Abstract
In this paper, a 2D numerical modeling of sound wave propagation in a shallow water medium that acts as a waveguide, are presented. This modeling is based on the method of characteristic which is not constrained by the Courant–Friedrichs–Lewy (CFL) condition. Using this method, the Euler time-dependent equations have been solved under adiabatic conditions inside of a shallow water waveguide which is consists of one homogeneous environment of water over a rigid bed. In this work, the stability and precision of the method of characteristics (MOC) technique for sound wave propagation in a waveguide were illustrated when it was applied with the semi-Lagrange method. The results show a significant advantage of the method of characteristics over the finite difference time domain (FDTD) method.Keywords:
wave propagation, shallow water, MOC method, waveguide, transmission lossReferences
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19. Jewell J. (2019), Higher-order Runge-Kutta type schemes for the method of characteristics, UVM Student Research Conference, https://scholarworks.uvm.edu/src/2019/program/355.
20. Jiang T., Guo P., Wu J. (2020), One‐sided on‐demand communication technology for the semi‐Lagrange scheme in the YHGSM, Concurrency and Computation: Practice and Experience, 32(7): e5586, https://doi.org/10.1002/cpe.5586.
21. Jihui W., GuijuanL., Bing, J., Zhenshan, W., Rui W. (2020), Numerical computation on the scattering sound field distribution of rigid sphere in shallow water waveguide, IOP Conference Series: Materials Science and Engineering, 780: 032058, https://doi.org/10.1088/1757-899X/780/3/032058.
22. Kauffmann T., Kocar I., Mahseredjian J. (2018), New investigations on the method of characteristics for the evaluation of line transients, Electric Power Systems Research, 160: 243–250, https://doi.org/10.1016/j.epsr.2018.03.004.
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24. Khalilabadi M.R. (2016b), The effect of meteorological events on sea surface height variations along the northwestern Persian Gulf, Current Science (00113891), 110(11): 2138–2141, https://doi.org/10.18520/cs/v110/i11/2138-2141.
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26. Khalilabadi M.R., Sadrinassab M., Chegini V., Akbarinassab M. (2015), Internal wave generation in the Gulf of Oman (outflow of Persian Gulf), Indian Journal of Geo-Marine Sciences, 44(03): 371–375, http://nopr.niscair.res.in/handle/123456789/34692.
27. Kirby R., Duan W. (2018), Modelling sound propagation in the ocean: A normal mode approach using finite elements, [In:] Australian Acoustical Society Annual Conference, AAS 2018, pp. 530–539, Australian Acoustical Society, http://hdl.handle.net/10453/139710.
28. Lax P.D. (2013), Stability of difference schemes, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 1–7, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_1.
29. LeFloch P.G. (2013), A Framework for Late-Time/Stiff Relaxation Asymptotics, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 119–137, Birkhäuser, https://doi.org/10.1007/978-0-8176-8394-8_8.
30. Li C., Campbell B.K., Liu Y., Yue D.K. (2019), A fast multi-layer boundary element method for direct numerical simulation of sound propagation in shallow water environments, Journal of Computational Physics, 392, 694–712, https://doi.org/10.1016/j.jcp.2019.04.068.
31. Li N., Zhu H., Wang X., Xiao R., Xue Y., Zheng G. (2021), Characteristics of very low frequency sound propagation in full waveguides of shallow water, Sensors, 21(1), 192, https://doi.org/10.3390/s21010192.
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33. Mahpeykar O., Khalilabadi M.R. (2021), Numerical modelling the effect of wind on water level and evaporation rate in the Persian Gulf, International Journal of Coastal and Offshore Engineering, 6(1): 47–53.
34. Matsumura Y., Okubo K., Tagawa N., TsuchiyaT., Ishizuka, T. (2015), Hybrid MM-MOC-based numerical simulation of acoustic wave propagation with non-uniform grid and perfectly matched layer absorbing boundaries, 2015 IEEE International Ultrasonics Symposium (IUS), pp. 1–4, https://doi.org/10.1109/ULTSYM.2015.0443.
35. Matsumura Y., Okubo K., Tagawa N., Tsuchiya T., Ishizuka T. (2017), Evaluation of numerical simulation of acoustic wave propagation using method of characteristics-based constrained interpolation profile (CIP-MOC) method with non-uniform grids, Acoustical Science and Technology, 38(1): 31–34, https://doi.org/10.1250/ast.38.31.
36. Mazumdar T., Gupta A. (2018), Application of Krylov acceleration technique in method of characteristics–based neutron transport code, Nuclear Science and Engineering, 192(2): 153–188, https://doi.org/10.1080/00295639.2018.1499340.
