Diffraction Of Plane SH-waves Research Articles

Scattering of plane SH waves by a semi-circular discontinuity in an elastic half-plane

A curved discontinuity near the ground can cause the scattering of seismic waves and significantly affect the ground and underground motions, posing some challenges in seismic design. In this paper, a series solution for the scattering and diffraction of plane SH waves from a semi-circular discontinuity in an elastic half-plane is first presented using the wave function expansion method, aiming to reveal the effects of the discontinuity properties on wave scattering and seismic motion. Based on the correspondence principle of viscoelasticity, the influence of the damping ratio of the medium on the seismic motion is also discussed. Finally, by combining the series solution and the inverse Fourier transform, the ground and underground motions induced by a transient plane SH wave are analyzed by exhibiting the wave field snapshots at different times and the waveform curves of different measurement points on the ground. The results indicate that both the stiffness of the discontinuity and the damping ratio of the medium have a great influence on the propagation of seismic waves and play an important role in seismic calculations.

Effects of a V-shaped canyon with a circular underground structure on surface ground motions under SH wave propagation

The two-dimensional scattering and diffraction of plane SH waves by a symmetrical V-shaped canyon with a circular underground structure in a half-space is studied by using the wavefunction expansion method. The underground structure involved covers three common types: a cavity, a lined tunnel and a geological inclusion. By combining a domain decomposition strategy and the method of images, the wavefields in corresponding subregions are constructed in terms of infinite series of wavefunctions with unknown complex coefficients. The unknown coefficients are obtained by applying the Graf's addition formula and solving a matrix that represents the boundary conditions along subregion interfaces. To show effects of the surficial canyon and underground structure combined irregularity on ground motions, a parametric study is executed in both the frequency and time domain. A zone of ground motion amplification is found at the vicinity of the corner of the canyon with a lined tunnel buried nearby.

A note on response of tunnels to incident SH-waves near hillsides

In this study, diffraction of plane SH waves by a cylindrical tunnel embedded in homogeneous, isotropic and linear elastic quarter-space is investigated. Analytical solution techniques are used to solve the two dimensional wave propagation problem. When the excitation is assumed to be harmonic, the governing equation would be the Helmholtz equation. By applying separation of variables method to Helmholtz equation for cylindrical coordinates, general solutions are obtained in terms of Fourier-Bessel series. Unknown complex constants of the Fourier-Bessel series are to be determined from boundary conditions. Boundary conditions at inner and outer side of the tunnel are satisfied directly since they are defined in cylindrical coordinates. Stress-free boundary conditions at the ground and the hillside surfaces are satisfied in closed form via imaging method and addition theorems. Numerical examples are compared with earlier studies and the effect of the tunnel is discussed.

Dynamic Response Analysis of Periodic Alluvial Valleys under Incident Plane SH-Waves

Combined with the region-matching technique, a periodic indirect boundary element method (PIBEM) is developed to investigate the scattering and diffraction of plane SH-waves by periodic alluvial valleys embedded in a layered half-space. By introducing Green’s functions for equivalent uniformly distributed loads acting on an inclined line to simulate the scattered fields outside the periodic valleys, the discretization effort is reduced to a single valley. And by using only a single irregularity as the computation cell to solve wave scattering problem of periodic irregularities (an infinite number of irregularities), the presented new method has the merits of high accuracy, low computational cost in CPU time, and low memory requirement. The implementation of the method is described in detail, and the accuracy of the method is verified by comparing its degenerated results with the analytical results of a single semi-elliptic valley, the semi-analytical results of two valleys and the IBEM results of multiple valleys. Numerical simulations are first performed in the frequency domain and then in the time domain using the inverse FFT. In addition, the time domain results reinforce the validity and reliability of the new method. The presented PIBEM can be used to study the various wave scattering problems of plane SH-waves by periodic topographies in a layered half-space.

