Dynamics of nonlinear three-dimensionalwaves on the interface between two fluids in a channel with low-sloping bottom and top

A combined approach is proposed to describe the transformation of three-dimensional disturbances of the interface between two incompressible immiscible fluids of different densities contained in a channel with fixed rigid top and bottom. It is assumed that the wavelengths are moderately large, the amplitudes are small but finite, the top and the bottom can be gently sloping, and capillary effects are small. The system of equations derived is applicable for modeling disturbances simultaneously scattering in arbitrary horizontal directions. Some typical wave problems are numerically solved and the effect of governing parameters is shown.     

High-frequency scattering from a metal-like dielectric lens

The scattering of a plane electromagnetic wave by a dielectric lens which behaves like a metal reflector is considered. At short wavelengths, the leading term of the backscattered field cannot be determined entirely through simple geometrical optics considerations; instead, it is obtained by means of a modified Watson transformation of the exact solution. The difficulties that arise in applying this technique to other lenses are discussed.This research was sponsored by the U.S. Air Force Cambridge Research Laboratories under Contract F 19628-68-C-0071.     

SH波绕界面孔的散射

用波函数展开方法研究了SH波绕界面孔的散射问题。由入射、反射和透射波组成的自由波场与孔的散射场叠加成总波场。按照一定方式将两个半平面散射波场延拓于全平面，通过Hankel-Fourier展开方法求得了任意形状孔散射场的级数解。以椭圆形孔为例计算了孔边缘的动应力集中系数。     

覆水饱和双相介质圆弧场地P波激励下散射问题的理论解

采用波函数展开法对平面P波入射复杂水域地形的空间变异性地震动场进行研究,该水域地形具有覆水层、饱和双相介质、场地非平坦以及第二类分层(场地跨越分层界面)等属性.首先,依据地震波反射和透射特性推导直角坐标系下的自由波场分布;然后,根据场地属性并引入大圆弧法分析极坐标系下的含有待定系数的散射波场;进而,结合土-水分界面和饱和土层分界面边界条件,求解散射波场中的待定系数;最后,通过自由波场和散射波场得到覆水饱和双相介质圆弧场地波函数理论解.基于理论解,通过算例验证了理论推导的合理性及可靠性,分析了地表位移在不同入射条件下的差异性.结果 表明,相对于均匀介质,饱和双相介质会显著影响地表位移分布.此外,入射波频率和角度对地震地面运动特性也有较大的影响.     

Scattering of harmonic waves by a disk-shaped rigid inclusion in a three-dimensional elastic matrix

We consider a three-dimensional problem on the interaction of harmonic waves with a thin rigid movable inclusion in an infinite elastic body. The problem is reduced to solving a system of two-dimensional boundary integral equations of Helmholtz potential type for the stress jump functions on the opposite surfaces of the inclusion. We propose a boundary element method for solving the integral equations on the basis of the regularization of their weakly singular kernels. Using the asymptotic relations between the amplitude-frequency characteristics of the wave farzone field and the obtained boundary stress jump functions, we determine the amplitudes of the shear plane wave scattering by a circular disk-shaped inclusion for various directions of the wave incident on the inclusion and for a broad range of wave numbers.     

On three-dimensional acoustic-gravity waves in model non-isothermal atmospheres

The propagation of acoustic-gravity waves is studied in a family of model non-isothermal atmospheres, including temperature profiles with any initial and asymptotic temperature and adjustable maximum temperature gradient. The equation for the vertical velocity of linear acoustic-gravity waves is solved exactly in terms of hypergeometric functions, the wave field being described to all orders in the scattering parameter kL, at all frequencies and distances, including wavelengths comparable to the scale of temperature change and atmospheric layers with a large temperature gradient. It is found that since the temperature is bounded, in the asymptotic regime waves grow in amplitude exponentially and phase increases linearly with altitude. The growth in amplitude is larger than exponential and the phase increases faster than linearly for atmospheres whose temperature increases with altitude, the effect being more marked for high frequency waves in regions of large temperature gradients. The accumulation of these effects leads to a wave field which is equivalent to the isothermal case at asymptotic temperature modified by a constant amplitude factor and phase shift which account for the history of propagation of the wave through the temperature gradients.     

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphist’s equation. The two relevant quantities are the scattering operators—transmission and reflection operators—which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.     

Scattering and depolarization of electromagnetic waves by a horizontally weakly inhomogeneous medium

A general theory of wave scattering from a weakly inhomogeneous medium is developed for the case where the inhomogeneity varies parallel to the boundary plane. The method of small perturbation is used and terms are carried up to and including the second order. It is found that the scattered waves are depolarized and present in all directions. In the special case of forward- or backscattering the depolarized fields are of the second order and are seen to result from a multiple scattering process; while in other directions, these fields could be of the first order, and result from a single scattering process.     

