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Acoustic energy have found many applications in areas such as the medical ultrasonic diagnostics, seismic exploration and so on. With the advent in computer technology, it is possible to conduct quantitative analysis of the complex physics of acoustic waves interaction with arbitrary complicated media on a computer. Pure brute-force numerical methods such as the time-domain finite- different method and the finite-element method thougt extremely powerful for solving general acoustic transient problems, are restricted to a 2-D geometry running on a workstation or a high- end PC. Therefore we seek for an efficient semi-analytical method to handle a 3-D geometry. The most complex 3-D structure that can still be solved by an 1-D method in the frequency wavenumber domain is multiple spherically layered medium. In this thesis we present numerical study of point-source impulse responses of acoustic waves scattered from a spherically layered medium. Our method is based on the well known Mie solution which gives the exact frequency domain solution of a plane electro-magnetic wave impinging upon a spherical dielectric object of any size and the time-domain solution can be obtained by taking the inverse Fourier transform. Furthermore, by matrix numerical method Mie' s technique can be extended to a spherically layered structure. To obtain good numerical solutions one needs to have high- quality numerical algorithms for computing spherical Bessel functions. In this paper we propose to solve this problem by using the upward/downward recurrence formula to compute spherical Bessel functions. It is known to be accurate and stable. However, one is likely to encounter floating point number overflow and under- flow in the process when both arguments and orders of Bessel functions are very large. In that case, extended range floating point arithmetic must be used.
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