Analytical and Numerical Study of the Reflection/Transmission Coefficients in Slight Viscoelastic Medium

The study done in this paper brings out the effect of the viscoelasticity on the reflection/transmission coefficients. The knowledge of this effect can be useful for several applications, such as enhancing the resolution of the seismic sections, fluid and fracture detection. It can also have other applications different from the geophysical domain, as the study of the bonding between the materials in the civil engineering domain. We use the complex Lame coefficients in the continuity equations at the boundary layers to get the analytical expressions of the reflection/transmission coefficients in viscoelastic media. The coefficients can be divided into two parts, the first part is independent from the quality factor, and it corresponds to the elastic reflection/transmission coefficients. The second part is dependent on the quality factor contrast and it represents the contribution of the viscoelasticity on the reflection/transmission coefficients. From the numerical study it appears that the effect of the viscoelasticity is significant near to the critical angles. This effect is not clear and it is difficult to interpret and we do not know if it has a physical meaning or it is only a mathematical artifact that is why it is better to be far from the critical angles for seismic investigation.

An accurately fast algorithm of calculating reflection/transmission coefficients

For the boundary between transversely isotropic media with a vertical axis of symmetry (VTI media), the interface between a liquid and a VTI medium, and the free-surface of an elastic half-space of a VTI medium, an accurately fast algorithm was presented for calculating reflection/transmission (R/T) coefficients. Specially, the case of post-critical angle incidence was considered. Although we only performed the numerical calculation for the models of the VTI media, the calculated results can be extended to the models of transversely isotropic media with a horizontal axis of rotation symmetry (HTI media). Compared to previous work, this algorithm can be used not only for the calculation of R/T coefficients of the boundary between ellipsoidally anisotropic media, but also for that between generally anisotropic media, and the speed and accuracy of this algorithm are faster and higher. According to the anisotropic parameters of some rocks given by the published literature, we performed the calculation of R/T coefficients by using this algorithm and analyzed the effect of the rock anisotropy on R/T coefficients. We used Snell's law and the energy balance principle to perform verification for the calculated results.

Seismic reflection and transmission coefficients of a single layer sandwiched between two dissimilar poroelastic solids

The seismic reflection and transmission characteristics of a single layer sandwiched between two dissimilar poroelastic solids saturated with two immiscible viscous fluids are investigated. The sandwiched layer is modeled as a porous solid with finite thickness. The propagation of waves is represented with potential functions. The displacements of particles in different phases of the aggregate are defined in terms of these potential functions. Due to the presence of viscosity in pore fluids, the reflected and transmitted waves are inhomogeneous in nature, i.e., with different directions of propagation and attenuation. The closed-form analytical expressions for reflection and transmission coefficients are derived theoretically for appropriate boundary conditions. These expressions are calculated as a non-singular system of linear algebraic equations and depend on the various parameters involved in this non-singular system. Hence,numerical examples are studied to determine the effects of various properties of the sandwich layer on reflection and transmission coefficients. The essential features of layer thickness, incident direction, wave frequency, liquidsaturation and capillary pressure of the porous layer on reflection and transmission coefficients are depicted graphically and discussed. The analysis shows that reflection and transmission coefficients are strongly associated with incident direction and various properties of the porous layer.

Intelligent Reflecting Surface Assisted Transmission Optimization Strategies in Wireless Networks

Wireless Power Transfer(WPT)technology can provide real-time power for many terminal devices in Internet of Things(IoT)through millimeterWave(mmWave)to support applications with large capacity and low latency.Although the intelligent reflecting surface(IRS)can be adopted to create effective virtual links to address the mmWave blockage problem,the conventional solutions only adopt IRS in the downlink from the Base Station(BS)to the users to enhance the received signal strength.In practice,the reflection of IRS is also applicable to the uplink to improve the spectral efficiency.It is a challenging to jointly optimize IRS beamforming and system resource allocation for wireless energy acquisition and information transmission.In this paper,we first design a Low-Energy Adaptive Clustering Hierarchy(LEACH)clustering protocol for clustering and data collection.Then,the problem of maximizing the minimum system spectral efficiency is constructed by jointly optimizing the transmit power of sensor devices,the uplink and downlink transmission times,the active beamforming at the BS,and the IRS dynamic beamforming.To solve this non-convex optimization problem,we propose an alternating optimization(AO)-based joint solution algorithm.Simulation results show that the use of IRS dynamic beamforming can significantly improve the spectral efficiency of the system,and ensure the reliability of equipment communication and the sustainability of energy supply under NLOS link.

