Intersections of the Lamb mode dispersion curves of free isotropic plates.

The intersections between Lamb mode dispersion curves of free isotropic plates at real values of frequency and wave number are examined for the full allowed range of Poisson's ratio σ. The generic intersections between the dispersion curves for symmetric and anti-symmetric branches are classified into three types. Type F intersections are conditioned by the two additional real solutions of Rayleigh's cubic equation that occur for σ<0.26308. Types I and II intersections occur for all values of σ, and are distinguished by the vanishing or divergence of the tangent functions in the defining equations for the Lamb modes. A brief discussion is provided of intersections between branches of like symmetry and additional intersections between unlike symmetry branches that occur for special values of σ.

Optimized determination of elastic constants of crystals and their uncertainties from surface Brillouin scattering.

Surface Brillouin scattering of light allows the angular-dependent velocities of Rayleigh surface acoustic waves (SAW), pseudo-SAW and longitudinal lateral waves (L) on the surface of an opaque crystal to be measured, and the elastic constants thereby determined. Closed form expressions exist for the surface wave velocities in high symmetry directions on crystallographic symmetry planes, and these have been exploited in the past for obtaining the values of the elastic constants. This paper describes a procedure for obtaining an optimized set of elastic constants from SAW, pseudo-SAW and L velocities measured in arbitrary directions in the (001) and (110) surfaces of cubic crystals. It does so by affecting a linearization of the numerically determined angular-dependent SAW and pseudo-SAW velocities near the best fit, and using analytic expressions for the L velocity. The method also generates covariance ellipsoids, from which the uncertainties in the determined values of the elastic constants can be read off. The method is illustrated using surface Brillouin scattering data to obtain the room-temperature elastic constants C11, C12 and C44 of the cubic crystals VC0.75 and Rh3Nb.

Supersonic surface acoustic waves on the 001 and 110 surfaces of cubic crystals.

Criteria are reported here for the existence of supersonic surface acoustic waves (SSAW) on the (001) and (110) surfaces of cubic crystals. These are the common crystal cuts for which SSAW have been observed experimentally using surface Brillouin scattering and other techniques. Two categories of SSAW are distinguished. Symmetry protected SSAW exist by virtue of being located in high symmetry crystallographic directions for which the coupling to the phase matched bulk wave, which would otherwise result in their attenuation, is suppressed by symmetry. Secluded SSAW occur in lower-symmetry directions, where the reason for the vanishing of their coupling to their phase matched bulk wave is less evident. The stability domain for the elastic constant ratios a=C11/C44 and b=C12/C44 is subdivided into a number of regions in which various symmetry protected and secluded SSAW exist. Some of the boundaries between these regions are expressible in analytical form, others have been established purely numerically.

Dispersion of an acoustic pulse passing through a large-grained polycrystalline film.

Propagation of a short acoustic pulse through a polycrystalline film comprised of large randomly oriented elastically anisotropic grains is analyzed theoretically. For average grain size much larger than the film thickness, a short acoustic pulse launched normally into the film will traverse each grain in a time determined by the acoustic slowness in the direction normal to the film, which will depend on the local grain orientation. A typical measurement averages over a large number of grains resulting in the broadening of the composite output pulse. The resulting pulse shape is characterized by distinct features related to stationary values of the directionally dependent acoustic slowness of the crystalline material. Maxima and minima in the slowness yield discontinuities in the pulse shape, while saddle points yield logarithmic singularities. For cubic and hexagonal crystals, power law singularities result from cones of directions in which the slowness is a maximum or minimum. Numerical results, taking into account Gaussian broadening of the input pulse, are presented for thin film materials commonly encountered in picosecond ultrasonic experiments, such as copper, gold, and aluminum.

Acoustic field radiated into a transversely isotropic solid from a small aperture spherical surface.

