Scattering of plane P waves by circular-arc alluvial valleys with saturated soil deposits
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 x–y 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 asin 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).
References (16)
- et al.
Surface motion of shallow circular alluvial valleys for incident plane SH waves: analytical solution
Soil Dyn Earthquake Eng
(1991) - et al.
Scattering and diffraction of plane P waves by circular–cylindrical canyons with variable depth-to-width ratio
Soil Dyn Earthquake Eng
(1990) - et al.
Some aspects of source characteristics of the 19 September 1985 Michoacan earthquake and ground motion amplification in and near Mexico city from strong motion data
Bull Seismol Soc Am
(1988) Surface motion of semi-cylindrical alluvial valley for incident plane SH waves
Bull Seismol Soc Am
(1971)- et al.
Surface motion of semi-elliptical alluvial valley for incident plane SH waves
Bull Seismol Soc Am
(1974) - et al.
Scattering of plane SH wave by a cylindrical alluvial valley of circular-arc cross-section
Earthquake Eng Struct Dyn
(1995) - et al.
Surface motion of circular-arc layered alluvial valleys for incident plane SH waves
Chin J Geotech Eng
(2000) - et al.
Response of circular-arc alluvial valleys under incident plane P waves
Rock Soil Mech (In Chinese)
(2001)
Cited by (30)
A semi-analytical solution to incident plane P waves scattering by saturated river valley with arbitrary shapes containing water
2024, Soil Dynamics and Earthquake EngineeringSurface motion of P/SV wave scattering by a semicircular canyon in saturated half-space by complex function and conformal mapping
2023, Soil Dynamics and Earthquake EngineeringThe scattering of seismic waves from saturated river valley with water layer: Modelled by indirect boundary element method
2023, Engineering Analysis with Boundary ElementsThree-dimensional IBEM solution to seismic wave scattering by a near-fault sedimentary basin
2022, Engineering Analysis with Boundary ElementsCitation Excerpt :In the past decades, the sedimentary basin effect investigated analytically and numerically has been a popular topic in earthquake engineering, seismology, and geophysics [10,26,28,43,60,67,72,75]. The analytical method is suitable for analyzing the seismic wave scattering mechanism of the sites with simple geometrical and material conditions [35,36,74]. However, it is challenging to obtain the analytical solution for seismic response in complex sites such as 3-D basins.
Nonlinear seismic response and amplification effect of 3D sedimentary basin based on bounding surface constitutive model
2022, Soil Dynamics and Earthquake EngineeringCitation Excerpt :These effects altogether aggravate the seismic hazards in the basin, which motivated significant interest in investigating the basin effects under strong earthquake as an essential task in seismic zoning, urban planning and engineering seismic design of cities located in sedimentary basins. The basin effects have been extensively examined employing two-dimensional (2D) analytical method [2–6] and numerical models [7–15]. Trifunac et al. [2] pioneered investigating the amplification effect of 2D sedimentary valleys using analytical method.