Scattering and reflection of SH waves around a slope on an elastic wedged space
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摘要
The scattering and reflection of SH waves by a slope on an elastic wedged space is investigated. A series solution is obtained by using the wave function expansion method. The slope on a wedged space is divided into two subregions by an artificial, auxiliary circular arc. The wave fields with unknown complex coefficients within each sub-region are derived. Applying Graf addition theorem, the scattered waves in the sub-regions are expressed in a global coordinate system. Fourier transform is adopted to derive a consistent form of standing waves in the inner region using the orthogonality of the cosine functions. The boundary-valued problem is solved by stress and displacement continuity along the artificial, auxiliary arc to obtain the unknown complex coefficients. Parametric studies are next performed to investigate how the topography from the slope on the wedged space will affect the scattering and diffraction, and hence the amplification and de-amplification of the SH waves. Numerical results show that the surface motions on the slope of the wedged space is influenced greatly by the topography. Amplification of the surface motions near the slope vertex is significant. The corresponding phases along the wedged space surfaces are consistent with the direction that the SH waves are propagating.
The scattering and reflection of SH waves by a slope on an elastic wedged space is investigated. A series solution is obtained by using the wave function expansion method. The slope on a wedged space is divided into two subregions by an artificial, auxiliary circular arc. The wave fields with unknown complex coefficients within each sub-region are derived. Applying Graf addition theorem, the scattered waves in the sub-regions are expressed in a global coordinate system. Fourier transform is adopted to derive a consistent form of standing waves in the inner region using the orthogonality of the cosine functions. The boundary-valued problem is solved by stress and displacement continuity along the artificial, auxiliary arc to obtain the unknown complex coefficients. Parametric studies are next performed to investigate how the topography from the slope on the wedged space will affect the scattering and diffraction, and hence the amplification and de-amplification of the SH waves. Numerical results show that the surface motions on the slope of the wedged space is influenced greatly by the topography. Amplification of the surface motions near the slope vertex is significant. The corresponding phases along the wedged space surfaces are consistent with the direction that the SH waves are propagating.
The scattering and reflection of SH waves by a slope on an elastic wedged space is investigated. A series solution is obtained by using the wave function expansion method. The slope on a wedged space is divided into two subregions by an artificial, auxiliary circular arc. The wave fields with unknown complex coefficients within each sub-region are derived. Applying Graf addition theorem, the scattered waves in the sub-regions are expressed in a global coordinate system. Fourier transform is adopted to derive a consistent form of standing waves in the inner region using the orthogonality of the cosine functions. The boundary-valued problem is solved by stress and displacement continuity along the artificial, auxiliary arc to obtain the unknown complex coefficients. Parametric studies are next performed to investigate how the topography from the slope on the wedged space will affect the scattering and diffraction, and hence the amplification and de-amplification of the SH waves. Numerical results show that the surface motions on the slope of the wedged space is influenced greatly by the topography. Amplification of the surface motions near the slope vertex is significant. The corresponding phases along the wedged space surfaces are consistent with the direction that the SH waves are propagating.
引文
Abo-Zena AM and King CY(1973),“SH Pulse in an Elastic Wedge,”Bull.Seismol.Soc.Am.,63:1571?1582.
Aboudi J(1971),“The Motion Excited by an Impulsive Source in an Elastic Half-Space with a Surface Obstacle,”Bull.Seismol.Soc.Am.,61:747?763.
Abramowitz M and Stegun IA(1972),Handbook of Mathematical Functions,with Formulas,Graphs,and Mathematical Tables,Dover,New York.
Achenbach JD(1973),Wave Propagation in Elastic Solids,North-Holland,Amsterdam.
Alielahi H,Kamalian M and Adampira M(2015),“Seismic Ground Amplification by Unlined Tunnels Subjected to Vertically Propagating SV and P Waves Using BEM,”Soil Dynam.Earthq.Eng.,71:63?79.
Amornwongpaibun A,Luo H and Lee VW(2016),“Scattering of Anti-Plane(SH)Waves by a Shallow Semi-Elliptical Hill with a Concentric Elliptical Tunnel,”J.Earthq.Eng.,20:363?382.
