Slowness Curves Research Articles

Two-Dimensional Wave Front Shape Caused by a Point Impulsive Source.

The partial differential equation describing the shape of the wave front caused by a two-dimensional point impulsive source in transversely isotropic elastic media is formulated and solved. The important role played by the slowness curve in establishing key features of the wave front is discussed. Recent research on the existence of slowness curve inflection points, experimental verification of anisotropic elastic wave front shapes in crystals, and the wave front from a circular distributed source are briefly discussed.

An approximation to the planar harmonic Green’s function at branch points in wave-number domain

An arbitrary anisotropic piezoelectric half-space is considered, and harmonic wave field in both time and space on its surface. A planar harmonic Green’s function, dependent on the wave number, is involved in the matrix relation between complex amplitudes of the applied traction and the resulting particle displacement on the surface of the half-space. An approximation to this function is derived, being valid in the vicinity of the cutoff slownesses of bulk waves, for certain cases of the shape of the slowness curves at the cutoffs. The Stroh formalism is adopted and a new theorem is proved for this purpose. This approximation is useful in any analysis of the wave field carried out by means of the methods of complex variable, for example in finding the singular points or determining the type of branch points (at the cutoff wave numbers of bulk waves) on the complex plane of a wave number. The branch points of square and cubic root type are discussed. The approximation parameters are derived from the matrix dependences at a given branch point. The computed examples are presented for effective surface dielectric permittivity of quartz substrate.

Anisotropy of sound velocity for a new PbB 4 O 7 crystal as determined by the laser ultrasonic technique

O7 crystal is a new nonlinear optical crystal. For the first time the anisotropy of the velocities of its longitudinal and surface acoustic waves (SAWs) are determined by laser ultrasonic technique. The velocities of surface waves for X-, Y-, and Z-cut crystals are also calculated. The theoretical calculations of slowness curves are in good agreement with experimental results. The SAW slowness curve is elliptical for Y- or Z-cut crystal wafers, and circular for an X-cut wafer.

Two-dimensional elastic wave front shape caused by a circular source in transversely isotropic media

The two-dimensional wave front emitted by a circular impulsive source in a transversely isotropic elastic solid is analyzed with respect to its geometric shape. It is found that the concept of slowness curve, previously used to explain features of a point source wave front, can be generalized so as to include a circular source. This generalized slowness curve is time-dependent.

Modelling surface waves in anisotropic structures I. Theory

Surface-wave theory in generally-anisotropic laterally-homogeneous media is partially reformulated in order to obtain intuitively-expected extensions of classic body-wave ideas such as Maslov plane-wave summation, the geometrical-ray/WKBJ limit and source-receiver reciprocity. This is done using the ‘reversal’ symmetry of Chapman [Chapman, C.H., 1994. Reflection/transmission coefficient reciprocities in anisotropic media. Geophys. J. Int. 116, 498–501] to generalize the point-source treatment of Kennett [Kennett, B.L N., 1983. Seismic Wave Propagation in Stratified Media. Cambridge Univ. Press, Cambridge, UK] to a stack of anisotropic layers with complex elastic parameters, lateral slowness and frequency. The 2D-integral representation over horizontal slownesses is reduced by residues to a 1D integral over each mode's slowness or dispersion curve at fixed frequency and this may be considered a 1D Maslov summation over local plane waves tangential to the phase front. The residue calculation involves a modified form of the usual variational principle, in which the Lagrangian now contains reversed modes. The 1D slowness integral may be reduced by stationary-phase arguments to the geometrical-ray or WKBJ limit, provided the dispersion-surface curvature does not vanish. This limit satisfies reciprocity, as the reversal symmetry shows. Dimples on the dispersion surface will correspond to folds on the phase front and multiple arrivals. An appendix contains a discussion of orthogonality of the surface-wave modes in relation to the various wave-equation symmetries.

Foundations of anisotropy for exploration seismics : K. Helbig, (Elsevier Science/Pergamon; Handbook of Geophysical Exploration, Seismic Exploration, Vol. 22), ISBN (hardback) 0 08 037224 4, price £75.00 (US$120.00), 1994, 486 pp, index

Fundamentals tools for the description of wave propagation under piecewise homogeneous anisotropic conditions elasticity elastic waves - the dispersion relation and some generalities about slowness and wave surfaces stability constraints one-parameter expressions for the slowness surfaces of transversely isotropic media and the slowness curves in the planes of symmetry of orthorhombic media one-parameter expressions for the wave curves in the symmetry planes of orthorhombic media squared slowness surfaces and squared slowness curves causes of anisotropy - periodic fine layering anisotropy and seismic exploration eigentensors of the elastic tensor and their relationship with material symmetry.

