important to account for the contribution of the leaky Rayleigh wave in the problem of scattering from rough water/ice interfaces. The leaky Rayleigh wave is very susceptible to the presence of surface cracks in the ice. The reported findings provide physical insights into arctic acoustics and may contribute to the interpretation of arctic acoustic data. [Work supported by ONR.]

Investigation of the conditions for the existence of a leaky Rayleigh wave

The Rayleigh wave, an inhomogeneous surface wave, exists for all isotropic elastic solids. When the free surface of the solid is bounded by a liquid, a leaky Rayleigh wave, or inhomogeneous damped interface wave, may exist. In the limit that the density of the liquid goes to zero, the leaky Rayleigh wave tends to the free Rayleigh wave. However, the leaky Rayleigh wave does not exist for all liquid/isotropic solid systems. A well‐known condition for the existence of the leaky Rayleigh wave is that the velocity of sound in the liquid must be less than that of the shear in the elastic solid. Upon investigation of the secular equation for the leaky Rayleigh wave velocity, other necessary conditions for existence become apparent. The conditions are influenced by the density ratio and the ratios of the various velocities in the liquid/solid system. [Work supported by ONR.]

The disturbance due to a line source in a semi-infinite elastic medium with a single surface layer

It has long been recognized that the simple ray theory provides only a very incomplete picture of the disturbance at a point due to a sudden localized movement in an elastic medium. In this paper an investigation is made of the disturbance created by a cylindrical pulse (of and 5-type) emitted from a line source in a surface layer of elastic material overlying a semi-infinite medium of different elastic constants and density. An exact formal description of the motion is obtained in terms of a succession of pulses; the double integrals corresponding to each are evaluated by approximate methods. It is found that at a remote point (at or near the surface) there should be felt pulses corresponding to travel by each one of the minimum-time-paths predicted by the ray theory, and, in addition, a whole series of diffraction effects. Ray-path pulses are of the same type as the initial pulse, showing the same ‘jerk’ in the displacements (or in the rate-of-change of these); diffraction pulses are in general ‘blunt’, but certain of them become sharper as the surface is approached until, at the surface, they become part of a minimum-time-path disturbance. The apparent 5- and 5^-anomalies are considered in the light of these results. At a certain range interference between pulses becomes important, and at very great range the dispersive Rayleigh wave-train becomes the dominant feature. A further study of the propagation of free Rayleigh waves shows that an infinite number of modes of vibration are possible. The degree to which each is excited and the resultant motion is determined in part I I ; the importance of the Airy phases is demonstrated. The pulse representation has a natural extension to systems of any number of layers; before the corresponding interference pattern at great range can be determined it will be essential to extend our knowledge of the dispersion of free surface waves to such multilayered systems.