The flat-bottom hole: An ultrasonic scattering model

A new model is developed that predicts the pulse-echo compressional wave transducer response from a flat-bottom hole whose axis is aligned with the axis of the transducer. This model uses the Schoch solution for the waves incident on the hole, the Kirchhoff approximation at the hole, and high frequency asymptotics to obtain an approximate analytical expression for the measured response. This solution significantly extends previous scattering models for this problem. For a small hole in the far-field of the transducer this new model is shown to reduce to the more restrictive existing models and numerical results are obtained to illustrate where differences with these earlier models are important. This solution is also used to obtain distance-gain-size (DGS)-like curves that can predict the significant response variations in both the near- and far-field.     

The Flat-Bottom Hole: An Ultrasonic Scattering Model

Abstract

A new model is developed that predicts the pulse-echo compressional wave transducer response from a flat-bottom hole whose axis is aligned with the axis of the transducer. This model uses the Schoch solution for the waves incident on the hole, the Kirchhoff approximation at the hole, and high frequency asymptotics to obtain an approximate analytical expression for the measured response. This solution significantly extends previous scattering models for this problem. For a small hole in the far-field of the transducer this new model is shown to reduce to the more restrictive existing models and numerical results are obtained to illustrate where differences with these earlier models are important. This solution is also used to obtain distance-gain-size (DGS)-like curves that can predict the significant response variations in both the near-and far-field.     

Modeling the Response of Ultrasonic Reference Reflectors

Reference reflectors characterized as flat-bottom hole (FBH), side-drilled hole (SDH), and spherical void (SPH) are commonly used in ultrasonic nondestructive evaluation (NDE) for calibration purposes. Here the measurement models currently available to simulate the A-scan response from those types of reference reflectors are examined. Measurement models suitable for both large and small reflectors are described and used to study the effect of beam variations over the surface of the reflector. The adequacy of the Kirchhoff approximation for predicting the waves scattered by these reference reflectors is also studied by comparing the results of that approximation to those obtained from more exact scattering solutions. The waveforms predicted by these various models are compared with experimentally determined responses in a pulse-echo immersion setup, and the accuracy of the models is discussed.     

Modeling Angle Beam Ultrasonic Testing Using Multi-Gaussian Beams

This paper proposes a new approach to modeling angle beam ultrasonic testing that can predict pulse-echo signals, in an absolute and computationally efficient manner, from various reflectors in steel welded joints. This approach relies on a model of the entire ultrasonic measurement process, a model which requires one to solve three subsidiary problems; 1) determination of a system efficiency factor, 2) evaluation of the ultrasonic beam field around the flaw, and 3) calculation of the scattering from the reflector. Here, solutions are offered for each of those three subsidiary problems. To solve the first problem we employ the specular reflection from the cylindrical part of a STB-A1 (Standard Test Block in compliance with Japanese Industrial Standards Z 2347) (equivalently IIW (International Institute of Welding) type 1) standard block to determine the system efficiency factor. To solve the second problem, we calculate the ultrasonic wave field at the flaw with a highly efficient multi-Gaussian beam model. For the third problem, we treat the scattering from a reflector by high frequency approximations. We explicitly give the solutions to all three of these subsidiary problems for counter bore, crack, and side-drilled hole reflectors. Experimental results that validate this approach are also given.     

Kirchhoff Approximation Revisited—Some New Results for Scattering in Isotropic and Anisotropic Elastic Solids

Through a series of numerical studies that compare the Kirchhoff approximation to more exact scattering theories, it is demonstrated that the Kirchhoff approximation can accurately predict the pulse–echo peak-to-peak responses of spherical pores and circular cracks in isotropic media over a very wide range of cases that extend well beyond the limits normally associated with this approximation. The reason for this good agreement is shown to lie in the ability of the Kirchhoff approximation to model accurately the very early time response of the flaw. It is also shown that in the Kirchhoff approximation the pulse–echo response of an arbitrary traction-free scatterer in an isotropic elastic solid is identical to the same response obtained using a scalar (fluid) scattering model. This leads to simple analytical expressions for the pulse–echo far-field scattering amplitude of some canonical geometries (circular cracks, spherical voids, cylindrical holes) and to simplified numerical expressions for more general scatterers. For general anisotropic volumetric flaws in a anisotropic elastic solid, it is shown that a high-frequency asymptotic evaluation of the Kirchhoff approximation yields an explicit analytical expression for the pulse–echo leading-edge response of the flaw. Explicit expressions are also given for the pitch–catch response of an elliptical-shaped flat crack in a general anisotropic solid.     