The elastic and piezoelectric properties of a lithium triborate single crystal

From the velocity of sound measurements, obtained using Brillouin scattering and ultrasound techniques, the elastic and piezoelectric coefficients of lithium triborate (LiB3O5; LBO) single crystals have been determined. Based upon the sound propagation equations and above results, the slowness curves in (100), (010), and (001) crystalline planes are calculated and presented. Some properties of sound propagation and electromechanical coupling in the crystal are also discussed.

Forbidden directions for inhomogeneous pure shear waves in dissipative anisotropic media

In this work we investigate the wave‐propagation properties of pure shear, inhomogeneous, viscoelastic plane waves in the symmetry plane of a monoclinic medium. In terms of seismic propagation, the problem is to describe SH‐waves traveling through a fractured transversely isotropic formation where we assume that the waves are inhomogeneous with amplitudes varying across surfaces of constant phase. This assumption is widely supported by theoretical and experimental evidence. The results are presented in terms of polar diagrams of the quality factor, attenuation, slowness, and energy velocity curves. Inhomogeneous waves are more anisotropic and dissipative than homogeneous viscoelastic plane waves, for which the wavenumber and attenuation directions coincide. Moreover, the theory predicts, beyond a given degree of inhomogeneity, the existence of “stop bands” where there is no wave propagation. This phenomenon does not occur in dissipative isotropic and elastic anisotropic media. The combination of anelasticity and anisotropy activates these bands. They exist even in very weakly anisotropic and quasi‐elastic materials; only a finite value of Q is required. Weaker anisotropy does not affect the width of the bands, but increases the threshold of inhomogeneity above which they appear; moreover, near the threshold, lower attenuation implies narrower bands. A numerical simulation suggests that, in the absence of material interfaces or heterogeneities, the wavefield is mainly composed of homogeneous waves.

Remote sensing of sheet metal texture in aluminum alloys

Metal forming operations such as deep drawing, stretching, bending, hemming, etc. are all significantly influenced by the degree of crystallographic anisotropy (texture) present within the workpiece. An effort is underway at Sandia National Laboratories/CA to develop two noncontact ultrasonic systems for the measurement of texture in aluminum sheets. Although the effect of stress wave velocities on elastic anisotropy is well known, its ability to predict plastic behavior has yet to be firmly established. Ultrasonic measurements, which hold the potential for remote, in-process screening, will be shown to exhibit a high degree of correlation with current techniques for measuring texture, which are off-line, destructive, and often inaccurate. Results from an electromagnetic acoustic transducer (EMAT) system are used as a baseline comparison for a more promising noncontact technique involving the optical generation and detection of ultrasound, commonly referred to as laser ultrasonics (LU). In addition to displaying the level of anisotropy, slowness curves obtained from these systems suggest the ability to distinguish between different types of texture, as predicted from theory.

The elastic anisotrophy of shales

Shales constitute about 75% of the clastic fill of sedimentary basins and have a decisive effect on fluid flow and seismic wave propagation because of their low permeability and anisotropic microstructure. The elastic stiffnesses of a shale with partially oriented clay particles is expressed in terms of the coefficients Wlmn in an expansion of the clay‐particle orientation distribution function in generalized Legendre functions. Application is made to the determination of the anellipticity of shales. For transverse isotropy the anellipticity quantifies the deviation of the P wave slowness curve from an ellipse and is shown to depend on a single coefficient W400 in the expansion of the clay‐particle orientation distribution function. If W400 is small, the anellipticity may be neglected, as is apparently the case for a near‐surface late Tertiary shale studied by Winterstein and Paulson. Strongly aligned clay particles result in a positive value of W400 and a positive anellipticity, in agreement with the majority of field measurements. However, less well ordered shales could have a significantly positive second moment W200 but only a small positive or even negative value of W400 For such shales the anellipticity would be small or negative despite a preferred alignment of clay particles in the bedding plane. Numerical examples of clay particle orientation distribution functions leading to zero or negative anellipticity are given.

