Jane B. Lawrie

Mode-matching without root-finding: Application to a dissipative silencer

This article presents an analytic mode-matching approach suitable for modelling the propagation of sound in a two-dimensional, three-part, ducting system. The approach avoids the need to find roots of the characteristic equation for the middle section of the duct (the component) and is readily applicable to a broad class of problems. It is demonstrated that the system of equations, derived via analytic mode-matching, exhibits certain features which ensure that they can be recast into a form that is independent of the roots of the characteristic equation for the component. The precise details of the component are irrelevant to the procedure; it is required only that there exists an orthogonality relation, or similar, for the eigenmodes corresponding to the propagating wave forms in this region. The method is applied here to a simple problem involving acoustic transmission through a dissipative silencer of the type commonly found in heating ventilation and air-conditioning ducts. With reference to this example, the silencer transmission loss is computed, and the power balance for the silencer is investigated and is shown to be an identity that is necessarily satisfied by the system of equations, regardless of the level of truncation.

Edge waves and resonance on elastic structures: An overview:

Over 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.

On tuning a reactive silencer by varying the position of an internal membrane

A mode-matching method is used to investigate the performance of a two-dimensional, modified reactive silencer. The modification takes the form of a membrane which is attached to the internal walls of the expansion chamber parallel to the axis of the inlet/outlet ducts. The height of the membrane above the level of the inlet/outlet ducts can be varied and, by this means, the device is tuned. It is shown that the stopband produced by the silencer can be broadened and/or shifted depending upon the height to which the membrane is raised. Attention is focused on the efficiency of the device at low-frequencies—the regime where dissipative silencers are usually least effective. The potential use of the device as a component in a hybrid silencer for heating ventilation and air-conditioning (HVAC) ducting systems is discussed.

Acoustic scattering in waveguides that are discontinuous in geometry and material property

Abstract The scattering of acoustic waves at the discontinuity between two ducts of different heights is considered. At least one of the ducts is bounded by a membrane and, thus, the underlying eigenproblem is non-Sturm–Liouville . A mode-matching procedure, based on an appropriate orthogonality relation, reduces the problem to that of truncating and solving an infinite system of linear equations. The distribution of power between the fluid regions and the membrane(s) is analysed. Further, it is shown that a fundamental property of the truncated system is that the expression for power balance is always satisfied.

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

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.

Orthogonality relations for fluid-structural waves in a three-dimensional, 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.

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.

On the propagation and scattering of fluid-structural waves in a three-dimensional duct bounded by thin elastic walls

The propagation of acoustic waves along ducts or pipes has long been of interest to scientists and engineers. Acoustic scattering is a feature that becomes relevant whenever there is an abrupt change in duct geometry or material property. Most of the analytic work relating to scattering in waveguides concerns ducts with two-dimensional or circular cylindrical geometries. The Wiener-Hopf technique has proved a powerful tool in those cases where the geometry is uniform but the material properties change discontinuously. Alternatively, where there is more than one change in the material properties or the geometry undergoes abrupt change (e.g. in duct height), eigenfunction expansions and their associated orthogonality relations are often an effective means of reducing the problem to a system of linear algebraic equations that can be truncated and solved numerically. Problems involving wave propagation and mode-conversion/scattering are much more difficult when the duct walls are compliant, i.e. when fluid-structural interactions need to be taken into account. In particular, the authors are interested in problems involving compressible fluids contained within ducts which have wave-bearing surfaces, such as membranes or elastic plates. It is a feature of such problems that, due to the presence of high order derivatives in the boundary conditions, the relevant eigen-sub-problems are not Sturm-Liouville in type. Nevertheless, appropriate orthogonality relations can be derived and these enable the successful solution of many such models.

An infinite, elastic, cylindrical shell with a finite number of ring constraints

Abstract The axisymmetric excitation of an infinite, elastic, cylindrical shell with a finite number of ring constraints is discussed. Exact solutions are presented for the cases of one and two constraints. These are then examined in an asymptotic limit that corresponds to light fluid-loading and small curvature.

Acoustic radiation from two opposed semi-infinite coaxial cylindrical waveguides. I: overlapping edges

Abstract This paper is the first part of an investigation into the axisymmetric radiation of sound waves from two semi-infinite circular cylinders. Both pipes have a common axis, measured by the Cartesian variable z, but the smaller cylinder extends to infinity in the direction z → -∞ whilst the larger extends in the direction z → ∞. The boundary value problem is formulated as a matrix Wiener-Hopf equation and a solution is obtained here for the case when the ducts overlap. The reflected, transmitted, and radiated far-field waves are determined for a range of values of the inner and outer pipe diameters, and the overlap spacing. A full discussion is made of the relevance of this model to physical problems in electromagnetic and acoustic wave propagation, and this is backed-up by conclusions drawn from the presented data.