Rigorous solution to second harmonic generation considering transmission and reflection of light at air–crystal interface

Considering the transmission and reflection of TE-polarized pump light at the air–crystal interface,the second harmonic generation(SHG)in a lithium niobate(LN)crystal is investigated theoretically and systematically in this work.In previous studies,the theoretical analyses of reflection and transmission of incident wave in the process of nonlinear frequency conversion were not considered in LN crystal on account of the complicated calculations.First,we establish a physical picture describing that a beam of light in TE mode transports in the LN crystal considering transmission and reflection at the crystal surface and generates nonlinear second-order optical polarization in crystal.Then we analytically derive the reflection coefficient and transmission coefficient of pump light by using the dispersion relationships and electromagnetic boundary conditions.We construct the nonlinear coupled wave equations,derive and present the small signal approximation solution and the general large signal solution exactly.Under the transmission model and reflection model,we find that the conversion efficiency of the second-harmonic wave is obviously dependent on transmission coefficient and other general physical quantities such as the length of LN crystal and the amplitude of pump light.Our analytical theory and formulation can act as an accurate tool for the quantitative evaluation of the SHG energy conversion efficiency in an LN crystal under practical situations,and it can practically be used to treat other more complicated and general nonlinear optics problems.

Reflection coefficients of P-SV waves in weak anisotropic media

We introduce the Thomsen anisotropic parameters into the approximate linear reflection coefficient equation for P-SV wave in weakly anisotropic HTI media. From this we get a new, more effective, and practical reflection coefficient equation. We performed forward modeling to AVO attributes, obtaining excellent results. The combined AVO attribute analysis of PP and PS reflection data can greatly reduce ambiguity, obtain better petrophysical parameters, and improve parameter accuracy.

Wave Reflection by Rectangular Breakwaters for Coastal Protection

In this study,we focus on the numerical modelling of the interaction between waves and submerged structures in the presence of a uniform flow current.Both the same and opposite senses of wave propagation are considered.The main objective is an understanding of the effect of the current and various geometrical parameters on the reflection coefficient.The wave used in the study is based on potential theory,and the submerged structures consist of two rectangular breakwaters positioned at a fixed distance from each other and attached to the bottom of a wave flume.The numerical modeling approach employed in this work relies on the Boundary Element Method(BEM).The results are compared with experimental data to validate the approach.The findings of the study demonstrate that the double rectangular breakwater configuration exhibits superior wave attenuation abilities if compared to a single rectangular breakwater,particularly at low wavenumbers.Furthermore,the study reveals that wave mitigation is more pronounced when the current and wave propagation are coplanar,whereas it is less effective in the case of opposing current.

Reflection and transmission of bottom simulating reflectors in gas hydrate-bearing sediments: Two-phase media models

The bottom simulating reflector （BSR） in gas hydrate-bearing sediments is a physical interface which is composed of solid, gas, and liquid and is influenced by temperature and pressure. Deep sea floor sediment is a porous, unconsolidated, fluid saturated media. Therefore, the reflection and transmission coefficients computed by the Zoeppritz equation based on elastic media do not match reality. In this paper, a two-phase media model is applied to study the reflection and transmission at the bottom simulating reflector in order to find an accurate wave propagation energy distribution and the relationship between reflection and transmission and fluid saturation on the BSR. The numerical experiments show that the type I compressional （fast） and shear waves are not sensitive to frequency variation and the velocities change slowly over the whole frequency range. However, type II compressional （slow） waves are more sensitive to frequency variation and the velocities change over a large range. We find that reflection and transmission coefficients change with the amount of hydrate and free gas. Frequency, pore fluid saturation, and incident angle have different impacts on the reflection and transmission coefficients. We can use these characteristics to estimate gas hydrate saturation or detect lithological variations in the gas hydrate-bearing sediments.

Theoretical analysis of JMC effect on stress wave transmission and reflection

Taking the joint matching coefficient(JMC) which represents the contact area ratio of the joint in rock masses as the key parameter, a one-dimensional contacted interface model(CIM-JMC) was established in this study to describe the wave propagation across a single joint. According to this model, the reflected and transmitted waves at the joint were obtained, and the energy coefficients of reflection and transmission were calculated. Compared with the modified Split Hopkinson pressure bar(SHPB) experiment, it was validated by taking the incident wave of the SHPB test as the input condition in the CIM-JMC, and the reflected and transmitted waves across the joint were calculated by the model. The effects of four sets of JMCs(0.81, 0.64, 0.49, and 0.36) on the transmission and reflection of the stress wave propagation across the joint were analyzed and compared with the experimental results. It demonstrated that the values of CIM-JMC could represent both the transmission and reflection of the stress wave accurately when JMC > 0.5, but could relatively accurately represent the reflection rather than the transmission when JMC < 0.5. By contrasting energy coefficients of joints with different JMCs, it was revealed that energy dissipated sharply along the decrease of JMC when JMC > 0.5.

