An explicit solution of the pseudo-hyperbolic initial boundary value problem in a multiply connected region

An explicit solution of the pseudo-hyperbolic initial boundary value problem with a mixed boundary condition has been constructed. The problem describes the propagation of non-stationary internal waves in a stratified and rotational fluid. The generation of waves is caused by small oscillations of double-sided plates beginning at time t = 0. Dynamic pressure is specified on one set of plates and this yields the first boundary condition. Normal velocities are specified on another set of plates and this leads to an analogue of the second boundary condition with time derivatives. The solution has been obtained by the method of non-classical time-dependent dynamic potentials. The uniqueness of the solution has been studied.     

Sur les équations de la magnétothermoélasticité

Summary Writing the MTE equations in non-dimensional variables, the author points out the essential parameters and the structure of the equations. The equation of the MTE system characteristics is determined, as well as the displacement velocities of the waves of weak discontinuity. Furthermore, the equation which is satisfied by the current density vectorJ is determined.The paper considers the Cauchy problem for the infinite perfectly conductive plane, in the presence of a temperature and of a magnetic field. The general solution for the current density and the magnetic field are determined from Maxwell's equation. The solution obtained permits one to determine the displacement and the temperature fields by means of the classical thermoelasticity.The method of solving the Cauchy problem is based upon the representation of the general solution by a continous superposition of plane waves.     

Exact geophysical waves in stratified fluids

When studying water waves travelling over an inviscid fluid at the Earth's surface there are additional Coriolis and centrifugal forces which influence the motion of the fluid particles. In particular, for waves propagating near the Equator the geophysical wave problem can be modelled by the so-called f-plane approximation. In this paper, we provide an explicit exact solution to the edge wave problem for stratified geophysical flows in the equatorial f-plane approximation.     

Uniqueness via the adjoint problems for systems of conservation laws

We prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws that includes in particular the p-system of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy Condition, or WEC for short, introduced in this paper. The main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution. For the proof of uniqueness, we derive an existence result (respectively a uniqueness result) for the backward (respectively forward) adjoint problem associated with the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients. © 1993 John Wiley & Sons, Inc.     

Anisotropic transient wave motions in a conducting fluid under uniform magnetic and current fields

An initial value investigation into the development of two-dimensional anisotropic surface waves generated by a harmonically oscillating pressure distribution acting on the undisturbed free surface of an inviscid, incompressible homogeneous and electrically conducting fluid is made in this paper in considerable detail. The problem is solved by the use of generalized function treatment in conjunction with asymptotic methods. An asymptotic solution of the problem related to some physically realistic pressure distributions is presented. It is shown that an ultimate steady state is set up in the limit. Two limiting cases such as (i) very deep fluid and (ii) very shallow fluid, which are of particular interest have been examined with some emphasis. Finally, the effects of the imposed magnetic and current fields as well as the surface tension on the wave motions has been examined in some detail. Additionally, it is shown that the present method of solution provides an interesting example of the applicability of the generalized function method in problems of magnetohydrodynamics     

Numerical analysis of the dispersion of electroelastic waves in a cylinder

The wave problem of electroelastic waves in a cylinder is reduced to a spectral problem for a system of eight ordinary differential equations. We give an algorithm for numerical solution based on reducing the original boundary-value problem to four Cauchy problems and determining the roots of the dispersion equation by the method of bisection. One figure. Bibliogrpahy: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 149–153.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

弹性波在平面多连通域中的绕射与动应力集中

本文用复变函数论方法研究了弹性波在平面多连通域中的绕射问题,给出了这一问题解的完备逼近序列及边备条件的一般表示。问题归结为无穷代数方程组的求解,使用电子计算机可直接求得解答。特别是,对弱耦合问题,本文提出了渐近求解方法并且使用这个方法详细地讨论了P波对圆孔群的绕射问题。基于绕射波场的解,文中给出了任意形状空腔动应力集中系数的一般算式。     

Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green’s function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found.     

