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One-dimension nonlinear and dispersive seismic wave modeling in solid media

Zhou Cong, Wang Qing-Liang
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• 摘要

非线性理论是解决地学问题的重要手段, 充分认识非线性波动特征有助于解释实际观测资料中的一些特殊地震现象. 本文基于Hokstad改造的非线性本构方程, 利用交错网格有限差分法实现了固体介质中一维非线性地震波数值模拟; 采用通量校正传输方法消除非线性数值模拟中波形振荡, 提高模拟精度. 通过与解析解的对比验证了模拟结果的正确性. 研究结果显示了非线性系数对地震波的传播有重要影响, 并且当取适当的非线性和频散系数时, 地震波表现出孤立波的传播特性. 最后分析了不同的非线性地震波在固体介质中的传播演化特征.

Abstract

The nonlinear theory in Earth Science is very important for solving the problems of the earth. When considering some of the nonlinear properties of the medium, solitary wave (a special wave with a finite amplitude and a single peak or trough) may appear. Previous studies showed that it may be related to the rupture in the earthquake process. Therefore, it would be very helpful to explain some special phenomena in actual observation data if we fully understand the characteristics of nonlinear waves.#br#In this paper, based on the nonlinear acoustic wave equation, we first perform 1-D nonlinear acoustic wave modeling in solid media using a staggered grid finite difference method. To get the stable and accurate results, a flux-corrected transport method is used. Then we analyze several different types of nonlinear acoustic waves by setting different parameters to investigate their nonlinear characteristics in the solid media. Compared with the linear wave propagation, our results show that the nonlinear coefficients have important influences on the propagation of the acoustic waves. When the equations contain only a third-order nonlinear term (consider the case β 1 ≠ 0, β 2=0, α =0), the main lobe of the wave is tilted backward and its amplitude gradually attenuates with the wave spreading, and the amplitude of its front side-lobe attenuates slowly while the back side-lobe attenuates quickly. The whole shape and amplitude of the wave remain unchanged after propagating a certain distance. When the equations contain only a fourth-order nonlinear term (consider the case β 2 ≠ 0, β 1=0, α =0), the main lobe and the two side-lobes of the wave are all slowly damped, but the shape of the whole wave is unchanged with the wave spreading.#br#In addition, for some combinations of nonlinear and dispersive parameters (consider the case β 1 ≠ 0, α ≠ 0, β 2=0), the wave acts like the linear wave, and the nonlinear acoustic wave is equal to solitary wave which is usually obtained by Kortewegde de Vries (KdV) equation. We validate our modeling method by comparing our results with the analytic solitary solutions. Solitary wave propagates with a fixed velocity slightly less than that of the linear compressional wave, which is probably due to the balance between nonlinear and dispersion effects, making the stress-strain constitutive relations show the nature of linear wave.

作者及机构信息

通信作者: 周聪, zhoucong323@126.com
• 基金项目: 地震行业重大科研专项(批准号: 201508009)资助的课题.

Authors and contacts

Corresponding author: Zhou Cong, zhoucong323@126.com
• Funds: Project supported by the Special Earthquake Research Project, which is grant-aided by the China Earthquake Administration (Grant No. 201508009).

