# Scattering of Antiplane SH Waves by Complex Landforms

Authors: Menghan Sun; Hui Qi
Journal: Shock and Vibration
 (2021)

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## Abstract

The multiple scattering of SH waves by isosceles triangular hill, semicircle depression, and isosceles trapezoidal hill in the solid half-space is studied. The complex model is divided into multiple subdomains by using the region matching method, then the wave functions in each subdomain are constructed by using the fractional-order Bessel function, and finally, the infinite algebraic equations for solving the unknown coefficients in the wave function are established by using the multipolar coordinate technique and the complex function method according to the boundary conditions. Fourier series is used to solve the unknown undetermined coefficients. The results show that due to the multiple reflections of the incident wave between complex landforms, surface displacement amplitude is affected by the incident angle, incident frequency, and the distance between the isosceles triangular hill, semicircle depression, and isosceles trapezoidal hill. It is found that when the incident frequency increases, there is a certain amplification effect between the hills and the depression. When the wave is incident horizontally, there is a certain “barrier” effect between hills and depression, and when the distance between the hills and depression reaches a certain level, the “barrier” effect will reach a stable value.


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## Body

## 1. Introduction

In recent years, the research [1–4] on seismology has been widely valued by seismologists at home and abroad, which continuously and vigorously develops in the fields of crustal upper mantle structure and tectonic detection, natural earthquake prediction and disaster prevention, engineering building detection, energy survey, and other fields, which profoundly affects the politics, economy, and military of human society. Among all the factors affecting earthquakes, it has been confirmed that local landform is the most important factor causing seismic hazard or earthquake local changes, so correctly studying and analyzing the effect rule of landform on earthquake has essential engineering application value in seismology and earthquake. How to enhance comprehensive seismic capacity, effectively reduce the seismic hazard, and ensure the safety of life and property in the event of an earthquake to the greatest extent becomes an important topic for the majority of seismology researchers.At present, based on the classical elastic wave theory [5–7], previous studies mainly focus on single landform aspects, such as single sag, deposit, and raised landform. Many scholars have studied these problems deeply and gained many valuable results. These studies have a reference significance of engineering to a certain extent. Trifunac [8–10] studied the nature of surface motion in and around a semicylindrical canyon, a semicylindrical alluvial valley, and a semielliptical alluvial valley under the case of incident plane SH waves. This study qualitatively explained some vibrating characteristics of long and deep alluvial valleys. Lee [11] studied the 2D scattering and diffraction of plane SH waves by a semiparabolic cylindrical canyon in an elastic half-space for the cases of SH waves coming from the side of the half-space toward the front of the canyon. Lee [12] presented a closed-form analytic solution of two-dimensional scattering and diffraction of plane SH waves by a semicylindrical hill with a semicylindrical concentric tunnel inside an elastic half-space by using the cylindrical wave function expansion method. Qiu [13] studied the antiplane response of an isosceles triangular hill to incident SH waves based on the method of complex function and by using a moving coordinate system. Yuan [14, 15] studied the problem of scattering of plane SH waves by a semicylindrical hill in an otherwise homogeneous, isotropic, and elastic two-dimensional half-space by using the series of wave functions and a new expansion technique and then presented a closed-form solution of two-dimensional scattering of plane SH waves by a cylindrical canyon of circular-arc cross section in a half-space by using the wave functions expansion method. Franciso [16] explained a boundary method developed by Herrera in connection with wave scattering and studied the application to the scattering of the SH wave by surface irregularities. Qi [17, 18] studied the problem of the scattering of SH wave by the isosceles triangular hill in the right-angle plane given by using the methods of complex function and multiple coordinate and the scattering of SH wave in a half-space with oblique semielliptical notches. Tsaur [19] studied the dynamic antiplane problem of an elliptic-arc canyon in the comer of an elastic quarter space by using the method of wave function expansions and the method of images. Wang [20] studied the seismic response of a tunnel lining structure embedded in a thick expansive soil stratum. Shyu [21] proposed a novel strategy for the investigation of displacement amplitude near and along symmetric dikes comprising a trapezoidal structure with a circular-arc foundation when subjected to SH waves.However, in recent years, many complex buildings such as major traffic projects and large-scale communication equipment and ultrahigh arch dam are planned in the coupled areas of alpine canyons and basins with complex topographical conditions, and their seismic fortification is in urgent need of reliable ground motion analysis [22, 23]. The combination of mountains and valleys in practical engineering can be reduced to a combination of triangles or trapezoids and semicircles. Mountains or valleys in previous studies are mostly taken as triangles or semicircles, but there are few studies on trapezoidal mountains. Rodriguez [24] studied the multiple scattering of elastic waves by subsurface fractures and cavities by using the indirect boundary element method to simulate the 2D scattering of elastic P and SV waves. Gao [25] proposed the series solution of wave functions for 2D scattering and diffraction of cylindrical SH waves induced by a symmetrical V-shaped canyon to account for near-source topographic effects. Yang [26] studied the antiplane response of two scalene triangular hills and a semicylindrical canyon under SH waves by using wave function expansion and the complex function method. Chen [27] studied the surface motion of multiple alluvial valleys for incident plane SH waves by using a semianalytical approach. Liu [28] studied the multiple scattering of SH waves by two hills of different geometries (a triangle and a semicircle) on a solid half-space. Du [29] studied the surface displacement function of two isosceles triangular hills during the incidence of SH wave by using wave function expanding. Ba [30] investigated the scattering and diffraction of plane SH waves by periodic alluvial valleys embedded in a layered half-space by using the region matching technique and periodic indirect boundary element method. Han [31] studied the scattering of SH waves on multiple hills and canyons by using wave function.In this paper, based on the accurate region matching method, the model is divided into four regions as shown in Figure1: region 1 contains three semicircular canyons at the half-space boundary, region 2 contains an isosceles triangular hill and a semicircular bottom region, region 3 contains a fan-shaped domain formed by a trapezoidal side boundary, and region 4 contains a semicylindrical domain formed by the trapezoidal upper boundary. Because of many subdomains, the multipolar coordinate technique is used to construct the wavefield expressions in each subdomain. In region 1, the wave functions wB, wC, and wD which satisfy the condition of no traction at the horizontal boundary are constructed. In region 2, the fractional-order Bessel function standing wave wA satisfying the condition of no traction at the triangular isosceles boundary was constructed. In region 3, the divergent wave wE and the cohesive wave wF satisfying the condition of no traction at the isosceles trapezoidal side were constructed. Finally, in region 4, the cohesive wave wG satisfying the condition of no traction at the upper horizontal boundary was constructed. According to the continuous condition of displacement and stress of boundary sb, sd, and se and the nontraction conditions of boundary sc, a set of infinite algebraic equations will be obtained by introducing multipolar coordinates and complex function method. The amplitude of surface displacement of complex terrain is obtained by a numerical example. In this paper, the effects of different incident angles, incident frequency, size of the isosceles triangular mountain and isosceles trapezoid mountain, size of semicircle basin, and distance between mountains and basin on ground motion are studied. This study provides a theoretical reference for seismic mitigation and engineering construction planning of complex landforms.Figure 1
The two-dimensional complex landforms model containing an isosceles triangular hill, semicircle depression, and isosceles trapezoidal hill in half-space.


