Scattering of plane P waves by circular-arc alluvial valleys with saturated soil deposits

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Abstract

This paper presents an analytical solution for two-dimensional scattering and diffraction of plane P waves by circular-arc alluvial valleys with shallow saturated soil deposits. The solution is based on Biot's dynamic theory for saturated porous media, and derived by employing Fourier–Bessel series expansion technique. In this analysis, soil deposits in the circular-arc valley are modeled as saturated porous media based on Biot's dynamic theory, and the circular-arc valley is assumed to be imbedded in an infinite half-space, filled with elastic single-phase media. Numerical results from this solution show that the amplitudes of displacement at the surface of an alluvial valley are mainly relative to the angle of incidence, the dimensionless frequency of incident P wave, the degree of saturation and porosity of soil deposits, and the stiffness and Poisson's ratio of the solid skeleton of the soil deposits. Furthermore, the proposed solution is compared with the previous solution, in which the soil deposit was modeled as an elastic single-phase solid.

Introduction

It has been well recognized that an alluvial valley might amplify ground motions significantly and lead to concentrated damage to both above and underground structures during earthquakes [1]. The amplification effects of alluvial valleys on incident seismic waves have received extensive attention by both seismologists and earthquake engineers in recent decades. Various methods for numerical modeling have been developed. However, very few analytical solutions are available so far, which are limited to the incidence of plane SH waves into circular-arc alluvial valleys [2], [3], [4], [5], [6]. When considering the P or SV wave incidence, the problem becomes much more complicated than the SH wave incidence because of the wave mode conversion during the reflection process. Recently, Liang and Lee [7] did some research on the scattering and diffraction of plane P waves in circular-arc alluvial valleys by means of series of Bessel function expansions.

It is noticed that soil deposit are modeled as an elastic single-phase medium in all the aforementioned solutions. In reality, soil deposits, especially those in soft alluvial valley, are often present in the form of porous solid saturated with fluid in nature. Therefore, the study of wave propagation in fluid saturated porous media has been of considerable interest. Biot [8] developed the propagation theory of elastic waves in fluid saturated porous media. However, there has been no analytical solution available for the scattering and diffraction of waves by irregular surface topography conditions in saturated porous media.

The purpose of the present study is, therefore, to investigate the two-dimensional scattering and diffraction of plane P waves in circular-arc alluvial valleys with shallow saturated soil deposits. Based on Biot's dynamic theory [8], an analytical solution is developed using the Fourier–Bessel series expansion technique. Because Biot's theory is applied to water saturated sands generally [16], the solution presented in this paper is only applicable to shallow alluvial deposits mainly filled with fluid saturated sandy or gravelly deposits. To illustrate the result of this solution, the surface displacement amplitudes are presented, and the effects of wavelength, angle of incidence, and ratio of depth to width of the valley on the dynamic response are discussed. In further, the present solution is compared with the solution, in which the soil deposit is assumed as an elastic single-phase solid. The methodology and analytical solution developed in this paper may provide a start point for further analysis of the scattering and diffraction of P, SV and Rayleigh waves by the irregular topography conditions in an infinite half-space.

Section snippets

The model and basic equations

A cross-section of the two-dimensional model to be analyzed is shown in Fig. 1. The circular-arc alluvial valley with saturated soil deposits is embedded in an infinite half-space(y>0). The valley is bounded by a flat ground surface, and the shape of the circular-arc is characterized by its center, o1, radius, a1, depth, h, and width, 2a as shown in Fig. 1 Two rectangular coordinate systems are defined: one is originated from o with its coordinates denoted as (x,y) and the other is originated

Boundary value problem

It is assumed that a plane P wave with its displacement and propagation vector situated in the xy plane incidents into the infinite half-space, as defined in last section. The potential function of the incident P wave, in the (x,y) coordinate system, can be expressed asΦ(i)=exp[ik1(xsinθαycosθα)iωt]in which, θα and ω are the angle of incidence and circular frequency of the incident P wave, respectively. kl is the longitudinal wave number, i is the unit of imaginary number with its value

Numerical results and discussions

From the point of view of earthquake engineering and strong motion seismology, an important aspect of above analysis is the description of displacement amplitudes at various points along the surface of the valley and the half-space in the vicinity of the valley. Once the wave potential functions have been determined, the displacement vector can be obtained by using Eq. (29), for the half-space, Eqs. (30), (31), for the alluvial valley.

The transformation from the cylindrical coordinates (r1,θ1)

Conclusions

Based on Biot's dynamic theory for saturated porous media, an analytical solution for the two-dimensional scattering and diffraction of plane P waves by circular-arc alluvial valleys with shallow (h/a≤1.0) saturated soil deposits is obtained by means of wave function expansion technique. A comparison of the present solution is also made, with the one in which the deposits are assumed as an elastic single-phase solid. From the numerical results, the following conclusions can be made associated

Acknowledgements

The study was financially supported by the Natural Science Foundation of P.R. China (Grant No. 50178005).

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