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Scattering of SH waves induced by a symmetrical V-shaped canyon: a unified analytical solution

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Abstract

This paper reports a series solution of wave functions for two-dimensional scattering and diffraction of plane SH waves induced by a symmetrical V-shaped canyon with different shape ratios. A half-space with a symmetrical V-shaped canyon is divided into two sub-regions by using a circular-arc auxiliary boundary. The two sub-regions are represented by global and local cylindrical coordinate systems, respectively. In each coordinate system, the wave field satisfying the Helmholtz equation is represented by the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then, the two wave fields are described in the local coordinate system using the Graf addition theorem. Finally, the unknown coefficients are sought by satisfying the continuity conditions of the auxiliary boundary. To consider the phase characteristics of the wave scattering, a parametric analysis is carried out in the time domain by assuming an incident signal of the Ricker type. Surface and subsurface transient responses demonstrate the characteristics and mechanisms of wave propagating and scattering.

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References

  • Abramowitz M and Stegun IA (1972), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York.

    Google Scholar 

  • Boore DM (1972), “A Note on the Effect of Simple Topography on Seismic SH Waves,” Bulletin of the Seismological Society of America, 62(1): 275–284.

    Google Scholar 

  • Boore DM (1973), “The Effect of Simple Topography on Seismic Waves: Implications for the Accelerations Recorded at Pacoima Dam, San Fernando Valley, California,” Bulletin of the Seismological Society of America, 63(5): 1603–1609.

    Google Scholar 

  • Bouchon M (1973), “Effect of Topography on Surface Motion,” Bulletin of the Seismological Society of America, 63(3): 615–632.

    Google Scholar 

  • Bouchon M (1985), “A Simple, Complete Numerical Solution to the Problem of Diffraction of SH Waves by an Irregular Surface,” Journal of the Acoustical Society of America, 77(1): 1–5.

    Article  Google Scholar 

  • Cao H and Lee VW (1989), “Scattering of Plane SH Waves by Circular Cylindrical Canyons with Variable Depth-to-width Ratio,” European Earthquake Engineering, 3(2): 29–37.

    Google Scholar 

  • Hirai H (1988), “Analysis of Transient Response of SH Wave Scattering in a Half Space by the Boundary Element Method,” Engineering Analysis, 5(4): 189–194.

    Article  Google Scholar 

  • Hough SE, Altidor JR, Anglade D, Given D, Janvier MG, Maharrey JZ, Meremonte M, Mildor BS, Prepetit C and Yong A (2010), “Localized Damage Caused by Topographic Amplification During the 2010. M 7.0. Haiti Earthquake,” Nature Geoscience, 3(11): 778–782.

    Article  Google Scholar 

  • Komatitsch D and Vilotte J-P (1998), “The Spectral Element Method: an Efficient Tool to Simulate the Seismic Response of 2D and 3D Geological Structures,” Bulletin of the Seismological Society of America, 88(2): 368–392.

    Google Scholar 

  • Lee VW (1982), “A Note on the Scattering of Elastic Plane Waves by a Hemispherical Canyon,” Soil Dynamics and Earthquake Engineering, 1(3): 122–129.

    Article  Google Scholar 

  • Lee VW (1990), “Scattering of Plane SH-waves by a Semi-parabolic Cylindrical Canyon in an Elastic Halfspace,” Geophysical Journal International, 100(1): 79–86.

    Article  Google Scholar 

  • Lee VW, Luo H and Liang JW (2004), “Diffraction of Anti-plane SH Waves by a Semi-circular Cylindrical Hill with an Inside Concentric Semi-circular Tunnel,” Earthquake Engineering and Engineering Vibration, 3(2): 249–262.

    Article  Google Scholar 

  • Lee VW, Luo H and Liang JW (2006), “Antiplane (SH) Waves Diffraction by a Semicircular Cylindrical Hill Revisited: An Improved Analytic Wave Series Solution,” Journal of Engineering Mechanics, ASCE, 132(10): 1106–1114.

    Article  Google Scholar 

  • Lee VW and Wu X (1994), “Application of the Weighted Residual Method to Diffraction by 2-D Canyons of Arbitrary Shape: I. Incident SH waves,” Soil Dynamics and Earthquake Engineering, 13(5): 355–364.

    Article  Google Scholar 

  • Liang JW, Luo H and Lee VW (2004), “Scattering of Plane SH Waves by a Circulararc Hill with a Circular Tunnel,” Acta Seismologica Sinica, 17(5): 549–563.

    Article  Google Scholar 

  • 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,” Bulletin of the Seismological Society of America, 100(3): 1306–1319.

    Article  Google Scholar 

  • Liu DK and Han F (1991), “Scattering of Plane SHwave by Cylindrical Canyon of Arbitrary Shape,” Soil Dynamics and Earthquake Engineering, 10(5): 249–255.

    Article  Google Scholar 

  • Lin Shuzhi, Qiu Faqiang and Liu Diankui (2010), “Scattering of SH Waves by a Scalene Triangular Hill,” Earthquake Engineering and Engineering Vibration, 9(1): 23–38.

    Google Scholar 

  • Moeen-Vaziri N and Trifunac MD (1988), “Scattering and Diffraction of Plane SH-waves by Two-dimensional Inhomogeneities: Part I,” Soil Dynamics and Earthquake Engineering, 7(4): 179–188.

    Article  Google Scholar 

  • Qiu Faqiang and Liu Diankui (2005), “Antiplane Response of Isosceles Triangular Hill to Incident SH Waves,” Earthquake Engineering and Engineering Vibration, 4(1): 37–43.

