Tsung-Jen Teng

On Evaluation of Lamb's Integrals for Waves in a Two-Dimension Elastic Half-Space

In this paper, a modified version of the method of steepest descent is proposed for the evaluation of Lambs integrals which can be considered as basis functions dealing with the development of the transition matrix method which can be used to study the wave scattering in a two-dimensional elastic half-space. The formal solutions of the generalized Lambs problem are studied and evaluated on the basis of the proposed method. After defining a phase function which presents in wavenumber integral, an exact mapping and an inverse mapping can be obtained according to the phase function. Thus, the original integration path can be deformed into an equivalent admissible path, namely, steepest descent path which passed through the saddle point, and then mapped onto a real axis of mapping plane, finally, resulted in an integral of Hermite type. This integral can be efficiently evaluated numerically in spite of either near- to far-field or low to high frequency. At the same time, the asymptotic value can easily be obtained by applying the proposed method. The numerical results for generalized Lambs solutions are calculated and compared with analytic, asymptotic or other existing data, the excellent agreements are found. The properties of generalized Lambs solutions are studied and discussed in details. Their possible applications for wave scattering in elastic half-space are also pointed out.

Scattering of Elastic Waves by a Buried Tunnel Under Obliquely Incident Waves Using T Matrix

This paper first studies the transition matrix formulation for the analysis of responses of an elastic halfspace with a buried tunnel subjected to obliquely incident waves. The basis functions are constructed using the moving P-, SV-, and SH-wave source potentials and to represent the scattered and refracted wave fields in series forms. The associated T-matrix expression of elastic inclusion is derived using Bettis third identity. Second, this study proposes a technique for calculating the integral representation of basis functions in the wave-number domain using the method of steepest descent. Finally, typical numerical results obtained under incident plane waves are presented for verification.

On formulation of a transition matrix for poroelastic medium and application to analysis of scattering problem

Based on the approach of Pao (1978) for elastic medium, we propose a set of the basis functions and an associated relationship of the material properties of the dilatational wave in the poroelastic medium. A transition matrix, which relates the coefficients of scattered waves to those of incident waves, is then derived through the application of Betti’s third identity and the associated orthogonality conditions for the poroelastic medium. To illustrate the application, we consider a simple case of the scattering problem of a spherical inclusion, either elastic or poroelastic, embedded within the surrounding poroelastic medium subjected to an incident plane compressional wave.

A Hybrid Method for Analyzing the Dynamic Responses of Cavities or Shells Buried in an Elastic Half-Plane

In this paper, based on a variational formalism which originally proposed by Mei [1] for infinite elastic medium and extended by Yeh, et al. [2,3] for elastic half-plane, a hybrid method which combines the finite element and series expansion method is implemented to solve the diffraction of plane waves by a cavity buried in an elastic half-plane. The finite domain which encloses all inhomogeneities including the cavity can be easily formulated by finite element methods. The unknown boundary data obtained by subtracting the known free fields from the total fields which include the boundary nodal displacements and tractions at the interface between the finite domain and the surrounding elastic half-plane are not independent of each other and can be correlated through a series representation. Due to the continuity condition at the interface, the same series representation is still valid for the exterior elastic half-plane to represents the scattered wave. The unknown coefficients of this series are treated as generalized coordinates and can be easily formulated by the same variational principle. The expansion function of the series is composed of basis function. Each basis function is constructed from the basis function for an infinite plane by superimposing an additional homogeneous reflective term to satisfy both traction free conditions at ground surface and radiation conditions at infinity. The numerical results are made against those obtained by boundary element methods, and good agreements are found.

The transition matrix for the scattering of elastic waves in a half‐space

Abstract In this paper, the transition matrix for elastic waves scattering from an alluvium on an elastic half‐space is developed. Bettis third identity is employed to establish orthogonality conditions among basis functions that are Lambs singular wave functions. The total displacements and associated tractions for both the surrounding half‐space and alluvium are expanded in a Rayleigh series. After the boundary conditions are applied, the T‐matrix can be obtained. Explicit forms of the basis functions are derived for the two‐dimensional anti‐plane and in‐plane problems. The linear transformation is utilized to construct a set of orthogonal basis functions. The transformed T‐matrix is related to the scattering matrix and it is shown that the scattering matrix is symmetric and unitary and that the T‐matrix is also symmetric. Some typical scattering cases induced by incident plane waves are illustrated for verification.

