Wiener-Hopf Research Articles

Theoretical model of acoustic scattering from a flat plate with serrations

A theoretical model is proposed in this work to study the scattering of sound waves from a serrated flat plat in the presence of a uniform flow, which is of both scientific significance and practical importance. The key contribution is the analytic and rigorous description of the scattering from the laterally periodic serrations by incorporating Fourier series expansions and the Wiener–Hopf method, which collectively give a closed-form analytical solution. To validate and demonstrate the model, a couple of test cases with some representative sinusoidal- and sawtooth-shaped serrations are studied by comparing with a commercial finite element solver. The comparisons show quite good agreement for various set-ups. The subsequent parametric studies further demonstrate the efficiency of the model and the effect of the serrations for noise control. Overall, the proposed theoretical model should be able to assist in studies of low-noise aerofoils and the silent flying capabilities of owls.

THE LINEAR SYSTEMS APPROACH TO LINEAR RATIONAL EXPECTATIONS MODELS

This paper considers linear rational expectations models from the linear systems point of view. Using a generalization of the Wiener-Hopf factorization, the linear systems approach is able to furnish very simple conditions for existence and uniqueness of both particular and generic linear rational expectations models. To illustrate the applicability of this approach, the paper characterizes the structure of stationary and cointegrated solutions, including a generalization of Granger’s representation theorem.

Wiener–Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators

We provide new formulas for the Wiener–Hopf factorization indices of a rational matrix function R which has neither poles nor zeros on the unit circle. In addition, we recover recent results on the Fredholm characteristics of the Toeplitz operator with symbol R via the method of matricial coupling. Furthermore, we present an alternative formula for the index in terms of the Fourier coefficients of R.

Efficient Impedance Eductions for Liner Tests in Grazing Flow Incidence Tube

This paper presents an efficient impedance eduction method for grazing flow incidence tube by using a surrogating model along with the Wiener–Hopf method, which enables rapid acoustic predictions and effective impedance eductions over a range of parametric values and working conditions. The proposed method is demonstrated by comparing to the theoretical results, numerical predictions, and experimental measurements, respectively. All the demonstrations clearly suggest the capability and the potential of the proposed solver for parametric studies and optimizations of the lining methods.

Plane wave scattering by a dielectric loaded slit in a thick impedance screen

The electromagnetic scattering of the plane wave by a slit loaded with dielectric in a thick impedance screen is considered for a rather general case where the surface impedances of the screen planes and vertical sides of the slit have different values. Because of the symmetry of the scattering structure, the image bisection principle is used and the original problem is splitted up into even and odd excitation cases. By introducing the Fourier transform for each case yields a modified Wiener–Hopf equation of the second kind for each excitation. After decoupling these equations by standard transformation, each modified Wiener–Hopf equation is reduced to a pair of coupled Fredholm integral equations of the second kind. The solution of each integral equation involves two sets of infinitely many constants satisfying two infinite systems of linear algebraic equations and Branch-cut integrals which are evaluated approximately. Numerical solutions of these systems are obtained for various values of the parameters such as the surface impedance of the screen, the vertical wall impedance of the slit, the width and the thickness of the slit and the permittivity of the filling material of the slit which permit one to study the effect of these parameters on the diffraction phenomenon. In order to validate the present Wiener–Hopf solution, experimental verification of the dielectric loaded thick metallic slit has been implemented in microwave frequencies with good agreements.

Influence of the Lateral Process Zone Near the Tip of an Interface Crack on the Contact of its Faces

Under conditions of plane deformation, by using the Wiener–Hopf method, we compute a small-scale contact zone near the tip of an interface crack under the condition of formation of the lateral process zone in its vicinity. It is assumed that the dry friction of the faces according to the Coulomb law is realized in the contact zone. We determine the sizes of the contact zone and deduce an expression for the contact stresses and an equation for the determination of the index of singularity of stresses near the crack tip. The influence of the load, friction coefficient, and elastic characteristics of the joined materials on the sizes of the contact zone is investigated. It is shown that, in the presence of the process zone, the length of the contact zone begins to depend not only on the configuration of the external load but also on its value.

