V Denis

F Gautier

A Pelat

J Poittevin

Measurement and modelling of the reflection coefficient of an Acoustic Black Hole termination

Keywords: reflection coefficient, wave method, flexural vibration, damping, acoustic black hole, inhomogeneous structural waveguide, correction length

The flexural waves propagating in a beam can be efficiently absorbed if one extremity is tapered with a power law profile and covered by a thin layer of additional damping material. Such a termination induces the so-called "acoustic black hole effect" (ABH): if the thickness decreases locally, flexural waves slow down and the amplitude of the displacement field increases, leading to efficient energy dissipation if an absorbing layer is placed where the thickness is minimum. This paper presents a specific study of the reflection coefficient of ABH beam terminations. A Kundt-like measurement method of the reflection coefficient of a beam termination is proposed. The method is validated using theoretical results in the case of a beam free end.

Results for several ABH extremities show a clear decrease of the modulus of the reflection coefficient R. The phase of R due to the decreasing thickness profile is also investigated and is interpreted by defining a correction length for the tapered termination. These experimental results are compared with several models: geometrical acoustics based model, beam waveguide model

Introduction

The reduction of unwanted vibrations is important for structural security, stability and comfort in many industrial applications. A classical vibration damping method consists in using surface damping treatments. Indeed, these methods have proven to be very efficient [START_REF] Ross | Damping of plate flexural vibrations by means of viscoelastic laminae[END_REF][START_REF] Nashif | Vibration Damping[END_REF], although they often result in an increased mass of the treated structure. This may be unacceptable for ecological and economical reasons in the transport industry for example. It is then of a great interest to develop lightweight vibration damping methods.

It is well known that the sharpness of the resonances of a structure depends on both the damping properties of the constitutive material and the boundary conditions [START_REF] Vemula | Attenuation of waves in plates and bars using a graded impedance interface at edges[END_REF]. Therefore, for a given beam structure made of a particular material, a reduction of the resonant behaviour can be obtained by attenuating bending wave reflections at the edges. In this matter, a method using a graded impedance interface at the edges of a beam is proposed by Vemula et al. [START_REF] Vemula | Attenuation of waves in plates and bars using a graded impedance interface at edges[END_REF], and consists in assembling several pieces of beams made with different materials. The energy dissipation within the composite material at the free end, coupled with relatively large amplitude vibrations caused by the impedance gradation, results in a smaller reflectivity. The so-called Acoustic Black Hole (ABH) effect can also be used to reduce the reflection coefficient of a beam free end [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF][START_REF] Krylov | Experimental investigation of the acoustic black hole effect for flexural waves in tapered plates[END_REF][START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF]. The ABH effect can be obtained from the bending wave properties in a beam of decreasing thickness: assuming geometrical acoustics hypothesis, Mironov [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF] shows that in the neighbourhood of a beam edge whose thickness decreases smoothly to zero, flexural waves slow down and stop without being reflected. The condition of sufficient smoothness can be fulfilled by a power-law thickness profile h(x) in the form:

h(x) = x m , ( 1
)
where x is the spatial coordinate and m ≥ 2. If the thickness is strictly zero at the edge [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF], it can be shown that the time taken by a wave to reach the edge becomes infinite. For a practical structure with a finite thickness, the reflection coefficient tends to zero with the decrease of the residual thickness at the tip of the tapered profile. It is however shown that manufacturing processes are such that this residual thickness can never be small enough for the effect to be attractive.

Krylov et al. [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF][START_REF] Krylov | Experimental investigation of the acoustic black hole effect for flexural waves in tapered plates[END_REF] show, also in the framework of geometrical acoustics (i.e. ignoring attenuating flexural waves in the medium), that covering the tapered profile by a thin damping layer compensates the finite thickness at the edge of experimental structures. The obtained analytical reflection coefficient has drastically reduced values compared to a free uniform extremity.

