Modified Wiener-Hopf Equation Research Articles

Rigorous Wiener–Hopf analysis of a horn radiator with thick walls

The radiation from a two-dimensional horn radiator with thick walls is studied rigorously when illuminated with a dominant TEM mode. The method consisting of the mode matching method and the Fourier transform technique together gives rise to a scalar modified Wiener–Hopf equation of the third kind (MWHE-3). The formulation includes branch-cut integrals, which are evaluated approximately, and an infinite number of unknown constants that satisfy an infinite system of linear equations that can be treated numerically. The effects of the wall thickness of a horn radiator, as well as the horn width and length, on the radiation from the horn are also presented in comparison with similar structures that are available in the literature.

Scattering of Sound Waves in Two Stepped Non-uniformly Lined Duct

Scattering of sound waves in two stepped cylindrical duct which walls are coated with different acoustically absorbent materials is investigated by using Wiener-Hopf technique directly and by determining scattering matrices. First, by using Fourier transform technique we obtain a couple of modified Wiener-Hopf equations whose solutions involve four sets of infinitely many unknown expansion coefficients providing systems of linear algebraic equations. Then we determine scattering matrices of the problem and we state the total transmitted field by using generalized scattering matrix method. Numerical results are compared for different parameters.

Modified Wiener-Hopf equation in identification problems

Modified Wiener-Hopf equation in identification problems

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.

TEM wave radiation from a dielectric-filled coaxial waveguide with a large circumferential gap on its outer wall

TEM wave radiation from a dielectric-filled coaxial waveguide with a large circumferential gap on its outer wall is investigated rigorously by applying direct Fourier transform and reducing the problem into the solution of a modified Wiener–Hopf equation of the first type. Classical Wiener–Hopf procedure is applied by considering two different approaches for the factorization of the kernel function, and the Wiener–Hopf equation is solved via a set of Fredholm integral equations of the second type. The problem is also analyzed by applying a combination of Fourier transform and simple series representation, and the results are compared numerically. Besides, the effects of radii of the walls, relative permittivity, frequency, and the gap width on the scattered fields are illustrated graphically.

Scattering of a TEM wave by a large circumferential gap on a coaxial waveguide

In the present work, the scattering of a TEM wave by a gap on the outer wall of a coaxial waveguide is analyzed rigorously. The formulation of the related boundary-value problem in terms of Fourier integrals leads to a modified Wiener–Hopf equation which is first reduced into a pair of Fredholm integral equations of the second kind and the diffraction coefficients related to the reflected, transmitted, and radiated fields are determined explicitly for large gap width compared to the wavelength. At the end of the analysis, numerical results illustrating the effects of the cross-sectional area of the coaxial cylindrical waveguide and the gap width on the fields are presented.

TEM Wave Scattering by a Step Discontinuity on the Outer Wall of a Coaxial Waveguide

In this paper, the propagation of TEM waves along a coaxial waveguide with a step discontinuity on its outer wall is investigated rigorously by applying the direct Fourier transform and reducing the problem into the solution of a modified Wiener-Hopf equation. The solution for the field terms are determined in terms of an infinite number of unknown coefficients, which satisfy an infinite set of linear algebraic equations. These equations are solved numerically and the effect of area ratio is presented graphically at the end of the analysis. The same problem is also analyzed by applying the mode-matching technique and the results of the two approaches are compared. It is observed numerically that the Wiener-Hopf technique provides a better convergence than the mode-matching technique.

Investigation of Thickness Effects on the TM-Wave Radiation from a Thick Impedance-Loaded Parallel-Plate Radiator

In this work, the influence of the plate thickness on the lowest order TM-wave radiation is investigated by using the mode-matching method in conjunction with the Fourier transform technique. While the classical formulation leads to an intractable matrix Wiener-Hopf factorization, the hybrid method adopted here reduces the related boundary value problem to a scalar modified Wiener-Hopf equation of the second kind. The solution involves infinitely many unknown constants satisfying an infinite system of linear algebraic equations susceptible to a numerical treatment. Some computational results illustrating the effect of the plate thickness and the plate impedances on the radiation phenomenon are revealed.

