Wave scattering by a circular cylinder half-immersed in water with an ice-cover

The problem of scattering of water waves obliquely incident on a fixed long circular cylinder half-immersed in deep water with an ice-cover is investigated here. The ice-cover is modelled as an elastic plate of very small thickness. The problem is formulated using the method of multipoles. This leads to an infinite system of linear equations which are solved numerically by truncation. The reflection and transmission coefficients are obtained and depicted graphically against the wave number for various values of the angle of incidence and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover on these quantities. The effect of ice-cover is seen to increase the reflection coefficient and to decrease the transmission coefficient.     

Wave scattering by a thin elastic plate floating on a two-layer fluid

The hydroelastic interaction between an incident gravity wave and a thin elastic plate floating on a two-layer fluid of finite depth is analyzed with the aid of the method of matched eigenfunction expansions. The fluid is assumed to be inviscid and incompressible. A two-dimensional problem is formulated within the framework of linear potential theory for small-amplitude waves. The fluid domain is divided into two and three regions for semi-infinite and finite plates, respectively, with the matching relations representing the continuities of the pressure and velocity. A new inner product involving two single integrals is proposed, in which the vertical eigenfunctions in the open water region of the two-layer fluid are orthogonal. Then the orthogonality of the eigenfunctions with respect to the newly defined inner product is used to obtain a set of simultaneous equations for the expansion coefficients of the velocity potentials, and the edge conditions are included as a part of the equation system. The effects of the fluid density ratio and the position of interface on the wave reflection and transmission are discussed. Numerical analysis shows that the method proposed herein is effective with a higher rate of convergence.     

Wave drift force in a two-layer fluid of finite depth

Based on the momentum and energy conservation principles, a compact calculation formula is analytically derived for the wave-drift force on a 2-D body floating in a two-layer fluid of finite depth. In a two-layer fluid, two different wave modes (the surface-wave mode with longer wavelength and the internal-wave mode with shorter wavelength) exist not only in the incident wave but also in the body-scattered wave, and these wave characteristics are properly incorporated in the obtained formula. It is noted that, unlike the single-layer case, the wave-drift force can be negative in the incident wave of surface-wave mode, if the transmitted wave with internal-wave mode is large. Numerical computations are implemented for a Lewis-form body by means of the boundary-integral-equation method with Green’s function for the two-layer fluid problem. The effects of density ratio, interface position, and body motions on the wave-drift force are studied, and some important features are found for two-layer fluids.     

Water-wave scattering by two submerged plane vertical barriers—Abel integral-equation approach

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207–215, 1971) who employed a completely different approach involving a Schwarz–Christoffel transformation of complex-variable theory to solve the problem.     

Oblique wave scattering by a circular cylinder submerged beneath an ice-cover

When a train of small-amplitude surface water waves is normally incident on a very long horizontal circular cylinder fully submerged in deep water with a free surface, it is well known that it passes over and below the cylinder with a change of phase without experiencing any reflection. However the cylinder does experience reflection for oblique incidence of the surface wave train. It is shown here that the same phenomenon also holds good when the deep water has an ice-cover instead of a free surface, the ice-cover being modelled as a thin elastic plate. Here, for oblique incidence, the reflection and transmission coefficients are obtained approximately and depicted graphically against the wave number in a number of figures.     

Light scattering by an infinite circular cylinder immersed in an absorbing medium

Analytic solutions are developed for the single-scattering properties of an infinite dielectric cylinder embedded in an absorbing medium with normal incidence, which include extinction, scattering and absorption efficiencies, the scattering phase function, and the asymmetry factor. The extinction and scattering efficiencies are derived by the near-field solutions at the surface of the particle. The normalized scattering phase function is obtained by use of the far-field approximation. Computational results show that, although the absorbing medium significantly reduces the scattering efficiency, it has little effect on absorption efficiency. The absorbing medium can significantly change the conventional phase function. The absorbing medium also strongly affects the polarization of the scattered light. However, for large absorbing particles the degrees of polarization change little with the medium's absorption. This implies that, if the transmitting lights are strongly weakened inside the particle, the scattered polarized lights can be used to identify objects even when the absorption property of the host medium is unknown, which is important for both active and passive remote sensing.     

On liquid motion in a circular cylinder with horizontal axis

Summary The flow pattern of the slosh motion of a homogeneous, nonviscous (inviscid) and incompressible fluid with a free surface, contained in a rigid circular canal, has been dealt with analytically and experimentally. The axis of the canal is perpendicular to the direction of gravity. Considered are transverse (lateral to axis) oscillations of the liquid. The shape of the free liquid surface is determined numerically by means of a simple procedure. Theoretically calculated streamlines (path lines) are found to be in good qualitative agreement with experimentally observed trajectories of small spheres, made of plastic material and immersed in the liquid. The plastic particles and the liquid (water with a solution of salt added) have the same density.     

