Scattering of scalar waves from a rough interface using a single integral equation

Using Green's function methods we consider the problem of scattering from a rough interface separating two semi-infinite homogenous media. We derive a single coordinate-space integral equation of the first kind for the generalized reflection coefficient R. A second integral equation of the first kind is derived for the generalized transmission coefficient T. The two equations are new results. In the limiting cases corresponding to Dirichlet and Neumann boundary conditions, we recover the usual boundary integral equations from the R-equation. In the flat surface limit, R and T reduce to the usual Fresnel reflection and transmission coefficients. The latter are derived from these more general results rather than assumed as in perturbation methods.     

Two capacitive windows in a rectangular wave guide

Summary The behaviour of two capacitive windows in a rectangular wave guide, located at small distances apart in comparison with the wave length, is investigated. The three-dimensional problem is reduced to a two-dimensional one. Two simultaneous integral equations for the electric field intensities in the apertures are derived. A solution for these unknown quantities can be obtained, representing them by a Fourier series and approximating the kernel functions. Expressions for the reflection and transmission coefficients are given. Absolute values of the transmission coefficient are obtained, both theoretically and experimentally.     

Diffraction of a mode close to its cut-off by a transversal screen in a planar waveguide

The problem of diffraction of a waveguide mode by a thin Neumann screen is considered. The incident mode is assumed to have frequency close to the cut-off. The problem is reduced to a propagation problem on a branched surface and then is considered in the parabolic approximation. Using the embedding formula approach, the reflection and transmission coefficients are expressed through the directivities of the edge Green’s function of the propagation problem. The asymptotics of the directivities of the edge Green’s functions are constructed for the case of small gaps between the screen and the walls of the waveguide. As the result, the reflection and transmission coefficients are found. The validity of known asymptotics of these coefficients is studied.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Acoustic radiation force on a rigid cylinder in an off-axis Gaussian beam near an impedance boundary

The exact equations of the axial and transverse acoustic radiation force functions of a Gaussian beam arbitrarily incident on an infinite rigid cylinder close to an impedance boundary and immersed in an ideal fluid are deduced by expressing the incident wave, the scattering wave and the boundary reflected wave in terms of the cylindrical wave function. The effects of the beam waist, the sound reflection coefficient, the cylinder position and the distance from the impedance boundary on the acoustic radiation force are studied using numerical simulations. The simulation results show that the amplitude of the acoustic radiation force function increases with beam width. Moreover, the values of the acoustic radiation force in both the axial and transverse directions reach those of a plane wave when the beam width is considerably larger than the wavelength of the Gaussian beam. The properties of the impedance boundary and the position of the cylinder in the Gaussian beam have a considerable effect on the magnitude and direction of the force. The simulation results, particularly in the case of a transverse force, indicate the presence of a negative acoustic radiation force that is related to the nondimensional frequency and position of the cylinder in the Gaussian beam.     

On the diffraction of acoustic waves by a quarter-plane

This paper follows the work of A.V. Shanin on diffraction by an ideal quarter-plane. Shanin’s theory, based on embedding formulae, the acoustic uniqueness theorem and spherical edge Green’s functions, leads to three modified Smyshlyaev formulae, which partially solve the far-field problem of scattering of an incident plane wave by a quarter-plane in the Dirichlet case. In this paper, we present similar formulae in the Neumann case, and describe a numerical method allowing a fast computation of the diffraction coefficient using Shanin’s third modified Smyshlyaev formula. The method requires knowledge of the eigenvalues of the Laplace-Beltrami operator on the unit sphere with a cut, and we also describe a way of computing these eigenvalues. Numerical results are given for different directions of incident plane wave in the Dirichlet and the Neumann cases, emphasising the superiority of the third modified Smyshlyaev formula over the other two.     

Transmission and reflection of acoustic waves by an obstacle in a waveguide

By an extension of the null field approach introduced by Waterman the transmission and reflection of acoustic waves by an obstacle in a waveguide are considered. The waveguide is assumed to have a constant cross section but otherwise the geometry is left arbitrary. The scattered field is obtained as a complicated mode sum containing the transition matrix of the obstacle, the reflection matrix of the waveguide wall and the transformation relations between the cylindrical and spherical basis functions. For a circular cross section of the waveguide relatively explicit expressions are given for the transmission and reflection coefficients, and numerical results are shown for spherical and spheroidal obstacles in rotationally symmetric configurations. Several natural extensions of the present results are finally recognized.     