The acoustic field modelling reported in this paper finds application in the design of a scanning probe tip for measuring the near-surface elastic properties of solids and surface structures at high frequencies and with high spatial resolution. The underlying concept is for a longitudinally polarized pulse to be launched from a spherically-shaped portion of the upper surface of the pyramidal or conical shaped tip, and focused towards the narrow lower end. The change in the reflectivity when the narrow end is brought into contact with a solid will provide a measure of the local frequency dependent compliance of that solid. The calculations assume the material from which the tip is fabricated to be transversely isotropic, with symmetry axis coinciding with the axis of the tip. The main issue addressed in this paper is the role of the curvature of the radiating surface and anisotropy of the medium in determining the focal length and focal spread of the radiated field. Two complementary approaches are taken, firstly the discretization of the equations of motion on an irregular mesh of around 3×10(5) triangular elements and solution using the commercial FE package ABAQUS/Explicit, and secondly an analytical approach based on ray tracing and a Green's function method exploiting the angular spectrum method and stationary phase approximation in its evaluation. Consistency is achieved between these approaches regarding the characteristics of the focal region. With the combination of the two approaches it is thus possible to model the wave field from low frequencies, where the FE method is computationally economical and able to handle complex geometries, to high frequencies, where advantage increasingly lies with ray tracing and the Green's function method.

Ray splitting in the reflection and refraction of surface acoustic waves in anisotropic solids.

This paper examines the conditions for, and provides examples of, ray splitting in the reflection and refraction of surface acoustic waves (SAW) in elastically anisotropic solids at straight obstacles such as edges, surface breaking cracks, and interfaces between different solids. The concern here is not with the partial scattering of an incident SAW's energy into bulk waves, but with the occurrence of more than one SAW ray in the reflected and/or transmitted wave fields, by analogy with birefringence in optics and mode conversion of bulk elastic waves at interfaces. SAW ray splitting is dependent on the SAW slowness curve possessing concave regions, which within the constraint of wave vector conservation parallel to the obstacle allows multiple outgoing SAW modes for certain directions of incidence and orientation of obstacle. The existence of pseudo-SAW for a given surface provides a further channel for ray splitting. This paper discusses some typical material configurations for which SAW ray splitting occurs. An example is provided of mode conversion entailing backward reflection or negative refraction. Experimental demonstration of ray splitting in the reflection of a laser generated SAW in GaAs(111) is provided. The calculation of SAW mode conversion amplitudes lies outside the scope of this paper.

Effect of spatial dispersion on transient acoustic wave propagation in 3D.

Spatial dispersion is the variation of wave speed with wavelength. It sets in when the acoustic wavelength approaches the natural scale of length of the medium, which could, for example, be the lattice constant of a crystal, the repeat distance in a superlattice, or the grain size in a granular material. In centrosymmetric media, the first onset of dispersion is accommodated by the introduction of fourth order spatial derivatives into the wave equation. These lead to a correction to the phase velocity which is quadratic in the spatial frequency. This paper treats the effect of spatial dispersion on the point force elastodynamic Green's functions of solids. The effects of dispersion are shown to be most pronounced in the vicinity of wave arrivals. These lose their singular form, and are transformed into wave trains known as quasi-arrivals. The step and ramp function wave arrivals are treated, and it is shown that their unfolded quasi-arrival forms can be expressed in terms of integrals involving the Airy function.

On interference effects in the elastodynamic Green's functions G33(x, omega) of Si and GaAs.

In this paper, we analyze interference effects present in the elastodynamic Green's functions G33(x,omega) of the cubic crystals Si and GaAs, which are associated with folded portions of the wave surface of the slow transverse (ST) acoustic mode. G33(x,omega) represents the three dimensional extension of the amplitude distribution imaged in the transmission acoustic microscopy of these crystals. The intensity contrast for oscillations of a particular wave vector k in the interference pattern is determined essentially by the 3D Fourier transform of G33(x,omega)G33*(x,omega). According to the Fourier autocorrelation theorem, that transform is equivalent to the autocorrelation function of the corresponding distribution G(33)(k,omega) in k-space. We show that due to the linear mapping between k-space and the slowness vector s-space, the interference phenomena discussed here are related to geometrical features of the slowness surface of the ST mode. We present calculations of these effects based on the angular spectrum technique.

The study of guided waves in surfaces and thin supported films using surface Brillouin scattering and acoustic microscopy.

