Cite this article as:

Manenkov S. А. Two Approaches to the Solution of the Scalar Problem of Diffraction on the Plane Two-periodic Lattice From Bodies of Revolution Located in the Liquid Layer. Izvestiya of Saratov University. New series. Series Physics, 2018, vol. 18, iss. 1, pp. 46-63. DOI: https://doi.org/10.18500/1817-3020-2018-18-1-46-63


UDC: 
534.23:537.874.6
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Russian

Two Approaches to the Solution of the Scalar Problem of Diffraction on the Plane Two-periodic Lattice From Bodies of Revolution Located in the Liquid Layer

Abstract

Background, Objectives and Methods: The problem of diffraction of acoustic waves on the lattices located in the layered media is of great scientific interest in hydroacoustics. There are many methods of the solution of this diffraction problem, such as the method of the surface integral equations, finite element method, boundary element method, etc. One of universal method of solution of diffraction problems is the modified method of discrete sources (MMDS). Earlier this method was applied to the solution of the problems of wave scattering on a single body of revolution, on a group of bodies and on lattices located in free space. The purpose of this study is to develop the numerical algorithms based on MMDS for solution of the scalar problem of diffraction of acoustic waves on the planar grating consisting of identical impedance bodies of revolution which is immersed in a liquid layer.

Results: Based on MMDS two techniques of the solution of the scalar three-dimensional problem of diffraction on the planar lattice consisting of identical impedance bodies of revolution located in a liquid layer are developed. The correctness of MMDS is illustrated on the example of diffraction of the plane wave on the single strongly elongated superellipsoid of revolution. Comparison of the results of calculation of the reflection and transmission coefficients for the lattice consisting of spherical elements obtained by means of both techniques offered in the paper is carried out. For validation of MMDS the check of the accuracy of fulfillment of the energy conservation law is executed and the dependence of the residual of the boundary condition on the contour of axial section of the central element of the lattice is plotted. It is shown that the residual has the order 1.5∙10−5 at the chosen model parameters. Frequency dependences for various geometries of the elements of the lattice (for two types of boundary conditions) and dependences of the absolute value of reflection and transmission coefficients on the filling factor of the lattice are obtained.

Conclusion: There is an essential difference between the behavior of the scattered field under diffraction on the lattices located in homogeneous medium and the behavior of the scattered field under diffraction on the lattices immersed in layered medium.

References

1. Kobelev Yu. A. Scattering of a plane sound wave by spherical particles performing monopole oscillations and positioned at the sites of an infi nite plane lattice with identical cells. Acoustical Physics, 2014, vol. 60, no. 1, pp. 1–10. DOI: https://doi.org/10.1134/S1063771013060092

2. Kobelev Yu. A. Multiple sound wave scattering by spherical particles performing monopole oscillations and located at the sites of a three-dimensional lattice with identical cells. Acoustical Physics, 2015, vol. 61, no. 4, pp. 392–401. DOI: https://doi.org/10.1134/S1063771015030094

3. Papkova Y. I. The fi eld of a point source in an inhomogeneous hydroacoustic waveguide with a body drifting on the surface. Acoustical Physics, 2015, vol. 61, no. 4, pp. 440–445. DOI: https://doi.org/10.1134/S1063771015040077

4. Kudasheva O. A., Sevryugova N. V. Izluchenie zvuka beskonechnoi periodicheskoi reshetkoi s zazorami [Sound radiation an infi nite periodic lattice with gaps]. Akusticheskii Zhurnal, 1976, vol. 22, no. 3, pp. 385–392 (in Russian). 

5. Vovk I. V. Izluchenie zvuka periodicheskoi reshetkoi iz sterzhnevykh preobrazovatelei, zvukoizolirovannoi s tyl’noi storony sloem [Sound radiation the periodic lattice from rod converters soundproofed from the back by a layer]. Akusticheskii Zhurnal, 1980, vol. 26, no. 4, pp. 522–527 (in Russian). 

6. Vovk I. V. Difraktsiia zvuka na reshetke iz lent konechnoi prozrachnosti [Diffraction of a sound on a lattice from stripes of fi nite transparency]. Akusticheskii Zhurnal, 1985, vol. 32, no. 3, pp. 378–381 (in Russian). 

7. Shestopalov V. P. Metod zadachi Rimana–Gil’berta v teorii difraktsii i rasprostraneniia elektromagnitnykh voln [Method of Riman Hylbery problem in theory of diffraction and electromagnetic wave propagation]. Kharkov, Izd-vo Khar'k. un-ta, 1971. 400 p. (in Russian). 

8. Shestopalov V. P., Litvinenko L. N., Masalov S. A., Sologub V. G. Difraktsiia voln na reshetkakh [Wave diffraction on lattices]. Kharkov, Izd-vo Khar'k. un-ta, 1973. 288 p. (in Russian). 

9. Yasumoto K., Toyama H., Kushta T. Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique. IEEE Transactions on antennas and propagation, 2004, vol. 52, iss. 10, pp. 2603–2611. DOI: https://doi.org/10.1109/TAP.2004.834440

10. Lapin A. D. Pogloshchenie zvuka reshetkoi rezonatorov s dissipatsiei [Absorption of a sound a lattice of resonators with dissipation]. Akusticheskii Zhurnal, 2002, vol. 48, no. 3, pp. 428–429 (in Russian). 