Bard PY(1982),“Diffracted Waves and Displacement Field over Two-Dimensional Elevated Topographies,”Geophys.J.Int.,71:731?760.
Boore DM(1972),“A Note on the Effect of Simple Topography on Seismic SH Waves,”Bull.Seism.Soc.Am.,62:275?284.
Bouchon M(1973),“Effect of Topography on Surface Motion,”Bull.Seism.Soc.Am.,63:615?632.
Bouchon M and Sánchez-Sesma FJ(2007),“Boundary Integral Equations and Boundary Elements Methods in Elastodynamics,”Adv.Geophys.,48:157?189.
Chang K,Tsaur D and Wang J(2015),“Response of a Shallow Asymmetric V-Shaped Canyon to Antiplane
Elastic Waves,”Philos.Trans.R.Soc.A.,471:20140215.
Fu L(2005),“Rough Surface Scattering:Comparison of Various Approximation Theories for 2D SH Waves,”Bull.Seism.Soc.Am.,95:646?663.
Gao Y and Zhang N(2013),“Scattering of Cylindrical SH Waves Induced by a Symmetrical V-Shaped Canyon:Near-Source Topographic Effects,”Geophys.J.Int.,193:874?885.
Gao Y,Zhang N,Li D,Liu H,Cai Y and Wu Y(2012),“Effects of Topographic Amplification Induced by a U-Shaped Canyon on Seismic Waves,”Bull.Seism.Soc.Am.,102:1748?1763.
Geli L,Bard PY and Jullien B(1988),“The Effect of Topography on Earthquake Ground Motion:A Review and New Results,”Bull.Seismol.Soc.Am.,78:42?63.
Hartzell SH,Carver DL and King KW(1994),“Initial Investigation of Site and Topographic Effects at Robinwood Ridge,California,”Bull.Seism.Soc.Am.,84:1336?1349.
Hough SE,Altidor JR,Anglade D,Given D,Janvier MG,Maharrey JZ,Meremonte M,Mildor BS-L,Prepetit C and Yong A(2010),“Localized Damage Caused by Topographic Amplification during the 2010 M 7.0 Haiti Earthquake,”Nature Geosci.,3:778?782.
Jeffery GB(1921),“Plane Stress and Plane Strain in Bipolar Co-ordinates,”Philos.Trans.R.Soc.London,Ser.A,221:265?223.
LaméG(1859),Lecons sur les Coordonnés Curvilignes,Paris.
Le T,Lee VW and Trifunac MD(2016),“SH Waves in a Moon-Shaped Valley,”Earthq.Eng.Struct.Dyn.,101:162?175.
Lee VW(1990),“Scattering of Plane SH-Waves by a Semi-Parabolic Cylindrical Canyon in an Elastic HalfSpace,”Geophys.J.Int.,100:79?86.
Lee VW,Luo H and Liang JW(2006),“Antiplane(SH)Waves Diffraction by a Semicircular Cylindrical Hill Revisited:an Improved Analytic Wave Series Solution,”J.Eng.Mech.ASCE,132:1106?1114.
Lee VW and Sherif RI(1996),“Diffraction Around Circular Canyon in Elastic Wedge Space by Plane SH-Waves,”J.Eng.Mech.ASCE,122:539?544.
Liu DK,Gai B and Tao G(1982),“Applications of the Method of Complex Functions to Dynamic Stress Concentrations,”Wave Motion,4:293?304.
Liu DK and Han F(1991),“Scattering of Plane SH-Wave by Cylindrical Canyon of Arbitrary Shape,”Soil Dyn.Earthq.Eng.,10:249?255.
Liu G,Chen HT,Liu DK and Khoo BC(2010),“Surface Motion of a Half-Space with Triangular and Semicircular Hills under Incident SH Waves,”Bull.Seism.Soc.Am.,100:1306?1319.
Liu Q,Zhang C and Todorovska MI(2016),“Scattering of SH Waves by a Shallow Rectangular Cavity in an Elastic Half Space,”Soil Dyn.Earthq.Eng.,90:147?157.
