Wiener-Hopf Approach Research Articles

Sound radiation from semi-infinite elliptical ducts with uniform subsonic jets: An analytical approach

This paper presents an analytical method for modeling the acoustic field radiation from a semi-infinite elliptical duct with a uniform subsonic jet. Elliptical ducts are often used as inlets for turbofan engines to take full advantage of the pre-compression effect of the fuselage and improve the stealth performance of the aircraft. The method accounts for the instability wave inside the vortex sheet induced by the shearing of jet and ambient flow and its effect to sound radiation. The method uses Mathieu functions to describe the incident and scattered sound in the elliptic cylindrical coordinates. A semi-analytical Wiener-Hopf approach with low computational cost is applied to obtain the near and far field solutions. The near field is illustrated by acoustic pressure maps at different modes and circumferential angles. The far field is shown by directivity patterns at single tones. Numerical simulations based on a finite element method are conducted to validate the accuracy of the analytical method. The results show good agreement between the analytical and numerical solutions. The proposed method can be used to analyze the acoustic characteristics of elliptical ducts with jets, which are relevant for noise control and optimization of turbofan engine inlets.

Analytical study on the propagation of semi-infinite crack due to SH-wave in pre-stressed magnetoelastic orthotropic strip

This study deals with the propagation of semi-infinite crack impacted by SH-wave propagation in a pre-stressed magnetoelastic orthotropic strip (PMOS). The proposed analytical model makes use of the Wiener-Hopf (WH) approach and two-sided Fourier integral transforms (FIT). The exact formulation of the stress intensity factor (SIF) for constant concentrated force (CCF) has been established in closed form. The effect of influencing factors, including the crack length, crack speed, magnetoelastic parameter, horizontal compressive pre-stress (HCP)/horizontal tensile pre-stress (HTP), vertical compressive pre-stress (VCP)/vertical tensile pre-stress (VTP), and anisotropy parameter on the SIF in the considered strip has been unraveled. The proposed model is compared with the case of a pre-stressed magnetoelastic isotropic strip (PMIS) and some peculiarities, along with the influence of orthotropy have been reported. The significant results reported in the present manuscript reveal that magnetoelasticity and pre-stress is a prominent factor impacting crack propagation in elastic material with orthotropy configuration properties and has a significant impact on the SIF, which has not been taken into account in the existing literature on the subject of fracture mechanics. The present investigation helps with structural and reliability analysis, fracture propagation, material durability and failure of engineering structures.

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.

Reducing aerofoil–turbulence interaction noise through chordwise-varying porosity

Abstract

Wiener–Hopf approach for the coaxial waveguide with an impedance-coated groove on the inner wall

In the present work, the band-pass filter characteristics of a coaxial waveguide with an impedance coated groove on the inner wall and perfectly conducting outer wall is analyzed rigorously through the Wiener–Hopf technique. By using the direct Fourier transform, the related boundary value problem is reduced to the Wiener–Hopf equation whose solution contains infinitely many constants satisfying an infinite system of linear algebraic equations. These equations are solved numerically and the field terms, which depends on the solution obtained numerically, are derived explicitly. The problem is also analyzed by applying a mode-matching technique and the results are compared numerically. Besides, some computational results illustrating the effects of parameters such as depth of the groove, surface impedances and radii of the walls are also presented.

Fast and spectrally accurate numerical methods for perforated screens (with applications to Robin boundary conditions)

AbstractThis paper considers the use of compliant boundary conditions to provide a homogenized model of a finite array of collinear plates, modelling a perforated screen or grating. While the perforated screen formally has a mix of Dirichlet and Neumann boundary conditions, the homogenized model has Robin boundary conditions. Perforated screens form a canonical model in scattering theory, with applications ranging from electromagnetism to aeroacoustics. Interest in perforated media incorporated within larger structures motivates interrogating the appropriateness of homogenized boundary conditions in this case, especially as the homogenized model changes the junction behaviour considered at the extreme edges of the screen. To facilitate effective investigation we consider three numerical methods solving the Helmholtz equation: the unified transform and an iterative Wiener–Hopf approach for the exact problem of a set of collinear rigid plates (the difficult geometry of the problem means that such methods, which converge exponentially, are crucial) and a novel Mathieu function collocation approach to consider a variable compliance applied along the length of a single plate. We detail the relative performance and practical considerations for each method. By comparing solutions obtained using homogenized boundary conditions to the problem of collinear plates, we verify that the constant compliance given in previous theoretical research is appropriate to gain a good estimate of the solution even for a modest number of plates, provided we are sufficiently far into the asymptotic regime. We further investigate tapering the compliance near the extreme endpoints of the screen and find that tapering with $\tanh $ functions reduces the error in the approximation of the far field (if we are sufficiently far into the asymptotic regime). We also find that the number of plates and wavenumber has significant effects, even far into the asymptotic regime. These last two points indicate the importance of modelling end effects to achieve highly accurate results.

