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1.
Using two-dimensional linear water wave theory, we consider the problem of normal water wave (internal wave) propagation over
small undulations in a channel flow consisting of a two-layer fluid in which the upper layer is bounded by a fixed wall, an
approximation to the free surface, and the lower one is bounded by a bottom surface that has small undulations. The effects
of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed
to calculate the first-order corrections to the velocity potentials in the two-layer fluid by using Green’s integral theorem
in a suitable manner and the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions
for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having
the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient
of twice the interface wave number and the ripple wave number. When this quotient approaches one, the theory predicts a resonant
interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples.
High reflection of the incident wave energy occurs if this number is large. Again, when a patch of sinusoidal ripples having
two different wave numbers for two consecutive stretches is considered, the interaction between the bed and the interface
near resonance attains in the neighborhood of two (singular) points along the x-axis (when the ripple wave number of the bottom undulation become approximately twice as large as the interface wave number).
The theoretical observations are presented in graphical form. 相似文献
2.
S. Mohapatra 《地球物理与天体物理流体动力学》2013,107(5):347-374
Scattering of oblique incident waves by small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated within the framework of linearized water wave theory. The ice-cover is being modeled as an elastic plate of very small thickness. There exist two modes of time-harmonic waves–one with lower wave number propagating along the ice-cover (ice-cover mode) and the other with higher wave number along the interface (interfacial mode). A perturbation analysis is employed to solve the corresponding boundary value problem governed by modified Helmholtz equation and thereby evaluating the reflection and transmission coefficients approximately up to first order for both modes. A patch of sinusoidal ripples, having two different wave numbers over two consecutive stretches, is considered as an example and the related coefficients are determined. It is observed that when the wave is incident on the ice-cover surface we always find energy transfer to the interface, but for interfacial incident waves there are parameter ranges for which no energy transfer to the ice-cover surface is possible. Also it is observed that for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. These results are demonstrated in graphical form. From the derived results, the solutions for problems with free surface can be obtained as particular cases. 相似文献
3.
J. D. Achenbach M. Kitahara Y. Mikata D. A. Sotiropoulos 《Pure and Applied Geophysics》1988,128(1-2):101-118
Reflection and transmission of elastic wave motion by a layer of compact inhomogeneities has been analyzed. For identical inhomogeneities whose geometrical centers are periodically spaced, the problem has been formulated and solved rigorously. The reflected and transmitted longitudinal and transverse wave motions have been expressed as superpositions of wavemodes, where each wavemode has its own cut-off frequency. At its cut-off frequency a mode converts from a standing into a propagating wavemode. The standing wavemodes decay exponentially with distance to the plane of the centers of the inhomogeneities. At small frequencies only the lowest order modes of longitudinal and transverse wave motion are propagating. Reflection and transmission coefficients have been defined in terms of the coefficients of the zeroth-order scattered wavemodes. These coefficients have been computed by a novel application of the Betti-Rayleigh reciprocal theorem. They are expressed as integrals over the surface of a single inhomogeneity, in terms of the displacements and tractions on the surface of the inhomogeneity. The system of singular integral equations for the surface fields has been solved numerically by the boundary integral equation method. Curves show the reflection and transmission coefficients for the reflected and transmitted longitudinal and transverse waves as functions of the frequency. Some results are also presented for planar distributions of cracks whose spacing and size are random variables. Finally, dispersion relations are discussed for solids which are completely filled with periodically spaced inhomogeneities. 相似文献
4.
M. R. Sarangi 《地球物理与天体物理流体动力学》2013,107(3):303-325
ABSTRACTA hydro-elastic frame has been considered to investigate the proliferation of waves over small base deformation on an infinitely extended flexible seabed. The flexible base surface is assumed as a thin elastic plate of very small thickness and it depends on the Euler–Bernoulli beam equation. For any particular frequency, there are two different modes of time-harmonic propagating wave exists rather than one mode of propagating wave along the positive horizontal direction. The waves with smaller wavenumber spread along the free-surface of the sea (say, free-surface mode) and the waves with higher wavenumber spread along the flexible base surface (say, flexural mode). A simplified perturbation approach is utilised to bring down the entire equations which govern the original boundary value problem (bvp) to a less complex bvp for the first-order velocity potential function. The first-order potential function along with the first-order reflection and transmission coefficients for both modes are calculated by a procedure based upon Fourier transform approach. A shape of sinusoidal swells flexible base surface is taken as an example to approve the scientific results. It is observed that when the train of normal incident propagating wave spreads over base distortion because of either the free-surface unsettling influence or the flexural wave movement in the sea, the reflected and transmitted energy are always feasible to be exchanged from one particular wave mode to another wave mode. Furthermore, we notice that the realistic changes in the flexural rigidity behaviour on the flexible base surface of the sea have a significant effect on the problem of water wave proliferation over small base deformation. Moreover, the energy conservation equation is derived with the help of the Green's integral theorem. The results for the values of reflection and transmission coefficients obtained for both the free-surface unsettling influence as well as flexural wave movement in the fluid are found to satisfy the energy conservation equation almost accurately. 相似文献
5.