Love AE 4th Edition,(1944),A Treatise on the Mathematical Theory of Elasticity,Dover,New York.
Luzón F,Sánchez-Sesma FJ,Rodriguez-Zuniga JL,Posadas AM,Garcia JM,Martin J,Romacho MD and Navarro M(1997),“Diffraction of P,S and Rayleigh Waves by Three-Dimensional Topographies,”Geophys.J.Int.,129:571?578.
MacDonald HM(1902),Electric Waves,Cambridge University Press,London,England.
Sánchez-Sesma FJ(1985),“Diffraction of Elastic SH Waves by Wedges,”Bull.Seism.Soc.Am.,75:1435?1446.
Sánchez-Sesma FJ and Campillo M(1991),“Diffraction of P,SV,and Rayleigh Waves by Topographic Features:A Boundary Integral Formulation,”Bull.Seism.Soc.Am.,81:2234?2253.
Sánchez-Sesma FJ and Rosenblueth E(1979),“Ground Motion at Canyons of Arbitrary Shape under Incident SH Waves,”Earthq.Eng.Struct.Dyn.,7:441?450.
Shah AH,Wong KC and Datta SK(1982),“Diffraction of Plane SH Waves in a Half-Space,”Earthq.Eng.Struct.Dynam.,10:519?528.
Sherif RI and Lee VW(1996),“Diffraction of a Circular Alluvial Valley in a Wedge Shaped Elastic Half Space due to Plane SH Waves,”Eur.J.Earthq.Eng.,10:21?28.
Sills LB(1978),“Scattering of Horizontally-Polarized Shear Waves by Surface Irregularities,”Geophys.J.R.astr.Soc.,54:319?348.
Smith WD(1975),“The Application of Finite Element Analysis to Body Wave Propagation Problems,”Geophys.J.R.Astr.Soc.,42:747?768.
Sohrabi-Bidar A,Kamalian M and Jafari MK(2010),“Seismic Response of 3-D Gaussian-Shaped Valleys to Vertically Propagating Incident Waves,”Geophys.J.Int.,183:1429?1442.
Takemiya H and Fujiwara A(1994),“SH-Wave Scattering and Propagation Analyses at Irregular Sites by Time Domain BEM,”Bull.Seism.Soc.Am.,84:1443?1455.
Trifunac MD(1971),“Surface Motion of a SemiCylindrical Alluvial Valley for Incident Plane SHWaves,”Bull.Seism.Soc.Am.,61:1755?1770.
Trifunac MD and Hudson DE(1971),“Analysis of the Pacoima Dam accelerogram-San Fernando,California,earthquake of 1971,”Bull.Seism.Soc.Am.,61:1393?1141.
Trifunac MD(1973),“Scattering of Plane SH Waves by a Semi-Cylindrical Canyon,”Earthq.Eng.Struct.Dyn.,1:267?281.
Tsaur DH(2011),“Scattering and Focusing of SH Waves by a Lower Semi Elliptic Convex Topography,”Bull.Seism.Soc.Am.,101:2212?2219.
Tsaur DH and Chang KH(2008),“An Analytical Approach for the Scattering of SH Waves by a Symmetrical V-Shaped Canyon:Shallow Case,”Geophys.J.Int.,174:255?264.
Tsaur DH and Chang KH(2009),“Scattering and Focusing of SH Waves by a Convex Circular-Arc Topography,”Geophys.J.Int.,177:222?234.
Tsaur DH,Chang KH and Hsu MS(2010),“An Analytical Approach for the Scattering of SH Waves by a Symmetrical V-Shaped Canyon:Deep Case,”Geophys.J.Int.,183:1501?1511.
Wong HL and Jennings PC(1975),“Effects of Canyon Topography on Strong Ground Motion,”Bull.Seism.Soc.Am.,65:1239?1257.
Wong HL and Trifunac MD(1974),“Scattering of Plane SH Waves by a Semi-Elliptical Canyon,”Earthq.Eng.Struct.Dyn.,3:157?169.
Yuan X and Liao Z(1994),“Scattering of Plane SHWaves by a Cylindrical Canyon of Circular-arc CrossSection,”Soil Dyn.Earthq.Eng.,13:407?412.
