Simultaneous Wiener-Hopf Equations Research Articles

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.

Plane wave diffraction by a finite plate with impedance boundary conditions.

In this study we have examined a plane wave diffraction problem by a finite plate having different impedance boundaries. The Fourier transforms were used to reduce the governing problem into simultaneous Wiener-Hopf equations which are then solved using the standard Wiener-Hopf procedure. Afterwards the separated and interacted fields were developed asymptotically by using inverse Fourier transform and the modified stationary phase method. Detailed graphical analysis was also made for various physical parameters we were interested in.

Diffraction From A Material Loaded Tandem Slit

In this paper, we consider plane wave diffraction by a tandem slit loaded with a homogenous material. The boundary value problem is formulated into a pair of simultaneous Wiener-Hopf equations via Fourier transformation. After decoupling these equations by elementary transformation, each modified Wiener-Hopf equation is reduced to a Fredholm integral equation of the second kind. The integral equations are then solved approximately to yield the Fourier transform of the diffracted fields. The inverse transform is evaluated asymptotically to obtain the far field expressions. Measurements and numerical simulations are also performed for several different geometry and material configurations. The analytic solutions compare well with measured and simulated results. The possibility of reducing beamwidth and increasing power coupled through the loaded tandem slit is explored.

DIFFRACTION BY A SEMI-INFINITE PARALLEL-PLATE WAVEGUIDE WITH SINUSOIDAL WALL CORRUGATION: COMBINED PERTURBATION AND WIENER-HOPF ANALYSIS

The difiraction by a semi-inflnite parallel-plate waveguide with sinusoidal wall corrugation is analyzed for the E-polarized plane wave incidence using the Wiener-Hopf technique together with the perturbation method. The problem is formulated in terms of the simultaneous Wiener-Hopf equations by introducing the Fourier transform for the unknown scattered fleld and applying approximate boundary conditions in the transform domain. Employing the factorization and decomposition procedure together with a perturbation series expansion, the zero- and flrst-order solutions of the Wiener-Hopf equations are obtained. Explicit expressions of the scattered fleld inside and outside the waveguide are derived analytically by taking the inverse Fourier transform and applying the saddle point method. Far fleld scattering characteristics of the waveguide are discussed in detail via representative numerical examples.

COMBINED WIENER-HOPF AND PERTURBATION ANALYSIS OF THE H-POLARIZED PLANE WAVE DIFFRACTION BY A SEMI-INFINITE PARALLEL-PLATE WAVEGUIDE WITH SINUSOIDAL WALL CORRUGATION

The difiraction by a semi-inflnite parallel-plate waveguide with sinusoidal wall corrugation is analyzed for the H-polarized plane wave incidence using the Wiener-Hopf technique combined with the perturbation method. Introducing the Fourier transform for the unknown scattered fleld and applying an approximate boundary condition together with a perturbation series expansion of the scattered fleld, the problem is formulated in terms of the zero-order and flrst-order simultaneous Wiener-Hopf equations. The Wiener-Hopf equations are solved via the factorization and decomposition procedure leading to the exact solutions. Explicit expressions of the scattered fleld inside and outside the waveguide are derived analytically by taking the inverse Fourier transform and applying the saddle point method. Far fleld scattering characteristics of the waveguide are discussed in detail via representative numerical examples.

DIFFRACTION BY A TERMINATED, SEMI-INFINITE PARALLEL-PLATE WAVEGUIDE WITH FOUR-LAYER MATERIAL LOADING: THE CASE OF H POLARIZATION

The diffraction by a terminated, semi-infinite parallel- plate waveguide with four-layer material loading is rigorously analyzed for the H-polarized plane wave incidence by means of the Wiener- Hopf technique. Introducing the Fourier transform for the unknown scattered field and applying boundary conditions in the transform domain, the problem is formulated in terms of the simultaneous Wiener-Hopf equations. The Wiener-Hopf equations are solved via the factorization and decomposition procedure together with the use of the edge condition leading to exact and approximate solutions. The scattered field inside and outside the waveguide is evaluated by taking the inverse Fourier transform and applying the saddle point method. Numerical examples on the radar cross section (RCS) are presented for various physical parameters, and the backscattering characteristics of the waveguide are discussed.