This paper reviews the use of surface Brillouin scattering (SBS) and acoustic microscopy (AM) in studying the surface dynamics of solids in order to obtain information about the near-surface elastic properties of solids and thin supported films. The vibrational modes that are probed by these means include Rayleigh surface and pseudo-surface acoustic waves, longitudinal lateral waves (surface skimming bulk longitudinal waves) and various thin film guided modes, such as Sezawa and Love waves. SBS is the inelastic scattering of light, mediated by thermodynamic fluctuations in the surface elevation and near surface elastic strains. The scattering cross-section is conveniently expressed in terms of Fourier domain elastodynamic Green's functions. AM depends on the insonification of a surface through a coupling fluid, and the resulting excitation and subsequent decay of the various surface modes. The complex reflectivity of the fluid-loaded surface, and the line and point force surface Green's functions are invoked in the interpretation of different modalities of AM, yielding much the same information about the surface dynamics. The focus in this paper is on the Green's function approach. A number of illustrative examples, drawn from the authors' research, are provided.

Inversion of acoustic diffraction fields in anisotropic solids.

We show that the fast Fourier transform (FFT) technique provides a computationally efficient method of calculating 2D amplitude and phase images of complex wave fields generated and measured in elastically anisotropic solids by phase sensitive acoustic microscopy. Further, we discuss how this technique can be used to treat inverse problems such as source reconstruction, image quality assessment, and the determination of elastic constants.

Principal surface wave velocities in the point focus acoustic materials signature V(z) of an anisotropic solid.

This paper deals with the point focus beam (PFB) acoustic materials signature V(z) of an anisotropic solid, and in particular how it tends to be dominated by a limited number of principal surface rays. These rays are associated with propagation directions in which the Rayleigh wave (RW), pseudo-surface acoustic wave (PSAW) or a lateral wave slowness has an extremum. The phenomenon is interpreted in terms of the complex azimuthally averaged reflectance function of the surface, and also explained on the basis of a ray model. We illustrate the phenomenon with a number of examples, pertaining to the surfaces of single crystal copper and a carbon-fibre epoxy composite. In the case of copper, which has a much larger acoustic impedance than the water couplant, the oscillations in V(z) are dominated by principal RW and PSAW, whereas for the composite there is no RW or pseudo-SAW to be discerned with acoustic microscopy (AM), and V(z) is dominated by principal lateral waves. The utility of PFB AM in the study of anisotropic solids is further elaborated with examples showing how V(z) is sensitive to surface orientation, and how V(z) is affected by the presence of a surface over layer. The phenomena examined in this paper expand the scope for determining materials characteristics, such as elastic constants, crystallographic orientation, residual stress and over layer properties, from PFB V(z) measurements.

Characteristic features of the velocity dispersion of surface acoustic waves in anisotropic coated solids.

Basic patterns of the velocity versus wavenumber dispersion of the surface waves in solids coated by a relatively light or dense, "slow" or "fast" layer are discussed in the general case of an arbitrary anisotropy of substrate and coating materials. The onset of the subsonic wave branch, characterized by either a speeding or a slowing trend, is examined. Competitive tendencies, which pertain to the low-frequency dispersion in the case of dense "fast" layer, are revealed.

Tomography of joint P-wave traveltime and polarization data: a simple approach for media with low to moderate velocity gradients.

An elastic wave tomography method utilizing joint traveltime and polarization data is proposed that is computationally simpler than the existing methods [Hu and Menke, Geophys. J. Int. 110, 63 (1992); Farra and Begat, Geophys. J. Int. 121, 371 (1995)]. In the linearization problem for the use of polarization data, we start with ray perturbation theory and assume that the medium is weakly inhomogeneous. Then the problem formulation for polarization data is approximately expressed as a linear integral of the gradient of the medium slowness perturbation along a reference ray. We call this a quasi-linear approximation which ignores the effect of the perturbation of the ray position on the first-order perturbation of the ray slowness vector. To efficiently obtain the solution for multi-data sets, a quadratic objective functional is constructed by including the data misfit terms and a model constraint term. Then a new conjugate gradient type of iterative reconstruction algorithm is developed to solve this minimization problem. This algorithm is also an extension of the conjugate gradient approach for standard least-squares problems. The feasibility and capability of the proposed tomography method is illustrated by conducting both noise-free and noisy synthetic experiments in a cross-hole geometry. The numerical results demonstrate that the additional use of polarization data not only improves the image quality, but also has a stabilizing effect on the iterative tomography solution. However, the limitation of the method is that it becomes inaccurate if the velocity variations in the medium change rapidly with position.