Reflection of Oblique Incident Waves by Breakwaters with Partially-Perforated Wall

The reflection of oblique incident waves from breakwaters with a partially-perforated front wall is investigated. The fluid domain is divided into two sub-domains and the eigenfunction expansion method is applied to expand velocity potentials in each domain. In the eigen-expansion of the velocity potential, evanescent waves are included. Numerical results of the present model are compared with experimental data. The effect of porosity, the relative chamber width, the relative water depth in the wave absorbing chamber and the water depth in front of the structure are discussed.

The force of oblique incident wave on the breakwater with a partially perforated wall

Wave forces induced by the interaction between the oblique incident wave and the breakwater with a partially perforated front wall is investigated. The fluid domain is divided into two sub-domains and the eigen-function expansion method is applied to expanding velocity potentials in each domain. In the eigen-expansion of the velocity potential, evanescent waves are included. Numerical results of the present model are compared with other theories and a good agreement can be found between them. Experimental data have been compared with the present theoretical results. The effect of the traverse wall on wave forces has been discussed in detail. On the basis of the linear wave theory, it is shown that in the range Of engineering practice, the incident angle of wave has small influence on wave forces on the unit length of perforated caisson.

Seismic response analysis of GRPS embankment under oblique incident P wave

In order to investigate the seismic performance of geosynthetic reinforced and pile supported(GRPS) embankment under seismic loads, an input method for three-dimensional oblique incidence of P wave was proposed. This method is based on the explicit finite element method while considering the viscous-spring artificial boundary(VSAB) condition. Using the proposed method, a numerical study was conducted, and the influence of oblique incidence on the seismic response of GRPS embankment under the oblique incident P waves was analyzed. The results indicate that in comparison with vertical incidence, the oblique incidence can significantly increase the displacement, velocity and acceleration of key locations in the GRPS embankment. The existence of geosynthetics can alleviate the impact of seismic load on the response of the embankment to a certain degree. Moreover, the number of reinforcement layers and modulus of geogrid also greatly influence the seismic performance of GRPS embankment.

First-order approximate analytical expressions of oblique incident elastic wave at an interface of porous media saturated with a nonviscous fluid

The analysis technology of Amplitude Variation with Offset(AVO)is one of the important methods for oil and gas reservoir prediction.Zoeppritz equation and its approximations are the theoretical basis of AVO analysis,which assumes that the upper and lower media of a horizontal interface are single-phase media.Limited by this assumption,AVO analysis has limited prediction and identification accuracy for complex porous reservoirs.In view of this,the first-order approximate analytical expressions of oblique elastic wave at an interface of porous media are derived.Firstly,the incident and scattering characteristics of various waves at the interface of porous media are analyzed,and the displacement vectors generated by these elastic waves are described by exponential function.Secondly,the kinematic and dynamic boundary conditions at the interface of porous media are discussed.Thirdly,by substituting the displacement vectors of incident and scattered waves into boundary conditions,the exact analytical equation is derived.Then,considering the symmetry of scattering matrix in the equation,the exact analytical expressions of each scattered wave are obtained.Furthermore,under the assumptions of small incident angle,weak elasticity at an interface of porous media,and ignoring the second-and higherorder terms,the first-order approximate analytical expressions are derived.Establishing a model of sandstone porous media with different porosity in upper and lower media,the correctness of the approximate analytical expressions is verified,and the elastic wave response characteristics of lithology and pore fluids are analyzed.

2.5-dimension soil seismic response to oblique incident waves based on exact free-field solution

With the three dimensional(3D)oblique incident waves exactly determined for the free field,the soil seismic responses in both frequency and time domains are studied by the 2.5 dimension(2.5D)finite/infinite element method.First,the free-field responses in frequency domain are solved exactly for 3D arbitrary incident P and SV waves,which requires no coordinate conversion or extra effort for SV waves with super-critical incident angles.Next,the earthquake spectra are incorporated by the concept of equivalent seismic forces on the near-field boundary,based only on the displacements input derived for unit ground accelerations of each frequency using the 2.5D approach.For the asymmetric 2.5D finite/infinite element model adopted,the procedure for soil seismic analysis is presented.The solutions computed by the proposed method are verified against those of Wolf’s and de Barros and Luco’s and for inversely calculated ground motions.Of interest is that abrupt variation in soil response occurs around the critical angle on the wave propagation plane for SV waves.In addition,the horizontal displacements attenuate with increasing horizontal incident angle,while the longitudinal ones increase inversely for 3D incident P and SV waves.