Diffraction of Love waves by a stress-free crack of finite width in the plane interface of a layered composite

A rigorous theory of the diffraction of Love waves by a stress-free crack of finite width in the interface of a layered composite is presented. The incident wave is taken to be either a bulk wave or a Love-wave mode. The resulting boundary-value problem for the unknown jump in the particle displacement across the crack is solved by employing the integral equation method. The unknown quantity is expanded in terms of a complete sequence of expansion functions in which each separate term satisfies the edge condition. This leads to an infinite system of linear, algebraic equations for the coefficients of the expansion functions. This system is solved numerically. The scattering matrix of the crack, which relates the amplitudes of the outgoing waves to the amplitudes of the incident waves, is computed. Several reciprocity and power-flow relations are obtained. Numerical results are presented for a range of material constants and geometrical parameters.     

Flexural wave scattering in a quarter-infinite thin plate with circular scatterers

Scattering of flexural waves by circular scatterers in a quarter-infinite thin plate is formulated using the wave expansion method together with the method of images. The scattered waves are expressed as a summation series of wave functions and the unknown scattering coefficients are determined by enforcing boundary conditions at the scatterers. Both holes and rigid scatterers are studied. Simply-supported and roller-supported boundary conditions on the quarter-infinite thin plate are also considered. The analysis can be used to determine the stress concentration caused by circular scatterers in quarter-infinite thin plates.     

Off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid sphere

In this paper, the off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid immovable (fixed) sphere is investigated. It is shown here that shifting the sphere off the axis of wave propagation induces a dependence of the scattering on the azimuthal angle. Theoretical expressions for the incident and scattered field from a rigid immovable sphere are derived. The near- and far-field acoustic scattering fields are expressed using partial wave series involving the spherical harmonics, the scattering coefficients of the sphere, the half-conical angle of the wave number components of the beam, its order and the beam-shape coefficients. The scattering coefficients of the sphere and the 3D scattering directivity plots in the near- and far-field regions are evaluated using a numerical integration procedure. The calculations indicate that the scattering directivity patterns near the sphere and in the far-field are strongly dependent upon the position of the sphere facing the incident high-order Bessel vortex beam.     

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphists equation. The two relevant quantities are the scattering operators—transmission and reflection operators—which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.     

Scattering of a Rayleigh wave by an elastic wedge whose angle is less than 180°

The steady-state problem of scattering of an incident Rayleigh wave by an elastic wedge whose angle is less than 180° is considered. The problem is reduced to the numerical solution of a pair of Fredholm integral equations of the second kind whose kernels are continuous functions. Numerical results are given for the amplitude and phase of the Rayleigh waves transmitted and reflected by the corner.     

P波对界面部分脱胶的刚性圆柱夹杂物的散射

本文求解了弹性Ｐ波对界面部分脱胶的可动刚性圆柱夹杂物的散射问题。将脱胶区看作表面不相接触的弧形界面裂纹，借助波函数展开法并利用边界条件将问题转化为一组对偶级数方程。然后通过引入裂纹面的位错密度函数，将其化为一组具有Ｈｉｌｂｅｒｔ核的第二类奇异积分方程，并进一步化为Ｃａｕｃｈｙ型奇异积分方程组，数值求解方程组可获得动应力强度因子，夹杂物刚体振动位移和散射截面等重要参量。结果显示该类结构在较低的频率上发生共振，此低频共振特性与脱胶区大小，入射波方向、材料组合等多种参数有关。与已有方法相比，本文的方法更具一般性，适用于任意材料组合。     

Spectral method for multiple edge diffraction by a flat strip

A spectral iteration scheme is employed to analyze time-harmonic and transient scattering of an E- or H-polarized incident plane wave by a perfectly conducting plane strip. The scattered field is synthesized by successive interactions between the edges, with each interaction modeled by half-plane diffraction. The plane wave spectrum generating a particular order of diffraction consists, in addition to the incident plane wave excitation, of a portion determined from the previous diffraction. The multiple integral spectral representations constructed in this manner satisfy the edge condition, and they are in a form suitable for inversion into the time domain by the modified Cagniard-deHoop method. Asymptotic reductions for special cases yield agreement with results from other methods, when available. Numerical calculations including up to triple diffraction have been performed for H- and E-polarized impulse and Gaussian pulse scattering. The results are clearly seen to repair the deficiencies of wavefront approximations at longer observation times, and from comparison with data generated independently by eigenfunction expansion, they describe accurately the total scattered response, owing to the high damping rate of higher-order diffractions.     

Finite-amplitude waves in a homogeneous fluid with a floating elastic plate

Equations for three nonlinear approximations of a wave perturbation in a homogeneous ideal incompressible fluid covered by a thin elastic plate are obtained using the method of multiple scales and taking into account that the acceleration of vertical flexural displacements of the plate is nonlinear. Based on the obtained equations, asymptotic expansions up third-order terms are constructed for the fluid velocity potential and the perturbations of the plate-fluid interface (plate bending) caused by a traveling periodic wave of finite amplitude. The wave characteristics are analyzed as functions of the elastic modulus and thickness of the plate and the length and tilt of the initial fundamental harmonic wave.     

Scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space

A rigorous theory of the scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space is presented. The boundary value problem (which is of the third kind) is solved by employing two suitably chosen Green functions. One of them is represented as a Fourier type of integral, the other is taken to be the Bessel function of the second kind and order zero. The procedure leads to a system of three, coupled, integral equations. This system is solved numerically. In case of an incident bulk wave, the amplitude of the launched surface wave is computed; in case of an incident surface wave, its transmission and reflection factor are computed. For both cases, an expression for the far-field radiation pattern of the scattered bulk wave is derived. A reciprocity relation is shown to exist between the amplitude of the launched surface wave and the far-field bulk wave radiation pattern. Numerical results are presented for a triangularly-prismatic protrusion; they are compared with the results pertaining to a corresponding indentation in the mass-loaded boundary, that have been obtained in a previous paper.     

Sound scattering and its cancellation by an elastic spherical shell in free space and near a free surface

Sound scattering by an elastic spherical shell is analysed using linear acoustics and linear structural dynamics. It is suggested to utilize the shell’s structural dynamics to reduce or even eliminate the scattered sound field, thus making it practically acoustically invisible. This can be achieved using a prescribed external pressure distribution acting on the shell’s wall. Exact analytical solutions are found for that external pressure distribution, eliminating the scattered wave when the sphere is in free space or near a free surface and is subject to an incoming planar monochromatic sound wave. The latter is assumed to propagate in a direction perpendicular to the free surface (if it exists). The case of a few pressure-actuators acting on the shell’s wall is also modelled and an optimal solution which reduces the sound scattering by these actuators is found. An aluminium shell of 1 m radius and 5 mm thickness, situated in fresh water is analysed for sound frequencies of up to 10 kHz. The scattered wave fields are presented as well as the external pressure distributions that eliminate these scattered sound field, i.e. achieving acoustic cloaking. Significant reduction in the scattered wave energy and the target strength of more than 10 dB are also realized using a few pressure-actuators as long as the distance between the actuators is no more than three times the incident wave length for the investigated cases.     

The penetrable coated sphere embedded in a point source excitation field

A low frequency acoustic wave field emanates from a given point and fills up the whole space. A penetrable lossy sphere with a coeccentric spherical core, which is also penetrable and lossy but characterized by different physical parameters, disturbs the given point source field. We obtain zeroth- and first-order low frequency solutions of this scattering problem in the interior of the spherical core, within the spherical shell, and in the exterior medium of propagation. We also derive the leading nonvanishing terms of the normalized scattering amplitude, the scattering cross-section as well as the absorption cross-section. The special case of a penetrable sphere is recovered either by equating the physical parameters that characterize the media in the shell and in the exterior, or by reducing the radius of the core sphere to zero. By letting the compressional viscosity of the medium in the interior sphere, or in the shell, go to zero, we obtain corresponding results for the lossless case. The incident point source field is so modified as to be able to obtain the corresponding results for plane wave incidence in the limit as the source point approaches infinity. It is observed that a small scatterer interacts stronger with a point source generated field than with a plane wave. A detailed analysis of the influence that the geometrical and the physical parameters of the problem have on the scattering process is also included. An interesting conclusion is that if the point source is located at a distance more than five radii of the scatterer away from it, then no significant changes with the plane excitation case are observed.     

Scattering of plane monochromatic waves from a heterogeneous inclusion of arbitrary shape in a poroelastic medium: An efficient numerical solution

Scattering of plane longitudinal monochromatic waves from a heterogeneous inclusion of arbitrary shape in an infinite poroelastic medium is considered. Wave propagation in the medium is described by Biot’s equations of poroelasticity. The scattering problem is formulated in terms of the volume integral equations for displacements of the solid skeleton and fluid pressure in the pore space in the region occupied by the inclusion. An efficient numerical method is applied to solve these equations. In the method, Gaussian approximating functions are used for discretization of the problem. For regular node grids, the matrix of the discretized problem has Toeplitz’s properties, and the Fast Fourier Transform technique can be used for the calculation of matrix–vector products. The latter accelerates substantially the process of iterative solution of the discretized problem. For material parameters of typical sedimentary rocks, the system of differential equations of poroelasticity contains a differential operator with a small parameter. As the result, the wave field in the inclusion region is split up into a slowly changing part, and boundary layer functions concentrated near the inclusion interface. The method of matched asymptotic expansions is used for the numerical solution in this case. For a spherical inclusion, the results of the numerical and matched asymptotic expansion methods are compared with a semi-analytical series solution. For a non-spherical heterogeneous inclusion, an example of the numerical solution is presented.