参考文献

 [1] Zheng H S, Zhang Z J, Yang B J 2004 Acta Seis. Sin. 26 77 (in Chinese) [郑海山, 张中杰, 杨宝俊 2004 地震学报 26 77] [2] Johnson P A, McCall K R 1994 Geophys. Res. Lett. 21 165 [3] Johnson P A 1996 J. Geophys. Res. 101 11553 [4] Van den Abeele K E-A 1996 J. Acoust. Soc. Am. 99 3334 [5] Miles J W 1980 Ann. Rev. Fl. Mech. 12 11 [6] Wang Z D 2005 Mechanics in Engineering 27 86 (in Chinese) [王振东 2005 力学与实践 27 86] [7] Sharon E, Cohen G, Fineberg J 2001 Nature 410 68 [8] Zhou C, Wang Q L, Wang S X 2014 Earthquake 34 112 (in Chinese) [周聪, 王庆良, 王双绪 2014 地震 34 112] [9] Wu Z L, Chen Y T 1998 Nonlinear Processes in Geophysics 5 121 [10] Bykov V G 2008 Acta Geophys. 56 270 [11] Bykov V G 2014 J. Seismol. 18 497 [12] McCall K R 1994 J. Geophys. Res. 99 2591 [13] Cheng N 1996 Geophysics 61 1935 [14] Hokstad K 2004 Geophysics 69 840 [15] Mandafu, Naranmandula 2009 Chinese J. Solid Mech. 30 614 (in Chinese) [满达夫, 那仁满都拉 2009 固体力学学报 30 614] [16] Mandafu, Naranmandula 2010 Acta Phys. Sin. 59 60 (in Chinese) [满达夫, 那仁满都拉 2010 物理学报 59 60] [17] Han H Y, Naranmandula, Shuang S 2012 Acta Phys. Sin. 61 059101 (in Chinese) [韩海英, 那仁满都拉, 双山 2012 物理学报 61 059101] [18] Qian Z W 2014 Chin. Phys. B 23 064301 [19] Zheng H S, Zhang Z J 2005 Chinese J. Geophys. 48 660 (in Chinese) [郑海山, 张中杰 2005 地球物理学报 48 660] [20] Boris J P, Book D L 1973 J. Comput. Phys. 11 38 [21] Fei T, Larner K 1995 Geophysics 60 1830 [22] Yang D H, Liu E, Zhang Z J, Teng J 2002 Geophys. J. Int. 148 320

施引文献

•  [1] Zheng H S, Zhang Z J, Yang B J 2004 Acta Seis. Sin. 26 77 (in Chinese) [郑海山, 张中杰, 杨宝俊 2004 地震学报 26 77] [2] Johnson P A, McCall K R 1994 Geophys. Res. Lett. 21 165 [3] Johnson P A 1996 J. Geophys. Res. 101 11553 [4] Van den Abeele K E-A 1996 J. Acoust. Soc. Am. 99 3334 [5] Miles J W 1980 Ann. Rev. Fl. Mech. 12 11 [6] Wang Z D 2005 Mechanics in Engineering 27 86 (in Chinese) [王振东 2005 力学与实践 27 86] [7] Sharon E, Cohen G, Fineberg J 2001 Nature 410 68 [8] Zhou C, Wang Q L, Wang S X 2014 Earthquake 34 112 (in Chinese) [周聪, 王庆良, 王双绪 2014 地震 34 112] [9] Wu Z L, Chen Y T 1998 Nonlinear Processes in Geophysics 5 121 [10] Bykov V G 2008 Acta Geophys. 56 270 [11] Bykov V G 2014 J. Seismol. 18 497 [12] McCall K R 1994 J. Geophys. Res. 99 2591 [13] Cheng N 1996 Geophysics 61 1935 [14] Hokstad K 2004 Geophysics 69 840 [15] Mandafu, Naranmandula 2009 Chinese J. Solid Mech. 30 614 (in Chinese) [满达夫, 那仁满都拉 2009 固体力学学报 30 614] [16] Mandafu, Naranmandula 2010 Acta Phys. Sin. 59 60 (in Chinese) [满达夫, 那仁满都拉 2010 物理学报 59 60] [17] Han H Y, Naranmandula, Shuang S 2012 Acta Phys. Sin. 61 059101 (in Chinese) [韩海英, 那仁满都拉, 双山 2012 物理学报 61 059101] [18] Qian Z W 2014 Chin. Phys. B 23 064301 [19] Zheng H S, Zhang Z J 2005 Chinese J. Geophys. 48 660 (in Chinese) [郑海山, 张中杰 2005 地球物理学报 48 660] [20] Boris J P, Book D L 1973 J. Comput. Phys. 11 38 [21] Fei T, Larner K 1995 Geophysics 60 1830 [22] Yang D H, Liu E, Zhang Z J, Teng J 2002 Geophys. J. Int. 148 320
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出版历程
• 收稿日期:  2015-05-14
• 修回日期:  2015-08-04
• 刊出日期:  2015-12-05

考虑频散效应的一维非线性地震波数值模拟

• 1. 中国地震局地质研究所, 北京 100029;
• 2. 中国地震局第二监测中心, 西安 710054
• 通信作者: 周聪, zhoucong323@126.com
基金项目: 地震行业重大科研专项(批准号: 201508009)资助的课题.

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