## 2. Problem Description

The half-space model composed of an isosceles triangle, an isosceles trapezoid, and a semicircular depression is shown in Figure1. The horizontal surface boundary is s, the convex vertex of the isosceles triangle is o5, the hypotenuse boundary is sa, the height is h1, the center point of the bottom side is o1, and the length of the bottom side is 2L1. The center of the semicircle depression is o and the radius is R. The intersection point of isosceles trapezoid oblique edge is o4, the oblique edge is sf, the upper boundary is sg, the center point of the upper boundary is o3, the length of the upper edge is 2L3, the center point of the bottom edge is o2, the length of the bottom edge is 2L2, the height is h4, and the height of the intersection point of isosceles trapezoid oblique edge is h3 from the upper boundary. Based on the idea of dividing regions, three auxiliary boundaries sb, sd, and se are introduced, which satisfy the continuity of stress and displacement at the auxiliary boundary. The model is divided into four regions as shown in Figure 1: region 1 is a half-space with three semicircular depressions at the horizontal boundary; region 2 is an isosceles triangle convex and semicircular bottom region; region 3 is a closed fan-shaped region formed by two semicircular curves; region 4 is an elastic semicircle. The six coordinate systems are x,y,x1,y1,x2,y2,x3,y3,x4,y4,andx5,y5, respectively.


## 3. Basic Theory

According to the small deformation theory, without considering the physical force, the elastic dynamic equation can be expressed as(1)λ+μ∇∇⋅u+μ∇2u=ρ,u¨(2)u=Reu,v,wexp−iωt,where ∇ is the vector differential operator, u is the displacement vector, ρu¨ is the inertia term, and λ and μ are Lame constants.In this paper, the SH wave is studied. Considering the antiplane displacement, thenu=v=0. By substituting formula into (2) and (1) and omitting the time harmonic factor exp−iωt, the Helmholtz equation can be obtained as(3)∇2w+k2w=0.Three polar coordinate systemsr,θ,r1,θ1,andr2,θ2 are defined in region 1, with a center point of o,o1,ando2 as shown in Figure 2. The total wavefield can be decomposed into the total scattered wavefield ws and free displacement field wi+r by auxiliary boundaries sb and sd and semicircular sag boundary sc. The total wavefield can be expressed as follows:(4)w1=wi+r+ws.Figure 2
Surface displacement amplitude of degraded isosceles triangle hill.The incident wave fieldwi and the reflected wavefield wr generated by the horizontal boundary s are as follows:(5)wi=w0expikysinα−xcosα,(6)wr=w0expikysinα+xcosα.The total scattered wave field can be divided into(7)ws=wB+wC+wD.At the horizontal boundary, the three scattered wavefields satisfy the condition of no traction, which are expressed as follows:(8)wBr1,θ1=w0∑m=0∞BmHm1kr1expimθ1+−1mexp−imθ1,(9)wCr,θ=w0∑m=0∞CmHm1krexpimθ+−1mexp−imθ,(10)wDr2,θ2=w0∑m=0∞DmHm1kr2expimθ2+−1mexp−imθ2.Among them, whenθ1=±π/2, then τθ1z1B=μ/r1∂wB/∂θ1=0. When θ=±π/2, then τθzC=μ/r∂wC/∂θ=0. When θ=±π/2, then τθ2z2D=μ/r2∂wD/∂θ2=0. Hm1⋅ is the first m-order Hankel equation and satisfies the radiation conditions. Bm,Cm,andDm are the unknown undetermined coefficients.The polar coordinate systemr5,θ5 is defined in the region and the center point is o5. Region 2 is a closed region consisting of auxiliary boundary sb and isosceles triangle boundary sa. The scattered wave wA is constructed in the closed region, and the condition of no traction at the free boundary sa is satisfied; that is, when θ5=±arctanL1/h1, then τθ5z5A=μ/r5∂wA/∂θ5=0. The scattered wave is as follows:(11)wAr5,θ5=w0∑m=0∞AmJmp1kr1expimp1θ5+−1mexp−imp1θ5,where p1=π/2arctanL1/h1. Am is the unknown undetermined coefficient.The polar coordinate systemr4,θ4 is defined in region 3 and the center point is o4. Due to the construction of two semicircular auxiliary boundaries sd and se, there are cohesive wave wF and divergent wave wE in the closed region. Therefore, the cohesive wave and the divergent wave need to satisfy the traction-free boundary condition of the isosceles trapezoid oblique edge sf; that is, when θ4=±θ0, then τθ4z4F=μ/r4∂wd/∂θ4=0, and τθ4z4E=μ/r4∂we/∂θ4=0. The cohesive wave wF is shown in formula (12) and the divergent wave wE is shown in formula (13):(12)wFr4,θ4=w0∑m=0∞FmJmp2kr4expimp2θ4+−1mexp−imp2θ4,(13)wEr4,θ4=w0∑m=0∞EmHmp21kr4expimp2θ4+−1mexp−imp2θ4,where p2=π/2θ0. EmandFm are the unknown undetermined coefficients.The polar coordinate systemr3,θ3 is defined in region 4 and the center point is o3. Region 4 is a closed region composed of a semicircular auxiliary boundary se and a horizontal boundary sg. A cohesive wave wG is constructed in the closed region, and the cohesive wave needs to satisfy the traction-free boundary condition of the horizontal boundary sg; that is, when θ3=±π/2, then τθ3z3G=μ/r3∂wG/∂θ3=0. The cohesive wave wG is as follows:(14)wG=w0∑m=0∞GmJmkr3expimθ3+−1mexp−imθ3,where Gm is the unknown undetermined coefficient.According to the stress and displacement continuity of the auxiliary boundariessb, sd, and se and the traction-free condition of the semicircular depression boundary se, the boundary equations for solving the unknown undetermined coefficients Am,Bm,Cm,Dm,Em,Fm,andGm are determined. The boundary conditions are as follows:(15)wAr5,θ5=wi+rr,θ+wBr1,θ1+wCr,θ+wDr2,θ2,r1=L1,τrzAr5,θ5=τrzi+rr,θ+τrzBr1,θ1+τrzCr,θ+τrzDr2,θ2,r1=L1,wdr4,θ4+wer4,θ4=wi+rr,θ+wBr1,θ1+wCr,θ+wDr2,θ2,r2=L2,τrzdr4,θ4+τrzer4,θ4=τrzi+rr,θ+τrzBr1,θ1+τrzCr,θ+τrzDr2,θ2r2=L2,wdr4,θ4+wer4,θ4=wGr3,θ3,r3=L3,τrzdr4,θ4+τrzer4,θ4=τrzGr3,θ3,r3=L3,τrzi+rr,θ+τrzBr1,θ1+τrzCr,θ+τrzDr2,θ2=0,θ=−π2,π2.The infinite algebraic equations with unknown undetermined coefficients can be solved by substituting the wavefield functions in each subdomain into the boundary conditions (15).