    Article  Google Scholar 

  • Sanchez-Sesma FJ, Herrera I and Aviles J (1982), “A Boundary Method for Elastic Wave Diffraction: Application to Scattering of SH Waves by Surface Irregularities,” Bulletin of the Seismological Society of America, 72(2): 473–490.

    Google Scholar 

  • Sanchez-Sesma FJ, Palencia VJ and Luzon F (2002), “Estimation of Local Site Effects During Earthquakes: an Overview,” ISET Journal of Earthquake Technology, 39(3): 167–193.

    Google Scholar 

  • Sanchez-Sesma FJ and Rosenblueth E (1979), “Ground Motion at Canyons of Arbitrary Shape under Incident SH Waves,” Earthquake Engineering and Structural Dynamics, 7(5): 441–450.

    Article  Google Scholar 

  • Shah AH, Wong KC and Datta SK (1982), “Diffracton of Plane SH Waves in a Half-space,” Earthquake Engineering and Structural Dynamics, 10(4): 519–528.

    Article  Google Scholar 

  • Takemiya H and Fujiwara A (1994), “SH-wave Scattering and Propagation Analyses at Irregular Sites by Time Domain BEM,” Bulletin of the Seismological Society of America, 84(5): 1443–1455.

    Google Scholar 

  • Trifunac MD (1973), “Scattering of Plane SH Waves by a Semi-cylindrical Canyon,” Earthquake Engineering and Structural Dynamics, 1(3): 267–281.

    Article  Google Scholar 

  • Trifunac MD and Hudson DE (1971), “Analysis of the Pacoima Dam Accelerogram-San Fernando, California, Earthquake of 1971,” Bulletin of the Seismological Society of America, 61(5): 1393–1141.

    Google Scholar 

  • Tsaur DH and Chang KH (2008), “An Analytical Approach for the Scattering of SH Waves by a Symmetrical V-shaped Canyon: Shallow Case,” Geophysical Journal International, 174(1): 255–264.

    Article  Google Scholar 

  • Tsaur DH and Chang KH (2009a), “Scattering of SH Waves by Truncated Semicircular Canyon,” Journal of Engineering Mechanics, ASCE, 135(8): 862–870.

    Article  Google Scholar 

  • Tsaur DH and Chang KH (2009b), “Scattering and Focusing of SH Waves by a Convex Circular-arc Topography,” Geophysical Journal International, 177(1): 222–234.

    Article  Google Scholar 

  • 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,” Geophysical Journal International, 183(3): 1501–1511.

    Article  Google Scholar 

  • Wong HL and Jennings PC (1975), “Effects of Canyon Topography on Strong Ground Motion,” Bulletin of the Seismological Society of America, 65(5): 1239–1257.

    Google Scholar 

  • Wong HL and Trifunac MD (1974), “Scattering of Plane SH Waves by a Semi-elliptical Canyon,” Earthquake Engineering and Structural Dynamics, 3(2): 157–169.

    Article  Google Scholar 

  • Yuan XM (2010), “A Complete-domain Transform Formula of Cylindrical Wave Functions for Scattering Problems,” Journal of Earthquake Engineering and Engineering Vibration, 30(4): 1–7. (in Chinese)

    Google Scholar 

  • Yuan XM and Liao ZP (1994), “Scattering of Plane SH Waves by a Cylindrical Canyon of Circular-arc Crosssection,” Soil Dynamics and Earthquake Engineering, 13(6): 407–412.

    Article  Google Scholar 

  • Yuan XM and Liao ZP (1996), “Surface Motion of a Cylindrical Hill of Circular-arc Cross-section for Incident Plane SH Waves,” Soil Dynamics and Earthquake Engineering, 15(3): 189–199.

    Article  Google Scholar 

  • Yuan XM and Men FL (1992), “Scattering of Plane SH Waves by a Semi-cylindrical Hill,” Earthquake Engineering and Structural Dynamics, 21(12): 1091–1098.

    Article  Google Scholar 

  • Zhang C and Zhao C (1988), “Effects of Canyon Topography and Geological Conditions on Strong Ground Motion,” Earthquake Engineering and Structural Dynamics, 16(1): 81–97.

    Article  Google Scholar 

  • Zhou H and Chen XF (2006), “A New Approach to Simulate Scattering of SH Waves by an Irregular Topography,” Geophysical Journal International, 164(2): 449–459.

    Article  Google Scholar 

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Authors and Affiliations

  1. Geotechnical Research Institute, Hohai University, Nanjing, 210098, China

    Ning Zhang, Yufeng Gao, Yongxin Wu & Fei Zhang

  2. Key Laboratory of Geomechanics and Embankment Engineering (Hohai University), Ministry of Education, Nanjing, 210098, China

    Ning Zhang, Yufeng Gao, Yongxin Wu & Fei Zhang

  3. College of Civil Engineering & Architecture, Shandong University of Science and Technology, Qingdao, 266590, China

    Dayong Li

Authors
  1. Ning Zhang
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  2. Yufeng Gao
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  3. Dayong Li
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  4. Yongxin Wu
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  5. Fei Zhang
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Corresponding author

Correspondence to Yufeng Gao.

Additional information

Supported by: National Natural Science Foundation of China Under Grant No. 51278382

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Zhang, N., Gao, Y., Li, D. et al. Scattering of SH waves induced by a symmetrical V-shaped canyon: a unified analytical solution. Earthq. Eng. Eng. Vib. 11, 445–460 (2012). https://doi.org/10.1007/s11803-012-0135-z

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  • DOI: https://doi.org/10.1007/s11803-012-0135-z

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