A Numerical Method for Calculating Transmission Coefficients Across Arbitrary Potential Barriers with High Accuracy

We report a new method for calculating transmission coefficients across arbitrary potential barriers based on the Runge-Kutta method. A numerical solution of the Schrodinger equation is calculated using the Runge-Kutta method,and a new model is established to analyze the numerical results to find the transmission coefficient. This technique is applied to various cases, such as parabolic potential barrier and double-barrier structures. Transmission probability with high precision is obtained and discussed. The tunnelling current density through a MOS structure is also explored and the result coincides with the Fowler-Nordheim model,which indicates the applicability of our method.

Reflection and Transmission of Regular Waves from/Through Single and Double Perforated Thin Walls

In this paper, reflection and transmission coefficients of regular waves from/through perforated thin walls are investigated. Small scale laboratory tests have been performed in a wave flume firstly with single perforated thin Plexiglas plates of various porosities. The plate is placed perpendicular to the flume with the height from the flume bottom to the position above water surface. With this thin wall in the flume wave overtopping is prohibited and incident waves are able to transmit. The porosities of the walls are achieved by perforating the plates with circular holes. Model settings with double perforated walls parallel to each other forming so called chamber system, have been also examined. Several parameters have been used for correlating the laboratory tests’ results. Experimental data are also compared with results from the numerical model by applying the multi-domain boundary element method （MDBEM） with linear wave theory. Wave energy dissipation due to the perforations of the thin wall has been represented by a simple yet effective porosity parameter in the model. The numerical model with the MDBEM has been further validated against the previously published data.

Reflection and transmission properties of double submerged rectangular blocks

An analytic method is used to study the reflection and transmission coefficients of the double submerged rectangular blocks （DSRBs） in oblique waves. The scattering potentials are obtained by means of the eigenfunction expansion method, and expressions for the reflection and transmission coefficients are determined. The boundary element method is employed to verify the correctness of the present analytical method. The DSRBs have better performance than the single submerged rectangular block （SSRB） in certain cases. The reflection and transmission properties of the DSRBs are investigated for some specific cases, and the influences of the geometric parameters are also presented.

Design formulas of transmission coefficients for permeable breakwaters

New empirical formulas of the transmission coefficient for permeable breakwaters were suggested based on available experimental data regarding the low-crest structure （LCS）, including the permeable rubble mound breakwater and pile-type breakwater. The rationality of the present formulas was verified by their comparison with existing empirical and analytical formulas. Numerical flume results were obtained by solving the modified Boussinessq-type wave equations （MBEs）, and a new expression relating the friction coefficient of to the relative submerged depth Rt/H8 was also derived. Comparative analysis shows that the results of the present formulas agree with the numerical flume results as well as available experimental data, and the present formulas are superior to the existing empirical and analytical expressions in estimating the transmission coefficient. The present formulas can provide references for estimation of the transmission coefficient in engineering practice.

Reflection and transmission of elastic waves through a couple-stress elastic slab sandwiched between two half-spaces

The reflection and transmission of elastic waves through a couple-stress elastic slab that is sandwiched between two couple-stress elastic half-spaces are studied in this paper. Because of the couple-stress effects, there are three types of elastic waves in the couple-stress elastic solid, two of which are dispersive. The interface conditions between two couple-stress solids involve the surface couple and rotation apart from the surface traction and displacement. The nontraditional interface conditions between the slab and two solid half-spaces are used to obtain the linear algebraic equation sets from which the amplitude ratios of reflection and transmission waves to the incident wave can be determined. Then,the energy fluxes carried by the various reflection and transmission waves are calculated numerically and the normal energy flux conservation is used to validate the numerical results. The special case, couple-stress elastic slab sandwiched by the classical elastic half-spaces, is also studied and compared with the situation that the classical elastic slab sandwiched by the classical elastic half-spaces. Incident longitudinal wave(P wave) and incident transverse wave(SV wave) are both considered. The influences of the couplestress are mainly discussed based on the numerical results.It is found that the couple-stress mainly influences the transverse modes of elastic waves.

