Article Surface Motion of a Half-Space Containing an Elliptical-Arc Canyon under Incident SH Waves
Hui Qi, Fuqing Chu *, Jing Guo * and Runjie Yang College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China; [email protected] or [email protected] (H.Q.); [email protected] (R.Y.) * Correspondence: [email protected] (F.C.); [email protected] or [email protected] (J.G.)
Received: 21 September 2020; Accepted: 23 October 2020; Published: 30 October 2020
Abstract: 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.
Keywords: SH wave; elliptical-arc canyon; Mathieu function; Mathieu function addition theorem; dynamic response
1. Introduction Scattering and diffraction occur when local seismic waves propagate in media with irregular topography (such as canyons, valleys, hills). This discovery plays a vital role in ground vibration during earthquakes, known as the “topographic effect”. When the wavelength of the seismic wave is comparable to the size of the irregular terrain, amplification or attenuation can be observed. This phenomenon of topographic effects was first observed during the San Fernando earthquake on a steep ridge near the abutment of the Pacoima dam, and was subsequently widely recognized [1–5]. The understanding of topographic effects can provide profound insights into the interpretation of strong ground motions and is conducive to seismic design of large-span structures. In order to understand and reveal the properties of seismic wave topographic effect, a lot of research work on seismic wave scattering has been done by numerical and analytical methods. The numerical methods used mainly include the finite element method [6,7] and the boundary element method [8–11], and the analysis methods used are mainly the wave function expansion method. However, in a relatively simple case, the use of the analysis method is of great value not only for revealing the basic mathematical and physical characteristics of the solution to the wave scattering problem but also for testing the accuracy of all numerical methods. The wave function expansion method can provide a closed-form solution, and many two-dimensional scattering problems of plane SH waves are solved by this method, such as horizontally stratified surface layers, semi-cylindrical canyon [12], semi-elliptical
Mathematics 2020, 8, 1884; doi:10.3390/math8111884 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1884 2 of 11 canyon [13], multi-layered inhomogeneous semi-cylindrical canyon [14], semi-cylindrical hill [15], cylindrical canyon of the circular-arc cross section [2], cylindrical hill of the circular-arc cross section [16], elliptic-arc canyon in the corner [17], semi-elliptical hill [3,18], and semi-elliptical hill with a concentric elliptical tunnel [19]. From the SH wave scattering problem solved by the wave function expansion method mentioned above, it can be seen that the scattering problem of cylindrical terrain has experienced a process from semi-cylindrical valley to semi-cylindrical hill, circular-arc canyon to circular-arc hill. This is a process from simple to complex. Compared with the circular section, the elliptical section has more geometric generality and can be changed by changing its axial ratio, so the elliptical cylinder can effectively approach kinds of geometric shapes. However, the research on the scattering problem of elliptical cylindrical terrain is not as smooth as that of cylindrical terrain. Due to mathematical difficulties, the research only progressed from semi-elliptical canyon to semi-elliptical hills. The Mathieu function is an effective method to solve the elliptical boundary value problem. With the maturity of numerical calculation and computing ability, it has been widely used in physics, electromagnetism, and microwave technology. The Mathieu function addition theorem is the bridge of function transformation between different elliptic coordinate systems. The theorem can be divided into extra-domain theorems and intra-domain theorems, and have different expressions [20]. The former is usually used to study the scattering of multi-envelope elliptical structures, while the latter is usually used to study the scattering of multi-elliptical structures [21–24]. In order to improve the understanding of the influence of the elliptical cylinder topography, this article takes the elliptical-arc canyon under the action of incident plane SH (shear-horizontal) waves as the research object, and proposes a strict analytical solution. Through the application of the extra-domain Mathieu function addition theorems, the problem of transforming wave functions between elliptic coordinate systems is dealt with. Due to the geometric generality of the elliptical section, the model can be adapted to more practical problems. Additionally, the accurate solution proposed can be used as a new benchmark for numerical methods.
