任意方向震源二维兰姆问题的一个精确解
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摘要
均匀弹性半空间表面或内部震源产生的地震波场的解析解属于Lamb问题,采用Cagniard-deHoop方法,对与水平面呈任意夹角的表面线源,求解了其作用于弹性半空间时的拉普拉斯-傅里叶双积分变换解.以δ-脉冲函数为例,给出了任意方向作用力下波场的构成,并定量解出了P波、S波、首波和Rayleigh波等各波的位移表达式,分析了不同作用方向下各波位移的相对大小.建立数值模型并进行数值模拟,模拟结果验证了理论研究的正确性.研究成果为近地表地震波场的研究提供了理论依据.
The analytic solution of seismic wave field caused by surface or inner seismic source in homogeneous elastic half-space belongs to Lamb problems.With the Cagniard-deHoop method,we developed a Laplace-Fourier double integral transformed solution for a surface line source which is in arbitrary angle with horizontal plane on elastic half-space.Some quantitative relations of displacements of P-wave,S-wave,head wave and Rayleigh wave under(-impulse are shown,and the relationship between amplitudes is analyzed.A digital model is built and the results of wave equation modeling are used to verify the accuracy of the theoretical achievements.Due to the particularity of near surface,the achievement on arbitrary direction source provides theoretical foundation to wave filed analysis.
The analytic solution of seismic wave field caused by surface or inner seismic source in homogeneous elastic half-space belongs to Lamb problems.With the Cagniard-deHoop method,we developed a Laplace-Fourier double integral transformed solution for a surface line source which is in arbitrary angle with horizontal plane on elastic half-space.Some quantitative relations of displacements of P-wave,S-wave,head wave and Rayleigh wave under(-impulse are shown,and the relationship between amplitudes is analyzed.A digital model is built and the results of wave equation modeling are used to verify the accuracy of the theoretical achievements.Due to the particularity of near surface,the achievement on arbitrary direction source provides theoretical foundation to wave filed analysis.
引文
[1]Lamb H.On the propagation of tremors over the surface ofan elastic solid.Philosophical Transactions of the RoyalSociety(London),1904,203:1-42.
[2]Cagniard L.Reflexion et Refraction des Ondes SismiquesProgressives.Paris:Cauthiers-Villars,1939;translated intoEnglish and revised by Flinn E A,Dix C H.Reflection andRefraction of Progressive Seismic Waves.New York:McGraw-Hill,1962.
[3]deHoop A T.A modification of Cagniard′s method forsolving seismic pulse problem.Applied Science Research,1960,8(1):349-356.
[4]Pekeris C L.The seismic buried pulse.Proceedings of theNational Academy of Sciences,1955,41(9):629-639.
[5]Chao C C.Dynamical response of an elastic half-space totangential surface loadings.ASME Journal of AppliedMechanics,1960,27(3):559-567.
[6]Johnson L R.Green′s function for Lamb′s problem.Geophysics Journal of Royal Astronomy Society,1974,37(1):99-131.
[7]王可成,王贻荪.弹性半空间内的突加竖向集中力所产生的表面位移.固体力学学报,1983,4(3):427-434.Wang K C,Wang Y S.Surface displacement of an elastichalf-space due to a vertically buried point-source load.Chinese Journal of Solid Mechanics(in Chinese),1983,4(3):427-434.
[8]郑建龙.弹性半空间动力问题的基本解.固体力学学报,1988,9(1):76-81.Zheng J L.Fundamental solutions of elastic half-space fordynamic problems.Chinese Journal of Solid Mechanics(inChinese),1988,9(1):76-81.
[9]刘凯欣,刘广裕.垂直点源问题的一个精确解.科学通报,2004,49(5):419-423.Liu K X,Liu G Y.An exact solution for a normal pointsource problem.Chinese Science Bulletin(in Chinese),2004,49(5):419-423.
[10]刘广裕,刘凯欣.区域脉冲载荷下二维Lamb问题的精确求解.固体力学学报,2007,28(4):341-346.Liu G Y,Liu K X.Exact solution for a two-dimensionalLamb′s problem due to a strip impulse loading.ActaMechanica Solida Sinica(in Chinese),2007,20(3):341-346.
[11]何樵登,韩立国,朱建伟等.地震波理论.长春:吉林大学出版社,2005:91-94.He Q D,Han L G,Zhu J W,et al.Theory of Seismic Waves(in Chinese).Changchun:Jilin University Press,2005:91-94.
[12]Forrestal M J,Fugelso L E,Neidharllt G L.Response of ahalf-space to transient loads.//Proceedings of EngineeringMechanics Division Specially Conference.The American Society ofCivil Engineers,1966:SN 1-SN 6.
[2]Cagniard L.Reflexion et Refraction des Ondes SismiquesProgressives.Paris:Cauthiers-Villars,1939;translated intoEnglish and revised by Flinn E A,Dix C H.Reflection andRefraction of Progressive Seismic Waves.New York:McGraw-Hill,1962.
[3]deHoop A T.A modification of Cagniard′s method forsolving seismic pulse problem.Applied Science Research,1960,8(1):349-356.
[4]Pekeris C L.The seismic buried pulse.Proceedings of theNational Academy of Sciences,1955,41(9):629-639.
[5]Chao C C.Dynamical response of an elastic half-space totangential surface loadings.ASME Journal of AppliedMechanics,1960,27(3):559-567.
[6]Johnson L R.Green′s function for Lamb′s problem.Geophysics Journal of Royal Astronomy Society,1974,37(1):99-131.
[7]王可成,王贻荪.弹性半空间内的突加竖向集中力所产生的表面位移.固体力学学报,1983,4(3):427-434.Wang K C,Wang Y S.Surface displacement of an elastichalf-space due to a vertically buried point-source load.Chinese Journal of Solid Mechanics(in Chinese),1983,4(3):427-434.
[8]郑建龙.弹性半空间动力问题的基本解.固体力学学报,1988,9(1):76-81.Zheng J L.Fundamental solutions of elastic half-space fordynamic problems.Chinese Journal of Solid Mechanics(inChinese),1988,9(1):76-81.
[9]刘凯欣,刘广裕.垂直点源问题的一个精确解.科学通报,2004,49(5):419-423.Liu K X,Liu G Y.An exact solution for a normal pointsource problem.Chinese Science Bulletin(in Chinese),2004,49(5):419-423.
[10]刘广裕,刘凯欣.区域脉冲载荷下二维Lamb问题的精确求解.固体力学学报,2007,28(4):341-346.Liu G Y,Liu K X.Exact solution for a two-dimensionalLamb′s problem due to a strip impulse loading.ActaMechanica Solida Sinica(in Chinese),2007,20(3):341-346.
[11]何樵登,韩立国,朱建伟等.地震波理论.长春:吉林大学出版社,2005:91-94.He Q D,Han L G,Zhu J W,et al.Theory of Seismic Waves(in Chinese).Changchun:Jilin University Press,2005:91-94.
[12]Forrestal M J,Fugelso L E,Neidharllt G L.Response of ahalf-space to transient loads.//Proceedings of EngineeringMechanics Division Specially Conference.The American Society ofCivil Engineers,1966:SN 1-SN 6.
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