John G. Harris

Measurements of coupled Rayleigh wave propagation in an elastic plate

At frequencies where the thickness of an elastic plate is more than a wavelength, the propagation of the two lowest Rayleigh–Lamb modes in an elastic plate can be viewed as the propagation of a Rayleigh surface wave over two weakly coupled, surface-wave waveguides. That is, a Rayleigh wave launched on one surface gradually transfers to the other and then back. It does so in a length we call the beatlength. Measurements of the beatlength for brass plates are reported as a function of frequency and thickness. This phenomenon is readily excited and persists over a wide range of thicknesses and frequencies

An Asymptotic Calculation of the Acoustic Signature of a Cracked Surface for the Line Focus Scanning Acoustic Microscope

An asymptotic description of the acoustic signature of a crack breaking the surface of an otherwise homogeneous, isotropic elastic material for a line focus, scanning acoustic microscope is constructed. The Debye approximation is used to calculate an incident focused beam whose profile falls off continuously at its edges. The wavefields scattered from the surface are constructed as Fourier integrals that are approximated asymptotically. Included in the asymptotic approximations are the leaky Rayleigh waves, which play a crucial part in the acoustic signature. Explicit expressions for the incident and scattered wavefields are given. The acoustic signature is calculated by using an electromechanical reciprocity identity to relate the wavefields in the coupling fluid to the voltage at the terminals of the microscope’s transducer. Several ways of evaluating this identity for an unbroken surface are explored and are shown to be asymptotically equivalent. The acoustic signature of a surface-breaking crack is then calculated by assigning to the crack reflection and transmission coefficients for the leaky Rayleigh wave and then using geometrical elastodynamics to construct the scattered wavefields. Explicit expressions for the acoustic signature of the cracked surface are given. Moreover, an explicit expression for the reflection coefficient of a Rayleigh wave reflected from a surface-breaking crack is given.

Numerical analysis of the acoustic signature of a surface-breaking crack

The boundary element method is used to calculate the acoustic signature, produced by a line focus scanning acoustic microscope, of an elastic object containing a surface-breaking crack. The acoustic signature has a vertical (z) and horizontal (x) dependence. A model of the microscope developed earlier is used and extended to take account of the crack. The mathematical formulation of the scattering problem for the cracked object leads to hypersingular integral equations. A suitable technique is employed to solve such equations by the boundary element method. An electromechanical reciprocity identity is used to relate the received voltage to the acoustic wavefields collected by the lens. The acoustic wavefield scattered from the cracked object is investigated, and curves are presented that display the acoustic signature, as functions of (x,z), for cracks of various depths and orientations. A method to measure the depth of a surface-breaking crack using the acoustic signature is suggested.<<ETX>>

Wave analysis of the acoustic material signature for the line focus microscope

A model is presented for the computation of the acoustic material signature for a line focus scanning acoustic microscope, based on a boundary element calculation and an electromechanical reciprocity identity. This identity is used to relate the voltage at the terminals of the microscopes transducer to the acoustic wavefields at the interface between the specimen and the coupling fluid. A Gaussian beam, launched in the buffer rod, is tracked through the lens and its matching layer. A model for the matching layer that is convenient for use with the boundary element technique is presented. The wavefields scattered from the surface of the specimen, including the leaky Rayleigh wave, are then calculated. Knowing the wavefields incident on and scattered from the specimen, the acoustic signature is calculated using the reciprocity relation. Results are presented for a defect free halfspace, and are compared with those of an analytical model and an experimental measurement.<<ETX>>

The acoustic signature for a surface-breaking crack produced by a point focus microscope

The acoustic signature of a crack, breaking the surface of an otherwise homogeneous, isotropic elastic material, produced by a point focus scanning acoustic microscope is constructed theoretically. This work is patterned after a similar calculation carried out for the line focus microscope. The incident axisymmetric focused beam is constructed as a Fourier integral that produces a specified profile in the focal plane. The wavefields scattered from the specimen are also represented as Fourier integrals. Because the lens of the acoustic microscope is characterized by a large Fresnel number and an F number of order one, the Fourier integrals can be asymptotically approximated to obtain explicit expressions for the incident wavefield and for the wavefield scattered from a defect-free surface. The latter wavefield contains the leaky Rayleigh wave that is incident to the surface-breaking crack. The surface-breaking crack is characterized by assigning it reflection and transmission coefficients. The wavefield scattered from the crack is estimated by tracing the leaky Rayleigh rays reflected and transmitted by the crack. The net wavefield scattered from the surface is then constructed by adding this crack scattered wavefield to that calculated for a defect-free surface. Lastly, the acoustic signature is calculated by using the appropriate incident and scattered wavefields in an electromechanical reciprocity identity that links the voltage measured at the microscope’s transducer to the scattered acoustic wavefields at the surface of the specimen. Expressions for acoustic signatures made using the line focus and point focus microscopes are compared. Moreover, from the expression for the acoustic signature, the Rayleigh wave reflection and transmission coefficients can be partly extracted.

