Wiener-Hopf Research Articles

Diffraction by a Right-Angled No-Contrast Penetrable Wedge: Analytical Continuation of Spectral Functions

Summary We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener–Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener–Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in C2. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem, and that we can reformulate the physical diffraction problem as a functional problem using this concept.

The strain gradient viscoelasticity full field solution of mode-III crack problem

The size and viscosity effects are noticeable at the micro-/nano scale. In the present work, the strain gradient viscoelastic solution of the mode-III crack in an infinite quasi-brittle advanced material is proposed based on the strain gradient viscoelasticity theory using the Wiener–Hopf method. The solutions to the gradient-dependent viscoelastic crack problem are obtained directly by using the correspondence principle between the strain gradient viscoelasticity and strain gradient elasticity in Maxwell’s standard linear solid model. In this model, the stress near the crack tip is time-dependent and size-dependent. Besides, the stress near the crack tip is more significant than that based on gradient elasticity theory. Compared with the elastic strain gradient effect, the viscous gradient effect makes the stress field at the crack tip harden. The location and the value of maximum stress change with time, which differs from the case in strain gradient elasticity theory. The time that the normalized stress takes to stabilize also changes with the distance from the crack tip. When the viscosity effect is neglected or time tends to infinity, the strain gradient viscoelasticity theory can be reduced to the classical strain gradient elasticity theory.

On the stability of transient bimaterial contact in the presence of dry friction and slip

An analytic formulation for the study of planar elastodynamic contact in bimaterial interfaces subject to dry friction and slip is presented. Using the Wiener–Hopf technique, explicit analytical expressions for the elastic waves scattered by a planar contact interface are derived. Two cases are studied: (1) the uncoupled problem where it is assumed that normal and tangential loads cause, respectively, no tangential and vertical displacements; (2) the fully coupled problem, where they do. In both cases, an analytic formulation for determining the magnitude of the regions of stick and slip expected at the interface is offered. This provides a complete analytic account of the interfacial tractions and, as a results, serves to model the elastic wave scattering by contact interfaces, a problem of interest in fields as disparate as geophysics, non-destructive testing, and fracture mechanics. The uncoupled problem is shown to be inconsistent: the magnitude of the reciprocal displacements caused by the normal/tangential loads is non-negligible, and of the same magnitude as the interfacial slip distribution itself. The coupled problem is shown to lead to a matricial Wiener–Hopf problem the scattering kernel of which is non-commutative; an Abrahams approximation reliant on the Padé approximants is used to study the problem. It is shown that in many circumstances, the bimaterial contact interface is bound to detach locally, even when the far-field, steady-state solutions would predict otherwise. This is brought about by interfacial loading mismatches brought about by the coupling, and is shown to be affected by various factors: weak pressure loads, the friction coefficient, the disparity in the elastic constants of the media under contact. An useful analytic criteria for detachment is offered.

Plane Potential Field Outside a Symmetric T-Shaped Profile

We consider a new plane problem of the potential theory outside a symmetric T-shaped profile. In view of the symmetry of the profile, we reduce the analyzed problem to a mixed boundary-value problem in a half plane with perpendicular cut whose solution is constructed by the method of partial domains with the use of polar coordinates and the Mellin integral transformation. The original problem is reduced to a system of two Wiener–Hopf equations whose solution is reduced to a completely regular infinite system of linear algebraic equations. The solution of the infinite system is approximately found by the method of reduction. The obtained explicit expressions for the unknown harmonic function make it possible to efficiently determine its values for all possible values of the arguments.

On the Domino Shuffle and Matrix Refactorizations

This paper is motivated by computing correlations for domino tilings of the Aztec diamond. It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is, the two-periodic Aztec diamond. One of the methods, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener–Hopf factorization for two-by-two matrix-valued functions, involves the Eynard–Mehta Theorem. For arbitrary weights, the Wiener–Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. This paper shows that, for arbitrary weightings of the Aztec diamond, the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. In particular, these dynamics can be used to find the inverse of the LGV matrix in the Eynard–Mehta Theorem.