In the framework of Euler-Bernoulli beam modelling, a wave-based approach that takes into account the effect of attenuating waves has been proposed by Georgiev et al. [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF]. From a wave-vector formulation, a Riccati equation is numerically solved to compute a mechanical impedance matrix along the tapered beam and to derive a reflection matrix. Some practical rules for determining the optimal geometrical and material properties of the damping layer are found using this model [START_REF] Georgiev | Numerical and experimental investigation of the acoutic black hole effect for vibration damping in beams and elliptical plates[END_REF]. Further investigations from Denis et al. [START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF] show that a two-dimensional behaviour has to be taken into account to model the tapered zone: the small thickness of the extremity induces a local plate behaviour in a beam with an ABH extremity, and local transverse eigenmodes can be found in the beam structure. Note that two-dimensional ABH have also been described in the literature: circular ABH used as plate vibration damper has been firstly proposed by Gautier et al. [START_REF] Gautier | Experimental investigation of the acoustic black hole effect for vibration damping in elliptical plates[END_REF] and studied both experimentally [START_REF] Gautier | Experimental investigation of the acoustic black hole effect for vibration damping in elliptical plates[END_REF][START_REF] Bowyer | Effect of geometrical and material imperfections on damping flexural vibrations in plates with attached wedges of power law profile[END_REF][START_REF] Bowyer | Experimental inverstigations of damping flexural vibrations in plates containing tapered indentations of power-law profile[END_REF] and theoretically [START_REF] O'boy | Damping of flexural vibrations in circular plates with tapered central holes[END_REF][START_REF] Aklouche | Scattering of flexural waves from an acoustic black hole in an infinite thin plate[END_REF][START_REF] Aklouche | Modle de la diffusion des ondes de flexion par un trou noir acoustique bi-dimensionnel[END_REF].

A claim of the previous studies is that the ABH effect is a way to obtain a quasi-anechoic termination; yet, most of the experimental studies observe frequency responses and not an actual reflection coefficient, which can be considered as a more direct indicator. Although this method has been developed for ABH termination, it remains applicable to any arbitrary beam end. This article focuses on the measurement of the reflection coefficient for flexural waves in a beam which carries an ABH on its extremity. First, the experimental setup is described and the measurement method is detailed in Sec. 2. In Sec. 3, the experimental results for the reflection coefficient of a uniform beam and several ABH beams are shown and commented. Sec. 4

presents different models of the literature and compares them with the experimental results. A conclusion is given in Sec. 5.

Measurement method

Reflection in beams has been thoroughly studied [START_REF] Lin | A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels[END_REF][START_REF] Mace | Wave reflection and transmission in beams[END_REF]. Several methods allow the estimation of reflection properties of beam extremities. Linjama et al. [START_REF] Linjama | Estimation of bending wave intensity in beams using the frequency response technique[END_REF][START_REF] Linjama | Measurement of bending wave reflection and impedance in a beam by the structural intensity technique[END_REF] propose to measure the reflection of bending waves using a structural intensity technique. Vemula et al. [START_REF] Vemula | Attenuation of waves in plates and bars using a graded impedance interface at edges[END_REF] estimate the reflected energy of bending waves in a graded impedance interface using the half power bandwidth of the resonance peaks of the transfer function. Carniel et al. [START_REF] Carniel | Vibrational power flow measurement: structure characterization and wave reflection coefficient measurement[END_REF] and Piaud et al. [START_REF] Piaud | La mesure de l'impdance mcanique[END_REF] both propose expressions for the far field reflection coefficient in a beam. Moulet et al. [START_REF] Gautier | Reflection, transmission and coupling of longitudinal and flexural waves at beam junctions. part I: Measurement methods[END_REF][START_REF] Moulet | Reflection, transmission and coupling of longitudinal and flexural waves at beam junctions. part ii: Experimental and theoretical results[END_REF] use a Kundt tube-like technique in order to estimate the scattering properties of a junction between two beams from the measurements of velocities along the beam. A method inspired by [START_REF] Gautier | Reflection, transmission and coupling of longitudinal and flexural waves at beam junctions. part I: Measurement methods[END_REF][START_REF] Moulet | Reflection, transmission and coupling of longitudinal and flexural waves at beam junctions. part ii: Experimental and theoretical results[END_REF] is used in the current paper.