This study investigates the reflection and transmission of plane SH-waves in two semi-infinite anisotropic magnetoelastic media. The lower half-space is considered as initially stressed and inhomogeneous. The density of lower half-space is taken exponentially varying with depth. The solutions for half-spaces are obtained analytically. The expressions for reflection and transmission coefficient are obtained in the closed form subject to continuity conditions at the interfaces of anisotropic magnetoelastic half-spaces and the Snell’s law. It is found that these coefficients depend on the initial stress, inhomogeneity parameter, the magnetoelastic coupling parameter, and the angle at which wave crosses the magnetic field of the half-spaces. Numerical computations are performed for these coefficients for a specific model of two different anisotropic magnetoelastic half-spaces. The numerical results are illustrated by the graph of reflection and transmission coefficient versus the angle of incidence. In general, as the initial stress increases the reflection and transmission coefficient increases, the affect is more prominent for more than 10 GPa. Inhomogeneity in the density of the material also increases the reflection and transmission coefficient. The anisotropic magnetoelastic parameter and the angle at which the wave crosses the magnetic field for both the half-spaces have a quite significant effect on the reflection and transmission coefficient. 相似文献
6.
Oblique wave scattering by undulating porous bottom in a two-layer fluid: Fourier transform approach
Srikumar Panda 《地球物理与天体物理流体动力学》2014,108(6):587-614
The problem involving scattering of oblique waves by small undulation on the porous ocean bed in a two-layer fluid is investigated within the framework of linearised theory of water waves where the upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different wave numbers (modes): wave with lower wave number propagates along the free surface whilst that with higher wave number propagates along the interface. When an oblique incident wave of a particular mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in conjunction with Fourier transform technique is used to derive the first-order corrections of velocity potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both modes. One special type of undulating bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the transformation of water wave energy between the two modes. Comparisons between the present results with those in the literature are made for particular cases and the agreements are found to be satisfactory. In addition, energy identity, an important relation in the study of water wave theory, is derived with the help of the Green’s integral theorem. 相似文献
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9.
There are several important wavenumber sampling issues associated with 2.5D seismic modelling in the frequency domain, which need careful attention if accurate results are to be obtained. At certain critical wavenumbers there exist rapid disruptions in the mainly smooth oscillatory spectra. The amplitudes of these disruptions can be very large, and this affects the accuracy of the inverse Fourier transformed frequency-space domain solution. In anisotropic elastic media there are critical wavenumbers associated with each wave mode—the quasi-P (qP) wave, and the two quasi-shear (qS1 and qS2) waves. A small wavenumber sampling interval is desirable in order to capture the highly oscillatory nature of the wavenumber spectrum, especially at increasing distance from the source. Obviously a small wavenumber sampling interval adds greatly to the computational effort because a 2D problem must be solved for every wavenumber and every frequency. The discretisation should be carried out up to some maximum wavenumber, beyond which the field becomes evanescent (exponentially decaying or diffusive). For receivers close to the source, activity persists beyond the critical wavenumber associated with the minimum shear wave velocity in the model. Fortunately, for receivers well removed from the source, the contribution from the evanescent energy is negligible and so there is no need to sample beyond this critical wavenumber. Sampling at Gauss–Legendre spacings is a satisfactory approach for acoustic media, but it is not practical in elastic media due to the difficulty of partitioning the integration around the different critical wavenumbers. We found to our surprise that in transversely isotropic media, the critical wavenumbers are independent of wave direction, but always occur at those wavenumbers corresponding to the maximum phase velocities of the three wave modes (qP, qS1 and qS2), which depend only on the elastic constants and the density. Additionally, we have observed that intermediate layers between source and receiver can filter out to a large degree, the sharp irregularities around the critical wavenumbers in the ω–k y spectra. We have found that, using the spectral element method, the singularities (poles) at the critical wavenumbers which exist with analytic solutions, do not arise. However, the troublesome spike-like behaviour still occurs and can be damped out without distorting the spectrum elsewhere, through the introduction of slight attenuation. 相似文献
10.