Yuan X and Liao Z(1996),“Surface Motion of a Cylindrical Hill of Circular-Arc Cross-Section for Incident Plane SH Waves,”Soil Dyn.Earthq.Eng.,15:189?199.
Yuan X and Men F(1992),“Scattering of Plane SHWaves by a Semi Cylindrical Hill,”Earthq.Eng.Struct.Dyn.,21:1091?1098.
Zhang C and Zhao C(1988),“Effects of Canyon Topography and Geological Conditions on Strong Ground Motion,”Earthq.Eng.Struct.Dynam.,16:81?97.
Zhang N,Gao Y,Cai Y,Li D and Wu Y,(2012),“Scattering of SH Waves Induced by a Non-Symmetrical V-Shaped Canyon,”Geophys.J.Int.,191:243?256.
Zhang N,Gao Y,Wu Y and Zhang F(2018),“A Note on Near-Field Site Amplification Effects of Ground Motion from a Radially Inhomogeneous Valley,”Earthq.Eng.Eng.Vib.,17(4):707?718.
Zhang C,Liu Q and Deng P(2017),“Surface Motion of a Half-Space with a Semicylindrical Canyon under P,SV,and Rayleigh Waves,”Bull.Seism.Soc.Am.,107:809?820.
Zhang C,Liu Q and Deng P(2015),“Antiplane Scattering of SH Waves by a Trapezoidal Valley with a Circular-Arc Alluvium in an Elastic Half Space,”J.Earthq.Tsunami,9,Article ID 1550008.
Aboudi J(1971),“The Motion Excited by an Impulsive Source in an Elastic Half-Space with a Surface Obstacle,”Bull.Seismol.Soc.Am.,61:747?763.
Abramowitz M and Stegun IA(1972),Handbook of Mathematical Functions,with Formulas,Graphs,and Mathematical Tables,Dover,New York.
Achenbach JD(1973),Wave Propagation in Elastic Solids,North-Holland,Amsterdam.
Alielahi H,Kamalian M and Adampira M(2015),“Seismic Ground Amplification by Unlined Tunnels Subjected to Vertically Propagating SV and P Waves Using BEM,”Soil Dynam.Earthq.Eng.,71:63?79.
Amornwongpaibun A,Luo H and Lee VW(2016),“Scattering of Anti-Plane(SH)Waves by a Shallow Semi-Elliptical Hill with a Concentric Elliptical Tunnel,”J.Earthq.Eng.,20:363?382.
Bard PY(1982),“Diffracted Waves and Displacement Field over Two-Dimensional Elevated Topographies,”Geophys.J.Int.,71:731?760.
Boore DM(1972),“A Note on the Effect of Simple Topography on Seismic SH Waves,”Bull.Seism.Soc.Am.,62:275?284.
Bouchon M(1973),“Effect of Topography on Surface Motion,”Bull.Seism.Soc.Am.,63:615?632.
Bouchon M and Sánchez-Sesma FJ(2007),“Boundary Integral Equations and Boundary Elements Methods in Elastodynamics,”Adv.Geophys.,48:157?189.
Chang K,Tsaur D and Wang J(2015),“Response of a Shallow Asymmetric V-Shaped Canyon to Antiplane
Elastic Waves,”Philos.Trans.R.Soc.A.,471:20140215.
Fu L(2005),“Rough Surface Scattering:Comparison of Various Approximation Theories for 2D SH Waves,”Bull.Seism.Soc.Am.,95:646?663.
Gao Y and Zhang N(2013),“Scattering of Cylindrical SH Waves Induced by a Symmetrical V-Shaped Canyon:Near-Source Topographic Effects,”Geophys.J.Int.,193:874?885.
Gao Y,Zhang N,Li D,Liu H,Cai Y and Wu Y(2012),“Effects of Topographic Amplification Induced by a U-Shaped Canyon on Seismic Waves,”Bull.Seism.Soc.Am.,102:1748?1763.