Seismic analysis of underground tunnels subjected to 3D oblique incident P and SV waves by 2.5D approach

This paper presents,for the first time,the consideration of three-dimensional(3D)oblique incident P and SV waves in calculating the 3D seismic response of a lined tunnel embedded in a half-space by the 2.5D finite/infinite element method(FIEM).Firstly,the applicability of the 2.5D FIEM for 3D seismic analysis is summarized.With the exact solutions obtained for the free field in the Appendix,the equivalent seismic forces are rationally computed for the near-field boundary,considering the horizontal and vertical excitations of the Chi-Chi Earthquake.By performing seismic analysis of the half space embedded with a tunnel using the 2.5D FIEM,the time-domain responses of the tunnel are obtained.The accuracy of the present solutions is verified against those of de Barros and Luco.Conclusions drawn from the parametric study include:(1)Stress concentration for the principal stress under oblique incident seismic waves occurs at the polar angles of 0(vault),90,180(inverted arch),and 270of the lining wall.(2)The vault and inverted arch are the weakest parts of the tunnel during earthquakes.(3)The accelerations of the tunnel during earthquakes can be regarded as of the rigid body type.(4)The responses of the tunnel lining caused by SV waves of an earthquake are much more critical than those by P waves.(5)For arbitrary seismic waves,the maximum longitudinal acceleration azmax is of the same order of magnitude as the maximum horizontal acceleration axmax.

Acoustic evaluation of cementing quality using obliquely incident ultrasonic signals

Ultrasonic cement bond logging is a widely used method for evaluating cementing quality. Conventional ultrasonic cement bond logging uses vertical incidence and cannot accurately evaluate lightweight cement bonding. Oblique incidence is a new technology for evaluating cement quality with improved accuracy for lightweight cements. In this study, we simulated models of acoustic impedance of cement and cementing quality using ultrasonic oblique incidence, and we obtained the relation between cementing quality, acoustic impedance of cement, and the acoustic attenuation coeffi cient of the A0-mode and S0-mode Lamb waves. Then, we simulated models of different cement thickness and we obtained the relation between cement thickness and the time difference of the arrival between the A0 and A0′ modes.

REFLECTION OF OBLIQUE INCIDENT WAVES BY PERFORATED CAISSONS WITH TRAVERSE WALL

The interaction of oblique incident waves with infinite number of perforated caissons is investigated. The fluid domain is divided into infinite sub-domains by the caissons, and eigen-function expansion is applied to expand velocity potentials in each domain. A phase relation is introduced for wave oscillation in each caisson, and the structure geometry is considered in constructing the models of reflection waves. The reflected waves with the present analysis include all of the waves traveling in different directions when incident wave period is short. Numerical examinations show that velocities at the inner and outer sides of the front walls of caissons ase close to each other, and reflection coefficients satisfy the energy conservation relation very well when porous effect parameter is infinite. Numerical results show that the reflection coefficients of oblique incident waves are smaller for shorter caissons at low frequency, and decrease with the increase of wave incident angle.

Soil seismic analysis for 2D oblique incident waves using exact free-field responses by frequency-based finite/infinite element method

The seismic analysis of a viscoelastic half-space under two-dimensional(2D)oblique incident waves is carried out by the finite/infinite element method(FIEM).First,the frequency-domain exact solutions for the displacements and stresses of the free field are derived in general form for arbitrary incident P and SV waves.With the present formulation,no distinction needs to be made for SV waves with over-critical incident angles that make the reflected P waves disappear,while no critical angle exists for P waves.Next,the equivalent seismic forces of the earthquake(Taft Earthquake 1952)imposed on the near-field boundary are generated by combining the solutions for unit ground accelerations with the earthquake spectrum.Based on the asymmetric finite/infinite element model,the frequency-domain motion equations for seismic analysis are presented with the key parameters selected.The results obtained in frequency and time domain are verified against those of Wolf’s,Luco and de Barros’and for inversely computed ground motions.The parametric study indicated that distinct phase difference exists between the horizontal and vertical responses for SV waves with over-critical incident angles,but not for under-critical incident angles.Other observations were also made for the numerical results inside the text.

