Diffraction Of Shear Waves Research Articles

Mathematical analysis on the diffraction of shear waves in an initially stressed dry sandy medium by a rigid strip

The present study investigates the diffraction of horizontally polarized shear wave (SH-wave) by a rigid strip in an initially-stressed dry sandy infinite-medium. The far-field solution for the diffracted displacement due to shear wave is obtained in closed-form and studied. The patterns of diffraction for displacement in an initially-stressed dry sandy infinite-medium are numerically calculated, and its dependency on wave number is illustrated graphically. The present analysis also describes a noticeable effect of different prevalent parameters, viz., horizontal and vertical initial stresses, sandiness parameter, shear wave speed, and incident angle on diffracted displacement. Furthermore, some particular cases are examined from the obtained expression of diffracted displacement of the shear wave at a significant distance from the rigid strip. Comparative analysis is performed for far-field solution for the diffracted displacement with distinct configurations. Furthermore, the obtained results are matched with pre-established results as a particular case and validation of the problem.

3D Ultrasound Shear Wave Imaging Simulation of Heterogeneous Tissue

Shear wave imaging can quantitatively measure tissue elasticity in vivo using a 2D algebraic Helmholtz inversion (AHI) equation. The 2D AHI equation reconstructs tissue elasticity by assuming that the shear wave diffraction in the elevational direction (out-of-plane direction) is not substantial. However, the application of 2D AHI equation is limited in assessing heterogeneous tissues, as the shear wave diffraction in the elevational direction critically affects the shear modulus reconstruction. A 3D AHI equation can improve the accuracy of shear modulus reconstruction for heterogeneous tissues. In this study, a finite element (FE) based shear wave simulation was performed on an early stage of cancer and an advanced cancer tissue model. The shear modulus of FE model computed by the proposed 3D AHI equation was compared to the 2D AHI equation. The 3D AHI equation substantially improved accuracy of shear modulus computation especially for the early stage of cancer model. The results from the shear modulus of inclusion by the 3D AHI equation produced the same as the input value of 150 kPa and two times lower variation than the shear modulus obtained by the 2D AHI equation.

Shear wave transmissivity measurement by color Doppler shear wave imaging

Shear wave elastography is a useful method for evaluating tissue stiffness. We have proposed a novel shear wave imaging method (color Doppler shear wave imaging: CD SWI), which utilizes a signal processing unit in ultrasound color flow imaging in order to detect the shear wave wavefront in real time. Shear wave velocity is adopted to characterize tissue stiffness; however, it is difficult to measure tissue stiffness with high spatial resolution because of the artifact produced by shear wave diffraction. Spatial average processing in the image reconstruction method also degrades the spatial resolution. In this paper, we propose a novel measurement method for the shear wave transmissivity of a tissue boundary. Shear wave wavefront maps are acquired by changing the displacement amplitude of the shear wave and the transmissivity of the shear wave, which gives the difference in shear wave velocity between two mediums separated by the boundary, is measured from the ratio of two threshold voltages required to form the shear wave wavefronts in the two mediums. From this method, a high-resolution shear wave amplitude imaging method that reconstructs a tissue boundary is proposed.

Shear wave scattering generated by the interaction of a circular half-cylinder with a wavefield originated by a line source.

The problem of shear wave diffraction caused by a circular half-cylinder was considered. In this case, the incident field was assumed to be generated by a line source, which was set to be parallel to the axis of the cylinder. The solution of this eigenvalue problem was obtained through the application of the superposition principle. In this case, the wavefield outside of the half-cylinder was expressed in terms of the free field and the diffracted field. On the other hand, the wavefield inside of the cylinder was defined in terms of refracted fields. Since the scatterer and the source possess cylindrical symmetry, all the wavefields could be expressed in terms of series of Bessel and Hankel functions. The unknown coefficients of these series were obtained by evaluating boundary conditions. With the solution of this problem, the seismic response of a semi-circular alluvial valley was studied.