Geli L,Bard PY and Jullien B(1988),“The Effect of Topography on Earthquake Ground Motion:A Review and New Results,”Bull.Seismol.Soc.Am.,78:42?63.
Hartzell SH,Carver DL and King KW(1994),“Initial Investigation of Site and Topographic Effects at Robinwood Ridge,California,”Bull.Seism.Soc.Am.,84:1336?1349.
Hough SE,Altidor JR,Anglade D,Given D,Janvier MG,Maharrey JZ,Meremonte M,Mildor BS-L,Prepetit C and Yong A(2010),“Localized Damage Caused by Topographic Amplification during the 2010 M 7.0 Haiti Earthquake,”Nature Geosci.,3:778?782.
Jeffery GB(1921),“Plane Stress and Plane Strain in Bipolar Co-ordinates,”Philos.Trans.R.Soc.London,Ser.A,221:265?223.
LaméG(1859),Lecons sur les Coordonnés Curvilignes,Paris.
Le T,Lee VW and Trifunac MD(2016),“SH Waves in a Moon-Shaped Valley,”Earthq.Eng.Struct.Dyn.,101:162?175.
Lee VW(1990),“Scattering of Plane SH-Waves by a Semi-Parabolic Cylindrical Canyon in an Elastic HalfSpace,”Geophys.J.Int.,100:79?86.
Lee VW,Luo H and Liang JW(2006),“Antiplane(SH)Waves Diffraction by a Semicircular Cylindrical Hill Revisited:an Improved Analytic Wave Series Solution,”J.Eng.Mech.ASCE,132:1106?1114.
Lee VW and Sherif RI(1996),“Diffraction Around Circular Canyon in Elastic Wedge Space by Plane SH-Waves,”J.Eng.Mech.ASCE,122:539?544.
Liu DK,Gai B and Tao G(1982),“Applications of the Method of Complex Functions to Dynamic Stress Concentrations,”Wave Motion,4:293?304.
Liu DK and Han F(1991),“Scattering of Plane SH-Wave by Cylindrical Canyon of Arbitrary Shape,”Soil Dyn.Earthq.Eng.,10:249?255.
Liu G,Chen HT,Liu DK and Khoo BC(2010),“Surface Motion of a Half-Space with Triangular and Semicircular Hills under Incident SH Waves,”Bull.Seism.Soc.Am.,100:1306?1319.
Liu Q,Zhang C and Todorovska MI(2016),“Scattering of SH Waves by a Shallow Rectangular Cavity in an Elastic Half Space,”Soil Dyn.Earthq.Eng.,90:147?157.
Love AE 4th Edition,(1944),A Treatise on the Mathematical Theory of Elasticity,Dover,New York.
Luzón F,Sánchez-Sesma FJ,Rodriguez-Zuniga JL,Posadas AM,Garcia JM,Martin J,Romacho MD and Navarro M(1997),“Diffraction of P,S and Rayleigh Waves by Three-Dimensional Topographies,”Geophys.J.Int.,129:571?578.
MacDonald HM(1902),Electric Waves,Cambridge University Press,London,England.
Sánchez-Sesma FJ(1985),“Diffraction of Elastic SH Waves by Wedges,”Bull.Seism.Soc.Am.,75:1435?1446.
Sánchez-Sesma FJ and Campillo M(1991),“Diffraction of P,SV,and Rayleigh Waves by Topographic Features:A Boundary Integral Formulation,”Bull.Seism.Soc.Am.,81:2234?2253.
Sánchez-Sesma FJ and Rosenblueth E(1979),“Ground Motion at Canyons of Arbitrary Shape under Incident SH Waves,”Earthq.Eng.Struct.Dyn.,7:441?450.
Shah AH,Wong KC and Datta SK(1982),“Diffraction of Plane SH Waves in a Half-Space,”Earthq.Eng.Struct.Dynam.,10:519?528.
Sherif RI and Lee VW(1996),“Diffraction of a Circular Alluvial Valley in a Wedge Shaped Elastic Half Space due to Plane SH Waves,”Eur.J.Earthq.Eng.,10:21?28.
Sills LB(1978),“Scattering of Horizontally-Polarized Shear Waves by Surface Irregularities,”Geophys.J.R.astr.Soc.,54:319?348.
