两斜交柱坐标系间谐波函数坐标变换的辅助平面方法
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摘要
针对利用波函数展开法进行三维地震响应研究中的坐标变换问题,提出了在两斜交柱坐标系间谐波函数空间坐标变换的辅助平面方法.通过建立与空间点对应的一系列辅助平面,将地震波三维散射研究中的谐波函数表达式从柱坐标系变换至辅助平面内的极坐标系,然后在辅助平面内运用Graf加法公式,将柱坐标系下的波函数表达式变换至与该柱坐标系斜交的另一柱坐标系下,从而将三维问题转换成二维问题进行处理,最终得到了斜交柱坐标系下内域问题和外域问题的坐标变换公式.
Aiming at the coordinate transformation in studying a 3-D seismic response with wave function expansion method, this paper presents an auxiliary plane method for harmonic function transformation between two skew cylindrical coordinate systems. By creating a series of auxiliary plane corresponding to spatial points, the 3-D harmonic wave function is transformed from cylindrical coordinate system to planar polar coordinate. Then the wave function under a cylindrical coordinate system is transformed into another one, which is skew to the former, by using the Graf’s addition formula on the auxiliary plane. Thus the 3-D problem is turned into two-dimensional. Finally, the coordinate transformation formula between two skew cylindrical coordinate systems is given.
Aiming at the coordinate transformation in studying a 3-D seismic response with wave function expansion method, this paper presents an auxiliary plane method for harmonic function transformation between two skew cylindrical coordinate systems. By creating a series of auxiliary plane corresponding to spatial points, the 3-D harmonic wave function is transformed from cylindrical coordinate system to planar polar coordinate. Then the wave function under a cylindrical coordinate system is transformed into another one, which is skew to the former, by using the Graf’s addition formula on the auxiliary plane. Thus the 3-D problem is turned into two-dimensional. Finally, the coordinate transformation formula between two skew cylindrical coordinate systems is given.
引文
董俊,赵成刚.2005.半球形凹陷饱和土半空间对入射平面SV波三维散射问题的解析解[J].地球物理学报,48(6):1412-1421.
李伟华,赵成刚.2004.饱和土沉积谷场地对平面SV波的散射问题的解析解[J].地球物理学报,47(5):911-919.
梁建文,纪晓东.2006.地下衬砌洞室对Rayleigh波的放大作用[J].地震工程与工程振动,26(4):24-31.
梁建文,魏新磊,LeeVW.2009.圆弧形沉积谷地对Rayleigh波三维散射解析解[J].天津大学学报,42(1):24-34.
梁建文,严林隽,LeeVW.2001.圆弧形层状沉积河谷地对入射平面P波的散射解析解[J].地震学报,23(2):167-184.
张郁山.2008.圆弧状多层沉积谷地在平面P波入射下稳态响应的解析解[J].地球物理学报,51(3):869-880.
赵成刚,韩铮.2007.半球形饱和沉积谷场地对入射平面Rayleigh波的三维散射问题的解析解[J].地球物理学报,50(3):905-914.
Abramowitz M,Stegun I A.1970.Handbook of Mathematical Functions with Formulas,Graphs,and Mathematical Table[M].9th ed.New York:Dover Publication:355-389.
Cao H,Lee V W.1990.Scattering and diffraction of plane P waves by circular-cylindrical canyons with variable depth-to-width ratio[J].Soil Dyn Earthquake Eng,9(3):141-150.
Davis C A,Lee V W,Bardet J P.2001.Transverse response of underground cavities and pipes to incident SV waves[J].Earthquake Eng Struct Dyn,30(3):383-410.
Lee V W.1982.A note on the scattering of elastic plane waves by a hemispherical canyon[J].Soil Dyn Earthquake Eng,1(3):122-129.
Lee V W.1984.Three-dimensional diffraction of plane P,SV and SH waves by a hemispherical alluvial valley[J].Soil Dyn Earthquake Eng,3(3):133-144.
Lee V W,Cao H.1989.Diffraction of SV waves by circular canyons of various depths[J].J Engrg Mech,115(9):2035-2056.
Lee V W,Karl J.1992.Diffraction of SV waves by underground,circular,cylindrical cavities[J].Soil Dyn Earthquake Eng,11(8):445-456.