37. Mollaesmaeilpour S., Mohammad Mahdizadeh M., Hasanzade S., Khalilabadi M.R. (2019), The study of hydrophysical properties of the northern Arabian Sea during monsoon: A numerical study, Hydrophysics, 5(1): 47–59.
38. Nakamura T., Tanaka R., Yabe T., Takizawa K. (2001), Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique, Journal of Computational Physics, 174(1): 171–207, https://doi.org/10.1006/jcph.2001.6888.
39. Oshima T., Hiraguri Y., Imano M. (2014), Geometry reconstruction and mesh generation techniques for acoustic simulations over real-life urban areas using digital geographic information. Acoustical Science and Technology, 35(2): 108–118, https://doi.org/10.1250/ast.35.108.
40. Piao X., Kim P., Kim D. (2018), One-step L(α)-stable temporal integration for the backward semi-Lagrangian scheme and its application in guiding center problems, Journal of Computational Physics, 366: 327–340, https://doi.org/10.1016/j.jcp.2018.04.019.
41. Rhebergen S., Cockburn B. (2013), Space-time hybridizable discontinuous Galerkin method for the advection–diffusion equation on moving and deforming meshes, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery C.A. de Moura, C.S. Kubrusly (Eds.), pp. 45–63, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_4.
42. Saadat M.H., Bösch F., Karlin I.V. (2020), Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes, Physical Review E, 101(2): 023311, https://doi.org/10.1103/PhysRevE.101.023311.
43. Schneider K., Kolomenskiy D., Deriaz E. (2013), Is the CFL condition sufficient? Some remarks, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 139–146, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_9.
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47. Verlinden C.M.A., Sarkar J., Cornuelle B.D., Kuperman W.A. (2017), Determination of acoustic waveguide invariant using ships as sources of opportunity in a shallow water marine environment, The Journal of the Acoustical Society of America, 141(2): EL102–EL107, https://doi.org/10.1121/1.4976112.
48. Wang W., Yang D., Shi J. (2020), A prediction method for acoustic intensity vector field of elastic structure in shallow water waveguide, Shock and Vibration, 2020: article ID 5389719, https://doi.org/10.1155/2020/5389719.
49. Yabe T., Xiao F., Utsumi T. (2001), The constrained interpolation profile method for multiphase analysis, Journal of Computational Physics, 169(2): 556–593, https://doi.org/10.1006/jcph.2000.6625.
2. Ara Y., Okubo K., Tagawa N., Tsuchiya T., Ishizuka T. (2011), A novel numerical simulation of sound wave propagation using sub-grid CIP-MOC method, 2011 IEEE International Ultrasonics Symposium, pp. 760–763, https://doi.org/10.1109/ULTSYM.2011.6293349.
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4. Ayas H., Chabaat M., Amara L. (2019), Dynamic analysis of a cracked bar by the method of characteristics, International Journal of Structural Integrity, 10(4): 438–453, https://doi.org/10.1108/IJSI-01-2018-0001.
5. Cao F., Liu J. (2020), Nonlinear partial differential equation model-based coordination control for a master-slave two-link rigid-flexible manipulator with vibration repression, Journal of Computational and Nonlinear Dynamics, 16(2): 021007, https://doi.org/10.1115/1.4049219.
6. Cho S.Y., Boscarino S., Russo G., Yun S.-B. (2021), Conservative semi-Lagrangian schemes for kinetic equations. Part I: Reconstruction, Journal of Computational Physics, 432: 110159, https://doi.org/10.1016/j.jcp.2021.110159.
7. Costa G., Montemurro M., Pailhès J. (2021), NURBS hyper-surfaces for 3D topology optimization problems, Mechanics of Advanced Materials and Structures, 28(7): 665–684, https://doi.org/10.1080/15376494.2019.1582826.
8. Domingues M.O., Gomes S.M., Roussel O., Schneider K. (2013), Space-time adaptive multiresolution techniques for compressible Euler equations, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), Birkhäuser, Boston, pp. 101–117, https://doi.org/10.1007/978-0-8176-8394-8_7.
9. Duan W., Kirby R. (2019), Guided wave propagation in buried and immersed fluid-filled pipes: Application of the semi analytic finite element method, Computers Structures, 212: 236–247, https://doi.org/10.1016/j.compstruc.2018.10.020.
10. Duan W., Kirby R. (2020), A numerical approach for calculation of characteristics of edge waves in three-dimensional plates, Journal of Theoretical and Computational Acoustics, 29(02): 2050014, https://doi.org/10.1142/S2591728520500140.
11. Fievisohn R.T., Yu K.H. (2016), Steady-state analysis of rotating detonation engine flowfields with the method of characteristics, Journal of Propulsion and Power, 33(1): 89–99, https://doi.org/10.2514/1.B36103.
