Rawlins' problem for half-plane diffraction: its generalized eigenfunctions with real wave numbers

This paper deals with a famous diffraction problem for a single half-plane Σ: x>0, y=0 as an obstacle and for some time-harmonic plane incident wave field. Rawlins in 1975 was the first to solve the mixed (Dirichlet/Neumann) boundary value problem for the scalar Helmholtz equation. He also was the first to solve the equivalent pair of coupled Wiener–Hopf equations explicitly by factoring their discontinuous 2×2 Fourier matrix symbol in 1980. Although for real wave numbers k the usual factorization procedure fails it will serve as the basis: Following the lines given by Ali Mehmeti in his habilitation thesis [1] for the (Dirichlet/Dirichlet) boundary value problem we combine the idea of integral path deforming along the branch cuts of the characteristic square root √(ξ²−k²) given in Meister's book [13] with the modern Wiener–Hopf method solution derived by Speck [24] explicitly in a H^1+ε, ε⩾0, Sobolev space setting. The symmetry of the intermediate spaces H^s, H^-s, ∣s∣<1 2, which is due to generalized factorization, plays a key role in deforming the Fourier integral paths in order to get Laplace transform representations of the generalized eigenfunctions of the problem. As a remarkable fact 0<ε<¼ must hold here. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

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.     

Finite element method for the diffraction problem in a waveguide

The convergence of the finite element method (FEM) as applied to the diffraction problem in a waveguide in the case when there is no damping in the medium filling the waveguide is proved. A functional space that takes into account the partial radiation conditions is introduced to carry out the proof. A highly accurate approximation method for the partial radiation conditions is considered.     

Optimal control problem for steady-state equations of acoustic wave diffraction

An optimal control problem is considered for the steady-state equations of acoustic wave diffraction caused by a three-dimensional inclusion in an unbounded homogeneous medium. The task is to minimize the L ²-deviation of the pressure field inside the inclusion from a certain prescribed value due to changing the field sources in the external medium. The solvability of the problem is proved. A solution algorithm is proposed, and its convergence is proved.     

用Backus-Gilbert方法求解声波散射问题

利用位势理论将散射问题的外边界问题转化为第一类边界积分方程求解,再利用Backus-Gilbert方法给出了二维空间的数值结果,与Tikhonov正则化方法比较,虽然精度稍差一些,但是计算方法和计算机实现比较简单.     

Finite difference formulas in the complex plane

Numerical Algorithms - Among general functions of two variables f(x, y), analytic functions f(z) with z = x + iy form a very important special case. One consequence of analyticity turns out to be...     

On diffraction in a piezoelectric medium by a half-plane: The Sommerfeld problem

This paper is concerned with the diffraction problem in a transversely isotropic piezoelectric medium by a half-plane. The half-plane obstacle considered here is a semi-infinite slit, or a crack; both its surfaces are traction free and electric absorbent screens. In a generalized sense, we are dealing with the Sommerfeld problem in a piezoelectric medium.¶The coupled diffraction fields between acoustic wave and electric wave are excited by both incident acoustic wave as well as incident electric wave; and the sound soft and electric "blackness" conditions on the screens are characterized by a system of simultaneous Wiener-Hopf equations. Closed form solutions are sought by employing special techniques. Some interesting results have been obtained, such as mode conversions between acoustic wave and electric wave, novel diffraction patterns in the scattering fields, and the effect of electroacoustic head wave, as well as of surface wave-Bleustein-Gulyaev wave.¶Unlike the classical Sommerfeld problem, in which the only concern is the scattering field of electric wave, the strength of material, e.g. material toughness, is another concern here. From this perspective, relevant dynamic field intensity factors at the crack tip are derived explicitly.     

Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies

The scalar problem of plane wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space ?³ but rather everywhere except for the screen edges. The original boundary value problem for the Helmholtz equation is reduced to a system of weakly singular integral equations in the regions occupied by the bodies and on the screen surfaces. The equivalence of the integral and differential formulations is proven, and the solvability of the system in the Sobolev spaces is established. The integral equations are approximately solved by the Bubnov-Galerkin method. The convergence of the method is proved, its software implementation is described, and numerical results are presented.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

The difference method in a diffraction problem: The technique of interior boundary condition

The diffraction problem for a plane wave on a half-plane covered by thin layer with an interface is solved by the difference method. The system of difference equations is derived from the variational principle. A boundary solution at infinity must be imposed; this is a radiation condition, which is used in the form of the limit absorption principle. The arising infinite system of difference equations is reduced to a finite part of the boundary (the interface) by using the technique of so-called interior boundary conditions in the sense of Ryaben’kii. The real conditions are found by the Fourier method with respect to one spatial variable in the form of Fourier or Laurent series in the corresponding variable, which converge either inside, outside, or on the unit circle. Above the upper boundary of the layer, all unknowns are eliminated by using the so-called grid Green function, that is, the resolving function for the half-plane satisfying the radiation condition at infinity. For the unknowns on the upper boundary of the layer, an equation in terms of a function of a complex variable of Wiener-Hopf type is obtained, which is solved by factorization. Factorization is performed numerically: the logarithm of the function is expanded in a bi-infinite series, which is replaced by a discrete Fourier series. The closing system in a neighborhood of the interface has order proportional to the number of points on the interface. Solving this system yields all of the required characteristics of the solution.     

High-frequency projection method for plane problems of wave diffraction on dielectric bodies

A version of the Galerkin incomplete projection method is described for plane problems of wave diffraction on dielectric bodies of arbitrary shape. The proposed method generalizes the Sommerfeld method, which constructs diffraction series rapidly converging at high frequencies for circular and spherical bodies. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 58–67.     

Pseudodifferential operator method in a problem on the diffraction of an electromagnetic wave on a dielectric body

We study the problem on the diffraction of electromagnetic waves on a solid body in free space. To analyze the integro-differential equations describing this phenomenon, we use the theory of pseudodifferential operators. We evaluate the asymptotic expansion of the symbol and prove the ellipticity and Fredholmness with index zero of the problem operator.     

The diffraction problem for a half plane with different face impedances revisited

The problem of the diffraction of an electromagnetic wave by a half plane with different face impedances is dealt with, following a rigorous approach based on the [L₂⁺( ]₂ theory of systems of Wiener-Hopf equations with piecewise continuous presymbols. The corresponding operator is defined in spaces of physically admissible solutions, the Sobolev spaces H_α⁺( )×H_α−1⁺( ) for , and its Fredholm characteristics are determined. For it is shown that the operators are invertible and their inverses are calculated. In the final section the inverse of a related operator presented by Meister and Speck is also obtained.     

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B‐spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.     

On wave diffraction by a half‐plane with different face impedances

The impedance wave diffraction problem by a half‐plane screen is revisited in view of its well‐posedness upon different impedance and wave parameters. The problem is analysed with the help of potential and pseudo‐differential operators. Seven conditions between the impedance and wave numbers are found under which the problem will be well‐posed in Bessel potential spaces. In addition, an improvement of the regularity of the solutions is shown for the previous seven conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

The Hele-Shaw problem with surface tension in a half-plane

In this paper, we consider the Hele-Shaw problem in a 2-dimensional fluid domain Ω(t) which is constrained to a half-plane. The boundary of Ω(t) consist of two components: Γ₀(t) which lies on the boundary of the half-plane, and Γ(t) which lies inside the half-plane. On Γ(t) we impose the classical boundary conditions with surface tension, and on Γ₀(t) we prescribe the normal derivative of the fluid pressure. At the point where Γ₀(t) and Γ(t) meet, there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure. We prove that the problem has a unique solution with smooth free boundary Γ(t) for some small time interval.     

The edge wave in the problem of diffraction on a boundary with jump of curvature

The problem of the diffraction of creeping waves on a point of transition of the convex boundary to the straight boundary of a domain is investigated. It is assumed that at the point of jump of curvature, the tangent to the boundary is continuous and its derivative has a jump. An expression for the edge wave is obtained and investigated. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 274–287. Translated by N. Ya. Kirpichnikova.