Exact Solution of the Wiener–Hopf Equation on the Segment for Contact Problems of the Theory of Cracks in a Layered Medium

Exact Solution of the Wiener–Hopf Equation on the Segment for Contact Problems of the Theory of Cracks in a Layered Medium

Diffraction by a Semi-Infinite Parallel-Plate Waveguide with Five-Layer Material Loading: The Case of H-Polarization

In this paper, the plane wave diffraction from a semi-infinite parallel-plate waveguide with five-layer material loading is rigorously analyzed for H-polarization using the Wiener–Hopf technique. The Fourier transform of the scattered field is introduced and boundary conditions are applied in the transform domain to formulate the problem as simultaneous Wiener–Hopf equations, which are solved by the factorization and decomposition procedure leading to exact and approximate solutions. The scattered field in real space is explicitly derived by taking the Fourier inverse of the solution in the transform domain. For the region inside the waveguide, the scattered field is represented by the waveguide TM modes, and the field outside the waveguide is evaluated asymptotically by applying the saddle-point method to obtain a far-field expression. Numerical examples of the radar cross section (RCS) for various physical parameters are presented, and far-field scattering characteristics of the waveguide are discussed in detail.

Wiener–Hopf Difference Equations and Semi-Cardinal Interpolation with Integrable Convolution Kernels

Let Hsubset {mathbb {Z}}^d be a half-space lattice, defined either relative to a fixed coordinate (e.g. H = {mathbb {Z}}^{d-1}!times !{mathbb {Z}}_+), or relative to a linear order preceq on {mathbb {Z}}^d, i.e. H = {jin {mathbb {Z}}^d: 0preceq j}. We consider the problem of interpolation at the points of H from the space of series expansions in terms of the H-shifts of a decaying kernel phi . Using the Wiener–Hopf factorization of the symbol for cardinal interpolation with phi on {mathbb {Z}}^d, we derive some essential properties of semi-cardinal interpolation on H, such as existence and uniqueness, Lagrange series representation, variational characterization, and convergence to cardinal interpolation. Our main results prove that specific algebraic or exponential decay of the kernel phi is transferred to the Lagrange functions for interpolation on H, as in the case of cardinal interpolation. These results are shown to apply to a variety of examples, including the Gaussian, Matérn, generalized inverse multiquadric, box-spline, and polyharmonic B-spline kernels.

The Sommerfeld Integral in Problems of Simulating the Diffraction of Acoustic Waves Using a Triangular Lattice

Analytical solutions are obtained for two problems for a discrete analogue of the Helmholtz equation on a triangular lattice: (1) the problem of radiation from a point source on a plane; (2) the problem of diffraction on a half-line with Dirichlet boundary conditions. It is shown that in these problems, the complete field can be represented as an integral of an algebraic function over a family of contours located on some complex manifold. The solution to the first problem is found as an integral of some differential form over this manifold, and the asymptotics of the far field for this solution is obtained. The second problem is solved using an analogue of the Sommerfeld integral. It is checked that the obtained solution coincides with the solution of this problem by the Wiener–Hopf method.

Mathematical modeling of electromagnetic radiations incident on a symmetric slit with Leontovich conditions in an-isotropic medium

This article presents the rigorous analysis of the an-isotropic medium affecting the diffraction process of EM-signals of high frequency by a symmetric slit of finite width. Leontovich boundary conditions for the present model are taken into consideration to show the existence of impedance. The ionosphere having non-thermal plasma contains the geomagnetic field which turns the medium into an an-isotropic medium. The well-known Helmholtz equation for electromagnetic (EM) waves is formulated through Maxwell’s equations with consideration of dielectric permittivity tensor for an an-isotropic medium due to non-thermal plasma. The Fourier transform is employed for the equations formulated for this model and then a pair of Wiener–Hopf equations (coupled form) is devised. These Wiener–Hopf equations are then tackled and an asymptotic method is used to achieve the target. Results for the isotropic medium can be obtained by assigning the values to permittivity elements controlling the medium. Graphical results are sketched for better visualization of the separated field for the an-isotropic medium and isotropic medium as well.