12. Fukuda A., Okubo K., Oshima T., Tsuchiya T., Kanamori M. (2018), Numerical analysis of three-dimensional acoustic field with background flow using constrained interpolation profile method, Japanese Journal of Applied Physics, 57(7S1): 07LC09, https://doi.org/10.7567/jjap.57.07lc09.
13. Gao W., Veeresha P., Prakasha D.G., Baskonus H.M. (2021), New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques, Numerical Methods for Partial Differential Equations, 37(1): 210–243, https://doi.org/10.1002/num.22526.
14. Gendre F., RicotD., Fritz, G., Sagaut P. (2017), Grid refinement for aeroacoustics in the lattice Boltzmann method: A directional splitting approach, Physical Review E, 96(2): 023311, https://doi.org/10.1103/PhysRevE.96.023311.
15. Hersh R. (2013), Mathematical intuition: Poincaré, Pólya, Dewey, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 9–30, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_2.
16. Hosseini S.H., Akbarinasab M., KhalilabadiM. R. (2018), Numerical simulation of the effect internal tide on the propagation sound in the Oman Sea, Journal of the Earth and Space Physics, 44(1): 215–225, https://doi.org/10.22059/jesphys.2018.221834.1006867.
17. Jeltsch R., Kumar H. (2013), Three-dimensional plasma arc simulation using resistive MHD, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 31–43, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_3.
18. Jena R.M., Chakraverty S., Baleanu D. (2019), On new solutions of time-fractional wave equations arising in shallow water wave propagation, Mathematics, 7(8): 722, https://doi.org/10.3390/math7080722.
19. Jewell J. (2019), Higher-order Runge-Kutta type schemes for the method of characteristics, UVM Student Research Conference, https://scholarworks.uvm.edu/src/2019/program/355.
20. Jiang T., Guo P., Wu J. (2020), One‐sided on‐demand communication technology for the semi‐Lagrange scheme in the YHGSM, Concurrency and Computation: Practice and Experience, 32(7): e5586, https://doi.org/10.1002/cpe.5586.
21. Jihui W., GuijuanL., Bing, J., Zhenshan, W., Rui W. (2020), Numerical computation on the scattering sound field distribution of rigid sphere in shallow water waveguide, IOP Conference Series: Materials Science and Engineering, 780: 032058, https://doi.org/10.1088/1757-899X/780/3/032058.
22. Kauffmann T., Kocar I., Mahseredjian J. (2018), New investigations on the method of characteristics for the evaluation of line transients, Electric Power Systems Research, 160: 243–250, https://doi.org/10.1016/j.epsr.2018.03.004.
23. Khalilabadi M. R. (2016a), A numerical study of internal tide generation due to interaction of barotropic tide with bottom topography in the Oman Gulf, Journal of the Earth and Space Physics, 42(3): 645–656, https://dx.doi.org/10.22059/jesphys.2016.57903.
24. Khalilabadi M.R. (2016b), The effect of meteorological events on sea surface height variations along the northwestern Persian Gulf, Current Science (00113891), 110(11): 2138–2141, https://doi.org/10.18520/cs/v110/i11/2138-2141.
25. Khalilabadi M.R. (2016c), Tide–surge interaction in the Persian Gulf, Strait of Hormuz and the Gulf of Oman, Weather, 71(10): 256–261, https://doi.org/10.1002/wea.2773.
26. Khalilabadi M.R., Sadrinassab M., Chegini V., Akbarinassab M. (2015), Internal wave generation in the Gulf of Oman (outflow of Persian Gulf), Indian Journal of Geo-Marine Sciences, 44(03): 371–375, http://nopr.niscair.res.in/handle/123456789/34692.
27. Kirby R., Duan W. (2018), Modelling sound propagation in the ocean: A normal mode approach using finite elements, [In:] Australian Acoustical Society Annual Conference, AAS 2018, pp. 530–539, Australian Acoustical Society, http://hdl.handle.net/10453/139710.
28. Lax P.D. (2013), Stability of difference schemes, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 1–7, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_1.
29. LeFloch P.G. (2013), A Framework for Late-Time/Stiff Relaxation Asymptotics, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 119–137, Birkhäuser, https://doi.org/10.1007/978-0-8176-8394-8_8.
30. Li C., Campbell B.K., Liu Y., Yue D.K. (2019), A fast multi-layer boundary element method for direct numerical simulation of sound propagation in shallow water environments, Journal of Computational Physics, 392, 694–712, https://doi.org/10.1016/j.jcp.2019.04.068.
31. Li N., Zhu H., Wang X., Xiao R., Xue Y., Zheng G. (2021), Characteristics of very low frequency sound propagation in full waveguides of shallow water, Sensors, 21(1), 192, https://doi.org/10.3390/s21010192.