Propagation of surface waves in anisotropic solids: theoretical calculation and experiment

This paper utilizes the Stroh-Barnett's integral formalism of surface waves in anisotropic solids to calculate the variation of surface wave speeds of a unidirectional fibre-reinforced composite in an efficient way. The decaying pattenr of the displacements and tractions along the depth were also calculated using the determined surface wave speeds. To study the propagations of surface waves in the fibre-reinforced composite plate experimentally, point-source/point-receiver (PS/PR) experiments were conducted. A pulsed laser was utilized to generate a point disturbance in the composite plate and the transient waves were recorded by a PZT transducer with an acting diameter of about 1.5 mm. The slowness curves of the surface wave energy velocity were determined from the experimental data and the results were in accordance with those calculated theoretically.

Acoustic microscope study of the elastic properties of fluorapatite and hydroxyapatite, tooth enamel and bone

Measurements of Rayleigh velocity and attenuation were taken in single mineral crystals of hydroxyapatite and fluorapatite at angular intervals relative to their c axes, using an acoustic microscope. These results are compared with the values that were calculated using the elastic constants of apatite from Yoon and Newnham [(1969) Am. Miner., 54, 1193–1197.] and Katz and Ukraincik [(1971) J. Biomechanics, 4, 221–227.]. The slowness curves of various wave modes are plotted and discussed in relation to cross-coupling effects that were found to cause instability in measurements of attenuation for the c axes direction. Velocity measurements were taken in specimens of tooth enamel and bone. Here comparisons are made with the values that were calculated by modelling the ‘ z’ scan response of the microscope, using published data for the elastic and acoustic properties. Comparisons are also made with the measurements on single crystals, since apatite is a major component of enamel and bone.

Zero curvature transonic states and free surface waves in anisotropic elastic media

Present surface wave existence theory rests on the assumption that the slowness curve has nonzero curvature at the point corresponding to the limiting wave. The existence theory is extended to take the possibility of zero curvature transonic states into account. It is concluded that a subsonic free surface wave solution always exists in the case of a zero curvature transonic state.

Influence of anisotropy on the dispersion characteristics of guided ultrasonic plate modes

Dispersion curves are developed for elastic wave propagation in an anisotropic plate of monoclinic or higher symmetry. Emphasis is placed on analytic expressions for various features. Generalization of the isotropic Rayleigh–Lamb dispersion relations are derived for the cases of (a) propagation along a material symmetry axis and (b) propagation in a general direction. Examination of the high-frequency limit of the lowest symmetric and antisymmetric mode dispersion curves yields expressions for the half-space surface or Rayleigh wave velocity. It is shown that the dispersion curves for these modes can exhibit multiple crossings in approaching this limit, and an analytic solution is presented for the constant crossing interval that occurs for propagation along symmetry directions. The analytic results are illustrated by extensive numerical calculations for a variety of degrees of anisotropy with emphasis placed on the relationship between the slowness curves governing partial wave propagation and various features of the dispersion curves.

RAYS AND WAVEFRONT CHARTS IN GRADIENT MEDIA1

AbstractWavefront charts in anisotropic gradient media are a useful tool in ray geometric constructions, particular in shear‐wave exploration. They can be constructed by: (i) a family of wavefronts that contains a vertical plane as member ‐ it is convenient to choose constant time increments; (ii) tracing one ray that makes everywhere the angle with the normal to the wavefront that is required by the anisotropy of the medium; (iii) scaling this ray to obtain a set of rays with different ray parameters; (iv) shifting these rays (with wavefront elements attached) so that they pass through a common source point; (v) interpolating the wavefronts between the elements.The construction is particularly simple in linear‐gradient media, since here all members of the family of wavefronts are planes. Since the ray makes everywhere the angle prescribed by the anisotropy with the normal of the (plane) wavefronts, the ray has the shape of the slowness curve rotated by −π/2.For isotropic media the slowness curve is a circle, and thus rays are circular arcs. The circles themselves intersect in the source point and in a second point above the surface of the earth. This provides a simple proof that wavefronts emanating from a point source in an isotropic linear‐gradient medium are spheres: inversion of the set of circular rays with the source as centre maps the pencil of circular rays into a pencil of straight lines passing through a point. A pencil of concentric spheres around this point is perpendicular to the pencil of straight lines. On inverting back the pencil of spheres is mapped into another pencil of spheres that is perpendicular to the circular rays.

Measurement of surface acoustic wave slowness curves with a scanning laser acoustic microscope

The measurement of the velocity and power flow angle of surface acoustic waves (SAWs) using a scanning laser acoustic microscope is described. Data obtained from a single scan by the microscope and from a single direction of insonification are Fourier transformed from the spatial domain into the spatial frequency domain. The incident cw surface waves scatter from the sides of the sample into all possible wave vectors. Thus, the Fourier transform in the spatial frequency domain is the inverse velocity or slowness curve for surface waves propagating on the material. From this curve, the phase velocity, group velocity, and power flow angle versus propagation direction can be obtained. Experimental results for YZ lithium niobate, ST quartz, and (111) silicon are in good agreement with theoretical calculations.