It is known that the reflection and transmission coefficients used in the zeroth order approximation of asymptotic ray theory (ART) are identical to those obtained for the plane wave impinging on a plane interface separating two perfectly elastic half-spaces. We have used ART to compute reflection and transmission coefficients for two viscoelastic media separated by a plane interface. Our method is different from the plane-wave approach because the ART approach requires only a local application of the boundary conditions both for the eikonal and the ray amplitudes. Several types of viscoelastic media were studied. For a given model, the elastic case was emulated by setting all the quality factors Q equal to each other. Several anelastic cases were computed by keeping the same velocities and densities while changing the Qs. The quality factor is a relatively difficult parameter to measure exactly. Hence elastic coefficients are used in most synthetic seismogram computations, and the quality factors are chosen from experimental measurements or simply estimated. From these computations, amplitude and phase differences between elastic coefficients and coefficients for dissipative media are observed in some cases. These differences show the importance of knowing the exact values of Q. Incorrect Q values can lead to unrealistic moduli and to noticeable phase differences of these viscoelastic coefficients. 相似文献
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12.
A problem of reflection and transmission of elastic waves at a plane interface between a uniform elastic solid half-space and a porous elastic half-space containing two immiscible fluids is investigated. The theory developed by Lo, Sposito and Majer for porous media containing two immiscible fluids is employed to find out the reflection and transmission coefficients. The incident wave is assumed to propagate through the uniform elastic half-space and two cases are considered. In the first case, a beam of plane longitudinal wave is assumed to be incident and in the second case, a beam of transverse wave is assumed to be incident at the interface. By taking granite as impervious elastic medium and columbia fine sandy loam containing air-water mixture as porous medium, reflection and transmission coefficients are obtained. By neglecting the inertial coupling coefficients, these coefficients are reduced to those obtained by Tomar and Arora using the theory of Tuncay and Corapcioglu. It is found that the inertial coupling parameters significantly affect the phase speeds and the amplitude ratios of the transmitted waves. 相似文献
13.
Based on the modified Biot＇s theory of two-phase porous media, a study was presented on seismic reflection and transmission coefficients at an air-water interface of saturated porous soil media. The major differences between air-saturated soils and water-saturated soils were theoretically discussed, and the theoretical formulas of reflection and transmission coefficients at an air-water interface were derived. The characteristics of propagation and attenuation of elastic waves in air-saturated soils were given and the relations among the frequency, the angle of incidence and the reflection, transmission coefficients were analyzed by using numerical methods. Numerical results show that the propagation characteristic of the wave in air-saturated soils is great different from that in water-saturated soils. The frequency and the angle of incidence can have great influences on the reflection and transmission coefficients at interface. Some new cognition about the wave propagation is obtained and the study suggests that we may carefully pay attention to the influence of air on the dynamic analysis of seismic wave. 相似文献
14.
弹性孔隙介质分界面上的反透射系数特征,在岩性划分、流体识别、储层边界判识等方面有重要的应用.本文研究上层为含两项不混合黏性流体孔隙介质、下层为含单项黏性流体孔隙介质分界面上的反透射理论.首先根据两种孔隙介质分界面上的能量守恒得到边界条件,再将波函数、位移、应力与应变关系代入边界条件,推导出完全连通孔隙情况下,第一类纵波入射到孔隙介质分界面上的反透射系数方程.通过建立砂岩孔隙介质模型,分别分析不同孔隙流体类型、不同含油饱和度及不同入射角情况下,各类波的反透射系数特征.研究表明,第二、三类纵波反透射系数数值比第一类纵波小多个数量级,且两者对入射角的变化不敏感,但对孔隙流体性质、含油饱和度的变化较敏感,而横波反透射系数特征恰好与此相反;第一类纵波反透射系数特征比较复杂,入射角、孔隙流体的性质及含油饱和度的变化都对其产生影响.不同孔隙流体弹性物性的差异、孔隙介质中含油饱和度的变化及不同入射角引起垂向和切向应力分量的变化都会影响各类波的反透射系数特征,分析这些特征可以为研究储层含油气性提供理论基础. 相似文献
15.