Experimental setup

Setup

The estimation of the reflection coefficient is based on mobility transfer functions measurements. In order to avoid static deformation due to gravity, the beam to be tested is hanged vertically (see Fig. 1a andb). The extremity whose reflection coefficient is estimated is oriented towards the bottom. An measurement points.

Samples

Two constitutive materials are used for the beams: aluminium and polymer (Vero White Plus FullCure 835). For each material, two beams are tested: a uniform beam (i.e. constant cross section) used as a reference for the estimation method, and a beam with an ABH termination, whose geometry is described on Fig. 2. A layer of damping material made with viscoelastic tape is stuck on the flat side of the tapered profile. The dimensions and material characteristics (Young modulus, mass density, loss factor and Poisson coefficient) of the samples are given in Tab. 1.

The ABH termination for the aluminium beam is manufactured using a conventional machining process, allowing to obtain a minimum thickness at the extremity of about 20 µm. The process requires high precision work and often generates irregularities and tearing at the tip of the profile. The loss factor for aluminium has an arbitrarily chosen low value reflecting a very lightly damped beam and allowing to obtain convincing results. The polymer beam is manufactured using UV polymerisation, which is a rapid prototyping process. This manufacturing process allows a thinner minimum thickness of about 16 µm to be obtained and enables a better control on the profile but restricts the choice of the material. For this material, a standard measurement of the complex Young modulus versus frequency has been done.

The results of the Dynamical Mechanical Analysis (DMA) [START_REF] Menard | Dynamic mechanical analysis: a practical introduction[END_REF] between 0.4 and 40 Hz have been extrapolated on the 0-12 kHz range using the Williams- Landel-Ferry model [START_REF] Ferry | Viscoelastic properties of Polymers[END_REF]. They provide the indicative values given in Tab. 1.

x x (a) (b) d d 0 0 h 0 h 0 l t l ABH l ABH h t h t h l h l h(x) = h 0 (x + l t ) m /l m t h(x) = h t + (x + l ABH ) m (h 0 -h t )/l m ABH
Variable complex Young modulus from the DMA results has been actually used in the following and is plotted on Fig. 3. Frequency (kHz) Table 1: Geometrical and material characteristics of the sample beams and damping layer.

Im(E) (GPa) (b)

Geometrical characteristics Characteristics of material

Aluminium beam

L=0.8 m, b=0.02 m, E 1 =70 GPa, η 1 =0.1 %, l ABH =0.06 m, l t =0.07 m, ρ 1 =2700 kg.m -3 , ν=0.3. h 0 =1.5 mm, m=2.
Polymer beam

L=0.6 m, b=0.02 m, E 1 3.5 GPa, η 1 2.8 %, l ABH =0.06 m, h t =0.16 mm ρ 1 =1175 kg.m -3 , ν=0.3. h 0 =1.5 mm, m=2.
Damping layer

h l =0.1 mm, E 2 =7 GPa, η 2 =40 %, d=0.05 m. ρ 2 =1000 kg.m -3 .
points at the right and left hand side of the neutral axis, with distances a l = A l A 0 and a r = A r A 0 , respectively. Let E and F (E) be the excitation point at abscissa x f and the force applied in E, respectively. The studied reflection coefficient concerns only flexural waves. The measured velocity Ẇi at a given abscissa x i can be written as

130 Ẇi = Ẇif + Ẇit , (2)
where Ẇif and Ẇit correspond to pure beam flexural motion and beam torsional motion, respectively. Only the flexural contribution should be estimated. The mobility on the neutral axis, where there is no torsional contribution, is defined as