## 4. Complex Function

The expression of the wavefield in each subdomain has been given, and the boundary equations for solving unknown undetermined coefficients have been determined. It is worth noting that the wave field expressions of each subdomain are established in their respective polar coordinates, but the solution of each boundary equation needs to be unified between the polar coordinatesr,θ,r1,θ1,andr2,θ2. At present, some studies use the addition formula to transform the coordinates. However, the physical model of complex landforms studied in this paper is relatively complex, so the addition formula theorem in isosceles triangle mountain, isosceles trapezoid mountain, and semicircle depression need to be derived again, which will bring a lot of calculation. Therefore, we use the complex function method to transform the coordinates of each subdomain, which will not only ensure the accuracy of the calculation but also greatly reduce the amount of calculation. In this paper, while defining the main coordinate system z=x+iy,z¯=x−iy, five subcoordinate systems zj=xj+iyj=rjeiθj,z¯j=xj−iyj=rje−iθjj=1,2,3,4,5 are also defined. However, by using the multipolar coordinate method, the transformation between each complex coordinate system can be easily obtained. The transformation equations are as follows:(16)z=z1−iL4=z2+iL5=z3−h4+iL5=z4−h2+iL5=z5−h1−iL4.In the above equation,h1,h2,andh4, respectively, represent the vertical distance between the subcoordinate system z5,z¯5,z4,z¯4,andz3,z¯3 and the main coordinate system z,z¯. L4 is the horizontal distance between the subcoordinate system z5,z¯5,z1,z¯1 and the main coordinate system z,z¯. L5 is the horizontal distance between the subcoordinate system z4,z¯4,z3,z¯3,andz2,z¯2 and the main coordinate system z,z¯.The expression of Helmholtz equation (3) in the main complex coordinate system is as follows:(17)∂2w∂z∂z¯+14k2w=0.The corresponding stress expressions are as follows:(18)τrz=μ∂w∂zeiθ+∂w∂z¯e−iθ,τθz=iμ∂w∂zeiθ−∂w∂z¯e−iθ.In the complex coordinate systemz,z¯, according to the transformation relationship between coordinates in equation (16), the incident wavefield in equation (5) and the reflected wavefield in equation (6) can be expressed as(19)wiz,z¯=w0∑m=−∞∞−imJmkzeimαzzm.wrz,z¯=w0∑m=−∞∞imJmkze−imαzzm.The total free fieldwi+r can be expressed as(20)wi+rz,z¯=w0∑m=−∞∞imJmkz−1meimα+e−imαzzm.Under the condition that the total free field radial stress satisfies the condition of no traction on the horizontal boundary, the expression of the total free field radial stress in a complex coordinate systemz,z¯ is as follows:(21)τrzi+rz,z¯=μkw02∑m=−∞∞imJm−1kz−Jm+1kz−1meimα+e−imαzzm.The scattered wavefieldwB in equation (8) can be expressed in the complex coordinate system z,z¯ as(22)wBz,z¯=w0∑m=0∞BmHm1kz+iL4z+iL4z+iL4m+−1mz+iL4z+iL4−m.The corresponding radial stress is as follows:(23)τrzBz,z¯=μkw02∑m=0∞BmHm−11kz+iL4z+iL4z+iL4m−1eiθ−−1mHm+11kz+iL4z+iL4z+iL4−m−1eiθ+−1mHm−11kz+iL4z+iL4z+iL4−m+1e−iθ−Hm+11kz+iL4z+iL4z+iL4m+1e−iθ.The scattered wavefieldwC in equation (9) can be expressed in the complex coordinate system z,z¯ as(24)wCz,z¯=w0∑m=0∞CmHm1kzzzm+−1mzz−m.The corresponding radial stress is as follows:(25)τrzCz,z¯=μkw02∑m=0∞CmHm−11kzzzm−1eiθ−−1mHm+11kzzz−m−1eiθ+−1mHm−11kzzz−m+1e−iθ−Hm+11kzzzm+1e−iθ.The scattered wavefieldwD in equation (10) can be expressed in the complex coordinate system z,z¯ as(26)wDz,z¯=w0∑m=0∞DmHm1kz−iL5z−iL5z−iL5m+−1mz−iL5z−iL5−m.The corresponding radial stress is as follows:(27)τrzDz,z¯=μkw02∑m=0∞DmHm−11kz−iL5z−iL5z−iL5m−1eiθ−−1mHm+11kz−iL5z−iL5z−iL5−m−1eiθ+−1mHm−11kz−iL5z−iL5z−iL5−m+1e−iθ−Hm+11kz−iL5z−iL5z−iL5m+1e−iθ.In the complex coordinate systemz1,z¯1, according to the transformation relationship between coordinates in equation (16), the total free field wi+r in formula (21) can be expressed as(28)wi+rz1,z¯1=w0∑m=−∞∞imJmkz1−iL4−1meimα+e−imαz1−iL4z1−iL4m.The