Keywords: R-matrix method, Trapped mode, Fano resonance, Avoided crossing, Exceptional point, Sound attenuation, Bloch waves, Slow sound matrix, Mode piégé, Résonance de Fano, Croisement évité, Point exceptionnel, Atténuation du R-matrix method, Trapped mode, Fano resonance, Avoided crossing, Exceptional point, Sound

guidance helped me a lot in the thesis work. Thanks to Bi's recommendation and Aurégan's acceptance, it was such a

vi Résumé Le thème central de cette thèse est l'utilisation des effets de couplage de modes pour l'amplification de l'atténuation du son dans les conduits acoustiques. Les applications potentielles concernent les silencieux pour les systèmes de ventilation et les traitements de réacteurs pour l'aviation. Deux stratégies différentes sont présentées.

La première stratégie proposée est de coupler le mode incident propagatif dans le guide d'onde avec un mode localisé dans la région diffusive. Cette stratégie est présentée et validée dans un système conduit-cavité et un guide d'onde partiellement traité en paroi avec un matériau à réaction locale. La méthode ! R-matrix " est introduite pour résoudre le problème de propagation d'onde dans le guide d'onde. Une annulation de la transmission se produit quand un mode piégé est excité dans le système ouvert. Le zero de transmission suivi immédiatement d'un pic de résonance forment une résonance de Fano. Ce mode piégé est formé par les interférences de deux modes voisins avec des fréquences de résonance complexes. Ce phénomène est aussi lié au croisement évité des valeurs propres et à un point exceptionnel (point dans un espace 2D où les valeurs propres et vecteurs propres d'un opérateur non-hermitien se confondent). La matrice de diffusion est exprimée en termes d'une matrice H eff qui décrit les résonances complexes dans la région diffusive. Avec l'aide des valeurs et fonctions propres de la matrice H eff , la formule de diffusion résonante acoustique traditionnelle peut être étendue pour décrire les effets de couplage entre la région diffusive et les parties rigides du guide d'onde.

Dans la seconde stratégie, un réseau d'inclusions rigides périodiques est intégré dans une couche poreuse pour améliorer la'tténuation du son à basse fréquence. Le couplage de modes est du à la présence de ces inclusions. Le théorème de Floquet-Bloch est proposé pour analyser l'atténuation du son dans un guide d'onde périodique en 2D. Un croisement de l'atténuation de deux ondes de Bloch est observé. Au voisinage de la fréquence de croisement, un pic d'atténuation est observé. Ce phénomène est utilisé pour expliquer le pic de pertes en transmission observé expérimentalement et numériquement dans un guide 3D avec une portion de paroi traitée par un matériau poreux avec des inclusions périodiques.

Enfin, le comportement acoustique d'un liner purement réactif dans un conduit rectangulaire avec et sans écoulement est étudié. Les résultats montrent un comportement acoustique inhabituel : dans une certaine gamme de fréquence, aucune onde ne peut se propager à contre sens de l'écoulement. Par analyse des différent modes à l'aide de la relation de dispersion, il est démontré que le son peut être ralenti et même arrêté.

Mots-clés

Méthode R-matrix, Mode piégé, Résonance de Fano, Croisement évité, Point exceptional, Atténuation du son, Ondes de Bloch, Son lent Chapter 1

General introduction 1.1 Background

Acoustic liners [1] are widely used in ducts to reduce aircraft engine noise emission. Two main different acoustic liners are commonly used: the locally-reacting liner and the bulk-reacting liner. The locally reacting liners are generally made with a perforated sheet backed by honeycomb and permit propagation only in the direction normal to the duct wall. This kind of materials have good absorption properties only in a narrow frequency band but their main advantages are their mechanical robustness and their capability to resist to harsh conditions. Bulk-reacting liners may consist of isotropic or anisotropic porous material. They generally offer a wider absorption/attenuation band. However, they suffer from a lack of absorption efficiency at low frequencies, comparing to their efficiency at higher frequencies.

The use of uniform acoustic duct liners to suppress the noise is well established, research efforts have been focused on the prediction of the acoustic liner performances and the design of optimum liners. The minimum transmission (maximum attenuation) coefficient for each mode in an infinite uniform waveguide or in a duct lined with locally reactive impedance, is Cremer's optimum attenuation [2]. The corresponding lined impedance is called Cremer's optimum impedance. Each mode has one and only one corresponding optimum impedance [3,4], in the absence of mean flow or with uniform flow [3]. Soon after, axially and circumferentially segmented liners have also been considered in the design of more effective noise-attenuation systems, for review, see for example Refs. [5,6,7] and references therein. The basic idea of these non-uniform liners is: one less attenuated mode may be scattered into higher attenuated modes at the non-uniform parts (interfaces) by mode couplings.

To date, due to the growth of fan diameter, the reduction of the rotation speed and of the number of blades, sound attenuation at low frequencies still represents a challenge. To solve that problem, other approaches must be investigated and new concepts of acoustic absorbers dedicated to the reduction of turbomachinery noise must be developed.

Chapter 1. General introduction

Motivated by this situation, the objective of this thesis is to use some new mechanisms to reduce sound transmission or improve the sound attenuation in acoustic ducts. Two different strategies will be presented.

For the first strategy, we will focus on the coherent mode couplings in a duct-cavity system and in a rigid duct partially treated with uniform locally reacting liner, and their effects on the sound propagation behaviours. The idea is to use the interference effect between the incoming propagative mode in the rigid regions and the trapped mode in the cavity/lined region, to produce a transmission zero or a narrow band of dip. This trapped mode localises in the lined/cavity region, which is regarded as an open cavity opened to the infinities through the rigid regions, in both transverse and axial directions, and it is also called (real or complex) resonance. We will use trapped mode(s) or resonance(s) alternatively in this thesis. In the point of view of resonance, the above mentioned interference effect is a resonance scattering process. In contrast to the usual resonance scattering (see, e.g., Uberall [8]), the interferences between the incoming wave and local resonance produce a Fano resonance. As trapped mode, Fano resonances are some relatively new concepts in acoustics, and we will show, in this thesis, that they are closely connected with "exceptional points" and "avoided crossings". We will explain these concepts by some simple examples in the following sections.

In the second strategy, to improve the liner efficiency at low frequencies, material including periodic subwavelength resonators are studied in analogy to optical metamaterials [9,10,11]. In this view, metaporous materials, made of periodic rigid inclusions embedded in the porous medium, have been proposed to enhance the absorption properties at frequencies lower than the quarter wavelength frequency [12,13]. The principle is that the energy is trapped between periodic rigid inclusions embedded in the porous medium and the rigid backing or in the inclusions themselves. In the second strategy, we deals with the bulk-reacting liner. The coupling of the modes due to the presence of the rigid inclusions embedded in the porous material is investigated and introduced to enhance the sound attenuation in the waveguide at low frequencies.

The use of periodic inclusion in the waveguide has also been studied by Nennig et al. [14]. The Floquet-Bloch theorem is used to reduce the computation on one periodic cell. It has been shown that open shape inclusions (e.g. U-shape) are able to enhance the attenuation when compared with a homogeneous liner. Thanks to Floquet-Bloch formalism assuming periodic heterogeneities [12,13,14,15], very interesting results have been obtained in acoustics. The Floquet theorem is applicable to ordinary differential equations with periodic coefficients and shows that the solutions are such that ψpx `L0 q " e jk B L 0 ψpxq, where L 0 is the period of the coefficients, k B is the Bloch wave number, and x is the propagation axis. Such functions are known as Bloch wave functions. Transverse modes in a 2D uniform (coordinate px, yq) normalized rigid duct (see Fig. 1.1) can be defined by Bφpyq By " ´α2 φpyq,

with boundary conditions Bφpyq By ˇˇˇy "0,1 " 0.

(1.

2)

The solutions are φ n pyq " Λ cospα n yq and α n " nπ with Λ " 1 if n " 0, else ?

where n " 0, 1, 2, ¨¨¨, . labels the mode. In the infinite uniform waveguide (x Ñ ˘8), waves ppx, yq with transverse modes φ n pyq and wavenumber in x´direction k x n propagate as ppx, yq " φ n pyqe ´jωt˘jk x n x , (1.4) where k x n " a k 2 ´α2 n .

x However, some open systems (see e.g. Fig. 1.2) may sustain such kind of mode, which mathematically corresponds to real eigenvalues of the relevant operator in an unbounded domain. In the literature, they are often called "trapped modes" to stress Chapter 1. General introduction their localization properties in open system. They are also called bound states in quantum mechanics [16]. If a part of energy is radiated to the infinity, the eigenvalues are complex, they are also called complex resonances in open cavities or leaky modes in waveguides [17]. Therefore, discrete trapped modes with eigenfunctions ϕpx, yq and real eigenvalues k, which satisfy eigenvalue problem B 2 ϕ{Bx 2 `B2 ϕ{By 2 " ´k2 ϕ (1.5) and boundary conditions along y and radiation conditions along x, and have finite energy,

ij s dy ż 8
´8 dx|ϕ| 2 is finite, (1.6) where s is waveguide cross section, in general do not exist. However, discrete trapped modes may exist below the first cut-off frequency of the transverse modes (e.g. in waveguides with pressure release boundary conditions (Dirichlet waveguides, see e.g. Fig. 1.3), or antisymmetric transverse modes in rigid waveguides (Neumann waveguides)), provided some kinds of defect or variations of geometry exist [17,18]. Discrete trapped modes may also exist above the first cut-off frequency for specific parameter combinations, called embedded trapped modes or bound states in continuum (BIC) in quantum mechanics [19].

Here "continuum" means the eigenvalues (or spectrum in maths and quantum mechanics) k are distributed continuously and the corresponding eigenfunctions are in general not square integrable due to the radiation of energy to infinity. Although BIC are embedded in continuum spectrum, their eigenfunctions are still square integrable, i.e., their energy is finite. The basic mechanism of trapping is illustrated by using a simple model of the potential well by Pagneux [18]. The existence of trapped mode problem in the waveguides has been investigated by many authors in various physical contexts theoretically and experimentally. In locallyperturbed acoustic waveguides, it was first observed experimentally by Parker in 1966 [20], who also made the first theoretical calculations of resonance frequencies (eigenvalues of trapped modes, in this thesis, we will use resonance or trapped mode alternatively) in the following year [21]. Since then, more investigations have been done, for example, embedded trapped mode frequencies for a variety of configurations have been calculated by Duan et al. [17]. Recently, S. Hein et al. [22] used the finite-element method to numerically compute the acoustic resonances in 2D acoustical duct-cavity systems. Three types of trapped modes are introduced: antisymmetric (about duct axis) trapped modes below the first cut-off frequency, embedded trapped modes linking with avoided crossings of resonances, and trapped modes associated with Fabry-Pérot interference between cavities. The second type of trapped mode is of our first interest, see chapters 2 and 3.

Trapped modes (resonaces, or BIC) may be linked closely with avoided crossings of eigenvalues and Exceptional Points (EPs) [23,24]. By Feshbach's theory of resonance, Friedrich and Wintgen [25] demonstrated that BICs in atomic physics can occur, due to the interference of resonance belonging to different channels. By varying a continuous parameter, avoided crossings of resonances are observed. At a particular value of the parameter, one resonance has exactly vanishing width and becomes a BIC, the same phenomenons are also observed in acoustical duct-cavity system and acoustic lined duct, see Chap. 2 and Chap. 3, respectively. In quantum systems, Almas F. Sadreev et al. [26] showed that the BIC also appears in open quantum billiards by varying their shape continuously. They also found that the BICs are close to the points of degeneracy [27] of the closed quantum system, which are the points in a two-dimensional parameter space where only the eigenvalues coalesce, while the corresponding eigenfunctions are still orthogonal. When the system is opened by attaching to two infinite leads, the coupling of the two resonances is turned on, an avoided crossing occurs with the variation of the parameters [24]. This avoided crossing is associated with the Exceptional P oints(EPs) in the parameter plane. At the EPs, the two resonances will coalesce, not only the eigenvalues but also the eigenfunctions [23,24]. An example about the existence of EP in acoustics is the Cremer's optimum impedance, see Chap.1.4 in the following.

Exceptional points and avoided crossings: a simple illustration example

We consider the simple example (see Fig. 1.4), two pendulums with the same mass m are coupled through a spring k. The length l of the unperturbed pendulum is changed for the second pendulum to the value lp1 `σq. The force on pendulum 1 in the direction of motion due to the coupling is ´kl pθ 1 ´p1 `σqθ 2 q and the force on pendulum 2 due to the coupling is ´kl pp1 `σqθ 2 ´θ1 q. The differential equations for the angular motions

d 2 dt 2 θ 1 `ω2 1 θ 1 `k m θ 1 ´k m p1 `σqθ 2 " 0, d 2 dt 2 θ 2 `ω2 2 θ 2 `k m θ 2 ´k m 1 p1 `σq θ 1 " 0, (1.7)
where ω 1 " b g l and ω 2 " b g lp1`σq are the natural frequencies of the two uncoupled pendulums. The differential equations can be written in a matrix form

d 2 dt 2 θ `H θ " 0, (1.8)
where

H " H 0 `k m H 1 " " ω 2 1 0 0 ω 2 2  `k m " 1 ´p1 `σq ´1 1`σ 1  (1.9)
with H 1 the coupling matrix due to the presence of the spring. The eigenvalues λ are the solutions of H θ n " λ n θ n , i.e., ∆pλ, σq " detpH ´λIq,

where n " 1, 2, λ " ω 2 , and I is the identity matrix.

When there is no coupling between the two pendulums, i.e. k " 0, the two eigenvalues coalesce if ∆pλ, σq " pω 2 1 ´λqpω 2 2 ´λq " 0 and B∆pλ, σq{Bλ " 0. We find that at σ " 0, λ 1,2 " ω 2 1 " g{l. This is called a degeneracy [27]. The two eigenvalues coalesce, while the two eigenfunctions, such as cos ω 1 t and sin ω 1 t, are orthogonal.

When k ‰ 0, there is coupling between the two pendulums, the eigenvalues are

λ 1,2 " 1 2 " ω 2 1 `ω2 2 `2 k m ˘Rc  , with R c " c ω 4 1 `ω4 2 ´2ω 2 1 ω 2 2 `4p k m q 2 . (1.10)
We can see that when R c ‰ 0, there exists avoided crossing between the two eigenval-1.3. Exceptional points and avoided crossings: a simple illustration example 7 ues(see Figs. 1.5 and 1.6). However, the two eigenvalues will coalesce if R c " 0, this occurs at the two complex conjugate σ (Here the σ " σ r `iσ i has been assumed to be analytically continued in the complex plane). We can find that at

σ ep " ´4C 4 ˘j2C 2 1 `4C 4 , (1.11)
we have

λ ep " 1 2 " ω 2 1 `ω2 2 `2 k m  , (1.12)
where C 2 " k{m g{l is the coupling strength. σ ep is an exceptional point (EP) in the complex parameter plane σ, where we have λ 1 " λ 2 " λ ep . Trajectories of the eigenvalues in the complex plane with three different σ i , i.e., ´0.2pII and IVq, ´0.19pI and IIIq, and ℑmpσ ep q(crossing), respectively. Arrows indicate the increase with σ r . The other parameters are ω 1 " 1 and k{m " 0.1.

As an example, we choose ω 1 " 1 and k{m " 0.1, leading to σ ep " ´0.0385 ˘j0.1923. We present a topological argument why either the real parts or the imaginary parts must cross when eigenvalue trajectories pass the vicinity of an EP [24]. When σ i p" ´0.2q ă ℑmpσ ep q, crossing for the real parts and avoided crossing for the imaginary are observed, (II and IV in Fig. 1.5). While when σ i p" ´0.19q ą ℑmpσ ep q, avoided crossing for the real parts and crossing for the imaginary are observed, ( I and III in Fig. 1.5). Only in the special case (σ i " ℑmpσ ep q) where the parameter σ moves straight through an EP will both, the real and imaginary part, cross. Using this crossing, we can divide the complex plane into four quadrants, labelled by I, II, III, and IV in Fig. 1.5, respectively.

The details of the crossings and avoided crossings in Fig. 1.5 are also considered again in Fig. 1.6, in which we plot the real and imaginary parts of the eigenvalues λ 1,2 as a function of the real parameter σ r for two different values of σ i (´0.2(upper figure) and ´0.19(bottom figure), respectively). Crossing is observed for the real part with Chapter 1. General introduction σ i " ´0.2, while for the imaginary part with σ i " ´0. 19. Other choices lead qualitatively to the same result. From Eq. (1.11), the σ ep is a complex conjugate pair. The other σ ep can also have the same effects on λ by taking σ i " 0.19 and 0.2. Exceptional points were first introduced by Kato [28], and were extensively developed by Heiss [23,24,29], Rotter [30,31,32], and Berry [33]. Extensions of this concepts in acoustics is shown in [34]. The mathematically topological structures of Riemann sheets of an EP, which are a square root branch point of the coalescing eigenvalues and a fourthorder branch point of the coalescing eigenfunctions depending on a complex or two real parameters, have been proved physical reality [35]. EPs have attracted much attention in non-Hermitian Hamiltonian quantum and optical systems (see, e.g. Refs. 30, 36 and the references therein). They have been found in different domains: laser-induced ionization states of atoms [37], atom waves in crystals of light [38], electronic circuits [39], atoms in cross magnetic and electric fields [40], and in microwave billiards [41]. Experiments in laboratories have been carried out with resonances in microwave cavities [35,41,42].

EPs are closely related to avoided crossings of eigenvalue curves. If the parameter σ is real, the system H in Eq. (1.9) is Hermitian. The typical behaviour of the eigenvalues of matrix H as a function of real σ when they approach each other, is repulsion, or an avoided crossing. This is the famous rule of von Neumann and Wigner [START_REF] Von Neumann | Uber das verhalten von eigenwerten bei adiabatischen prozessen[END_REF] in quantum mechanics who proved that in systems without any symmetry we have to vary two parameters in order to create a (conventional) degeneracy. The analytic continuation into the complex σ-plane yields a complex conjugate pair of EPs where the two coalescing 1.4. Exceptional points and avoided crossings: applications in lined waveguides modes are analytically connected by a square root branch point [29]. Avoided crossings of eigenvalues play an important role in quantum mechanics [START_REF] Landau | Quantum Mechanics[END_REF]. They have been also found in the area of structural dynamics [START_REF] Leissa | On a curve veering aberration[END_REF][START_REF] Kuttler | On curve veering[END_REF][START_REF] Perkins | Comments on curve veering in eigenvalue problem[END_REF] and related to mode localization in disordered structures [START_REF] Pierre | Mode localisation and eigenvalue loci veering phenomena in disordered structures[END_REF][START_REF] Triantafyllou | Frequency coalescence and mode localisation phenomena: a geometric theory[END_REF]. In Chap. 4, we will see that they also play an important role in the sound attenuation in a waveguide lined by a porous material embedded with periodic inclusions. A maximum sound attenuation is observed near a crossing (or avoided crossing) of the mode attenuations between two Bloch waves.

Exceptional points and avoided crossings: applications in lined waveguides

In the following, we will see that for the applications in the lined intakes of an aeroengine, EP [34] exists in complex admittance (impedance) plane and plays an important role in the sound attenuation [34]. An example of the EPs in acoustics is the Cremer's optimum impedance. The Helmholtz equation " 0

B 2 ψ By 2 `k2 y ψ " 0 (1.
is solved by Multimodal Method [START_REF] Pagneux | A study of wave propagation in varying cross-section waveguides by modal decomposition. part i. theory and validation[END_REF][START_REF] Bi | Modelling of sound propagation in a non-uniform lined duct using a multi-modal propagation method[END_REF], where k y is the corresponding eigenvalue. Function ψpyq is projected in the basis of the rigid modes (Eq. (1.3))

ψ y pyq " ÿ A n φ n pyq.
Multiplying the Helmholtz equation of ψpyq by φ n pyq on both sides, and then integrating with respect to y over r0, 1s, we can write the upper eigenvalue problem in a matrix form

k 2 y A " Γ A ´jKY ¨M A,
where M mn " φ m p0qφ n p0q, and Γ is a diagonal matrix with elements α 2 n . The eigenvalue problem of the lined waveguide can be regard as solving the eigenvalue problem of the matrix H " Γ ´jKY M. (labeled by "Mode1" and "Mode2", respectively ) as a function of ℜepKY q under two different values of ℑmpKY q " ´1.5 ('*') and ´2 ('+') respectively. (c)Trajectories of the wavenumber in complex plan, arrow shows the direction of the movements with increasing the real part of KY . the work of F. P. Mechel [4], the first two exceptional points for the first three modes are KY EP 1 " 2.05998 ´j1.65061 and KY EP 2 " 5.33471 ´j2.05785, with expp´jωtq considered. Here, the first point is investigated, i.e. the behaviours of the first two eigenvalues are considered. In Fig. 1.7, we show the motions of the first two eigenvalues k y (labeled by "Mode1" and "Mode2", respectively) as a function of ℜepKY q under two different values of ℑmpKY q. The amplitude of its eigenfunction from the decreases exponentially away from the lined wall, which is 9 expp´|k y |yq. With increasing the absolute value of ℑmpKY q, the |k y | will increase, resulting in that the eigenfunction of transverse mode decreases faster from the lined wall, these properties are also discussed in Chap. 5 of this thesis.

Two corresponding different kind of avoided crossings are found in Fig. 1.7 for ℑmpKY q " ´1.5 and ´2, respectively. The transition of the two types of avoided crossings indicates that there exist an exceptional value of pℜepKY q, ℑmpKY qq, at which the two eigenvalues curves will coalesce, this value is called the Exceptional Point(EP) in the complex KY parameter plane. To see the EP location in the parameter plane clearly, the real and imaginary parts of k y as a function of both ℜepKY q and ℑmpKY q are also given, see Fig. 1.8. We can see that the two eigenvalues are the values of one single analytical function on two Riemann sheets. The point (k y1 " k y2 ) is called branch point singularity [23].

Fano resonance

Fano resonance was firstly suggested by Ugo Fano [START_REF] Fano | Effects of configuration interaction on intensities and phase shifts[END_REF] for the description of autoionizing atomic states. In contrast to the conventional isolated resonances, e.g., resonance in a harmonic oscillator with periodic forcing, for which the spectral dependence can be described by the Lorentzian or Breit-Wigner formula -a symmetric spectral line; Fano resonance was explained as a phenomenon of constructive and destructive interferences between a bound state and the continuum [START_REF] Fano | Effects of configuration interaction on intensities and phase shifts[END_REF][START_REF] Yong S Joe | Classical analogy of fano resonances[END_REF]. The corresponding spectral lines are asymmetric. Since its discovery, Fano resonance has been attracting much attention in different research fields, for review articles see, for example, [START_REF] Luk'yanchuk | The fano resonance in plasmonic nanostructures and metamaterials[END_REF][START_REF] Miroshnichenko | Fano resonances in nanoscale structures[END_REF] and the references therein. Fano resonances have been observed experimentally in different domains of physics [START_REF] Luk'yanchuk | The fano resonance in plasmonic nanostructures and metamaterials[END_REF][START_REF] Miroshnichenko | Fano resonances in nanoscale structures[END_REF]. Extensive studies were carried out in Fano resonance induced by the interferences between bound states (or quasi-bound states, complex resonances) and the continuum, or between quasi-bound states through the continuum, for example, see Refs. 56, 57, 58, 59, 60, 61. Fano resonance in acoustic scattering and their relations with trapped modes in waveguides including obstacles or in duct-cavity systems have been studied by Hein et al. [22,[START_REF] Hein | Fano resonances in acoustics[END_REF]. Y. Joe et al. [START_REF] Yong S Joe | Classical analogy of fano resonances[END_REF] gave a classical analogy of appearance of the Fano resonances based on a simple harmonic oscillator system model. In this section, the same system as used in Sec. 1.3 is applied here to give the explanation of the physical nature of Fano resonance. The equations of angular motion for the two coupled pendulums with an external driving Chapter 1. General introduction

m m k θ 1 θ 2 l l(1+σ) Force 1 2
Figure 1.9: Schematic view of two coupled pendulums with a driving force applied to one of them.

force on pendulum 1 may be written as

d 2 dt 2 θ 1 `ω2 1 θ 1 `k m θ 1 ´k m p1 `σqθ 2 " a 1 e jωt , and
d 2 dt 2 θ 2 `ω2 2 θ 2 `k m θ 2 ´k m 1 p1 `σq θ 1 " 0, (1.16)
where a 1 e ´jωt written in a complex form is the term due to the external force. After some manipulation, one can obtain that the steady-state solutions for the angular of the pendulums are also harmonic such that

θ 1 " c 1 e ´jωt , θ 2 " c 2 e ´jωt .
The amplitudes are

c 1 " `ω2 2 ´ω2 `k m ω2 1 ´ω2 `k m ˘`ω 2 2 ´ω2 `k m ˘´p k m q 2 a 1 , (1.17)
c 2 " k m 1 1`σ `ω2 1 ´ω2 `k m ˘`ω 2 2 ´ω2 `k m ˘´p k m q 2 a 1 . (1.18)
The amplitudes of the two pendulums as a function of the frequency of an external force are shown in Fig. 1.10(a) and (b), respectively, where we have used σ " ´0.1, k{m " 0.1, and a 1 " 0.1. In the figure, the frequency ω is normalised by ω 1 . Two resonant peaks appear, the location of the resonant peaks corresponds to the real parts of the complex eigen-frequencies ω1,2 " a λ 1,2 (Eq.(1.10)). In Fig. 1.10(a), the reason why the second resonant peak is asymmetric is due to the existence of the zero-frequency at ω zero " b ω 2 2 `k m " 1.0954, which is right near the peak position(see the inset), and depends on the coupling strength k{m. The tendency of the resonance of the second pendulum as a function of the frequency is rather straightforward, see Fig. 1.10(b).