Y (A 0 , E) = Ẇif F (E) . (3)
It can be shown that

Y (A 0 , E) = a l Y (A l , E) + a r Y (A r , E) a l + a r = αY (A l , E) + Y (A r , E) 1 + α , (4)
where α = a l /a r . The α coefficient is adjusted in order to minimize the amplitude of the first torsional peak, allowing to find Y (A 0 , E) from the measurements of Y (A l , E) and Y (A r , E). The effect of torsion on the wave field is therefore efficiently reduced on a wide frequency range for each abscissa. Fig. 4b shows the mobility transfer function between the excitation and a random measurement abscissa for the uniform aluminium beam at left and right hand sides of the centreline (full and dashed gray lines). The transfer function on the neutral axis (thick black line) is also plotted and shows that the first torsional peak at 197 Hz vanishes. Mobilities of the uniform and ABH beams made of aluminium are plotted on Fig. 4c; the torsional peaks are minimized and the smoothing effect of the ABH treatment can be noticed.

The quality of the transfer function measurements is estimated using ordinary coherence function. It is easily shown that excitation level has to be large enough to get coherence function close to 1. To make sure that this condition is valid over a wide frequency range, the whole useful band 0-12 kHz is split in two bands: 0-2 kHz and 2-12 kHz for which two different levels are applied for the swept sine excitation. Hence, for each couple (E, A 0 ), two sets of measurements are performed in order to compute the mobility.

x x i+1 x i x i-1 A l × A r × × A 0 a l a r

Methodology for the estimation of the reflection coefficient

Wave field solution for beams

Let us consider a beam of constant cross section whose extremity x=0 bears an ABH profile seen as an unknown boundary condition (see Fig. 5).

Under the Euler-Bernoulli assumptions, the equation of harmonic flexural motion W (x, ω) of such a beam in the absence of excitation, and assuming a harmonic motion with time dependency e -jωt , is [START_REF] Mace | Wave reflection and transmission in beams[END_REF]

E * I ∂ 4 W (x, ω) ∂x 4 -ρSω 2 W (x, ω) = 0 ∀x ∈ [0, L], (5)
where ω is the angular frequency, E * = E(1 + jη) is the complex Young modulus (with E the storage modulus and η the loss factor), I is the moment of inertia of the cross-section, ρ is the mass density, S is the cross-section area and L is the length of the uniform part of the beam.

The general solution for the displacement can be written as the sum of four waves [START_REF] Mace | Wave reflection and transmission in beams[END_REF]:

W (x, ω) = A(ω)e -jk f x +B(ω)e +jk f x +C(ω)e -k f x +D(ω)e +k f x ∀x ∈ [0, L], (6)
where the flexural wavenumber k f is such that

k 4 f = ρS E * I ω 2 . ( 7
)
The scalar coefficients A, B, C and D represent the magnitude of backward and forward propagating and attenuating waves, respectively (see Fig. 5).

At the ABH extremity or the free extremity (x=0), the boundary conditions associated to the equation of motion ( 5) can be written as a reflection matrix R such that

175   A C   = R   B D   , (8)
where

R =   R pp R ap R pa R aa   , (9)
with R ij the reflection coefficient between incident wave i and reflected wave j; i and j symbolize p and a standing for propagating and attenuating respectively.

180

We are interested in measuring terms of R. The boundary condition at x=L has to be defined for the experiment but do not play any role in the measurements of R. In practice, free boundary conditions are supposed at x=L.

Unknown boundary condition

Excitation

0 x f x N x 2 x 1 x 0 x L A D C B

Far field assumption

Far from the excitation point and from the boundaries, attenuating waves vanish and the wave field can be well approximated as the sum of two propagating waves only:

W (x, ω) = A(ω)e -jk f x + B(ω)e +jk f x . ( 10
)
The boundary condition can then be described with a far field reflection coefficient R that corresponds to R pp in Eq. ( 9). It is assumed that the far field assumption is valid at a given frequency when the attenuating waves coming from an excitation point or a boundary condition lose 90 % of their magnitude over the propagation length l. Therefore, there should be

     x N > l, x f -x 0 > l, (11)
where the distance l satisfies [START_REF] Gautier | Reflection, transmission and coupling of longitudinal and flexural waves at beam junctions. part I: Measurement methods[END_REF] e -k f l = 0.1.

