Excitation of a layered sphere by $N$ acoustic sources: Exact solutions, low-frequency approximations, and inverse problems
Authors:
Andreas Kalogeropoulos and Nikolaos L. Tsitsas
Journal:
Quart. Appl. Math. 81 (2023), 141-173
MSC (2020):
Primary 74J20, 35P25, 78A48, 78A46
DOI:
https://doi.org/10.1090/qam/1632
Published electronically:
October 18, 2022
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Acoustic excitation of a layered sphere by $N$ external and internal point sources is considered. The direct problem is solved by developing a T-Matrix method leading to the analytical determination of all the involved acoustic fields. Low-frequency far-field approximations are derived for different source distributions and scatterer’s characteristics. The behaviour of the scattering cross sections is investigated. Several inverse problems are formulated and solved analytically; thus the respective unknown quantities are recovered explicitly. These problems include localization of the sources, determination of their number, identification of the core’s type and extraction of the parameters of the scatterer’s layers.
References
- G. Dassios and A. S. Fokas, Electro-magneto-encephalography for a three-shell model: dipoles and beyond for the spherical geometry, Inverse Problems 25 (2009), no. 3, 035001, 20. MR 2480171, DOI 10.1088/0266-5611/25/3/035001
- M. J. Povey, Ultrasound particle sizing: A review, Particuology 11 (2013), 135–147.
- V. Pinfield and D. M. Forrester, Multiple scattering in random dispersions of spherical scatterers: Effects of shear-acoustic interactions, Journal of the Acoustical Society of America, 141 (2017), 649–660.
- George Dassios and Gregory Kamvyssas, Point source excitation in direct and inverse scattering: the soft and the hard small sphere, IMA J. Appl. Math. 55 (1995), no. 1, 67–84. MR 1349743, DOI 10.1093/imamat/55.1.67
- G. Dassios, M. Hadjinicolaou, and G. Kamvyssas, Direct and inverse scattering for point source fields. The penetrable small sphere, ZAMM Z. Angew. Math. Mech. 79 (1999), no. 5, 303–316. MR 1695282, DOI 10.1002/(SICI)1521-4001(199905)79:5<303::AID-ZAMM303>3.0.CO;2-A
- Nikolaos L. Tsitsas and Christodoulos Athanasiadis, Point source excitation of a layered sphere: direct and far-field inverse scattering problems, Quart. J. Mech. Appl. Math. 61 (2008), no. 4, 549–580. MR 2448644, DOI 10.1093/qjmam/hbn017
- Prokopios Prokopiou and Nikolaos L. Tsitsas, Direct and inverse low-frequency acoustic excitation of a layered sphere by an arbitrarily positioned point source, Math. Methods Appl. Sci. 41 (2018), no. 3, 1040–1046. MR 3762328, DOI 10.1002/mma.4088
- N. L. Tsitsas and P. A. Martin, Finding a source inside a sphere, Inverse Problems 28 (2012), no. 1, 015003, 11. MR 2864500, DOI 10.1088/0266-5611/28/1/015003
- Xiaodong Liu, Bo Zhang, and Guanghui Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems 26 (2010), no. 1, 015002, 14. MR 2575347, DOI 10.1088/0266-5611/26/1/015002
- J. H. Wu, A. Q. Liu, H. L. Chen and T. N. Chen, Multiple scattering of a spherical acoustic wave from fluid spheres, Journal of Sound and Vibration 290 (2006), 17–33.
- N. S. Grigorieva and G. M. Fridman, Scattering of sound by an elastic spherical shell immersed in a waveguide with a fluid bottom, Acoustical Physics 59 (2013), 373–381.
- Roland Potthast, Filippo M. Fazi, and Philip A. Nelson, Source splitting via the point source method, Inverse Problems 26 (2010), no. 4, 045002, 17. MR 2601318, DOI 10.1088/0266-5611/26/4/045002
- G. Ping, E. Fernandez-Grande, P. Gerstoft and Z. Chu, Three-dimensional source localization using sparse Bayesian learning on a spherical microphone array, Journal of Acoustical Society of America 147 (2020), 3895–3904.