corresponding total free field radial stress is as follows:(29)τrzi+rz1,z¯1=μkw02∑m=−∞∞imJm−1kz1−iL4−Jm+1kz1−iL4−1meimα+e−imαz1−iL4z1−iL4m.The scattered wavefieldwA in equation (11) can be expressed in the complex coordinate system z1,z¯1 as(30)wAz1,z¯1=w0∑m=0∞AmJmp1kz1+h1z1+h1z1+h1mp1+−1mz1+h1z1+h1−mp1.The corresponding radial stress is as follows:(31)τrzA=μkw02∑m=0∞AmJmp1−1kz1+h1z1+h1z1+h1mp1−1eiθ1−−1mJmp1+1kz1+h1z1+h1z1+h1−mp1−1eiθ1+−1mJmp1−1kz1+h1z1+h1z1+h1−mp1+1e−iθ1−Jmp1+1kz1+h1z1+h1z1+h1mp1+1e−iθ1.The scattered wavefieldwB in equation (8) can be expressed in the complex coordinate system z1,z¯1 as(32)wBz1,z¯1=w0∑m=0∞BmHm1kz1z1z1m+−1mz1z1−m.The corresponding radial stress is as follows:(33)τrzBz1,z¯1=μkw02∑m=0∞BmHm−11kz1z1z1m−1eiθ1−−1mHm+11kz1z1z1−m−1eiθ1+−1mHm−11kz1z1z1−m+1e−iθ1−Hm+11kz1z1z1m+1e−iθ1.The scattered wavefieldwC in equation (9) can be expressed in the complex coordinate system z1,z¯1 as(34)wCz1,z¯1=w0∑m=0∞CmHm1kz1−iL4z1−iL4z1−iL4m+−1mz1−iL4z1−iL4−m.The corresponding radial stress is as follows:(35)τrzCz1,z¯1=μkw02∑m=0∞CmHm−11kz1−iL4z1−iL4z1−iL4m−1eiθ1−−1mHm+11kz1−iL4z1−iL4z1−iL4−m−1eiθ1+−1mHm−11kz1−iL4z1−iL4z1−iL4−m+1e−iθ1−Hm+11kz1−iL4z1−iL4z1−iL4m+1e−iθ1.The scattered wavefieldwD in equation (10) can be expressed in the complex coordinate system z1,z¯1 as(36)wDz1,z¯1=w0∑m=0∞DmHm1kz1−iL4+L5z1−iL4+L5z1−iL4+L5m+−1mz1−iL4+L5z1−iL4+L5−m.The corresponding radial stress is as follows:(37)τrzDz1,z¯1=μkw02∑m=0∞DmHm−11kz1−iL4+L5z1−iL4+L5z1−iL4+L5m−1eiθ1−−1mHm+11kz1−iL4+L5z1−iL4+L5z1−iL4+L5−m−1eiθ1+−1mHm−11kz1−iL4+L5z1−iL4+L5z1−iL4+L5−m+1e−iθ1−Hm+11kz1−iL4+L5z1−iL4+L5z1−iL4+L5m+1e−iθ1.In the complex coordinate systemz2,z¯2, according to the transformation relationship between each coordinate in equation (16), the total free field wi+r in formula (21) can be expressed as(38)wi+rz2,z¯2=w0∑m=−∞∞imJmkz2+iL5−1meimα+e−imαz2+iL5z2+iL5m.The corresponding total free field radial stress is as follows:(39)τrzi+rz2,z¯2=μkw02∑m=−∞∞imJm−1kz2+iL5−Jm+1kz2+iL5−1meimα+e−imαz2+iL5z2+iL5m.The scattered wavefieldwB in equation (8) can be expressed in the complex coordinate system z2,z¯2 as(40)wBz2,z¯2=w0∑m=0∞BmHm1kz2+iL4+L5z2+iL4+L5z2+iL4+L5m+−1mz2+iL4+L5z2+iL4+L5−m.The corresponding radial stress is as follows:(41)τrzBz2,z¯2=μkw02∑m=0∞BmHm−11kz2+iL4+L5z2+iL4+L5z2+iL4+L5m−1eiθ2−−1mHm+11kz2+iL4+L5z2+iL4+L5z2+iL4+L5−m−1eiθ2+−1mHm−11kz2+iL4+L5z2+iL4+L5z2+iL4+L5−m+1e−iθ2−Hm+11kz2+iL4+L5z2+iL4+L5z2+iL4+L5m+1e−iθ2.The scattered wavefieldwC in equation (9) can be expressed in the complex coordinate system z2,z¯2 as(42)wCz2,z¯2=w0∑m=0∞CmHm1kz2+iL5z2+iL5z2+iL5m+−1mz2+iL5z2+iL5−m.The corresponding radial stress is as follows:(43)τrzCz2,z¯2=μkw02∑m=0∞CmHm−11kz2+iL5z2+iL5z2+iL5m−1eiθ2−−1mHm+11kz2+iL5z2+iL5z2+iL5−m−1eiθ2+−1mHm−11kz2+iL5z2+iL5z2+iL5−m+1e−iθ2−Hm+11kz2+iL5z2+iL5z2+iL5m+1e−iθ2.The scattered wavefieldwD in equation (10) can be expressed in the complex coordinate system z2,z¯2 as(44)wDz2,z¯2=w0∑m=0∞DmHm1kz2z2z2m+−1mz2z2−m.The corresponding radial stress is as follows:(45)τrzDz2,z¯2=μkw02∑m=0∞DmHm−11kz2z2z2m−1eiθ2−−1mHm+11kz2z2z2−m−1eiθ2+−1mHm−11kz2z2z2−m+1e−iθ2−Hm+11kz2z2z2m+1e−iθ2.The cohesive wave fieldwF in equation (12) can be expressed in the complex coordinate system z2,z¯2 as(46)wFz2,z¯2=w0∑m=0∞FmJmp2kz2+h2z2+h2z2+h2mp2+−1mz2+h2z2+h2−mp2.The corresponding radial stress is as follows:(47)τrzFz2,z¯2=μkw02∑m=0∞FmJmp2−1kz2+h2z2+h2z2+h2mp2−1eiθ2−−1mJmp2+1kz2+h2z2+h2z2+h2−mp2−1eiθ2+−1mJmp2−1kz2+h2z2+h2z2+h2−mp2+1e−iθ2−Jmp2+1kz2+h2z2+h2z2+h2mp2+1e−iθ2.The divergent wavefieldwE in equation (13) can be expressed in the complex coordinate system z2,z¯2 as(48)wEz2,z¯2=w0∑m=0∞EmHmp21kz2+h2z2+h2z2+h2mp2+−1mz2+h2z2+h2−mp2.The corresponding radial stress is as follows:(49)τrzEz2,z¯2=μkw02∑m=0∞EmHmp2−11kz2+h2z2+h2z2+h2mp2−1eiθ2−−1mHmp2+11kz2+h2z2+h2z2+h2−mp2−1eiθ2+−1mHmp2−11kz2+h2z2+h2z2+h2−mp2+1e−iθ2−Hmp2+11kz2+h2z2+h2z2+h2mp2+1e−iθ2.In