REFLECTION AND RADIATION OF A WAVE SYSTEM AT THE OPEN END OF A SUBMERGED ELASTIC PIPE

The reflection and radiation of a wave system at the open end of a submerged.semi-infinite elastic pipe are studicd.This wave system consists of a flexural wave in the pipe,anacoustic surface wave in the fluid exterior to the pipe and an acoustit wave in the pipe’sinterior.Fourier transfrom techniques are used to formulate this semi-infinite geometryproblem rigorously as a Wiener-Hopf type equation.An approximate solution is obtainedby using a perturbation method in which the ratio of the massdensities of the fluid and thepipe material is regarded as a small parameter.The calculation of the reflection coefficientis emphasized,and the polar plots of the radiation coefficient are also presented.     

PROPAGATION OF A L0NG WAVE IN A CURVED DUCT(I) BASIC ANALYSIS OF LONG WAVE PROPAGATION IN A CURVED DUCT WITH VARIABLE CROSS SECTION

The propagation of a long wave in a three-dimensional curvedduct with variable cross section is studied in this paper.It isshown that a three-dimensional Helmholtz equation can be decom-posed into a two-dimensional Laplace(or Poisson)equation and aone-dimensional Webster equation by the curvilinear orthogonalcoordinate system,non-dimensionization of reduced wave equationand regular perturbation with small parameter ka,where k isthe wave number and a is the characteristic radius of the duct.The influences of the duct’s geometric parameters(the area varia-tion of the cross section,the curvature and torsion of the cen-tral line)on the asymptotic expansion of the solution are analy-sed.It is concluded that the effects of the variation of thecross sectional area first appear in the first term of the asym-ptotic ezpansion,and when the cross section shape has certainsymmetric properties,the effects of the curvature and torsion ofthe central line first appear in the third and the fourth terms,respectively.An e     

THE FULLY PLASTIC J-INTEGRAL OF CORNER CRACKS AT A HOLE IN A PLATE

The line-spring model of surface cracks is applied to the fully plastic analysis of cornercracks at a hole in a plate.The generalized fully plastic constitutive relations and the fully plastic J-in-tegral,as well as its coefficients in polynomial expressions are given.The model obtained is incorpo-rated into a finite element program.The corner cracks at a hole in a plate subjected pure tensions arecalculated by the present model.The fully plastic J-integral is then estimated.The results obtainedshow that the line-spring model is effective for the analysis of corner cracks.The influence of thecrack depth and the hardening exponent on the fully plastic J-integral is also discussed.     

Reflection and radiation of a wave system at the open end of a submerged elastic pipe

The reflection and radiation of a wave system at the open end of a submerged semi-infinite elastic pipe are studied. This wave system consists of a flexural wave in the pipe, an acoustic surface wave in the fluid exterior to the pipe and an acoustic wave in the pipe’s interior. Fourier transform techniques are used to formulate this semi-infinite geometry problem rigorously as a Wiener-Hopf type equation. An approximate solution is obtained by using a perturbation method in which the ratio of the massdensities of the fluid and the pipe material is regarded as a small parameter. The calculation of the reflection coefficient is emphasized, and the polar plots of the radiation coefficient are also presented.     

WAVE PROPAGATION WITH MASS-COUPLING EFFECT IN FLUID-SATURATED POROUS MEDIA

This article utilizes the theory of mixtures to formulate a general theory of wavepropagation with mass-coupling effect in fluid-saturated porous media.An attempt is madeto discuss the physical interpretation and the thermodynamic restriction of the coefficientsappearing in the equations obtained.By the comparison it is shown that Biot’s classicaltheory and the present one are essentially consistent.Also,wave velocities in some specialcases are calculated,from which it is concluded that mass-coupling and permeability ofmedia greatly affect wave propagation behavior.     

THE ESTIMATE AND MEASUREMENT OF TRANSVERSE WAVE INTENSITY

Transverse waves are a type of structural waves and should be considered in the analy-sis of high frequency vibration because the energy carried by transverse waves increases with the in-crease of frequency and becomes important at high frequencies. This paper studies the estimate theoryand measuring technique of the transverse wave intensity in two dimensional homogeneous structures.In general, the intensity vector is the sum of the effective intensity vector and the intensity variationvector. Each axial intensity component is proportional to two imaginary parts of cross spectral densitiesand its estimate is complicated. For the special case where transverse waves propagate in one direction,the intensity variation is zero and the estimate of the intensity is simplified. The intensity technique isformed based on the finite difference principle. Transverse wave intensity can be measured using a pairof two-transducer arrays lying in the orthogonal direction for the general case or a two-transducer ar-ray lying in the propagating direction for the special case. In order to assess the measurement accuracyof transverse wave intensity, the coupling loss factors from bending to transverse waves in buildingstructures were measured using the intensity technique and compared with the results predicted andmeasured using the conventional method. It is shown that the agreement between the results measuredusing the intensity technique and that by the conventional method is good.     