Scattering and diffraction of plane SH-waves by periodically distributed canyons

A new method is presented to study the scattering and diffraction of plane SH-waves by periodically distributed canyons in a layered half-space. This method uses the indirect boundary element method combined with Green’s functions of uniformly distributed loads acting on periodically distributed inclined lines. The periodicity feature of the canyons is exploited to limit the discretization effort to a single canyon, which avoids errors induced by the truncation of the infinite boundary, and the computational complexity and the demand on memory can be significantly reduced. Furthermore, the total wave fields are decomposed into the free field and scattered field in the process of calculation, which means that the method has definite physical meaning. The implementation of the method is described in detail and its accuracy is verified. Parametric studies are performed in the frequency domain by taking periodically distributed canyons of semi-circular and semi-elliptic cross-sections as examples. Numerical results show that the dynamic responses of periodically distributed canyons can be quite different from those for a single canyon and significant dynamic interactions exist between the canyons.

Diffraction of Plane SH Waves by a Cylindrical Cavity in an Infinite Wedge

Diffraction of plane harmonic SH waves by a cylindrical cavity in a homogeneous, isotropic and linear elastic infinite wedge is investigated. Analytical closed form solution is obtained by utilizing Wave Function Expansion Method and Image Technique. Governing equation of this two dimensional steady-state wave motion problem is the Helmholtz equation. Helmholtz equation for polar coordinates is solved by using Separation of Variables Method. Displacement fields are expressed in terms of Fourier-Bessel series in complex form. Complex coefficients of Fourier-Bessel series are to be determined from boundary conditions. Zero-stress boundary condition at cavity inner surface is satisfied directly since polar coordinate systems are used. Free stress conditions at flat surfaces of the infinite wedge are satisfied in closed form via imaging method. In this technique, imaginary cavities symmetric with respect to flat surfaces are considered. With this consideration, scattered waves from imaginary cavities would represent reflections from flat surfaces so that stress free boundary condition on flat surfaces would be automatically satisfied. This technique would generate two additional displacement fields with two additional set of unknown complex Fourier-Bessel constants. These unknowns could be determined from boundary conditions of imaginary cavities. The cylindrical scattered waves from each cavity are defined in different polar coordinate systems. Addition Theorems are used for adequate coordinate transformation. Since Image Technique is used, to maintain angular symmetry, analytical solutions are obtained for a specific wedge whose internal angle is 120 degrees. Also, direction of incident waves has to be either horizontal or on the symmetry line of the wedge and cavity center has to be on the symmetry line. In numerical results, parameters in length dimension are normalized with respect to cavity radius.

Scattering of SH waves induced by a symmetrical V-shaped canyon: a unified analytical solution

This paper reports a series solution of wave functions for two-dimensional scattering and diffraction of plane SH waves induced by a symmetrical V-shaped canyon with different shape ratios. A half-space with a symmetrical V-shaped canyon is divided into two sub-regions by using a circular-arc auxiliary boundary. The two sub-regions are represented by global and local cylindrical coordinate systems, respectively. In each coordinate system, the wave field satisfying the Helmholtz equation is represented by the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then, the two wave fields are described in the local coordinate system using the Graf addition theorem. Finally, the unknown coefficients are sought by satisfying the continuity conditions of the auxiliary boundary. To consider the phase characteristics of the wave scattering, a parametric analysis is carried out in the time domain by assuming an incident signal of the Ricker type. Surface and subsurface transient responses demonstrate the characteristics and mechanisms of wave propagating and scattering.

Scattering of SH waves induced by a non-symmetrical V-shaped canyon

SUMMARY Topographies have significant effects on seismic waves. The wavefunction expansion method has been frequently employed to study the topographic effect because this method can reveal the physics of the wave scattering and can testify the accuracy of numerical methods. The 2-D scattering and diffraction of plane SH waves induced by a non-symmetrical V-shaped canyon is examined here by using the wavefunction expansion method. Through a new domain decomposition strategy, the half-space containing a V-shaped canyon is divided into three subregions. Then the wavefield for every subregion is constructed in terms of an infinite series of wavefunctions with unknown coefficients in three coordinate systems, respectively. After that, three wavefields are all represented in the same coordinate system using the Graf addition theorem. The unknown coefficients are obtained by satisfying the continuity conditions of the auxiliary boundary. The proposed series solution of wavefunctions for a non-symmetrical V-shaped canyon can be reduced to a symmetrical case. Results in this study for the symmetrical case agree very well with those in the published literature. To show the effects of the non-symmetrical-geometry topography on the surface ground motion, a parametric study is carried out in the frequency domain. Surface and subsurface transient responses in the time domain demonstrate the phenomenon of wave propagating and scattering. Finally, the proposed model is successfully applied to two idealized cross sections of Pacoima canyon and Feitsui canyon.