the complex coordinate systemz3,z¯3, according to the transformation relationship between coordinates in equation (16), the cohesive wave field wF in equation (12) can be expressed as(50)wFz3,z¯3=w0∑m=0∞FmJmp2kz3+h3z3+h3z3+h3mp2+−1mz3+h3z3+h3−mp2.The corresponding radial stress is as follows:(51)τrzFz3,z¯3=μkw02∑m=0∞FmJmp2−1kz3+h3z3+h3z3+h3mp2−1eiθ3−−1mJmp2+1kz3+h3z3+h3z3+h3−mp2−1eiθ3+−1mJmp2−1kz3+h3z3+h3z3+h3−mp2+1e−iθ3−Jmp2+1kz3+h3z3+h3z3+h3mp2+1e−iθ3.The divergent wavefieldwE in equation (13) can be expressed in the complex coordinate system z3,z¯3 as(52)wEz3,z¯3=w0∑m=0∞EmHmp21kz3+h3z3+h3z3+h3mp2+−1mz3+h3z3+h3−mp2.The corresponding radial stress is as follows:(53)τrzEz3,z¯3=μkw02∑m=0∞EmHmp2−11kz3+h3z3+h3z3+h3mp2−1eiθ3−−1mHmp2+11kz3+h3z3+h3z3+h3−mp2−1eiθ3+−1mHmp2−11kz3+h3z3+h3z3+h3−mp2+1e−iθ3−Hmp2+11kz3+h3z3+h3z3+h3mp2+1e−iθ3.The cohesive wavefieldwG in equation (14) can be expressed in the complex coordinate system z3,z¯3 as(54)wGz3,z¯3=w0∑m=0∞GmJmkz3z3z3mp2+−1mz3z3−mp2.The corresponding radial stress is as follows:(55)τrzGz3,z¯3=μkw02∑m=0∞GmJm−1kz3z3z3m−1eiθ3−−1mJm+1kz4z3z3−m−1eiθ3+−1mJm−1kz3z3z3−m+1e−iθ3−Jm+1kz3z3z3m+1e−iθ3.In equation (15), the continuous conditions of displacement and stress between regions 1 and 2, the continuous conditions of displacement and stress between regions 1 and 3, the continuous conditions of displacement and stress between regions 3 and 4, and the condition of no traction in the boundary of the semicircle depression can be expressed in the complex coordinate system as follows:(56)wAz1,z¯1=wi+rz1,z¯1+wBz1,z¯1+wCz1,z¯1+wDz1,z¯1,z1=L1,τrzAz1,z¯1=τrzi+rz1,z¯1+τrzBz1,z¯1+τrzCz1,z¯1+τrzDz1,z¯1,z1=L,wdz2,z¯2+wez2,z¯2=wi+rz2,z¯2+wBz2,z¯2+wCz2,z¯2+wDz2,z¯2,z2=L2,τrzdz2,z¯2+τrzez2,z¯2=τrzi+rz2,z¯2+τrzBz2,z¯2+τrzCz2,z¯2+τrzDz2,z¯2,z2=L2,wdz3,z¯3+wez3,z¯3=wGz3,z¯3,z3=L3τrzdz3,z¯3+τrzez3,z¯3=τrzGz3,z¯3,z3=L3,τrzi+rz,z¯+τrzBz,z¯+τrzCz,z¯+τrzDz,z¯=0.z=R,θ=−π2,π2.By substituting the displacement field and stress field in each complex coordinate system into the boundary (56), the infinite algebraic equations can be obtained:(57)∑m=0∞Amξm11+∑m=0∞Bmξm12+∑m=0∞Cmξm13+∑m=0∞Dmξm14=∑m=−∞∞ζm1,∑m=0∞Amξm21+∑m=0∞Bmξm22+∑m=0∞Cmξm23+∑m=0∞Dmξm24=∑m=−∞∞ζm2,∑m=0∞Bmξm32+∑m=0∞Cmξm33+∑m=0∞Dmξm34+∑m=0∞Emξm35+∑m=0∞Fmξm36=∑m=−∞∞ζm3,∑m=0∞Bmξm42+∑m=0∞Cmξm43+∑m=0∞Dmξm44+∑m=0∞Emξm45+∑m=0∞Fmξm46=∑m=−∞∞ζm4,∑m=0∞Emξm55+∑m=0∞Fmξm56+∑m=0∞Gmξm57=∑m=−∞∞ζm5,∑m=0∞Emξm65+∑m=0∞Fmξm66+∑m=0∞Gmξm67=∑m=−∞∞ζm6,∑m=0∞Bmξm72+∑m=0∞Cmξm73+∑m=0∞Dmξm74=∑m=−∞∞ζm7,where the expressions of ξml,ζmk, l = 11, 12, 13, 14, 21, 22, 23, 24, 32, 33, 34, 35, 36, 42, 43, 44, 45, 46, 55, 56, 57, 65, 66, 67, 72, 73, 74, k = 1, 2, 3, 4, 5, 6, 7 in equation (58) are given in Appendix.The Fourier series expansion method is used to solve the infinite algebraic equations. Multiply both sides of the infinite algebraic equation system bye−imθ and integrate on −π/2,π/2; then(58)∑n=−∞∞∑m=0∞Amξmn11+∑n=−∞∞∑m=0∞Bmξmn12+∑n=−∞∞∑m=0∞Cmξmn13+∑n=−∞∞∑m=0∞Dmξmn14=∑n=−∞∞∑m=−∞∞ζmn1,∑n=−∞∞∑n=−∞∞Amξmn21+∑m=0∞Bmξmn22+∑n=−∞∞∑m=0∞Cmξmn23+∑n=−∞∞∑m=0∞Dmξmn24=∑n=−∞∞∑m=−∞∞ζmn2,∑n=−∞∞∑m=0∞Bmξmn32+∑n=−∞∞∑m=0∞Cmξmn33+∑n=−∞∞∑m=0∞Dmξmn34+∑n=−∞∞∑m=0∞Dmξmn35+∑n=−∞∞∑m=0∞Dmξmn36=∑n=−∞∞∑m=−∞∞ζmn3,∑n=−∞∞∑m=0∞Bmξmn42+∑n=−∞∞∑m=0∞Cmξmn43+∑n=−∞∞∑m=0∞Dmξmn44+∑n=−∞∞∑m=0∞Dmξmn45+∑n=−∞∞∑m=0∞Dmξmn46=∑m=−∞∞ζmn4,∑n=−∞∞∑m=0∞Emξmn55+∑n=−∞∞∑m=0∞Fmξmn56+∑n=−∞∞∑m=0∞Gmξmn57=∑n=−∞∞∑m=−∞∞ζmn5,∑n=−∞∞∑m=0∞Emξmn65+∑n=−∞∞∑m=0∞Fmξmn66+∑n=−∞∞∑m=0∞Gmξmn67=∑n=−∞∞∑m=−∞∞ζmn6,∑n=−∞∞∑m=0∞Bmξmn72+∑n=−∞∞∑m=0∞Cmξmn73+∑n=−∞∞∑m=0∞Dmξmn74=∑n=−∞∞∑m=−∞∞ζmn7,where(59)ξmn11=12π∫−π/2π/2Jmp1kz1+h1z1+h1z1+h1mp1+−1mz1+h1z1+h1−mp1e−inθ1dθ1,(60)ζmn1=12π∫−π/2π/2imJmkz1−iL4−1meimα+e−imαz1−iL4z1−iL4me−inθ1dθ1.The Fourier integral expression of the other functions is similar in form to equations (60) and (61). The system of equations (59) is an infinite algebraic equation system for solving unknown coefficients Am,Bm,Cm,Dm,Em,Fm,andGm.