Transition of axial segregation patterns in a long rotating drum

Axial segreganon or a bidisperse mixture of particles in a long rotating drum is studied using the discrete element method. Simulation results show that particle interaction is responsible for axial segregation, the patterns of which are influenced by the end wall effect. Axial segregation patterns transform under competing influences of the end walls and the particle interaction forces. The two influential factors vary with various rotational speeds and end wall friction levels. The result is the transition of different axial segregation patterns： two large-particle bands at both ends, two small-particle bands at both ends, or a random segregation pattern where either a large-particle band or small-particle band may appear at either end.     

THE BOUNDARY ELEMENT METHOD (BEM) FOR WAVE EQUATION

The formula of BEM suited to solve the problems of wave propagation in boundlessmedium is obtained from numerical treatment of Kirchhoff integral equation.Afer quotingthe coefficients of refraction and reflection of wave at surface or interface,the expression ofBEM which is suitable for the problems of wave propagation in multi-isotropic mediums isalso given.     

A THEORETICAL INVESTIGATION ON THE WAVE FORCES ON THE MULTIPLE CYLINDERS IN SHALLOW WATER

The diffraction problem of two kinds of shallow water wave,cnoidal wave and solita.wave, around a group of cylinders is discussed. A Bessel corrdinate transformation (?)employed to uniform the coordinate system, and thus the boundary condition on eachcylinder＇s surface can be satisfied by determining the coefficients in the solution.Severalexamples are calculated for two kinds of incident wave and various arrangement of thecylinders,and the results are discussed and compared with the available experimental data.     

A NUMERICAL METHOD OF MODELING ELASTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

The velocity-stress finite-difference method is adopted to simulate the elastic wave propa-gation in azimuthal anisotropic media.The difference grids are completely staggered in the numerical im-plementation.To reduce the computational work,the absorbin8 boundary conditions for anisotropic mediaare introduced first and the corner points are specially treated.Examples show that more accurate resultscan be obtained from the modeling algorithm,which cost much less computational time than the conven-tional methods.Therefore,the algorithm has broad application prospects in engineering.     

Propagation of a long wave in a curved duct (I) basic analysis of long wave propagation in a curved duct with variable cross section

The propagation of a long wave in a three-dimensional curved duct with variable cross section is studied in this paper. It is shown that a three-dimensional Helmholtz equation can be decomposed into a two-dimensional Laplace (or Poisson) equation and a one-dimensional Webster equation by the curvilinear orthogonal coordinate system, non-dimensionization of reduced wave equation and regular perturbation with small parameterka, wherek is the wave number anda is the characteristic radius of the duct. The influences of the duct's geometric parameters (the area variation of the cross section, the curvature and torsion of the central line) on the asymptotic expansion of the solution are analysed. It is concluded that the effects of the variation of the cross sectional area first appear in the first term of the asymptotic expansion, and when the cross section shape has certain symmetric properties, the effects of the curvature and torsion of the central line first appear in the third and the fourth terms, respectively. An example of long wave propagation in a curved circular duct is also given at the end of this paper.     

A long wave low frequency motion model for a finitely sheared elastic layer

We consider two-dimensional long wave low frequency motion in a pre-stressed layer composed of neo-Hookean material. Specifically, the pre-stress is a simple shear deformation. Derivation of the dispersion relation associated with traction-free boundary conditions is briefly reviewed. Appropriate approximations are established for the two associated long wave modes. From these approximations it is clear that there may be either two, one or no real long wave limiting phase speeds. These approximations are also used to establish the relative asymptotic orders of the displacement components and pressure increment. Using these relative orders to motivate the introduction of appropriate a scales, an asymptotically consistent model long wave low frequency motion is established. It is shown that in the presence of shear there is neither bending nor extension, or analogues of their previously established pre-stressed counterparts. In fact, both the in-plane and normal displacement components have the same asymptotic orders and the derived governing equation is of vector form.     

Abnormal long wave dispersion phenomena in a slightly compressible elastic plate with non-classical boundary conditions

A two parameter asymptotic analysis is employed to investigate some unusual long wave dispersion phenomena in respect of symmetric motion in a nearly incompressible elastic plate. The plate is not subject to the usual classical traction free boundary conditions, but rather has its faces fixed, precluding any displacement on the boundary. The abnormal long wave behaviour results in the derivation of non-local approximations for symmetric motion, giving frequency as a function of wave number. Motivated by these approximations, the asymptotic forms of displacement components established and long wave asymptotic integration is carried out.     