Fano resonance in an acoustic waveguide with a single defect

Analogous to the single defect in a quasi-one-dimensional wire [START_REF] Yong S Joe | Classical analogy of fano resonances[END_REF][START_REF] Philip | Evanescent modes and scattering in quasi-one-dimensional wires[END_REF][START_REF] Tekman | Fano resonances in quasi-one-dimensional electron waveguides[END_REF], we also consider a single defect with Y pxq P r0, x 1 s in a acoustic duct with normalized height h " 1. The defect is introduced by an normalized admittance Y pxq with very small length x 1 , see Fig. 1.11. When the length x 1 is very small and tends to be zero, the admittance can be expressed by a δ function, i.e., Y pxq " Y 0 δpxq, with δpxq " 1 if and only if x " 0, else δpxq " 0. Here and in the following, all quantities are nondimensionalized. The sound ppx, yq can propagate along x direction, the full wave equation is

ˆB2 Bx 2 `B2 By 2 `K2 ˙ppx, yq " 0, (1.19)
where K " ωh{c 0 is the dimensionless frequency, ω is the circular frequency, and c 0 is Chapter

B 2 c m pxq Bx 2 `pK 2 ´α2 m qc m pxq " ´jKY pxq ÿ n c n pxqφ n p0qφ m p0q. (1.23)
The upper equation can be written in a matrix form

B 2 c m pxq Bx 2 `pK 2 ´α2 m qc m pxq " ÿ n M 0 mn c n pxq, (1.24)
where M 0 mn " ´jKY pxqφ m p0qφ n p0q. In the regions where the admittance is zero, the solutions to Eq.(1.24) are c n pxq "

#

A n e jβnx `Bn e ´jβnx , x ă 0

C n e jβnpx´x 1 q `Dn e ´jβnpx´x 1 q , x ą x 1 (1.25) for the propagating modes, where β n " a K 2 ´α2 n , and

c n pxq " # A n e ´|βn|x `Bn e |βn|x , x ă 0 C n e ´|βn|px´x 1 q `Dn e |βn|px´x 1 q , x ą x 1 (1.26)
for the evanescent modes. ´"

ÿ n M 0 mn c n p0q, (1.28)
we have usedc m p0 `q " c m p0 ´q.

We suppose that plane mode is incident, and |KY 0 | ă |α 0 ´α1 | and K ă α 1 . In this case, only the plane mode is propagating, and it is surfficient to consider only one evanescent mode n " 1. We assume that the waves are incident from the left, resulting in D n " 0 for all the modes, and A n " 0 for all the evanescent modes. We can write

A 1 `B1 " C 1 , B 2 " C 2 , jβ 1 C 1 ´jβ 1 pA 1 ´B1 q " M 0 11 C 1 `M0 12 C 2 , and ´|β 2 |C 2 ´|β 2 |B 2 " M 21 C 1 `M22 C 2 .
We otain

p2jβ 1 ´M0 11 qC 1 ´M0 12 C 2 " 2jβ 1 A 1 , and M 0 21 C 1 `pM 0 22 `2|β 2 |qC 2 " 0.
(1.29)

From the above two equations, we can have the amplitudes for the transmitted modes

C 1 " 2jβ 1 pM 0 22 `2|β 2 |q p2jβ 1 ´M0 11 qpM 0 22 `2|β 2 |q `pM 0 12 q 2 A 1 , (1.30)
C 2 " ´2jβ 1 M 0 21 p2jβ 1 ´M0 11 qpM 0 22 `2|β 2 |q `pM 0 12 q 2 A 1 .
(1.31)

From Eq.(1.30), the transmission and reflection coefficients for the plane mode are obtained as

t 11 " C 1 A 1 " 2jβ 1 pM 0 22 `2|β 2 |q p2jβ 1 ´M0 11 qpM 22 `2|β 2 |q `pM 0 12 q 2 ,
(1.32)

r 11 " B 1 A 1 " M 0 11 pM 0 22 `2|β 2 |q ´pM 0 12 q 2 p2iβ 1 ´M11 qpM 0 22 `2|β 2 |q `pM 0 12 q 2 .
(1. 33) We can see that if M 0 22 `2|β 2 | " 0, we have t 11 " 0. Since

M 0 22 " ´jKY 0 φ 2 p0qφ 2 p0q " ´2jKY 0 , Chapter 1. General introduction we can have ´2jKY 0 `2b α 2 2
´K2 " 0, this results in

K 2 zero " α 2 2 1 ´Y 2 0 . (1.34)
If Y 0 is purely imaginary negative value, we can have K zero ă α 2 . Let us take a look at the reflection coefficient, if

M 0 11 pM 0 22 `2|β 2 |q ´pM 0 12 q 2 " 0,
we will have reflection zero, namely, transmission is one, and because M 0 11 M 0 22 ´pM 0 12 q 2 " 0, so that if |β 2 | " 0, namely, the incident frequency equals to cut-on frequency of the second mode, then the maximum transmission is reached. The transmission zero followed by a transmission one constructs the Fano resonance line-shape [START_REF] Yong S Joe | Classical analogy of fano resonances[END_REF], see Fig. 1.12(a). For the sake of simplification, we assume that the admittance Y " ´0.5i is independent of frequency. The wave functions c 1 pxq and c 2 pxq are plotted in Fig. 1.12(b). We can see that wave c 2 pxq is localized at the defect position, and decreases to infinite. Comparing the two cases: an impedance in the acoustic waveguide and an attractive potential in the quantum waveguide [START_REF] Yong S Joe | Classical analogy of fano resonances[END_REF], they both have the same form for the transmission and reflection coefficients when the impurity can be represented by a δ function.

1.6. R-matrix method

R-matrix method

In this thesis, an efficient method, the Reaction matrix (R-matrix) method [START_REF] Wigner | Higher angular momenta and long range interaction in resonance reactions[END_REF], is used to study the wave propagation in perturbed acoustic waveguide, the use of this method in this thesis can be found in chapters 2 and 3. The R-matrix method can treat easily the non-separable problem along transverse and axial directions. It was developed by Wigner and Eisenbud [START_REF] Wigner | Higher angular momenta and long range interaction in resonance reactions[END_REF] in scattering processes in nuclear physics in the late 1940s. Recently, it was extended by Racec et al [START_REF] Racec | Evanescent channels and scattering in cylindrical nanowire heterostructures[END_REF] to investigate the scattering phenomena in cylindrical nanowire heterostructures. The basic idea of this method is similar to the Multimodal method [START_REF] Pagneux | A study of wave propagation in varying cross-section waveguides by modal decomposition. part i. theory and validation[END_REF][START_REF] Bi | Modelling of sound propagation in a non-uniform lined duct using a multi-modal propagation method[END_REF]. For Multimodal method, the wave function is expanded in terms of a convenient transverse complete basis. The R-matrix method decomposes the whole system into a scattering region (cavity or acoustic liner in this thesis), and two semi-infinite rigid ducts. The wave function in the scattering region is expanded in terms of any convenient complete set of modes of a closed cavity with convenient boundary conditions. It is noted that the basis includes the transverse and axial components. Using the continuity conditions of the pressure and the normal particle velocity at the interfaces between the regions, we can write the scattering matrix S in terms of the R-matrix. Similar studies using R-matrix to calculate S matrix can be found in Refs. 67, 68, 69. By the scattering matrix, the matrix H eff of the open system is derived [START_REF] Stöckmann | Effective hamiltonian for a microwave billiard with attached waveguide[END_REF]. Matrix H eff describes totally the properties of the open cavity/lined section. Its eigenvalues are complex and give the poles of the S matrix. Its eigenfunctions are used for the calculation of the coupling matrix between the open cavity/lined section and uniform waveguide. There are some important advantages of this method. H eff is frequency dependent, therefore its eigenvalues or the complex poles of the S matrix and its eigenfunctions or the coupling matrix are frequency dependent. The frequency dependence of poles may be not important in the case of isolated resonances, but it is very important when the resonances are overlapping [START_REF] Rotter | Fano resonances in the overlapping regime[END_REF][START_REF] Stöckmann | Effective hamiltonian for a microwave billiard with attached waveguide[END_REF]. Another advantage of this method is that the S matrix is related clearly to the coupling matrix between the open cavity/lined section and uniform waveguide as shown in Eq. (3.20) in Chap. 3. The calculation method is inspired by Stöckmann et al. [START_REF] Stöckmann | Effective hamiltonian for a microwave billiard with attached waveguide[END_REF], who developed a relation of S matrix with an effective Hamiltonian for a microwave billiard with attached waveguide.

Thesis organization

The central theme of this thesis is to use the mode coupling effects to reduce the sound transmission in an acoustic duct for potential applications, for example silencers for ventilation systems, wall treatments for aircraft engines, and silencers for industrial gas turbines. This dissertation is organized into six chapters. Besides the current chapter (which intends to give a brief introduction of some basic and important concepts) and the last chapter with the conclusions, the other four chapters are organized as follows:

Chapter 1. General introduction In chapter 2, the first strategy is first introduced and presented by using a simple duct-cavity system. The trajectories of the modes in a closed cavity with varying the cavity length are first reviewed. After the closed cavity is opened and attached to rigid duct, the R-matrix method is introduced to solve the wave propagation problem in the system. Trapped modes can be found at some particular parameter combinations and near the avoided crossing of eigenvalues between two neighboured modes.

In chapter 3, the same strategy introduced and presented in chapter 2 is applied here to an acoustic waveguide lined with locally reacting liner, in order to produce an additional transmission zero, besides the one due to the resonance frequency of the liner. The scattering matrix is also derived by using the R-matrix method. For that, we project the Helmholtz equation over the eigenfunctions of the rigid closed counterpart of the scattering region which form an orthogonal and complete function basis. We express the scattering matrix S in terms of an effective matrix H eff . By using matrix H eff , the traditional acoustic resonance scattering formula is extended to describe the coupling effects between the open lined section and the rigid parts of the waveguide.

In chapter 4, the second strategy is presented to deal with the porous material. A set of periodic rigid inclusions are embedded in a porous lining to enhance sound attenuation in an acoustic duct at low frequencies. Floquet-Bloch theorem is introduced to investigate the mode attenuation in an infinite waveguide lined with periodic inclusions embedded in porous material. An analysis is first given for a 2D infinite periodic waveguide, crossing is observed for the mode attenuations between two Bloch waves, resulting in the maximum attenuation. Experimental measurements and numerical simulations are performed to obtain the transmission loss for 3D configurations. The transmission loss can be enhanced by the embedment of the rigid inclusions.

In chapter 5, we investigate the the acoustical behaviours of a purely reacting liner in a rectangular duct in both absence and presence of flow. Multi-Model Method is proposed to solve the problem in the absence and presence of uniform flow. The classical Ingard-Myers boundary condition is used in the presence of flow. The results exhibit an unusual acoustical behaviour: for a certain range of frequencies, no wave can propagate against the flow. The effect of shear flow is investigated by the Chebyshev Spectral Method, which provides detailed information near the walls. The effects of the Mach number and the frequency on the eigenvalues of the modes are also studied.

Chapter 2

Fano resonance scattering in a duct-cavity system

In this chapter, the trapped mode due to the interaction of two neighboured modes in an open system is first investigated in a simple duct-cavity system. The eigenvalues of the modes depend on the geometry of the cavity. By varying one parameter continuously, two types of avoided crossings between the eigenvalues of two modes in the complex plane are observed under two different values of the other parameter. The transition between these two different avoided crossings indicates the existence of the exceptional point in the parameter plane. In the vicinity of the avoided crossing, a trapped mode with almost vanishing imaginary part is always found. This trapped mode couples with the incoming propagating mode, resulting in the Fano resonance which give raises to an asymmetric transmission line-shape. A transmission zero is observed in the vicinity of this asymmetric line-shape.

Introduction

A duct-cavity system is often used to study the scattering of sound due to the expansion chamber in an acoustic waveguide, especially with the presence of flow [1]. Typical applications are internal combustion engine exhaust silencers and silencers in industrial duct systems. Similar resonant-cavity configurations have also been used in other various wave propagation systems and led to important applications, see for example Refs. [2,3,4,5,6] .

The purpose of this chapter is to investigate the mechanism associated with trapped modes that cause transmission dips in acoustic waveguide when avoided crossings are formed between two modes, by using the simple duct-cavity system. S. Hein et al. [7] 25 used FEM to numerically compute the complex modes of the similar system. However, to our knowledge, there is no theory to describe the coupling of the modes of the closed system via the transverse modes of the attached ducts in acoustics. Motivated by this situation, in our work, a matrix H eff [8,9] is derived to explicitly describe the coupling effects.

The existence of trapped mode problem in waveguides has been investigated by many authors in various physical contexts theoretically and experimentally. Trapped modes were introduced more than fifty years ago (see for example, Ursell [10] and Jones [11]), and since then have induced an important amount of works in acoustics, electromagnetism, elasticity, quantum mechanics and water waves, for review articles see, for example, Refs. 12, 13, and the references therein. In this work, we will focus on the discrete trapped modes existing above the first cut-off frequency for specific parameter combinations, which are called embedded trapped modes [13] or bound states in continuum (BICs) in quantum mechanics [14,15]. By Feshbach's theory of resonance, Friedrich and Wintgen [16] demonstrated that BICs in atomic physics can occur, due to the interference of resonances (or modes in acoustic vocabulary) belonging to different channels. By varying the separation of the two resonances as a function of a continuous parameter, avoided crossings of resonances are observed. At a particular value of the parameter, one resonance has exactly vanished width (the imaginary part of the eigenvalue is zero), and becomes a BIC. In quantum systems, Almas F. Sadreev et al. [2] showed that the BIC also appears in open quantum billiards by varying their shape continuously. They also found that the BICs are close to the points of degeneracy [17] of the closed quantum system. When the system is opened, the coupling of the two resonances is turned on, an avoided crossing occurs with the variation of the parameters [18,19]. The avoided crossing is associated with the Exceptional P oints(EPs) [18,19] in the complex plane. At the EPs, the two resonances will coalesce, not only the eigenvalues but also the eigenfunctions.

Trapped (quasi-trapped) modes or bound (quasi-bound) states are crucial ingredient of Fano resonance scatterings in waveguides with impurities [20] or quantum dots with an attractive potential [21,22]. Fano resonance was firstly suggested by Ugo Fano [23] for the description of autoionizing atomic states. In contrast to the conventional isolated resonances, e.g., resonance in a harmonic oscillator with periodic forcing, whose spectral dependence can be described by the Lorentzian or Breit-Wigner formula -a symmetric spectral line; Fano resonance was explained as a phenomenon of constructive and destructive interferences between a bound state and the continuum [23]. The corresponding spectral lines are asymmetric. Since its discovery, Fano resonance has been attracting much attention in different research fields, for review articles see, for example, [24,25] and the references therein.

Fano resonances scatterings in acoustics and their relations with trapped modes in waveguides including obstacles or in duct-cavity systems have been shown by Hein, Koch and Nannen [7,26]. They use finite-element method to numerically compute the acoustic modes as well as transmission and reflection for an incoming fundamental duct mode. For the resonance problem, they obtained complex resonance frequencies and corresponding eigenfunctions, i.e., the homogeneous solutions of the Helmholtz equation with a perfectly matched layer as absorbing boundary conditions. The complex resonance frequencies are the positions of the poles of scattering matrix of the corresponding scattering problem.

The Reaction matrix (R-matrix) method [27] is used to study the wave propagation in the duct-cavity system. The R-matrix formulation was developed by Wigner and Eisenbud [27] in scattering processes in nuclear physics in the late 1940s. Recently, it was extended by Racec et al. [28] to investigate the scattering phenomena in cylindrical nanowire heterostructures. The basic idea of this method is similar to the Multimodal method [29,30]. For Multimodal method, the wave function is expanded in terms of a convenient transverse complete basis. The R-matrix method decomposes the whole system into a scattering region, and an asymptotic scattering region (here, two semiinfinite rigid ducts). The wave function in the scattering region is expanded in terms of any convenient complete set of modes of a closed cavity with convenient boundary conditions. It is noted that the basis includes the transverse and axial components. Using the continuity conditions of the pressure and the normal particle velocity at the interfaces between the regions, we can have the scattering matrix. Once the scattering matrix is obtained, the transmission and reflection coefficients can be obtained easily. By the scattering matrix, the matrix H eff of the open system is derived. Its eigenvalues are complex and give the poles of the S matrix. This chapter is organized as follows. In Chap.2.2, the formulation of the R-matrix method is described, the scattering matrix S in terms of and matrix H eff are also derived. The relation between the eigenvalues of the matrix H eff and the poles of the matrix S is also discussed. The conception of exceptional point is introduced by studying the motions of the eigenvalues of H eff with varying the cavity length. In Chap.2.3, we present that trapped modes are observed due to the coupling of two neighboured modes. The interaction of the trapped mode with the incoming propagating mode, results in the Fano resonance. We give the conclusion in the last section.

Formulation and method of the problem

We consider the acoustic scattering problem in a two-dimensional infinite duct-cavity system, see Fig. 2.1. The sound pressure ppx, yq in the waveguide satisfies the nondimensional governing equation

ˆB2 Bx 2 `B2 By 2 `K2 ˙ppx, yq " 0, (2.1)
with K " pω ˚{c 0 qh ˚the dimensionless incident frequency, where ω ˚is the circular frequency, time dependence expp´jω ˚t˚q is applied. Here the asterisk denotes dimensional quantities. All quantities are made dimensionless in the following: lengths with the uniform duct height h ˚(h " 1 in the following), velocities with sound speed c 0 , densities with the ambient density ρ 0 , and pressures with ρ 0 c 0 2 .

To solve the sound scattering problem, the whole waveguide is split into three regions: the two semi-infinite uniform ducts x ă 0 and x ą a (Region I and III, respectively), and the corresponding scattering region 0 ă x ă a (Region II). The sound pressure in the whole waveguide system is then piecewise solved. Out of the scattering region, the sound pressure can be expressed as a sum over the transverse modes of uniform ducts with rigid boundary conditions. In the numerical example, only the plane mode will be incident, but the analysis presented here is more general by assuming that one single duct mode m is incident from the left side of the waveguide. For x ă 0, the sound pressure is written as a sum of the incident and reflected modes, and only transmitted modes for x ą a:

p m px, yq " $ ' ' ' ' ' & ' ' ' ' ' % e jKmx φ m pyq `M´1 ÿ m 1 "0 R m 1 ,m e ´jK m 1 x φ m 1 pyq, x ď 0, M ´1 ÿ m 1 "0 T m 1 ,m e jK m 1 px´aq φ m 1 pyq, x ě a, (2.2)
where

K 2 m " K 2 ´α2 m , (2.3)
with ℜetK m u ą 0, ℑmtK m u ą 0, mpm 1 q " 0, 1, 2, ¨¨¨, M ´1. M is the truncation number of the transverse modes in the uniform ducts, pα m , φ m q are the eigenvalues and the eigenfunctions of the transverse modes in the uniform duct with rigid boundary conditions, with α m " mπ, and φ m pyq satisfying the orthogonality relation

ż 1 0 φ m pyqφ n pyqdy " δ mn . (2.4)
In the expression of the acoustic pressure, Eq. (2.2), R and T are matrices linking incoming and outgoing wave components. They are also used to define the scattering matrix S, as given in Eq. (2.5),

S " ˜R T 1 T R 1 ¸. (2.5)
Matrices R and T (respectively, R 1 and T 1 ) correspond to wave incident from the left (respectively, from the right). In this work, due to the symmetry of the geometry R " R 1 , and due to the reciprocity T " T 1 .

R-matrix method

According to the R-matrix method [28], the sound pressure ppx, yq in the scattering region (Region II) is projected on the complete basis of functions ψ µν px, yq p m px, yq "

Nx´1 ÿ µ"0 Ny´1 ÿ ν"0 a m,µν ψ µν px, yq, 0 ă x ă a, (2.6)
where the sums have been truncated by N x and N y . We choose the ψ µ,ν to be the eigenfunctions of closed cavity with length a and width d (d " h `b) defined by

∇ 2 ψ µν " ´γ2 µν ψ µν , (2.7)
Bψµν Bx ˇˇx "0,a " 0, Bψµν By ˇˇy "´b,h " 0.

(2.8)

The eigenvalues γ µν and eigenfunctions ψ µν px, yq are classified by two integers pµ, νq, where µ p0, 1, 2, ¨¨¨, N x ´1q labels the longitudinal mode, and ν p0, 1, 2, ¨¨¨, N y ´1q the transverse mode.

It is shown in Appendix 2.A.1 that a relation between the coefficients, a m,µν , and the first derivative of the pressure with respect to x (Bp m {Bx) at the interfaces can be written in the form of

a m K N " ż h 0 Bp m Bx ψ ˇˇˇx "0 x"a dy, (2.9)
where a m " pa m,1 , a m,2 , a m,3 , ¨¨¨a m,n , ¨¨¨q, ψ " pψ 1 , ψ 2 , ψ 3 , ¨¨¨, ψ n , ¨¨¨q, and K N is a N ˆN diagonal matrix with elements K n " K 2 ´γ2 n . Here and in the following, n " 1, 2, 3, ¨¨¨correspond to pµ, νq " p0, 0q, p0, 1q, p0, 2q, ¨¨¨, p1, 0q, p1, 1q, p1, 2q, ¨¨¨, and N " N x ¨Ny is the number of the eigenfunctions ψ µν taken into consideration.

To construct the scattering matrix S, the continuity conditions of the pressure and the normal velocity at the discontinuity interfaces (x " 0 and a) are used. By Eqs. (2.9) Chapter 2. Fano resonance scattering in a duct-cavity system and (2.2), the scattering matrix S is given (see Appendix 2.A.2)

S T " ! I ´2 rI `jK M Qs ´1) , (2.10)
where

Q " Q 1 K ´1 N Q 1 T , (2.11)
Q 1 is the coupling matrix between the modes of the closed cavity and the transverse modes of the duct induced at the interfaces. Elements of Q 1 are shown in Appendix 2.A.2 Eq. (2.23). "T " indicates the transposition. K M is a 2M ˆ2M diagonal matrix with elements K m , in which the upper M rows correspond to the situation when the mode is incident from the left side, the lower ones correspond to the opposite situation. Matrix K M characterizes the uniform ducts.

Matrix H eff

After attaching to the uniform ducts (Regions I and II), the modes of the closed cavity turn over in complex resonant modes. A matrix H eff [8] is derived here to indirectly compute the eigenvalues K R of the open system. From Eq. (2.27) in Appendix 2.A.3, the scattering matrix S is rewritten as

S " ´I `2jQ 1 1 K 2 I ´Heff Q T 1 K M , (2.12)
where

H eff " H cc ´jQ 1 T K M Q 1 (2.13)
describes the closed cavity which is partially opened at x " 0 and a by attaching to the uniform ducts. Using the continuity conditions at interfaces x " 0 and a. H cc denotes the closed cavity, it is a diagonal matrix with elements γ 2 n . The second term of H eff is due to the mode coupling introduced at the interfaces, and K M is K dependent.

From Eq. (2.13), the matrix H eff is complex, so do its eigenvalues (denoted by K 2 λ ) and eigenvectors Ψ, which can be obtained by solving the eigenvalue problem

H eff Ψ " K 2 λ Ψ. (2.14)
Due to the dependence of K M on K in Eq. (2.13), matrix H eff is also frequency dependent through the parameter, it also dependents on the geometry size of the configuration. The coupling of the modes can be analyzed by using matrix H eff .