In the experiment described in Sec. 2.1, the closest measurement point is at l=0.1 m of the tapered zone of the aluminium ABH beam, so that the far field assumption is theoretically valid above 200 Hz.

Estimation of the wave and reflection coefficients

Let us consider N +1 points along the beam satisfying the far field assumption (see Fig. 5). The displacement measured for all abscissas x i (i ∈ [0, N ]) can be related to the coefficients A and B through the matrix equation

           W (x 0 , ω) W (x 1 , ω) W (x 2 , ω) ... W (x N , ω)            =            e -jk f x 0 e jk f x 0 e -jk f x 1 e jk f x 1 e -jk f x 2 e jk f x 2 ... ... e -jk f x N e jk f x N            M   A(ω) B(ω)   , (13)
with M a (N × 2) matrix. A least-squares estimation of the [A, B] T vector can be effectively obtained with

  A(ω) B(ω)   = M †            W (x 0 , ω) W (x 1 , ω) W (x 2 , ω) ... W (x N , ω)            , (14)
where M † = (M * M) -1 M * is the Moore-Penrose generalized inverse [START_REF] Penrose | A generalized inverse for matrices[END_REF] of

205

M and M * is the conjugate transpose of M.

From this estimation of the wave coefficients, the far field reflection coefficient of the extremity can be derived:

R(ω) = A(ω) B(ω) . ( 15
)
3. Measurement of the reflection coefficient

Validation on the uniform beam

It can be shown that the reflection coefficient for a free end is given by

[17] R = A B = -j. ( 16
)
This result is used to validate the proposed measurement method. Fig. 6 presents the reflection coefficient of the free end at x=0 measured on aluminium (full black line), estimated with the method described in Sec. 2. It is checked that the theoretical result is experimentally obtained: modulus and phase are constant with respect to frequency and equal to 1 and -π 2 , respectively. It is remarkable to see that the experimental result of Fig. 6 is conform to the theoretical result of Eq. ( 16) on a wide frequency range.

Coherences of the measurements are close to 1 up to 12 kHz and make this estimation reliable. This verification of the theoretical result assesses the measurement method. Fig. 6 also shows the reflection of the free end in the polymer case with the assumed material characteristics of Tab. 1 (full light gray line). The unexpected phase increase suggests an incorrect knowledge of the wavenumber. Indeed, modulus and phase of the reflection coefficient are related to the imaginary and real parts of the wavenumber, respectively. The wavenumber has then to be carefully corrected in order to fit the measured reflection coefficient with the free end theoretical result, in modulus and phase. The fit made on the characteristics of the polymer material consists in increasing by 2 % the real part of the wavenumber calculated with characteristics of Tab. 1, in order to obtain the theoretical reflection coefficient of a free end (E 1 3.8 GPa). Modulus and phase of R with fitted wavenumber (see Fig. 6, dashed dark gray line) are constant and show that the estimation method relies on a precise knowledge of the wavenumber.

However, it can be noticed that the reflection coefficient for the polymer beam behaves poorly after 7 kHz. This can be expected since the structure cannot be considered anymore as a beam above this frequency and is likely to display a two-dimensional plates behaviour, because the wavelength (λ Polymer f =7kHz 0.026 m) neighbours the width of the cross-section, that is

λ Polymer f =7kHz 1.3b. (17)
Consequently, the estimation based on one-dimensional wave decomposition does not make sense above 7 kHz for the polymer beams. This issue does not appear with the aluminium beam in which the wavelength is much larger than the beam width in the 0-12 kHz range (λ Aluminium f =12kHz

1.7b 0.034 m).