11. Lapin A. D., Mironov M. A. Sound absorption by a planar array of monopole-dipole scatterers. Acoustical Physics, 2006, vol. 52, no.4, pp. 425–428. DOI: https://doi.org/10.1134/S1063771006040087

12. Manenkov S. A. Diffraction of an electromagnetic wave by a three dimensional planar lattice. J. Commun. Technol. Electron., 2010, vol. 55, no. 4, pp. 375–384. DOI: https://doi.org/10.1134/S1064226910040029

13. Manenkov S. A. Two approaches to the solution of the problem of diffraction by a plane grating of dielectric bodies of revolution. J. Commun. Technol. Electron., 2015, vol. 60, no. 8, pp. 809–821. DOI: https://doi.org/10.1134/S1064226915080148

14. Manenkov S. A. Solution of the three dimensional problem of plane wave diffraction by a two-period plane grating. Acoustical Physics, 2016, vol. 62, no. 2, pp. 133–142. DOI: https://doi.org/10.7868/S0320791916020118

15. Maurel A., Mercier J.-F., Félix S. Wave propagation through penetrable scatterers in a waveguide and through a penetrable grating, J. Acoust. Soc. Amer., 2014, vol. 135. iss.1, pp. 165-174. DOI: https://doi.org/10.1121/1.4836075

16. Karimi M., Croaker P., Kessissoglou N. Acoustic scattering for 3D multi-directional periodic structures using the boundary element method. J. Acoust. Soc. Am., 2017, vol. 141, iss.1, pp. 313–323. DOI: https://doi.org/10.1121/1.4973908

17. Hassan A. Kalhor, Mohammad Ilyas. Scattering of plane electromagnetic waves by a grating of conducting cylinders embedded in a dielectric slab over a ground plane. IEEE Transactions on antennas and propagation, 1982, vol. AP-30, no. 4, pp. 576–579. 

18. Hennion A. C., Bossut R., Decarpigny J. N., Audoly C. Application of the fi nite element method to analyze the scattering of a plane acoustic wave from doubly periodic structures. Physical acoustics. Eds. O. Leroy, M. A. Breazeale. New York, Plenum Press, 1991. P. 359–364. 

19. Panin S. B., Poyedinchuk A. Ye. Electromagnetic-wave diffraction by a grating with a chiral layer. Izv. Vuzov. Radiophysics and Quantum Electronics, 2002, vol. 45, no. 8, pp. 629–639.

20. Groby J.-P., Duclos A., Dazel O., Boeckx L., Lauriks W. Absorption of a rigid frame porous layer with periodic circular inclusions backed by a periodic grating. J. Acoust. Soc. Amer., 2011, vol. 129, iss. 5, pp. 3035–3046 . DOI: https://doi.org/10.1121/1.3561664

21. Lagarrigue C., Groby J.-P., Tournat V., Dazel O., Umnova O. Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions. J. Acoust. Soc. Amer., 2013, vol. 134, iss. 6, pp. 4670–4680. DOI: https://doi.org/10.1121/1.4824843

22. Petoev I. M., Tabatadze V. A., Zaridze R. S. Application of the method of auxiliary sources to the problems of diffraction of electromagnetic wave on some metal-dielectric structures. J. Commun. Technol. Electron., 2013, vol. 58, no. 5, pp. 404–416. DOI: https://doi.org/10.7868/S0033849413050069

23. Kurkchan A. G., Manenkov S. A., Negorozhina E. S. Diffraction of a plane wave by a multiserial lattice located in a dielectric layer. J. Commun. Technol. Electron., 2016, vol. 61, no. 3, pp. 224–233. DOI: https://doi.org/10.1134/S1064226916030104

24. Abawi A. T., Krysl P., España A., Kargl S., Williams K., Plotnick D. Modeling the acoustic response of elastic targets in a layered medium using the coupled fi nite element/ boundary element method. J. Acoust. Soc. Am., 2016, vol. 140, iss. 4, pp. 2968–2977. DOI: https://doi.org/10.1121/1.4969182

25. Kupradze V. O priblizhennykh metodakh resheniia zadach matematicheskoi fiziki [About approximate methods of the solution of problems of mathematical physics]. Uspekhi mat. nauk, 1967, vol. 22, no. 2, pp. 59–107 (in Russian). 

26. Kurkchan A. G., Minaev S. A., Soloveichik A. L. A modifi cation of the method of discrete sources based on prior information about the singularities of the diffracted fi eld. J. Commun. Technol. Electron., 2001, vol. 46, no. 6, pp. 615–621. 

27. Kyurkchan A. G., Smirnova N. I. Mathematical modeling in diffraction theory based on a priori information on the analytic properties of the solution. Amsterdam, Elsevier, 2015. 268 p. 

28. Kyurkchan A. G., Manenkov S. A. Application of different orthogonal coordinates using modifi ed method of discrete sources for solving a problem of wave diffraction on a body of revolution. Journal of Quantitative Spectroscopy and Radiative Transfer, 2012, vol. 113, pp. 2368–2378. DOI: https://doi.org/10.1016/j.jqsrt.2012.05.010

29. Manenkov S. A. A new version of the modifi ed method of discrete sources in application to the problem of diffraction by a body of revolution. Acoustical Physics, 2014, vol. 60, no. 2, pp. 127–133. DOI: https://doi.org/10.1134/S1063771014010102

30. Brekhovskikh L. M. Volny v sloistykh sredakh [Waves in layered media]. Moscow, Nauka, 1973. 343 p. (in Russian). 

31. Tikhonov A. N., Samarskii A. A. Uravneniia matematicheskoi fi ziki [Equations of Mathematical Physics]. Moscow, Izd-vo Mosk. un-ta, 1999. 798 p. (in Russian). 

32. Kleev A. I., Kyurkchan A. G. Application of the pattern equation method in spheroidal coordinates to solving diffraction problems with highly prolate scatterers. Acoustical Physics, 2015, vol. 61, no. 1, pp. 19–27. DOI: https://doi.org/10.1134/S1063771014060104.

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