JOURNAL OF APPLIED MATHEMATICS AND DECISION SCIENCES, 5(3), 165-180 Copyright() 2001, Lawrence Erlbaum Associates, Inc. Mathematical Modelling of Ultrasonic Non-Destructive Evaluation LARISSA JU FRADKIN* fradkil@sbu.ac.uk Centre for Waves and Fields, School of EEIE, South Bank University, London SE1 OA A England VICTOR ZALIPAEV zalipav@liverpool, ac.uk Department of Mathematical Sciences, University of Liverpool, M 0 Building, Peach Street, Liverpool L69 7ZL, United Kingdom DMITRI GRIDIN gridindsbu.ac.uk Centre for Waves and Fields, School of EEIE, South Bank University, London SE1 OAA, England Abstract. High-frequency asymptotics have been used at our Centre to develop codes for modelling pulse propagation and scattering in the near-field of the ultrasonic trans- ducers used in NDE (Non-Destructive Evaluation), particularly of walls of nuclear reac- tors. The codes are hundreds of times faster than the direct numerical codes but no less accurate. 1. Introduction Our Centre specialises in mathematical modelling of NDE based on high- frequency asymptotics. We have been the first 1-3 to produce the complete asymptotic description of the time-harmonic near field of a circular com- pressional transducer which is directly coupled to isotropic solid, both its geometrical regions and boundary layers in between the geometrical regions (see Fig. 1). Pulse propagation, rectangular transducers and transducers of complex apodisation have been also modelled using this approacha-6. The crux of the method is approximation of integrals containing an ex- ponential factor, such that when observation point moves across the near field, the factor undergoes many oscillations while the amplitudes varies slowly. The main contributions to the integrals of this type come from the critical points: singularities of the amplitude, PSP (the stationary points Due to space limitation this article was omitted from theprevious special issue on McNabb Symposium. Requests for reprints should be sent to Larissa Ju Fradkin, Centre for Waves and Fields, School of EEIE, South Bank University, London SE1 0AA, England