The scattering of water waves by an array of circular cylinders in a channel

The full linear problem of the scattering of water waves by an array of N bottom-mounted vertical circular cylinders situated in a channel of constant depth and width is solved using the method of multipoles. A simple formula is derived for the velocity potential in the vicinity of a cylinder, and in particular on the cylinder surfaces, which allows hydrodynamic quantities such as forces to be easily evaluated. The simplicity of the solution makes the evaluation of quantities of interest straightforward and extensive results are given. An approximate solution for the forces on the cylinders, based on the assumption that the wavelength of the incident wave is long compared with the cylinder radii, is also given, and this is compared with results from the exact linear solution.     

Free-surface flow of a fluid body with an inner circular cylinder in impulsive motion

An analytical theory and numerical computations are developed for the two-dimensional free-surface flow of an initially circular layer of inviscid fluid surrounding a rigid circular cylinder. The two cylinders are initially concentric. The fluid packet is released from rest and the flow suddenly starts forced by gravity and by the simultaneous impulsive motion of the inner body. A small-time expansion of the fully nonlinear free-surface problem is developed and a closed-form solution is found up to third order for an arbitrary radius of the rigid cylinder. For the gravitational flow around the body at rest, the solution is extended up to fourth order. Free-surface profiles and hydrodynamic forces on the cylinder are calculated and discussed against numerical solutions of the exact unsteady nonlinear problem. Some basic features, such as the formation of an almost uniform layer surrounding the upstream side of the body, are captured by the theory quite well and only later on in time significant quantitative differences appear. Similarly, the behaviour of hydrodynamic loads is rather well predicted during initial stages preceding larger fluctuations observed on a longer time-scale.     

An analytical solution for long-wave scattering by a submerged circular truncated shoal

An analytical solution of the long-wave (or shallow-water) equation in closed-form is obtained for simple harmonic waves scattered by a submerged circular truncated shoal. This analytical solution is firstly validated against Longuet-Higgins’s classical analytical solution for a submerged cylinder, and then validated against numerical solutions obtained by using the DRBEM (dual reciprocity boundary-element method) model for a submerged circular truncated cone. Finally, the analytical solution is used to investigate the changing trend of maximum wave amplification, the trace pattern of focal position of wave-energy versus the wave period and the influence of shoal submergence on wave-energy trapping.     

Transverse electric scattering by a dielectric biconvex cylinder loaded with a shallow circular trough in a ground plane

The scattering problem of transverse electric wave from a dielectric biconvex cylinder buried in a shallow circular trough of a ground plane is investigated and a rigorous series solution is also derived. Based on the region-matching method, the analysed region is decomposed into two subregions by introducing a semi-circular auxiliary boundary. The magnetic field of each subregion is expressed in terms of cylindrical wave functions with unknown expansion coefficients. After imposing the matching conditions and the boundary condition on the trough surface with the aid of Graf's addition theorem, the unknown coefficients are determined. Comparisons with published data for a dielectric circular cylinder case show very good agreement. Visible effects of depth-to-half-width ratios of a dielectric biconvex cylinder on echo width, far- field pattern and near-field distribution are illustrated in graphical form.     

Transformation of flexural gravity waves by heterogeneous boundaries

The transformation of flexural gravity waves due to wave scattering by heterogeneous boundaries is investigated under the assumption of the linearized water-wave theory. The heterogeneous boundaries include step-type bottom topography as well as heterogeneity in the material property of a floating ice-sheet. By applying the generalized expansion formulae along with the corresponding orthogonal mode-coupling relations, the boundary-value problem (BVP) is reduced to linear system of algebraic equations. The system of equations is solved numerically to determine the full solution of the problem under consideration. Energy relations are derived and used to check the accuracy of the computational results of the scattering problem. Explicit relations for the shoaling and scattering coefficients due to the change in water depth and heterogeneous ice-sheet are derived. These derivations are based on the law of conservation of energy flux under the assumptions of the linearized shallow-water theory. The change in water depth and the structural characteristics of the medium significantly contribute to the change in the scattering and shoaling coefficients and the deflection of the structure. The present results are likely to play a significant role in the analysis of flexural gravity-wave propagation in problems of variable topography for which a direct computational approach is being utilized.     

