On the factorization of a class of Wiener-Hopf kernels

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The definitive publisher-authenticated version: Abrahams, I.D. & Lawrie, J.B. (1995) “On the factorisation of a class of Wiener-Hopf kernels.” I.M.A. J. Appl. Math., 55, 35-47. is available online at: http://imamat.oxfordjournals.org/cgi/content/abstract/55/1/35.The Wiener-Hopf technique is a powerful aid for solving a wide range of problems in mathematical physics. The key step in its application is the factorization of the Wiener-Hopf kernel into the product of two functions which have different regions of analyticity. The traditional approach to obtaining these factors gives formulae which are not particularly easy to compute. In this article a novel approach is used to derive an elegant form for the product factors of a specific class of Wiener-Hopf kernels. The method utilizes the known solution to a difference equation and the main advantage of this approach is that, without recourse to the Cauchy integral, the product factors are expressed in terms of simple, finite range integrals which are easy to compute

An orthogonality condition for a class of problems with high order boundary conditions: Applications in sound/structure interaction

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in The Quarterly Journal of Mechanics and Applied Mathematics following peer review. The definitive publisher-authenticated version: Lawrie, J.B. & Abrahams, I.D. (1999) “An orthogonality condition for a class of problems with high order boundary conditions; applications in sound/structure interaction.” Q. Jl. Mech. Appl. Math., 52(2), 161-181. is available online at: http://qjmam.oxfordjournals.org/cgi/content/abstract/52/2/161There are numerous interesting physical problems, in the fields of elasticity, acoustics and electromagnetism etc., involving the propagation of waves in ducts or pipes. Often the problems consist of pipes or ducts with abrupt changes of material properties or geometry. For example, in car silencer design, where there is a sudden change in cross-sectional area, or when the bounding wall is lagged. As the wavenumber spectrum in such problems is usually discrete, the wave-field is representable by a superposition of travelling or evanescent wave modes in each region of constant duct properties. The solution to the reflection or transmission of waves in ducts is therefore most frequently obtained by mode-matching across the interface at the discontinuities in duct properties. This is easy to do if the eigenfunctions in each region form a complete orthogonal set of basis functions; therefore, orthogonality relations allow the eigenfunction coefficients to be determined by solving a simple system of linear algebraic equations. The objective of this paper is to examine a class of problems in which the boundary conditions at the duct walls are not of Dirichlet, Neumann or of impedance type, but involve second or higher derivatives of the dependent variable. Such wall conditions are found in models of fluid/structural interaction, for example membrane or plate boundaries, and in electromagnetic wave propagation. In these models the eigenfunctions are not orthogonal, and also extra edge conditions, imposed at the points of discontinuity, must be included when mode matching. This article presents a new orthogonality relation, involving eigenfunctions and their derivatives, for the general class of problems involving a scalar wave equation and high-order boundary conditions. It also discusses the procedure for incorporating the necessary edge conditions. Via two specific examples from structural acoustics, both of which have exact solutions obtainable by other techniques, it is shown that the orthogonality relation allows mode matching to follow through in the same manner as for simpler boundary conditions. That is, it yields coupled algebraic systems for the eigenfunction expansions which are easily solvable, and by which means more complicated cases, such as that illustrated in figure 1, are tractable

A brief historical perspective of the Wiener-Hopf technique

It is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener–Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener–Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems

Scattering of flexural waves by a semi-infinite crack in an elastic plate carrying an electric current

Copyright @ 2011 Sage Publications LtdSmart structures are components used in engineering applications that are capable of sensing or reacting to their environment in a predictable and desired manner. In addition to carrying mechanical loads, smart structures may alleviate vibration, reduce acoustic noise, change their mechanical properties as required or monitor their own condition. With the last point in mind, this article examines the scattering of flexural waves by a semi-infinite crack in a non-ferrous thin plate that is subjected to a constant current aligned in the direction of the crack edge. The aim is to investigate whether the current can be used to detect or inhibit the onset of crack growth. The model problem is amenable to an exact solution via the Wiener–Hopf technique, which enables an explicit analysis of the bending (and twisting) moment intensity factors at the crack tip, and also the diffracted field. The latter contains an edge wave component, and its amplitude is determined explicitly in terms of the current and angle of incidence of the forcing flexural wave. It is further observed that the edge wave phase speed exhibits a dual dependence on frequency and current, resulting in two distinct asymptotic behaviours