Point source diffraction from a semi-infinite perfect electromagnetic conductor half plane

The Wiener-Hopf (WH) technique is employed to study the problem of spherical wave diffraction (emanating due to a point source) from a semi-infinite perfect electromagnetic conductor (PEMC) half plane. The importance of the PEMC solution is due to the fact that: (i) it is the generalization of both perfect electric conductor (PEC) solution and perfect magnetic conductor (PMC) solution, (ii) the reflected electromagnetic wave from a PEMC boundary has cross polarized component in addition to the co-polarized component. Some graphs showing the effects of various parameters on the diffracted field are plotted and discussed.

Quaternionic Wiener Algebras, Factorization and Applications

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem for the quaternionic matrix-valued Wiener algebras (discrete and continuous) and explore the connection to the Riemann-Hilbert problem in that setting. As applications, we characterize solvability of two classes of quaternionic functional equations and give an explicit formula for the canonical factorization of quaternionic rational matrix functions via realization.

Combining Rational Approximation with Asymptotic Wiener–Hopf Factorization Algorithm for Matrix Functions: Implementation and Testing

This paper discusses how an asymptotic Wiener–Hopf factorization can be implemented for a wide class of functions. Asymptotic Wiener–Hopf factorization was discussed [19] and the convergence for matrices sufficiently close to the identity matrix is shown. We demonstrate how the algorithm can be successfully implemented with the help of the rational approximations. The idea is to simplify the matrix first by rationally approximating and then perform the approximate factorisation. There is no compromise in accuracy since the factorisation is approximate anyway and rational approximations are very precise usually. There is also a mapping of real line discussed and implemented to make the rational approximation more optimal. The code is tested against some easy examples which are calculated by hand. The use of this code is illustrated with some more complicated matrix functions motivated by applications. The method has been implemented for 2 × 2 and 4 × 4 matrices but can easy be adapted for any size matrix. The code will be made available with the publication of this paper. We note that to date there are very few implemented Wiener–Hopf factorisation available due to instabilities, so this paper will make an important contribution to this area.

Propagation of Plastic Strips from a Corner Point of the Interface of Rigidly Coupled Materials

We consider a symmetric problem of development of plastic strips from the corner point of the interface of isotropic rigidly coupled media. Plastic strips are modeled by the lines of discontinuity of tangential displacements. The exact solution of the corresponding problem of the theory of elasticity is constructed by the Wiener–Hopf method. On the basis of this solution, we determine the length and the direction of propagation of plastic strips.

Two-dimensional model of the interaction of a plane acoustic wave with nozzle edge and wing trailing edge.

The interaction of a plane acoustic wave with two-dimensional model of nozzle edge and trailing edge is investigated theoretically by means of the Wiener-Hopf technique. The nozzle edge and the trailing edge are simulated by two half-planes with offset edges. Shear layer behind the nozzle edge is represented by a vortex sheet supporting Kelvin-Helmholtz instability waves. The considered configuration combines two well-known models (nozzle edge and trailing edge), and reveals additional interesting physical aspects. To obtain the solution, the matrix Wiener-Hopf equation is solved in conjunction with a requirement that the full Kutta condition is imposed at the edges. Factorization of the kernel matrix is performed by the combination of Padé approximation and the pole removal technique. This procedure is used to obtain numerical results. The results indicate that the diffracted acoustic field may be significantly intensified due to scattering of hydrodynamic instability waves into sound waves provided that the trailing edge is close enough to the vortex sheet. Similar mechanism may be responsible for the intensification of jet noise near a wing.

A Wiener-Hopf factorization approach for pricing barrier options in the Heston model

A Wiener-Hopf factorization approach for pricing barrier options in the Heston model

Large-N correlation functions in mathcal{N} = 2 superconformal QCD

We study extremal correlation functions of chiral primary operators in the large-N SU(N ) mathcal{N} = 2 superconformal QCD theory and present new results based on supersymmetric localization. We discuss extensively the basis-independent data that can be extracted from these correlators using the leading order large-N matrix model free energy given by the four-sphere partition function. Special emphasis is given to singletrace 2- and 3-point functions as well as a new class of observables that are scalars on the conformal manifold. These new observables are particular quadratic combinations of the structure constants of the chiral ring. At weak ’t Hooft coupling we present perturbative results that, in principle, can be extended to arbitrarily high order. We obtain closed-form expressions up to the first subleading order. At strong coupling we provide analogous results based on an approximate Wiener-Hopf method.