- H. Ding, Y. Bao, Q. Huang, C. Li and G. Chai, Three-dimensional localization of point acoustic sources using a planar microphone array combined with beamforming, Royal Society Open Science 5 (2018), 181407.
- L. Zhang, D. Ding, D. Yang, J. Wang and J. Shi, Sound source localization using non-conformal surface sound field transformation based on spherical harmonic wave decomposition, Sensors 17 (2017), 1087.
- E. Fernandez-Grande, Sound field reconstruction using a spherical microphone array, Journal of the Acoustical Society of America 139 (2016), 1168–1178.
- Andreas Kalogeropoulos and Nikolaos L. Tsitsas, Excitation of a layered medium by $N$ sources: scattering relations, interaction cross sections and physical bounds, Quart. Appl. Math. 79 (2021), no. 2, 335–356. MR 4246495, DOI 10.1090/qam/1581
- Andreas Kalogeropoulos and Nikolaos L. Tsitsas, Electromagnetic interactions of dipole distributions with a stratified medium: power fluxes and scattering cross sections, Stud. Appl. Math. 148 (2022), no. 3, 1040–1068. MR 4429080, DOI 10.1111/sapm.12469
- A. Kalogeropoulos and N. L. Tsitsas, Analysis of interaction scattering cross sections and their physical bounds for multiple-dipole stimulation of a three-dimensional layered medium, IEEE Open Journal of Antennas and Propagation 2 (2021), 506–520.
- George Dassios and Ralph Kleinman, Low frequency scattering, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1858914
- Milton Abramowitz (ed.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1965. Superintendent of Documents. MR 0177136
- George Arfken, Mathematical methods for physicists, Academic Press, New York-London, 1966. MR 0205512
- J. P. McKelvey, Simple transcendental expressions for the roots of cubic equations, Amer. J. Phys. 52 (1984), no. 3, 269–270. MR 736875, DOI 10.1119/1.13706
References
- G. Dassios and A. S. Fokas, Electro-magneto-encephalography for a three-shell model: dipoles and beyond for the spherical geometry, Inverse Problems 25 (2009), no. 3, 035001, 20. MR 2480171, DOI 10.1088/0266-5611/25/3/035001
- M. J. Povey, Ultrasound particle sizing: A review, Particuology 11 (2013), 135–147.
- V. Pinfield and D. M. Forrester, Multiple scattering in random dispersions of spherical scatterers: Effects of shear-acoustic interactions, Journal of the Acoustical Society of America, 141 (2017), 649–660.
- George Dassios and Gregory Kamvyssas, Point source excitation in direct and inverse scattering: the soft and the hard small sphere, IMA J. Appl. Math. 55 (1995), no. 1, 67–84. MR 1349743, DOI 10.1093/imamat/55.1.67
- G. Dassios, M. Hadjinicolaou, and G. Kamvyssas, Direct and inverse scattering for point source fields. The penetrable small sphere, ZAMM Z. Angew. Math. Mech. 79 (1999), no. 5, 303–316. MR 1695282, DOI 10.1002/(SICI)1521-4001(199905)79:5<303::AID-ZAMM303>3.0.CO;2-A
- Nikolaos L. Tsitsas and Christodoulos Athanasiadis, Point source excitation of a layered sphere: direct and far-field inverse scattering problems, Quart. J. Mech. Appl. Math. 61 (2008), no. 4, 549–580. MR 2448644, DOI 10.1093/qjmam/hbn017
- Prokopios Prokopiou and Nikolaos L. Tsitsas, Direct and inverse low-frequency acoustic excitation of a layered sphere by an arbitrarily positioned point source, Math. Methods Appl. Sci. 41 (2018), no. 3, 1040–1046. MR 3762328, DOI 10.1002/mma.4088
- N. L. Tsitsas and P. A. Martin, Finding a source inside a sphere, Inverse Problems 28 (2012), no. 1, 015003, 11. MR 2864500, DOI 10.1088/0266-5611/28/1/015003
- Xiaodong Liu, Bo Zhang, and Guanghui Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems 26 (2010), no. 1, 015002, 14. MR 2575347, DOI 10.1088/0266-5611/26/1/015002
- J. H. Wu, A. Q. Liu, H. L. Chen and T. N. Chen, Multiple scattering of a spherical acoustic wave from fluid spheres, Journal of Sound and Vibration 290 (2006), 17–33.
