Analytic mode-matching for acoustic scattering in three dimensional waveguides with flexible walls: Application to a triangular duct

This is the post-print version of the Article. The official published version can be accessed from the links below - Copyright @ 2012 ElsevierAn analytic mode-matching method suitable for the solution of problems involving scattering in three-dimensional waveguides with flexible walls is presented. Prerequisite to the development of such methods is knowledge of closed form analytic expressions for the natural fluid–structure coupled waveforms that propagate in each duct section and the corresponding orthogonality relations. In this article recent theory [J.B. Lawrie, Orthogonality relations for fluid–structural waves in a 3-D rectangular duct with flexible walls, Proc. R. Soc. A. 465 (2009) 2347–2367] is extended to construct the non-separable eigenfunctions for acoustic propagation in a three-dimensional rectangular duct with four flexible walls. For the special case in which the duct cross-section is square, the symmetrical nature of the eigenfunctions enables the eigenmodes for a right-angled, isosceles triangular duct with flexible hypotenuse to be deduced. The partial orthogonality relation together with other important properties of the triangular modes are discussed. A mode-matching solution to the scattering of a fluid–structure coupled wave at the junction of two identical semi-infinite ducts of triangular cross-section is demonstrated for two different sets of “junction” conditions

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

The Cauchy Problem for Wave Maps on a Curved Background

We consider the Cauchy problem for wave maps u: \R times M \to N for Riemannian manifolds, (M, g) and (N, h). We prove global existence and uniqueness for initial data that is small in the critical Sobolev norm in the case (M, g) = (\R^4, g), where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe for proving global existence and uniqueness of small data wave maps u : \R \times \R^d \to N in the critical norm, for d at least 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru.Comment: Fixed minor typos in previous version. To appear in Calculus of Variations and Partial Differential Equation

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 acoustic propagation in three-dimensional rectangular ducts with flexible walls and porous linings

This is the post-print version of the Article. The official published version can be accessed from the links below - Copyright @ 2012 Acoustical Society of AmericaThe focus of this article is toward the development of hybrid analytic-numerical mode-matching methods for model problems involving three-dimensional ducts of rectangular cross-section and with flexible walls. Such methods require first closed form analytic expressions for the natural fluid-structure coupled waveforms that propagate in each duct section and second the corresponding orthogonality relations. It is demonstrated how recent theory [Lawrie, Proc. R. Soc. London, Ser. A 465, 2347–2367 (2009)] may be extended to a wide class of three-dimensional ducts, for example, those with a flexible wall and a porous lining (modeled as an equivalent fluid) or those with a flexible internal structure, such as a membrane (the “drum-like” silencer). Two equivalent expressions for the eigenmodes of a given duct can be formulated. For the ducts considered herein, the first ansatz is dependent on the eigenvalues/eigenfunctions appropriate for wave propagation in the corresponding two-dimensional flexible-walled duct, whereas the second takes the form of a Fourier series. The latter offers two advantages: no “root-finding” is involved and the method is appropriate for ducts in which the flexible wall is orthotropic. The first ansatz, however, provides important information about the orthogonality properties of the three-dimensional eigenmodes

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

Comments on a class of orthogonality relations relevant to fluid-structure interaction

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