It is also noted that in the two beams, the wavelength is much larger than the beam thickness, justifying the use of the classical Euler-Bernoulli beam theory.

Small singularities appears on the curves, especially for the aluminium beam. This is due to a drop of coherence at anti-resonances. A more damped structure such as the polymer beam shows less coherence drops and thus, a smoother curve for the reflection coefficient.

Results on ABH beams

Typical results

Fig. 7 shows the estimated reflection coefficient for the ABH extremity in modulus and phase, in aluminium (full black line) and polymer (dashed dark gray line) beams. Compared to the free end, it is very clear that the modulus of R decreases with increased frequency. Values in modulus can be as small as 0.1. Moreover, oscillations can be observed on the modulus. The phase of the reflection coefficient displays a significant additional roll down with frequency.

Qualitative variations of |R| are similar for the aluminium and polymer cases. The modulus of the reflection coefficient is smaller for the polymer beam, showing that the ABH is more efficient in this material. The phase also rolls down faster in the polymer case below 7 kHz.

Phase of the reflection coefficient and equivalent length of the tapered profile

Special attention is paid to the phase of the reflection coefficient. Instead of estimating the reflection coefficient at x = 0 (see Fig. 6) for the uniform aluminium beam (named R Ref ), it is estimated at distance l ABH (see Fig. profile because it is slowed down. An alternative point of view is to consider a uniform profile of larger length instead of the tapered profile, in which the bending wave speed remains constant. This leads to define a correction length as it can be done in acoustic pipes to take into account acoustic radiation at the open end of the pipe [START_REF] Kinsler | Fundamentals of acoustics[END_REF]. Hence, the correction length ∆l provided by the ABH profile corresponds to the length of uniform cross-section that should be added to l ABH in order obtain the same phase. The reflection coefficient of ABH termination can be written as

R ABH = |R ABH |e jk f (l ABH +∆l) , (18)
where k f is the wavenumber in the beam of uniform cross-section. The argument of Eq. ( 18) yields

Arg(R ABH ) = 2Re{k f }l ABH (1 + ∆l l ABH ), (19)
where ∆l/l ABH is the dimensionless correction length.

From the experimental results, the calculated correction length is plotted on Fig. 8 for aluminium and polymer ABH beams. The correction length for the aluminium case is remarkably constant on the 0-9 kHz range. The value close to 0.8 m means that a tapered and covered profile of 0.06 m is equivalent to a uniform profile of 0.11 m (i.e. the length is approximately doubled) from the phase viewpoint. This observation even less conclusive in the polymer case which still yields a quasi-constant correction length close to 1 on the 0-4 kHz range. This correction length phenomenon has been theoretically shown by Mironov [START_REF] Mironov | One-dimensional acoustic waves in retarding structures with propagation velocity tending to zero[END_REF] in the case of a retarding tube termination, constituting the acoustical analogy of the ABH termination for flexural

Comparison with theoretical results

Three models of the literature are used in this section in order to represent the reflection coefficient of an ABH beam extremity.

Wave model based on geometrical acoustics assumptions

The model proposed by Krylov et al. [START_REF] Krylov | Acoustic 'black holes' for flexural waves as effective vibration dampers[END_REF][START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF] is the first to consider the use of a damping layer in the ABH framework. Considering geometrical acoustics assumptions, the reflection coefficient of the truncated wedge is calculated using the expression

|R(0, ω)| = exp - 0 -l ABH Im(k(x, ω))dx , (20)
where k(x, ω) is the flexural wavenumber and l ABH the length of truncated profile (see Fig. 2). The reflection coefficient is evaluated at x=0. Note that [START_REF] Krylov | Acoustic 'black holes' for flexural waves as effective vibration dampers[END_REF][START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF] give only an information about the modulus of the reflection coefficient.

