scholarly journals Surface Motion of a Half-Space Containing an Elliptical-Arc Canyon under Incident SH Waves

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1884
Author(s):  
Hui Qi ◽  
Fuqing Chu ◽  
Jing Guo ◽  
Runjie Yang
Keyword(s):  
Wave Function ◽  
Half Space ◽  
Wave Scattering ◽  
Scattering Problem ◽  
Expansion Method ◽  
Numerical Results ◽  
Mathieu Function ◽  
Axis Ratio ◽  
Function Expansion ◽  
Wave Function Expansion Method

The existence of local terrain has a great influence on the scattering and diffraction of seismic waves. The wave function expansion method is a commonly used method for studying terrain effects, because it can reveal the physical process of wave scattering and verify the accuracy of numerical methods. An exact, analytical solution of two-dimensional scattering of plane SH (shear-horizontal) waves by an elliptical-arc canyon on the surface of the elastic half-space is proposed by using the wave function expansion method. The problem of transforming wave functions in multi-ellipse coordinate systems was solved by using the extra-domain Mathieu function addition theorem, and the steady-state solution of the SH wave scattering problem of elliptical-arc depression terrain was reduced to the solution of simple infinite algebra equations. The numerical results of the solution are obtained by truncating the infinite equation. The accuracy of the proposed solution is verified by comparing the results obtained when the elliptical arc-shaped depression is degraded into a semi-ellipsoidal depression or even a semi-circular depression with previous results. Complicated effects of the canyon depth-to-span ratio, elliptical axis ratio, and incident angle on ground motion are shown by the numerical results for typical cases.

scholarly journals Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 246
Author(s):  
Hui Qi ◽  
Fuqing Chu ◽  
Yang Zhang ◽  
Guohui Wu ◽  
Jing Guo
Keyword(s):  
Wave Function ◽  
Stress Concentration ◽  
Wave Scattering ◽  
Dynamic Stress ◽  
Scattering Problem ◽  
Soil Layer ◽  
Expansion Method ◽  
Dynamic Stress Concentration ◽  
Sh Wave ◽  
Wave Function Expansion Method

Wave diffusion in the composite soil layer with the lined tunnel structure is often encountered in the field of seismic engineering. The wave function expansion method is an effective method for solving the wave diffusion problem. In this paper, the wave function expansion method is used to present a semi-analytical solution to the shear horizontal (SH) wave scattering problem of a circular lined tunnel under the covering soil layer. Considering the existence of the covering soil layer, the great arc assumption (that is, the curved boundary instead of the straight-line boundary) is used to construct the wavefield in the composite soil layer. Based on the wave field and boundary conditions, an infinite linear equation system is established by adding the application of complex variable functions. The finite term is intercepted and solved, and the accuracy of the solution is analyzed. Although truncation is inevitable, due to the Bessel function has better convergence, a smaller truncation coefficient can achieve mechanical accuracy. Based on numerical examples, the influence of SH wave incident frequency, soil parameters, and lining thickness on the dynamic stress concentration factor of lining is analyzed. Compared with the SH wave scattering problem by lining in a single medium half-space, due to the existence of the cover layer and the influence of its stiffness, the dynamic stress of the lining can be increased or inhibited. In addition, the lining thickness has obvious different effects on the dynamic stress concentration coefficient of the inner and outer walls of different materials.

scholarly journals Multiple Scattering of P1 Waves by Arbitrarily Arranged Cavities in Saturated Soils

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Miaomiao Sun ◽  
Huajian Fang ◽  
Xiaokang Zheng ◽  
Ru Zhang ◽  
Shimin Zhang ◽  
...  
Keyword(s):  
Wave Function ◽  
Multiple Scattering ◽  
Expansion Method ◽  
Wave Theory ◽  
Saturated Soil ◽  
Displacement Amplitude ◽  
Function Expansion ◽  
Wave Function Expansion Method ◽  
Hexagonal Arrangement ◽  
Complex Coefficients

Based on Biot’s saturated soil wave theory, using wave function expansion method, theoretical solutions of multiple scattering of plain P1 waves are achieved by rows of cavities as barrier with arbitrarily arranged cavities in saturated soil. Undetermined complex coefficients after wave function expansion are obtained by cavities-soil stress and displacement free boundary conditions. Numerical examples are used to investigate variation of dimensionless displacement amplitude at the back and force of cavities barrier under P1 wave incident, and it is also discussed that the main parameters influenced isolation effect such as scattering orders, separation of cavities, distances between cavity rows, numbers of cavities, and arrangement of barriers. The results clearly demonstrate optimum design proposals with rows of cavities: with the multiple scattering order increases, the displacement amplitude tends to converge and the deviation caused by subsequent scattering cannot be neglected; it will obtain higher calculation accuracy when the order of scattering is truncated at m=4; it is considered to select 2.5≤sp/as≤3.0 and 2.5≤h/as≤3.5, while designing cavity spacing and row-distance, respectively. The isolation properties of elastic waves with rectangular arrangement (counterpoint) are weaker than that with hexagonal arrangement (counterchanged) when the row-distance of barrier is uniform.