Surface Wave Diffraction on a Floating Elastic Plate

Plane surface wave diffraction by a floating semi-infinite plate is studied. An analytic solution of the problem is constructed by the Wiener-Hopf technique. Analytic formulas for the reflection and transmission coefficients and their shortwave and longwave asymptotics are obtained. An explicit representation for the fluid velocity potential is found. The displacement, strain, and pressure distributions over the plate are investigated as functions of a dimensionless parameter, namely, the reduced rigidity of the plate, and the asymptotic distribution is studied for long and short waves.     

求解Helmholtz方程基于核重构思想的最小二乘配点法

基于核重构思想构造近似函数，将配点法和最小二乘原理相结合对微分方程进行离散， 建立了Helmholtz方程的最小二乘配点格式，并分别研究了Helmholtz方程的波传播问题和 边界层问题. 通过数值算例可以发现，给出的数值计算结果非常接近于精确解，计算精度明显高于SPH 法的数值结果，且随着节点数目的增加，其精确度越来越高，具有良好的收敛性.     

Acoustic radiation force on an elastic cylinder in a Gaussian beam near an impedance boundary

Acoustic radiation force (ARF) is studied by considering an infinite elastic cylinder near an impedance boundary when the cylinder is illuminated by a Gaussian beam. The surrounding fluid is an ideal fluid. Using the method of images and the translation-addition theorem for the cylindrical Bessel function, the resulting sound field including the incident wave, its reflection from the boundary, the scattered wave from the elastic cylinder, and its image are expressed in terms of the cylindrical wave function. Then, we deduce the exact equations of the axial and transverse ARFs. The solutions depend on the cylinder position, cylinder material, beam waist, reflection coefficient, distance from the impedance boundary, and absorption in the cylinder. To analyze the effects of the various factors intuitively, we simulate the radiation force for non-absorbing elastic cylinders made of stainless steel, gold, and beryllium as well as for an absorbing elastic cylinder made of polyethylene, which is a well-known biomedical polymer. The results show that the impedance boundary, cylinder material, absorption in the cylinder, and cylinder position in the Gaussian beam significantly affect the magnitude and direction of the force. Both stable and unstable equilibrium regions are found. Moreover, a larger beam waist broadens the beam domain, corresponding to non-zero axial and transverse ARFs. More importantly, negative ARFs are produced depending on the choice of the various factors. These results are particularly important for designing acoustic manipulation devices operating with Gaussian beams.     

General stability of memory-type thermoelastic Timoshenko beam acting on shear force

In this paper, we consider a linear thermoelastic Timoshenko system with memory effects where the thermoelastic coupling is acting on shear force under Neumann–Dirichlet–Dirichlet boundary conditions. The same system with fully Dirichlet boundary conditions was considered by Messaoudi and Fareh (Nonlinear Anal TMA 74(18):6895–6906, 2011, Acta Math Sci 33(1):23–40, 2013), but they obtained a general stability result which depends on the speeds of wave propagation. In our case, we obtained a general stability result irrespective of the wave speeds of the system.     

Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives

A modified form of Green's integral theorem is employed to derive the energy identity in any water wave diffraction problem in a single-layer fluid for free-surface boundary condition with higher-order derivatives. For a two-layer fluid with free-surface boundary condition involving higher-order derivatives, two forms of energy identities involving transmission and reflection coefficients for any wave diffraction problem are also derived here by the same method. Based on this modified Green's theorem, hydrodynamic relations such as the energy-conservation principle and modified Haskind-Hanaoka relation are derived for radiation and diffraction problems in a single as well as two-layer fluid.     

A note on the diffraction of an obliquely incident surface wave by a partially immersed fixed vertical barrier

The problem of the diffraction of surface waves, obliquely incident on a partially immersed fixed vertical barrier in deep water, is solved approximately by reducing it to the solution of an integral equation, for small angle of incidence of the incident wave. The corrections to the reflection and transmission coefficients over their normal incidence values for small angle of incidence are obtained and presented graphically for some intermediate values of wave numbers.     

Generalization of the Kirchhoff theory to elastic wave diffraction problems

The Kirchhoff approximation in the theory of diffraction of acoustic and electromagnetic waves by plane screens assumes that the field and its normal derivative on the part of the plane outside the screen coincides with the incident wave field and its normal derivative, respectively. This assumption reduces the problem of wave diffraction by a plane screen to the Dirichlet or Neumann problems for the half-space (or the half-plane in the two-dimensional case) and permits immediately writing out an approximate analytical solution. The present paper is the first to generalize this approach to elastic wave diffraction. We use the problem of diffraction of a shear SH-wave by a half-plane to show that the Kirchhoff theory gives a good approximation to the exact solution. The discrepancies mainly arise near the screen, i.e., in the region where the influence of the boundary conditions is maximal.     