Inhomogeneous wave generation and propagation in lossy anisotropic solids. Application to the characterization of viscoelastic composite materials

This article develops a method for investigating some anisotropic media, such as composites, by the use of ultrasonic waves transmitted through a plate-shaped sample immersed in water. The discussion begins with Christoffel’s equations for plane linear anelastic waves under the assumptions that for small angles of incidence the wave modes are plane and inhomogeneous and that the anisotropy is representable by hexagonal symmetry. The water–sample interface is treated using the law of Snell–Descartes for nonabsorbing media and takes into account mode conversion and the generation of acoustic surface waves. The method produces viscoelastic constants and relative attenuation coefficients as a function of the angle of refraction. The experimental measurement apparatus is described and data are given for the 25-layer unidirectional Gr/epoxy composite. Results are presented in terms of slowness, damping vector, and attenuation curves. The results are significant in that they demonstrate the anisotropy both for the elastic stiffness and the attenuation. The method appears to hold promise for characterizing some classes of anisotropic media, including 2-D composites, in terms of their anelastic behavior.

LOVE‐TYPE SEAM‐WAVES IN WASHOUT MODELS OF COAL SEAMS*

ABSTRACTThe propagation of Love seam‐waves across washouts of coal seams was studied by calculating synthetic seismograms with a finite‐difference method. Seam interruption, seam end and seam thinning models were investigated. The horizontal offset, the dip of the discontinuities and the degree of erosion served as variable parameters. Maximum displacement amplitudes, relative spectral amplitudes and phase and group slowness curves were extracted from the synthetic seismograms.Both seam interruption and seam thinning reduce the maximum displacement amplitudes of the transmitted Love seam‐waves. The degree of amplitude reduction depends on the horizontal offset and the degree of erosion. It is four times greater for a total seam interruption than for an equivalent seam thinning with a horizontal offset of four times the seam thickness. In a seam cut vertically, the impedance contrast between the coal and the washout filling determines the maximum displacement amplitudes of the reflected Love seam‐waves. They diminish by a maximum factor of four in oblique interruption zone discontinuities with a dip of maximum 27°, and by a maximum factor of ten in a seam thinning with a degree of erosion of at least 22%.The analysis of the relative spectral amplitudes indicates a preferential transmission of those phases with frequencies below, and a preferential reflection of those phases with frequencies above the first mode Airy‐phase. The relative spectral amplitudes of the reflected Love seam‐waves show a distinct interference pattern of the waves reflected at both interruption zone discontinuities.The dispersion analysis reveals a flattening of the phase and group slowness curves with increasing frequencies, horizontal offset and degrees of erosion.These results imply that a detection of washouts in‐mine will be possible in a frequency range including at least the first mode Airy‐phase. An interference pattern and a flattening of the dispersion curve indicate a washout rather than other seam obstructions and leads to an estimate of the washout dimension.

Analysis of trapped energy resonators with circular electrodes

A simple theoretical model of trapped energy resonators with circular electrodes utilizing monoclinic crystal plates is presented. The effects due to ana nisotropic slowness curve for guided thicknesswaves in piezoelectric plates are included in this model so that it can be generally applied to the rotated Y-cut plates of quartz, LiTa03, or LiNb03. The theory is based on the angular spectrum of plane waves representation of two-dimensional thickness-wave solutions in the vicinity of the cutoff frequencies. Experimental measurements of resonant frequencies, vibrational patterns, and motional inductances on AT-cut quartz resonators are provided, and it is shown that calculations based on the theory are in excellent agreement with the results of experiments.

Analytical and graphical evaluation of local quasi-stable angular points during ion bombardment at oblique incidence

An implied analytical condition, connecting the incidence angles θo and θe between the two intersecting microfacet normals and the ion beam direction, has been inferred for the existence of local quasi-stable intersections (angular points) during ion erosion. This condition, which can also be deduced on the basis of the erosion slowness curve, leads to a higher degree algebraic equation with a parameter, applicable for any dependence S=S (θ), expressed as an algebraic polynomial expression in cos θ. A new graphical method for evaluation of orientations θe compatible with a given orientation θo, based on a known polar diagram cursor-type, has also been proposed.