On formulation of a transition matrix for electroporoelastic medium and application to analysis of scattered electroseismic wave

On the basis of Prides theory (1994) which couples Biots theory for poroelastic medium (1956) and Maxwell equations via flux/force transport equations, we extend Yeh et al. (2004) approach for poroelastic medium to develop a transition matrix for electroporoelastic medium. The transition matrix, which relates the coefficients of scattered waves to those of incident waves, is then derived through the application of Bettis third identity and the associated orthogonality conditions for the electroporoelastic medium. To illustrate the application, a simple case of the scattering problem of a spherical electroporoelastic inclusion, embedded within the surrounding electroporoelastic medium subjected to an incident plane compressional wave is considered.

SH wave scattering at a trapezoid hill and a semi-cylindrical alluvial basin by hybrid method

The response of surface motion inside and near an irregular area embedded into an elastic half-plane is investigated for the case of incident plane SH wave. The irregular areas include a trapezoid hill and semi-cylindrical alluvial basin. The results of simple geometric shapes such as semi-circular or semi-elliptical canyons are obtained by using separation of variables and expansion of the solution in a basis of orthogonal functions. In this paper, based on a variational formalism which proposed by Mei(1980), a hybrid method which combing the finite element and series expansion method is implemented to solve the scattering problems. We define a substructure which enclosing the irregular area can easily be formulated by finite element method. The unknown boundary data called the scattered waves can be formulated through a series representation with unknown coefficients. Due to the continuity condition at the interface, therefore, the unknown coefficients of this series representation are treated as generalized coordinates and can be easily formulated by the same variational principle. The expansion function of the series representation is constituted of basis function, each basis function is constructed by Lambs solution and satisfies both traction free condition at ground surface and radiation condition at infinity. The merit of the hybrid method is that the flexibility of finite elements offers the greatest advantage to model the irregular area. We use a simple mapping function to calculate the coordinates of the irregular region. The node numbers of the finite elements and the arrangement of the elements are the same as different areas.

Resonance analysis of a 2D alluvial valley subjected to seismic waves

The T-matrix formalism and an ultrasonic experiment are developed to study the scattering of in-plane waves for an alluvial valley embedded in a two-dimensional half-space. The solution of the in-plane scattering problem can be determined by the T-matrix method, where the basis functions are defined by the singular solutions of Lambs problems with surface loading in both horizontal and vertical directions. In the experiment, a thin steel plate with a semicircular aluminum plate attached on the edge is used to simulate the two-dimensional alluvial valley in the state of plane stress. Based on the spectra of displacement signals measured at the free edge of the scatterer, the resonance frequencies where the peaks appear can be identified. It can be shown that the nondimensional resonance frequency is one of the characteristic properties of the scattering system. Furthermore, it is noted that the nondimensional resonance frequencies measured experimentally are in good agreement with those calculated theoretically.

Dynamic Response of a Hemispherical Foundation Embedded in an Elastic Half-Space

The dynamic response of a massless rigid hemispherical foundation embedded in a uniform homogeneous elastic half-space is considered in this study. The foundation is subjected to external forces, moments, plane harmonic P and SH waves, respectively. The series solutions are constructed by three sequences of Lamb’s singular solutions which satisfy the traction-free conditions on ground surface and radiation conditions at infinity, automatically, and their coefficients are determined by the boundary conditions along the soil-foundation interface in the least square sense. The fictitious eigen-frequencies, which arise in integral equation method, will not appear in the numerical calculation by the proposed method. The impedance functions which characterize the response of the foundation to external harmonic forces and moments at low and intermediate frequencies are calculated and the translational and rocking responses of the foundation when subjected to plane P and SH waves are also presented and discussed in detail.

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

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.