Scattering by Rough Surfaces: the Dirichlet Problem for the Helmholtz Equation in a Non-locally Perturbed Half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.     

The dynamics of the motion of a flat punch on the boundary of an elastic half-plane

The dynamic contact problem of the motion of a flat punch on the boundary of an elastic half-plane is considered. During motion, the punch deforms the elastic half-plane, penetrating it in such a manner that its base remains parallel to the boundary of the half-plane at each instant of time. In movable coordinates connected to the moving punch, the contact problem reduces to solving a two-dimensional integral equation, whose two-dimensional kernel depends on the difference between the arguments for each of the variables. An approximate solution of the integral equation of the problem is constructed in the form of a Neumann series, whose zeroth term is represented in the form of the superposition of the solutions of two-dimensional integral equations on the coordinate semiaxis minus the solution of the integral equation on the entire axis. This approach provides a way to construct the solution of the two-dimensional integral equation of the problem in four velocity ranges of motion of the punch, which cover the entire spectrum of its velocities, as well as to perform a detailed analysis of the special features of the contact stresses and vertical displacements of the free surface on the boundary of the contract area. An approximate method for solving the integral equation, which is based on a special approximation of the integrand of the kernel of the integral equation in the complex plane, is proposed for obtaining effective solutions of the problem that do not contain singular quadratures.     

Green functions of a quasi-static problem

In this paper Green functions are constructed in analytic form for a deformable half-plane of a quasi-static problem of thermoelasticity when the heat flow on the boundary x₂=0 of the half-plane is zero. To construct the Green functions, certain integral representations are used whose kernels are known Green functions of the corresponding problems of elasticity theory. The functions constructed make it possible to obtain a wide class of new solutions of boundary-value problems of thermoelasticity, in particular solutions for a piecewise homogeneous half-plane. Bibliography: 6 titles. Translated fromObchyslyuwval’na ta Pryklandna Matematyka, No. 77, 1993, pp. 97–104.     

The impact of a plane punch on an elastic half-plane

The transient dynamic contact problem of the impact of a plane absolutely rigid punch on an elastic half-plane is considered. The solution of the integral equation of this problem in terms of the unknown Laplace transform of the contact stresses at the punch base is constructed by a special method of successive approximations. The solution of the transient dynamic contact problem is obtained after applying an inverse Laplace transformation to the solution of the integral equation over the whole time range of the impact process, and the law of the penetration of the punch into the elastic medium is determined from a Volterra-type integrodifferential equation. The conditions for the punch to begin to separate from the elastic half-plane are formulated from the solution obtained, and all the stages of the separation process are investigated in detail. The law of the punch motion on the elastic half-plane and the width of the contact area, which varies during the separation, are then determined from the solution of the Volterra-type integrodifferential equation when an additional condition is satisfied.     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

The Helmholtz equation with impedance in a half-plane

This Note gives answers to the uniqueness and existence questions for solutions of the Helmholtz equation in an half-plane with an impedance or mixed boundary condition. We deal with unbounded domains which boundaries are unbounded too. The radiation conditions are different from the ones that we found in an usual exterior problem due to the appearance of surface waves. We first compute and study the half-plane Green's function to see how the solutions behave at infinity, and second obtain integral representation for these solutions. To cite this article: M. Duran et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Summatory and integral equations in periodic diffraction problems for elastic waves at defects in stratified media

We consider the diffraction problem for an elastic wave on a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a pair summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the pair equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.     

Slow viscous flow past a cavity with infinite depth

Two-dimensional slow viscous flow on infinite half-plane past a perpendicular infinite cavity is considered on the basis of the Stokes approximation. Using complex representation of the two-dimensional Stokes flow, the problem is reduced to solving a set of Fredholm integral equations of the second kind. The streamlines and the pressure and vorticity distribution on the wall are numerically determined.     