Sound radiation from a semi-infinite lined duct

In this paper we consider the radiation properties of a pair of semi-infinite, parallel-plate ducts in which the inner duct is buried inside the outer duct. A Robin condition is applied to one of the inner walls (to represent an acoustic lining), while Neumann conditions are applied on all other surfaces. This leads to a matrix Wiener-Hopf problem, which requires the factorisation of a 3 × 3 matrix of a form which, to our knowledge, has not previously been considered in the literature and which is not directly amenable to standard pole-removal techniques. We derive the exact factorisation of this matrix here, and present results for the far-field scattered sound which show the effect of varying the properties of the wall lining

A surprising observation in the quarter-plane diffraction problem

In this paper, we revisit Radlow's highly original attempt at a double Wiener-Hopf solution to the canonical problem of wave diffraction by a quarter-plane. Using a constructive approach, we reduce the problem to two equations, one containing his somewhat controversial ansatz, and an additional compatibility equation. We then show that despite Radlow's ansatz being erroneous, it gives surprisingly accurate results in the far-field, particularly for the spherical diffraction coefficient. This unexpectedly good result is established by comparing it to results obtained by the recently established modified Smyshlyaev formulae.The first author was supported by EPSRC/UKRI grant EP/N013719/1. The second author was supported by EPSRC/UKRI grants EP/K032208/1 and EP/R014604/1

High-contrast approximation for penetrable wedge diffraction

Abstract The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is different for acoustic and electromagnetic waves for example. Based on this assumption, a new asymptotic iterative scheme is constructed. The solution to the penetrable wedge is written in terms of infinitely many solutions to (possibly inhomogeneous) impenetrable wedge problems. Each impenetrable problem is solved using a combination of the Sommerfeld–Malyuzhinets and Wiener–Hopf techniques. The resulting approximated solution to the penetrable wedge involves a large number of nested complex integrals and is hence difficult to evaluate numerically. In order to address this issue, a subtle method (combining asymptotics, interpolation and complex analysis) is developed and implemented, leading to a fast and efficient numerical evaluation. This asymptotic scheme is shown to have excellent convergent properties and leads to a clear improvement on extant approaches.</jats:p

Multiple waves propagate in random particulate materials

© 2019 Society for Industrial and Applied Mathematics Publications. All rights reserved. For over 70 years it has been assumed that scalar wave propagation in (ensembleaveraged) random particulate materials can be characterized by a single effective wavenumber. Here, however, we show that there exist many effective wavenumbers, each contributing to the effective transmitted wave field. Most of these contributions rapidly attenuate away from boundaries, but they make a significant contribution to the reflected and total transmitted field beyond the low-frequency regime. In some cases at least two effective wavenumbers have the same order of attenuation. In these cases a single effective wavenumber does not accurately describe wave propagation even far away from boundaries. We develop an efficient method to calculate all of the contributions to the wave field for the scalar wave equation in two spatial dimensions, and then compare results with numerical finite-difference calculations. This new method is, to the best of the authors' knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions

Analytical methods for perfect wedge diffraction: A review

© 2019 Elsevier B.V. The subject of diffraction of waves by sharp boundaries has been studied intensively for well over a century, initiated by groundbreaking mathematicians and physicists including Sommerfeld, Macdonald and Poincaré. The significance of such canonical diffraction models, and their analytical solutions, was recognised much more broadly thanks to Keller, who introduced a geometrical theory of diffraction (GTD) in the middle of the last century, and other important mathematicians such as Fock and Babich. This has led to a very wide variety of approaches to be developed in order to tackle such two and three dimensional diffraction problems, with the purpose of obtaining elegant and compact analytic solutions capable of easy numerical evaluation. The purpose of this review article is to showcase the disparate mathematical techniques that have been proposed. For ease of exposition, mathematical brevity, and for the broadest interest to the reader, all approaches are aimed at one canonical model, namely diffraction of a monochromatic scalar plane wave by a two-dimensional wedge with perfect Dirichlet or Neumann boundaries. The first three approaches offered are those most commonly used today in diffraction theory, although not necessarily in the context of wedge diffraction. These are the Sommerfeld–Malyuzhinets method, the Wiener–Hopf technique, and the Kontorovich–Lebedev transform approach. Then follows three less well-known and somewhat novel methods, which would be of interest even to specialists in the field, namely the embedding method, a random walk approach, and the technique of functionally-invariant solutions. Having offered the exact solution of this problem in a variety of forms, a numerical comparison between the exact solution and several powerful approximations such as GTD is performed and critically assessed