2. The Model The two-dimensional model to be studied is shown in Figure1. It represents an elastic, isotropic, and homogeneous half-space from which a portion of an elliptical cylinder was removed to form a canyon. The half-space defined by the horizontal boundary Γ and the elliptical-arc boundary L is the region to be analyzed. For the elliptical-arc canyon, the elliptical coordinate system can greatly simplify the free surface boundary conditions around the canyon, so it is more suitable. Therefore, in addition to the global elliptical coordinate system (ξ, η) defined at the center of the elliptical cylinder, another elliptical coordinate system ξq, ηq is defined at the midpoint of the horizontal boundary extension line above the canyon. The corresponding Cartesian coordinate systems (x, y) and xq, yq are established, in which the x axis, y axis, xq axis, and yq axis are along the η = 0, η = π/2, ηq = 0, and ηq = π/2, respectively. n o n o Therefore, the horizontal boundary Γ can be defined as xq, yq yq = 0 or ξq, ηq ηq = 0 ηq = π , ∪ and the elliptical-arc boundary L can be defined as (ξ, η) ξ = ξ with yq 0. The hypothesis parameters | 0 ≤ are as follows: The rigidity and the shear wave velocity of the medium: µ, β; The lengths of the semi-major axis and semi-minor axis of the ellipse where the elliptical-arc is located are: A, B; The focal length of the ellipse where the elliptical-arc is located is: c; Span of the canyon: 2l; Depth of the canyon: h; Distance between the origins of the two coordinate systems: d. MathematicsMathematics2020, 20208, 1884, 8, x FOR PEER REVIEW 3 of 113 of 11
Figure 1. The two-dimensional half-space model containing an elliptical-arc canyon. Figure 1. The two-dimensional half-space model containing an elliptical-arc canyon. An incident plane wave excites, with time dependence and frequency ω, at an angle α with An incident plane wave excites, with time dependence and frequency ω, at an angle α with respect to the positive direction of the xq -axis, while the direction of propagation is perpendicular respect to the positive direction of the x -axis, while the direction of propagation is perpendicular to to the z-axis, i.e.: q the z-axis, i.e.: αω+− i = ik(cos xqq y sin) a i t i uuez ik(0xq cos α+yq sin a) iω, t (1) uz = u0e − , (1) ω β where u0 is the amplitude of the incident wave and k is the shear wavenumber ( / ). It can be where u0 is the amplitude of the incident wave and k is the shear wavenumber (ω/β).− ω It can be represented in the elliptical coordinates system by Mathieu functions [13] (the time factor e itwas represented in the elliptical coordinates system by Mathieu functions [13] (the time factor e iωt was omitted here): − omitted here): ∞∞ im=+ηξα(1) m ηξα (1) uz 2 i cemqq (,) q Mc m (,) qq q ce m (,)2 q q i se mqq (,) q Ms m (,) qq q se m (,) q q, (2) X mm==01X i ∞ m (1) ∞ m (1) uz = 2 = i 22cem(ηq, qq=)Mcm (ξq, qq)ce= m(α, qq) + 2 i sem(ηq, qq)Msm (ξq, qq)sem(α, qq), (2) where qckqq/4, ccq 0.1 , and u0 1 . In addition, cem () and sem () are an even angular m=0 m=1 (1) Mathieu function and an odd angular Mathieu function, respectively, with order m; Mcm () and (1) 2 2 whereMsqqm =() c qarek /the4, cqeven= 0.1 andc, and oddu functions= 1. In addition,of the firscet mmodified( ) and se Mathieum( ) are anfunction even angularof order Mathieum, 0 · · respectively. (1) (1) function and an odd angular Mathieu function, respectively, with order m; Mcm ( ) and Msm ( ) are · y = 0 · the even andIn the odd absence functions of a canyon, of the the first incident modified wave Mathieu is reflected function on the of horizontal order m, free respectively. surface ( q ), ui u r uir+ Inand the the absence incident of wave a canyon, z and the the incident reflected wave wave is reflectedz interfere on to thegive horizontal the final wave free field surface z ( yinq = 0), uir+ i r i+r and thethe incidenthalf-space. wave z u canz and be identified the reflected