The elastoplastic stressed state of a half-plane under local nonstationary heating

We solve the thermoplastic problem for a semi-infinite plate under local nonstationary heating by heat sources. The physical equations are taken to be the relations of the nonisothermic theory of plastic flow associated with the Mises fluidity condition. The solution of the problem is constructed by the method of integral equations and the self-correcting method of sequential loading, where time is taken as the loading parameter. We carry out numerical computations of the stresses in the case of heating a plate with heat output by normal-circular heat sources. We study the problem of optimization of heating regimes in order to introduce favorable residual compressive stresses (from the point of view of hardness) in a given region of a half-plane. Two figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 29–34.     

The motion of a screw dislocation in a wedge-shaped region

The problem of antiplane deformation for a wedge-shaped region containing a uniformly moving screw dislocation is considered. A general solution of the problem is obtained using Laplace and Kontorovich-Lebedev integral transformations. It is shown during the solution that the method is suitable for a wedge apex angle greater than n. The limiting case, when the wedge-shaped region is a half-plane, is also considered. For this case the solution can be simplified considerably.     

局部扰动半平面上的散射问题

该文讨论半平面上有局部扰动情况下的散射问题．通过位势理论,应用边界积分方程的方法研究了该问题解的存在与唯一性．主要方法是运用对称反射,使该无界区域上的散射问题变成一个有界区域上的散射问题,只是这一有界区域的边界不光滑．通过仔细分析相应的边界积分算子,作者得到了其解的存在与唯一性．     

Mixed problem of plane orthotropic elasticity in a half-plane

We consider a mixed problem of plane isotropic elasticity in a half-plane in which the displacement vector and the normal component of the stress tensor are alternately specified on successive intervals of the real axis. We derive a closed-form expression for the solution of this problem, which is similar to the well-known Keldysh–Sedov formula for the half-plane.     

The influence of frictional forces on the interaction between an anisotropic half-plane and an absolutely rigid body with a cavity in its surface

By applying functions of a complex variable we reduce the problem of incomplete mechanical contact with friction between an anisotropic half-plane and a rigid body to solving a singular integral equation of second kind with respect to the height of the gap between the interacting bodies. We give the results of numerical analysis of the geometric contact characteristics of the gap for a particular cavity shape. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

On a forward-backward parabolic equation

Boundary value problems for the equation $$\operatorname{sgn} (x) \cdot u_y - u_{xx} + ku = f(x,y)$$ (where k is a positive constant and ? is a given function) are investigated. The domain of the solutions will be the whole upper half-plane y>0, or the half-plane y>0 cut along the positive y-axis. We are interested in square integrable solutions u, with square integrable generalized derivatives u_y and u_xx. Existence theorems are proved, with an integral equations technique. Thus a theory is developed of Wiener-Hopf integral equations of the first kind with solutions belonging to Sobolev spaces.     

Optimization of the stress tensor in an elastic anisotropic half-plane

We consider the problem of optimizing the components of the stress tensor and their integral characteristics. The normal and tangential forces prescribed on the boundary of the elastic anisotropic half-plane y≥0 are chosen from certain function classes of curvilinear strip type. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 110–116.     

The sommerfeld half-plane problem revisited I: The solution of a pair of coupled Wiener-Hopf integral equations

A study and the solution of an extension of the classical Sommerfeld half-plane problem which leads to a pair of integral equations of the Wiener-Hopf type is given. The method of solution is function theoretic in character and employs a combination of the ideas of Wiener and Hopf and Carleman.     

A Class of Quadratic Integral-Differential Equations

A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform.     

Determination of a harmonic function with minimal gradient deviation

We find a function u(x, y) harmonic in the upper half-plane y>0 satisfying the conditions grad u(x₀,y₀) b|₂ min, . The solution of the problem is expressed in terms of the Poisson integral for the upper half-plane.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 55–57.     

Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

Computing numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions

In this article we compute numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions. A compactly perturbed half-plane Helmholtz problem is used to motivate this calculation, by treating it through integral equation techniques. These require the knowledge of the calculated Green’s function, and lead to a boundary element discretization. The Green’s function is computed using the inverse Fourier operator of its spectral transform, applying an inverse FFT for the regular part, and removing the singularities analytically. Finally, some numerical results for the Green’s function and for a benchmark resonance problem are shown.