An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating

An a priori error estimate using a so called α,β‐ periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β‐quasi periodic solution are used to transform the associated Helmholtz problem. The well‐posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β‐quasi periodic method converges at a far lower number of degrees of freedom than the α,0‐quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer. Copyright © 2012 John Wiley & Sons, Ltd.     

Propagation of kelvin waves from a channel into a semibounded tank

The problem of propagation of Kelvin waves from a channel into a semibounded tank is considered. An exact solution of the problem is constructed using the Wiener — Hopf method. The solution is analyzed asymptotically and numerically. The Wiener — Hopf method was used in /1,2/ to solve the problem of diffraction of the Kelvin waves in tanks bounded by the infinite and semiinfinite parallel walls. Below a generalized method of matching /3/ is used to solve the problem of diffraction of Kelvin waves in the case when the walls confining the fluid meet at the right angles.     

Self-similar problem of Cauchy-Poisson waves at an inclined shore : PMM vol. 37, n6, 1973, pp. 1035–1039

We consider a two-dimensional problem concerning Cauchy-Poisson waves at an inclined shore in the case of an initial disturbance concentrated near the shore edge. We study the behavior of the solution near the shore and at large distances from it.Numerous investigations, devoted to the study of standing and progressive waves on an inclined shore, are described in [1]. A two-dimensional problem concerning nonstationary waves on a shore with an angle of inclination γ = π/2n, where n is an integer, was analyzed in [2, 3]. We consider below a case in which the angle of inclination is commensurable with λ/2, subject to the condition that the initial disturbance is concentrated in the vicinity of the shore edge, so that the problem may be considered self-similar.     

Diffraction of a plane wave by a step discontinuity in an impedance plane

In the present paper, the high-frequency diffraction of a plane wave by a right-angle step discontinuity in an impedance plane is analyzed with the help of the uniform geometric theory of diffraction, beginning with the Maluzhenets solution. The principal term of an asymptotic solution of the problem, which is uniform with respect to the angle of incidence of the plane wave and the angle of observation, is derived. The excitation of primary and multiply diffracted fields radiated from the upper edge, as well as surface waves, is considered (the lower edge does not radiate cylindrical or surface waves owing to a right-angle step). For simplicity, the details of computation are given here for a right-angle step discontinuity. A similar procedure is applied to other examples with more complicated geometry. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 97–108. Translated by V. V. Zalipaev.     

Erratum to: Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.     

A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

In this paper, we consider the propagation of waves in a closed full or half waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition that assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, eg, considered recently by Fliss and Joly and allows an extension to open wave guides. After application of the Floquet‐Bloch transform, we consider the Bloch variable α as a parameter in the resulting quasiperiodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result, which goes back to Colton and Kress.     

INVERSE SCATTERING BY AN INHOMOGENEOUS PENETRABLE OBSTACLE IN A PIECEWISE HOMOGENEOUS MEDIUM

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play important roles for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.     

An Integral Simplex Algorithm for Solving Combinatorial Optimization Problems

In this paper a local integral simplex algorithm will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex algorithm creates and solves the problems in a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc. The solution to at least one of these subproblems is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is an optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n.A method for programming the algorithms in this paper to run on parallel computers is discussed briefly.     

Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play important roles for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.     

Comparison of displacement and stress fields upon diffraction of elastic waves at different defects: Cracks and thin rigid intrusions

This paper considers the problem of diffraction of elastic waves at rectilinear defects (cracks or thin rigid intrusions) in an unbounded elastic medium. A discontinuous solution of the Helmholtz equation is used to reduce the problem to a singular integral equation in unknown jumps. There is a detailed analysis and comparison of the stressed state close to the defects and the diffraction field at large distances from the defect for defects of different types. Translated fromDinamicheskie Sistemy. Vol. 12, pp. 14–23, 1993.