as: wave uz interfere to give the final wave field uz in the half-space. ui+r can be identified as: ∞ z ir+ =+= i r m ηξα(1) uz uzz u4 i ce mqqmqqm (,) q Mc (,) q ce (,) q q. (3) m=0 X∞ Close to the canyon,i+r ithe incidentr wavem ui and the( 1reflected) wave u r are scattered and uz = uz + uz = 4 i cem(zηq, qq)Mcm (ξq, qq)cem(α, qqz). (3) diffracted by the elliptical-arc surface, mand=0 one more group of plane waves is generated. The waves s are scattered and diffracted outgoing waves uz , which can be expressed as the series of elliptic wave i r Closefunctions, to the as follows: canyon, the incident wave uz and the reflected wave uz are scattered and diffracted by the elliptical-arc surface, and∞∞ one more group of plane waves is generated. The waves are scattered us =+ a ces (,)ηξ q Mc(3) (,) q b se (,) ηξ q Ms (3) (,) q and diffracted outgoingzmmqqmqqmmqqmqq wavesu , which can be expressed as the series of elliptic, wave functions,(4) mm==01z as follows: a b Mc(3) () Ms(3) () where m and mX are the unknown coefficients to beX determined. m and m are the s ∞ (3) ∞ (3) even and odduz = functionsamce ofm (theηq, qthirdq)Mc kindm (ξ qMathieu, qq) + functionbmse mof( ηorderq, qq) Msm, mrespectively.(ξq, qq) , The total (4) m=0 total m=1 displacement field uz is the superposition of the incident wave, the reflected wave, and the scattered wave, i.e., uuuutotal=++ i r s . utotal should satisfy the following( 3boundary) conditions:(3) where am and bm are the unknownzzz coezfficientsz to be determined. Mc ( ) and Ms ( ) are the even m · m · and oddTraction functions free at ofΓ: the third kind Mathieu function of order m, respectively. The total displacement total field uz is the superposition of the incident wave, the reflected wave, and the scattered wave, total i r s total i.e., uz = uz + uz + uz. uz should satisfy the following boundary conditions: Traction free at Γ: total total ση z = ∂uz /∂ηq = 0, (5) q Γ Γ Mathematics 2020, 8, 1884 4 of 11
Traction free at L: total total σ = ∂uz /∂ξ = 0, (6) ξz L L where J2 = (cosh 2ξ cos 2η)/2. 0 − s Since dsem(ηq, qq)/dηq , 0, the unknown coefficients bm must be equal to 0. Then, u can ηq=0, π z be expressed as: ± X s ∞ (3) uz = amcem(ηq, qq)Mcm (ξq, qq). (7) m=0
3. Application of Mathieu Function Addition Theorem
total To adapt to the boundary conditions, the total displacement field uz in the local coordinate (ξq, ηq) must be represented by the global coordinate (ξ, η). The Mathieu function addition theorem is an effective tool for transforming functions between different elliptical coordinate systems. The Mathieu function addition theorem can be divided into two types: intra-domain and extra-domain. According to [20,22], the extra-domain Mathieu function addition theorem is adopted, which is expressed as follows:
X∞ X∞ ( ) (t)( ) = ( ) (t)( ) + ( ) (t)( ) cem ηq, qq Mcm ξq, qq WEelmcel η, q Mcl ξ, q WOelmsel η, q Msl ξ, q , (8) l=0 l=1
X∞ X∞ ( ) (t)( ) = ( ) (t)( ) + ( ) (t)( ) sem ηq, qq Msm ξq, qq WEolmcel η, q Mcl ξ, q WOolmsel η, q Msl ξ, q , (9) l=0 l=1 where:
l m P P j+p j = πi − ∞ ∞ ( ) m( ) l ( )( ( ) + ( ) ( ) +) WEelm Ne (q) i Aj qq Ap q Jp j kd cos ϕ− 1 Jp+j kd cos ϕ , l j=0 p=0 − − − l m P P j+p j = πi − ∞ ∞ ( ) m( ) l ( )( ( ) ( ) ( ) +) WOelm No (q) i Aj qq Bp q Jp j kd sin ϕ− 1 Jp+j kd sin ϕ , − l j=0 p=0 − − − − l m P P j+p j = πi − ∞ ∞ ( ) m( ) l ( )( ( ) + ( ) ( ) +) WEolm Ne (q) i Bj qq Ap q Jp j kd sin ϕ− 1 Jp+j kd sin ϕ , l j=0 p=0 − − − l m P P j+p j = πi − ∞ ∞ ( ) m( ) l ( )( ( ) ( ) ( ) +) WOolm No (q) i Bj qq Bp q Jp j kd cos ϕ− 1 Jp+j kd cos ϕ , l j=0 p=0 − − − − + ϕ = jϕq + pϕ, where Jp+j( ) denotes Bessel function, ϕ is the angle between line ooq and the positive x axis, and ϕq · m l is the angle between line oqo and the positive xq axis. The coefficients Aj and Bp are the Fourier series coefficients of the Mathieu functions, and Nol and Nel are normalized constants. In addition, although WEelm, WOelm, WEolm, and WOolm