For the duct-cavity system, the motions of the 3rd and the 4th K λ as a function of cavity length a with b " 1 are investigated, as shown in Fig. ). If we vary the two parameters a and K more precisely, at some critical values, a cri and K cri , the two eigenvalues and eigenvectors of H eff will coalesce. These critical values are called exceptional pointpsq (EP(s)) [18,19] in the parameter plane. In our example, a cri " 4.74 and K cri " 1.51 are found. If K ă K cri , crossing for the real (Fig. 2.2(a)) and avoided crossing for the imaginary (Fig. 2.2(b)) parts are observed; Inversely, avoided crossing (Fig. 2.2(d)) for the real and crossing (Fig. 2.2(e)) for the imaginary parts if K ą K cri . We can conclude that there always exists avoided crossing in the vicinity of the EP (see Figs. 2.2(c) and (f)). Re-examine the expression of H eff , Eq. (2.13), it is three parameters dependent, i.e. K, a, and b. If only two of them are used to control H eff , by fixing the third one, the EPs are isolated points in the plane of the two parameters, such as the example shown in Fig. 2.2. However, in the three-parameter space, the EPs will form a curve.

Based on the Q matrix representation of the S matrix, Eq. (2.10), the poles of the S matrix are found as the solutions of the equation detrI `jK M Qs " 0.

(2.15)

Chapter 2. Fano resonance scattering in a duct-cavity system

If the above equation is satisfied, the singularities of the scattering matrix S appear, which are also associated with the eigenvalues K R of the open system. Instead of solving the eigenvalue problem of the open system directly, we return to locate the positions of the poles of matrix S in the complex plane.

However, the representation of the S matrix in terms of the matrix H eff , Eq. (2.12), has the advantage that it yields the positions of the poles, as well as K R , by solving the eigenvalue problem of matrix H eff , with satisfying the fixed-point equation [8]

K R " K λ pK " RepK R qq, (2.16)
which can be solved numerically by using an iterative procedure. Due to the relation between K R and K λ , the EPs of modes K R in the two-parameter plane is one isolate point from the EPs curve of K λ in the three-parameter space.

In the following, the matrix H eff will be used to study the interference between the modes and to predict the parameter combinations of the trapped mode by making K approach to the trapped mode frequency.

Results and discussions

Degeneracy and avoided crossing

First, we would like to briefly review the eigenvalue problem of the corresponding closed cavity with vanishing normal velocity conditions toward the interfaces. The eigenvalues of the closed cavity with rigid boundary conditions can be obtained analytically,

γ µν " c ´µπ a ¯2 `´νπ d ¯2.
(2.17)

For a closed and conservative cavity, the eigenvalues γ µν show degeneracy [17] when the length a is varied, i.e. two modes have the same eigenvalues, meanwhile the eigenfunctions are still orthogonal. We assume that modes pp, qq and pm, nq have the same eigenvalues, which means that ´pπ a

¯2 `´qπ d ¯2 " ´mπ a ¯2 `´nπ d ¯2 .
We can have the condition on a{d, a d "

d p 2 ´m2 n 2 ´q2 , (2.18)
at which the two modes degenerate. We take mode pm, nq " p0, 1q as an example, which results in a d " p a 1 ´q2 .

a{d being real results in that the condition for modes pp, qq exist is q " 0, which demonstrates that there exists degeneracy between modes p0, 1q and pp, 0q when a{d " p. We will have degeneracies between the following modes (see Fig.

b + h = 2 ( 0,2) ( 3,2) ( 0,0) ( 0,3) ( 0,1) ( 1,0) ( 2,2) ( 3,1) ( 2,0) ( 3,0) ( 2,1) ( 1,1) ( 4,1) ( 1,3) ( 4,3) ( 3,3) ( 2,3) ( 4,0) C A B ( 1,2) ( 4,2)
Figure 2.3: Motions of the eigenvalues of the closed rigid cavity as a function of cavity length ap" a ˚{h ˚q, the width of the cavity is d " b `h " 2. Degeneracies are found for two modes having the same eigenvalues, for example in the vicinity of crossing "A" , "B", and "C" labeled in the figure.

However, once the closed system is opened by attaching to the ducts, the eigenvalues of the open system are generally complex. The trajectories of the eigenvalues of the two modes that have crossing in Fig. 2.3 will have avoided crossing in the complex plane. For the open system, we first calculated the K λ of H eff as a function of cavity length a with b " 1 and K " 1.6, see Fig. 2.4(a). For the sake of comparison, Fig. 2.4(b) shows the motions of the real part of eigenvalues K R of the open system by solving the fixed-point equation Eq. (2.16). Here and in the following, the indices pµ, νq for the closed cavity are also used to identify the eigenvalues K λ and K R . In Figs. 2.4(a) and (b), solid lines label the modes which are symmetric in x, and symbols " ◭ " for the modes which are antisymmetric. Avoided crossings between the two modes who have the same symmetry in x are observed in both figures, for example between modes (0,1) and (2,0) (labelled by "A"), modes (1,1) and (3,0) (labelled by "B"), etc.. We have reviewed before that for the corresponding closed cavity, those modes will just cross without any coupling ("A", "B" , and "C" in Fig. 2.3). Only the same parity in x-direction has to be satisfied, it is

1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 (0,1) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1) (4,0) (4,1) (5,0) (5,1) (6,0) (a) a ℜe(K λ ) A B C 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 (0,1) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1) (4,0) (4,1) (5,0) (5,1) (6,0) (b) a Re(K R ) A B C Figure 2.4: (a)Real part of eigenvalues of H eff , Re(K λ ), as a function of a with b " 1, K " 1.6. (b) Motions of Re(K R
) of the open system with varying a by solving the fixed-point equation Eq. (2.16). Solid lines mark x-symmetric modes; symbols " ◭ " mark x-antisymmetric modes.

because the system we considered is symmetric for x " a{2. By the comparison between Fig. 2.4(a) and (b), we can see that, with suitable value of K, the eigenvalues K λ of H eff can be used to analyze the coupling effects between two neighboured modes of the open system, for example between mode (0,1) and mode (2,0) by taking K " 1.6. One of the advantages to use matrix H eff , instead of solving the fixed-point equation, is that the eigenvalues K λ can be obtained easily by solving the eigenvalue problem of the matrix.

To study the interference between two modes, we take avoided crossing "A" in Fig. 2.4(a) as an example, the motions of modes (0,1) and (2,0) as a function of the geometry size are investigated. The results are summarized in Fig. 2.5. By varying cavity length a, crossing and avoided crossing of the real and imaginary parts of K R for the two modes can be traced under different conditions, i.e. for different values of b. When b " 0.95, there exists crossing for the real and avoided crossing for the imaginary parts, see Figs. 2.5(a)(b). Inversely, avoided crossing for the real and crossing for the imaginary parts are observed in Figs. 2.5(d)(e) when b " 1.05. The figures also demonstrate that the degeneracy discussed before for the closed cavity now shows avoided crossing for the open system, see Figs. 2.5(c)(f). The similar duct-cavity system has also been investigated by Hein and Koch et al. [7], but only the second type of avoided crossing is found (see Fig. 4 [7]). The occurrences of the two types of avoided crossings indicate the existence of EP in the parameter plane pa, bq, i.e. a cri and b cri , at which the two modes will coalesce, not only the eigenvalues but also the eigenfunctions. It is exactly due to the shift of the value b from b cri , there always exists avoided crossing in the vicinity of EP with varying the other parameter a, as shown in Figs. 2.5(c)(f).

a I m ( KR) 1.2 1.4 1.6 1.8 2 2.2 -0.3 -0.2 -0.1 0 (0,1) (2,0) (c) R e (K R ) I m ( KR) 3.5 4 4.5 5 5.5 1.2 1.4 1.6 1.8 2 (0,1) (2,0) (d) a R e ( KR) 3.5 4 4.5 5 5.5 -0.3 -0.2 -0.1 0 (0,1) (2,0) (e) a I m ( KR) 1.2 1.4 1.6 1.8 2 2.2 -0.3 -0.2 -0.1 0 (0,1) (2,0) (f) R e (K R ) I m ( KR)

Trapped mode and Fano resonance

By looking at the trajectories of the eigenvalues of the two modes in the complex plane, we can see that before reaching the EP, the two modes move in a opposite direction and approach to each other; in the vicinity of the EP, a strong repulsion between the two modes is observed, which makes them avoid crossing and move away from each other. However, the important and interesting point is that before the repulsion, the imaginary part of one mode is almost zero (see Figs. 2.5(b)(e)), it means that this mode will not have radiation loss, which is the so-called embedded trapped mode [15] (we will just call it "trapped mode").

In Fig. 2.6(a), we show the trajectories of K R of mode (0,1) as a function of a P r3.73, 4.22s with b " 1 (circle with solid line). A trapped mode is observed at a " 3.93 with b " 1, marked by the "b" in Fig. 2.6(a). For comparison, the motions of K λ as a function of a with b " 1, K " 1.597 are also shown in Fig. 2.6(a) (square with dashed line). From the two curves, we can see that the matrix H eff can be well used to predict the appearances of the trapped modes. To qualitatively characterize the effects of the trapped mode on the sound propagation, the transmission coefficient for the plane wave as a function of K is computed, with a " 3.93 and b " 1. The results are plotted in Fig. 2.6(b). A transmission peak and dip is observed around K " 1.597, it first reaches to "1" and then decreases to "0" sharply (shown clearly in the inset of Fig. the absolute value of the imaginary part of the trapped mode is, sharper and faster the coefficient changes from "1" to "0". This asymmetric line profile is one of the main features of the so-called Fano resonance [23], known from atomic physics. This phenomenon is due to the interaction of the trapped mode with the incoming propagating mode.

From the field distribution in the waveguide, Fig. 2.6(c), we see that the pressure field in the cavity is much higher than in the ducts, the trapped mode is confined in the scattering region, a similar pressure profile for the trapped mode is also given in Fig. 7(a) by Hein and Koch et al. [7]. The eigenfunction of the trapped mode is a strong mixing of the eigenfunctions of mode (0,1) and mode (2,0) of the closed cavity with the superposition coefficients 0.865 and 0.498, respectively. The superposition coefficients of several eigenfunctions ψ µ,ν , 0 ď µ ď 9, 0 ď ν ď 3, for the trapped mode are shown in Fig. 2.6(d). The contributions of the other modes are much smaller than of the modes (0,1) and (2,0). In Fig. 2.8, we plot the transmission coefficient as a function of both frequency K and cavity length a, with b " 2. The eigenvalues of the resonance K R of the open system are also given in the same figure, labelled by "˝". It is remarkable that near the avoided crossing, one can observe the abrupt changes of the transmission coefficients. They appear at the points where the imaginary parts of the modes are zero, see Fig. 2.7(d).

Multimodes interference

Figure 2.9 shows the transmission coefficients of the plane mode as a function of frequency K with different values of cavity length a, in order to investigate its effects on the transmission spectrum. In Figs. 2.9(a) and (c), two kinds of Fano antisymmetric lineshape in the transmission spectrum are observed. In Fig. 2.9(a), transmission coefficient |T | decreases to "0" first and then increases to "1" sharply. However, it changes in the opposite way in Fig. 2.9(c). We can see from Fig. 2.7(d) that trapped modes are formed around a " 5.2, and 5.6, respectively. Once the incoming propagating mode interfers with the trapped modes, the Fano resonances results. In Fig. 2.9(b), we show the transmission coefficients with the cavity length a " 5.4, where the interference between the modes is weak. We can see the sharp transmission dips disappear, while a symmetrical Breit-Wigner (BW) (or Lorentzian) line shape shows up near the frequency of mode (2,2).

Conclusions

In summary, the couplings of the modes in the open duct-cavity systems are investigated by means of varying a parameter continuously. Instead of computing the complex resonance frequencies of the systems directly, an effective matrix H eff is introduced to describe the coupling of the modes of the closed system via the attached ducts successfully and efficiently. Consequently, we have obtained various interesting avoided crossings for both the eigenvalues of H eff and the resonance frequencies of the open system which have the same parity in x´direction. We have shown that the avoided crossings are in the vicinity of the exceptional points.

By matrix H eff , we can predicts the parameter combinations of the trapped modes very well. Our computation on the transmission coefficients shows very interesting Fano asymmetric line-shape, due to the interference of the trapped modes and the incoming propagating mode. Strong localization of the pressure field in the scattering region is observed, it is a mixing of the eigenfunctions of the two resonances. We have also observed that the sharp dips disappear if the coupling of the modes becomes weak.

In the next chapter, the same mechanism is introduced in a waveguide lined with locally reactive liner, to produce another transmission zero at the frequency of the trapped mode, in addition to the transmission zero observed at the quarter-wavelength frequency of the liner. Appendix 2.A Derivation of Eqs.(2.9)(2.10)(2.12) (2.20)

The eigenfunctions ψ µν of the closed cavity satisfy the rigid boundary conditions on all the walls, Eq. (3.6). Substituting the rigid boundary conditions for both ppx, yq and ψ µν into Eq. (3.25), results in the following equation

ż h 0 ˆψµν Bp Bx ˙ˇˇˇx "0
x"a dy " pK 2 ´γ2 µν q ż ż ψ µν pdxdy, (2.21)

Using the orthogonality property of eigenfunctions ψ µν and Eq. (3.4), the upper equation can be written in a matrix form, i.e. Eq. (2.9).

2.A.2 Derivation of Eq. (2.10)

By Eq. (2.2), the first derivative of the pressure with respect to x at the interfaces, Bp m {Bx is taken, and then substituted into the right side of the Eq. (2.9). The resulting equations are written in a matrix form,

AK N " jpI ´ST qK M Q 1 , (2.22)
where A is a 2M ˆN matrix consisting of the vectors a m , the upper M rows correspond to wave incident from the left side, and the remaining from the right side. The elements

of matrix Q 1 are Q 1 " ¨ş1 0 φ 1 py 1 qψ 1 p0, y 1 q dy 1 . . . ş 1 0 φ M py 1 qψ N p0, y 1 q dy 1 ş 1 0 φ 1 py 1 qψ 1 pa, y 1 q dy 1 . . . ş 1 0 φ M py 1 qψ N pa, y 1 q dy 1 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' . ( 2

.23)

By matrix A, the pressure in the scattering region is then written in a vector form,

ppx, yq " jpI ´ST qK M Q 1 K ´1 N ψpx, yq. (2.24)
where p " pp 1 , p 2 , ¨¨¨, p m , ¨¨¨q T . By the continuity condition of the pressure at the interfaces x " 0, a, results in two equations. Multiplying the two equations by φ m , writing them in a matrix form, we have

jrI ´ST sK M Q 1 K ´1 N Q T 1 " rI `ST s. (2.25)
By Eq. (2.25), we end up with the scattering matrix of Eq. (2.10).

2.A.3 Derivation of Eq. (2.12)

Due to the symmetry property of the matrix S, Eq. (2.10) can be rewritten as

S " ´I ´jQK M I `jQK M " ´I `2jQK M I `jQK M . (2.26)
Now we expand the denominator in Eq. (2.26) into a geometric series [8],

S " ´I `2jQ 1 1 K N Q T 1 K M 8 ÿ q"0 ˆ´jQ 1 1 K N Q T 1 K M ˙q " ´I `2jQ 1 1 K N 8 ÿ q"0 ˆ´jQ T 1 K M Q 1 1 K N ˙q Q T 1 K M " ´I `2jQ 1 1 K N 1 1 `jR T 1 K M R 1 1 K N Q T 1 K M " ´I `2jQ 1 1 K N `jR T 1 K M R 1 Q T 1 K M (2.27)
By the expression of K N , the upper equation results in the Eq. (2.12).

Chapter 3

Fano resonance scatterings in waveguides with impedance boundary conditions

The firs strategy we introduced in the previous chapter is now applied here to a waveguide lined with a locally reacting material.

In this chapter 1 , the resonance scattering theory is used to study the sound propagation in a waveguide with a portion of its wall lined by a locally reacting material. The objective is to understand the effects of the mode coupling in the lined portion on the transmission. It is shown that a zero in the transmission is present when a real resonance frequency of the open system, i.e. the lined portion of the waveguide which is coupled to the two semi-infinite rigid ducts, is equal to the incident frequency. This transmission zero occurs as a Fano resonance -due to the excitation of a trapped mode in the open system. The trapped mode is formed by the interferences of two neighbored modes with complex resonance frequencies. It is also linked to the avoided crossing of eigenvalues of these two modes, which occurs near an exceptional point (a subject which has attracted much attention in recent years in different physical domains). The real and complex resonance frequencies of the open system are determined by an equivalent eigenvalue problem of matrix H eff , which describes the eigenvalue problem defined in the finite lined portion (scattering region). With the aid of the eigenvalues and eigenfunctions of matrix H eff , the usual acoustic resonance scattering formula can be extended to describe the coupling effects between the scattering region and the rigid parts of the waveguide. 48 Chapter 3. Fano resonance scatterings in waveguides with impedance boundary conditions

Introduction

Acoustic liners are commonly used in noise control devices for duct systems. Typical applications include silencers for ventilation systems, wall treatments for aircraft engines, and silencers for industrial gas turbines, for review articles see, e.g., Refs. 1 and 2. To increase the liner efficiency, various strategies can be used. One is to find a new material design for which the impedance is close to its optimal value in the targeted range of frequencies [3,4]. Another strategy is to take advantage of the acoustic impedance changes (like discontinuities) in axial [5,6,7] or circumferential [8,9] segments, or both of them [10,11,12,13]. The strategy used in this paper is different: The idea is to couple the incoming propagative mode in the waveguide with the modes localized in the lined region. As an example, a very simple 2D model with a portion of the waveguide lined by a uniform liner is analyzed in this paper.

Such interferences between resonances and scattering appear in Fano resonances (for review articles see, for example, Refs. 14 and 15, and the references therein). In contrast to the conventional isolated resonances scattering, Fano resonance is explained by constructive and destructive interferences between a resonance (trapped mode) or a complex resonance (quasi-trapped mode) and the background or nonresonant scattering [16,17]. A transmission zero is produced when a real resonance frequency is equal to the incident frequency [18].

One of the crucial ingredients to form a Fano resonance is to have trapped mode with a real resonance frequency. Trapped modes are localized oscillations in unbounded media and do not radiate energy to infinity. They were first observed experimentally in acoustics by Parker in 1966 [19]. Discrete trapped modes may exist below the first cutoff frequency of the transverse modes, provided that some kinds of defect or variations of geometry exist [20,21]. Discrete trapped modes may also exist above the first cutoff frequency for specific parameter combinations, they are called embedded trapped modes [20] or bound states in continuum (BIC) in quantum mechanics [22]. Friedrich and Wintgen [23] demonstrated that BIC is a natural feature of common physical situations, and can occur due to the interference of resonances. They have linked BICs directly to the phenomenon of an avoided crossing of neighbored resonance states (modes with complex eigenvalues). Recently, BICs in the vicinity of exceptional points were also found in open billiards [24]. Exceptional points are the points in a two-dimensional parameter space, where not only the eigenvalues but also the eigenfunctions of a non-Hermitian operator coalesce [25,26,27]. Sadreev et al. [24] found that the BICs are close to the points of degeneracy [28] of the closed system, which are the points in a two-dimensional parameter space where only the eigenvalues of the Hermitian operator coalesce, while the corresponding eigenfunctions are still orthogonal. When the system is opened, the degeneracies are lifted and avoided crossings occur.

Fano resonance scatterings in acoustics and their relations with trapped modes in waveguides including obstacles or in duct-cavity systems have been studied by Hein 3.2. Model et al. [18,29]. They used a finite-element method to compute numerically the acoustic resonances as well as the transmission and reflection for an incoming duct mode. They obtained complex resonance frequencies and the corresponding eigenfunctions. The complex resonance frequencies are the positions of the poles of scattering matrix of the corresponding scattering problem. Fano resonance scatterings were related to three types of trapped modes: antisymmetric (about duct axis) trapped modes below the first cutoff frequency, embedded trapped modes linked to avoided crossings of resonances, and trapped modes associated with Fabry-Pérot interferences between cavities.

In this paper, we study the sound propagation in a waveguide lined on a portion with a locally reacting material by the resonance scattering theory [30]. The objective is to understand the effects of the mode coupling on the transmission of the lined portion. We show that by varying a control parameter (the section length or the product of the lined admittance and the frequency), two neighbored modes with complex resonance frequencies interfere in the scattering region: the lined portion opened to the two semiinfinite rigid waveguides. In the vicinity of an exceptional point, where the eigenvalues and eigenfunctions coalesce, one mode turns to be trapped, the corresponding resonance frequency (eigenvalue) is real. A transmission zero is present when the real resonance frequency is equal to the incident frequency. This transmission zero occurs as a Fano resonance, due to the excitation of a trapped mode in the open system (section 3.3).

In section 3.2, we derive the Scattering matrix (S matrix). For that, we project the Helmholtz equation over the eigenfunctions of the rigid closed counterpart of the scattering region which form an orthogonal and complete function basis. The used eigenfunctions include the transverse and axial components, thus this method generalizes the Multimodal method [11] in which the wave function is expanded only in terms of transverse eigenfunctions. We express the scattering matrix S in terms of an effective matrix H eff . The matrix H eff describes the complex resonances of the scattering region. Its eigenvalues are complex and give the poles of the S matrix. With the help of its eigenfunctions, we can extend the traditional acoustic resonance scattering formula [30] to describe the coupling effects between the scattering region and the rigid waveguide.

Model

We consider the acoustic scattering problem in a two-dimensional infinite waveguide lined over a finite length a ˚with a locally reacting material, as shown in Fig. 3.1. The waveguide is decomposed into three parts: two semi-infinite rigid waveguides x ˚ď 0 and x ˚ě a ˚(regions I and III, respectively), and one scattering region (region II). Here, asterisk denotes dimensional quantities. In the following, all the quantities are nondimensionalized. Sound pressure p and coordinates (x and y) are normalized by taking p " p ˚{pρ 0 c 0 2 q and px, yq " px ˚{h ˚, y ˚{h ˚q, respectively, where ρ 0 is the air density, c 0 is the sound velocity, and h ˚is the height of the waveguide. Time dependence is assumed as expp´jω ˚t˚q and will be omitted in the following, where ω ˚is the circular frequency. The sound pressure ppx, yq in the waveguide satisfies the non-dimensional Helmholtz equation

ˆB2 Bx 2 `B2 By 2 `K2 ˙ppx, yq " 0, (3.1)
where K " pω ˚{c 0 qh ˚refers to dimensionless frequency. " 0 (for region II), (

respectively, where Y is the normalized admittance of the liner material.

Inspired by the R-matrix method [32,33], the sound pressure ppx, yq in region II is expanded in terms of an orthogonal and complete set of functions ψ µν px, yq ppx, yq "

Nx´1 ÿ µ"0 Ny´1 ÿ ν"0 a µν ψ µν px, yq " ψ T a, (3.4)
where the sums have been truncated by N x and N y , " T " refers to transpose, and a µ,ν are the expansion coefficients. ψ is a column vector, its elements are arranged as pµ, νq " p0, 0q, p0, 1q, ..., p0, N y ´1q, p1, 0q, p1, 1q, ..., p1, N y ´1q, ..., pN x ´1, 0q, pN x ´1, 1q, ..., pN x ´ 51 1, N y ´1q. We choose ψ µ,ν to be the eigenfunctions of rigid closed cavity defined by By solving the eigenproblem of Eqs. (3.5) and (3.6), the eigenfunctions ψ µν and eigenvalues γ µν are given as ψ µν " p1{ a Λ x Λ y q cospµπx{aq cospνπyq and γ µν " a pµπq 2 {a 2 `pνπq 2 , respectively, where Λ x " a 2 ´δ0,ν { ? a and Λ y " a 2 ´δ0,ν are normalization coefficients, with δ i,j " 1 if i " j, else δ i,j " 0.

∇ 2 ψ µν " ´γ2 µν ψ µν , (3.5)
It needs to be noted that the eigenfunctions ψ µν in Eq. (3.4) include the transverse and axial components, instead of only transverse component as used in Multimodal method [11], and that any complete set of functions can be used but the present choice defined in Eqs. ( 5) and ( 6) is one of the most convenient. In this paper, we use "modes" to refer to the 2D cavity modes, and "transverse modes" to refer to the 1D duct modes.

Multiplying Eq. (3.1) by ψ, integrating over x and y, applying Green's theorem, and using Eq. (3.5) and the boundary conditions Eq. (3.6), we obtain ppx, yq "

ż 1 0 " Qpx, y, 0, y 1 q Bppx 1 , y 1 q Bx 1 ˇˇx 1 "0 ´Qpx, y, a, y 1 q Bppx 1 , y 1 q Bx 1 ˇˇx 1 "a  dy 1 , (3.7)
(see Appendix I.1), where Qpx, y, x 1 , y 1 q " ψ T px, yq "

K 2 I ´Hlcc ‰ ´1 ψpx 1 , y 1 q. (3.8)
In Eq. (3.8), I refers to identity matrix, H lcc " Γ ´jKY C lcc , where Γ is a diagonal matrix with elements γ 2 µν , "lcc" refers to lined closed cavity, and C lcc is a block diagonal matrix, its elements can be calculated analytically by

C lcc µν,µ 1 ν 1 " ż a 0 ψ µν px 1 , 0qψ µ 1 ν 1 px 1 , 0qdx 1 . (3.9)
With the help of Eq. (3.26) in Appendix I.1, it is clear that the eigenvalue problem of matrix H lcc is equivalent to the eigenvalue problem of the lined closed cavity with the admittance on the bottom wall (defined in Appendix II for clarity).