Splitting and time reversal for Markov additive processes

We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into the behaviour of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener–Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Restricting to the case of no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line. The latter result is obtained using three quite different approaches based on the established path decomposition theory, which further demonstrates its applicability.

A queueing/inventory and an insurance risk model

AbstractWe study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω(x) when the inventory is at level x>0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω(x) is constant and ω(x) = ax. The key tool is the Wiener–Hopf factorization technique. When ω(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory.

Sound Transmission in a Duct with Sudden Area Expansion, Extended Inlet, and Lined Walls in Overlapping Region

The transmission of sound in a duct with sudden area expansion and extended inlet is investigated in the case where the walls of the duct lie in the finite overlapping region lined with acoustically absorbent materials. By using the series expansion in the overlap region and using the Fourier transform technique elsewhere we obtain a Wiener-Hopf equation whose solution involves a set of infinitely many unknown expansion coefficients satisfying a system of linear algebraic equations. Numerical solution of this system is obtained for various values of the problem parameters, whereby the effects of these parameters on the sound transmission are studied.

Investigation of the Process Zone Near the Tip of an Interface Crack in the Elastic Body in Shear Within the Framework of the Complex Model

Under the conditions of plane deformation, by using the Wiener–Hopf method, we perform the numerical analysis of a narrow small-scale process zone in the elastic body subjected to shear near the tip of an interface crack whose faces are in contact with friction. The process zone originates at the crack tip and makes a certain angle with the interface of the media. It is modeled by the straight line of discontinuities of displacements and consists of two segments. In the segment adjacent to the crack tip, which models the region of destruction of the material, both normal and tangential displacements have discontinuities. At the same time, in the second segment, only the normal component of displacements suffers discontinuities. The angle between the process zone and the interface of the media is determined from the condition of maximum of circumferential tensile stresses. We determine the sizes of the entire process zone and the region of destruction, the crack-tip opening displacements, and maximum opening displacement of the initial process zone. We also investigate the influence of the friction coefficient on the parameters of the process zone. By using the deformation criterion of fracture, we analyze the role played by the zone of destruction in the onset of crack propagation. It is shown that the crack starts as a result of the relative shear of its faces near the tip. The comparison of the obtained results with the data of other researchers is presented.

Line source diffraction by perfectly conducting successive steps

A rigorous investigation for the high frequency diffraction of the cylindrical waves by the perfectly conducting successive step discontinuities is considered by using the Fourier transform technique in conjunction with the mode-matching method. The corresponding boundary value problem is formulated into a modified scalar Wiener–Hopf equation of the third kind whose approximate solution, which is obtained by iterations of a pair of Fredholm integral equations of the second kind, contains five sets of infinite number of unknown constants satisfying five infinite systems of linear algebraic equations and two types of branch-cut integrals which can be evaluated approximately and numerically due to their types. The effects of the characteristic parameters of the problem such as width and height of the steps on the diffraction phenomenon are shown graphically.

Lévy flights between absorbing boundaries: Revisiting the survival probability and the shift from the exponential to the Sparre-Andersen limit behavior.

We revisit the problem of calculating the survival probability of Lévy flights in a finite interval with absorbing boundaries. Our approach is based on the master equation for discrete Lévy fliers, previously considered to treat the semi-infinite domain. We argue that, although the semi-infinite case can be treated exactly due to Wiener-Hopf factorization, the approximation involved in the problem with the finite interval is actually fairly good. We evidence the shift in the universal behavior of the long-term survival probability from the exponential decay in the presence of two absorbing barriers to the Sparre-Andersen power-law dependence in the single-barrier limit. In some cases, we also calculate the short- and intermediate-term behavior and present the explicit dependence of the survival probability on the Lévy flier's starting position. Our analytical results are confirmed by numerical simulations.

H 2-optimization of sampled-data systems with a linear periodic plant. II. H 2-optimization of system S t based on the Wiener–Hopf method

Using the concept of the parametric transfer matrix and the Wiener---Hopf method, part II of the paper develops a constructive algorithm for solving the H2-optimization problem for the system St defined in part I. An illustrative example is given.