are not all used in this article, they are listed for the sake of formula integrity. The theorem can be accurate in a region outside the circle C of radius d within the circle Cq of radius Aq, that is, the region outside the shadow in Figure2. By reasonably changing ϕ and ϕq, it is possible to obtain an elliptical-arc canyon at any angle to the horizontal boundary. Because it is tedious to discuss the elliptical-arc canyon at various angles, only the case where the long axis of the ellipse where the elliptical-arc canyon is located is parallel to the horizontal boundary is discussed in this article. The case is ϕ = 3π/2 and ϕq = π/2. Mathematics 2020, 8, 1884 5 of 11 Mathematics 2020, 8, x FOR PEER REVIEW 5 of 11
Figure 2. The scope of application of the extra-domain Mathieu function addition theorem. Figure 2. The scope of application of the extra-domain Mathieu function addition theorem. By applying the extra-domain Mathieu function addition theorem, we can obtain the expressions i+r s of uz Byand applyinguz in the the globalextra-domain coordinate Mathieu (ξ, η), function as follows: addition theorem, we can obtain the expressions + of uir and u s in the global coordinate (ξ,η ), as follows: z z + X∞ ∞∞X∞ (1) X ∞ (1) i r + m uz = 4ir=+i cem m(α, qαηξηξq) WEelmcel(η, q)Mc (1)(ξ, q) + WOelmsel(η, q)Ms (1) (ξ, q), (10) uzmqelmllelmll4 iceq (, ) WEceqMcq (,)l (,) WOseqMsq (,) (,)l , (10) m=0 ml==00l=0l = 11 X ∞∞XX ∞ s u∞s =+ a∞ WE ce(,)ηξ q Mc(3)(3) (,) q∞ WO se (,) ηξ q Ms (3)( (,)3) q . uz = zmam WEelm elmllcel(η, q)Mc (ξ, q) + WO elmllelmsel(η, q)Ms (ξ, q). (11) ml==00l l = 1 l m=0 l=0 l=1
4. Algebra Equations ui+ir+r us s The sum ofof waveswaves uz z andand uz zsatisfying satisfying the the stress-free stress-free boundary boundary condition condition Equation Equation (6) (6) can can be writtenbe written as: as: ∞∞" ∞ # P∞ P∞a WE( ce(,)ηξ) q( Mc3) (3)( ′′ ( ,)) q ++P∞ WO se (,)( ηξ q) Ms (3)(3) (( ,)= q ) = ammelmllWEelmcel η, q Mcl 0 ξ0, q00 WO elmllelmsel η, q Msl 0 ξ0, q m=0 ml==00l=0 l= l =1 1 " , # (12) P∞ m∞∞P∞ (1) P∞ ∞ (1) m( ) ( ) (1)( ) + ( ) (1) ( ) −+4i ce4mi ceα, q (,qαηξηξ q )WE WEelmcel ceη, (,)q Mc qMcl 0 ′′ξ (0, q ,) q WOseWOelmse (,)l η qMs, q Msl (0 ,)ξ q0, q −m=0 mql=0 elml l00l =1 elml l ml==00 l = 1 where ‘,’‘,’ onon thethe radialradial MathieuMathieu functionsfunctions designatesdesignates didifferentiationfferentiation withwith respectrespect toto ξξ.. MultiplyingMultiplying η ()π Equation (12)(12) withwithce nce(nη(,), q) qfor forn = n 0,1,2,3= 0,1,2,3…,... , integrating integrating over over the the interval interval( 0,0, 2π 2) gives: gives:
∞∞ X∞ (3)′′ξαξ=−X∞ m (1) am WE enm( Mc3) n(,)00 q 4 im ce m (,) q q WE enm Mc n ( (,)1) q . (13) amWEenmMc 0(ξ , q) = 4i cem(α, qq)WEenmMc 0(ξ , q). (13) mm==00n 0 − n 0 m=0 m=0 Equation (13) can be combined in an infinite matrix form as follows: Equation (13) can be combined in an infinite matrix= form as follows: [ Ax][ ] [ y] , (14) where [x] and [ y] are the unknown coefficients[A][x] = and[y], excitation column vector, respectively, (14)and can be given by: where [x] and [y] are the unknown coefficients and excitation column vector, respectively, and can be []= [ ]T given by: xaa12 am , (15) h iT [x] = a a a []= 1 [2 m ]T , (15) y yy12··· yn , (16) h iT ∞ [y] = y y yn , (16) = m αξ(1)′ 1 2 ··· with yiceqWEMcqnmqenmn 4(,) (,)0 . m=0 P∞ m (1) with yn = 4i cem(α, q[qA)WE] enmMcn 0(ξ0, q). The infinitem=0 matrix can be given by:
Mathematics 2020, 8, 1884 6 of 11
The infinite matrix [A] can be given by: A11 A12 A1m ··· A A A 21 22 2m [A] = . . ··· . , (17) . . .. . . . . . A An Anm n1 2 ··· (3) with Anm = WEenmMcn 0(ξ0, q). In theory, if the matrix equation of infinity can be solved, the exact solution of the problem can be obtained by substituting the solved unknowns into the expressions of stress and displacement. However, in practice, this is obviously impossible, so the matrix equation should be solved by truncation.