Seismic stability of jointed rock slopes under obliquely incident earthquake waves

Seismic stability of slopes has been traditionally analyzed with vertically propagated earthquake waves.However,for rock slopes,the earthquake waves might approach the outcrop still with a evidently oblique direction.To investigate the impact of obliquely incident earthquake excitations,the input method for SV and P waves with arbitrary incident angles is conducted,respectively,by adopting the equivalent nodal force method together with a viscous-spring boundary.Then,the input method is introduced within the framework of ABAQUS software and verified by a numerical example.Both SV and P waves input are considered herein for a 2 D jointed rock slope.For the jointed rock mass,the jointed material model in ABAQUS software is employed to simulate its behavior as a continuum.Results of the study show that the earthquake incident angles have significance on the seismic stability of jointed rock slopes.The larger the incident angle,the greater the risk of slope instability.Furthermore,the stability of the jointed rock slopes also is affected by wave types of earthquakes heavily.P waves induce weaker responses and SV waves are shown to be more critical.

3D Diffraction of obliquely incident SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space

Abstract This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The approach is validated by comparison with the literature, and the effects of cavity interval, incident frequency, and boundary drainage condition on the diffraction are studied through numerical examples. It is shown that, the interaction between two cavities is significant and surface displacement peaks become large when two cavities are close, and the surface displacement may be significantly amplified by twin cavities, and the influence range with large amplification can be as wide as 40 times of the cavity radius. Surface displacements in dry poroelastic case and saturated poroelastic cases with drained and undrained boundaries are evidently different under certain circumstances, and the differences may be much larger than those in the free-field response.

Properties of Ternary One Dimensional Plasma Photomic Crystals for an Obliquely Incident Electromagnetic Wave Considering the Effect of Collisions in Plasma Layers

An analytical study is presented on the modal dispersion characteristics, group velocity, and effective group, as well as the phase index of a ternary one dimensional plasma photonic crystal for an obliquely incident electromagnetic wave considering the effect of collisions in plasma layers. The dispersion relation is derived by using the transfer matrix method and the boundary conditions based on electromagnetic theory. The dispersion curves are plotted for both the normal photonic band gap structure and the absorption photonic band gap structure. It is found that the increase in the angle of incidence shifts the photonic band gap toward higher frequencies. Also, the cutoff frequency is independent of collisions.

Time domain method for calculating free field motion of a layered half-space subjected to obliquely incident body waves

In numerical simulation of wave scattering under oblique incident body waves using the finite element method, the free field motion at the incident lateral boundary induced by the background layered half-space complicates the computational area. In order to replace the complex frequency domain method, a time-domain method to calculate the free field motion of a layered half-space subjected to oblique incident body waves is developed in this paper. The new method decouples the equations of motion used in the finite element method and offers an interpolation formula of the free field motion. This formula is based on the fact that the apparent horizontal velocity of the free field motion is constant and can be calculated exactly. Both the theoretical analysis and numerical results demonstrate that the proposed method offers a high degree of accuracy.

Three-dimensional Scattering of Obliquely Incident Plane SH Waves by an Alluvial Valley in a Layered Half-space

The indirect boundary element method （IBEM） is used to study three-dimensional scattering of obliquely incident plane SH waves by an alluvial valley embedded in a layered half-space. The free-field response of the layered half-space is calculated by the direct stiffness method, and dynamic Green＇s functions of moving distributed loads acting on inclined lines in a layered half-space are calculated to simulate the scattering wave field. The presented method yields very accurate results since the three-dimensional dynamic stiffness matrix is exact and the moving distributed loads can act directly on the valley boundary without singularity. Numerical results and analyses are performed for amplification of obliquely incident plane SH waves around an alluvial valley in a uniform half-space and in single layer over half-space. The results show that the three-dimensional responses are distinctly different from the two-dimensional responses, and the displacement amplitudes around alluvial valleys in a uniform haft-space are obviously different from those in a layered half-space.

The Interaction of Oblique Flexural Gravity Waves With a Small Bottom Deformation on a Porous Ocean-Bed： Green＇s Function Approach

The interaction of oblique incident water waves with a small bottom deformation on a porous ocean-bed is examined analytically here within the framework of linear water wave theory. The upper surface of the ocean is assumed to be covered by an infinitely extended thin uniform elastic plate, while the lower surface is bounded by a porous bottom surface having a small deformation. By employing a simplified perturbation analysis, involving a small parameter c^（〈〈l ）, which measures the smallness of the deformation, the governing Boundary Value Problem （BVP） is reduced to a simpler BVP for the first-order correction of the potential function. This BVP is solved using a method based on Green＇s integral theorem with the introduction of suitable Green＇s function to obtain the first-order potential, and this potential function is then utilized to calculate the first-order reflection and transmission coefficients in terms of integrals involving the shape function c（x） representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number propagating just below the elastic plate and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the surface below the elastic plate. Again, for small angles of incidence, the reflected wave energy is more as compared to the other angles of incidence. It is also observed that the reflected wave energy is somewhat sensitive to the changes in the flexural rigidity of the elastic plate, the porosity of the bed and the ripple wave numbers. The main advantage of the present study is that the results for the values of reflection and transmission coefficients obtained are found to satisfy the energy-balance relation almost accurately.