PLANE WAVE DIFFRACTION BY A FINITE PARALLEL-PLATE WAVEGUIDE WITH FOUR-LAYER MATERIAL LOADING: PART I - THE CASE OF E POLARIZATION

The diffraction by a finite parallel-plate waveguide with four-layer material loading is rigorously analyzed by means of the Wiener-Hopf technique for the H-polarized plane wave incidence. Taking the Fourier transform for the unknown scattered field as well as the Helmholtz equation and applying boundary conditions in the transform domain, the problem is formulated in terms of the simultaneous Wiener-Hopf equations. The Wiener-Hopf equations are solved via the factorization and decomposition procedure together with the use of rigorous approximation procedures leading to an efficient approximation solution. The scattered field in the real space is evaluated explicitly by taking the inverse Fourier transform. Illustrative numerical examples on the radar cross section (RCS) are presented and the far field scattering characteristics of the waveguide are discussed.

Plane wave diffraction by two oppositely placed, parallel two-part planes

The problem of plane wave diffraction by two oppositely placed, parallel two-part planes, consisting of the junction of perfectly conducing and resistive half-planes, is solved with the Fourier transform technique. The problem is formulated as two simultaneous Wiener–Hopf equations that are uncoupled by the analytical properties of the functions that occur. The solution involves an infinite number of expansion coefficients satisfying an infinite system of linear algebraic equations. The expansion coefficients are determined numerically, and some graphical results showing the influence of the surface resistivity and the distances between the two-part planes on the diffracted and transmitted waves are presented.

DIFFRACTION BY A TERMINATED SEMI-INFINITE PARALLEL PLATE WAVEGUIDE WITH TWO-LAYER MATERIAL LOADING AND IMPEDANCE BOUNDARIES

The plane wave diffraction by a terminated semi infinite parallel-plate waveguide with two-layer material loading and impedance boundaries is rigorously analyzed for E polarization using the Wiener-Hopf technique. Introducing the Fourier transform for the scattered field and applying boundary conditions in the transform domain, the problem is formulated in terms of the simultaneous Wiener-Hopf equations, which are solved via the factorization and decomposition procedure. The scattered field is evaluated by taking the inverse Fourier transform and applying the saddle point method. The numerical examples of the radar cross section (RCS) are represented for various physical parameters and backscattering characteristics, of considered geometry for open ended cavity and discussed in detail.

Scattering at the Junction Formed by a PEC Half-Plane and a Half-Plane with Anisotropic Conductivity

In this study, scattering of plane electromagnetic waves at the junction formed by a PEC half-plane and a half-plane with anisotropic conductivity is investigated. By using a Fourier transform technique, the problem is formulated into a matrix Wiener-Hopf system, and a rigorous solution is obtained. Also, to show the effect of planar physical discontinuity explicitly, four different special cases are considered where the problems are reduced into pairs of simultaneous Wiener-Hopf equations. They are decoupled via polynomial transformations and solved through the standard procedure. The diffracted fields are expressed in a form suitable for GTD applications.

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.

Diffraction by a terminated semi-infinite parallel-plate waveguide with three-layer material loading

The plane wave diffraction by a terminated semi-infinite parallel-plate waveguide with three-layer material loading is rigorously analyzed for E polarization using the Wiener-Hopf technique. Introducing the Fourier transform for the scattered field and applying boundary conditions in the transform domain, the problem is formulated in terms of the simultaneous Wiener-Hopf equations, which are solved via the factorization and decomposition procedure. The scattered field is evaluated by taking the inverse Fourier transform and applying the saddle point method. Representative numerical examples of the radar cross section (RCS) are given for various physical parameters and the backscattering characteristics are discussed in detail. Some comparisons with a high-frequency technique are also given to validate the present method.

Diffraction coefficient for a half-plane with anisotropic conductivity

The diffraction of obliquely incident plane waves by a half-plane having anisotropic conductivity is investigated. The half-plane is assumed to be perfectly conducting in one direction and to have finite conductivity in the transverse direction. The material of the half-plane is simulated by a resistive-type boundary condition with the resistivity being a constant tensor. By using the Fourier transform technique, the problem is formulated into a pair of simultaneous Wiener-Hopf equations which are decoupled by means of a polynomial transformation and then solved by standard techniques.