5. Model Verification Since boundary Γ is a traction-free boundary, the accuracy of the analysis method can be checked by the residual error of dimensionless stress on the canyon surface. According to Wong [13] and Yuan [2], if the maximum residual stress is limited to less than 5%, the accuracy of displacement can be guaranteed. In this paper, in the case of ETA = 1, the truncation term m = n = 50 is sufficient 3 to achieve the accuracy of 10− for the residual error of dimensionless stress. The following results are obtained when the maximum stress residue is reduced to less than 1% by using the appropriate truncation term m. To verify the model and the analytical solution, two past exact elastic solutions for a semi-cylindrical canyon and a semi-elliptical canyon are employed. It is convenient to define displacement amplitude q 2 2 uz as uz = Re(uz) + Im(uz) . Additionally, the results can be displayed with the dimensionless | | | | parameter ETA, which is defined as follows: ETA = 2A/λ. d = 0 can make the local coordinate system coincide with the global coordinate system, which makes the elliptical-arc canyon model simplified to a semi-elliptical canyon model. The results at this time can be compared with the classic results of the semi-ellipse model obtained by Wong [13]. Figure3 depicts the comparison between the results obtained under conditions ETA = 1.0, B/A = 0.7, and d = 0 and the results of a semi-elliptical canyon. The general trend is that our numerical results are consistent with those given by Wong [13]. Due to the geometric generality of the ellipse, when ξ0 tends to infinity, the semi-elliptical canyon tends to a semi-circular one. At this time, our numerical results can be compared with the results obtained by Trifunac [12]. The displacement amplitudes for ETA = 1.25, B/A = 0.7, and d = 0 when a = 0◦, 30◦, 60◦, and 90◦ are presented in Figure4. It can be clearly seen that their trends are almost the same. After careful comparison, it can be seen that the maximum value of the displacement difference is only very small: less than 1%. The above differences may be due to the different functions used in each problem on the one hand, and because the axis ratio of the ellipse is 0.99 instead of 1 on the other hand. The results in other cases are equally consistent but are not listed here for brevity. Thus, the accuracy and usability of the method proposed in this paper were verified. Mathematics 2020, 8, 1884 7 of 11 Mathematics 2020, 8, x FOR PEER REVIEW 7 of 11
FigureFigure 3. 3.Comparison Comparison with with the the case case of of a semi-ellipticala semi-elliptical canyon canyon by by Wong Wong [13 [13].].
FigureFigure 4. 4.Comparison Comparison with with the the case case of of a a semi-circular semi-circular canyon canyon by by Trifunac Trifunac [12 [12].]. 6. Numerical Results and Analysis 6. Numerical Results and Analysis In order to study the effects of local terrain irregularities, the amplitude of ground motion at In order to study the effects of local terrain irregularities, the amplitude of ground motion at different points is important. With the aid of numerical results, we can examine the magnification different points is important. With the aid of numerical results, we can examine the magnification mode of the SH-wave in the elliptical-arc canyon under the effect of terrain, so that we can have a mode of the SH-wave in the elliptical-arc canyon under the effect of terrain, so that we can have a clearer understanding of such problems. As expected, the level of magnification and terrain changes clearer understanding of such problems. As expected, the level of magnification and terrain changes in canyons or near canyons are strongly influenced. Since we assume that the excitation consists of in canyons or near canyons are strongly influenced. Since we assume that the excitation consists of an infinite SH-wave with an amplitude of 1, in the absence of a canyon, the amplitude of the ground an infinite SH-wave with an amplitude of 1, in the absence of a canyon, the amplitude of the ground motion is 2. However, in the case of canyons, due to the interference of scattered waves, the amplitude motion is 2. However, in the case of canyons, due to the interference of scattered waves, the amplitude of ground motion may be different from that of 2. of ground motion may be different from that of 2. Therefore, we would like to analyze and discuss the amplitude of ground motion in the elliptical-arc Therefore, we would like to analyze and discuss the amplitude of ground motion in the elliptical- canyonarc canyon or near or near the canyon. the canyon. The Th calculatione calculation accuracy accuracy of theof the Mathieu Mathieu function function calculation calculation program program arc canyon or near9 the canyon. The calculation accuracy of the Mathieu function calculation program5 in this paper is 10− -9and the program compiled with the addition formula has an accuracy of 10− -5. in this paper is 10-9 and the program compiled with the addition formula has an accuracy of 10-5 . The elliptical minor-to-major axes ratio, the depth-to-span of the elliptical-arc canyon, and incident angle