Diffraction by a crack of a cylindrical longitudinal pulse

The diffraction of a cylindrical longitudinal pulse by a semi-infinite crack is studied analytically using integral transform techniques and the Wiener-Hopf method. The inverse transformations are performed by an extension of the Cagniard-de Hoop technique that enables three successive inversions to be performed in one step rather than the usual two successive inversions. Examination of both the reflected and diffracted displacements near the transverse-wave reflection boundary indicates the presence of a transition disturbance. Such a disturbance is not found if the incident pulse is plane or if the elastic solid is replaced by a medium supporting only one wave type such as an inviscid fluid. Asymptotic approximations to the most important diffracted pulses, that are valid near their wave fronts, are given and are compared with those predicted by the geometrical theory of diffraction.

The propagation of ultrasonic waves through a bubbly liquid into tissue: a linear analysis

The steady-state response induced by a harmonically driven, ultrasonic wave in a structure comprised of two layers, the first a bubbly liquid, and the second a viscoelastic solid with a rigid boundary, is studied in the linear approximation. This structure is intended to model a cavitating liquid in contact with tissue. The upper surface of the liquid is driven harmonically and models the source. The lower surface of the solid is rigid and models the bone. While cavitation is inherently nonlinear, the propagation process is approximated as linear. The transient response is not calculated. The model of the bubbly liquid is a simple continuum one, supplemented by allowing for a distribution of different equilibrium bubble radii and for the relaxation of the oscillations of each bubble. The model contains three functions, the probability distribution describing the distribution of bubble radii, and two functions modeling the mechanical response of the individual bubble and the tissue, respectively. Numerical examples are worked out by adapting data taken from various published sources to deduce the parameters of these functions. These examples permit an assessment of the overall attenuation of the structure, and of the magnitudes of the pressure and particle velocity in the bubbly liquid and of the traction and the particle displacement in the tissue. >

Focusing of an ultrasonic beam by a curved interface

Abstract We calculate the wavefields near the caustics, and their cusps, formed when a well collimated, ultrasonic beam scatters from a concave fluid-solid interface. The radius of curvature of the interface is assumed to be sufficiently large and the angle of incidence sufficiently small that only reflection and transmission of the beam need be considered. The incident beam is modeled as a two-dimensional wavefield whose initial profile is rectangular. The aperture is assumed to be sufficiently large (in wavelengths) that a well collimated beam is radiated, and the interface is assumed to lie close enough to the aperture that it is struck by a wavefield that has not yet evolved into a cylindrical wave. The scattered wavefields are represented as multiple integrals and are evaluated using a combination of asymptotic and numerical analysis. Special attention is given to the sometimes competing effects of the shadow boundary of each scattered beam with its corresponding caustic, and cusp if one is formed.

An acoustic lens design using the geometrical theory of diffraction

The design of an acoustic lens that reduces the diffraction spreading from an ultrasonic transducer to produce a narrow focused beam is proposed. The design is based upon a description of the radiated field using the geometrical theory of diffraction.

Modeling Scanned Acoustic Imaging of Defects at Solid Interfaces

This is an expository summary of my and my collaborators work building mathematical models of scanned acoustic imaging of defects such as cracks or voids that break the surface of a solid or form along solid-solid interfaces. We construct explicit models both of a high frequency, scanned acoustic microscope operating in a reflection mode, and of a lower frequency, scanned confocal acoustic imaging system operating in a transmission mode. The acoustic microscope can operate from 100 megahertz to several gigahertz. One of its most interesting imaging modes is the detection of small surface- breaking cracks, whose traces at the surface of a solid are smaller than an acoustic wavelength, even at high megahertz frequencies. It does so by using a leaky Rayleigh wave as part of its imaging mechanism. The confocal imaging system operates in a neighborhood of 10 megahertz, a lower frequency. It is used to image complicated solid- solid interfaces comprised of scatterers at numerous length scales, many of which are less than a wavelength. For both cases, we explain how the sound scattered from the defects is mapped into the sound collected by the transducers and hence into the voltages they produce. The models are approximate, make use of reciprocity relations and depend upon asymptotic evaluations of Fourier integrals.