Equation (3.7) links the sound pressure in the scattering region (II) with its first x-derivative at the interfaces between the scattering region and the other two regions (I and III).

The sound pressure is written as a sum of the incident (amplitudes c m ) and reflected conditions transverse modes for region I, and only transmitted transverse modes for region III ppx, yq "

$ ' ' ' ' ' & ' ' ' ' ' % M ´1 ÿ m"0 c m e jK x m x φ m pyq `M´1 ÿ m"0 M ´1 ÿ m 1 "0 R m,m 1 c m e ´jK x m 1 x φ m 1 pyq, x ď 0, M ´1 ÿ m"0 M ´1 ÿ m 1 "0 T m,m 1 c m e jK x m 1 px´aq φ m 1 pyq, x ě a, (3.10)
where R m,m 1 and T m,m 1 refer to the reflection and transmission coefficients, and M is the truncation number.

K x m " a K 2 ´α2
m are axial wavenumbers of transverse mode m in the rigid ducts. α m and φ m are the eigenvalues and eigenfunctions of transverse modes in regions I and III, they are given as α m " mπ and φ m " Λcospα m yq, with Λ " a 2 ´δ0,m the normalization coefficients. Equation (4.22) is written in matrical form as ppx, yq "

# φ T E 0 c `φT E 0 Rc, x ď 0, φ T E à Tc, x ě a, (3.11)
where E 0 , E 0 , and E à are (M ˆM ) diagonal matrices with the elements e jK x m x , e ´jK x m x , and e jK x m px´aq , respectively. φ is a (M ˆ1) column vector, its elements are the eigenfunctions φ m . Substituting Eq. (3.11) into Eq. (3.7) at x " 0 and x " a, we obtain

I M `R " jC 0 T pK 2 I ´Hlcc q ´1pC 0 K x pI M ´Rq ´Ca K x Tq, (3.12)
T " jC a T pK 2 I ´Hlcc q ´1pC 0 K x pI M ´Rq ´Ca K x Tq,

where K x is a (M ˆM ) diagonal matrix with elements K x m , and I M is a (M ˆM ) identity matrix. The elements of matrices C 0 and C a are

C 0paq,µν,m " ż 1 0 ψ µν px 1 " 0paq, y 1 qφ m py 1 qdy 1 , (3.14)
where C 0 and C a have dimensions N ˆM with N " N x ¨Ny . Due to the orthogonality between the term cospνπyq in ψ µν and the function φ m , elements C 0paq,µν,m will be zero if ν ‰ m. From Eqs. (3.12) and (3.13), we obtain

" R T  " rI 2M `jGK x 2M s ´1 ´rI M , 0 M s T `jC 0a T rK 2 I ´Hlcc s ´1C 0 K x ( , (3.15)
where

G " C 0a T pK 2 I´H lcc q ´1C 0a , C 0a " rC 0 , C a s is a (N ˆ2M ) matrix, I 2M is a (2M ˆ2M ) identity matrix, K x
2M is a (2M ˆ2M ) block diagonal matrix with two (M ˆM ) diagonal matrices K x on its main diagonal, and 0 M is a (M ˆM ) zero matrix.

If we assume that the problem is symmetric, the scattering matrix can be written

3.2. Model 53 as S " " R T T R  " rI 2M `jGK x 2M s ´1 r´I 2M `jGK x 2M s . (3.16)
The scattering matrix S can be expressed as [34] S " ´I2M `2jC 0a T "

K 2 I ´Heff ‰ ´1 C 0a K x 2M , (3.17)
(see Appendix I.2), under the condition that K 2 I ´Heff can be inverted, where

H eff " H lcc ´jC 0a K x 2M C 0a T . (3.18)
The eigenvalues K λ and eigenfunctions φλ " ψ T V λ are defined by the eigenproblem of matrix H eff ,

H eff V λ " K 2 λ V λ . (3.19)
They describe the complex resonances of scattering region II, which is opened to infinities through regions I and III, and truncated at the interfaces, x " 0 and x " a. The elements of vector ψ are the eigenfunctions of the rigid closed cavity defined by Eqs. (3.5) and (3.6). Because the eigenfunctions φλ in scattering region II are non-separable in x and y, we use only one index λ to describe the eigenvalues K λ and the eigenfunctions φλ .

With the help of eigenvalues K λ and eigenvectors V λ , Eq. (3.17) can be written as a resonance form

S " ´I2M `2jC 0a T VrK 2 VV ´1 ´VH eff V ´1s ´1V T C 0a K x 2M (3.20) " ´I2M `2j CT 0a rK 2 I ´Kλ s ´1 C0a K x 2M ,
where K λ is a (N ˆN ) diagonal matrix with K 2 λ its main elements. V is a (N ˆN ) matrix with its columns the eigenvectors V λ of matrix H eff . H eff is a symmetric non-Hermitian matrix, its eigenvectors are bi-orthogonal. V ´1 " V T has been used to obtain Eq. (3.20). Matrix C0a is defined as

C0a " V T C 0a " ż h 0 V T rψpx 1 " 0, y 1 qφ T py 1 q, ψpx 1 " a, y 1 qφ T py 1 qsdy 1 (3.21) " ż h 0 r φpx 1 " 0, y 1 qφ T py 1 q, φpx 1 " a, y 1 qφ T py 1 qsdy 1 ,
where φpx 1 " 0, y 1 q and φpx 1 " a, y 1 q are pN ˆ1q vectors with elements the eigenfunctions φλ of scattering region II at x 1 " 0 and x 1 " a, respectively. Matrix C0a describes the couplings of the scattering region II with regions I and III. Equation (3.20) is not valid at exceptional points at which the eigenvalues and eigenfunctions coalesce, and therefore V ´1 is singular. Equation (20) shows that the poles K pole of the scattering matrix S are the eigen-conditions

values of H eff K pole " K λ (3.22)
when ℜepK λ q " K. It is well known [30,31] that the poles of scattering matrix are precisely the complex resonance frequencies of the scatter. Therefore, using Eq. ( 20), we show that the original complex resonance problem of the open system defined in Fig. 3) with the radiation conditions ppx, yq9e ˘jK x m x at x Ñ ˘8 is reduced to an eigenvalue problem of matrix H eff , which describes the eigenvalue problem defined in the scattering region (region II). It is noted that the complex resonances are calculated in Refs. 18 and 29 by finite element method with absorbing boundary conditions. Equation (3.20) extends the traditional acoustic resonance scattering formula [30,31], in which the complex eigenfrequencies of scattering region give the resonance poles, to include the coupling effects between scattering region and rigid waveguides.

Results and discussions

First, we show how the trapped modes with real resonance frequencies occur in the vicinity of exceptional points and how they are linked to avoided crossings of the eigenvalues of matrix H eff . Then, we show that a transmission zero is present when the real resonance frequency is equal to the incident frequency. Finally, we consider the effects of the dissipation in acoustic absorbing material. The liner can be described by an impedance model (Y " 1{Z):

Z " Re `j cotpKd l q, (3.23)
where d l is the normalized depth of the liner, and Re is the resistance. Z is assumed to be uniform. All the numerical results are computed with the truncation numbers M " 30, N x " 30, and N y " 30. Although Eqs. (3.17) and (3.20) are valid for multimode propagation in the waveguide, for the sake of simplicity, we assume that only the plane wave is propagative in regions I and III in the following. It is noted that there is multimode propagating in region II.

We first consider the mode behaviours in the lined closed cavity as defined in Appendix II. It can be described by the eigenvalues (denoted β 2 µ,ν ) of matrix H lcc . Matrix H lcc depends on parameters a and KY when Re " 0, so do β µ,ν . The trajectories of ℜepβ µ,ν q as a function of |KY | are shown in Fig. 3.2 with a " 4 and Re " 0. Due to the fact that there is no dissipation in the liner (Re " 0), β µ,ν are either purely real (plot in Fig. 3.2) or purely imaginary. When the eigenvalue curves cross, the modes are degenerate [28]. At the degeneracies, the eigenvalues coalesce, while the eigenfunctions are still bi-orthogonal [28] and there is no interaction between the modes.

We then consider the mode behaviours when the region II is opened. It can be

|K Y | ℜe(β µ ,ν ) (0,0) (1,0) (5,0) (6,0) (3,0) (2,0) (4,0) (7,0) (8,0) (9,0)(10,0) (11,0) 1 2 A 3 4 (0,1) (1,1) (3,1)
(2,1) modelled by the matrix H eff which depends on parameters: a, KY , and K when Re " 0.

In Fig. 3.3, we plot the eigenvalue trajectories as a function of |KY | with a " 4, K " 2.5, and Re " 0. The curves with the same symmetry about x, which have crossings in Fig. 3.2, now have either crossings for the real and avoided crossings for the imaginary parts, e.g., A, 3, and 4 in Fig. 3.3, or avoided crossings for the real and crossings for the imaginary parts, e.g., 1 and 2 in Fig. 3.3. Due to the energy leakage in the rigid duct, the K λ are now complex. The crossings and avoided crossings of the eigenvalues K λ in Fig. 3.3, e.g., A, 1, 2, 3, and 4, occur in the vicinity of the degeneracies of the lined closed cavity, as labelled by A, 1, 2, 3, and 4, respectively in Fig. 3.2. Avoided crossings, already known in structural dynamics [35,36,37], are less used in cavities lined with impedance.

The above crossing and avoided crossing behaviours may be changed when we change the liner length a. Taking the two modes in the vicinity of "A" in Figs. 3.2 and 3.3 as an example, we decrease the liner length a, the results are shown in Fig. 3.4. When a " 3.8, there is a crossing for ℜe(K λ ) and an avoided crossing for ℑm(K λ ) (see Fig. 3.4 (a) and (b)), while for a " 3.6, there is an avoided crossing for ℜe(K λ ) and a crossing for ℑm(K λ ) (see Fig. 3.4 (d) and(e)). The trajectories of K λ in the complex plane are also shown in Fig. 4 (c) and (f) with a " 3.8 and 3.6, respectively, where avoided crossings are always observed.

These findings and the type change of the avoided crossings from Fig. 3.4(c) to Fig. 3.4(f) show that there exists a critical value a cri {h, for which the two curves of K λ in the complex plane as a function of |KY | will cross at a critical value |KY | cri . These critical values for the two eigenvalues in Fig. 3.4 are |KY | cri " 4.18 and a cri " 3.7 when K " 2.5. (|KY | cri , a cri ) is called an Exceptional Point (EP) in the parameter plane (|KY |, a) when K is fixed. In this case not only the eigenvalues but also the eigenfunctions of the two modes coalesce. A strong mixing of the eigenfunctions of the two modes occurs near the EP. The two modes which participate in the avoided crossing exchange their identities [35,36].

Avoided crossing occurs in the vicinity of an EP. EPs were first introduced by Kato [38], and were extensively developed by Heiss [25,26,27], Rotter [39,40,41], and Berry [42]. The mathematically topological structures of Riemann sheets at an EP, depend on a complex parameter or on two real parameters. A typical EP distribution of H eff in the plane of |KY | and a, with K fixed, is shown in Fig. 3.5. They occur It is remarkable that at |KY | " 4.5 in Fig. 3.4 (b) and |KY | " 4.8 in Fig. 3.4 (e), ℑm(K λ ) goes to zero. They are real eigenvalues of H eff in the scattering region II. It is the same kind of mode that Friedrich et al [23] found in nuclear reaction, and Sadreev et al. [24] found in quantum billiards. The corresponding trapped modes play a crucial role in the transmission zeros in the lined waveguide.

ℜe ( K λ ) ℑm ( K λ ) (f) (d) (a) (b) (c) (e) 0
To realize a practical design of transmission zero, we need to understand the mode behaviours in the parameters space (a, d l , K). We re-produce Fig. 3.4 in Fig. 3.6 using d l as a varying parameter. It is quite surprising that the eigenvalue trajectories in the two different parameter spaces of Figs. 3.4 and 3.6 are very similar to each other. We find that at (a " 3.8, d l " 0.425) and (a " 3.6, d l " 0.436), real eigenvalues (trapped modes) occur, with K " 2.5. Having the above information, we can obtain the real resonance frequencies of the open system defined in Fig. 3.1 and by Eqs. (3.1)-(3.3) with the radiation conditions ppx, yq9e ˘jK x m x at x Ñ ˘8 by computing the real poles of the scattering matrix given in Eq.(3.17). The real resonance frequency we seek for corresponds to the point where the curves y " K λ pK, d l , a " 3.8q cross the line y " K. We take the EP "C1" in Fig. 5 as an example. We find that at d l " 0.35 and a " 3.8 with K " 2.85, a real resonance (trapped mode) occurs. This can be verified by Fig. 3.7, in which we plot 3.3. Results and discussions 59 the motions of the eigenvalues K λ as a function of d l with K " 2.85, a " 3.8, and Re " 0. It is clear to show that at d l " 0.35 the mode labelled by solid line in Fig. 3.7(b) has ℑmpK λ q " 0. Using this group of parameters (a " 3.8, d l " 0.35, Re " 0), we compute the transmission and reflection coefficients of the plane mode by Eq. (3.16), and we plot it in Fig. 3.8(a). A transmission zero occurs near K " 2.853, where the reflection coefficient has a corresponding resonance peak. It needs to stress that in the vicinity of this resonance peak, a reflection zero is also present. They clearly show the Fano resonance structure. The corresponding sound pressure field at the frequency of the transmission zero is shown in Fig. 3.8(b). The high sensitivity on the dissipation is due to the fact that the crucial ingredient to form Fano resonance in the transmission and reflection coefficients -the trapped mode with real resonance frequency, turns to be quasi-trapped mode with complex resonance frequency. The sharp asymmetric profile becomes smoother, as shown in Fig. 3.9.

ℜe ( K λ ) ℑm ( K λ ) 0

Conclusions

We have shown that the acoustic scattering matrix in a waveguide with a portion of its wall lined by a locally reacting material can be efficiently described with the help of an effective matrix H eff , whose eigenvalues give the poles of the scattering matrix. The real poles are in the vicinity of the exceptional points of matrix H eff . Using this effective matrix, the original acoustic resonances problem of the lined portion in the infinite waveguide is reduced to an equivalent eigenvalue problem of matrix H eff , which describes the eigenvalue problem defined in the finite lined portion (scattering region).

There are an infinite number of exceptional points of H eff in the parameter plane (|KY |, a) with K fixed, at which the eigenvalues and eigenfunctions of two modes coalesce. In the vicinity of each exceptional point, crossing or avoided crossing of the real and imaginary parts of the complex resonances (eigenvalues) occur. By varying one of the parameters, say |KY |, one mode turns to be a trapped mode, its resonance frequency becomes real. When a plane mode is incident, a transmission zero is present when the real resonance frequency is equal to the incident frequency. This transmission zero occurs as a Fano resonance, due to the excitation of a trapped mode in the open system. EPs, real resonance frequencies, and transmission zeros can also be obtained for parameters (a, d l , K), which can suit the practical need of noise mitigation. We have also shown that the transmission zeros and reflection peaks are highly sensitive to the dissipation.

With the aid of the eigenvalues and eigenfunctions of matrix H eff , the traditional acoustic resonance scattering formula is extended to include the coupling effects between the open lined portion and the rigid parts of the waveguide.

In this paper, the numerical calculations only show the situation where plane mode is propagating in the rigid ducts, and thus "zero transmission" means that the total sound field is stopped. However, the model is valid for multimode propagation in the waveguides. The possibility of stopping the total sound field when several modes propagate is still an open question.

Appendix I: Derivation of Eq. (3.7) and (3.17 where rF s x"a x"0 " F paq ´F p0q works for any function F , I is an identity matrix, and Γ is a diagonal matrix with elements γ 2 µν . Substituting the boundary conditions, Eqs. (3.3) and (3.6), into Eq. (3.25), results in

ż 1 0 " ψ Bp Bx  x"0 x"a dy ´jKY ż a 0 ψpx, 0qppx, 0qdx " pK 2 I ´Γq ż 1 0 ż a 0 ψp dxdy. (3.26)
Replacing the pressure function p inside the scattering region by Eq. (3.4), and using the orthogonality property of eigenfunctions ψ µν , the expansion coefficients a can be written as

a " ˆK2 I ´Γ `jKY ż a 0 ψpx, 0qψ T px, 0qdx ˙´1 ż 1 0 " ψ Bp Bx  x"0
x"a dy.

(3.27)

Substitute the upper expression of a into Eq. (3.4), we end up with Eq. (3.7).

Chapter 3. Fano resonance scatterings in waveguides with impedance boundary conditions

Derivation of Eq. (3.17)

Due to the symmetry property of the matrix S, Eq. (3.16) can be rewritten as

S " ´I2M `2jC 0a T pK 2 I ´Hlcc q ´1C 0a K x 2M I 2M `jC 0a T pK 2 I ´Hlcc q ´1C 0a K x 2M . (3.28)
Now we expand the denominator in Eq. (3.28) into a geometric series [34],

S " ´I2M `2jC T 0a 1 K 2 I ´Hlcc C 0a K x 2M 8 ÿ q"0 ˆ´jC T 0a 1 K 2 I ´Hlcc C 0a K x 2M ˙q " ´I2M `2jC T 0a 1 K 2 I ´Hlcc 8 ÿ q"0 ˆ´jC 0a K x 2M C T 0a 1 K 2 I ´Hlcc ˙q C 0a K x 2M " ´I2M `2jC T 0a 1 K 2 I ´Hlcc 1 1 `jC 0a K x 2M C T 0a 1 K 2 I´H lcc C 0a K x 2M " ´I2M `2jC T 0a 1 K 2 I ´Hlcc `jC 0a K x 2M C T 0a C 0a K x 2M .
(3.29)

By the expression of K N , the upper equation results in the Eq. (3.17).

Appendix II

The eigen-problem of the closed cavity with the admittance on the bottom wall, called lined closed cavity can be expressed by

∇ 2 ϕ µν " ´β2 µν ϕ µν , (3.30

Use of metaporous materials in acoustic ducts

The second strategy is presented and validated in this chapter. A set of periodic rigid inclusions are embedded in a porous lining to enhance the sound attenuation in an acoustic duct at low frequencies. Floquet-Bloch theorem is introduced to investigate the sound attenuation in a 2D infinite waveguide lined with periodic inclusions embedded in porous material. Crossing is observed between the mode attenuations of two Bloch waves. Here the mode coupling is due to the presence of the inclusions embedded in the porous material. The most important and interesting figure is that near the frequency where the crossing of the mode attenuations appears, an attenuation peak is observed. This phenomenon can be used to explain the transmission loss peak observed numerically and experimentally in a 3D waveguide with a finite portion of its wall lined by a porous material embedded with periodic inclusions.

Introduction

The acoustic treatment we discussed in the previous chapter is generally made with a perforated sheet backed by honeycomb [1]. This kind of material has good absorption properties only in a narrow frequency band but their main advantages are their mechanical robustness and their capability to resist to harsh conditions i.e. they constitute the reference solution in a turbofan engine. For some other applications porous material are often used to reduce the noise emission, e.g., in the ventilation systems and the Auxiliary Power Units (APU) of aircrafts. They generally offer a wider absorption/attenuation band. However, they suffer from a lack of absorption efficiency at low frequencies, comparing to their efficiency at higher frequencies. To solve that problem, other approaches must be investigated and new concepts of acoustic absorbers dedicated to the reduction 68 Chapter 4. Use of metaporous materials in acoustic ducts of turbo-machinery noise at low frequencies must be developed.

To improve the liner capability at the low frequency range, in the applications of acoustic gratings and panels, the usual way is by multi-layering [2,3,4]. Another alternative solution, material including periodic subwavelength resonators, is studied in analogy to optical metamaterials [5,6,7]. In this view, metaporous materials, made of periodic rigid inclusions embedded in the porous medium, have been proposed to enhance the absorption properties at frequencies lower than the quarter wavelength frequency [8,9,10]. The principle is that the energy is trapped and dissipated between periodic rigid inclusions embedded in the porous medium and the rigid backing or in the inclusions themselves. A nearly total absorption can be obtained for a frequency lower than the quarter-wavelength resonance frequency due to the excitation of trapped modes. Local resonance and trapped modes are another possibility to localize the field. Using the trapped mode to produce additional transmission zero is also discussed in the previous two chapters.

In this chapter, the influence of the mode coupling due to the periodic embedment of rigid inclusions in porous material is studied for the waveguide applications instead of acoustic gratings [8,9] numerically and experimentally. The use of periodic inclusion in the waveguide has also been studied by Nennig et al. [11], in which a finite element (FE) method, inspired from Ref. [12], was chosen to tackle a quadratic eigenvalue problem. The Floquet-Bloch theorem is used to reduce the computation on one periodic cell. It has been shown that open shape inclusions (e.g. U-shape) are able to enhance the attenuation when compared with a homogeneous liner. Thanks to Floquet-Bloch formalism assuming periodic heterogeneities [8,9,11,13], very interesting results have been obtained in acoustics. The Floquet theorem is applicable to ordinary differential equations with periodic coefficients and shows that the solutions are such that ψpx `L0 q " e jk B L 0 ψpxq, where L 0 is the period of the coefficients, k B is the Bloch wave number, and x is the propagation axis. Such functions are known as Bloch wave functions.

In this work, the Floquet-Bloch theorem is applied to analyse the sound attenuation in a 2D infinite periodic waveguide lined by a porous material embedded with rigid inclusions. The mode attenuation can be explicitly described by the imaginary part of the wave number k B . The wave numbers and the wave functions of the Bloch waves in the given waveguide can be defined by the solutions of a generalized eigenvalue problem [14,15]. The propagating mode of a periodic waveguide are given by the Bloch waves with unitary eigenvalues.

The present work is organised as follows. In Chap.4.2, the problem of a 3D waveguide partially lined with a porous material embedded with periodic inclusions is first described. The experimental and numerical methods to consider the transmission be-haviours are also introduced. Floquet-Bloch theorem is proposed to investigate the sound attenuation in a similar 2D infinite waveguide lined with periodic inclusions embedded in porous material(Chap.4.3). In Chap.4.4, the experimental and numerical results for the 3D waveguide with different inclusion shapes and configurations are presented and discussed. The test facility used in this study is schematically depicted in Fig. 4.1 [16]. A two sources method is used: the acoustic waves are produced by two loudspeakers 3a and 3b and propagate in a rectangular duct (width W=100mmˆheight H=15mm). Two anechoic terminations are used to avoid resonant conditions in the duct.

Description of the problem

The acquisition of signals is performed by Agilent VXI 1432 hardware platform which drives the source excitation synchronously with the acoustic pressure signals recording. A swept-sine over the frequency range 30-3500 Hz is used with a frequency increment of 10 Hz. The test materials are put in position 1.

Two series of four microphones mounted in the upstream (2a) and downstream (2b) of the test material section are used to measure the acoustical pressure. Those microphones are located at the positions: x u1 ´xu2 " x d1 ´xd2 " 63.5mm, x u1 ´xu3 " x d1 ´xd3 " 211.5mm and x u1 ´xu4 " x d1 ´xd4 " 700mm. The use of 2ˆ4 microphones allows an over-determination of the transmitted and reflected waves on both sides of the test material and avoids the problems in the precision of measurement when the acoustic wavelength is close to half the distance between two microphones.

Chapter 4. Use of metaporous materials in acoustic ducts

In the frequency range 30-3400Hz, only 2 acoustic modes can propagate in the rigid ducts: the plane wave and the first-order mode along dimension W. The microphones being located just at the centre of dimension W, this second mode is not measured and the microphones only capture the plane waves.

Measuring technique

The aim of the experimental apparatus is to measure the transfer matrix or the scattering matrix of a test material [17]. The scattering matrix for the plane wave relates the scattered pressure amplitudes p 2 and p 1 (see Fig. 4.1) to the incident pressure amplitudes p 1 and p 2 by ˜p1

p 2 ¸" « R `T T `R´ff ˜p1 p 2 ¸" S ˜p1 p 2 ¸, (4.1)
where T `and T ´are the anechoic transmission coefficients, R `and R ´are the anechoic reflection coefficients, and superscripts ˘indicate the direction of wave propagation along the x axis, and the subscripts '1, 2' indicate the upstream and downstream of the duct.

The method of measurement used in the present study is called "the 2 sources method". Two measurements are made in two different states of the system. These different states are obtained by switching on the upstream source, the downstream source being switched off (measurement I), and vice versa (measurement II).

