An ultrasonic array is a single transducer that contains a number of individually connected elements. Recent years have seen a dramatic increase in the use of ultrasonic arrays for non-destructive evaluation. Arrays offer great potential to increase inspection quality and reduce inspection time. Their main advantages are their increased flexibility over traditional single element transducer methods, meaning that one array can be used to perform a number of different inspections, and their ability to produce immediate images of the test structure. These advantages have led to the rapid uptake of arrays by the engineering industry. These industrial applications are underpinned by a wide range of published research which describes new piezoelectric materials, array geometries, modelling methods and inspection modalities. The aim of this paper is to bring together the most relevant published work on arrays for non-destructive evaluation applications, comment on the state-of the art and discuss future directions. There is also a significant body of published literature referring to use of arrays in the medical and sonar fields and the most relevant papers from these related areas are also reviewed. However, although there is much common ground, the use of arrays in non-destructive evaluation offers some distinctly different challenges to these other disciplines.

Ultrasonic arrays for non-destructive evaluation: A review

1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

Methods of Mathematical Physics. By Harold Jeffreys and Bertha Swirles Jeffreys. Pp. viii, 679. 63s. 1946. (Cambridge University Press)

In previous work, we have shown that time-resolved 2-D transient elastography is a promising technique for characterizing the elasticity of soft tissues. It involves the measurement of the displacements induced by the propagation of low frequency (LF) pulsed shear waves in biological tissues. In this paper, we present a novel apparatus that contains a LF vibrating device surrounding a linear array of 128 ultrasonic transducers that performs ultrafast ultrasonic imaging (up to 10,000 frames/s) and that is able to follow in real time the propagation of a LF shear wave in the human body. The vibrating device is made of two rods, fixed to electromagnetic vibrators, that produce in the ultrasonic image area a large amplitude shear wave. The geometry has been chosen both to enhance the sensitivity and to create a quasi linear shear wave front in the imaging plane. An inversion algorithm is used to recover the shear modulus map from the spatio-temporal data, and the first experimental results obtained from tissue-equivalent materials are presented.

Shear modulus imaging with 2-D transient elastography

This paper presents an analytical model for support loss in clamped–free (C–F) and clamped–clamped (C–C) micromachined beam resonators with in-plane flexural vibrations. In this model, the flexural vibration of a beam resonator is described using the beam theory. An elastic wave excited by the shear stress of the beam resonator and propagating in the support structure is described through the 2D elastic wave theory, with the assumption that the beam thickness (h) is much smaller than the transverse elastic wavelength (λT). Through the combination of these two theories and the Fourier transform, closed-form expressions for support loss in C–F and C–C beam resonators are obtained. Specifically, closed-form expression for the support loss in a C–C beam resonator is derived for the first time. The model suggests lower support quality factor (Qsupport) for higher order resonant modes compared to the fundamental mode of a beam resonator. Through comparison with experimental data, the validity of the presented analytical model is demonstrated. © 2003 Elsevier B.V. All rights reserved.

An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations

The effect of topography on surface motion is investigated in the cases of incident SH , P and SV waves. Several types of topography ranging from a ridge to a valley are used. Different incidence angles are considered for a wavelength interval extending from 2 h to 20 h , where h is the vertical dimension of the anomaly. The computation is made by using a single frequency method developed by Aki and Larner and by synthesizing the results in order to get a solution in the time domain. A method of correction taking into account the residual stress is added. The surface displacement appears to be very much influenced by surface irregularities. In the case of a ridge, a zone of amplification takes place near the top, whereas, for a depression, a zone of attenuation occurs near the bottom. An application to the Pacoima Dam accelerograph site suggests that the high accelerations recorded there during the San Fernando earthquake could have been amplified between 30 and 50 per cent by the uneven topography.

Effect of topography on surface motion

The problem of wave-motion in a semi-infinite solid has been studied by several workers in connexion with seismological disturbances. In a classic paper (1904), Lamb investigated the propagation of vibrations over the surface of a semi-infinite isotropic solid, due to the application of force at a point or along a line in the free surface. A related problem is studied by Lapwood (I 949) who considers the case of an infinitely long dilating line-source situated parallel to and just below the free surface. Both authors obtain solutions in the form of definite integrals which they evaluate asymptotically to obtain the various components of the field at points on or near the surface at large distances from the source. In a more recent paper by Margery Newlands (I 952) this work has been extended to include the case of a source situated in a thin surface layer and propagating into a semi-infinite solid. The static problem of the distortion of a semi-infinite solid by stresses on the free surface was investigated by Boussinesq in a series of papers (1878-83) and by Love (1929). In the present paper Lamb's method has been adapted to furnish definite-integral representations of the field at an arbitrary point in a semi-infinite isotropic solid, due to prescribed periodic stresses on the free surface, the particular stress distributions considered here being such as would be produced by certain types of electro-mechanical transducer. Asymptotic formulae are obtained for the field at infinity, and the results for the various displacement components are presented in the form of polar diagrams. To obtain the radiation impedance integral expressions are derived for the mean displacement over the part of the surface which lies beneath the source; these expressions have been evaluated by numerical quadrature in a number of cases and a table shows the corresponding values of the radiation impedance.

The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid

Expressions are derived for the power radiated in the compressional, shear and surface waves set up by a circular disk vibrating normally to the free surface of a semi-infinite isotropic solid. The total radiated power is also calculated independently by integrating the displacement velocity over the area of the source. The theory is extended to a general type of multi-element radiator in the form of an array of elements on the circumference of a circle. The calculation of the total power here involves a ‘mutual admittance’ function, a table of which is given for the case when the Poisson’s ratio of the medium is equal to ¼. The theory is applied to a three-element radiator of a type used in a recent geophysical investigation, and it is shown that the efficiency of radiation in the compressional mode can be varied between wide limits by varying the distance between the elements. Finally, an approach is suggested for problems in which the most suitable idealized boundary condition is one of known displacement under the radiator, the stress being zero elsewhere on the free surface. It is shown that the stress under the radiator satisfies an integral equation whose kernel is derived from the mutual admittance function.

H. Pursey

Papers

The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid

On the partition of energy between elastic waves in a semi-infinite solid