Shear Wave Spectroscopy for In Vivo Quantification of Human Soft Tissues Visco-Elasticity

In vivo assessment of dispersion affecting the propagation of visco-elastic waves in soft tissues is key to understand the rheology of human tissues. In this paper, the ability of the supersonic shear imaging (SSI) technique to generate planar shear waves propagating in tissues is fully exploited. First, by strongly limiting shear wave diffraction in the imaging plane, this imaging technique enables to discriminate between the usually concomitant influences of both medium rheological properties and diffraction affecting the shear wave dispersion. Second, transient propagation of these plane shear waves in soft tissues can be measured using echographic images acquired at very high frame. In vitro and in vivo experiments demonstrate that dispersion curves, which characterize the rheological behavior of tissues by measuring the frequency dependence of shear wave speed and attenuation, can be recovered in the 75-600 Hz frequency range. Based on a phase difference algorithm, the dispersion curves are computed in 1 cm2 regions of interest from the acquired propagation movie. In vivo measurements in Biceps Brachii muscle and liver of three healthy volunteers show important differences in the rheological behavior of these different tissues. Liver tissue appears to be much more dispersive with a phase velocity ranging from approximately 1.5 m/s at 75 Hz to approximately 3 m/s at 500 Hz whereas muscle tissue shows an important anisotropy, shear waves propagating longitudinally to the muscular fibers are almost nondispersive while those propagating transversally are very dispersive with a shear wave speed ranging from 0.5 to 2 m/s between 75 and 500 Hz. The estimation of dispersion curves is local and can be performed separately in different regions of the organ. This signal processing approach based on the SSI modality introduces the new concept of in vivo shear wave spectroscopy (SWS) that could become an additional tool for tissue characterization. This paper demonstrates the in vivo ability of this SWS to quantify both local shear elasticity and dispersion in real time.

Diffraction of shear waves by a nanosized spherical cavity

Surface effects become important in the mechanical performances of nanostructured devices and materials, owing to the increasing ratio of surface to bulk volume. In the present paper, the diffraction of shear waves (S-wave) by a nanosized spherical cavity is studied theoretically based on the surface elasticity theory. It is found that when the cavity size reduces to nanometers, surface effects play a significant role in the diffraction of elastic waves. As an example, the surface effects on the dynamic stress concentration around the spherical cavity are discussed in detail.

Ultrafast imaging of transient shear waves for breast elastography: First clinical results

Transient elastography of soft human tissues has been developed in our lab for several years. This technique consists in following the shear wave propagation induced by a low frequency pulsed vibrator located at the surface of the body with an ultrasonic imaging system. As the shear wave speed can reach up to 10 m/s, an ultrafast imaging system made of 128 fully programmable and independent channels has been developed able to produce 5000 images/s. Such frame rates are attainable by limiting the emission to a plane wave illumination and achieving the beamforming only in the receive mode. The comparison of successive images by classical cross-correlation techniques allows us to produce a 2-D movie of the displacements induced by the shear wave deep inside the body. An inversion algorithm is then used to recover the shear modulus map from the spatiotemporal data. We present here the first clinical results obtained on 20 female patients. These preliminary results of quantitative imaging of breast elasticity are very promising. The limits of the system due to shear wave diffraction are discussed and the improvements planned for the next clinical experiments are discussed.

The shear wave diffraction on tunnel cavities in an elastic layer and a half-layer

The antiplane steady dynamic problem of the theory of elasticity for an isotropic layer and a half-layer weakened by tunnel cavities of arbitrary cross section is investigated. A radiating monochromatic shear wave (SH-wave) is considered as loading. Using the constructed Green's functions for a layer and a half- layer, the corresponding boundary problems are reduced to the Fredholm's integral equations of the second kind. The obtained algorithm is realized numerically by the quadrature method. Results specifying the influence of the openings configuration, their number and relative position, types of edge conditions and inertia effects on tangential stress concentration are given.