Utilization of prior flaw information in ultrasonic NDE: An analysis of flaw scattering amplitude as a random variable

Probabilistic approaches to flaw detection, classification, or characterization often assume prior knowledge of the flaw distribution. It is implicit that there is a scattering amplitude distribution associated with the flaw distribution. In a number of previously published probabilistic analyses, it has been assumed that scattering amplitude is an uncorrelated, Gaussian random variable with zero mean and known variance. In the work reported here, these assumptions are evaluated for the case of a lognormal distribution of spherical flaws. The correlation, mean, variance, and nature of the scattering amplitude distribution are considered as a function of frequency and as a function of the breadth of the assumed flaw distribution. It is shown for the assumed flaw distributions that scattering amplitude is not uncorrelated and does not have zero mean. It is shown that errors in estimating the flaw distribution variance affect both the scattering amplitude mean and variance. Using both analytical and numerical procedures, the scattering amplitude distribution is shown to be lognormal at long wavelength for a lognormal distribution of spherical scatterers. At high frequency, the distribution is shown to have a bimodal character.     

Prediction of Side-Drilled Hole Signals Captured by a Dual Crystal Contact Probe

The dual crystal transducer plays an important role in many practical ultrasonic inspections due to its outstanding near surface resolution and signal-to-noise ratio. In order to get a deeper understanding on the characteristics of this kind inspection system, it is essential to develop an ultrasonic measurement model for dual crystal probes. To address such a need, by combining a multi-Gaussian beam model and a scattering model with the separation of variables method, we provide an efficient ultrasonic measurement model for the dual element transducer. This measurement model can be used to determine the system efficiency factor and predict the responses of a side-drilled hole. Furthermore, the comparisons of model predictions with experimental results are presented to certify the accuracy of this provided model.     

Off-axis scattering of an ultrasound bessel beam by a sphere

In this paper, the scattering of an ultrasound zero-order Bessel beam by a rigid sphere in the off-axis configuration is studied. The beam is described through the partial wave expansion. The beam-shape coefficients which represent the amplitude of each multipole mode of the partial wave expansion are computed by numerical quadrature. Calculations are presented for both near- and far-field off-axis scattering. The far-field scattering is examined in both Rayleigh and geometrical acoustic limits. Results demonstrate that the scattered pressure in the off-axis case may significantly deviate from that in the on-axis configuration. In addition, the directive pattern of the scattered pressure is highly dependent on the relative position of the beam to the sphere.     

Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium

The radiative properties of bubbles or particles embedded in an absorbing medium are investigated. We aim first to determine the conditions under which absorption by the surrounding medium must be accounted for in the calculation of the efficiency factors by comparing results from Mie theory and the far-field and near-field approximations. Then, we relate these approximations for a single particle to the effective radiation characteristics required for solving the radiative transfer in an ensemble of scatterers embedded in an absorbing medium. The results indicate that the efficiency factors for a spherical particle can differ significantly from one model to another, in particular for large particle size parameter and matrix absorption index. Moreover, the effective scattering coefficient should be expressed based on the far-field approximation. Also, the choice of the absorption efficiency factor depends on the model used for estimating the effective absorption coefficient. However, for small void fractions, absorption by the matrix dominates, and models for the absorption coefficient and efficiency factor are unimportant. Finally, for bubbles in water, the conventional Mie theory can be used between 0.2 and 200 mum except at some wavelengths at which absorption by water must be accounted for.     

Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid

The theory of an arbitrarily oriented, shaped, and located beam scattered by a homogeneous spheroid is developed within the framework of the generalized Lorenz-Mie theory (GLMT). The incident beam is expanded in terms of the spheroidal vector wave functions and described by a set of beam shape coefficients (G(m)(n),(TM),G(m)(n),(TE)). Analytical expressions of the far-field scattering and extinction cross sections are derived. As two special cases, plane wave scattering by a spheroid and shaped beam scattered by a sphere can be recovered from the present theory, which is verified both theoretically and numerically. Calculations of the far-field scattering and cross sections are performed to study the shaped beam scattered by a spheroid, which can be prolate or oblate, transparent or absorbing.     

Single scattering by a small volume element

Starting from first principles, we present a detailed analysis of the concept of single scattering of light by a small volume element filled with sparsely and randomly positioned particles. We first derive the formulas of the far-field single-scattering approximation, which treats the volume element as a single scatterer, and discuss its range of applicability, using for illustration exact T-matrix results for randomly oriented two-sphere clusters. Our second approach is to treat the volume element as a small cloud of particles and apply the so-called first-order-scattering approximation. We demonstrate that although the two approaches are based on somewhat different sets of assumptions, they give essentially the same result for the electromagnetic response of a sufficiently distant polarization-sensitive detector.     

DGS diagrams and frequency response curves for a flat-bottom hole: A model-based approach

Recently, we have developed analytical scattering models for predicting the pulse-echo response of a flat-bottom hole in both contact and immersion testing. Here, we present the results of a series of experimental validations of the immersion case model. It is shown that the model can be used to develop distance-gain-size (DGS) curves and frequency response curves that compare favorably with experiments even in the very near field of the transducer. In fact, the model is shown to reproduce the entire scattered waveform from the hole quite accurately. The quality of these results suggest that the model can serve as an important new theoretical reference standard for a variety of ultrasonic calibration and sizing applications.     