The reflection and transmission of elastic waves at a interface between a transversely isotropi celastic soils and saturated soils

According to Biot′s wave equation of transversely isotropic saturated soil, this paper deduces the general equation of the reflection coefficients and transmission coefficients when qP 1 wave goes through from saturated soil to elastic media. The effects of anisotropies and boundary drainage condition on reflection coefficients and transmission coefficients are analyzed by numerical method. The idea of this paper can be applied to the case when qSV wave or qP 2 wave goes through from saturated soils to elastic soils.

Reflection and transmission of seismic waves at an interface betwen two saturated soils

Based on the modified Biot model for asturated soils, taking the compressibilities of the grains and the pore fluid as well as the viscous coupling into account, the reflection and transmission of seismic aves at an interface between two saturated soils are studied in this paper. A formula is derived for calculation of the amplitude reflection and transmission coefficients of various waves. A aumerical investigation of the dependence of the coefficients on the angle of incidence and the frequency is performed. This study is of a value for seismological studies and geophysical exploration.

A Multi-Domain Boundary Element Method to Analyze the Reflection and Transmission of Oblique Waves From Double Porous Thin Walls

In the present paper, we examine the performance of an efficient type of wave-absorbing porous marine structure under the attack of regular oblique waves by using a Multi-Domain Boundary Element Method(MDBEM). The structure consists of two perforated vertical thin barriers creating what can be called a wave absorbing chamber system. The barriers are surface piercing, thereby eliminating wave overtopping. The problem of the interaction of obliquely incident linear waves upon a pair of perforated barriers is first formulated in the context of linear diffraction theory. The resulting boundary integral equation, which is matched with far-field solutions presented in terms of analytical series with unknown coefficients, as well as the appropriate boundary conditions at the free surface, seabed, and barriers, is then solved numerically using MDBEM. Dissipation of the wave energy due to the presence of the perforated barriers is represented by a simple yet effective relation in terms of the porosity parameter appropriate for thin perforated walls. The results are presented in terms of reflection and transmission coefficients. The effects of the incident wave angles, relative water depths, porosities, depths of the walls, and other major parameters of interest are explored.

Reflection and transmission of Laguerre Gaussian beam from uniaxial anisotropic multilayered media

Based on angular spectrum expansion and 4 × 4 matrix theory, the reflection and transmission characteristics of a Laguerre Gaussian （LG） beam from uniaxial anisotropic multilayered media are studied. The reflected and transmitted beam fields of an LG beam are derived. In the case where the principal coordinates of the uniaxial anisotropic media coincide with the global coordinates, the reflected and transmitted beam intensities from a uniaxial anisotropic slab and three-layered media are numerically simulated. It is shown that the reflected intensity components of the incident beam, especially the TM polarized incident beam, are smaller than the transmitted intensity components. The distortion of the reflected intensity component is more evident than that of the transmitted intensity component. The distortion of intensity distribution is greatly affected by the dielectric tensor and the thickness of anisotropic media. We finally extend the application of the method to general anisotropic multilayered media.

Long Wave Reflection and Transmission over A Sloping Step

This investigation examines long wave reflection and transmission induced by a sloping step. Bellman and Kalaba's (1959) invariant imbedding is introduced to find wave reflection. An alternative method matching both the surface elevation and its surface slope of each region at the junction is applied to the determination of wave reflection and transmission. The proposed methods are compared with the accurate numerical results of Porter and Porter (2000) and those of Mei (1983) for a vertical step. The wave reflection obtained for a mildly sloping step differs significantly from the result of Mei. The wave reflection is found to fluctuate owing to wave trapping for the mild sloping step. The height and the face slope of the step are important for determining wave reflection and transmission coefficients.

POWER REFLECTION AND TRANSMISSION IN BEAM STRUCTURES CONTAINING A SEMI-INFINITE CRACK

Wave reflection and transmission in a beam containing a semi-infinite crack are studied analytically based on Timoshenko beam theory., Two kinds of crack surface conditions： non-contact （open） and fully contact （closed） cracks, are considered respectively for an isotropic beam. The analytical solution of reflection and transmission coefficients for a semi-infinite crack is obtained. The power reflection and transmission ratios depend on both the frequency and the position of the crack. Numerical results show the conservation of power transport. The transmitted energy among various wave modes is also investigated. A finite element method is used to verify the validity of the analytical results.