The scattering matrix is calculated from the two measurements using the following relation:

« pp 1 {p 1 q I pp 1 {p 2 q II pp 2 {p 1 q I pp 2 {p 2 q II ff " S « 1 pp 1 {p 2 q II pp 2 {p 1 q I 1 ff , (4.2)
This calculation is meaningful only if the two measurements are independent, i.e., if the determinant of the second right side matrix does not vanish:pp 2 {p 1 q I ‰ pp 1 {p 2 q II .

For the measurement I, the sound pressure in the waveguide is written (time dependence e ´jωt is omitted):

ppx 1 q " p 1 e jk `x1 `p1 e ´jk ´x1 " p 1 ´ejk `x1 `R1 e ´jk ´x1 ¯, (4.3)
where R 1 is the reflection coefficient from measurement I, and k `and k ´are the wavenumbers in the duct in the direction of `x and in the reverse direction. Then the expressions of the pressure at microphones u i and u j are obtained by ppu i q " p 1 e jk `xu i `p1 e jk `xu i " p 1 ´ejk `xu i `R1 e ´jk ´xu i ¯, and ppu j q " p 1 e jk `xu j `p1 e jk ´xu j " p 1 ´ejk `xu j `R1 e ´jk ´xu j ¯, 4.2. Description of the problem 71 respectively, where x u i is the position of the microphone u i relatively to the inlet of the measured element, the same for microphone u j .

The upper two equations result in ppu j q I ppu i q I " e jk `xu j `R1 e ´jk ´xu j e jk `xu i `R1 e ´jk ´xu i . (4.4)

A transfer function is defined to label the ratio of the pressure between microphones u j and u i ,

H I u j u i " ppu j q I ppu i q I . (4.5)
By Eq. (4.4), the reflection coefficient from measurement I can be found from the transfer function between two different microphones by a relation of the type

pp 1 {p 1 q I " H I u j u i e jk `xu i ´ejk `xu j e ´jk ´xu j ´HI u j u i e ´jk ´xu i . ( 4.6)
All the other matrix elements can be found by the same way (see [18] for details). Thus, the wavenumbers k `and k ´have to be known to calculate the scattering matrix.

For measurement II, we have ppd i q " p 2 e jk `xd i `p2 e ´jk ´xd i " p 2 ´R1 e jk `xd i `e´jk ´xd i ¯, and ppd j q " p 2 e jk `xd j `p2 e ´jk ´xd j " p 2 ´R1 e jk `xd j `e´jk ´xd j ¯.

The same as we did before, the upper two equations result in

pp 2 {p 2 q II " H II d j d i e
ppd i q ppu i q " p 2 ´ejk `xd i `1 R 1 e ´jk ´xd i p1 ´ejk `xu i `R1 e ´jk ´xu i ¯,
we define

H 21 d i u i " ppd i q ppu i q , H 12
u i d i " ppu i q ppd i q ,
the upper equations can lead to pp 2 {p 1 q I " H 21

d i u i e jk `xu i `R1 e ´jk ´xu i e jk `xd i `1 R 1 e ´jk ´xd i , pp 1 {p 2 q II " H 12 u i d i R 1 e jk `xd i `e´jk ´xd i 1 R 1 e jk `xu i `e´jk ´xu i .

Description of the configurations

The porous material is a metallic foam that has been chosen to avoid any skeleton vibrations. It is supplied as plates (200mmˆ100mm) of thickness 5mm. Five layers of this porous material are assembled together to form the uniform porous material, it is labeled by "5P". The parameters of the metallic foam used in the fluid equivalent model have been measured on another setup. The values are porosity Φ " 0.99, tortuosity α 8 " 1.17, viscous length Λ " 1 ˆ10 ´4 m, thermal length Λ 1 " 2.4 ˆ10 ´4m, and resistivity σ " 6.9 ˆ10 3 kg m ´3 s ´1.

In order to embed the inclusions in the porous material, holes are drilled in three of the porous material layers. Two types of inclusion shape are investigated: the first type is a metallic hollow cylinder with one closed end and one open end (labeled by "R1" in the following), the second (labeled "R2" ) is a closed hollow cylinder with a slit on the side (Helmholtz resonator), see Fig. 4.2(a). The inclusions are filled with air. The external diameter of R1 is D=22mm, and the inner diameter is d=21mm. Three material. As we did before for the R1, the other two lays of porous material can also be assembled in three different ways, e.g., as shown in Fig. 4.5(b).

Numerical method

Computations have been performed at Airbus using ACTIPOLE software developed by Airbus Group Innovations [19]. ACTIPOLE is designed to solve harmonic acoustic wave propagation problems by the Boundary Element Method. The parallel and out-ofcore direct solver has been used in order to predict the transmission loss (TL) and the pressure field in the whole computational domain for each computed frequency, taking into account porous liners by assuming a fluid equivalent behavior. The higher the value of the TL, the better the attenuation of the porous medium.

To study the influence of the two different inclusions and the porous material on the transmission loss, five different configurations are considered, as defined in Table 4.1. For "Air+R1_A+Air" and "Air+R2_A+Air", the inclusions are embedded in big rigid cavity filled with air. We consider here the acoustic wave propagation in a 2D infinite periodic waveguide lined with a porous material with embedded rigid inclusions, as described in Fig. 4 For the simplication, the shape of the inclusion is [-shape. In the air domain, i.e. y P r0, h a s, the sound pressure p a px, yq satisfies the governing equation in the 2D Cartesian coordinates px, yq ˜B2

Bx 2 `B2 By 2 `ˆω c a ˙2¸p a px, yq " 0, (4.7)
where ω is the circular frequency, and c a denotes the sound speed in the air. Time dependence is assumed as exp(-jωt) and will be omitted in the following. The acoustic velocity is obtained by

ρ a Bu a Bt " ´∇p a ,
where ρ a is the air density.

In the porous material, i.e. y P r´h p , 0s, the skeleton of the porous material is considered as infinitely rigid, thus the Champoux-Allard-Johnson equivalent fluid model [20] is used to get the effective compressibility κ eq and effective density ρ eq (see Appendix 4.A for details). The phase velocity is given by c eq " a κ eq pωq{ρ eq pωq. The sound pressure p p px, yq in the porous material satisfies the wave equation

˜B2 Bx 2 `B2 By 2 `ˆω c eq pωq ˙2¸p p px, yq " 0. (4.8)
The acoustic velocity in the porous can also be obtained by ρ eq Bu p Bt " ´∇p p .

On the rigid walls, the pressure satisfies the rigid boundary condition. At the surface y " 0 between the air and the porous material, the continuity of pressure and normal Chapter 4. Use of metaporous materials in acoustic ducts velocity leads to p p px, 0 ´q " p a px, 0 `q, 1 ρ eq Bp p By ˇˇˇy "0

´" 1 ρ a Bp a By ˇˇˇy "0 `.

(4.9)

For an infinite duct lined with porous material, the eigenvalues can be solved by Finite Difference Method (FDM) (see Appendix 4.B for details). The eigenvalue problem is rewritten as

`D2 `k2 I ˘P " k 2 x P , (4.10)
where D 2 is the 2nd order differential matrix, k contains the elements ω{c a and ω{c eq (see Appendix 4.B for details), I is an identity matrix, and k x is the wavenumber in x´direction.

By computing the eigenvalues of Eq.(4.10), the wavenumber k x can be obtained, with ℑmpk x q ą 0 for `x-direction and ℑmpk x q ă 0 for ´x-direction. With the help of the resulting wavenumbers k x and eigenvectors P , the pressure in ˘x -direction can be expressed by

p ˘pxq " n"N ÿ n"1 C n P n expp˘jk x,n xq " XEp˘xqC ˘, (4.11)
where n is the mode index, X is a matrix in which the columns are the the eigenvectors P n , Ep˘xq are diagonal matrices with expp˘jk x,n xq on the main diagonal, and C is a vector with mode amplitudes C n .

Bloch waves and eigenvalue problems

When the governing equations, the boundary conditions and the geometry are L 0periodic along x (see Fig. 4.6), it follows from the Floquet-Bloch theorem that the solution are Bloch waves [21] ppx, yq " ppx, yqe jk B x , (4.12)

i.e. the pressure function ppx, yq can be split into a L 0 -periodic function ppx, yq modulated by a plane wave with the Floquet-Bloch wavenumber k B . It needs to be noted that the Bloch wavenumber is common for both media as the axial wavenumber for classical guided wave problems. The concept of mode can be extended to periodic system thanks to the Floquet-Bloch theorem.

On the right and left boundaries of the elementary cell, respectively on Γ r and Γ l , the pressure function satisfies the condition ppx `L0 , yq " ppx, yqe jk B L 0 .

(4.13)

It follows from Eq. (4.13) that the real part of k B L 0 measures the change in phase across the cell and its imaginary part the attenuation. In addition, it can be directly shown from Eq. (4.12) that k B is defined modulo 2π{L 0 . The smallest values belongs to the irreducible Brillouin zone.

Taking an arbitrary cell (see Fig. 4.6), the scattering matrix (denoted as S) concept

ˆC0 C L0 ˙" " R t T r  loooomoooon S ˆC0 C Ĺ0 ˙(4.14)
remains valid, where Rprq and Tptq are the matrices containing reflection and transmission coefficients, C 0 and C L0 are the amplitudes of ˘x-direction waves for the pressures at

x " 0 and L 0 , respectively. Due to the symmetry of the geometry, we have R " r, and due to the reciprocity T " t. The scattering matrix S for an elementary cell can be obtained by using mode matching of the pressure and velocity at the discontinuity interfaces, the details are given in Appendix 4.C.

Moreover, the pressure functions obey the Bloch condition [22] ˆCL

0 C Ĺ0 ˙" e jk B L 0 ˆC0 C 0 ˙. (4.15)
The upper two equations, Eqs. (4.14) and (4.15), lead to the following eigenvalue problem

" T r 0 I  loooomoooon M 1 ˆC0 C Ĺ0 ˙" e jk B L 0 lo omo on λ " I 0 R t  loooomoooon M 2 ˆC0 C Ĺ0 ˙, (4.16)
where I and 0 are the identity and zero matrices. As the reflection and transmission matrices Rprq and Tptq are already known (see Appendix 4.C), the dispersion relation giving the solution for the Bloch wavenumbers k B can be solved numerically [14,15]. With the help of the eigenvalues λ from the generalized matrix eigenvalue problem, the Bloch wavenumber is obtained by

k B " ´j lnpλq{L 0 . (4.17)
The propagating Bloch waves are such that |λ| " 1; the other waves are evanescent and decay exponentially in the direction of propagation. With the help of the resulting eigenvectors pC 0 , C Ĺ0 q T from the eigenvalue problem of Eq. (4.16) and the scattering matrix S, the amplitudes of the incoming and outgoing modes at the entrance and exit can be given. Using the Eq.(4.26) in Appendix 4.C, the amplitudes of the modes at different segments can be obtained. The wave functions of the Bloch waves then can be expressed by using Eq.(4.11) for each segment.

Effects of inclusions and porous material

The aim of this section is to illustrate the effects of the inclusions and the porous material on the sound attenuation. The properties of a uniform waveguide lined with pure porous material is first investigated. The effects of the porous material on the transmission loss of a plane wave in a rigid duct is also considered. Then the inclusions are embedded in the porous material to enhance the sound attenuation at low frequencies.

When there is no inclusion embedded in the porous material, the eigenvalue problem in an infinite waveguide lined with a pure porous material (called porous lined duct in the following) can be solved by Finite Difference Method, as detailed in Appendix 4.B. The wavefunction is separable and can be expressed by ppx, yq " P e jkxx .

For the sake of convenience, Bloch waves introduced in Sec. 4.3.2 are also applied here to find the Bloch wavenumbers in the porous lined duct, with the same period L 0 as in Fig. 4.6 considered. In this particular case, the Bloch wavenumbers k B and the axial wavenumbers k x are identical, and ppx, yq in Eq. (4.13) is independent of x, which becomes the transverse eigen-modes. In Fig. 4.7, for this particular case, we plot the real and imaginary parts of Bloch wavenumbers k B as a function of frequency. From the imaginary parts of k B , we can have the sound attenuation of each mode along x as a function of frequency. Here we are only interested in the lower two modes. At low frequencies the distinction between the mode with the least attenuation and all other modes is fairly clear because the latter are, in general, highly attenuated cut-off modes. The fact that there is a mode with minimum attenuation is extremely useful because provided the source distribution does excite all the modes, then at some distance from the source the sound field will consist of just this one mode, and hence the sound attenuation at larger distances will be entirely determined by this minimum value [23]. From Fig. 4.7(b), we can see that around 4700Hz, the imaginary parts of the lower two waves cross. This is where the maximum attenuation of the least attenuated mode happens. For a waveguide lined with locally reacting liner, of special interest is the question of optimizing the sound attenuation by judiciously selecting the impedance characteristics of the liner. In this view, the concept of optimal liner was first defined and investigated by Cremer [24]. It has been shown that Cremer's optimum impedance is a branch point [23] and an Exceptional Point (EP) in complex admittance (impedance) plane (as we discussed in Chap. 1.4. ) and EPs play an important role in the sound attenuation [25].

However, if a finite length (L " 0.2 m) of such porous material is attached to two semi-infinite rigid ducts, in which the porous liner is truncated by rigid walls at the two ends (see e.g. Fig. 4.1), the transmission loss (TL) peak is observed around 3800Hz. The results are given in Fig. 4.8, with only plane wave incident in the rigid duct. Similar as described in Appendix 4.C, the scattering matrix can be derived by using mode matching at the interfaces x " 0 and L. Good agreements can be seen from the comparisons between the computation and experimental results [16]. The oscillations in the reflection coefficient are linked to the wave reflection at the end of the material px " Lq. The reflection coefficient oscillates around about a value of |R| " 0.41 which is close to the value p1 ´αq{p1 `αq " 0.45, where α " H{pH `Bq, valid at low frequencies for an area expansion without porous material [16]. To have a comparison and to see the effects from the discontinuity interfaces, the TL (i.e. e jk B L ) of the least attenuated mode with length L " 0.2 m is also plotted in Fig. 4.8(b) by the dashed line. We can see that the frequency of the TL peak is smaller with the two semi-infinite rigid ducts, comparing with the one for the least attenuated mode. To understand the mode coupling effects, we first analyse results without porous material. In Fig. 4.9(a), the band diagram is given for the waveguide with the [-shape inclusions embedded in air (no dissipation is included). These scatterers are located at the centre of the periodic cell with the dimensions: width D " 22mm, height h 0 " 15mm and periodic length L 0 " 24mm. For the sake of simplification, the inclusion thickness is neglected. At some point, the wavenumber k B becomes complex due to the interference with the reflection at each discontinuity, and the wave function is attenuated along the propagation direction, for example the mode labelled by "1" in Fig. 4.9(a) becomes complex when f ą 3400Hz. This can also be illustrated by Fig. 4.9(d)(e), where the real and imaginary parts of the two k B as a function of frequency are given. There exist a frequency band rf 1 , f 2 s, in which modes 1 and 2 both can propagate. When f ă f 1 , only mode 1 can propagate, mode 2 is evanescent. Then when f ą f 2 , the wavenumber of mode 1 becomes complex. The absolute value of the periodic functions ppx, yq for the lower 2 Bloch waves (labelled by 1 and 2 in Fig. 4.9(a)) at 2000Hz are plotted in Fig. 4.9(b)(c). From the function distribution in the unit cell, we can see that the pressure of mode 2 is mainly localized inside the inclusion with the help of the rigid backing, we will call it "Localized mode" in the following. While for mode 1, the pressure is mainly in the region ´5mm ă y ă 15mm, which will be called "Acoustic mode".

To have sound attenuation, the porous material has to be applied. The air filled inclusions are embedded in the porous material (see Fig. 4.6) to improve the sound attenuation at low frequencies. The mode attenuation in dB/m, given by Apk B q " 8.68 ¨ℑmk B , (4.18) can be shown in Fig. 4.12(a) for such configuration. The frequency (880Hz) of the first attenuation peak in a waveguide lined with periodic inclusions embedded in porous material (symbol "'") is much lower than the one (4700Hz) in a porous lined duct (symbol " ").

It indicates that due to the embedment of the rigid inclusions in the porous materials, the sound attenuation at 880Hz is enhanced. To explain this behaviour, the band diagram for the waveguide lined with periodic inclusions embedded in porous material (see Fig. 4.6), obtained by plotting the Bloch wavenumbers k B as function of the frequency, is presented in Fig. 4.10(a). We can see that due to the presence of the porous material, the wavenumbers become complex, and they are very different with those without porous material(Fig. 4.9). The imaginary parts of the lower two k B as a function of frequency, which give the essential information about the sound attenuation, are also plotted (see the bottom figure in Fig. 4.10(b)). Crossing is observed between mode 1 and mode 2. The maximum attenuation of the least attenuated mode then can be found near 880Hz, where the crossing of the imaginary parts between the lower two modes is observed. First of all, we can see that the results verify ppx, yq " ppx `L0 , yq. Mode 1 is the so-called localized mode which is also observed in Fig. 4.9(b), the pressure field is mainly localized inside the inclusions. However, for the periodic function of mode 2, it is a mixing of the acoustic mode and localized mode at 880Hz, which indicates the existence of the coupling between the two different kinds of mode. The periodic waveguide can interfere in the homogeneous waveguide framework. It is worth noting that the first attenuation peak is strongly connected to this coupling effect and to the crossing observed for the imaginary parts. Indeed, modes avoid each other thanks to a modification of the k B imaginary part leading to an enhancement of the attenuation.

To see the effect of the porous material embedded with inclusions on the transmission behaviours in a rigid waveguide, the transmission loss in a waveguide partially lined with such material, in which 8 cells are embedded in the centre of a porous material of length L " 0.2m, is computed, shown by the solid line in Fig. 4.12(b). The TL of the least attenuated Bloch waves is also plotted (symbol "'") for length L " 0.2m. The TL peak observed at 880Hz can be explained by the crossing of the mode attenuation between Figure 4.12: (a) Mode attenuations (dB/m) of the lower two Bloch waves in the uniform waveguides lined with periodic inclusions embedded in porous material (symbol "'") and in the porous lined duct (without inclusions)(symbol " "). (b) Symbol "'": TL of the lease attenuated Bloch mode with length L " 0.2 m (i.e. e jK B L ). Solid line: TL of the plane mode in an acoustic waveguide with a portion of its wall lined by a porous material embedded with 8 inclusions, the total length of the lined portion is L. Dashed line: Measured TL for a waveguide partially lined with a pure porous material of length L.

the lower two Bloch waves (see "'" in Fig. 4.12(a)), due to the coupling of the two lower Bloch waves. The TL in the waveguide embedded with inclusions (solid line) is enhanced between 500Hz and 1300Hz, comparing to the one without inclusions (dashed line).

As already shown by Groby et al. [8] that the inclusion position in the periodic cell can be a sensitive parameter. The effects of the positions of the inclusions in the porous material on the TL with length L " 0.2m of the least attenuated mode are also investigated, by moving the inclusions above (+ mm) or below ( -mm) the current centre position. The results are shown in Fig. 4.13. The frequency of the first peak decreases when the inclusions are moved both above and below the centre position. The opposite effect is observed for the second TL peak. These effects are not so evident when the inclusions are moved above the centre poison.

It need to be noted that under different inclusion positions, the Bloch wave numbers k B can move in a different way with varying the frequency. To illustrate this point, we plot the ℜepk B q and ℑmpk B q as a function of frequency, see Fig. 4.14, with three different inclusions positions: the inclusions are in the centre 0 mm (Fig. 4.14(a)(b)), below the centre 4 mm (Fig. 4.14(c)(d)) and 4.9 mm (Fig. 4.14(e)(f)). We can see that for the ℑmpk B q, the crossing becomes avoided crossing with moving the inclusions below the centre. This might be related to the exceptional point(see Chaps. 2 and 3).

ℑm ( k B ) (b) (a) (c) (d) (e) (f)

3D numerical and experimental results

Numerical simulations and experimental measurements are carried out for the configurations described in Chap. 4.2.4 and Chap. 4.2.3, respectively. It need to be noted that the material is put in a big rigid cavity.

Numerical results

Numerical calculations have been performed for various configurations which are reported in Table 4.1 and within the frequency range of the audible sound, particularly, at low frequencies. For all calculations, the top of the big cavity (which is used to put the porous material) as well as any non-rigid/empty surface (e.g. the surface between the open end of the inclusion and the porous material) are made of a resistive layer with very small resistivity: Re" 1e´4. The transmission loss (TL) for the 5 configurations given in Table 4.1 are shown in Fig. 4.15, when only the plane mode is incident. The numerical frequency range is 500Hz to 4000Hz. At very low frequencies (below 800 Hz), the results of the configurations with porous material are quite similar meaning that the inclusions do not have any effect in this frequency range. However, when the frequency increases, a mid-frequency peak (MFP) is observed in transmission losses for both P+R1_A+P and P+R2_A+P at 2400Hz and 1400Hz, respectively. The MFP of P+R2_A+P happens at a lower frequency than the MFP of P+R1_A+P. A second TL peak appears for P+R2_A+P at 3400Hz. It can can be noted that due to the presence of the inclusions, the attenuation is increased around the MFP but there is a small negative effect on TL at frequencies below MFP, and a large negative effect on TL at frequencies above MFP.

Air+R1_A+Air and Air+R2_A+Air are used to consider the effects of the two different inclusions when no porous material is added. At low frequencies, f ă1.2kHz, the two TL curves are quite similar. They are also close to the results of an empty cavity (not shown here). Thus, in this frequency range, the inclusions have no significant effects on the TL. Two TL peaks are found at 1.7 kHz and 3.6 kHz for Air+R2_A+Air. However, no obvious effects are found by correlating P+R1_A+P with Air+R1_A+Air, and the MFP of P+R1_A+P seems to be due to a more complex interaction of the inclusion R1 with the porous material. The porous material plays a very important role for the TL peak of P+R1_A+P. Comparing P+R2_A+P and Air+R2_A+Air, the TL of P+R2_A+P (porous material + inclusions R2) cumulates the resonant effects of inclusions (Air+R2_A+Air), with a shift of peaks to lower frequencies, and the effects of porous material (5P). Due to the presence of porous material, the TL increases, and the frequencies of the two TL peaks shift to lower frequencies. The objective for inclusions to improve the attenuation of the sole porous material in low frequency range (peak of attenuation can be tuned) seems to be reachable, at the price of a moderate degradation of the attenuation at other frequencies. The computed pressure fields for P+R1_A+P and P+R2_A+P at the frequencies of the TL peaks are given in Fig. 4.16 and Fig. 4.17, respectively. A plane wave is incident from the left side, the pressure decreases in the duct along the material. Large pressure is observed inside some inclusions for P+R1_A+P. Between the rigid bottom of R2 and the rigid backing, the pressure localization is observed at 3400Hz (Fig. 4.17(b)). 4.18 shows the experimental results when the closed ends of all the inclusions R1 are on the bottom (R1_B) near the rigid backing, as illustrated in the inset of Fig. 4.18(a). When there are only three layers of porous material embedded with inclusions (R1_B), due to the reduction of the porous materials, its TL is the smallest at all the frequency range of interest comparing to the other configurations. For f ą 2400Hz, more porous material layers are near the waveguide, higher transmission loss (TL) is observed, we have the following relation TL(R1_B+2P)ăTL(P+R1_B+P)ăTL(2P+R1_-B). It means that for the configurations with R1_B, the porous material plays the main role on the transmission loss. For f ă 2400Hz, the three configurations have similar transmission loss (transmission coefficient). For the reflection coefficient, big difference is observed when f ą 1200Hz, see Fig. 4.18(c).

The TL of configuration P+R1_B+P is quite similar to the one of 5P, which means that such configuration has even no enhancement on the sound attenuation. This can also be verified and explained by the presence of the Bloch waves in the periodic waveguide, as we introduced in Sec. 4.3. A maximum attenuation can be observed near the frequency where the mode attenuations of the lower two modes have crossing (or avoided crossing). For this 3D configuration, the Bloch wave computation is carried out by using a finite element discretisation [11]. The 3D periodic waveguide is an infinite duct lined with configuration P+R1_B+P (see the right one in Fig. 4.20(c)), it need to be noted that the bottom of the liner is a rigid wall. The TLs of the lower two modes for this periodic waveguide with length L " 0.2m as a function of frequency are shown by " " in Fig. 4.20(a) [26] 1 . At the frequency range we considered, there is no crossing between the two TLs. The TL of the least attenuated Bloch wave increases with the frequency slowly, leading to the slow increase of TL with frequency in a rigid duct partially lined with P+R1_B+P.