Smith WD(1975),“The Application of Finite Element Analysis to Body Wave Propagation Problems,”Geophys.J.R.Astr.Soc.,42:747?768.
Sohrabi-Bidar A,Kamalian M and Jafari MK(2010),“Seismic Response of 3-D Gaussian-Shaped Valleys to Vertically Propagating Incident Waves,”Geophys.J.Int.,183:1429?1442.
Takemiya H and Fujiwara A(1994),“SH-Wave Scattering and Propagation Analyses at Irregular Sites by Time Domain BEM,”Bull.Seism.Soc.Am.,84:1443?1455.
Trifunac MD(1971),“Surface Motion of a SemiCylindrical Alluvial Valley for Incident Plane SHWaves,”Bull.Seism.Soc.Am.,61:1755?1770.
Trifunac MD and Hudson DE(1971),“Analysis of the Pacoima Dam accelerogram-San Fernando,California,earthquake of 1971,”Bull.Seism.Soc.Am.,61:1393?1141.
Trifunac MD(1973),“Scattering of Plane SH Waves by a Semi-Cylindrical Canyon,”Earthq.Eng.Struct.Dyn.,1:267?281.
Tsaur DH(2011),“Scattering and Focusing of SH Waves by a Lower Semi Elliptic Convex Topography,”Bull.Seism.Soc.Am.,101:2212?2219.
Tsaur DH and Chang KH(2008),“An Analytical Approach for the Scattering of SH Waves by a Symmetrical V-Shaped Canyon:Shallow Case,”Geophys.J.Int.,174:255?264.
Tsaur DH and Chang KH(2009),“Scattering and Focusing of SH Waves by a Convex Circular-Arc Topography,”Geophys.J.Int.,177:222?234.
Tsaur DH,Chang KH and Hsu MS(2010),“An Analytical Approach for the Scattering of SH Waves by a Symmetrical V-Shaped Canyon:Deep Case,”Geophys.J.Int.,183:1501?1511.
Wong HL and Jennings PC(1975),“Effects of Canyon Topography on Strong Ground Motion,”Bull.Seism.Soc.Am.,65:1239?1257.
Wong HL and Trifunac MD(1974),“Scattering of Plane SH Waves by a Semi-Elliptical Canyon,”Earthq.Eng.Struct.Dyn.,3:157?169.
Yuan X and Liao Z(1994),“Scattering of Plane SHWaves by a Cylindrical Canyon of Circular-arc CrossSection,”Soil Dyn.Earthq.Eng.,13:407?412.
Yuan X and Liao Z(1996),“Surface Motion of a Cylindrical Hill of Circular-Arc Cross-Section for Incident Plane SH Waves,”Soil Dyn.Earthq.Eng.,15:189?199.
Yuan X and Men F(1992),“Scattering of Plane SHWaves by a Semi Cylindrical Hill,”Earthq.Eng.Struct.Dyn.,21:1091?1098.
Zhang C and Zhao C(1988),“Effects of Canyon Topography and Geological Conditions on Strong Ground Motion,”Earthq.Eng.Struct.Dynam.,16:81?97.
Zhang N,Gao Y,Cai Y,Li D and Wu Y,(2012),“Scattering of SH Waves Induced by a Non-Symmetrical V-Shaped Canyon,”Geophys.J.Int.,191:243?256.
Zhang N,Gao Y,Wu Y and Zhang F(2018),“A Note on Near-Field Site Amplification Effects of Ground Motion from a Radially Inhomogeneous Valley,”Earthq.Eng.Eng.Vib.,17(4):707?718.
Zhang C,Liu Q and Deng P(2017),“Surface Motion of a Half-Space with a Semicylindrical Canyon under P,SV,and Rayleigh Waves,”Bull.Seism.Soc.Am.,107:809?820.
Zhang C,Liu Q and Deng P(2015),“Antiplane Scattering of SH Waves by a Trapezoidal Valley with a Circular-Arc Alluvium in an Elastic Half Space,”J.Earthq.Tsunami,9,Article ID 1550008.