Diffraction of plane SH waves by a semi-circular cavity in half-space

This paper presents a closed-form solution for diffraction of plane SH waves by a semi-circular cavity in half-space by using wave function expansion method. Accuracy of the solution is checked by the displacement residual and stress residual along the boundaries. Numerical results show that there are notable differences for response amplitudes between a semi-circular cavity and a whole-circular cavity in a half-space.

On the relationship between the stress intensity factors and the far field of SH-waves diffracted on interface cracks

By using the Wiener-Hopf method, we solve the problem of diffraction of plane SH-waves on semiinfinite interface cracks and deduce the relationships between the radiation field in fixed directions and the stress intensity factors. The dependence of the stress intensity factors on the field of displacements scattered in the plane perpendicular to the crack tip is analyzed. It is shown that the ratio of the stress intensity factors can be represented via the ratio of the scattered far fields measured for various angles and frequencies of sounding.

Diffraction of anti-plane SH waves by a semi-circular cylindrical hill with an inside concentric semi-circular tunnel

A closed-form analytic solution of two-dimensional scattering and diffraction of plane SH waves by a semicylindrical hill with a semi-cylindrical concentric tunnel inside an elastic half-space is presented using the cylindrical wave functions expansion method. The solution is reduced to solving a set of infinite linear algebraic equations. Fourier expansion theorem with the form of complex exponential function and cosine function is used. Numerical solutions are obtained by truncation of the infinite equations. The accuracy of the presented numerical results is carefully verified.

Scattering of SH Waves by Fixed-Rigid Nearly Circular Cylinder

A solution for the two-dimensional scattering and diffraction of plane SH waves by fixed-rigid cylinder of arbitrary shape in an elastic infinite medium is presented. The wave field for arbitary geometry in this paper is computed numericaly by the method weighted residues (moment method). Dynamic stress concentration factors around the arbitrary obstacles are obtained with respect to various dimensionless wave numbers. When the cavity is circular, results obtained from the analytical solutions are the same.

Diffraction of SH-Waves by Subsurface Inclusions of Arbitrary Shape

The weighted residual method was applied to the problem of scattering and diffraction of plane SH-waves by an underground inclusion of a different elastic medium and of arbitrary shape in a two-dimensional half-space. Results from the application of the weighted residual approach to the case of an underground circular inclusion closely approximate those obtained using the closed-form solution for underground inclusions. To demonstrate the versatility of the method, it was applied to inclusions of elliptical and square shape. It was shown that significant ground-motion amplification, with respect to the amplitude of the incident waves, occurred due to the presence of an inclusion or cavity. Amplifications were dependent on the size, shape, and depth of the inclusion as well as the frequency and angle of incidence of the incident waves. Amplification profiles for the lower-frequency incident waves were simple with peak values that did not differ significantly from 2, the value expected on the surface of a uniform half-space. As the frequency is increased, amplification profiles become more complicated with worst-case values exceeding 7 for elliptical-shaped inclusions.

Application of the weighted residual method to diffraction by 2-D canyons of arbitrary shape: I. Incident SH waves

A solution for the two-dimensional scattering and diffraction of plane SH waves by canyons of arbitrary shape in an elastic half space is presented. The wave field for arbitrary geometry in this paper is computed numerically by the method of weighted residues (moment method). The wave displacement field computed by the present residual method for the case of a semi-circular canyon was shown to agree analytically and numerically with that computed by the exact closed form series solution. The same observations about ground amplifications, their dependence on frequencies and orientations of the incident waves, can be stated here for canyons of arbitrary shape as previously made for circular canyons.