However, for configuration 2P+R1_B, the transmission loss is improved for f ą 2400Hz comparing with 5P and P+R1_B+P. It might be because the frequency of the crossing between the lower two mode attenuations are lower than the one for configuration P+R1_B+P, due to the different inclusion positions in the porous material. The effect of the inclusion position in the porous material on the sound attenuation has also be investigated in the 2D waveguide, see Fig. in Fig. 4.20(c)), labelled by " " in Fig. 4.20(b), a crossing of the TLs between the two Bloch waves are observed. The most important is that the TL peak (solid line) of the lined duct appears just near this crossing. This is exactly what we discussed before in Sec.4.3 for a 2D configuration. The TL in a rigid duct partly lined with finite periodic liner is dominated by the mode attenuation of the least attenuated mode in the periodic waveguide. An attenuation peak is reached if there is a crossing for the mode attenuations between this least attenuated and the 2nd mode.

Taking a look at the results for R1_T in Fig. 4. 19, the open ends were closed by the big rigid cavity when the measurements were performed. At the frequency range we considered, no resonance effect from the inside of the inclusions appears, which should be at higher frequency. Highest TL peak is observed for R1_T around 2000Hz, comparing with R1_T+2P and P+R1_T+P. It is due to the reduction of porous materials, the amplitude of the TL peak is higher while the frequency band is narrower. Biggest absorption coefficient is observed for R1_T around 2200Hz, see Fig. 4. 19(d).

A different TL peak is observed around 1700Hz for configuration 2P+R1_T, which is lower than 2000Hz for configurations R1_T+2P and P+R1_T+P. It is because the inclusions are moved below the centre position, leading to a lower frequency at which the crossing between two mode attenuations appears. The effects of the inclusion position embedded in the porous material on the TL peak are also investigated, by moving the inclusion above or below the centre position, the ACTIPOLE results are given in Fig. 4.21(a). We can see that, comparing with the centre position, by moving the inclusions away from the centre, the frequency of the TL peak becomes lower. The shift of the frequency is obvious when the inclusions are moved below the centre position. The From the results in the Figs. 4.18 and 4.19, we can see that configuration 2P+R1_-B can improve the TL at high frequency (2500Hz, 3500Hz), while 2P+R1_T at low frequency (1000Hz, 2400Hz), comparing with 5P. These combined effects lead to the global enhancement of the TL at (1000Hz, 3500Hz) for the configuration 2P+R1_A, the results are shown in Fig. 4. 22(a). It takes the advantages of the closed bottom (2P+R1_-B) and the closed top (2P+R1_T), the first peak of 2P+R1_A is due to the maximum TL of 2P+R1_T near 2000Hz, and the TL enhancement for f ą 2500Hz is due to the high TL of 2P+R1_B. On one hand, the TL of 2P+R1_A takes the benefits of better TL of the two different arrangements at some frequencies, on the other hand, the amplitude of TL decreases. Fig. 4.22 shows the results for the configurations where the inclusions are arranged in an alternated way(R1_A). Different resonant frequencies of TL are observed for different configurations. R1_A+2P and P+R1_A+P have the resonant frequency around 2350Hz, while 2000Hz for R1_A and 2P+R1_A. This can also be explained by the sensitivity of the inclusion position in the porous material on the TL peak frequency. Between R1_A and 2P+R1_A, the addition of two layers of porous material makes obvious improvement for the TL of 2P+R1_A.

Experimental results for the configurations with R2

The effect of the second type of inclusion R2 on the transmission loss is also considered. The experimental results for different configurations with R2 are shown in served for the configurations R2_A+2P, R2_0 ˝+2P, and R2_90 ˝+2P. To see the effect of the orientation of the resonators on the frequency of the TL peaks, we compare the results in the same figure, see Fig. 4.26. By modulating the orientation of the resonators, the frequency of the first TL peak decreases slightly, while the amplitude of the TL becomes smaller. The frequency of the second TL peak does not change with varying the orientation of R2.

Taking Fig. 4.23(a) as an example, the effect of the other two layers of porous material on the sound propagation can also be seen, by arranging the porous material in different positions (see Fig. 4.5(b)). Fig. 4. 23(a) shows the TL of the three different configurations R2_A+2P, P+R2_A+P and 2P+R2_A. The first TL peak is observed for all the three configurations. With increasing the number of porous material layer on the top near the waveguide, the frequency of the first TL peak decreases slightly, which is the same phenomenon we discussed before for R1. The second TL peak around 2800Hz is observed only for R2_A+2P. From the Actipole results (see Fig. 4.15), the second TL peak for P+R2_A+P (P+R2_A+P here) is at 3400Hz. It means that the porous material has the opposite effects on the second TL peak, comparing with the first TL peak. Bigger the distance between the bottom of the R2 and the rigid backing, lower the frequency of the second TL peak is. The second TL peaks is closely related to the mode localization existing between the rigid bottoms of R2 and the rigid backing, this can be observed in Fig. 4.17(b). The same effects have also been discussed in Refs. [8,9], which is explained by the presence of the "underlying" periodic array formed by the inclusions images with respect to the rigid backing. The appearances of the two TL peaks for R2_A+2P lead to a globally enhancement of TL at the frequency range we considered. Figs. 4.23(b)(c) give the absolute values of the transmission and reflection coefficients from both upstream and downstream. The inclusion distribution in the porous material is symmetrical about y´axis, the results for upstream and downstream are supposed to be the same. Good agreement is observed for the results between the upstream and downstream. Fig. 4. 23(d) shows the absorption coefficient as a function of the frequency for the three different configurations. The highest absorption is observed for R2_A+2P.

The orientation of the R2 in the porous material is changed to see the effect. In order to avoid the coupling effect near the slits between two inclusions, the inclusions all orient on the same angle. As shown in Figs. 4.5(a), two other different inclusion distributions R2_0 ˝and R2_90 ˝are considered, the results are given in Fig. 4.24 and Fig. 4.25, respectively. The same as we did before, the effect of the porous material is considered. The same phenomenons as in Fig. 4

Comparison between numerical and experimental results

The test sample of P+R1_A+P is shown in Fig. 4.4(b). The comparison of the numerical and experimental results for P+R1_A+P and P+R2_A+P are performed, see Figs. 4.27 and 4.28. There is a very good agreement between the numerical and the experimental results, despite a general trend for ACTIPOLE to over-predict the peak of attenuation.

Conclusions

An analysis of a 2D infinite periodic waveguide with inclusions embedded in porous material is performed by using Floquet-Bloch theorem. A maximum sound attenuation for this infinite periodic waveguide can be reached near the crossing (or avoided crossing) of the mode attenuations between two lower Bloch waves. The avoided crossing is related to the exceptional point. A coupling between acoustic mode and localized mode is happened near this crossing. This is the second strategy we presented in this chapter.

The influence of the periodical array of 3D inclusions embedded in the porous material on the transmission loss of an acoustic duct is studied numerically and experimentally. Two different inclusion shapes are considered, an open cylinder and a Helmholtz resonator. When the rigid inclusions are embedded in the porous material, the low frequency behaviours of the transmission loss can be significantly changed, peaks are observed in the mid-frequency range. The effects of the inclusion position on the TL peak frequency are different for different inclusion shapes. For the configurations with open cylinder, the frequency of the TL peak becomes lower when the open cylinders are moved away from the centre. The peaks can also be explained by the crossing phenomenon, due to the mode coupling of the Bloch waves. For the configurations with Helmholtz resonator, the frequency of the first TL peak increases with increasing the distance between the inclusions and the rigid backing, while the frequency of second peak decreases. The second TL peak is due to the localized mode observed between the rigid bottoms of the resonator and the rigid backing. Good agreements are observed between the experimental and numerical results, which opens the way to the optimisation of metaporous material for better sound attenuation at low frequencies. Employing a second-order expansion for pressure function at y " 0, and considering the boundary conditions in Eq.(4.9), the eigenvalue problem in the waveguide with porous materials can be expressed by

1 ∆h 2 » - - - - - - - - - - - - - - - - - - ´1 1 1 ´2 1 . . . . . . . . . 1 ´2 1 G a G b G c G d M a M b M c M d 1 ´2

´1

. . .

P 2 P 1 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' `» -´ω ca ¯2 I Na O O ´ω ceq ¯2 I Np fi fl looooooooooooooooomooooooooooooooooon k 2 I ¨PN 2 P N 2 ´1
. . .

P N P `2 P N P `1 P N P P N P ´1
. . .

P 2 P 1 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' " k 2 x ¨PN 2 P N 2 ´1
. . .

P N P `2 P N P `1 P N P P N P ´1
. . . To obtain the scattering matrix S of one elementary cell, the cell is divided into 4 segments (see Fig. 4.29), according to Eq. (4.11), the pressures at different segments can be written as

P 2 P 1 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' . ( 4

4.C Scattering matrix of a unit cell

p 1 pxq " X 1 E 1 pxqC 1 `X1 E 1 p´xqC 1 , (4.22)
p 2 pxq " X 2 E 2 pxqC 2 `X2 E 2 pD ´xqC 2 , (4.23)
p 3 pxq " X 3 E 3 pxqC 3 `X3 E 3 pD ´xqC 3 , (4.24)
p 4 pxq " X 1 E 1 px ´DqC 4 `X1 E 1 pD ´xqC 4 , (4.25)
where 1, 2, 3, and 4 are used to label the 4 segments, and D is the width of the inclusion.

4.C. Scattering matrix of a unit cell 101

The continuity conditions of the p and Bp{Bx at the interfaces x " 0 and x " D lead to

X 1 1 pC 1 `C1 q " X 2 pC 2 `E2D C 2 q, X 3 1 pC 1 `C1 q " X 2 3 pC 3 `E3D C 3 q, X 1 1 K 1x pC 1 ´C1 q " X 2 K 2x pC 2 ´E2D C 2 q, X 3 1 K 1x pC 1 ´C1 q " X 2 3 K 3x pC 3 ´E3D C 3 q, X 2 1 K 1x pC 1 ´C1 q " 0, 0 " X 1 3 K 3x pC 3 ´E3D C 3 q, and
X 1 1 pC 4 `C4 q " X 2 pE 2D C 2 `C2 q, X 3 1 pC 4 `C4 q " X 2 3 pE 3D C 3 `C3 q, X 1 1 K 2x pC 4 ´C4 q " X 2 K 2x pE 2D C 2 ´C2 q, X 3 1 K 1x pC 4 ´C4 q " X 2 3 K 3x pE 3D C 3 ´C3 q, X 2 1 K 1x pC 4 ´C4 q " 0, 0 " X 1 3 K 3x pE 3D C 3 ´C3 q.
These continuity equations can be rewritten in a matrix form

M 1 ¨C1 C 4 C 2 C 2 C 3 C 3 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' " M 2 ˆC1 C 4 ˙, (4.26)
where M 1 and M2 are the matrices used to relate the amplitudes at different segments.

The scattering matrix (denoted as S D ) between x " 0 and D, which relates incoming (C 1 and C 4 ) and outgoing (C 1 and C 4 ) modes, can be truncated from matrix

M ´1 1 M 2 by S D " M ´1 1 M 2 p1 : 2N 2 , 1 : 2N 2 q.
For an porous lined duct with length l 2 , the scattering matrix is Chapter 4. Use of metaporous materials in acoustic ducts Following Furnell and Bies [31] the operation b can be defined by

S l 2 " " 0 E 1 pl 2 q E 1 pl 2 q 0  . ( 4
S p1`2q " S 1 b S 2 " " R p1`2q t p1`2q T p1`2q r p1`2q  with T p1`2q " T 2 ET 1 , t p1`2q " t 1 Ft 2 , R p1`2q " R 1 `t1 FR 2 T 1 , r p1`2q " r 2 `T2 Er 1 t 2 , E " pI ´r1 R 2 q ´1, F " pI ´R2 r 1 q ´1,
where I is the identity matrix, and pT 1 , t 1 , R 1 , r 1 q and pT 2 , t 2 , R 2 , r 2 q are the elements of matrices S 1 and S 2 , respectively. 108 Chapter 5. Acoustical behavior of purely reacting liners

The instability was demonstrated experimentally by Aurégan et al. [1] [2]: There is an increase in the transmission coefficient at the resonant frequency, and the authors explained that this behavior was caused by the hydrodynamic instability over the liner. Rienstra [3] analyzed the behavior of the acoustic modes of a lined duct with and without uniform mean flow, it is shown that there exists three types of modes: genuine acoustic modes, acoustic surface waves and hydrodynamic surface waves. Subsequently, Brambley and Peake [4] further investigated the behavior of the surface modes. It's well known that when the flow exists, the viscosity effects cannot be omitted especially near the walls, and so the boundary layer has to be considered. The Ingard-Myers [5] [6] boundary condition, which is derivated under the assumption that the partial displacement is continuous across the vorticity sheet [6], is commonly used for the analytical prediction of sound propagation in ducts with flow. But then, Aurégan et al. [7] pointed out that depending on the ratio of the acoustic and stationary boundary layer thicknesses, the kinematic wall condition changes gradually from continuity of normal acoustic displacement to continuity of normal acoustic mass velocity. Renou [8] performed an experimental investigation, the results show that the Ingard-Myers boundary condition fails to predict with accuracy the acoustic behavior in a lined duct with flow.

For the past few years, the investigators have done lots of work about the modes existing in the lined duct in the presence of flow, especially the surface modes, and the instability problem caused by the coupling of the flow and the lining. In the present work, we focus our attention on the propagation problem in a two-dimensional acoustic partially lined duct with presence of flow, the liner is purely reactive. For a certain range of frequencies and Mach numbers, no wave can propagate against the flow. In Sec. 5.2, the case without flow is first reviewed, in the vicinity of the resonant frequency, a zero in the transmission coefficient is observed. In Sec. 5.3, in the presence of uniform flow, the modes in the liner and the transmission coefficient in two directions are investigated. In Sec. 5.4, the modes in the rigid duct with shear flow are first investigated. Attention is focused on the modes in the lined duct with shear flow and the transmission behaviour.

Propagation in the reacting lined ducts without flow

Modes in the lined ducts without flow

Statement of the problem

First, we consider the modes in an infinite lined duct without flow. For the infinite duct, the upper wall is rigid and the lower wall is lined with a locally-reacting liner, see Fig. and ψpyq satisfies the Helmholtz equation

d 2 ψpyq dy 2 `β2 ψpyq " 0, (5.2)
where β 2 " ω 2 ´k2 is the eigenvalues of the transverse modes in the lined duct. It is noted that the equations are written in a dimensionless form with the pressure p normalized by ρc 2 0 , x and y by width H, t by H{c 0 , where ρ is the air density and c 0 is the sound velocity. Then ω " 2πf c 0 H is the dimensionless frequency or Helmholtz number. Time dependence is expp´jωtq, and will be omitted in the following.

The solution of Eq. (5.2) is given by ψpyq " A 1 cospβyq `B1 sinpβyq,

where A 1 and B 1 can be computed from boundary conditions. The boundary conditions have to be applied to solve the eigenvalue problem. For the rigid wall, y " 1, the acoustic particle velocity normal to the wall vanishes, i.e.

can be approximately by

Zpωq " R `j cotpωbq,

The rigid eigenfunction φ m pyq is orthogonal

ż 1 0 φ m pyqφ n pyqdy " " 1 m " n 0 m ‰ n .
With the help of the boundary conditions Eqs.(5.4)(5.5) and Eq.(5.10), the second integral term in Eq. (5.11)

" jωY Φp0qpp0q ´ α 2 ş Φpyqψpyqdy " jωY N 1 A ´N2 A , (5.12)
where I is the identity matrix, α is a column vector with elements α n , and the elements of matrix N 1 can be obtained by N 1m,n " φ m p0qφ n p0q.

The matrices N 1 and N 2 are written as

N 1 " ¨1 ? 2 ? 2 ¨¨? 2 2 2 ¨¨? 2 2 2 ¨¨. . . ¨¨¨‹ ‹ ‹ ' , N 2 " ¨α2 0 0 ¨¨0 α 2 1 . . . . . . ‹ '.
Equation (5.11) can be reformulated as a matrix eigenvalue problem

k 2 I A " " ω 2 I `ω tanpωbqN 1 ´N2 ‰ A.
(5.13)

By computing the matrix generalized eigenvalue problem of Eq.(5.13), for a given ω, we can obtain the eigenvalues λ and the corresponding eigenvectors X. The axial wavenumbers are calculated by k 2 " λ. A purely reacting liner is considered here, so the wavenumber k is either purely real or purely imaginary values.

The eigenvalues β and eigenfunctions ψpyq, Eq. (5.10), in the transverse direction can be obtained by solving the following eigenproblem mode in the rigid duct, α 0 " 0, turns to be purely imaginary value for 0 ă ω ă ω R , where ω R is the quarter-wavelength resonant frequency of the liner, i.e. ω R b " π{2. From the eigenfunction profile (solid line in Fig. 5.3(a)) of the mode 0, the amplitude decreases away from the lined wall, y " 0. If the frequency ω is closer to ω R , the amplitude of the mode 0 decreases faster, the wave becomes more and more concentrated near the lined wall, so-called surface mode [3]. This is also illustrated in Fig. 5.6 where the pressure field is computed. The profiles of the other three modes are also shown in the same Fig. 5.3(a). They can be classified by the number of the pressure node in transverse direction. By Eq. (5.13), we also plot the dispersion curves of the first mode in x-direction, see Fig. 5.3(b), the real and imaginary parts of the wavenumber k as a function of ω are shown. Purely real values of k indicate the propagative modes. ℜepkq ą 0 indicates that the mode propagates in `x-direction. It shows that there exists a frequency band gap, in which no mode can propagate. In fact, at this frequency range, the values of k become purely imaginary (see the bottom figure in Fig. 5.3(b)), they become the evanescent modes in x direction. The band gap starts with the resonant frequency ω R , quarter-wavelength frequency of the locally reacting liner, at which the wavenumber becomes infinite. Since k " 2π{λ x , then the wavelength in x direction, λ x , tends to be zero. These can also be explained by the expression of the admittance, Eq. (5.7), when 0 ă ωb ă π{2, ℑmpY q ă 0; when ωb " π{2, ℑmpY q Ñ 8; and when π{2 ă ωb ă π, ℑmpY q ą 0.

β 2 I A " rN 2 ´jωY N 1 s A. ( 5

Scattering matrix

For the sound propagation problem in the waveguide partially lined with a locally reacting liner, the scattering matrix S needs to be constructed to calculate the transmission coefficient. By using the previously obtained wavenumbers k and the eigenvectors X, the similar method as in Ref. [9] was used here to calculate the transmission coefficients and the pressure field. The configuration is shown in Fig. 5 The configuration for constructing the scattering matrix, L is the length of the liner, normalized by duct width H, A i , B i pi " 1, 2, 3q are all vectors, indicating the pressure amplitude of each mode in `x-direction and ´x-direction, respectively. The total pressure consists of the `x and ´x-direction sound pressures.

the incoming waves, A 1 and B 3 , are linked to the amplitudes of outgoing waves, A 3 and B 1 , by a scattering matrix .15) Being aware of the eigenvalues and the eigenvectors in Eq. (5.13), the pressure in the lined part can be written as a sum of the transverse modes in the lined duct,

˜ A 3 B 1 ¸" S ˜ A 1 B 3 ¸, where S " " T r R t  . ( 5
p 2 px, yq " Φ T " XD `pxq A 2 `XD ´pxq B 2 ı , (5.16)
where D `pxq and D ´pxq are diagonal matrices with elements e jknx and e ´jknpx´Lq , respectively, and X is a matrix with the column vectors are eigenvectors X. For simplicity, we assume that there are multimodes incident from the left side of the waveguide, i.e. B 3 " 0. The sound pressures in regions I and III can be written as

p 1 px, yq " Φ T " D R pxq A 1 `DR p´xq B 1 ı (5.17) and p 3 px, yq " Φ T " D R px ´Lq A 3 ı , (5.18)
respectively, where D R pxq is diagonal matrix with elements e jk R n x , and k R n " a pω 2 ´α2 n q are the axial wavenumbers in the rigid wall. As given in Appendix 5.B , we can have the transmission coefficient by

T " t " pFD L ´GF ´1GD L qpF ´GD L F ´1GD L q ´1, (5.19)
where D L " D `pLq " D ´p0q. Only plane wave is considered here, so the transmission coefficient is Tp1, 1q. The transmission coefficient as a function of frequency with B " 50mm, H " 15mm, L " 200mm is shown in Fig. 5.5(a). It shows that there exists a specific frequency range, at which no wave can transmit through the lined part, the transmission coefficient is zero, it can also be revealed in the bottom figure of At the frequency around ω " 0.7, the transmission coefficient starts to increase, and from Fig. 5.3(b) we can find that the mode appears again at that frequency. To see the effect of the liner length on the transmission behavior at low frequencies, we also plot the transmission coefficient as a function of the ratio of the length L to the wavelength in x direction, the results are shown in Fig. 5.5(b). We demonstrate that when the length L equals to the integer multiple of the half wavelength, the transmission coefficient achieves to maximum; When the length equals to the integer multiple of the quarter wavelength, the coefficient achieves to minimum. From Fig. 5.3(b), around the resonant frequency ω R " 0.471, the wavenumber k in x direction tends to be infinite, resulting in the wavelength λ x " 2π k tends to be zero, since λ x " c fx , the oscillate frequency f x tends infinite, as shown in Fig. 5.5(a). From the pressure filed distribution in the waveguide (see Fig. 5.6), we can find that the pressure field is concentrated on the liner, only surface wave is found, provided that the frequency is low enough that only plane wave exists in the rigid parts. Because of the rigid-lined interface, not all the sound can propagate through the lined part, some sound waves is reflected.

Thus in a lined duct with a purely reacting impedance, the sound propagates slower than in a rigid duct. It can be imagined that, if the sound is slow enough, a counter flow can stop the sound even if the Mach number is smaller than 1. That is what we are going to investigate in the next sections.

Propagation in the lined ducts with uniform flow

Basic equations

The problem considered in the present section is the sound propagation in an acoustic lined rectangular duct with the presence of fluid flow. To reduce the complexity of the problem, the geometry will be confined to two dimensions, and the mean shear flow will be in the x-direction and assumed to be only a function of y, as shown in Fig. 5.7. The upper wall py " Hq is rigid. The lower wall is treated with a purely locally liner for 0 ă x ă L and rigid elsewhere. Neglecting the viscosity, considering the isotropic fluid, with D Dt " ˆB Bt `M B Bx ˙.

(5.26)

If the pressure is written in the form of ppx, yq " F e ´jωt`jkx , then we can obtain the ordinary differential equation

d 2 F dy 2 `2kM 1 ω ´M k dF dy `"pω ´M kq 2 ´k2 ‰ F " 0, (5.27)
where prime " 1 " refers to the first derivative with respect to y. It is the well known Pridmore-Brown equation [10].

Concerning the boundary condition, for the lined wall in the presence of flow, we have to be careful, the effect of the boundary layer has to be considered. In the present work, the classical Ingard-Myers [5]

Uniform flow

With uniform flow, M is independent of y, then the governing equation Eq.(5.24) becomes more simple, the right term of the equal vanishes. The pressure can be projected on the complete basis of functions φ n pyq which are transverse modes in the rigid duct. Thus, ppx, yq can be written: ppx, yq " 8 ÿ n"0 p n pxqφ n pyq " Φ T P , where P and Φ are column vectors, with P the mode coefficients, and Φ the basis functions φ n pyq, which are given by Eq. (5.9).

Applying the same procedures as we did before (see Eqs. (5.11)(5.12)): substituting the pressure expression into the governing equation, Eq. (5.24), multiplying the resulting equation by Φ, integrating over the y axis, and using the boundary conditions, Eqs. (5.4)(5.28), we have We define Q " d P dx , then Eq. (5.30) can be written in a matrix form of

D 2 Dt 2 ˆ P `Y pxq ´jω N 1 P ˙´B 2 P Bx 2 `N2 P " 0, ( 5
d dx ˜ P Q ¸" ˆ0 I M ´1 1 M 3 M ´1 1 M 2 ˙˜ P Q ¸, (5.31)
where 0 and I are the zero and identity matrices. By computing the eigenvalues λ n and eigenvectors X n of the matrix in Eq. (5.31), the wave numbers k of x-direction in the liner can be obtained by k " ´jλ m . Those wave numbers k and the eigenvectors are split into two sets: k `and X `when ℑmpkq ą 0; k ánd X ´when ℑmpkq ă 0. The propagative waves are of interest. The wave numbers with real values are only considered. They are divided into two sets: k `and k ´, the signs "`" and "´" indicate the propagation direction. The wavenumbers of the propagative modes as a function of ω are given in Fig. 5.8. It can be seen that the two positive real parts of Mode 1 and Mode 3 merge at the resonant frequency of the liner pω " ω R q. When they merge, the wave number is equal to ω{M . The two negative solutions, merge at a frequency ω M which is lower than ω R .