Application of the Singularity Expansion Method to Elastic Wave Scattering

1990 ◽  
Vol 43 (10) ◽  
pp. 235-249 ◽  
Author(s):  
Herbert U¨berall ◽  
P. P. Delsanto ◽  
J. D. Alemar ◽  
E. Rosario ◽  
Anton Nagl
Keyword(s):  
Elastic Wave ◽  
Elastic Waves ◽  
Wave Scattering ◽  
Scattering Amplitude ◽  
Scattering Problem ◽  
Expansion Method ◽  
Complex Frequency ◽  
Elastic Wave Scattering ◽  
Physical Measurements ◽  
Scattering Of Elastic Waves

The singularity expansion method (SEM), established originally for electromagnetic-wave scattering by Carl Baum (Proc. IEEE 64, 1976, 1598), has later been applied also to acoustic scattering (H U¨berall, G C Gaunaurd, and J D Murphy, J Acoust Soc Am 72, 1982, 1014). In the present paper, we describe further applications of this method of analysis to the scattering of elastic waves from cavities or inclusions in solids. We first analyze the resonances that appear in the elastic-wave scattering amplitude, when plotted vs frequency, for evacuated or fluid-filled cylindrical and spherical cavities or for solid inclusions. These resonances are interpreted as being due to the phase matching, ie, the formation of standing waves, of surface waves that encircle the obstacle. The resonances are then traced to the existence of poles of the scattering amplitude in the fourth quadrant of the complex frequency plane, thus establishing the relation with the SEM. The usefulness of these concepts lies in their applicability for solving the inverse scattering problem, which is the central problem of NDE. Since for the case of inclusions, or of cavities with fluid fillers, the scattering of elastic waves gives rise to very prominent resonances in the scattering amplitude, it will be of advantage to analyze these with the help of the resonance scattering theory or RST (first formulated by L Flax, L R Dragonette, and H U¨berall, J Acoust Soc Am 63, 1978, 723). These resonances are caused by the proximity of the SEM poles to the real frequency axis, on which the frequencies of physical measurements are located. A brief history of the establishment of the RST is included here immediately following the Introduction.

Analytical solution to the SH wave scattering problem caused by a circular cavity in a half space with inhomogeneous modulus

Meccanica ◽  
2021 ◽  
Vol 56 (3) ◽  
pp. 705-709
Author(s):  
Jinlai Bian ◽  
Zailin Yang ◽  
Guanxixi Jiang ◽  
Yong Yang ◽  
Menghan Sun
Keyword(s):  
Analytical Solution ◽  
Half Space ◽  
Wave Scattering ◽  
Scattering Problem ◽  
Circular Cavity ◽  
Sh Wave

Elastic Wave Scattering From an Interface Crack in a Layered Half Space Submerged in Water: Part II: Incident Plane Waves and Bounded Beams

1986 ◽  
Vol 53 (2) ◽  
pp. 333-338 ◽  
Author(s):  
S. M. Gracewski ◽  
D. B. Bogy
Keyword(s):  
Elastic Wave ◽  
Half Space ◽  
Interface Crack ◽  
Wave Scattering ◽  
Ultrasonic Transducer ◽  
Plane Waves ◽  
Incident Beam ◽  
Numerical Results ◽  
Elastic Wave Scattering ◽  
Layered Half Space

This is Part II of a two part paper which analyzes time harmonic elastic wave scattering by an interface crack in a layered half space submerged in water. The analytic solution was derived in Part I. Also numerical results for uniform harmonic normal or shear traction applied to the liquid-solid interface were presented. These were compared with previously published results as a check on the computer program used to obtain the numerical results. Here in Part II, additional numerical results are presented. Plane waves incident from the liquid onto the solid structure are first considered to gain insight into the response characteristics of the structure. The solution for an incident beam of Gaussian profile is then presented since this profile approximates the output of an ultrasonic transducer.

SH wave scattering by a frozen porous inclusion in fluid-saturated porous media using two approaches: wave function expansion and boundary element method

2019 ◽  
Vol 219 (3) ◽  
pp. 2187-2197
Author(s):  
A Furukawa ◽  
T Saitoh ◽  
S Hirose
Keyword(s):  
Porous Media ◽  
Wave Function ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Wave Scattering ◽  
Saturated Porous Media ◽  
Sh Wave ◽  
Function Expansion ◽  
Fluid Saturated Porous Media ◽  
Element Method

Summary This paper presents SH wave scattering by a frozen porous inclusion embedded in fluid-saturated porous media. We propose two computational methods, wave function expansion (WFE) and boundary element method (BEM), for wave scattering analyses. In WFE formulation, the components of displacement and stress are expressed by the superposition of the Bessel functions. The unknown coefficients in the expression are obtained via boundary conditions. On the other hand, in BEM formulation, boundary values of the frozen porous media are expressed by generalized displacement and traction. The generalized displacement consists of displacement components of the solid skeleton and the ice matrix, and the generalized traction is composed of the traction components of the two solid phases. Several numerical examples provide the validity of the proposed methods and the properties of the scattered waves. The discussion of the scattering properties focuses on the effects of ice saturation parameter, frequency of harmonic incident wave, the incident angle of the harmonic wave and the shape of the inclusion.

Scattering of SH-Waves to Semi-Cylindrical Canyon and Rectangular Elastic Hill on the Ground

2012 ◽  
Vol 610-613 ◽  
pp. 2544-2551
Author(s):  
Wen Pu Shi
Keyword(s):  
Half Space ◽  
Function Method ◽  
Expansion Method ◽  
Free Boundaries ◽  
Sh Waves ◽  
Green Function Method ◽  
Common Boundary ◽  
Wave Function Expansion Method ◽  
The Common ◽  
The Given

Wave function expansion method and Green function method were employed to study thescattering problem of SH-waves to the semi-cylindrical canyon and rectangular hill on the gr ound. First, the displacements in the half space and rectangular hill were given which can santisfy the stress-free conditions on the free boundaries. Then, the first kind of Fredholm integration equation of the unknown distribution stress was obtained by using the displacement conditions on the common boundary between the half-space and the rectangular hill, and Gauss-Legendre integration formula was used to solve the equation. The given example results show the feasibility and practicability of the method here.

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