32. Liu Z. (2021), 5 – The method of characteristics, [In:] Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation, L. Cao, H. Wu (Eds.), pp. 73–108, Woodhead Publishing, https://doi.org/10.1016/B978-0-12-818221-5.00010-6.
33. Mahpeykar O., Khalilabadi M.R. (2021), Numerical modelling the effect of wind on water level and evaporation rate in the Persian Gulf, International Journal of Coastal and Offshore Engineering, 6(1): 47–53.
34. Matsumura Y., Okubo K., Tagawa N., TsuchiyaT., Ishizuka, T. (2015), Hybrid MM-MOC-based numerical simulation of acoustic wave propagation with non-uniform grid and perfectly matched layer absorbing boundaries, 2015 IEEE International Ultrasonics Symposium (IUS), pp. 1–4, https://doi.org/10.1109/ULTSYM.2015.0443.
35. Matsumura Y., Okubo K., Tagawa N., Tsuchiya T., Ishizuka T. (2017), Evaluation of numerical simulation of acoustic wave propagation using method of characteristics-based constrained interpolation profile (CIP-MOC) method with non-uniform grids, Acoustical Science and Technology, 38(1): 31–34, https://doi.org/10.1250/ast.38.31.
36. Mazumdar T., Gupta A. (2018), Application of Krylov acceleration technique in method of characteristics–based neutron transport code, Nuclear Science and Engineering, 192(2): 153–188, https://doi.org/10.1080/00295639.2018.1499340.
37. Mollaesmaeilpour S., Mohammad Mahdizadeh M., Hasanzade S., Khalilabadi M.R. (2019), The study of hydrophysical properties of the northern Arabian Sea during monsoon: A numerical study, Hydrophysics, 5(1): 47–59.
38. Nakamura T., Tanaka R., Yabe T., Takizawa K. (2001), Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique, Journal of Computational Physics, 174(1): 171–207, https://doi.org/10.1006/jcph.2001.6888.
39. Oshima T., Hiraguri Y., Imano M. (2014), Geometry reconstruction and mesh generation techniques for acoustic simulations over real-life urban areas using digital geographic information. Acoustical Science and Technology, 35(2): 108–118, https://doi.org/10.1250/ast.35.108.
40. Piao X., Kim P., Kim D. (2018), One-step L(α)-stable temporal integration for the backward semi-Lagrangian scheme and its application in guiding center problems, Journal of Computational Physics, 366: 327–340, https://doi.org/10.1016/j.jcp.2018.04.019.
41. Rhebergen S., Cockburn B. (2013), Space-time hybridizable discontinuous Galerkin method for the advection–diffusion equation on moving and deforming meshes, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery C.A. de Moura, C.S. Kubrusly (Eds.), pp. 45–63, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_4.
42. Saadat M.H., Bösch F., Karlin I.V. (2020), Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes, Physical Review E, 101(2): 023311, https://doi.org/10.1103/PhysRevE.101.023311.
43. Schneider K., Kolomenskiy D., Deriaz E. (2013), Is the CFL condition sufficient? Some remarks, [In:] The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After its Discovery, C.A. de Moura, C.S. Kubrusly (Eds.), pp. 139–146, Birkhäuser, Boston, https://doi.org/10.1007/978-0-8176-8394-8_9.
44. Song P., Zhang Z., Zhang Q., Liang L., Zhao Q. (2020), Implementation of the CPU/GPU hybrid parallel method of characteristics neutron transport calculation using the heterogeneous cluster with dynamic workload assignment, Annals of Nuclear Energy, 135: 106957, https://doi.org/10.1016/j.anucene.2019.106957.
45. Subbotina N.N., Krupennikov E.A. (2017), The method of characteristics in an identification problem, Proceedings of the Steklov Institute of Mathematics, 299(1): 205–216, https://doi.org/10.1134/S008154381709022X.
46. Twyman J. (2018), Transient flow analysis using the method of characteristics MOC with five-point interpolation scheme, Obras y Proyectos, 24, 62–70, https://doi.org/10.4067/s0718-28132018000200062.
47. Verlinden C.M.A., Sarkar J., Cornuelle B.D., Kuperman W.A. (2017), Determination of acoustic waveguide invariant using ships as sources of opportunity in a shallow water marine environment, The Journal of the Acoustical Society of America, 141(2): EL102–EL107, https://doi.org/10.1121/1.4976112.
48. Wang W., Yang D., Shi J. (2020), A prediction method for acoustic intensity vector field of elastic structure in shallow water waveguide, Shock and Vibration, 2020: article ID 5389719, https://doi.org/10.1155/2020/5389719.
49. Yabe T., Xiao F., Utsumi T. (2001), The constrained interpolation profile method for multiphase analysis, Journal of Computational Physics, 169(2): 556–593, https://doi.org/10.1006/jcph.2000.6625.