Scattering of the TEM mode at the junction of perfectly conducting and impedance coaxial waveguides

In the present work, a rigorous Wiener–Hopf approach is used to investigate the radiation characteristics of a two-part coaxial waveguide, whose left part is perfectly conducting while the surface impedances of the inner and outer cylinders of the right part are different from each other. The representation of the solution to the boundary-value problem in terms of Fourier integrals leads to two simultaneous modified Wiener–Hopf equations whose formal solution is obtained by using the factorization and decomposition procedures. The solution involves two infinite sets of unknown coefficients satisfying two infinite systems of linear algebraic equations. These systems are solved numerically, and some graphical results showing the influence of the spacing between the coaxial cylinders and the surface impedances on the reflection coefficient are presented.

Reflection and transmission of plane acoustic waves in an infinite annular duct with a finite gap on the inner wall

The diffraction of acoustic waves by an infinitely long annular duct having a finite gap on the inner wall is investigated rigorously. The related boundary-value problem is formulated into a modified Wiener–Hopf equation, which is then reduced to a pair of simultaneous Fredholm integral equations of the second kind. At the end of the analysis, numerical results illustrating the effects of the width of the coaxial cylindrical waveguide and the gap length on the diffraction phenomenon are presented. Copyright © 2010 John Wiley & Sons, Ltd.

Surface wave scattering by a material filled rectangular impedance groove in a reactive plane

The problem of TE-polarized surface wave scattering from a rectangular impedance groove located on an infinite reactive plane which is filled with dielectric material is considered for a rather general case where the impedances of the horizontal and vertical sides of the groove have different values. The multiple interactions up to the second order between the edges of the groove are obtained to yield diffracted field. The diffraction problem is first reduced into a modified Wiener-Hopf equation and then solved approximately. The solution contains branch-cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical evaluations of the corresponding integrals as well as the numerical solutions of the linear algebraic equation systems are obtained for various values of the parameters such as the surface reactance of the guiding plane, the vertical and horizontal wall impedances of the groove, the permittivity of the material loading, the width and the height of the groove which permit one to study the effect of these parameters on the diffraction phenomenon.

Influence of acoustic dominant mode propagation in a trifurcated lined duct with different impedances

In this study, we analyzed the diffraction of the acoustic dominant mode in a parallel-plate trifurcated waveguide with normal impedance boundary conditions in the case where surface impedances of the upper and lower infinite plates are different from each other. The acoustic dominant mode is incident in a soft/hard semi-infinite duct located symmetrically in the infinite lined duct. The solution of the boundary value problem using Fourier transform leads to two simultaneous modified Wiener–Hopf equations that are uncoupled using the pole removal technique. Two infinite sets of unknown coefficients are involved in the solution, which satisfy two infinite systems of linear algebraic equations. These systems are solved numerically. The new kernel functions are factorized. Some graphical results showing the influence of sundry parameters of interest on the reflection coefficient are presented.

Radiation characteristics of a coaxial waveguide with opposing dielectric filled grooves

The purpose of the present study is to provide a rigorous analysis to the coaxial waveguide with two opposing grooves in the outer and inner cylinders with different depths and filled with different dielectric materials. The representation of the solution to the three-part mixed boundary-value problem in terms of Fourier integrals leads to a couple of simultaneous modified Wiener-Hopf equations that are reduced to infinite systems of linear algebraic equations. These systems are solved numerically and the bandstop filter characteristics of the reflection coefficient are studied in terms of frequency, groove sizes and the parameters of the filling dielectric material.

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.