In the proposed approach, attenuating waves are not taken into account. In [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF], the author proposes two ways to take the damping layer into account in the expression of the wavenumber, depending on the relative thickness of the layer. We are interested here in the expression of the wavenumber k when the layer is relatively thick compared to the plate (see Eq. [START_REF] Aklouche | Modle de la diffusion des ondes de flexion par un trou noir acoustique bi-dimensionnel[END_REF][START_REF] Lin | A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels[END_REF] in [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF]):

k = 12 1/4 k 1/2 p (1 + ρ 2 h 2 ) 1/4 h 1/2 [1 + e 2 h 2 (3 + 6h 2 + 4h 2 2 )] 1/4
. 1 + j 1 4

η 1 + η 2 e 2 h 2 (3 + 6h 2 + 4h 2 2 ) 1 + e 2 h 2 (3 + 6h 2 + 4h 2 2 ) , (21)
where h is the thickness of the plate alone, η 1 and η 2 are the loss factors of the plate and the layer, e 2 the ratio of the Young moduli of the layer and the 315 beam, h 2 the ratio of the thicknesses of the layer and the beam, ρ 2 is the ratio of the mass density of the layer and the beam and

k p = ω ρ 1 (1 -ν 2 )/E 1 is
the wavenumber of a plate wave and ω is the circular frequency.

The reflection coefficient of Fig. 9 (dash-dotted line) is plotted using this

Inhomogeneous wave model under plane wave assumption

The ABH extremity can be considered as an heterogeneity in a structural waveguide constituted by the beam. Several techniques exist in the literature to describe the vibratory field and compute a reflection coefficient: transfer matrix method, scattering matrix method, impedance matrix method, etc. [START_REF] Lin | A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels[END_REF][START_REF] Norris | Reflection and transmission of structural waves at an inter-35 face between doubly curved shells[END_REF][START_REF] Felix | Multimodal analysis of acoustic propagation in three-dimensional bends[END_REF]. If one considers the attenuating waves, impedance matrix method is the most efficient one. Georgiev et al. [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF] propose a model that takes into account the attenuating part of the wave field to compute the reflection coefficient of an ABH beam extremity using an impedance matrix technique. Following the Euler-Bernoulli assumptions, the cross-section remains undeformed during the vibration. All points of the cross-section are thus in phase, which is the main feature of a plane wave. No transverse variations of the vibration field are considered. In [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF], the state vector X describing the kinematics and the efforts along the beam respects the state

equation ∂X ∂x = HX, ( 22
)
where H is a matrix describing the governing equations of the Euler-Bernoulli model. Such formulation allows to develop the non linear Riccati equation for the impedance Z along the beam

∂Z ∂x = -ZH 1 -jωZH 2 Z + H 3 jω + H 4 Z, (23)
where H 1 to H 4 are the four submatrices of H. A relation between the impedance matrix and a reflection matrix at a given point is then used.

The damping layer effect is taken into account using equivalent material properties, according to [START_REF] Ross | Damping of plate flexural vibrations by means of viscoelastic laminae[END_REF]; the equivalent complex bending stiffness EI * (x) of the beam covered with a layer is expressed (see Eq. (B1) in [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF])

EI * = E 1 I 1 (1 + jη 1 ) + e 2 h 3 2 (1 + jη 2 ) + 3(1 + h 2 ) 2 e 2 h 2 (1 -η 1 η 2 + j(η 1 + η 2 )) 1 + e 2 h 2 (1 + jη 2 ) , (24)
where E 1 I 1 is the bending stiffness of the beam alone, η 1 and η 2 are the loss factors of the plate and the layer, e 2 the ratio of the Young moduli of the layer and the beam and h 2 the ratio of the thicknesses of the layer and the beam. Note that Eq. ( 24) yields a wavenumber similar to Eq. ( 21).

