Acoustic propagation in 3-D, rectangular ducts with flexible walls

This article is posted here with the permission of the publishers, INCE/USA. Personal use of the material is permitted,; however, permission to reprint or republish any part of this article must be obtained from the publisher.24th National Conference on Noise Control Engineering 2010 (Noise-Con 10) Held Jointly with the 159th Meeting of the Acoustical Society of America. Baltimore, MD, USA, 19-21 April, 2010. INCE Conference Proceedings, 3: 1960-1968, Apr 2010. New York, NY, USA.In this article some analytic expressions for acoustic propagation in 3-D ducts of rectangular cross-section and with flexible walls are explored. Consideration is first given to the propagation of sound in an unlined 3-D duct formed by three rigid walls and closed by a thin elastic plate. An exact closed form expression for the fluid-structure coupled waves is presented. The effect of incorporating internal structures, such as a porous lining, into the duct is also discussed. Such configurations are directly relevant to the heating, ventilation and airconditioning industry

On eigenfunction expansions associated with wave propagation along ducts with wave-bearing boundaries

A class of boundary value problems, that has application in the propagation of waves along ducts in which the boundaries are wave-bearing, is considered. This class of problems is characterised by the presence of high order derivatives of the dependent variable(s) in the duct boundary conditions. It is demonstrated that the underlying eigenfunctions are linearly dependent and, most significantly, that an eigenfunction expansion representation of a suitably smooth function, say $f(y)$, converges point-wise to that function. Two physical examples are presented. It is demonstrated that, in both cases, the eigenfunction representation of the solution is rendered unique via the application of suitable edge conditions. Within the context of these two examples, a detailed discussion of the issue of completeness is presented

Orthogonality relations for fluid-structural waves in a 3-D rectangular duct with flexible walls

An exact expression for the fluid-coupled structural waves that propagate in a three-dimensional, rectangular waveguide with elastic walls is presented in terms of the non-separable eigenfunctions ψn(y,z). It is proved that these eigenfunctions are linearly dependent and that an eigenfunction expansion representation of a suitably smooth function f(y,z) converges point-wise to that function. Orthogonality results for the derivatives ψny(a,z) are derived which, together with a partial orthogonality relation for ψn(y,z), enable the solution of a wide range of acoustic scattering problems. Two prototype problems, of the type typically encountered in two-part scattering problems, are solved, and numerical results showing the displacement of the elastic walls are presented.Brunel Open Access Publishing Fun

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

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 a fluid-structure coupled wave at a flanged junction between two flexible waveguides

Copyright 2013 Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America. The following article appeared in the Journal of the Acoustical Society of America, 134(3), 1939 - 1949 and may be found at http://scitation.aip.org/content/asa/journal/jasa/134/3/10.1121/1.4817891.The scattering of a fluid-structure coupled wave at a flanged junction between two flexible waveguides is investigated. The flange is assumed to be rigid on one side and soft on the other; this enables a solution to be formulated using mode-matching. It is shown that both the choice of the edge conditions imposed on the plates at the junction and the choice of incident forcing significantly affect the transmission of energy along the duct. In particular, the edge conditions crucially affect the transmission of structure-borne vibration but have little effect on fluid-borne noise. Given the singular nature of the velocity field at the flange tip, particular attention is paid to the validity of the mode-matching method. It is demonstrated that the velocity field can be accurately reconstructed by incorporating the Lanczos filter into the truncated modal expansions. The mode-matching method is thus confirmed as an viable tool for this class of problem.The Higher Education Commission, Pakista

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

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

Edge waves and resonance on elastic structures: An overview

Copyright @ 2011 SAGE PublicationsOver 50 years have elapsed since the first experimental observations of dynamic edge phenomena on elastic structures, yet the topic remains a diverse and vibrant source of research activity. This article provides a focused history and overview of such phenomena with a particular emphasis on structures such as strips, rods, plates and shells. Within this context, some of the recent research highlights are discussed and the contents of this special issue of Mathematics and Mechanics of Solids on dynamical edge phenomena are introduced

Exact solution to a class of functional difference equations with application to a moving contact line flow

A new integral representation for the Barnes double gamma function is derived. This is canonical in the sense that solutions to a class of functional difference equations of first order with trigonometrical coefficients can be expressed in terms of the Barnes function. The integral representation given here makes these solutions very simple to compute. Several well-known difference equations are solved by this method and a treatment of the linear theory for moving contact line flow in an inviscid fluid wedge is given.https://journals.cambridge.org/action/displayAbstract?aid=231870