Wiener–Hopf analysis of the parallel plate wave guide with opposing rectangular dielectric-filled grooves

The purpose of the present work is to provide a rigorous analysis of the parallel plate wave guide with two opposing rectangular grooves of different depths and filled with different dielectric materials. This configuration may be used as a band-stop filter. The representation of the solution to the three-part mixed boundary-value problem in terms of Fourier integrals leads to a couple of simultaneous modified Wiener–Hopf equations. By using the analytical properties of the functions that occur, the simultaneous modified Wiener–Hopf equations are reduced to the solution of four infinite systems of linear algebraic equations. These systems are solved numerically, and the band-stop filter characteristics of the reflection coefficient are studied in terms of frequency, groove sizes, and the parameters of the filling dielectric material.PACS Nos.: 42.25.Bs, 42.25.Gy, 42.82.Et

PROPAGATION OF WAVES IN A BIFURCATED CYLINDRICAL WAVEGUIDE WITH WALL IMPEDANCE DISCONTINUITY

In the present work the radiation of sound from a bifurcated circular waveguide formed by a semi-infinite rigid duct inserted axially into a larger infinite tube with discontinuous wall impedance is reconsidered through an alternative approach which consists of using the mode matching technique in conjunction with the Wiener-Hopf method. By expressing the total field in the appropriate waveguide region in terms of normal modes and using the Fourier transform technique elsewhere, we end up with a single modified Wiener-Hopf equation whose solution involves an infinite system of algebraic equations. This system is solved numerically and the influence of some parameters on the radiation phenomenon is shown graphically. The equivalence of the direct method described in (1) and the present mixed method are shown numerically.

WIENER-HOPF ANALYSIS OF FINITE-LENGTH IMPEDANCE LOADING IN THE OUTER CONDUCTOR OF A COAXIAL WAVEGUIDE

A new canonical scattering problem consisting of the propagation of the dominant TEM mode at the finite-length impedance discontinuity in the outer conductor of a coaxial waveguide is solved. The contributions from the successive impedance discontinuities are accounted for through the solution of a modified Wiener-Hopf equation. Some graphical results displaying the reflection and transmission characteristic are presented.

A HYBRID METHOD FOR THE SOLUTION OF PLANE WAVE DIFFRACTION BY AN IMPEDANCE LOADED PARALLEL PLATE WAVEGUIDE

The diffraction of E-polarized plane waves by an impedance loaded parallel plate waveguide formed by a two-part impedance plane and a parallel half plane with different face impedances is investigated rigorously by using the Fourier transform technique in conjunction with the Mode Matching Method. This mixed method of formulation gives rise to a scalar Modified WienerHopf equation, the solution of which contains infinitely many constants satisfying an infinite system of linear algebraic equations. A numerical solution of this system is obtained for various values of the surface impedances and waveguide height.

Radiation from Dielectric-Filled Thick-Walled Parallel-Plate Waveguide Junction Loaded with a Dielectric Half-Plane

The radiation of a dielectric-filled thick-walled parallel-plate impedance waveguide junctioned with a thick dielectric half-plane is treated by using the mode-matching method in conjunction with the Wiener–Hopf technique. The hybrid method that we adopt here reduces the related boundary value problem to two scalar modified Wiener–Hopf equations of the second kind. This bypasses the most challenging stage of the classical formulations, which lead to an intractable matrix Wiener–Hopf factorization. The solution involves infinitely many unknown constants satisfying an infinite system of linear algebraic equations susceptible to a numerical treatment. Some computational results illustrating the various values of the dielectric constant, plate impedances, plate thickness, and plate to plate separation distance on the radiation phenomenon are also presented.

Diffraction by a rectangular groove with resistive walls

The plane wave diffraction from a rectangular groove having resistive vertical walls with material loading is analyzed for E polarization by using the Fourier transform technique. The problem is formulated into a modified Wiener–Hopf equation applying boundary conditions. The modified Wiener–Hopf equation is solved via the factorization and decomposition procedure. The scattered field is evaluated by taking the inverse Fourier transform and applying the saddle point method. Numerical results on the backscatter radar cross-section (RCS) patterns are presented for various physical parameters.