Elastodynamic response of a coplanar periodic layer of elastic spherical inclusions

Reflection and transmission spectra of a plane longitudinal wave normally incident on a coplanar periodic array of identical spherical elastic inclusions in a solid (polyester) matrix are measured at wavelengths that are comparable to the inter-particle distance. These spectra exhibit pronounced Wood's anomalies i.e., a sharp variations in amplitude at the onset of a shear wave diffraction order as well as a drop in the transmission and a corresponding maximum in the reflection coefficients due to an enhancement of the resonance of the particles (for the case reported herein, this is the rigid body translational “dipole” resonance). An approximate low frequency ( ka<1) self-consistent model is developed in which multiple scattering is explicitly taken into account by adding appropriate terms to the well-known solution for the scattering of a plane longitudinal wave by a single spherical particle in an unbounded matrix. The results of the numerical calculations show excellent agreement with the experimental data.

Acoustic response of a periodic layer of nearly rigid spherical inclusions in an elastic solid

Reflection and transmission spectra of a plane longitudinal wave normally incident on a periodic (square) array of identical spherical particles in a polyester matrix are measured at wavelengths which are comparable to the particle radius and the interparticle distance. The spectra are characterized by several resonances whose frequencies are close to the cutoff frequencies for the shear wave diffraction orders. Arrays of heavy particles (lead and steel) exhibit a pronounced resonance which occurs when the lattice resonant frequency is close to the frequency of the rigid-body translation (dipole) resonance of an isolated sphere in an unbounded matrix. An approximate low-frequency theory is developed which assumes that the inclusions are rigid, but which takes into account the multiple-scattering effect. The comparison between theory and the experiment is found to be good for arrays with particle area fraction as high as 32%.

A boundary element solution for a mode conversion study on the edge reflection of Lamb waves

The boundary element method, well known for bulk wave scattering, is extended to study the mode conversion phenomena of Lamb waves from a free edge. The elastodynamic interior boundary value problem is formulated as a hybrid boundary integral equation in conjunction with the normal mode expansion technique based on the Lamb wave dispersion equation. The present approach has the potential of easily handling the geometrical complexity of general guided wave scattering with improved computational efficiency due to the advantage of the boundary-type integral method. To check the accuracy of the boundary element program, vertical shear wave diffraction, due to a circular hole, is solved and compared with previous analytical solutions. Edge reflection factors for the multibackscattered modes in a steel plate are satisfied quite well with the principle of energy conservation. In the cases of A0, A1, and S0 incidence, the variations of the multireflection factors show similar tendencies to the existing results for glass. It is observed that the reflection of an incident wave becomes close to zero over a certain frequency range seen through energy interaction with other reflected modes, and increases again beyond this minimum point due to reverse mode conversion. The reflections of the higher symmetric incident modes, S1 and S2, are also investigated. It turns out that S1 mode is an unusual mode which is nearly unaffected by the mode conversion in the Lamb wave edge reflection.

Longitudinal and shear wave diffraction at the junction of two fluid loaded, doubly curved shells

A perturbation solution is developed to study diffracted membrane waves, both longitudinal and shear, originating from the junction of two doubly curved shells which are joined so that their tangent is continuous. The excitation is an acoustic wave incident obliquely to the tangent line of the junction. The diffracted membrane wavefield is dependent upon or forced by an outer solution which characterizes the acoustic interaction between two joined, flat plates and the surrounding fluid. To illustrate the simplicity of the approach, the outer solution is approximated by the acoustic diffraction from two massive plates. Explicit expressions are derived for the membrane wave diffraction coefficients which are related, by reciprocity, to the analogous membrane-to-acoustic diffraction coefficients.

Dynamical problems on longitudinal shear of a half-space with nonhomogeneities

Problems on diffraction of shear waves at cavities and rigid inclusions in a half-space with a fixed boundary and in one with a boundary which is free from forces are considered. The problem is reduced to a singular integral equation in the case of rigid inclusion in a half-space and to a Fredholm equation of the second kind in the case of a cavity.