The elliptical minor-to-major axes ratio, the depth-to-span of the elliptical-arc canyon, and incident are considered as the influencing factors on the ground motion. For easy calculation, the dimensionless angle are considered as the influencing factors on the ground motion. For easy calculation, the depth is h∗ = h/A, the dimensionless* distance between the origins of two coordinate systems d∗ = d/A dimensionless depth is hhA* =/ , the dimensionless distance between the origins of two coordinate (or d = B/A h ). systems∗ ddA* −=/∗ (or dBAh**=/ − ). systemsFirst atddA all,=/ we calculated (or dBAh=/ the displacement). amplitude of the elliptical-arc canyon that approximates First at all, we calculated the displacement amplitude of the elliptical-arc canyon that the circular-arc canyon (B/A = 0.99). Figure= 5 illustrates the displacement amplitudes for an approximates the circular-arc canyon ( BA/0.99= ). Figure 5 illustrates the displacement amplitudes almost-circular elliptical canyon (B/A = 0.99)= and the dimensionless parameter ETA equal to 1. for an almost-circular elliptical canyon ( BA/0.99= ) and the dimensionless parameter ETA equal to 1. Additionally, the figure shows three-dimensional plots of displacement amplitudes at different =° ° ° ° incident angles a =°0 , 30°, 60° , and 90° vs. the distance x/A on and around the canyon and the
Mathematics 2020, 8, 1884 8 of 11
Additionally, the figure shows three-dimensional plots of displacement amplitudes at different incident Mathematics 2020, 8, x FOR PEER REVIEW 8 of 11 angles a = 0◦, 30◦, 60◦, and 90◦ vs. the distance x/A on and around the canyon and the dimensionless distance d∗ varying within the range [0, 0.9]. It can be seen that when the waves are oblique incidence d* anddimensionless grazing incidence, distance the ground varying motion within near the the range illuminated [0, 0.9]. It surface can be of seen the canyonthat when terrain the waves produces are Mathematics 2020, 8, x FOR PEER REVIEW 8 of 11 violentoblique fluctuations. incidence and Obviously, grazing this incidence, is due to the the grou reflectionnd motion of the canyon’snear the illuminatedilluminated surfacesurface against of the canyon terrain produces violent fluctuations. Obviously, this is due to the reflection of the canyon’s the waves. In the area of x/Ad* > 0, due to the barrier effect of the canyon terrain, the ground motion is illuminateddimensionless surface distance against the varying waves. within In the the area range of [0,xA/0 0.9].> It ,can due be to seen the that barrier when effect the waves of the are canyon significantlyoblique weakened,incidence and and grazing the response incidence, is alsothe grou verynd smooth. motion Asnear the the value illuminated of d∗ increases, surface theof the barrier eterrain,ffect of thethe canyonground terrain motion still is exists,significantly but the weakened movement, ofand the the entire response surface istends also very to be smooth. gentle, and As thethe canyon* terrain produces violent fluctuations. Obviously, this is due to the reflection of the canyon’s amplitudevalueilluminated of d of theincreases, surface response against the becomes barrier the waves. smaller.effect In of the the area canyon of xA/0 terrain> , due still to theexists, barrier but effect the ofmovement the canyon of the entireterrain, surface the tends ground to motionbe gentle, is significantly and the amplitude weakened of, and the theresponse response becomes is also very smaller. smooth. As the value of d* increases, the barrier effect of the canyon terrain still exists, but the movement of the entire surface tends to be gentle, and the amplitude of the response becomes smaller.
= BA/0.99= = α =°α =°0 ° 30°° 60° FigureFigureFigure 5. 5.Displacement 5.Displacement Displacement amplitude amplitudeamplitude for for ETAETA =11 and and BBA//0.99A = 0.99 whenwhen when α = 00◦,, 3030◦, 60, 60◦, and, , and 90,◦ .and 90° ° . 90 . Then, we reduced B/A to 0.6. The three-dimensional plots of displacement amplitudes at different incidentThen, anglesThen, we awereduced= reduced0◦, 30 ◦B, 60/BA◦/ ,A andto to 0.6. 900.6.◦ Thevs.The the three-dimensionalthree-dimensional distance x/A on plots andplots aroundof ofdisplacement displacement the canyon amplitudes dimensionlessamplitudes at at =° ° ° ° distancedifferentdifferentd incident∗ varying incident angles within angles a the=°a 0 range0, ,30 30° , [0, , 6060 0.5]° ,, andand are shown9090° vs. vs. thein the Figuredistance distance6 .x/ ItA x can/onA onand be and around seen around that the thecanyon the change canyon * trenddimensionlessdimensionless of the entire distance surfacedistance d at* d thisvarying varying time within iswithin roughly thethe range the same [0, [0, 0.5] 0.5] as are whenare shown shownB/ inA Figurein= Figure0.99 6., butIt 6.can theIt becan displacement seen be thatseen that BA/0.99= amplitudethe changethe change decreases trend trend of the obviously.of the entire entire surface surface at at thisthis time isis roughly roughly the the same same as whenas when BA/0.99= , but ,the but the displacement amplitude decreases obviously. displacement amplitude decreases obviously.