Mathematical analysis of electromagnetic radiations diffracted by symmetric strip with Leontovich conditions in an-isotropic medium

The current article explores the meticulous analysis of the an-isotropic medium affecting the high-frequency signals diffracted by a conductible finite symmetric strip. A symmetric strip of finite length is considered here with impedance characteristics. The non-thermal plasma in the ionosphere contains a geomagnetic field which makes the surrounding medium as an an-isotropic medium. This idea is devised, mathematically, to analyze the propagation and diffraction pattern of EM-waves in the an-isotropic medium. Maxwell's equations modeled for non-thermal plasma are used to devise the EM-wave equation. Time is executed harmonically throughout the investigation. A coupled system of Wiener–Hopf equations is formulated and kernel functions are put to product decomposition to get the integral form of desired result. This integral form is then coped by employing the modified stationary phase method which yields an asymptotic expression of the separated field in the an-isotropic medium. The model can be turned out into a model for the isotropic medium by allocating the specific numeric values to the parameters controlling the medium. Physical interpretation of the separated field (diffraction by finite symmetric strip) is described for both the an-isotropic as well as the isotropic medium.

Contact of the Faces of an Interface Semiinfinite Crack

We consider the equilibrium of two rigidly connected elastic half planes made of different materials, containing a semiinfinite crack on their interface, and subjected to the action of normal and tangential concentrated forces applied to the crack faces. We take into account the friction contact of the crack faces near the crack tip and at a certain distance from the tip. The solution of integral equation of the analyzed problem is obtained in the closed form with the help of the Wiener–Hopf method. We determined the boundaries of contact zones between the crack faces and the distributions of stresses in these zones and on the interface of half planes outside the crack.

Utilization of the ExactMPF package for solving a discrete analogue of the nonlinear Schrödinger equation by the inverse scattering transform method

We are revisiting the problem of solving a discrete nonlinear Schrödinger equation by the inverse scattering transform method, by use of the recently developed ExactMPF package within MAPLE Software. ExactMPF allows for an exact Wiener–Hopf factorization of matrix polynomials regardless of the partial indices of the matrix. The package can be widely used in various problems, where Wiener–Hopf factorization as one of the effective mathematical tools is required, as its code has already been disclosed. The analysis presented in this paper contains not only numerical examples of its use, but is also supported by appropriate and accurate a priori estimations. The procedure itself guarantees that the ExactMPF package produces all computations arithmetically exactly , and a detailed numerical analysis of various aspects of the computational algorithm and approximation strategies is provided in the case of a finite initial impulse.

A neural network for solving the generalized inverse mixed variational inequality problem in Hilbert Spaces

<abstract><p>In this paper, we study and analyze the generalized inverse mixed variational inequality. The existence and uniqueness of the solution of such problem are proposed. The neural network associated with the generalized inverse mixed variational inequality is presented, and moreover, the Wiener-Hopf equation which the solution of the equation is equivalent to the solution of the generalized inverse mixed variational inequality, is considered. The stability and existence of solution of such neural network are proved. Finally, we introduce some algorithms which are constructed by the concept of the neural network and display a numerical example for understanding our results.</p></abstract>

Analysis of noise attenuation through soft vibrating barriers: an analytical investigation

<abstract><p>In this article, the impact of fluid flow and vibration on the acoustics of a subsonic flow is examined. Specifically, it focuses on the noise generated by a convective gust in uniform flow that is scattered by a vibrating plate of limited size. The study analyzes the interaction between acoustics and structures by considering the scattering of sound waves by a soft finite barrier. To achieve this, the Wiener-Hopf technique is utilized for the analytical treatment of the acoustic model. The approach involves performing temporal and spatial Fourier transforms on the governing convective boundary value problem, then formulating the resulting Wiener-Hopf equations. The product decomposition theorem, an extended version of Liouville's theorem, and analytic continuation are employed to solve these equations. Finally, the scattered potential integral equations are computed asymptotically. This study can be significant for understanding the acoustic properties of structures and how they interact with fluid flow in subsonic environments, which could have applications in fields such as aerospace engineering, noise reduction, and structural acoustics.</p></abstract>

Wiener–Hopf factorization indices of rational matrix functions with respect to the unit circle in terms of realization