Role of multiple scattering in cross-correlated light scattering with a single laser beam

Previous systems for measuring cross-correlated light scattering by small particles suspended in a liquid with multiple-scattering suppression have illuminated the particles with two laser beams. It is shown that multiple-scattering suppression should also occur in cross correlation for a system that employs a single laser beam and two closely spaced detectors with wide fields of view. The single-scattering, double-scattering, and single-double-scattering cross-term contributions to the intensity cross-correlation function are calculated. It is found that the two cross terms, when added together, are unimportant for both autocorrelation and cross correlation. The amplitude of the double-scattering term can be greatly diminished by judicious detector spacing because the spatial coherence area in the detector plane for double scattering is much smaller than that for single scattering.     

Numerical algorithm for defect reconstruction in elastic media with a circular ultrasonic scanning

A numerical reconstruction method is proposed, which is applied to image identification of defects detected in elastic solid samples, in the case when a circular Ultrasonic scanning provides a measurement of the scattering pattern over full interval of the incident polar angle. The problem is first formulated as a system of respective boundary integral equations whose solution is used to calculate the far-field scattering diagram. Then the inverse reconstruction problem is reduced to a minimization of a certain strongly nonlinear functional. The proposed numerical algorithm is tested on some examples of volumetric flaw. It is also evaluated the influence of the error in the input data on precision of the reconstruction.     

Estimation of electromagnetic scattering patterns from sinusoidalsurfaces using FFT

The scattering of electromagnetic waves from perfectly conducting sinusoidal surfaces as prototypes of periodic rough surfaces is studied. Physical optics and Fourier methods are used to calculate the scattered far field at a function of the observation angle. The patterns for several surface parameters are evaluated and plotted. The size of the illuminated surface area is varied. Homogeneous and Gaussian field amplitude perturbation of a laser beam in TEM₀₀-mode is considered. The results are compared with other methods and show a rather good agreement. Therefore, the method may be used for far-field computation if several criteria for the validity of the Kirchhoff approximation are not violated     

超声A扫描信号建模及其缺陷识别方法研究

针对超声检测中A扫描信号依靠经验评定结构缺陷、识别缺陷类型困难等问题,提出一种A扫描信号建模方法,计算出缺陷A扫描信号。该方法基于超声检测系统建立A扫描信号数学模型,应用多元高斯声束法计算模型中缺陷表面超声波传播质点速度,运用基尔霍夫近似理论描述模型中缺陷散射振幅,从而获得缺陷超声A扫描信号。应用CSII-1/20标准试块平底孔(缺陷)超声A扫描表明:该方法计算获得试块平底孔5#(缺陷)A扫描信号与试验测量结果在缺陷位置和幅度基本吻合,是识别结构孔洞类缺陷的一种有效方法。     

The disturbance of a plane dyadic wave by a small spherical cavity

Dyadic scattering offers a general setting for solving wave-obstacle interaction problems in Continuum Mechanics, because it eliminates the direction of polarization from the scattering formulation. Once the dyadic problem has been solved, any classical scattering problem for the displacement field is recoverable through a contraction with the given polarization. In the present work we solve the scattering problem of a plane dyadic incident field which is disturbed by a spherical cavity in the medium of propagation. The cavity is considered to be small in the sense that its characteristic dimension is much less than the wave length of the incident field. The zeroth and the first order low-frequency approximations of the near field as well as the leading approximation of the far-field (which is of the third order) are obtained explicitly via an appropriate generalization of the Papkovich representation for dyadic fields. The leading approximation of the scattering cross-section is also provided. The results are then used to check the credibility of related vector results obtained from the Boundary Element Method and an amazing coincidence is observed, at least for small enough frequencies.     

Virtual Excitation and Multiple Scattering Correction Terms to the Neutron Index of Refraction for Hydrogen

The neutron index of refraction is generally derived theoretically in the Fermi approximation. However, the Fermi approximation neglects the effects of the binding of the nuclei of a material as well as multiple scattering. Calculations by Nowak introduced correction terms to the neutron index of refraction that are quadratic in the scattering length and of order 10⁻³ fm for hydrogen and deuterium. These correction terms produce a small shift in the final value for the coherent scattering length of H₂ in a recent neutron interferometry experiment.     

Electromagnetic field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering

A previously developed theoretical procedure for determination of electromagnetic fields associated with the interaction of a higher-order Gaussian beam with a homogeneous spherical particle is used to investigate the effects of incident beam type on far-field scattering. Far-field scattering patterns are calculated for (0,0), (0,1), and (1,1) mode Hermite-Gaussian beams and for the helix doughnut mode beam. The effects of incident beam type on the angular distribution of far-field scattering, for both on-sphere-center and off-sphere-center focusing, are examined.     

Low frequency electromagnetic scattering by a perfectly conducting moving sphere

The paper investigates the propagation of a plane electromagnetic waves in the exterior of a moving obstacle. Under the assumption that the obstacle moves with uniform velocity and more slowly than the incident field, we apply the Lorentz transformation. In the object’s frame, where the scatterer is stationary, we introduce the low-frequency approximation technique. For the near electromagnetic field we obtain the Rayleigh low-frequency coefficients while in the far field we derive the leading non-vanishing terms for the scattering amplitude and scattering cross-section. Finally, using the inverse transformation we express the same quantities in the observer’s frame.