= α =° ° ° Figure 6. Displacement amplitude for ETA = 1 and BA/0.6 when 0 , 30 , 60 , and Figure° 6. Displacement amplitude for ETA = 1 and B/A = 0.6 when α = 0◦, 30◦, 60◦, and 90◦. 90 . = α =° ° ° Figure 6. Displacement amplitude for ETA = 1 and BA/0.6 when 0 , 30 , 60 , and ° 90 .
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To further demonstrate the novelty of this model and method, examples of different elliptical- arc ToMathematicscanyons further 2020with demonstrate, 8, thex FOR same PEER the REVIEWdepth-to-span novelty of this ratio model are pr andesented. method, As examplesshown inof Figure different 7, although elliptical-arc9 of 11 these canyonscanyons with have the the same same depth-to-span depth-to-span ratio ratio, are they presented. belong to As different shown ellipses. in Figure The7, although scattering these of SH To further demonstrate the novelty of this model and method, examples of different elliptical- canyonswaves by have these the canyons same depth-to-span with the same ratio, depth-to-span they belong ratio to and diff erentbelong ellipses.ing to different The scattering ellipses and of SH the arc canyons with the same depth-to-span ratio are presented. As shown in Figure 7, although these wavesdifferences by these between canyons them with are the exactly same depth-to-spanwhat can be solved ratio andby the belonging methodto and diff modelerent ellipsesin this paper. and the canyons have the same depth-to-span ratio, they belong to different ellipses. The scattering of SH differences between them are exactly what can be solved by the method and model in this paper. waves by these canyons with the same depth-to-span ratio and belonging to different ellipses and the differences between them are exactly what can be solved by the method and model in this paper.
FigureFigure 7. 7.Di Differentfferent elliptical-arc elliptical-arc canyons canyons with with the the same same depth-to-span depth-to-span ratio. ratio. Figure 7. Different elliptical-arc canyons with the same depth-to-span ratio. SinceSince the the ellipse ellipse where where the elliptical-arcthe elliptical-arc canyon canyon is located is located is different, is different, the value the of valueETA is of di ffETAerent is Since the ellipse where the elliptical-arc canyon is located is different, the value of ETA is underdifferent the sameunder wavenumber, the same sowavenumber, we further define so we the dimensionlessfurther define parameter the dimensionlessETA∗ = 2l/ λparameterand the different under the same wavenumber, so we further define the dimensionless parameter depth-to-spanETA* = 2/ l λ ratio and thel∗ = depth-to-spanh/2l. As shown ratio in Figurelhl* = /28, under. As shown the conditions in Figure of 8, lunder∗ = 0.3 theand conditionsETA∗ = 1 of, ETA* = 2/ l λ and the depth-to-span ratio lhl* = /2 . As shown in Figure 8, under the conditions of the* comparison= of displacement* = amplitudes of canyons with an axis ratio of 0.99, 0.8, and 0.6 are l l*0.3= 0.3 and ETAETA* =11, the comparison of displacement amplitudes of canyons with an axis ratio of presented. It and should be noted, the thatcomparison when B of/ Adisplacement= 0.6, the elliptical-arc amplitudes of canyon canyons is with exactly an axis a semi-elliptical ratio of 0.99,0.99, 0.8, 0.8, and and 0.6 0.6 are are presented. presented. It should be be noted noted that that when when BA /=0.6BA/=0.6, the, elliptical-arcthe elliptical-arc canyon canyon is is canyon, and when B/A = 0.99, the elliptical-arc canyon is an almost circular-arc canyon. It can be exactlyexactly a semi-elliptical a semi-elliptical canyon, canyon, and whenwhen BABA// =0.99 =0.99, the, the elliptical-arc elliptical-arc canyon canyon is an almostis an almost circular- circular- obviouslyarc arccanyon. canyon. found It canIt that can be thebe obviously obviously variation found trends thatthat of thethe the displacementvari variationation trends trends amplitude of the of thedisplacement displacement under diff amplitudeerent amplitude axial under ratios under are substantiallydifferentdifferent axial consistentaxial ratios ratios are atare thesubstantially substantially same incident consistentconsistent angle. at at However,the the same same incident the incident maximum angle. angle. However, value However, of the the maximum displacementthe maximum amplitudevaluevalue of theof and the displacement thedisplacement position amplitude ofamplitude the maximum andand the the positi valueposition areon of ofthe obviously the maximum maximum di valuefferent, value are which obviously are obviously is obvious different, different, when * = α ° = α ° = = * αwhich90which◦ ,is and obviousis obvious not so when obviouswhen α =90=90 when° ,, andα notnot0 ◦so so. obvious This obvious can when bewhen observed =0α =0. This° . at This canl∗ becan0.2 observed beeven observed more at l obviously, at0.2 l = 0.2 aseven showneven more more in Figureobviously, obviously,9. Therefore, as as shown shown the in diFigurefference 9. 9. Theref Theref betweenore,ore, the the the difference scattering difference between problems between the scattering causedthe scattering by problems the ellipticalproblems caused by the elliptical arc canyon with the same depth-to-span ratio and different axis ratio is worth arccaused canyon bywith the elliptical the same arc depth-to-span canyon with ratiothe same and didepth-to-spanfferent axis ratio ratio is and worth different noting. axis ratio is worth noting. noting.