## 5. Surface Displacement Amplitude

In order to study the interaction of antiplane SH waves among the isosceles triangular hill, semicircular basin depression, and the isosceles trapezoidal hill, it is necessary to discuss the influence of displacement amplitude on the frequency and incidence angle of incident SH wave and the influence of the distance between two hills and semicircle depression. The acceleration can be obtained if the displacement amplitude is obtained, which means that it can provide a better reference for seismic disaster mitigation for practical projects.In region 1, the total displacement field isw1, which can be expressed as(61)w1=wi+r+wB+wC+wD.In region 2, the total displacement fieldw2 can be expressed as(62)w2=wA.In region 3, the total displacement fieldw3 can be expressed as(63)w3=wE+wF.In region 4, the total displacement fieldw4 can be expressed as(64)w4=wG.Equations (62)–(65) can also be expressed as(65)wj=wjeiωt−φjj=1,2,3,4,where wj is the displacement amplitude and φj is the phase angle.The dimensionless frequency of an incident wave can be expressed as(66)η=2Rλ=kRπ,where λ is the wavelength of the incident wave.


## 6. Numerical Calculation and Analysis

Because horizontal boundarys and semicircular concave boundary sc are no traction boundary, the precision of the analytical method can be tested using the residual dimensionless stress on the surface of the basin. In this paper, taking the truncation of the Fourier series m=n=20 is enough to make the residual dimensionless stress reach 10−3. The amplitude of incident wave under all computation conditions is w0=1. What needs to be pointed out in our calculation model is that the incident wave, reflected wave, and scattered wave are all expressed at the origin o while the standing wave in the isosceles triangle is expressed at the origin o5, and the standing waves in the isosceles trapezoid are all expressed at the origin o3 and o4 expansion which means that there is an isosceles triangle hill on the left side of the semicircular depression and an isosceles trapezoid hill on the right side of the semicircular depression.To confirm the correctness of complex model wave function construction and calculation results in this paper, the model is first reduced to only contain the isosceles triangle hill bulge, and the isosceles triangle hill size data is set according to the literature [16]. h1/L1 that is the ratio of the height of the isosceles triangle to the base is 0.75, 0.5, and 0.25, respectively. The incident wave frequency is set to η=0.5, and the incidence angle is set to α=0o. As shown in Figure 2, the computation result of the complex landforms model reducing isosceles triangle hill model is nearly the same as the literature [16]. Then, the complex landforms model is reduced to only contain semicircular depression, and the semicircular depression size is set according to the literature [9]. The radius of the semicircular depression is set to R=1, the incident wave frequency is set to η=0.5, and the incidence angles are set to 0o, 30o, 60o, and 90o. As is shown in Figure 3, the computation result of the complex landforms model reducing semicircular depression is nearly the same as the literature [9]. Finally, the complex landforms model is reduced to only contain the isosceles trapezoid hill, and the isosceles trapezoid hill size data is set according to the literature [21]. The top side radius of the isosceles trapezoid is set to L3=0.5. The bottom side radius of the isosceles trapezoid is set to L4=1, and the incident angles are 0o, 30o, 60o, and 90o. As shown in Figure 4, the computation result of the complex landforms model reducing the isosceles trapezoid is nearly the same as the literature [21]. The abovementioned reducing model computation confirms the accuracy and applicability of the method in this article.Figure 3
Surface displacement amplitude of degraded semicircular depression.Figure 4
Surface displacement amplitude of degraded isosceles trapezoid hill.To research the interaction between complex landforms, the amplitude of surface motion at different points is very important. Through numerical calculation, we can judge how far the distance between the mountains is, and the influence between the mountains will be ignored. In this paper, because we suppose that encouragement contains an SH wave with an amplitude of 1, without the isosceles triangle hill, the semicircular depression, and the isosceles trapezoid hill, the amplitude of ground motion should be 2. But under the complex landforms interference, because of the interference of the scattered waves and the standing waves, the amplitude of the ground motion will fluctuate around 2.In Figure5, we set the radius of the base of the isosceles triangle hill, the radius of the semicircular depression, and the radius of the base of the isosceles trapezoid hill L1=R=L2=1. The distance between the center point of the isosceles triangle hill and the center point of the semicircular depression is L4=3. The distance between the center point of the isosceles trapezoid hill and the center point of the semicircle depression is L5=3. The top radius of the isosceles trapezoid hill is L3=0.5. The height of the isosceles triangle hill is h1=1, and the height of the isosceles trapezoid hill is h4=0.5. To research the surface displacement amplitude of complex landforms, four cases η=0.5,1,1.5,2 are taken to research incidence angle α=0o,45o,90o condition. It can be seen obviously in Figure 5 that complex landforms coupling site has a significant impact on seismic waves. Unity site reaction depends on the incidence frequency and incidence angle. As shown in Figure 5(a), when η=0.5, and under the circumstance that the wave is incident vertically α=0o, the displacement amplitude of the top of the isosceles triangle hill (near x/R=−3) is small, which is only around 2.5. When the wave is incident horizontally α=90o, the displacement amplitude of the top of the isosceles triangle hill (near x/R=−3) reaches 4, which shows that when the wave is incident horizontally, the displacement amplitude of the isosceles triangle hill will be affected by right semicircular depression and tends to increase. When the wave is incident vertically α=0o, the displacement amplitude of the semicircular (near x/R=0) is close to 0. The hills on the left and right sides show a “barrier” effect on the semicircular depression, the displacement amplitude in the semicircular depression will have a greater decline, and when the wave is incident horizontally α=90o, the displacement amplitude of the semicircular depression (near x/R=0) reaches about 3, and the hills on the left and right have an enlarged effect on it. When the wave is incident vertically α=0o, the maximum displacement amplitude of the isosceles trapezoid hill (near x/R=3) can reach about 3.5. When the wave is incident vertically α=90o, the displacement amplitude of the isosceles trapezoid hill (near x/R=3) has a slightly decreasing trend, and the displacement amplitude of the flat section on the right side of isosceles trapezoid hill gradually decreases to 2 and then reaches stabilization, which shows that when the wave level is incident from the left to the right, the isosceles triangular hill on the left and the semicircular depression have a “barrier effect” on the isosceles trapezoidal hill on the right.Figure 5
