The stable analytical solution for the wave fields in the sphere

  • Fatyanov Aleksei G., Institute of Computational Mathematics and Mathematical Geophysics, 6 Lavrent’ev Avenue, Novosibirsk 630090, Russia
Keywords: mathematical modeling in the sphere, stable analytical solution, full wave field, primary wave, new asymptotic behavior of Bessel functions, homogeneous and inhomogeneous waves for the sphere


We investigate the well-known analytical solution to the problem of the wave fields in the sphere. It is shown that the use of the standard asymptotic behavior of the Bessel functions leads to interference in the solution. A new asymptotic expression for the Bessel functions is found which gives a stable analytical solution that allows one to obtain the exact solution. The homogeneous and inhomogeneous waves for the sphere are detected. We present some examples of analytical calculation of the full wave fields and the primary wave for the sphere.


[1]Tikhonov A. N. and Samarskii A. A., Equations of mathematical physics, Nauka, Moscow, 1977  mathscinet
[2]James J., Faran Jr., “Sound scattering by solid cylinders and spheres”, J. Acoustic Soc. Amer, 3:4 (1951), 405–418  mathscinet
[3]Varadan V. V., Ma Y., Varadan V. K., Lakhtakia A., “Scattering of waves by spheres and cylinders”, Field representations and Introduction to Scattering, North-Holland, Amsterdam, 1991, 211–324  mathscinet
[4]Ávila-Carrera R., Sánchez-Sesma F. J., “Scattering and diffraction of elastic $P-$ and $S-$waves by a spherical obstacle: A review of the classical solution”, Geofys. Intern., 45:1 (2006), 3–21
[5]Aganyan G. M., Voevodin Vad. V., and Romanov S. Yu., “On applicability of layered models in solving 3D problems of ultrasonic tomography”, Vychisl. Metody i Programmirovanie, 14 (2013), 533–542  mathnet  elib
[6]Tolokonnikov L. A. and Rodionova G. A., “Diffraction of the spherical sonic wave on an elastic sphere with heterogeneous covering”, Izv. Tulsk. Gos. Univ., 2014, no. 3, 131–137
[7]Korneev V. A., Johnson L. R., “Scattering of elastic waves by a spherical inclusion”, Geophys. J. Int., 1993, Theory and numerical results, no. 115, 230–250  crossref  scopus
[8]Fatyanov A. G., Numerical modeling of wave fields in an inhomogeneous sphere, Preprint, AN SSSR. SO, Comp. Center, Novosibirsk, 1981, 22 pp.  mathscinet
[9]Shanjie Zhang, Jian-Ming Jin., Computation of special functions, Wiley, 1996  mathscinet  zmath
[10]Fatianov A. G., Mikhailenko B. G., “Numerically-analytical method for calculation of theoretical seismograms in layered-inhomogeneous inelastic media”, Geophys. data inversion methods and applications, Proc. 7th Intern. Math. Geophys. Seminar (Berlin, February 8–11), Theory and Practice of Appl. Geophys., 1989, 499–530
[11]Burmin V.Yu., Fat'yanov A. G., “Analytical modeling of wave fields at extremely long distances and experimental research of water waves”, Izvestiya, Physics of the Solid Earth, 45:4 (2009), 313–325  crossref  scopus
[12]Fatianov A. G., “A semi-analytical method to solve direct dynamic problems in layered media”, Dokl. Akad. Nauk, 310:2 (1990), 323–327  mathnet  mathscinet
[13]Fatyanov A. G., “Analytical Modeling of Superlong-Distance Wave Fields in the Media with Composite Subsurface Geometries”, Yak. Math. Journal, 22 (2015), 86–96  zmath  elib
[14]Aki K., Richards P. G., Quantitative seismology. Theory and methods, W. H. Freeman and Company, San Francisco, 1980
How to Cite
Fatyanov, A. (2016) “The stable analytical solution for the wave fields in the sphere”, Mathematical notes of NEFU, 23(3), pp. 91-103. Available at: (Accessed: 22September2020).
Mathematical Modeling