Asymptotic approximation of the Wiener-Hopf technique as applied to jet atomisation

An approximate Wiener-Hopf (WH) technique is developed for solving problems involving fine spatial structure. As an example of the application of this method we investigate the atomisation of a liquid jet. The jet exits from a nozzle into an ambient fluid. Short interfacial waves become unstable and break into small particles. This problem is treated as a potential flow under the influence of capillary effects at the interface and the pressure fluctuation at the nozzle wall. Two simultaneous WH equations are obtained. To solve them, the singular parts in each equation are separated from the regular ones, that leads to a linear system of algebraic equations for the residues. The response-wave amplitudes are evaluated numerically and the instability diagram is presented. It is found that resonance occurs at double roots of the dispersion relation. For a given azimuthal number m, the double roots form two curves parametrised by the Weber number β. They merge at a certain critical point, where an even stronger resonance occurs. This finally selects the dominant modes. By gauging one parameter, namely the velocity ratio U, the theoretical prediction agrees quite well with experimental results of the jet atomisation.

Plane Wave Diffraction by an Open-Ended Parallel Plate Waveguide Cavity

The diffraction of a plane electromagnetic wave by an open-ended parallel plate waveguide cavity with a planar termination is rigorously analyzed for the E -polarized case using the Wiener-Hopf technique. Introducing the Fourier transform for the unknown scattered field and applying boundary conditions in the transform domain, the problem is formulated in terms of the simultaneous Wiener-Hopf equations, which are solved via a factorization and decomposition procedure. The solution is exact but formal since branch-cut integrals with unknown integrands and an infinite number of unknowns are involved. Approximation procedures based on rigorous asymptotics are further presented and the approximate solution to the Wiener-Hopf equations is derived. The scattered field inside and outside the cavity is evaluated by taking the inverse Fourier transform and applying the saddle point method of integration. Based on these results, numerical computations of the bistatic scattered pattern as well as the radar cross sec...

Two-Dimensional Slow Viscous Flow from a Converging Nozzle

Two-dimensional slow viscous flow through a converging nozzle composed of two semi-infinite plates is investigated on the basis of Stokes approximation. The flow is caused by a pressure difference between inner and outer region of the nozzle of an arbitrary angle. A formal expression for the flow is obtained by solving a pair of simultaneous Wiener-Hopf equations. Streamlines and stress distributions on each plate are determined by evaluating the formal expression. The relation between the discharge through the nozzle and pressure difference is given as a function of the angle of nozzle. The asymptotic behaviors of the flow at far field are also discussed.

Analysis of a shear fracture in unidirectional fiber-reinforced composites.

A semi-infinite shear fracture initiated from notch roots and the other stress concentrators in unidirectional fiber-reinforced composites is considered on the basis of the two-dimensional theory of orthotropic elasticity. By use of the Wiener-Hopf technique, simultaneous Wiener-Hopf equations are derived for the semi-infinite shear fracture in an infinite strip of the composites under tensile loading at infinity. The stress intensity factors at the tip of the shear fracture are obtained through asymptotic properties of the equations with the aid of some numerical computations. As a consequence, it is revealed that the considerably large stress intensity factor for the opening mode appears in addition to the stress intensity factor for the in-plane-shear mode. Furthermore, when the results for the stress intensity factors is compared with one obtained by the well-known method of energy balance, it is found that the results do not coincide and the latter is inappropriate to estimate the stress intensity factors.

Slow Viscous Flow Due to Sliding of a Semi-Infinite Plate over a Plane

Two-dimensional slow viscous flow in a region bounded by a plane wall and an inclined semi-infinite flat plate at a distance is investigated on the basis of Stokes' approximation. The motion is caused by the translation of the plane wall parallel to itself. A formal expression for the flow is obtained by solving a pair of simultaneous Wiener-Hopf equations. Streamlines and stress distributions on the plate are determined by evaluating the formal expression. The case in which the flow is caused by a pressure difference between up- and down-stream infinity with the plane wall at rest is also considered. When the plate is not perpendicular to the plane, it is found that separation occurs at the leading edge of the plate for both cases and that for the flow due to pressure difference a viscous eddy of which size diminishes as the inclination angle approaches 90° appears adjacent to the broader side of the plate.

On the Solution of Two Coupled Wiener–Hopf Equations

Vector Wiener–Hopi equations previously solved in closed form by using different techniques, involve matrices having a particular structure. A general procedure is given to factorize these matrices; it leads to an approach which, even if sometimes cumbersome, is uniform and allows the solution of new problems. In addition a new method is proposed which transforms simultaneous Wiener–Hopf equations in the form of Fredholm integral equation of the second kind.

Slow Viscous Flow around an Inclined Fence on a Plane

Two-dimensional slow viscous flow around an inclined fence on an infinite plane is investigated on the basis of the Stokes approximation. A formal expression for the flow is obtained by solving a pair of simultaneous Wiener-Hopf equations. Force and moment exerted on the fence are calculated and features of the flow are discussed.