Application of Steepest Descent Path Method to Lamb’s Solutions for Scattering in Thermo-elastic Half-Plane

  • Po-Jen Shih
  • Sheng-Ping Peng
  • Chau-Shioung Yeh
  • Tsung-Jen Teng
  • Wen-Shinn Shyu
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 30)


When an incidence impinges an alluvial valley in half-plane, wave interactions of three inhomogeneities are considered on thermoelastic coupling effects, and the stress concentration along continuous interface is demonstrated. Because of the inhomogeneities, the scattering waves can be deduced by three part, the incidence sources in the half-plane, reflection waves simulated by the image sources in the mirror image half-plane, and the refraction wave inside the alluvial valley. For in-plane problem, two coupled longitudinal waves, of which one is predominantly elastic and the other is predominantly thermal, and a transversal wave are adopted to analyze scattering. This work uses a Rayleigh series of Lamb’s formal integral solutions as a simple basis set. The corresponding integrations of the basis set are calculated numerically by applying a modified steepest descent path integral method, which provides strongly convergence in numerical integrations. Moreover, Betti’s third identity and orthogonal conditions are applied to obtain a transition matrix for solving the scattering. The results at the surface of a semicircular alluvial valley embedded in half-plane are demonstrated to show the displacement fields and the temperature gradient fields. They also indicate that softer alluvial valley is associated with a substantially greater amplification at the interface of the alluvial valley.


Rayleigh Wave Integral Path Transversal Wave Alluvial Valley Thermoelastic Theory 
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Copyright information

© Springer Dordrecht Heidelberg London New York 2011

Authors and Affiliations

  • Po-Jen Shih
    • 1
  • Sheng-Ping Peng
    • 2
  • Chau-Shioung Yeh
    • 2
  • Tsung-Jen Teng
    • 3
  • Wen-Shinn Shyu
    • 4
  1. 1.National University of KaohsiungKaohsiungTaiwan
  2. 2.National Taiwan UniversityTaipeiTaiwan
  3. 3.National Center for Research on Earthquake EngineeringTaipeiTaiwan
  4. 4.National Pingtung University of Science and TechnologyPingtungTaiwan

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