Surface displacement amplitude with the change ofη. (a) Surface displacement amplitude at η=0.5. (b) Surface displacement amplitude at η=1. (c) Surface displacement amplitude at η=1.5. (d) Surface displacement amplitude at η=2.
(a)(b)(c)(d)In the caseη=2, as shown in Figure 5(c), as the frequency of the incident wave increases, the surface displacement distribution of complex landforms becomes more complicated. The displacement spatial shock gradually intensifies, and the mutual interference between the waves becomes more obvious. When the incident wave changes from vertical incidence to horizontal incidence, the surface displacement amplitude of the isosceles triangle hill (near x/R=−3) will be greatly decreased by the “barrier” effect, and the minimum surface displacement amplitude is 0.2. Compared with the low frequency η=0.5, it is reduced by nearly 90%. The surface displacement amplitude of the isosceles trapezoid hill (near x/R=3) is also greatly decreased, and the surface displacement amplitude is close to 2. Compared with the low frequency η=0.5, it is reduced by nearly 50%. There is no obvious increase or decrease in the amplitude of the surface displacement in the semicircular depression (near x/R=0). On the whole, when waves are incident horizontally α=90o from the left to the right, the left side of the isosceles triangle hill (x/R≤−4) is determined by the displacement response of the isosceles triangle hill itself, the displacement amplitude distribution on the right side of the isosceles trapezoid hill (x/R≥−4) is determined by the complex landform coupling “barrier” effect, and the displacement amplitude is significantly reduced. When the incident frequency is higher, the “barrier” effect is more obvious. When the wave is incident vertically α=0o, with the increase of the incident frequency, the amplification effect of the displacement amplitude of the isosceles triangle hill (near x/R=−3) and the isosceles trapezoid hill (near x/R=3) is more obvious, and the displacement amplitude distribution oscillation is intensified. However, the semicircular depression basin has a more obvious inhibitory effect.Figure6 shows the effect of changes in the distance between isosceles triangular hill, semicircular depression, and isosceles trapezoidal hill on the internal seismic response of complex landforms under different incident angles. In order to explore how far the distance between the landforms is, the influence between the landforms will be ignored. We give the distribution map of the displacement amplitude of the complex landforms under four conditions: the distance between the center point of the isosceles triangle hill and the center point of the semicircular depression and the distance between the center point of the isosceles trapezoid hill and the center point of the semicircular depression are L4=L5=3,4,6,10. We set the incidence frequency η=1, and the radius of the base of the isosceles triangle hill, the radius of the semicircular depression, and the radius of the base of the isosceles trapezoid hill are L1=R=L2=1. The top radius of the isosceles trapezoid hill is L3=0.5. The height of the isosceles triangle hill is h1=1, and the height of the isosceles trapezoid hill is h4=0.5. To research the surface displacement amplitude of complex landforms, first, we analyze the isosceles triangle hill. When the wave is incident vertically α=0o and the isosceles triangle hill is L4=3 away from the semicircular depression, the surface displacement amplitude of the isosceles triangle hill (near x/R=−3) reaches the maximum at the apex, and the maximum displacement amplitude is about 3.8. When the isosceles triangle hill is L4=4 away from the semicircular depression, the surface displacement amplitude of the isosceles triangle hill (near x/R=−4) reaches the maximum at the apex, and the maximum displacement amplitude is about 5. When the isosceles triangle hill is L4=6 away from the semicircular depression, the surface displacement amplitude of the isosceles triangle hill (near x/R=−6) reaches the maximum at the apex, and the maximum displacement amplitude is about 5. When the isosceles triangle hill is L4=10 away from the semicircular depression, the surface displacement amplitude of the triangular hill (near x/R=−10) reaches its maximum at the apex, and the maximum displacement amplitude is also about 5. In summary, when L4 reaches 4, the influence of terrain coupling between complex landforms on the isosceles triangular hill will be negligible.Figure 6
Surface displacement amplitude with the change ofL4 and L5. (a) Surface displacement amplitude at L4=L5=3. (b) Surface displacement amplitude at L4=L5=4. (c) Surface displacement amplitude at L4=L5=6. (d) Surface displacement amplitude at L4=L5=10.
(a)(b)(c)(d)When the wave is incident verticallyα=0o, the same is applied to the influence of terrain coupling between complex landforms on the semicircular depression (near x/R=0). It can be seen from Figure 6 that when L4=L5 reach 4, the surface displacement amplitude of the semicircular depression (near x/R=0) will reach about 1.8 and become stable. When the wave is incident vertically α=0o, we analyze the impact of terrain coupling between complex landforms on isosceles trapezoid hill. It can be seen from Figure 6 that when L4=L5 reach 4, the surface displacement amplitude of the isosceles trapezoid mountain will reach about 3.3 and become stable. On the whole, when L4=L5 reach 4 and the wave is incident vertically, the characteristics of the surface displacement between the landforms will depend on their own response, and the mutual influence between the terrains will be negligible.When the wave is incident horizontallyα=90o, because of the “barrier” effect, it is equally important to analyze the mutual influence of the distance between the landforms. It can be seen from Figure 6 that at that time L4=L5=3, the maximum displacement amplitude of the isosceles triangle hill (near x/R=−3) reached 2.5 and no longer increased with the increase of L4 and L5. At that time L4=L5=4, the maximum displacement amplitude of the semicircular depression (near x/R=0) reached 1.5 and no longer increased with the increase of L4 and L5. At that time L4=L5=4, the maximum displacement amplitude of the isosceles trapezoid hill (near x/R=4) reaches 0.8, it will no longer increase with the increase of L4 and L5, and the flat displacement on the right side of the isosceles trapezoid hill will reach about 1.5 and then tend to remain unchanged. On the whole, when the wave is incident horizontally, the complex landforms show a certain shielding effect. As the distance between the landforms increases, the shielding effect will tend to stabilize.