Summary On the basis of Alsop's method (1966), approximated reflection and transmission coefficients are obtained for surface waves obliquely incident at a vertical discontinuity. Rayleigh and Love waves are arranged in such a way that they form a homogeneous eigenfunction system and their mutual conversion into each other is compatible with this theory. The structure of the eigenvalue equation is responsible for the mathematical complications in many surface wave problems. On the basis of a computer routine, first result concerning the angular dependence of the reflection and transmission coefficients including mode conversion are presented. The problem of phase jump is discussed. A comparison of theoretical and experimental results shows the applicability of Alsop's method for 90°-corners, too. 相似文献
16.
本文研究了纵波垂直入射情况下两种介质分界面处的纵波反射和透射系数的频散特性,分界面上下两侧分别为层状双孔页岩介质和层状双孔砂岩介质.当纵波沿垂直于分界面的方向传播至分界面处时,会在上层双孔介质中产生三类反射纵波,在下层双孔介质中产生三类透射纵波.基于层状双孔介质的特性,给出了分界面处的六个边界条件.根据层状双孔介质的波动方程,利用平面波分析得到了纵波的反射和透射系数.结果表明：当多孔介质中存在流体时,纵波的反射和透射系数与频率相关,即存在频散现象.波致流体流动是造成纵波反射和透射系数频散的主要原因.此外,结果还表明局部流体流动引起地震频带内反射和透射系数的频散,宏观Biot流引起超声频带内反射和透射系数的频散.本文同时对岩石参数对反射和透射系数频散曲线的影响进行了研究. 相似文献
17.
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. 相似文献
18.
Rayleigh’s method of approximation is employed to find out the reflection and transmission coefficients due to an incident
plane SH wave at a corrugated interface between a laterally and vertically inhomogeneous anisotropic elastic solid half-space
and a laterally and vertically inhomogeneous isotropic visco-elastic solid half-space. The lateral and vertical inhomogeneities
are described by the exponential variations of elastic parameters. The formulae of reflection and transmission coefficients
are derived in closed form for the first-order approximation of the corrugation. The effects of the corrugation of the interface,
the inhomogeneity, the anisotropy, the visco-elasticity and the frequency of the incident wave on these coefficients are studied
analytically and numerically for a specific model containing a periodic interface. The results of earlier workers have been
reduced as particular cases from the present formulation. 相似文献
19.
The scattering of first mode linear baroclinic Rossby waves by a top-hat ridge in a continuously stratified ocean, with Brunt-Väisälä frequency that decays exponentially with depth below a surface mixed layer, is the subject of this study. A numerical mode matching technique is used to calculate the transmission coefficients for the propagating modes over the ridge. It is found that the scattered field depends crucially upon the stratification. For example, when the majority of the density variation is confined to a thin thermocline, corresponding to a small e-folding scale, gamma ?1, for the Brunt-Väisälä frequency, a large amount of the incident wave energy is reflected by a small amplitude ridge. Appreciable energy conversion between the propagating barotropic and baroclinic modes takes place in this case. An asymptotic analysis for a small amplitude ridge is presented that confirms these numerical results. In the limit gamma ?1→ 0, it is demonstrated that the scattered field in the continuously stratified ocean model differs markedly from the two-layer solution. The latter does not exhibit appreciable reflection of the incident wave energy for a small amplitude ridge. In conclusion, the application of a two-layer ocean model to describe Rossby wave scattering by ridges in place of a continuously stratified model cannot be recommended. 相似文献
20.
Summary Formulae are derived for the reflection and transmission coeficients of plane elastic waves for a transition layer. Haskell's technique and the so-called delta matrices[5, 7] are used for this purpose. No problems are encountered in deriving the reflections and transmission coefficients from Haskell's matrices[3]. However, in some cases Haskell's matrices do not guarantee the accuracy required. For this reason attention is mainly devoted to deriving the reflection and transmission coefficients from the delta matrices. In deriving the transmission coefficients use is made of the fact that some3×3 subdeterminants of the delta matrices are squares of the3×3 subdeterminants of Haskell's matrices. 相似文献