Oblique Wave Trapping by a Surface-Piercing Flexible Porous Barrier in the Presence of Step-Type Bottoms

The present study deals with the oblique wave trapping by a surface-piercing flexible porous barrier near a rigid wall in the presence of step-type bottoms under the assumptions of small amplitude water waves and the structural response theory in finite water depth.The modified mild-slope equation along with suitable jump conditions and the least squares approximation method are used to handle the mathematical boundary value problem.Four types of edge conditions,i.e.,clamped-moored,clamped-free,moored-free,and moored-moored,are considered to keep the barrier at a desired position of interest.The role of the flexible porous barrier is studied by analyzing the reflection coefficient,surface elevation,and wave forces on the barrier and the rigid wall.The effects of step-type bottoms,incidence angle,barrier length,structural rigidity,porosity,and mooring angle are discussed.The study reveals that in the presence of a step bottom,full reflection can be found periodically with an increase in(i)wave number and(ii)distance between the barrier and the rigid wall.Moreover,nearly zero reflection can be found with a suitable combination of wave and structural parameters,which is desirable for creating a calm region near a rigid wall in the presence of a step bottom.

Oblique incidence properties of locally resonant sonic materials with resonance and Bragg scattering effects

A locally resonant sonic material （LRSM） is an elastic matrix containing a periodic arrangement of identical local resonators （LRs）, which can reflect strongly near their natural frequencies, where the wavelength in the matrix is still much larger than the structural periodicity. Due to the periodic arrangement, an LRSM can also display a Bragg scattering effect, which is a characteristic of phononic crystals. A specific LRSM which possesses both local resonance and Bragg scattering effects is presented. Via the layered-multiple-scattering theory, the complex band structure and the transmittance of such LRSM are discussed in detail. Through the analysis of the refraction behavior at the boundary of the composite, we find that the transmittance performance of an LRSM for oblique incidence depends on the refraction of its boundary and the transmission behaviors of different wave modes inside the composite. As a result, it is better to use some low-speed materials （compared with the speed of waves in surrounding medium） as the LRSM matrix for designing sound blocking materials in underwater applications, since their acoustic properties are more robust to the incident angle. Finally, a gap-coupled LRSM with a broad sub-wavelength transmission gap is studied, whose acoustic performance is insensitive to the angle of incidence.

Interaction of Oblique Waves and A Rectangular Structure with An Opening Near A Vertical Wall

With the method of separation of variables and the eigenfunction expansion employed, an analytical solution is presented for the radiation and diffraction of a rectangular structure with an opening near a vertical wall in oblique seas, in which the unknown coefficients are determined by the boundary conditions and matching requirement on the interface. The effects of the width of the opening and the angle of incidence on the hydrodynamic characteristics of a rectangular structure with an opening near a vertical wall are mainly studied. The comparisons of the calculation results with wall-present and with wall-absent are also made. The results indicate that the variation trends of the heave added mass and excitation force with wall-present are almost the same as those with wall-absent, and that the peak values in the former case are obviously larger than those in the latter due to the reflection of the vertical wall.

Design of wideband graded-index antireflection coatings at oblique light incidence

We suggest a design method of graded-refractive-index （GRIN） antireflection （AR） coating for s-polarized or p- polarized light at off-normal incidence. The spectrum characteristic of the designed antireflection coating with a quintic effective refractive-index profile for a given state of polarization has been discussed. In addition, the genetic algorithm was used to optimize the refractive index profile of the GRIN antireflection for reducing the mean reflectance of s- and p-polarizations. The average reflectance loss was reduced to only 0.04% by applying optimized GRIN AR coatings onto BK7 glass over the wavelength range from 400 to 800 nm at the incident angle of θo = 70°.

Response Characteristics of Position Sensitive Detector under Oblique Incidence Condition

Based on Lucovsky equation,the response solution of 1-D position sensitive detector(PSD) is given under the condition of parallel light oblique incidence and Gaussian beam oblique incidence,and the response characteristics are analyzed by numerical calculation. The relation between oblique incident angle and output current is introduced and illustrated.