## 7. Conclusions

An analytical solution of steady-state SH wave scattering problem in semispace complex landforms (including an isosceles triangular hill, a semicircular depression, and an isosceles trapezoid hill) is presented by using region matching method, multipolar coordinate technique, and complex function method. A numerical example is given to calculate the surface displacement amplitude of complex landforms. The results of numerical examples show the following.When the incidence frequency increases from 0.5 to 2, the fluctuation of the surface displacement amplitude distribution over complex landforms becomes more and more severe, and the oscillation range of the surface displacement amplitude gradually increases. When the frequency reaches 2, the displacement amplitude of the hill surface is twice that of the depression surface, which indicates that there is a stronger wave interference between different landforms in the case of high frequency.When the wave is horizontally incident from left to right, the surface displacement amplitude of the left isosceles triangular hill is determined by its own displacement response. The right semicircle depression and isosceles trapezoidal hill are affected by the “barrier” effect, and the surface displacement amplitude is significantly reduced.When the wave is vertically incident, with the increase of the incident frequency, the surface displacement amplitude of the hills will increase by a certain value, and the surface displacement amplitude of the depression will be obviously suppressed.When the distance between different landforms reaches a certain value, the surface displacement amplitude between different landforms will reach a stable value under the influence of coupling.



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*Source: 9997230-2021-08-06.xml*