As in the paper Groenewald et al. (2017) our aim is to obtain explicitly the Wiener–Hopf indices of a rational m×m matrix function R(z) that has no poles and no zeros on the unit circle T but, in contrast with Groenewald et al. (2017), the function R(z) is not required to be unitary on the unit circle. On the other hand, using a Douglas–Shapiro–Shields type of factorization, we show that R(z) factors as R(z)=Ξ(z)Ψ(z), where Ξ(z) and Ψ(z) are rational m×m matrix functions, Ξ(z) is unitary on the unit circle and Ψ(z) is an invertible outer function. Furthermore, the fact that Ξ(z) is unitary on the unit circle allows us to factor as Ξ(z)=V(z)W∗(z) where V(z) and W(z) are rational bi-inner m×m matrix functions. The latter allows us to solve the Wiener–Hopf indices problem. To derive explicit formulas for the functions V(z) and W(z) requires additional realization properties of the function Ξ(z) which are given in the last two sections.

Physical and Spectral Analysis of a Semi-Infinite Grounded Slab Illuminated by Plane Waves

In this article, we study the scattering problem of a truncated grounded slab illuminated by an arbitrarily incident <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E_{\text{z}}$ </tex-math></inline-formula> -polarized plane wave. We present a solution using the novel semianalytical spectral method based on an original extension of the Wiener–Hopf (WH) technique that uses the concept of characteristic Green’s function and the Fredholm factorization technique. The combination of these mathematical tools allows to extend the capabilities of classical WH method to a new set of problems. One of the main benefits of the proposed semianalytical solution is that it allows the computation of field components similar to what is done with closed-form spectral solutions when available. Physical phenomena, such as the reflection, diffraction, and excitation of surface/leaky waves, are reported. Numerical results validate the proposed methodology.

Wiener–Hopf factorization for arithmetic Brownian motion with time-dependent drift and volatility

In this paper we obtain a Wiener–Hopf type factorization for a real-valued arithmetic Brownian motion with time-dependent drift and volatility. To the best of our knowledge, this paper is the very first step towards realizing the objective of deriving Wiener–Hopf type factorizations for (real-valued) time-inhomogeneous Lévy processes. In order to prove our main theorem, we derive some new results regarding time-inhomogeneous noisy Wiener–Hopf factorization. We demonstrate that in the special case of the arithmetic Brownian motion with constant drift and volatility our main result agrees with classical Wiener–Hopf factorization for this particular time-homogenous Lévy process.

Recognizing the aeroacoustic information of noise radiated by an unflanged duct based on convolutional neural networks.

Accurately recognizing the aeroacoustic information of noise propagating into and radiating out of an aero-engine duct is of both fundamental and practical interest. The aeroacoustic information includes (1) the acoustic properties of the noise source, such as the frequency (f) and the circumferential and radial mode numbers (m, n), and (2) the flight conditions, including the ambient flow speed (M0) and the jet flow speed (M1). In this study, a data-driven model is developed to predict the aeroacoustic information of a simplified aero-engine duct noise from the far-field sound pressure level directivity. The model is constructed by the integration of one-dimensional convolutional layers and fully connected layers. The training and validation datasets are calculated from the analytical model for noise radiation from a semi-infinite unflanged duct based on the Wiener-Hopf method. For a single-spinning mode source, a regression model is established for f, M0, and M1 prediction, and a classification model is built up for m and n prediction. Additionally, for a multi-spinning mode source, the regression model is used to predict the coefficient of each mode. Results show that the proposed data-driven model can effectively and robustly predict the acoustic characteristics of noise propagation in and radiation out of an aero-engine bypass duct.

Radiation of sound waves by semi-perforated duct

In this paper, the Wiener-Hopf method is studied to solve the radiation of acoustic waves from a perforated part through an infinite channel. At first the boundary value problem involving a Helmholtz equation with mixed-boundary conditions is formulated into a Wiener Hopf equation by applying the Fourier transform. After that the Wiener-Hopf kernel is split into two functions that are analytic in specified regions of the complexplane and equation is solved analytically. Finally, numerical results showing the effect of the problem’s parameters (channel radius, frequency, etc.) on the radiation phenomenon are graphically displayed.