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

The Journal of the Acoustical Society of America

2012

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The focus of this paper is towards the development of hybrid analytic-numerical modematching methods for model problems involving three-dimensional ducts of rectangular cross-section and with flexible walls. Such methods require firstly closed form analytic expressions for the natural fluid-structure coupled waveforms that propagate in each duct section and secondly the corresponding orthogonality relations. It is demonstrated how recent theory [Lawrie, Proc. R. Soc. A. 465, 2347-2367] may be extended to a wide class of three-dimensional ducts, for example, those with flexible walls and a porous lining (modelled 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 duct with flexible walls, whilst the second takes the form of a Fourier series. The latter has several advantages: in particular, no "rootfinding" 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 eigenmodes.

Section: Orthotropic Membranementioning

confidence: 56%

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

The Journal of the Acoustical Society of America

2012

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“…Howls & Trasler 1998) would be helpful in this respect. Alternatively, following Lawrie & Kirby (2006), it may be possible to construct a 'root-free' approach for certain duct geometries. In summary, this article presents an analytic investigation into a class of eigenfunctions that is potentially of use in mode-matching problems involving the propagation of sound in three-dimensional rectangular ducts.…”

Section: Discussionmentioning

confidence: 99%

“…The problems are drawn from a class often encountered when using mode-matching to solve a two-part problem involving acoustic propagation in a duct though normally for the two-dimensional case (e.g. Lawrie & Kirby 2006). For both prototype problems the duct is semi-infinite, lying in the region x > 0, but in all other respects is identical to that described at the beginning of §2.…”

Section: Two Prototype Scattering Problemsmentioning

confidence: 99%

Orthogonality relations for fluid-structural waves in a three-dimensional, rectangular duct with flexible walls

2009

Proc. R. Soc. A.

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 ψ n (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 ψ ny (a, z) are derived which, together with a partial orthogonality relation for ψ n (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.

“…In view of the fact that the systems of equations derived by mode-matching automatically conserve power, 15,14 it is worthwhile asking what global condition the normal velocities satisfy at x = ±ᐉ when h Ͼ a. For the parameters and frequency ranges considered herein, it has been verified numerically that the modematching solution satisfies ͑52͒ and…”

Section: ͑52͒mentioning

confidence: 99%

“…For a nondissipative system, such as this, root finding usually presents few problems. It is worthwhile mentioning, however, that the boundary value problem for the model problem falls within the class whereby the mode-matching equations can be recast into root-free form, 15 which bypasses the root-finding process. In contrast, the method employed by Huang 8,9 automatically avoids the need for root finding.…”

Section: Introductionmentioning

confidence: 99%

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

The Journal of the Acoustical Society of America

Guled

2006

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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 ͑HVAC͒ ducting systems is discussed.