- N. S. Grigorieva and G. M. Fridman, Scattering of sound by an elastic spherical shell immersed in a waveguide with a fluid bottom, Acoustical Physics 59 (2013), 373–381.
- Roland Potthast, Filippo M. Fazi, and Philip A. Nelson, Source splitting via the point source method, Inverse Problems 26 (2010), no. 4, 045002, 17. MR 2601318, DOI 10.1088/0266-5611/26/4/045002
- G. Ping, E. Fernandez-Grande, P. Gerstoft and Z. Chu, Three-dimensional source localization using sparse Bayesian learning on a spherical microphone array, Journal of Acoustical Society of America 147 (2020), 3895–3904.
- H. Ding, Y. Bao, Q. Huang, C. Li and G. Chai, Three-dimensional localization of point acoustic sources using a planar microphone array combined with beamforming, Royal Society Open Science 5 (2018), 181407.
- L. Zhang, D. Ding, D. Yang, J. Wang and J. Shi, Sound source localization using non-conformal surface sound field transformation based on spherical harmonic wave decomposition, Sensors 17 (2017), 1087.
- E. Fernandez-Grande, Sound field reconstruction using a spherical microphone array, Journal of the Acoustical Society of America 139 (2016), 1168–1178.
- Andreas Kalogeropoulos and Nikolaos L. Tsitsas, Excitation of a layered medium by $N$ sources: scattering relations, interaction cross sections and physical bounds, Quart. Appl. Math. 79 (2021), no. 2, 335–356. MR 4246495, DOI 10.1090/qam/1581
- Andreas Kalogeropoulos and Nikolaos L. Tsitsas, Electromagnetic interactions of dipole distributions with a stratified medium: power fluxes and scattering cross sections, Stud. Appl. Math. 148 (2022), no. 3, 1040–1068. MR 4429080, DOI 10.1111/sapm.12469
- A. Kalogeropoulos and N. L. Tsitsas, Analysis of interaction scattering cross sections and their physical bounds for multiple-dipole stimulation of a three-dimensional layered medium, IEEE Open Journal of Antennas and Propagation 2 (2021), 506–520.
- George Dassios and Ralph Kleinman, Low frequency scattering, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1858914
- Milton Abramowitz (ed.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1965. Superintendent of Documents. MR 0177136
- George Arfken, Mathematical methods for physicists, Academic Press, New York-London, 1966. MR 0205512
- J. P. McKelvey, Simple transcendental expressions for the roots of cubic equations, Amer. J. Phys. 52 (1984), no. 3, 269–270. MR 736875, DOI 10.1119/1.13706
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
74J20,
35P25,
78A48,
78A46
Retrieve articles in all journals
with MSC (2020):
74J20,
35P25,
78A48,
78A46
Additional Information
Andreas Kalogeropoulos
Affiliation:
School of Informatics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
MR Author ID:
1434482
ORCID:
0000-0001-6526-5606
Email:
akaloger@csd.auth.gr
Nikolaos L. Tsitsas
Affiliation:
School of Informatics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
MR Author ID:
779168
ORCID:
0000-0003-1409-2631
Email:
ntsitsas@csd.auth.gr
Keywords:
Acoustic waves,
scattering,
layered sphere,
point sources,
cross sections,
inverse problems
Received by editor(s):
July 13, 2022
Received by editor(s) in revised form:
August 22, 2022
Published electronically:
October 18, 2022
Additional Notes:
The first author was co-financed by Greece and the European Union (European Social Fund) through the Operational Program “Human Resource Development, Education and Lifelong Learning”, 2014–2020, in the context of the Action “Strengthening human resources through the implementation of Ph.D. Research Sub-Action 2: IKY Scholarship Program for Ph.D. Candidates at Greek Universities”
Article copyright:
© Copyright 2022
Brown University