The Mode 1 and the Mode 2 are close to the 2 propagative waves without flow(solid lines). At low frequencies, their wavenumbers can be found by k{ω " ˘?1 `b{p1 ?1 `bM q which converges to the no-flow solution when M Ñ 0. Thus they can be seen as "acoustical modes". The other two modes (3 and 4) exist only with flow. The Mode 4 is a mode with a negative phase velocity but with a positive group velocity (c g " dω{dk is linked to the slope of the curves in Fig. 5.8). Thus this mode propagates in the positive direction (direction of the flow). At low frequencies, the phase and the group velocities of the mode 3 are positive and this mode also propagates in the positive direction. Aurégan and Pagneux [11] demonstrate that in this frequency range, this mode is a Negative Energy Wave.

It can be noticed that when M ą 1{ ? 1 `b, the Mode 2 and 4 do not exist and in this case, only two waves going in the positive direction exist at low frequencies. It means that no wave can propagate against the flow even at very low frequencies.

The singular change between the case of a rigid duct (2 propagative waves) and the lined part of a duct (4 propagative waves) for low Mach number and low frequencies, can lead to mathematical problem when we try to compute the scattering of a finite length liner connected to two rigid ducts in the uniform flow case. To avoid these potential problems, we will study in the next section a shear flow with a vanishing velocity at the lined wall to avoid the use of the Ingard-Myers condition which is suspected to be responsible of this singular behavior.

Propagation in the lined ducts with shear flow

With the presence of shear flow, the governing equations are Eqs. (5.24)(5.25). For the sake of simplification, the flow profile we considered here is symmetrical about the central line of the duct, it can be expressed as

M pyq " M 0 2n `1 2n `1 ´y2n ˘, (5.32)
where the parameter n is used to regulate the gradient of the flow, the bigger n is, the closer the shear flow is to the uniform flow. The original point of the coordinate is located at the central line of the duct. All the length quantities in the geometry, see Fig. 5.7, are all normalized by half of the channel width, i.e. H{2, such that the dimentionlessly analytical width interval of the duct is r´1, 1s. An example of the flow profiles is given in Fig. 5.9 under different values of n with M 0 " 0.2.

The flow velocities on the walls are zero, resulting in the boundary condition on the " ´jωY p| y"´1 .

(5.33)

Furthermore, harmonic waves in the x-direction are assumed, giving the following complex forms, p " P pyqe jp´ωt`kxq , v " V pyqe jp´ωt`kxq , q " Qpyqe jp´ωt`kxq , (5.34) with q " ´j Bp Bx , (5.35) where k is the dimensionless axial wavenumber.

The spectral method [12,13] is proposed to solve the problem. Therefore, discretization in the y-coordinate is employed by the Chebyshev collocation points, the MATLAB program chebdif.m [13] generates these points for the interval r´1, 1s. The interpolation points on this interval are y i " cos ˆpi ´1qπ N ´1 ˙, i " 1, 2, ....N, (5.36) where N is the number of the discrete points in y-direction. We can note that these points in this interval are not evenly spaced but are clustered at the ends of the interval, which provides more detailed information near the wall.

Chebyshev points used to discretize an unknown function f pyq interpolated at N 5.4. Propagation in the lined ducts with shear flow 121 nodes f py k q can be approximated by using interpolant polynomials φ k pyq:

f pxq « N `1 ÿ k"1
f px k qφ k pyq.

(5.37) Thus, taking lth derivative of f pyq and evaluating the result at the nodes y i , results in

f py i q plq « N ÿ k"1
d l dy l pφ k py i qqf py k q.

(5.38)

The derivative operator may be represented by a matrix D plq , the differentiation matrix, with entries

D plq i,k "
d l dy l pφ k py i qq.

(5.39)

Finally, the numerical differentiation can be written as the matrix-vector product f plq " D plq f.

(5.40)

Whereas differentiation matrices provided by Weideman and Reddy [13] are used to approximate the differential operator. Then Eqs. (5.41) where I is the identity matrix. D 1 and D 2 are differential matrices giving the first respectively, second-order differential operator with respect to y, which can be obtained by Chebyshev polynomials interpolation [13]. Q, V , P are the column vectors whose elements are Qpy i q, V py i q, and P py i q, respectively.

In order to construct the eigenvalue problem, the boundary conditions are introduced in the matrix (5.41). The 1th, pN `1qth, p2N `1qth rows of Eq. (5.41) indicate the rigid wall, while N th, p2N qth, p3N qth rows indicate the lined wall. The 1th and N th rows are used to introduce rigid wall condition and lined wall condition, respectively.

Modes in the rigid duct with shear flow

For the rigid wall with Y " 0, the boundary conditions is D 1 P| y"1,´1 " 0.

(5.42)

By the combinations of the boundary condition and the matrix Eq. (5.41), we can obtain eigenvalues k and eigenvectors Q, V , P in which each column corresponds to each k. In total, 3 types of modes are found, which can generally be divided in acoustic modes propagating (or decaying) in the `x-direction, acoustic modes propagating (or decaying) in the ´x-direction, and hydrodynamic modes in the direction of the mean flow. The effects of Mach number on the modes are shown in Figs. 5.10. It can be seen that the hydrodynamic modes depend significantly on the Mach number. From Eq. (5.27), the hydrodynamic modes verifies ω ´kM " 0. As M P r0, M max s, it means that k P rω{M max , 8s and that the hydrodynamic modes form a continuum of mode that is discretized by taking a finite number of point N in the section.

Modes in the lined duct with shear flow

To study the modes in the lined duct with shear flow, the boundary conditions are applied to calculate the wave numbers in the lined duct. In vectorial form, the boundary conditions can be expressed as

D 1 P ˇˇy "1 " 0,
and pD 1 `iωY Iq P ˇˇy "´1 " 0, (5.43) where the admittance of the liner here is Y " ´j tanpωbq.

Performing a similar procedures as done for rigid duct with shear flow, the eigenvalues in the lined duct are shown below (Fig. 5.11). From the comparisons of Fig. 5.11 with Fig. 5.10, the first propagative modes in `x and ´x-direction dependent significantly on the Mach number, this is because there exists the coupling between the flow and the liner.

The effect of the boundary layer thickness

The Mode 1 for which 0 ă k ă ω{M 0 is practically unchanged by the presence and the shape of the boundary layer. A big difference between the uniform and shear flow cases is the presence of a continuum of hydrodynamic modes. k " ω{M is a singular solution of Eq. (5.27) for all M between 0 and M 0 . Thus all k ą ω{M 0 are solutions of the hydrodynamic modes, and Mode 3 is now embedded in the continuum. The black "˛" indicates n " 10, the blue "˝" n " 30, and the red "˚" n " 100.

For comparison, the results with uniform flow are given by "¨" The flow profiles for the three corresponding n are shown in Fig. 5.9.

Chapter 6

Conclusions

In this thesis, two different strategies are presented to enhance the sound attenuation in acoustic ducts by using the mode coupling effects. Trapped mode and transmission loss peak are observed near the avoided crossing of eigenvalues between two neighboured (Bloch) modes. We have shown that this avoided crossing is closely related to the Exceptional Point in the parameter plane.

First, the wave propagation problems in an acoustic duct-cavity system and a waveguide with a portion of its wall lined by a locally reacting material is studied by the Rmatrix method. We have shown that the acoustic scattering matrix can be efficiently described with the help of an effective matrix H eff , whose eigenvalues give the poles of the scattering matrix. The real poles are in the vicinity of the exceptional points of matrix H eff . Using this effective matrix, the original acoustic resonances problem in the infinite waveguide is reduced to an equivalent eigenvalue problem of matrix H eff , which describes the eigenvalue problem defined in the scattering region. When a plane mode is incident, a transmission zero is present when the real resonance frequency is equal to the incident frequency. This transmission zero occurs -due to the interference of the incoming propagative mode with the trapped mode, which is also called the Fano resonance. This is the idea of the first strategy. The trapped mode is formed by the interferences of two neighbored modes with complex resonance frequencies. With the aid of the eigenvalues and eigenfunctions of matrix H eff , the traditional acoustic resonance scattering formula is extended to include the coupling effects between the scattering region and the rigid parts of the waveguide.

Second, the mode coupling due to the embedment of the rigid inclusions in the porous material is used to enhance the sound attenuation in an acoustic lined duct at low frequencies. This strategy is validated numerically and experimentally in a 3D waveguide lined with periodic inclusions embedded in porous material. Two different inclusion shapes are considered, an open cylinder and a Helmholtz resonator. When rigid inclusions are embedded in the porous material, the low frequency behaviours of the transmission loss can be significantly changed, peaks are observed in the mid-frequency 133 Chapter 6. Conclusions range. The effects of the inclusion position on the frequency of the TL peak are different for different inclusion shapes. Good agreements are observed between the experimental and numerical results, which opens the way to the optimisation of metaporous material for better sound attenuation at low frequencies.

An analysis of a 2D infinite periodic waveguide with inclusions embedded in porous material is performed by using Floquet-Bloch theorem. A maximum sound attenuation for this infinite periodic waveguide can be reached near the crossing (or avoided crossing) of the mode attenuations between two lower Bloch waves, and it is related the exceptional point. A coupling between acoustic mode and localized mode is happened near this crossing.

Finally, the acoustical behaviors of a purely reacting liner in a rectangular duct in both absence and presence of flow are investigated. Multi-Model Method is proposed to solve the problem in the absence and presence of uniform flow. The results exhibit an unusual acoustical behavior: for a certain range of frequencies, no wave can propagate against the flow. The effect of shear flow is investigated by the Chebyshev Spectral Method, which provides detailed information near the walls. A negative group velocity is found in a certain range of frequencies, and it is demonstrated that the sound can be slowed down and even stopped.

THÈSE DE DOCTORAT

Lei XIONG

Use of mode coupling to enhance sound attenuation in acoustic ducts

Résumé

Deux stratégies sont présentées à utiliser des effets de couplage de modes pour l'amplification de l'atténuation du son dans les conduits acoustiques. La première est de coupler le mode incident propagatif avec un mode localisé, aussi appelé résonance de Fano. Cette stratégie est présentée et validée dans un système conduit-cavité et un guide d'onde partiellement traité en paroi avec un matériau à réaction locale. La méthode "R-matrix" est introduite pour résoudre le problème de propagation d'onde. Une annulation de la transmission se produit quand un mode piégé (qui est formé par les interférences de deux modes voisins) est excité dans le système ouvert. Ce phénomène est aussi lié au croisement évité des valeurs propres et à un point exceptionnel. Dans la seconde stratégie, un réseau d'inclusions rigides périodiques est intégré dans une couche poreuse pour améliorer la'tténuation du son à basse fréquence. Le couplage de modes est du à la présence de ces inclusions. Le théorème de Floquet-Bloch est proposé pour analyser l'atténuation du son dans un guide d'onde périodique en 2D. Un croisement de l'atténuation de deux ondes de Bloch est observé. Ce phénomène est utilisé pour expliquer le pic de pertes en transmission observé expérimentalement et numériquement dans un guide 3D partiellement traitée par un matériau poreux avec des inclusions périodiques. Enfin, le comportement acoustique d'un liner purement réactif dans un conduit rectangulaire avec et sans écoulement est étudié. Dans une certaine gamme de fréquence, aucune onde ne peut se propager à contre sens de l'écoulement. Par analyse des différent modes à l'aide de la relation de dispersion, il est démontré que le son peut être ralenti et même arrêté.

Abstract

Two strategies are presented to use the mode coupling effects to enhance sound attenuation in acoustic ducts. The strategy is to couple the incoming propagative mode with the localized mode, which is also called Fano resonance. This strategy is presented and validated in a duct-cavity system and a waveguide partially lined with a locally reacting material. The R-matrix method is introduced to solve the propagation problems. A zero in the transmission is present, due to the excitation of a trapped mode which is formed by the interferences of two neighboured modes. It is also linked to the avoided crossing of the eigenvalues and exceptional point. In the second strategy, a set of periodic rigid inclusions are embedded in a porous lining to enhance sound attenuation at low frequencies. The mode coupling is due to the presence of the embedded inclusions. Floquet -Bloch theorem is proposed to investigate the attenuation in a 2D periodic waveguide. Crossing is observed between the mode attenuations of two Bloch waves. The most important and interesting figure is that near the frequency where the crossing appears, an attenuation peak is observed. This phenomenon can be used to explain the transmission loss peak observed numerically and experimentally in a 3D waveguide partially lined by a porous material embedded with periodic inclusions. Finally, the acoustical behaviours of a purely reacting liner in a duct in absence and presence of flow are investigated. The results exhibit an unusual acoustical behaviour : for a certain range of frequencies, no wave can propagate against the flow. a negative group velocity is found, and it is demonstrated that the sound can be slowed down and even stopped.

Figure 1 . 1 :

 11 Figure 1.1: A 2D uniform rigid duct.

Figure 1 . 2 :

 12 Figure 1.2: A 2D open system, where the bottom wall is soft and the others are rigid.

Figure 1 . 3 :

 13 Figure 1.3: An example of trapped in a semi-infinite system (see Fig. 1.2), with Dirichlet condition on the bottom wall and rigid boundary conditions on the other walls.

2 Figure 1 . 4 :

 214 Figure 1.4: Schematic view of two coupled pendulums

  Figure 1.5: Trajectories of the eigenvalues in the complex plane with three different σ i , i.e., ´0.2pII and IVq, ´0.19pI and IIIq, and ℑmpσ ep q(crossing), respectively. Arrows indicate the increase with σ r . The other parameters are ω 1 " 1 and k{m " 0.1.

Figure 1 . 6 :

 16 Figure 1.6: Eigenvalues λ 1,2 as a function of σ r pP r´0.1, 0.1sq with (a)(b)(c)σ i " ´0.2 and (d)(e)(f)σ i " ´0.19. The other parameters are ω 1 " 1 and k{m " 0.1. Figures (c) and (f) give the trajectories of the eigenvalues in the complex plane.

(1. 15 ) 10 ChapterFigure 1 . 7 :

 151017 Figure 1.7: (a) Real and (b) imaginary part of wavenumbers k y of the first 2 modes(labeled by "Mode1" and "Mode2", respectively ) as a function of ℜepKY q under two different values of ℑmpKY q " ´1.5 ('*') and ´2 ('+') respectively. (c)Trajectories of the wavenumber in complex plan, arrow shows the direction of the movements with increasing the real part of KY .

Figure 1 . 8 :

 18 Figure 1.8: Eigenvalue surfaces of the first 2 in that complex KY plane: (a) real and (b) imaginary parts of k y . The EP is located in the complex KY plane, at where the two eigenvalues coalesce.

Figure 1 . 10 :

 110 Figure 1.10: The amplitudes of the (a) pendulum 1 and (b) pendulum 2 as a function of frequency, where the frequency is in units of the natural frequency ω 1 . The other parameters are ω 1 " 1, σ " ´0.1, a 1 " 0.1, and k{m " 0.1.

1 Figure 1 .

 11 Figure 1.11: A single defect introduced by a normalized admittance Y pxq in an acoustic waveguide. The length of the admittance is x 1 .

1. 5 .

 5 Fano resonance 15 Let the δ-function admittance be Y pxq " Y 0 δpxq. (1.27) Integrating Eq.(1.24) across the δ function gives

Figure 1 .

 1 Figure 1.12: (a) Transmission and reflection coefficients of the first mode as a function of frequency. (b)The wav functions c 1 pxq and c 2 pxq when |t 11 | " 0.

Chapter 2 .Figure 2 . 1 :

 221 Figure 2.1: A 2D acoustical duct-cavity system, the lengths are normalized by the duct height h ˚, i.e. a " a ˚{h ˚and b " b ˚{h ˚.

2 . 2 .

 22 Two parameters, K and a, are used to control the movements of the two K λ . Under different values of K, two different types of avoided crossings are observed (Figs. 2.2(c) and (f)), where

Figure 2 . 2 :

 22 Figure 2.2: Real (a)(d) and imaginary (b)(e) parts of the 3rd (solid line) and the 4th (dashed line) eigenvalues K λ as a function of length a for different values of K: (a)(b)(c) K " 1.5 and (d)(e)(f) K " 1.6, with b " 1, M " 20, N " 30 ˆ30. The trajectories of the eigenvalues in the complex plane are shown in (c)(f). The " " in (c)(f) indicates the starting point.

Figure 2 . 5 :

 25 Figure 2.5: Real (a)(d) and imaginary (b)(e) parts of the modes p0, 1q (solid line) and p2, 0q (dashed line) as a function of length a for different values of b: (a)(b)(c) b " 0.95 and (d)(e)(f) b " 1.05. The trajectories of the modes in the complex plane are shown in (c)(f). The " " in (c)(f) indicates the starting point. The two modes are in the vicinity of avoided crossing "A" in Fig.2.4.

Figure 2 . 6 :

 26 Figure 2.6: (a) Trajectories of resonance (0,1) (blue circle) as a function of a with b " 1. The resonance width vanishes at a " 3.93 (marked by " À "). The motions of K λ of H eff with K " 1.597 is shown by red square. (b) Transmission coefficients as a function of incident frequency K with a " 3.93, b " 1. (c) The absolute value of the pressure field in the waveguide when the transmission coefficient is zero. (d) The superposition coefficients of the eigenfunctions ψ µν px, yq for the trapped mode.

Figure 2 . 7 :Figure 2 . 8 :

 2728 Figure 2.7: Real and imaginary parts of K λ as a function of a with b " 2 for different K: (a)(b) K " 0.4 and (c)(d) K " 2.4. The resonance frequencies and widths of the system by solving the fixed-point equation are also shown in (c) and (d) by symbols "˝".

Figure 2 . 9 :

 29 Figure 2.9: Transmission coefficients as a function of frequency K for different cavity lengths (a) a " 5.2, (b) a " 5.4, (c) a " 5.6, with b " 2.

Chapter 3 .Figure 3 . 1 :

 331 Figure 3.1: A two-dimensional infinite waveguide lined with a locally reacting material of normalized length a. Y refers to the normalized admittance. The closed counterpart of the scattering region is called lined closed cavity, with rigid conditions at x " 0 and x " a.

3 . 1

 31 and by Eqs. (3.1), (3.2), and (3.

Figure 3 . 2 :

 32 Figure 3.2: Trajectories of the real parts of eigenvalues β µ,ν of the lined closed cavity as a function of |KY | with Re " 0 and a " 4. Solid lines label even modes in x-direction, dashed lines label odd modes in x-direction. The mode indices (µ, ν) are marked near each curve.

4 Figure 3 . 3 :

 433 Figure 3.3: Trajectories of ℜepK λ q and ℑmpK λ q of matrix H eff as a function of |KY | with K " 2.5, a " 4, and Re " 0: (a) real parts and (b) imaginary parts. Solid lines and dashed lines correspond to different symmetries as shown in Fig. 3.2. For comparison, we plot also the trajectories of ℜepβ µ,ν q of the closed cavity by dash-dot lines.

Figure 3 . 4 :

 34 Figure 3.4: Crossings and avoided crossings of K λ as a function |KY | with K " 2.5 under different values of a: (a), (b) and (c) for a " 3.8; (d), (e) and (f) for a " 3.6. They are the two modes in the vicinity of "A" in Figs. 3.2 and 3.3. EP occurs at |KY | " 4.18 and a " 3.7, labelled by "C 1 " in Fig. 3.5. At (b) |KY | " 4.5 and (e) |KY | " 4.8, ℑmpK λ ) goes to zero, K λ turns to be a real resonance frequency.

Figure 3 . 5 :

 35 Figure 3.5: The distribution of EPs of H eff in the (a, |KY |) plane with K " 2.5. The branches A i (i " 1, 2, 3, ¨¨¨) and C i (B i and D i ) correspond to the crossings between two modes in lined closed cavity with even (odd) symmetry about x. C 1 corresponds to EP p|KY | cri " 4.18, a cri " 3.7q when K " 2.5. It also corresponds to "A" in Figs. 3.2 and 3.3.

Figure 3 . 6 :

 36 Figure 3.6: Re-plot Fig. 3.4 using d l , the liner depth, as x-coordinate.

Figure 3 . 7 :

 37 Figure 3.7: Trajectories of eigenvalues K λ of matrix H eff as a function of d l with K " 2.85, a " 3.8, and Re " 0: (a) real parts and (b) imaginary parts.

Figure 3 . 8 :

 38 Figure 3.8: (a) Transmission and reflection coefficients as a function of K with d l " 0.35, a " 3.8, and Re " 0. (b)The absolute value of sound pressure field in the waveguide when real resonance frequency is crossed, i.e., |T | " 0 . Now we consider the effects of dissipation. A small resistance Re is added in the impedance model Eq. (3.23). In Fig. 3.9, we plot the reflection and transmission coefficients as a function of K with d l " 0.35 and a " 3.8 under different values of Re. The peaks of the reflection and transmission coefficients decrease rapidly with increasing Re.

Figure 3 . 9 :

 39 Figure 3.9: (a) Transmission and (b) reflection coefficients of the plane mode as a function of frequency under different values of resistance Re with d l " 0.35.

  )

1 .

 1 Derivation of Eq. (3.7) Multiplying Eq. (3.1) by ψ, integrating over the closed cavity, we obtain s theorem for Eq. (3.24), substituting Eq. (3.5) into the resulting equation

Figure 4 . 1 :

 41 Figure 4.1: Schematic view of the experimental setup. 1: material, 2a: four upstream microphones, 2b: four downstream microphones, 2c: array of 11 microphones, 3a: upstream source, 3b: downstream source. It need to be noted that the test material is put in a big rigid cavity.

Figure 4 . 2 :

 42 Figure 4.2: (a) Two different inclusion shapes: open cylinder(R1) and Helmholtz resonator(R2). (b) Picture of the measured sample.

Figure 4 . 3 :

 43 Figure 4.3: Vertical plane views and the definitions of the different configurations we considered. The inclusions R1 are embedded in 3 layers of porous material with three different arrangements (labeled by "R1_B", "R1_A", and "R1_T", respectively). One or two layers of porous materials are also assembled in different ways to consider the effects. A picture of P+R1_A+P is shown in Fig. 4.2(b).

Figure 4 . 4 :

 44 Figure 4.4: Distributions of the inclusions in the porous material. A horizontal plane of the porous material with inclusions.

Figure 4 . 5 :

 45 Figure 4.5: (a)Three different orientations of R2 in the porous materials(horizontal view). (b) The other two layers of porous material are assembled in three different ways.

Figure 4 . 6 :

 46 Figure 4.6: Periodic [ -shape inclusions embedded in a porous material, with h a " 15mm and h p " 25mm in the following. L 0 " 24mm is the length of the unit cell. D " 22mm and h 0 =15 are the width and height of the rigid inclusion.

Figure 4 . 7 :

 47 Figure 4.7: (a)Real and (b) imaginary parts of k B as a function of frequency when there is no inclusions in Fig. 4.6.

Figure 4 . 8 :

 48 Figure 4.8: (a) Transmission and reflection coefficients and (b) transmission loss (TL) for the plane wave in a rigid duct partially treated with porous material (length L=0.2m) as a function of frequency. The experimental measurements are also performed and compared with the computation results. The TL of the least attenuated mode with length 0.2m is also given by the dashed line.

Figure 4 . 9 :

 49 Figure 4.9: (a)Band diagram for the waveguide with inclusions embedded in the air, i.e. when there is no porous material in Fig. 4.6. (b) and (c) Absolute value of periodic function ppx, yq for the lower two Bloch waves (corresponding to "1" and "2" in Fig.4.9(a)) at 2000Hz, the unit of the axis is in millimetre. (d)Real and (e)imaginary parts of the lower two k B (`x-direction) as a function of frequency.

Figure 4 .

 4 Figure 4.10: (a)Band diagram for the waveguide with inclusions embedded in the porous material. (b)Real (upper) and imaginary (bottom) parts of k B as a function of frequency for the waveguide shown in Fig. 4.6. The periodic wave functions of the lower two modes at the frequencies 700Hz, 880Hz, 1000Hz, and 2000Hz ("˝" in Fig. 4.10(b)(upper)) are plotted in Fig. 4.11.

Figure 4 . 11 :

 411 Figure 4.11: Absolute value of periodic function ppx, yq for the mode 1 (upper) and mode 2 (bottom) ( modes 1 and 2 from Fig.4.10) at frequencies 700Hz, 880Hz, 1000Hz, and 2000Hz, respectively.

Figure 4 .

 4 Figure 4.13: (a)The effect of the inclusion position in the porous material on the TL of the least attenuated Bloch wave with length L " 0.2m. "0 mm" indicates that the inclusions are embedded in the centre, as in Fig.4.6. "1mm" or "´1mm" means that the inclusions are moved of 1mm above or below the centre position. (b)A zoom of Fig.4.13(a) for the low frequencies.

Figure 4 . 14 :

 414 Figure 4.14: Real and imaginary parts of k B as a function of frequency with different inclusion positions in the porous: the inclusions are (a)(b)in the centre( 0 mm), (c)(d)below the centre (´4 mm), and (e)(f)below the centre (´4.9 mm).