A uniform asymptotic analysis of dispersive wave motion

The signaling problem for the one dimensional Klein-Gordon equation with spatially varying coefficients is analyzed. A formal, uniformly valid, asymptotic expansion of the solution is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which we call the diffraction functions. The validity of the expansion is established when the coefficients in the Klein-Gordon equation are constants, and the results are applied to a signaling problem for a class of acoustic wave guides.     

Three-dimensional propagation of the transmitted shock wave in a square cross-sectional chamber

An investigation into the three-dimensional propagation of the transmitted shock wave in a square cross-section chamber was described in this paper, and the work was carried out numerically by solving the Euler equations with a dispersion-controlled scheme. Computational images were constructed from the density distribution of the transmitted shock wave discharging from the open end of the square shock tube and compared directly with holographic interferograms available for CFD validation. Two cases of the transmitted shock wave propagating at different Mach numbers in the same geometry were simulated. A special shock reflection system near the corner of the square cross-section chamber was observed, consisting of four shock waves: the transmitted shock wave, two reflection shock waves and a Mach stem. A contact surface may appear in the four-shock system when the transmitted shock wave becomes stronger. Both the secondary shock wave and the primary vortex loop are three-dimensional in the present case due to the non-uniform flow expansion behind the transmitted shock.PACS: 43.40.Nm     

Lower-dimensional long wave dynamic models for idealised fibre-reinforced elastic structures

The dispersion of harmonic waves in an idealised fibre-reinforced elastic layer is investigated. Guided by a numerical and asymptotic long-wave investigation of the dispersion relation, appropriate scales are introduced to help elucidate features of long wave high- and low-frequency motion. In the former case, the stress–strain-state is determined in terms of the long-wave amplitude, appropriate leading-order and refined second-order governing equations being obtained from the second- and third-order problems, respectively. At each order the dispersion relation associated with the governing equation agrees with the appropriate expansion of the exact dispersion relation. With respect to low-frequency motion, the long wave limit of anti-symmetric motion is non-zero. This contrasts with the classical case and also indicates that inextensible fibres preclude classical bending. The asymptotic long-wave low frequency stress–strain-state is determined in terms of the governing extensions and mid-surface deflection in the symmetric and anti-symmetric cases, respectively. Appropriate leading and second-order governing equations are also found for these functions. The second-order equations act both to refine the stress–strain-state and also provide the leading-order governing equation in the vicinity of the appropriate quasi wave front. This phenomenon is illustrated by considering a problem concerning shock edge loading of a semi-infinite layer.The work of the second author (R.M.) is supported by a grant from the University of Salford Research Promotion Fund. This award is very gratefully acknowledged.     

Asymptotic long wave models for a pre-stressed elastic layer with elastically restrained boundaries

Long wave dispersion phenomena is investigated in respect of a pre-stressed incompressible elastic layer subject to elastically restrained boundary conditions (ERBC). Such conditions can be treated as a generalisation of classical free and fixed-face boundary conditions, allowing investigating of the transition between the Neumann and Dirichlet statements of the problem. Symmetric elastically restrained boundary conditions are introduced, followed by both a numerical investigation and a multi-parameter asymptotic analysis of the dispersion relations. All possible asymptotic regimes are grouped into classes based on the magnitude of the associated restraint parameter. A long wave low frequency model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency models are developed describing asymptotic regimes connected with vibration within the vicinity of the thickness resonances.     

Diffraction of a plane electromagnetic wave by a perfectly conducting elliptic cylinder

The scattering of a plane electromagnetic wave by a perfectly conducting elliptic cylinder is investigated theoretically. The calculations are based upon the expansion of the scattered wave functions in terms of Mathieu functions. Both E- and H-polarized waves are considered. Numerical results, in particular for the scattering cross-section, are presented for cylinders the cross-sectional dimensions of which are up to many wavelengths (e.g. distance between the focal lines up to 20 wavelengths).     

Fluid flow through a network of thin pipes

We study the fluid flow through a network of intersected thin pipes with prescribed pressure at their ends. Pipes are either thin or long and the ratio between the length and the cross-section is considered as the small parameter. Using the asymptotic analysis with respect to that small parameter the effective behaviour of the flow is found. At each junction an explicit formula for computing the value of the pressure is found. The interior layer phenomenon in vicinity of the junction is studied. We generalize the junction formula on the case of adiabatic compressible flow.     

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovski to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.