Vol. 97

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues

High Frequency Scattering by an Impenetrable Sphere

By Wei E. I. Sha and Weng Cho Chew
Progress In Electromagnetics Research, Vol. 97, 291-325, 2009


The high frequency scattering of a scalar plane wave from an impenetrable sphere with a diameter of several thousand wavelengths is treated by the Sommerfeld-Watson transformation, the saddle-point technique (SPT), and the numerical steepest descent method (NSDM). Both the near and far fields for the sphere are computed within the observation angle range of 0 to 180 degree. First, with the aid of the Watson transformation, the fast-convergent residue series replacing the slow-convergent Mie series is derived. Second, a new algorithm for finding the zeros of the Hankel functions is developed. Third, a novel NSDM, which is adaptive to frequency and is hence frequency independent, is proposed to overcome the breakdown of the traditional SPT in the transition region. Numerical results show that when the observation angle is very small, the Mie series solution of the near-field will not be accurate due to error accumulation. Furthermore, using the proposed methods, the CPU times for both the near-field and far-field calculations are frequency independent with controllable error. This work can be used to benchmark future works for high-frequency scattering.


Wei E. I. Sha and Weng Cho Chew, "High Frequency Scattering by an Impenetrable Sphere," Progress In Electromagnetics Research, Vol. 97, 291-325, 2009.


    1. Mie, G., "Beitrage zur optik truber medien, speziell kolloidaler metallosungen," Annalen der Physik, Vol. 25, 377-445, 1908.

    2. Kong, J. A., Electromagnetic Wave Theory, Wiley-Interscience, New York, 1990.

    3. Watson, G. N., "The diffraction of electric waves by the earth," Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 95, 83-99.

    4. Sommerfeld, A., Partial Differential Equations in Physics, Academic Press, New York, 1964.

    5. Nussenzveig, H. M., "High-frequency scattering by an impenetrable sphere," Annals of Physics, Vol. 34, 23-95, 1965.

    6. Rumerman, M. L., "Application of the Sommerfeld-Watson transformation to scattering of acoustic-waves obliquely incident upon cylindrical-shells," Journal of the Acoustical Society of America, Vol. 91, 2502-2509, May 1992.

    7. Kim, H. T., "High-frequency analysis of EM scattering from a conducting sphere coated with a composite-material ," IEEE Transactions on Antennas and Propagation, Vol. 41, 1665-1674, Dec. 1993.

    8. Shim, J. and H. T. Kim, "An asymptotic solution of EM backscattering from a conducting sphere coated with a composite material," IEEE Transactions on Antennas and Propagation, Vol. 52, 1465-1472, Jun. 2004.

    9. Paknys, R., "Evaluation of Hankel functions with complex argument and complex order," IEEE Transactions on Antennas and Propagation, Vol. 40, 569-578, May 1992.

    10. Pakny, R. and D. R. Jackson, "The relation between creeping waves, leaky waves, and surface waves," IEEE Transactions on Antennas and Propagation, Vol. 53, 898-907, Mar. 2005.

    11. Li, M. K. and W. C. Chew, "A new Sommerfeld-Watson transformation in 3-D," IEEE Antennas and Wireless Propagation Letters, Vol. 3, 75-78, Dec. 2004.

    12. Valagiannopoulos, C. A., "An overview of the Watson transformation presented through a simple example," Progress In Electromagnetics Research, Vol. 75, 137-152, 2007.

    13. Sen, S. G. and M. Kuzuoglu, "Analysis of high frequency plane wave scattering from a double negative cylinder via the modi¯ed Watson transformation and Debye expansion," Progress In Electromagnetics Research, Vol. 84, 55-92, 2008.

    14. Langdon, S. and S. N. Chandler-Wilde, "A wavenumber independent boundary element method for an acoustic scattering problem," SIAM Journal on Numerical Analysis, Vol. 43, 2450-2477, 2006.

    15. Bruno, O. P. and C. A. Geuzaine, "An O(1) integration scheme for three-dimensional surface scattering problems," Journal of Computational and Applied Mathematics, Vol. 204, No. 2, 463-476, Jul. 2007.

    16. Davis, C. P. and W. C. Chew, "Frequency-independent scattering from a flat strip with TEz-polarized flelds," IEEE Transactions on Antennas and Propagation, Vol. 56, 1008-1016, Apr. 2008.

    17. Delves, L. M. and J. N. Lyness, "A numerical method for locating the zeros of an analytic function," Mathematics of Computation, Vol. 21, 543-560, Oct. 1967.

    18. Kravanja, P., T. Sakurai, and M. van Barel, "On locating clusters of zeros of analytic functions," Bit Numerical Mathematics, Vol. 39, 646-682, Dec. 1999.

    19. Kravanja, P. and M. van Barel, "A derivative-free algorithm for computing zeros of analytic functions," Computing, Vol. 63, 69-91, 1999.

    20. Kravanja, P., M. Van Barel, O. Ragos, M. N. Vrahatis, and F. A. Zafiropoulos, "ZEAL: A mathematical software package for computing zeros of analytic functions," Computer Physics Communications, Vol. 124, 212-232, Feb. 2000.

    21. Protopopov, V. V., "Computing first order zeros of analytic functions with large values of derivatives," Numerical Methods and Programming, Vol. 8, 311-316, 2007.

    22. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007.

    23. Huybrechs, D., Multiscale and hybrid methods for the solution of oscillatory integral equations , Ph.D. Dissertation, Department of Computer Science, Katholieke Universiteit Leuven, 2006.

    24. Chew, W. C., "Waves and Fields in Inhomogenous Media," Van Nostrand Reinhold, New York, 1990.

    25. Fock, V. A., "Diffraction of radio waves around the earth's surface," Journal of Physics-USSR, Vol. 9, 255-266, 1945.

    26. Nussenzveig, H. M., "Uniform approximation in scattering by spheres," Journal of Physics A: Mathematical and General, Vol. 21, 81-109, Jan. 1988.

    27. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1970.

    28. Wiscombe, W. J., "Improved Mie scattering algorithms," Applied Optics, Vol. 19, 1505-1509, 1980.

    29. Olver, F. W. J., "The asymptotic expansion of Bessel functions of large order," Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 247, 328-368, Dec. 1954.