Exterior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder: semiclassical scattering theory analysis

We use the semiclassical limit of electromagnetic wave scattering theory to determine the properties of the exterior caustics of a diagonally incident plane wave scattered by an infinitely long homogeneous dielectric circular cylinder in both the near zone and the far zone. The transmission caustic has an exterior/interior cusp transition as the tilt angle of the incident beam is increased, and each of the rainbow caustics has a farzone rainbow/exterior cusp transition and an exterior/interior cusp transition as the incident beam tilt angle is increased. We experimentally observe and analyze both transitions of the first-order rainbow. We also compare the predictions of the semiclassical approximation with those of ray theory and exact electromagnetic wave scattering theory.     

Light scattering by a coated infinite cylinder in an absorbing medium

The scattering formulation for a coated infinite cylinder in an absorbing medium is presented in this paper. The cylinder is subjected to an arbitrarily polarized plane wave propagating in a general direction at the cylinder. The refractive index and magnetic permeability of the host medium, as well as those for the core and coating of the cylinder, can be real or complex. The scattering and extinction efficiencies and the scattering amplitudes are derived for both the near field and the far field. As the medium is absorbing, the "true" extinction and scattering efficiencies are derived based on the radiative energy outflow at the surface of the cylinder. The radiative efficiencies in the far field are denoted as "apparent" properties because they include absorption by the intervening medium. The influence of the refractive index and permeability of the host medium on the scattering properties of a coated cylinder is illustrated by numerical examples.     

Evolution equations for nonlinear waves in a tapered elastic tube filled with a viscous fluid

In this work, employing the reductive perturbation method and treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube, the propagation of weakly nonlinear waves is investigated in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, depending on the viscosity and perturbation parameters we obtained various evolution equations as the extended Korteweg-de Vries (KdV), extended KdV Burgers and extended perturbed KdV equations. Progressive wave solutions to these evolution equations are obtained and it is observed that the wave speeds increase with the distance for negative tapering while they decrease for positive tapering.     

Scattering of antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface

Summary We present analytical solutions for the scattering of an antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface. We first consider the simple case in which the imperfection is homogeneous along the interface. Two typical imperfect interfaces are addressed: 1) mechanically compliant and dielectrically weakly conducting interface, and 2) mechanically compliant and dielectrically highly conducting interface. The expressions for the directivity pattern and scattering cross-section of the scattered shear waves are derived. We then investigate the more difficult problem in which the imperfection is circumferentially inhomogeneous along the interface. A concise expression for an inhomogeneously compliant and weakly conducting interface is derived by means of matrix notation. Numerical examples are presented to demonstrate the effect of the imperfection and the circumferential inhomogeneity of the interface on the directivity patterns and scattering cross-sections of the scattered shear wave. The circumferentially inhomogeneous interface is also utilized to model the interface where an arbitrary number of cracks exist. Results show that when every part of the interface is rather compliant, large low-frequency peaks of the scattered cross-sections, which correspond to the resonance scattering, can be observed no matter if the interface is homogeneous or inhomogeneous. The appearance of large low-frequency peaks can be well explained by estimating the natural frequency of the corresponding reduced mass-spring system where the cylinder is assumed as a rigid body. Peaks of the scattered cross-sections spanning from low frequencies to high frequencies can be observed for a cylinder with a partially debonded interface.     

A Lie-Group Adaptive Method for Imaging a Space-Dependent Rigidity Coefficient in an Inverse Scattering Problem of Wave Propagation

We are concerned with the reconstruction of an unknown space-dependent rigidity coefficient in a wave equation. This problem is known as one of the inverse scattering problems. Based on a two-point Lie-group equation we develop a Lie-group adaptive method (LGAM) to solve this inverse scattering problem through iterations, which possesses a special character that by using onlytwo boundary conditions and two initial conditions, as those used in the direct problem, we can effectively reconstruct the unknown rigidity function by aself-adaption between the local in time differential governing equation and the global in time algebraic Lie-group equation. The accuracy and efficiency of the present LGAM are assessed by comparing the imaged results with some postulated exact solutions. By means of LGAM, it is quite versatile to handle the wave inverse scattering problem for the image of the rigidity coefficient without needing any extra information from the wave motion.     

水声材料低频声性能的驻波管测量

介绍一种用于水声材料低频声性能测量的驻波管测试系统。该系统采用双水听器传递函数法来测量材料样品在背衬下的低频反射系数 ,测量频率低于 1kHz,管子的内径为 85mm。文中首先对测量方法和系统构成作必要的说明 ,然后对水 空气自由界面的反射系数进行测量来验证本系统的测量方法 ,通过对均匀橡胶材料样品的测量 ,分析了系统的测量不确定度