Figure 8. Comparison of displacement amplitudes of canyons belonging to ellipses with different Figure 8. Comparison of displacement amplitudes of canyons belonging to ellipses with different axial axial ratios when l* = 0.3 and ETA* = 1 . ratios when l = 0.3 and ETA = 1. ∗ ∗ Figure 8. Comparison of displacement amplitudes of canyons belonging to ellipses with different axial ratios when l* = 0.3 and ETA* = 1 .
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Mathematics 2020, 8, x FOR PEER REVIEW 10 of 11
FigureFigure 9. Comparison 9. Comparison of displacementof displacement amplitudes amplitudes of of canyons canyons belonging belonging toto ellipsesellipses withwith didifferentfferent axial axial ratios when l* = 0.2 and ETA* = 1 . ratios when l∗ = 0.2 and ETA∗ = 1.
7. Conclusions7. Conclusions An analyticalAn analytical solution solution using using the the method method of wave wave-function-function series series expansion expansion for the for scattering the scattering of of steady-statesteady-state SH wavesSH waves in half in half space space with with an elliptical-arc an elliptical-arc canyon canyon was was presented. presented. In theIn the process process of solvingof solving the problem, the extra-domain Mathieu function addition theorem was applied. The the problem, the extra-domain Mathieu function addition theorem was applied. The elliptical-arc elliptical-arc canyon was degraded into a semi-elliptical canyon and an almost semi-circular canyon canyonby wasutilizing degraded the ellipse’s into a geometric semi-elliptical characteristics. canyon and The anresults almost were semi-circular compared with canyon those by of utilizing Wong the ellipse’s[13] geometricand Trifunac characteristics. [12], respectively, The and results the werevalidity compared of the method with those was verified. of Wong By [13 comparing] and Trifunac the [12], respectively,displacement and theamplitudes validity caused of the by method the elliptical-arc was verified. canyon By under comparing the same the depth-to-span displacement ratio amplitudes and causeddifferent by the axial elliptical-arc ratios, it was canyon found under that the maxi samemum depth-to-span value of the ratio displacement and different amplitude axial ratios,and the it was foundposition that the of maximumthe maximum value value of the are displacement obviously different. amplitude This andwas theobvious position when of the maximumSH-wave is value are obviouslyincident horizontally. different. ThisAlthough was obviousthe truncation when of the the SH-wave infinite equation is incident is inevitable, horizontally. the accuracy Although of the truncationthe numerical of the results infinite was equation carefully is checked. inevitable, In additi theon, accuracy based on of the the method numerical proposed results in this was paper, carefully checked.it can In be addition, further used based to study on the the method more practical proposed problems in this paper,of shallow it can valleys be further or shallow used hills to study on the incident plane SH waves. more practical problems of shallow valleys or shallow hills on incident plane SH waves. Author Contributions: Data curation, H.Q. and F.C.; Formal analysis, F.C.; Funding acquisition, J.G.; Software, Author Contributions: Data curation, H.Q. and F.C.; Formal analysis, F.C.; Funding acquisition, J.G.; Software, F.C. and R.Y.;F.C. Writing—originaland R.Y.; Writing—original draft, F.C. draft, All authorsF.C. All haveauthors read have and read agreed and toagr theeed published to the published version version of the manuscript.of the manuscript. Funding: This research was funded by the Fundamental Research Funds for Central Universities, grant number 3072019CF0205.Funding: This research was funded by the Fundamental Research Funds for Central Universities, grant number 3072019CF0205. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. References References
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