On the T-matrix for water-wave scattering problems

A rigid cylinder of infinite length is floating in the free surface of deep water. The cylinder is held fixed and a given time-harmonic wave of small amplitude is incident upon it. The corresponding linear two-dimensional boundary-value problem for a velocity potential φ is treated using the null-field method, and an expression for the T-matrix is obtained. (The T-matrix connects the diffraction potential away from the cylinder to the given incident potential.) Fundamental properties of the T-matrix are derived from considerations of energy and reciprocity. For regular wavetrains incident from the right or from the left, there are well-known relations between the corresponding reflection and transmission coefficients; these relations are recovered by specialising the equations satisfied by the T-matrix. Two extensions to water of constant finite depth are described: one uses multipole potentials whilst the other uses Havelock wavemaker functions; this second approach also leads to a new method for treating the problem of waves in a semi-infinite channel with an end-wall of arbitrary shape.     

On Green's functions for the reduced wave equation in a circular annular domain with dirichlet, Neumann and radiation type boundary conditions

Summary In this paper Green's functions for the reduced wave equation (Helmholtz equation) in a circular annular domain with the Dirichlet, the radiation, and Neumann boundary conditions are derived. The convergence of the series representing Green's functions is then established. Finally it is shown that these functions reduce to Green's function for the exterior of a circle as given by Franz and Etiènne when the outer radius is moved towards infinity.     

Hydrodynamic forces and moments due to interaction of linear water waves with truncated partial-porous cylinders in finite depth

An exact analytical method is employed for studying the diffraction problems in an ocean due to the presence of a specific type of cylinders. In this current work, two models are studied: (i) a floating surface-piercing truncated partial-porous cylinder, (ii) a surface-piercing truncated partial-porous cylinder placed at the bottom. In both cases, the configuration of the composite cylinder is such that it consists of an impermeable inner cylinder rising above the free surface and a coaxial truncated porous cylinder around the lower part of the inner cylinder with the top of the porous cylinder being impermeable. By using linear water wave theory, a three-dimensional representation of the problem is developed based on eigenfunction expansion method. The condition on the porous boundary is defined by applying Darcy’s law. Pressure and velocity satisfy continuity conditions across the linear interface between the adjacent fluid domains. Hydrodynamic force, moment and wave run-up are calculated by using the velocity potentials. Comparisons are carried out with results of wave diffraction by a floating and bottom-mounted compound cylinder, i.e., when the whole cylinder is non-porous. Handy agreements are observed from these comparisons. Through numerical tests, various experiments are carried out to investigate the impact of various parameters, such as porous coefficients, draft ratio, the ratio of inner and outer radii, the water depth etc., on hydrodynamic force, moment and wave run-up. The results clearly indicate that an appropriate optimal ratio for various parameters may be considered in designing practical ocean structures with minimum adverse hydrodynamic effect. The appearance of resonance in the results and role of porosity in mitigating resonance effect are explained. Proposal to select various appropriate parameters for the best possible effect is put forward.     

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。     

Unsteady interaction of shock wave diffracting around a circular cylinder in air

The reflection and diffraction of a planar shock wave around a circular cylinder are a typical problem of the complex nonlinear shock wave phenomena in literature. It has long been studied experimentally, analytically as well as numerically. Takayama in 1987 obtained clear experimental pictures of isopycnics in shock tube under the condition that the impinging shock wave propagates as far as 3 diameters away from the cylinder. To know more completely the whole unsteady process, it is desirable to get experimental results in a region which is more than 10 diameters away from the cylinder. This is what has been done in this paper by using the pulsed laser holographic interferometry for several shock Mach numbers of the impinging shock. Results for several moments are shown, giving more knowledge about the whole unsteady flow field. This is useful for a reliable and complete understanding of the changing force acting on the cylinder, and provides interesting data to check the performance of many recently developed high resolution numerical methods for unsteady shock wave calculation. The project suported partially by National Natural Science Foundation of China     

Some analytic solutions for a simple model of microwave absorber

An absorber, whose permittivity is a continuous function of the wave propagation direction z only, is placed between two planes, z = 0 and z = L, in a straight waveguide with a uniform cross section and perfectly conducting walls. Solutions of Maxwell's equations for the region occupied by the absorber, are given as transverse electric and transverse magnetic waves, and expressed in terms of known functions for certain choices of permittivity. Expressions for reflection and transmission coefficients are deduced.