Diffraction by a truncated planar array of dipoles:A Wiener–Hopf approach

We present a rigorous solution to the problem of scattering of a semi-infinite planar array of dipoles, i.e., infinite in one direction and semi-infinite in the other direction, thus presenting an edge truncation, when illuminated by a plane wave. Such an arrangement represents the canonical problem to investigate the diffraction occurring at the edge-truncation of a planar array. By applying the Wiener–Hopf technique to the Z-transformed system of equations derived from the electric field integral equation, we provide rigorous close form expressions for the dipoles’ currents. We find that such currents are represented as the superposition of the infinite array solution plus a perturbation, which comprises both edge diffraction and bound surface waves excited by the edge truncation. Furthermore, we provide an analytical approximation for the double-infinite sum involved in the calculation which drastically reduces the computational effort of this approach and also provides physically-meaningful asymptotics for the diffracted currents.

The unified transform for mixed boundary condition problems in unbounded domains

This paper implements the unified transform to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener-Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. By approximating the known data and the unknown boundary values by suitable functions and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The method is illustrated for the modified Helmholtz and Helmholtz equations. In each case, comparisons between the traditional Wiener-Hopf approach, other spectral or boundary methods and the unified transform approach are discussed.

Solving Wiener–Hopf problems without kernel factorization

A new approach to solving problems of Wiener–Hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics. The new method adapts mathematical ideas underlying the so-called unified transform method due to A. S. Fokas and collaborators in recent years. The method has the key advantage of avoiding what is usually the most challenging part of the usual Wiener–Hopf approach: the factorization of kernel functions into sectionally analytical functions. Two example boundary value problems, involving both harmonic and biharmonic fields, are solved in detail. The approach leads to fast and accurate schemes for evaluation of the solutions.

T-Stresses for Edge Cracks and Vanishing Ligaments

An edge-cracked half-plane 0 < x < A and a half-plane x > 0 with a semi-infinite crack x > a perpendicular to the edge are examined in this paper. Uniform crack-face loading is thoroughly examined, with a thorough exposition of the Koiter Wiener–Hopf approach (Koiter, 1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374); an analytical expression for the corresponding T-stress is obtained. For the additional cases of (i) nonuniform edge-crack crack-face loading σ(x/A)k (ℜ(k)>-1), (ii) concentrated loading at the edge-crack crack mouth, the Wiener–Hopf solutions and analytical T-stress expressions are provided, and tables of T-stress results for σ(x/A)k and σ(1-x/A)k are presented. A Green's function for the edge-crack T-stress is developed. The differing developments made by Koiter (1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374, Wigglesworth (1957, “Stress Distribution in a Notched Plate,” Mathematika, 4, pp. 76–96), and Stallybrass (1970, “A Crack Perpendicular to an Elastic Half-Plane,” Int. J. Eng. Sci., 8, pp. 351–362) for the case of an edge-cracked half-plane are enhanced by deducing a quantitative relationship between the three different Wiener–Hopf type factorizations. An analytical universal T-stress expression for edge-cracks is derived. Finally, the case of a vanishing uncracked ligament in a half-plane is examined, and the associated Wiener–Hopf solution and analytical T-stress expression are again provided. Several limiting cases are examined.

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.

Simplified Formula for a Controller in Optimal Control Problems

We derive a simplified frequency-domain formula for a controller transfer matrix for LQ optimal output feedback control of stochastic systems. For this purpose we apply a generalization of the Wiener-Hopf method. The generalization is characterized by the following three properties: (1) We generalize the usual operator $\{\cdot\}_+$ on rational matrices in the traditional Wiener-Hopf approach. The same algorithms as for computation of $\{\cdot\}_+$ are applicable to compute the new operator. (2) The matrices of the Youla-Kuccera parametrization do not appear in the optimal controller transfer matrix ${\bf C}$ (they appear only in its derivation), even if the plant is unstable. (3) Unlike the traditional Wiener-Hopf method [D. C. Youla, H. A. Jabr, and J. J. Bongiorno, Jr., IEEE Trans. Automat. Control, AC-21 (1976), pp. 319-338], where the spectral factors in the two spectral factorizations are both stable and minimum phase, our spectral factors need to be minimum phase only. Finally, three state-space applications of the formula are presented.

Wiener-Hopf Solution for Impenetrable Wedges at Skew Incidence

A new Wiener-Hopf approach for the solution of impenetrable wedges at skew incidence is presented. Mathematical aspects are described in a unified and consistent theory for angular region problems. Solutions are obtained using analytical and numerical-analytical approaches. Several numerical tests from the scientific literature validate the new technique, and new solutions for anisotropic surface impedance wedges are solved at skew incidence. The solutions are presented considering the geometrical and uniform theory of diffraction coefficients, total fields, and possible surface wave contributions

Wiener–Hopf approach for predicting the transmission loss of a circular silencer with a locally reacting lining

The propagation of sound in an infinite rigid circular cylindrical duct with an inserted expansion chamber whose walls are treated with an acoustically absorbent, locally-reacting material is investigated rigorously through the Wiener–Hopf technique. The expansion chamber is separated from the central airway which contain a uniform main gas flow by a perforated cylindrical screen which also increases the silencing performance. The influence of the expansion chamber radius, lining impedances, the mean flow and the acoustical impedance of the central perforated tube on the transmission loss are displayed graphically.