Scattering of Plane Sh-Waves By A Semi-Parabolic Cylindrical Canyon In an Elastic Half-Space

The 2-D scattering and diffraction of plane SH-waves by a semi-parabolic cylindrical canyon in an elastic half-space is analysed for the cases of SH-waves coming from the side of the half-space towards the front of the canyon. the exact analytic series solution of the problem is presented. It shows that the surface topography can have significant effects on the incident waves, which become prominent at high frequencies, or when the wavelengths of the incident waves are short compared with the focal length of the canyon. Higher frequencies will also lead to greater complexity of the pattern of surface displacement amplitudes involving abrupt changes of amplification or deamplification from one point to another.

Scattering and diffraction of plane SH-waves by two-dimensional dimensional inhomogeneities: Part I

Scattering and diffraction of plane SH-waves by two-dimensional dimensional inhomogeneities: Part I

Scattering and diffraction of plane SH-waves by two-dimensional inhomogeneities: Part I

A wave expansion technique is presented for the analysis of scattered and diffracted plane SH-waves by two-dimensional inhomogeneities. Comparison of this approximate method with exact series solutions suggests that the proposed method may become a useful tool in predicting the local site amplification patterns in earthquake engineering applications which deal with the estimation of site specific spectra and in the problems involving earthquake response of long structures with multiple supports.

Diffraction of plane sh waves in a half‐space

AbstractScattering of antiplane shear waves (SH) in two dimensions by surface and near‐surface defects in a homogeneous, isotropic elastic semi‐infinite medium has been studied. Attention has been focused here in the range of medium to long wavelengths. A combined finite element and analytical technique has been used to study the problems of scattering by semi‐circular and triangular canyons. The results for the former case are compared with the known exact solution and those for the latter case are compared with some available approximate solutions. Finally a problem of multiple scattering by a triangular canyon and a nearby circular tunnel is studied. Numerical results are presented showing the effects of multiple scattering and different angles of incidence. These results are of interest in earthquake engineering.

Scattering of plane sh waves by a semi‐elliptical canyon

AbstractIn this paper we analyse the two‐dimensional scattering and diffraction of plane SH waves by a semi‐elliptical canyon. The exact series solution of the problem, for general angle of incidence of the plane SH waves, has been used to examine the dependence of surface amplifications inside and near the canyon. The nature of ground motion has been found to depend on two key parameters: (a) The angle of incidence. (b) The ratio of the canyon width to the wave length of incident SH waves.For short incident waves surface displacement amplitudes change rapidly from one point to another, while for the long waves and shallow canyons displacement amplitudes display only minor departure from the uniform half‐space amplification of 2. For shallow canyons and long incident waves, the angle of incidence introduces only minor changes into the overall behaviour of surface amplitudes. For deep canyons and nearly grazing incidences, a prominent shadow zone is realized behind the canyon.

Scattering of plane sh waves by a semi‐cylindrical canyon

AbstractThe two‐dimensional scattering and diffraction of plane SH waves by a semi‐cylindrical canyon is analysed for a general angle of wave incidence. The closed‐form solution of the problem shows that the surface topography can have prominent effects on incident waves only when the wavelengths of incident motion are short compared to the radius of a canyon. The surface amplification of displacement amplitudes around and in the canyon changes rapidly from one point to another, but the amplification is always less than 2. The over‐all trends of amplification pattern are determined by two principal parameters: (1) γ, the angle of incidence of plane SH waves, and (2) η, the ratio of radius of the canyon to one‐half wave length of incident waves. The higher η leads to greater complexity of the pattern of surface displacement amplitudes characterized by more abrupt changes of amplification from one point to another, while γ mainly determines the over‐all trends of displacement amplitudes. For grazing and nearly grazing incidences, for example, a strong shadow zone is developed behind the canyon.The qualitative analysis of the topographic effects on the Pacoima Dam accelerogram,1 based on the semi‐cylindrical canyon, suggests that this strong‐motion record was not seriously affected by surface topography of the recording site.