Figure 4 . 15 :

 415 Figure 4.15: Transmission loss (dB) for the 5 configurations when the plane mode is incident.

Figure 4 . 16 :

 416 Figure 4.16: Pressure field distribution in the duct for P+R1_A+P at 2400Hz (scale in dB SPL), a plane mode is incident with an intensity of 100 dB.

Figure 4 . 17 :

 417 Figure 4.17: Pressure field distribution in the duct for P+R2_A+P at (a)1700Hz and (b)3400Hz (scale in dB SPL), a plane mode is incident with an intensity of 100 dB.

4. 4 . 2

 42 Experimental results for the configurations with R1

Figure 4 .

 4 Figure 4.18: (a)Transmission loss, absolute value of (b) T `, (c) R `and (d) absorption coefficients when the closed send of all the R1 are on the bottom(R1_B) near the rigid backing, and also with the consideration of porous materials layers.

Fig.

  Fig.4.18 shows the experimental results when the closed ends of all the inclusions R1 are on the bottom (R1_B) near the rigid backing, as illustrated in the inset of Fig.4.18(a). When there are only three layers of porous material embedded with inclusions (R1_B), due to the reduction of the porous materials, its TL is the smallest at all the frequency range of interest comparing to the other configurations. For f ą 2400Hz, more porous material layers are near the waveguide, higher transmission loss (TL) is observed, we have the following relation TL(R1_B+2P)ăTL(P+R1_B+P)ăTL(2P+R1_-

  for the different configurations are also considered, as shown in Fig.4.18(d). Biggest absorption is observed for R1_B+2P for frequencies 1500Hză f ă 3000Hz, it is due to the smallest reflection coefficient, see Fig.4.18(c), we have the opposite relation for absorption coefficient Ab(R1_B+2P)ąAb(P+R1_B+P)ąAb(2P+R1_-B), comparing to the TL.

Figure 4 .

 4 Figure 4.19 gives the results for the configurations in which the closed ends of all the inclusions are on the top (R1_T) near the air filled duct. It is remarkable that there is transmission loss peaks around f " 2000Hz for configurations R1_T, R1_T+2P and P+R1_T+P. It means that the periodic waveguide with inclusions (R1_T) interferes in the porous lined duct at low frequencies, and results in the TL peak. From the TLs of the Bloch waves in a periodic waveguide with configuration P+R1_T+P (see the left on

Figure 4 .

 4 Figure 4.19: (a)Transmission loss, absolute value of (b) T `, (c) R `and (d) absorption coefficients when the inclusions are all closed on the top and open on the bottom(R1_T), and also with the consideration of different porous materials layers.

Figure 4 . 20 :

 420 Figure 4.20: Measured transmission loss (solid line) for configurations configurations (a) P+R1_B+P and (b) P+R1_T+P. " " labels the TLs [26] of the lower two Bloch waves in a periodic waveguide(length L " 0.2m) lined with the two 3D configurations. The ACTIPOLE results are also given by "˛" in figure (b). (c) The 3D meshes for the two 3D periodic cell [26].

Figure 4 .Fig. 4 .

 44 Figure 4.21: (a)The effects of the positions of the inclusions embedded in the porous material (ACTIPOLE results) on the TL. "0 mm" indicates that the inclusions are embedded in the centre, corresponding to P+R1_T+P. "1mm" or "-1mm" means that the inclusions are moved of 1mm above or below the centre position. (b) The pressure field distribution for three different positions of the inclusions at the frequencies of TL peaks.

Figure 4 .

 4 Figure 4.22: (a)Transmission loss, absolute value of (b) T `, (c) R `, and (d) absorption coefficients when the inclusions are alternated, arrangement (R1_A), and also with the consideration of porous materials layers.

Figs. 4 .

 4 23, 4.24, and 4.25. For all the three figures, two transmission loss peaks are ob-

Figure 4 .

 4 Figure 4.23: (a)Transmission loss, absolute value of (b) T `and T ´, (c) R `and R ´, and (d) absorption coefficients when the resonators 2 are embedded as R2_A in Fig.4.5(a), and also with different porous material layer arrangements.

Figure 4 . 24 :

 424 Figure 4.24: The results for the configurations when the resonators 2 are embedded as R2_0 ˝in Fig.4.5(a), and also with different porous material layer arrangements.

  .23(a) and (b) are observed in Fig. 4.24(a)(b) and Fig. 4.25(a)(b). However, for R2_0 ˝, due to the different distribution of the inclusions at the two interfaces x " 0 and x " L, the reflection coefficients from upstream and downstream are not the same, which also makes the absorption coefficients different, see Figs. 4.24(c)(d).

Figure 4 . 25 :Figure 4 . 26 :

 425426 Figure 4.25: The results for the configurations when the resonators 2 are embedded as R2_90 ˝in Fig.4.5(a), and also with different porous material layer arrangements.

Figure 4 . 27 :

 427 Figure 4.27: Comparisons between Actipole computation and the measurement results for P+R1_A+P (a) transmission loss, and (b) absorption coefficients.

Figure 4 . 28 :

 428 Figure 4.28: Comparisons between Actipole computation and the measurement results for P+R2_A+P (a) transmission loss, and (b) absorption coefficients.

2 Figure 4 . 29 :

 2429 Figure 4.29: Geometry of one elementary cell.

. 27 )

 27 With the addition of two uniform duct of finite length l 2 on both sides, the scattering matrix S of a elementary cell in Figs. 4.6 and 4.29 is obtained byS " S l 2 b S D b S l 2 .(4.28)

5 . 1 .Figure 5 . 1 :

 5151 Figure 5.1: Scheme of the infinite reacting lined duct. Z is the impedance of the liner with B the thickness.

4 )

 4 For the lined wall, y " 0, the admittance boundary condition used, where Y is the normalized admittance of the lined wall. It can be obtained by Y pωq " 1{Zpωq. Since purely reacting liner is considered here, we can derive the expression of the impedance analytically (see Appendix 5.A). With the consideration of the dissipation, the impedance of the liner in Fig.5.1

. 14 )

 14 In Fig.5.2, we plot the motions of the eigenvalues β as a function of frequency ω with b " B{H " 50{15 m. The dashed lines in Fig.5.2(a) label the eigenvalues of transverse modes in rigid duct, i.e. α n " nπ. Due to the purely imaginary value of the admittance Y , the eigenvalues β are purely real or imaginary values. The eigenvalue of the plane

Figure 5 . 2 :

 52 Figure 5.2: Trajectories of the (a) real parts and (b) imaginary parts of β as a function of frequency ω with b " B{H " 50{15 m.

5. 2 .-Figure 5 . 3 :

 253 Figure 5.3: (a) The eigenfunction profiles of the first 4 transverse modes in the lined duct with ω " 0.3 and b " B{H " 50{15 m, theirs eigenvalues are labelled by 'ˆ' in Fig. 5.2. (b) The real(upper) and imaginary(bottom) parts of the eigenvalue for the lowest mode in x-direction as a function of frequency ω.

Figure 5 . 4 :

 54 Figure 5.4: The configuration for constructing the scattering matrix, L is the length of the liner, normalized by duct width H, A i , B i pi " 1, 2, 3q are all vectors, indicating the pressure amplitude of each mode in `x-direction and ´x-direction, respectively. The total pressure consists of the `x and ´x-direction sound pressures.

Figure 5 . 5 :

 55 Figure 5.5: (a)Transmission coefficient of the plane mode as a function of (a) ω and (b) the ratio of the liner length L to the wavelength with B " 50mm, H " 15mm, L " 200mm and R " 0. Figure (b) is corresponding to the range 0 ă ω ă 0.4 in figure (a).

Fig. 5 .

 5 3(b).

Figure 5 . 6 :

 56 Figure 5.6: Pressure field distribution in the waveguide at ω " 0.412 with B " 50mm, H " 15mm, and L " 200mm

1 Figure 5 . 8 :

 158 Figure 5.8: Variation of the propagative wave numbers k as a function of the Helmholtz number ω for b " 10{3 and M 0 " 0.2.

Figure 5 . 9 :

 59 Figure 5.9: The profiles of the shear flow under different values of n in Eq. (5.32), with M 0 " 0.2.lined wallBp By ˇˇˇy

  (5.24)(5.25)(5.35) can be written in the form of matrix k ¨I ´M2 pyq ´2jM

Figure 5 . 10 :

 510 Figure 5.10: Axial wave numbers distributions in the complex k plane for different mach number M 0 with ω " 0.24 and n " 30. The duct is rigid.

5. 4 .Figure 5 . 11 :

 4511 Figure 5.11: Axial wave numbers distributions in the complex k{ω plane for different mach number M 0 with ω " 0.24, n " 30. The duct is lined.

Figure 5 . 12 :

 512 Figure 5.12: Wave number as a function of ω for 3 different values of n with b " 10{3 and M 0 " 2. The black "˛" indicates n " 10, the blue "˝" n " 30, and the red "˚" n " 100. For comparison, the results with uniform flow are given by "¨" The flow profiles for the three corresponding n are shown in Fig.5.9.

  1. General introduction the ambient sound speed. The admittance boundary condition can be written as

			Bppx, yq By	ˇˇˇy	"0		" ´jKY pxqppx, 0q,	(1.20)
	with time dependence e ´jωt used.
	We will expand the solutions of the full scattering problem (Eq.(1.19)) in terms of
	the rigid transverse eigenfunctions φ n pyq (see Eq. (1.3))
									8
				ppx, yq "	ÿ n"0	c n pxqφ n pyq.	(1.21)
	B 2 c m pxq Bx 2	`Bp By	φ m pyq ˇˇˇ1 0	´p Bφ m By	ˇˇˇ1 0	0 `ż h	p	B 2 φ

Multiplying Eq.

(1.19)

by φ m pyq, integrating over y, and substituting Eq.(1.21) into the resulting equation, we can have m pyq By 2 dy `K2 c m pxq " 0. (1.22) Applying the boundary conditions Eq.(1.20) and the Eq.(1.3) into the above equation Eq.(1.22), we obtain

  Multiplying Eq. (3.1) by ψ µν px, yq, integrating over the closed cavity, we obtain

	2.A.1 Derivation of Eq. (2.9)
		ż a 0	ż h 0	ψ µν	ˆB2 p Bx 2	`B2 p By 2 ˙dxdy " ´K2	0 ż a	0 ż h	ψ µν pdxdy,	(2.19)
	Applying an integration by parts for the left side of Eq. (3.24), substituting Eq. (3.5) into
	the resulting equation, we obtain
	ż h 0	ˆψµν	Bp Bx	´Bψ µν Bx ż ż	p	˙ˇˇˇx "a x"0	dy	`ż a 0	ˆψµν	Bp By	´Bψ µν By	p	˙ˇˇˇy "h y"0	dx
	" ´pK 2 ´γ2 µν q	ψ µν pdxdy.

  )

	Chapter 4
	Bϕ µν Bx	ˇˇx	"0,a	" 0,	Bϕ µν By	ˇˇy	"0	" jKY ϕ µν , and	Bϕ µν By	ˇˇy	"1	" 0.	(3.31)

Table 4 .

 4

		1: Definition of the configurations studied with Actipole
	CONF	Inclusion	Porous material	Objective
	P+R1_A+P P+R2_A+P Air+R1_A+Air	R1 R2 R1	ˆEffect of R1 ˆEffect of R2 The sole effect of R1 (without porous)
	Air+R2_A+Air	R2		The sole effect of R2 (without porous)
	5P	No inclusions	ˆReference with only porous material
	4.3 2D simple model analysis
	4.3.1 Problem statement

  then the eigenfunction in y-direction is P " rP N 2 , P N 2 ´1, . . . , P Np`2 , P Np`1 loooooooooooooooooomoooooooooooooooooon Np , P Np´1 , . . . , P 2 , P 1 looooooooooooomooooooooooooonFor the rigid walls, we can have P 0 " P 1 and P N 2 `1 " P N 2 , then

										s T .
								Np
	The 2nd-order derivatives are approximated by
		ˆB2 P By 2 ˙i "	P i`1 ´2P i `Pi´1 ∆h 2	.		(4.19)
	ˆB2 P By 2 ˙1 "	P 2 ´P1 ∆h 2 and	ˆB2 P By 2	˙N2	"	´PN 2 `PN 2 ∆h 2	´1	.	(4.20)

Na

, P

  [6] [8] boundary condition

	Bp By	ˇˇˇy	"0	"	D 2 Dt 2 ˆY ´jω	p	˙ˇˇˇy	"0	(5.28)
	is used (see Appendix 5.C)

  .29) where N1, N2 are shown as before in Sec. 5.2.1. Equation (5.29) can be written as "I´M 2 0 ´I `Y ´jω N 1 ¯, M 2 " ´2jωM 0 ´I `Y ´jω N 1 ¯, M 3 "

	M 1	d 2 P dx 2 " M 2	d P dx	`M3 P ,
	the above equation results in
	d 2 P dx 2 " M ´1 1 M 2	d P dx	`M´1 1 M 3 P ,	(5.30)
	where M 1 " N 2	ı ´ω2 pI `Y ´jω N 1 q	.

This chapter in its current form has been recently accepted by J. Acous. Soc. Am..

Part of this work is presented and published in 19th AIAA/CEAS Aeroacoustics Conference (AIAA 2013-2077): Y. Aurégan, L. Xiong, and W. Bi, "Acoustical behaviour of purely reacting liner".

ACKNOWLEDGEMENTS

I would like to thank my supervisors Yves Aurégan and Wenping Bi for the continuous support of my Ph.D study and related research. Their patience, enthusiasm, and

Appendix

4.A Equivalent fluid model

The porous material we used here is a metallic foam. Rigid frame porous material is modelled using the Johnson-Champoux-Allard model. The dynamic compressibilty κ eq and dynamic density ρ eq are [27,28] ρ eq " ρ a α 8 Φ

Here, Φ is the porosity, α 8 is the Johnson Koplik & Dashen tortsity factor [29], k 0 is Darcy's viscous permeability, Λ is the Johnson Koplik & Schwartz pore size parameter [29], k 1 0 (which can be approximated by k 1 0 " ΦΛ 1 2 {8 ) is the Lafarge et. al. thermal permeability [27] and Λ 1 is the Champoux & Allard pore size parameter [30]. Thermodynamic properties of the saturating air are given by: ρ a the ambient density, γ is the air specific heat ratio, P 0 is the atmospheric pressure, Pr is the Prandtl number. The relation between the resistivity σ and the kinematic viscosity ν can be written as σ " ρ a ν{k 0 .

4.B Eigenvalue problem in a porous lined duct

For a uniform infinite waveguide filled by air (with height h a ) and porous materials (with height h p ), the eigenvalue problem can be solved by Finite Difference Method (FDM). The principle of the FDM is that: Derivatives in the partial differential equation are approximated by linear combinations of function values at the grid points. There are N a " h a {∆h and N p " h p {∆h discrete points in the air and porous mediums, respectively, we denote N 2 " N a `Np , where ∆h is the spacing between interior points, and the first and last points are taken ∆h{2 from the the duct walls. The pressure values at different discrete points are

Chapter 5

Acoustical behavior of purely reacting liners

This chapter 1 investigates the acoustical behaviors of a rectangular duct lined with a purely locally-reacting liner in both absence and presence of flow. Multi-Model Method is proposed to solve the problem in the absence and presence of uniform flow. The results exhibit an unusual acoustical behavior: for a certain range of frequencies, no wave can propagate against the flow. The effect of shear flow is investigated by using Chebyshev Spectral Method, which provides detailed information near the walls. A negative group velocity is found in a certain range of frequencies and a zero group velocity is found at two critical frequencies. It is demonstrated that the sound can be slowed down and even stopped.

Introduction

Nowadays, acoustical liner are widely used to reduce aircraft engine noise emission by damping the acoustic modes propagating within the inlet or the exhaust ducts. Typically, the lining is usually assumed to be locally-reacting liner. Commonly, the interaction of acoustics with the lining is characterized by the impedance of the liner, which then is used as the boundary condition to construct the eigenvalue problem. In the absence of flow, the problem is straightforward. The acoustic transmission coefficient in a rigid duct partially lined with the locally-reacting liner has a minimum value at the resonant frequency of the liner impedance (when the depth of the liner is p2n `1qλ{4 with n a integer). In presence of the flow, an instability can appear.

where b " B{H is the normalized thickness of the liner, and R is the normalized resistance. In the following, we assume R " 0, which leads to Y pωq " ´j tanpωbq.

(5.7)

Substituting the pressure expression, Eq. (5.3), into the boundary conditions, Eqs. (5.4) and (5.5), results in B 1 " 0 and the eigenvalue equation β tan β " ´ω ¨tanpωbq.

(5.8)

In the limit of low frequency (ω ! 1 and β ! 1), from Eq.(5.8), we can have β 2 " ´bω 2 and ψpyq " A coshpβyq, and then the wavenumber in x-direction is approximated by k{ω " ˘p1 `bq 1{2 . The reduced phase velocity c φ " ω{k is then smaller than 1, which demonstrates that the phase velocity c φ is smaller than the sound velocity c 0 . The propagative waves are kind of waves which propagate in x-direction along with a exponentially decrease away from the lined wall.

Multimodal Method

In order to calculate the eigenvalues and eigenfunctions of the transverse modes in the lined duct more quickly and efficiently, in the present paper, the Multi-Model Method [9] is proposed to solve the dispersion relation and to obtain the transmission coefficient. For a rigid rectangular duct, the eigenvalues and the eigenfunctions are well known, namely,

where α n " nπ, n " 0, 1, 2, 3 ¨¨¨.

The eigenfunction ψpyq in the lined duct is projected on the complete basis of functions φ n pyq by

where A and Φ are column vectors, their elements are the coefficients A n of the basis functions and the basis function φ n pyq, respectively. N is the truncation number. The superscript " T " indicates the transpose.

Multiplying by Φpyq on both sides of Eq.(5.2), and integrating over y coordinate, wa can have

where D Dt " B Bt ` v ¨ ∇, ρ˚is the mass density, p˚is the pressure, v˚is the velocity vector and c is the speed of sound. The acoustic equations are obtained by considering small perturbations on a mean state ρ 0 , p 0 , and so that ρ˚" ρ 0 `ρ, p˚" p 0 `p, and v˚" U pyq e x `u e x `v e y , with e x , e y unit vectors in x-and y-direction, and U pyq the shear flow profile.

At first order in the perturbation, the acoustics equations are

)

The quantities are normalised by p Ñ p{c 2 0 , u, v Ñ u{c 0 , v{c 0 , Mach number M pyq " U pyq{c 0 , t Ñ t H{c 0 , and ω " 2πf c 0 H. We can obtain the dimensionless acoustic equations The behavior of the propagative waves against the shear flow is further investigated. To have a comparison with Fig. 5.8, the wave number as a function of ω for 3 different values of n is plotted in Fig. 5.12. The waves propagating against the flow are very sensitive to the shape of the boundary layer. The number of propagative modes is changed by the effect of the boundary layer. This effect has been already noted by Brambley [14]. For the shear flow with n " 30, there is only one propagative mode for 0 ă ω ă ω 1 . This wave is close to the Mode 2 in Fig. 5.8 with uniform flow(also shown by "¨" in Fig. 5.12). There are three propagative waves exist for ω 1 ă ω ă ω 2 . Two of them are close to the mode 2 and 4 in the uniform case and another wave appears with large value of k. For ω 2 ă ω ă ω R , there is only one wave with large ω.

By the definition of the group velocity c g " Bω Bk , we can find the group velocities at the two critical frequencies, ω 1 and ω 2 , are zero. The negative group velocity is also found for n " 30 and 100. As n increases, the shear flow profile tends to uniform flow, and the dispersion curve (the "˚" in Fig. 5.12) becomes similar with the uniform flow case (the "¨" in Fig. 5.12). When the boundary layer thickness increases (n decreases), the wave for which the group and the phase velocities are opposite in sign (Mode 4) will disappear (see the case n " 10 in Fig. 5.12). In this case, the points with a zero group velocity also disappear. Varying the frequency ω, trajectories of wave number for the wave in ´x-direction are plotted in complex k-plane, as shown in Fig. 5.13. By observing the movements, at low frequencies, one propagative mode exists (indicated by 1). With increasing the frequency, two evanescent modes (1a and 1b) move in opposite direction, and approach 5.4. Propagation in the lined ducts with shear flow 125 to the ℜepkq axial. At the critical frequency ω 1 , they have the same value. But then they repel each other, and split into two propagative modes (2a and 2b). Continuing increasing the frequency, two modes move along x-axis in opposite direction, shown as arrows 2a and 2b. When the frequency reach to another critical frequency ω 2 , the same phenomenon happens again, waves 1 and 2b meet and become evanescent modes 3a and 3b. Finally only one mode maintains (2a). The results are corresponding to Fig. 5.12, n " 30. Comparing Fig. 5.13(a) and (b), due to the consideration of the dissipation on the impedance, there exists obvious avoided crossing between the two modes.

Transmission and reflection in the lined liner with shear flow

We already calculated the transmission coefficient for the lined duct in the absence of flow, here we will study the effects of the shear flow on the transmission behavior. At the interfaces, i.e. x " 0 and x " L, the continuity of the pressure and normal velocity disturbance as well as continuity of the parameter q " Bp Bx are used to construct the scattering matrix.

The whole waveguide can be divided into 3 parts, the pressure expression in each part is consist of the incident and reflected waves, which can be written as respectively, where superscript "`" indicates the modes propagating in `x-direction (acoustic and hydrodynamic), "´" the opposite.

The continuity conditions at the two interfaces are applied, for x " 0, we have

and

˙.

(5.53)

The transmission and reflection coefficients of the plane wave can be obtained by the matrix S " S ´1 1 S 2 . They are defined by

, and R

´" A 3 B 3 .

(5.54)

From the upper definitions and the matrix S, the propagation behavior is discussed. Figure 5.14 shows an example of the transmission and reflection coefficients of the plane wave as a function of frequency f , with M 0 " 0.3, n " 30, L " 200mm, and B " 50mm. The height of the duct is H " 15mm. The dissipation of the impedance is considered by taking Re " 0.1. In Fig. 5.15, the mach number M 0 and the resistance Re are changed, the experimental results are also shown in the same figure.

Conclusion

The modes in a duct lined with locally reacting liner are computed and discussed, in both absence and presence of flow. The purely reacting liners produce surface waves which are concentrated along the liner. When a uniform flow is added the waves that propagate against the flow can be cut off. This will produce a large frequency band with transmission zero. Hydrodynamic modes are also observed. In the presence of shear flow, at a specific frequency band rω 1 , ω 2 s, there are three propagative modes against the flow direction. Negative group velocity can be induced. What the most important is, at the two critical frequencies, the group velocity is zero. This phenomenon is very sensitive to the thickness of the boundary layer. For the wave propagates through a lossless medium, the group velocity can be thought as the the velocity at which energy is conveyed along the wave. So it indicates that the energy cannot propagate, which might be used to slow down and even stop the sound.

Appendix

5.A The admittance of the liner

The liner we considered here is a honeycomb layer which is locally reactive and backed by a rigid wall. In the single tube, the pressure satisfies the Helmholtz equation where ω " 2πf H{c 0 is the normalised frequency. In the following, all quantities are non-dimensionalized: lengths with the waveguide height H, velocities with sound speed c 0 , densities with the ambient density ρ 0 , and pressures with ρ 0 c 2 0 . Time dependence is exp(-jωtq.

The solution of the above equation can be written as ppyq " C 1 cospωyq `C2 sinpωyq.

(

5.B Scattering matrix S

The amplitudes of incoming waves A 1 , B 3 are linked to the amplitudes of the outgoing waves A 3 , B 1 by a scattering matrix,

In order to solve the matrix, Mode-Matching method is proposed. Multypling Eqs. (5.16) (5.17) (5.18) by Φ, we can obtain the corresponding equations as below

)

x ą L P 3 " D R px ´Lq A 3 `DR pL ´xq B 3 .

(5.64)

where D `pxq, D ´pxq and D R pxq are diagonal matrices with elements e jknx , e ´jknpx´Lq and e jk R n x , respectively, and X is matrix with columns the eigenvectors in Eq. (5.13) Using the continuity conditions of the pressure and velocity at x " 0 and x " L, we can have

(5.68)

where D L " D `pLq " D ´p0q, and K R , K Y are diagonal matrices with elements the axial wave numbers k R n and k n , respectively. Since the duct is infinite and symmetric about x, we can have T " t and R " r. For simplicity, we assume that there are only multimodes incident from the left side, i.e. B 3 " 0. We denote

( (5.74)

5.C Ingard-Myers boundary condition

The expression of the impedance is given in Eq. (5.59). As there is no flow at the lined wall, v is linked to the acoustic displacements δ w at the wall by v ¨n " Bδ w Bt ¨n, at y " 0.

(5.75)

Assuming a thin shear layer close to the lined wall, the continuity conditions of the normal displacement and the acoustic pressure across the boundary layer need to be considered. Then, the transverse kinematic condition is written as