Todorovska M I,Lee V W.1990.A note on response of shallow circular valleys to Rayleigh waves:analytical approach[J].Earthquake Eng and Eng Vibration,10(1):21-34.
Todorovska M I,Lee V W.1991.Surface motion of shallow circular alluvial valleys for incident plane SH waves:analytical solution[J].Soil Dyn Earthquake Eng,10(4):192-200.
Trifunac M D.1972.Scattering of plane SH wave by a semi-cylindrical canyon[J].Earthquake Eng Struct Dyn,1(3):267-281.
Wong H L,Trifunac M D.1974a.Scattering of plane SH-wave by a semi-elliptical canyon[J].Soil Dyn Earthquake Eng,9(3):157-169.
Wong H L,Trifunac M D.1974b.Surface motion of semi-elliptical alluvial valley for incident plane SH waves[J].Bull Seism Soc Amer,64(1):1389-1408.
Yuan X M,Liao Z P.1995.Scattering of plane SH wave by a cylindrical alluvial valley of circular-arc cross-section[J].Earthq Eng Struct Dyn,24(1):1303-1313.
李伟华,赵成刚.2004.饱和土沉积谷场地对平面SV波的散射问题的解析解[J].地球物理学报,47(5):911-919.
梁建文,纪晓东.2006.地下衬砌洞室对Rayleigh波的放大作用[J].地震工程与工程振动,26(4):24-31.
梁建文,魏新磊,LeeVW.2009.圆弧形沉积谷地对Rayleigh波三维散射解析解[J].天津大学学报,42(1):24-34.
梁建文,严林隽,LeeVW.2001.圆弧形层状沉积河谷地对入射平面P波的散射解析解[J].地震学报,23(2):167-184.
张郁山.2008.圆弧状多层沉积谷地在平面P波入射下稳态响应的解析解[J].地球物理学报,51(3):869-880.
赵成刚,韩铮.2007.半球形饱和沉积谷场地对入射平面Rayleigh波的三维散射问题的解析解[J].地球物理学报,50(3):905-914.
Abramowitz M,Stegun I A.1970.Handbook of Mathematical Functions with Formulas,Graphs,and Mathematical Table[M].9th ed.New York:Dover Publication:355-389.
Cao H,Lee V W.1990.Scattering and diffraction of plane P waves by circular-cylindrical canyons with variable depth-to-width ratio[J].Soil Dyn Earthquake Eng,9(3):141-150.
Davis C A,Lee V W,Bardet J P.2001.Transverse response of underground cavities and pipes to incident SV waves[J].Earthquake Eng Struct Dyn,30(3):383-410.
Lee V W.1982.A note on the scattering of elastic plane waves by a hemispherical canyon[J].Soil Dyn Earthquake Eng,1(3):122-129.
Lee V W.1984.Three-dimensional diffraction of plane P,SV and SH waves by a hemispherical alluvial valley[J].Soil Dyn Earthquake Eng,3(3):133-144.
Lee V W,Cao H.1989.Diffraction of SV waves by circular canyons of various depths[J].J Engrg Mech,115(9):2035-2056.
Lee V W,Karl J.1992.Diffraction of SV waves by underground,circular,cylindrical cavities[J].Soil Dyn Earthquake Eng,11(8):445-456.
Todorovska M I,Lee V W.1990.A note on response of shallow circular valleys to Rayleigh waves:analytical approach[J].Earthquake Eng and Eng Vibration,10(1):21-34.
Todorovska M I,Lee V W.1991.Surface motion of shallow circular alluvial valleys for incident plane SH waves:analytical solution[J].Soil Dyn Earthquake Eng,10(4):192-200.
Trifunac M D.1972.Scattering of plane SH wave by a semi-cylindrical canyon[J].Earthquake Eng Struct Dyn,1(3):267-281.
Wong H L,Trifunac M D.1974a.Scattering of plane SH-wave by a semi-elliptical canyon[J].Soil Dyn Earthquake Eng,9(3):157-169.
Wong H L,Trifunac M D.1974b.Surface motion of semi-elliptical alluvial valley for incident plane SH waves[J].Bull Seism Soc Amer,64(1):1389-1408.
Yuan X M,Liao Z P.1995.Scattering of plane SH wave by a cylindrical alluvial valley of circular-arc cross-section[J].Earthq Eng Struct Dyn,24(1):1303-1313.
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