Shear wave diffraction loss for circular plane-polarized source and receiver

An integral expression is derived which gives the shear wave magnitude received by a circular plane-polarized receiver that is identical to and coaxial with a source on the surface of an isotropic solid half-space. Numerical evaluation of the expression shows that the shear wave diffraction profile (i.e., the received magnitude as a function of source/receiver separation) is very similar to that for compressional waves in a solid and acoustic waves in a fluid in the Kirchoff limit of large ka, where k is the wave number and a is the source and receiver radius. Under such conditions, simplified acoustic diffraction formulas may be applied to shear wave data with little loss of rigor. In the opposite limit of small ka, the shear diffraction profile differs significantly from that of compressional or acoustic waves and the shear wave diffraction formula presented herein should be employed.

Diffraction of shear waves on an elliptical cylinder

The problem of the state of stress around an elliptical cavity in an unbounded medium under the effect of a shear wave is reduced to the determination of unknown constants involving the application of the apparatus of Mathieu functions. The numerical investigation is carried out for different frequencies, angles of incidence of the shear wave, and ratios of the axes of the ellipse.

Diffraction of shear waves by an elastic cylindrical inclusion with two cuts on the phase boundary

A method /1/ similar to that used in the case of one cut /2/ is used to determine the stress and deformation at the boundary of a cylindrical inclusion with two cuts placed on the contact contour. The external perturbation varies sinusoidally and is a plane wave in an isotropic medium. At the boundary of the inclusion the shear wave is reflected as a shear wave.

Diffraction of shear waves on cylindrical inclusions and cavities in an elastic half space

Dynamic problems of the steady-state oscillations of a half space with different types of cylindrical homogeneities (cavities, rigid and stiff inclusions) are examined. The boundary of the half space is assumed to be fixed or free of forces. A harmonic shear wave radiating from infinity or a concentrated harmonic source may be radiators of the exciting wave field. Integral representations of displacement amplitudes, which automatically satisfy fixity conditions on the boundary of the half space and radiation conditions at infinity are constructed. The edge problems are reduced to Fredholm integral equations of the second kind and to singular integral equations. Selection of additional conditions for the latter is substantiated. Some computer-generated results are presented.

Diffraction of shear waves by a rigid strip in a medium of monoclinic type

The paper discusses the two dimensional problem of diffraction of shear waves by a rigid strip in an infinite medium of monoclinic type. This problem is reduced to a system of dual integral equations of which the solution provides the diffracted field. The method of steepest descent has been used in the determination of the diffracted fields at a large distance from the strip. Diffraction pattern for displacement and stress field have been computed and the effect of anisotropy is distinctly marked.

The Dynamic Stress Intensity Factor Due to Arbitrary Screw Dislocation Motion

The dynamic stress intensity factor for a stationary semi-infinite crack due to the motion of a screw dislocation is obtained analytically. The dislocation position, orientation, and speed are largely arbitrary. However, a dislocation traveling toward the crack surface is assumed to arrest upon arrival. It is found that discontinuities in speed and a nonsmooth path may cause discontinuities in the intensity factor and that dislocation arrest at any point causes the intensity factor to instantaneously assume a static value. Morever, explicit dependence on speed and orientation vanish when the dislocation moves directly toward or away from the crack edge. The results are applied to antiplane shear wave diffraction at the crack edge. For an incident step-stress plane wave, a stationary dislocation near the crack tip can either accelerate or delay attainment of a critical level of stress intensity, depending on the relative orientation of the crack, the dislocation, and the plane wave. However, if the incident wave also triggers dislocation motion, then the delaying effect is diminished and the acceleration is accentuated.

Diffraction of shear waves by an edge crack in an elastic wedge

Low frequency diffraction of plane harmonic shear ( SH) wave by an edge crack in an elastic wedge of arbitrary vertex angle is studied. Kontorowich-Lebedev transform is used to solve the mixed boundary value problem under consideration. For low frequency case, i.e. wavelength large compared to the length of the crack, the displacement field is obtained by successive approximation of the resulting Wiener-Hopf equation. For the limiting case of an elastic half space the results agree with those obtained by the method of matched asymptotic expansions.