Simon Félix

Sound attenuation in lined bends

In the present paper we are concerned with sound propagation and attenuation in two- or three-dimensional lined bends. First it is shown that the effect of locally reacting absorbing materials at the walls of a waveguide can easily be taken into account in the multimodal formulation proposed in earlier papers by the authors, and, for bends, algebraic solutions are carried out for the acoustic field and scattering properties. Then a study of the sound attenuation in lined bends is given using the multimodal formulation and the properties of such waveguides are shown and discussed, in particular, the presence of a plateau of attenuation at high frequencies and a whispering gallery effect that occurs in bends.

Experiments on metasurface carpet cloaking for audible acoustics

We present experiments on acoustic carpet cloaking by using a metasurface made of graded Helmholtz resonators. The thin metasurface, placed over the object to hide, is designed such that the reflection phase shifts of the resonators at the resonance frequency are tuned to compensate the shape of the object to cloak. Experimental as well as numerical results show the efficiency of the cloak at the resonance frequency. The reflection of a short pulse is also reported to inspect the broadband character of the cloak.

Sound propagation in rigid bends: A multimodal approach

The sound propagation in a waveguide with bend of finite constant curvature is analyzed using multimodal decomposition. Two infinite first-order differential equations are constructed for the pressure and velocity in the bend, projected on the local transverse modes. A Riccati equation for the impedance matrix is then derived, which can be numerically integrated after truncation at a sufficient number of modes. An example of validation is considered and results show the accuracy of the method and its suitability for the formulation of radiation conditions. Reflection and transmission coefficients are also computed, showing the importance of higher order mode generation at the junction between the bend and the straight ducts. The case of varying cross-section curved ducts is also considered using multimodal decomposition.

Multimodal analysis of acoustic propagation in three-dimensional bends

An exact multimodal formalism is proposed for acoustic propagation in three-dimensional rigid bends of circular cross-section. Two infinite systems of first-order differential equations are constructed for the components of the pressure and axial velocity in the bend, projected on the local transverse modes. These equations are numerically unstable, due to the presence of evanescent modes, and cannot be integrated directly. An impedance matrix is defined, which obeys a Riccati equation, numerically workable. With this nonlinear first-order differential equation, the impedance can be calculated everywhere in the bend, allowing a direct characterization of its acoustical properties or allowing the acoustic field to be integrated. An exact algebraic formulation of the reflection and transmission matrices is carried out to allow bends and more complex duct systems to be characterized. This result is applied to calculate the reflection and transmission of a typical bend, and also to obtain the resonance frequencies of closed tube systems.

On the use of leaky modes in open waveguides for the sound propagation modeling in street canyons

An urban, U-shaped, street canyon being considered as an open waveguide in which the sound may propagate, one is interested in a multimodal approach to describe the sound propagation within. The key point in such a multimodal formalism is the choice of the basis of local transversal modes on which the acoustic field is decomposed. For a classical waveguide, with a simple and bounded cross-section, a complete orthogonal basis can be analytically obtained. The case of an open waveguide is more difficult, since no such a basis can be exhibited. However, an open resonator, as displays, for example, the U-shaped cross-section of a street, presents resonant modes with complex eigenfrequencies, owing to radiative losses. This work first presents how to numerically obtain these modes. Results of the transverse problem are also compared with solutions obtained by the finite element method with perfectly mathed layers. Then, examples are treated to show how these leaky modes can be used as a basis for the modal decomposition of the sound field in a street canyon.

Wave propagation through penetrable scatterers in a waveguide and through a penetrable grating.

A multimodal method based on the admittance matrix is used to analyze wave propagation through scatterers of arbitrary shape. Two cases are considered: a waveguide containing scatterers, and the scattering of a plane wave at oblique incidence to an infinite periodic row of scatterers. In both cases, the problem reduces to a system of two sets of first-order differential equations for the modal components of the wavefield, similar to the system obtained in the rigorous coupled wave analysis. The system can be solved numerically using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed (convergence, reciprocity, energy conservation). Alternatively, the admittance matrix can be used to get analytical results in the weak scattering approximation. This is done using the plane wave approximation, leading to a generalized version of the Webster equation and using a perturbative method to analyze the Wood anomalies and Fano resonances.

A coupled modal-finite element method for the wave propagation modeling in irregular open waveguides.

In modeling the wave propagation within a street canyon, particular attention must be paid to the description of both the multiple reflections of the wave on the building facades and the radiation in the free space above the street. The street canyon being considered as an open waveguide with a discontinuously varying cross-section, a coupled modal-finite element formulation is proposed to solve the three-dimensional wave equation within. The originally open configuration-the street canyon open in the sky above-is artificially turned into a close waveguiding structure by using perfectly matched layers that truncate the infinite sky without introducing numerical reflection. Then the eigenmodes of the resulting waveguide are determined by a finite element method computation in the cross-section. The eigensolutions can finally be used in a multimodal formulation of the wave propagation along the canyon, given its geometry and the end conditions at its extremities: initial field condition at the entrance and radiation condition at the output.

Generalized method for retrieving effective parameters of anisotropic metamaterials

Electromagnetic or acoustic metamaterials can be described in terms of equivalent effective, in general anisotropic, media and several techniques exist to determine the effective permeability and permittivity (or effective mass density and bulk modulus in the context of acoustics). Among these techniques, retrieval methods use the measured reflection and transmission coefficients (or scattering coefficients) for waves incident on a metamaterial slab containing few unit cells. Until now, anisotropic effective slabs have been considered in the literature but they are limited to the case where one of the axes of anisotropy is aligned with the slab interface. We propose an extension to arbitrary orientations of the principal axes of anisotropy and oblique incidence. The retrieval method is illustrated in the electromagnetic case for layered media, and in the acoustic case for array of tilted elliptical particles.

Propagation in waveguides with varying cross section and curvature: a new light on the role of supplementary modes in multi-modal methods

We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems.

Effects of bending portions of the air column on the acoustical resonances of a wind instrument.

The need to keep long wind musical instruments compact imposes the bending of portions of the air column. Although manufacturers and players mention its effects as being significant, the curvature is generally not included in physical models and only a few studies, in only simplified cases, attempted to evaluate its influence. The aim of the study is to quantify the influence of the curvature both theoretically and experimentally. A multimodal formulation of the wave propagation in bent ducts is used to calculate the resonances frequencies and input impedance of a duct segment with a bent portion. From these quantities an effective length is defined. Its dependence on frequency is such that, compared to an equivalent straight tube, the shift in resonance frequencies in a tube with bent sections is not always positive, as generally stated. The curvature does not always increase the resonances frequencies, but may decrease them, resulting in a complex inharmonicity. An experimental measurement of the effect of the curvature is also shown, with good agreement with theoretical predictions.