Rayleigh waves modes

A dramatic hybridization splitting around the crossing between the dispersionless adlayer mode and the substrate Rayleigh wave (and a less dramatic one around the crossing with the co = CiQg line - due to the Van Hove singularity in the projected bulk phonon density of states). 

Experimental data of Gibson and Sibener appears to confirm qualitatively these predictions at least for monolayers. The phonon linewidths were broadened around T up to half of the Brillouin zone. The hybridization splitting could not be resolved, but an increase of the inelastic transition probability centered around the crossing with the Rayleigh wave and extending up to 3/4 of the zone has been observed and attributed to a resonance between the adatom and substrate modes. 

It is noteworthy that the phonon anomaly, due to the dynamical coupling between substrate Rayleigh wave and adlayer mode, is likewise present in the bi- and even the trilayer films. It is only the Q range of the anomaly which... 

In the surface of an anisotropic solid the situation is more complicated. Pure Rayleigh waves can exist only along certain symmetry directions in which pure SV waves exist. Away from these directions, however, the two quasi-shear polarizations are not pure SV and SH therefore, although the particle motions are orthogonal, at the surface they can be weakly coupled. If the SH mode has a higher velocity than the SV, then there can be no real solution to Snell s law... 

For propagation in an isotropic medium or along a pure-mode direction of a crystal (e.g., a plane of symmetry). Equation 3.38 reduces to a Rayleigh wave, characterized by having no transverse component Ux = 0. Since Uy and Uz are 90° out of phase, the particles move in an elliptical orbit in the sagittal plane die surface motion resembles that of the ocean under the influence of a passing wave. 

Rayleigh wave devices exhibit sensitivity, when calculated on a relative basis (Av/vo), that is proportional to frequency. In contrast, SH plate mode sensitivity displays no significant frequency dependence. This difference can be attributed to the fact that with surface waves, the acoustic energy becomes distributed closer to the surface as frequency increases. Like the TSM resonator, the energy density for each propagating SH mode is, dependent upon plate thickness rather than frequency. 

The seismic wave generated by an explosion is a compression wave the vibration is parallel to the direction of propagation of the wave. On the surface, the initial movement of this wave corresponds to a lifting of the ground. However, this is not the only propagation mode for seismic waves. Three other types are possible shear wave. Love wave and Rayleigh wave. 

Fig. I. Source-receiver paths for regional earthquake seismograms used in the studies of Qiu et al. (1996) and Priestley (1999) superimposed on the major crustal subdivisions of southern Africa. A, seismograph stations jlk, earthquake locations dotted lines denote paths for the events used in the regional waveform modelling. Events 1-8 were used by Qiu et al. (1996) event 9 was used by Priestley (1999). Fundamental mode Rayleigh wave phase velocity dispersion from Priestley (1999) was measured for the SUR-BOSA and BOSA-LBTB paths. The bold lines denote the extent of the South African array.

James Pouch (2002) showed intermediate period fundamental mode Rayleigh wave phase and group velocity data measured across the Kalahari Craton and a shear-wave velocity model from inversion of these data. They find evidence for a weak low-velocity zone below 120-130 km depth. The phase velocity data of James Pouch (2002) are not significantly different from the Rayleigh wave phase velocity measured by Priestley (1999). However, such intermediate period fundamental mode dispersion data do not provide stringent constraints on mantle velocities below c. 200 km depth. 

The harmonic-oscillator and elastic-continuum models can be used to explain the presence of surface phonons (Rayleigh waves and localized surface modes of vibration) and the larger mean-square displacement of surface atoms compared to that of atoms in the bulk. 

In most materials, however, the modification of the forces at the surface is such that the surface localized modes have frequencies which lie below the frequencies of an associated bulk band with the same symmetry they have the appearance of having been peeled down from this bulk band [24]. In the usual case, the lowest energy of all these peeled -down modes derives from the bulk transverse acoustic band and is normally sagittally polarized. This dispersion branch is called the Rayleigh wave (RW) because it was predicted by Lord Rayleigh from continuum wave theory over a century ago [38]. Helium atom scattering experiments on virtually every material so far investigated have detected the RW on clean crystalline surfaces. 

Figure 16. Surface phonon dispersion curves for LiF(OOl). The calculated bulk bands are indicated by the vertical-striped regions. The surface localized modes are shown by heavy solid lines, whereas the resonances lying within bulk bands are given by thinner solid lines. The mode label S refers to the Rayleigh wave, to the longitudinal resonance, Sg to the crossing resonance, and S2, S3, and S4 to optical modes. (Reproduced from Fig. 2 of Ref. 58, with permission of Elsevier Science Publishers.)...

Figure 18. Surface phonon dispersion for RbCl(OOl). The shaded regions correspond to the surface projected density of states, with the darker shades representing higher state densities. The Rayleigh wave, crossing resonance, and optical mode are indicated by RW, CR, and 2, respectively. (Reproduced from Ref. 119.)...

These three salts are isobaric because the masses of the anions and cations are nearly the same. Although crossing resonances were somewhat unexpectedly observed in the studies of KBr and to a lesser extent in Rbl, for these compounds an Sg mode, the folded extension of the Rayleigh wave as in Fig. 6, was anticipated [70]. For the isobaric case, this vibration appears to be a tme surface mode at least for part of the way across the SBZ and not just a resonance with a high density of states at the surface [70],... 

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