A generalization of the Wiener-Hopf approach to direct and inverse scattering problems connected with non-homogeneous half-spaces bounded by n-part boundaries

The classical Wiener-Hopf method connected with mixed two-part boundary-value problems is generalized to cover n-part boundaries. To this end one starts from an ad-hoc representation for the Green function, which involves n unknown functions having certain analytical properties. Thus the problem is reduced to a functional equation involving n unknowns, which constitutes a generalization of the classical Wiener-Hopf equation in two unknowns. To solve this latter which cannot be solved exactly when n > 3, one establishes a new method permitting one to obtain the asymptotic expressions valid when the wavelength is sufficiently small as compared with the widths of the inner strips of the boundary. The essentials of the method are elucidated through a concrete inverse scattering problem whose aim is to determine the constitutive electromagnetic parameters of a slab and a half-space bounded by an n-part impedance plane. Some illustrative numerical examples show the applicability as well as the accuracy of the method.

Wiener–Hopf approach to derivation of rational doubly coprime factorizations

This paper presents a method to construct a stable rational doubly coprime factorization (DCF) of a transfer matrix, proper or improper, from a given polynomial DCF. Wiener–Hopf factor polynomials are used to rationalize the polynomial DCF matrices. It is shown that the suggested construction procedure results in the stable rational DCF description that includes the normalized matrix fractional description (MFD) and inner coprime MFD as special cases.

The solution of a class of free boundary problems

Mellin transform and Wiener–Hopf techniques are used to solve several free boundary problems occurring in groundwater flow. This is a novel application of a technique not usually associated with free boundary problems; here it is advocated as a powerful alternative to methods relying on conformal mappings. In particular analytical methods can now be applied to classes of problems whose explicit solution is otherwise unobtainable. Relatively simple integral representations are also provided for a class of problems that otherwise require complicated analysis involving hypergeometric functions. The problems are formulated in an intermediate complex plane, and the Mellin transform is used to solve the problems directly in this plane. Alternative methods require a hodograph plane to be formulated; this is then conformally mapped to the intermediate plane. If the conformal mapping can be done then the full solution is determined. As a byproduct of this analysis the conformal mappings of some curvilinear polygons to a half plane are found. To perform such mappings directly for non‐trivial problems involving more than three singular points requires the solution of Fuchsian differential equations that involve accessory parameters and unknown points. Apart from degenerate cases there appears to be no constructive method of obtaining solutions to such Fuchsian differential equations, as a result few practical problems, whose representation in the hodograph plane involves curvilinear quadrangles, have been solved. This paper has the aim of developing, and demonstrating, a simple coherent strategy for tackling the free boundary problems that lead to a class of these more complicated situations, this substantially extends the range of problems that can be treated. The method presented sidesteps the difficulties associated with the free point and accessory parameter, and the solution for a problem where the hodograph plane is a curvilinear quadrangle with one vertex angle of $2\pi$ is found; in the physical plane this corresponds to a point of inflection on the free surface. Hence it is possible to conformally map a class of curvilinear quadrangles to an upper half plane using the analytic methods developed here and this resolves an open question in conformal mapping theory. Two illustrative examples are also solved, one involving three singular points and this provides a comparison with previous techniques, the other illustrates the occurrence of a point of inflection in a relatively simple situation. The alternative method advocated here is compared to the other analytical techniques available; these are briefly reviewed to place the Mellin transform, Wiener–Hopf approach in context.

A Wiener-Hopf approach to trade-offs between stability margin and performance in two- and three-degree-of-freedom multivariable control systems

Trade-offs between stability margin and performance are considered in two- and three-degree-of-freedom multivariable control systems using a Wiener-Hopf design approach. Maximum improvement in an approximate measure of stability margin is achieved at the expense of a prescribed increase in the quadratic cost functional measuring system performance. In order to attain an analytical solution to this fundamental trade-off problem, the approximate measure of stability margin chosen is also a quadratic cost function. A novel approach is introduced which allows structured perturbations in the coprime polynomial matrix fraction description of the plant transfer matrix to be taken into account. As a consequence, it is believed that the use of an approximate measure of stability margin is mitigated. Moreover, if needed, the solution obtained could serve as a very good initial one from which to search for better solutions iteratively.

X-ray edge singularity: Fredholm versus Wiener-Hopf method

Two methods, Fredholm and Wiener-Hopf, for solving the microscopic model of the X-ray edge singularity due to Mahan [Phys. Rev. 163 (1967) 612] and Nozières and De Dominicis [Phys. Rev. 178 (1969) 1097] are compared. We analyze the conditions under which these methods deliver a unique (exact) solution. Two large (infinite) time scales must be introduced to distinguish the methods, relaxation time T and an effective lifetime excited electron-hole pair τ. It is shown that the edge behavior depends on the ratio T/ τ being zero in the Fredholm and infinity in the Wiener-Hopf approach.

A Wiener-Hopf approach for the scattering by a disk

A Wiener-Hopf approach for the scattering by a disk