The Ricatti equation ( 23) is solved numerically using a fourth-order Runge-Kutta scheme. As it is seen on Fig. 9 (dashed line), this model provides a result that differs from the model of [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF] for the reflection coefficient. The geometrical model result is monotonously decreasing, without any oscillations. However, using the same parameters the same order of magnitude is obtained on the reflection coefficient. It can be assumed that the oscillations come from reflections throughout the ABH profile, suggesting that the sufficient smoothness condition [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF] is not applicable in the studied frequency range for this ABH geometry. This subject has been studied by Feurtado et al. [START_REF] Feurtado | A normalized wave number 525 variation parameter for acoustic black hole design[END_REF] who conclude that the sufficient smoothness condition is not verified at frequencies lower than 10 kHz for ABHs with dimensions on the order of several centimeters. An information of phase is given by [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF], while it is absent in the expression given by [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF]. The phase rolls down and can give a hint about the travel time of a wave in the tapered region of the beam, as said in Sec. 3.2.2.

Inhomogeneous plate model

In a previous work [START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF], the authors consider that the decreasing thickness in the ABH profile makes the beam locally behave like a plate even at low frequencies and investigate two-dimensional behaviour. A plate model with varying thickness along x is used, in which the displacement W (x, y) due to an harmonic excitation at x f of amplitude F and angular frequency ω satisfies the following equation of motion:

∇ 2 (D(x)∇ 2 W (x, y)) -(1 -ν) ∂ 2 D(x) ∂x 2 ∂ 2 W (x, y) ∂y 2 - ρ(x)h(x)ω 2 W (x, y) = F δ(x -x f ), (25)
where ρ(x) is the mass density, h(x) is the thickness, D(x) is the local complex bending stiffness and ν the Poisson coefficient. The damping layer effect is taken into account in the same way than in Sec. 4.2 and allows to define the equivalent bending stiffness for the plate covered with damping layer. The problem is completely defined by using free boundary conditions on every edge of the plate.

This model is numerically solved from a finite difference scheme. This method uses a spatial mesh that is adapted to the thickness variation, in order to reduce the numerical dispersion. Thus a transformed grid is defined on which the wavelength remains constant spatially. Finally, the discretized problem leads to the matrix problem

(K -ω 2 M)W = F, (26)
where K and M are respectively the complex stiffness matrix and the mass matrix of the structure, F is the force vector and W is the displacement field.

Using this numerical model, frequency responses are simulated and can be post-processed as described in Sec. 2. The result for the reflection coefficient is plotted on Fig. 9 (full line).

On Fig. 9, plane wave and plate models yield the same results when fed with the same parameters of Tab. 1. Actually, the phase of the reflection coefficient is slightly different; it can be attributed to a small numerical dispersion in the finite difference scheme of the plate model, leading to a difference between theoretical and actual wave speed in the structure. Considering that [START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF] provides a two-dimensional model and [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF] a beam model, it can be concluded that no two-dimensional effect is acting in this case: the trapped transverse modes described in [START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF] thus do not seem to play any role.

Comparisons and discussion

The result of the model of [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF] for the estimated parameters and the experimental results in the aluminium case are plotted on Fig. 10. The modulus of R is of the same order of magnitude up to 4000 Hz and displays oscillations in the two cases, although it is more irregular in the experimental case.

Above 4000 Hz, the experimental reflection coefficient is slightly smaller in modulus. The phases are very similar in their order of magnitude.

The tapered extremities are usually far from perfect and may present irregularities [START_REF] Bowyer | Effect of geometrical and material imperfections on damping flexural vibrations in plates with attached wedges of power law profile[END_REF]. Available models do not take into account these imperfections. They may allow the excitation of two-dimensional modes described in [START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF] by an incident wave and thus cause energy transfer and enhancement the estimation method is validated using comparisons with the theoretical result for a free end, for two different materials. It is observed that the reflection coefficient of the ABH termination decreases with increased frequency and presents oscillations. The phase of the reflection coefficient rolls down, indicating a relatively important travel time in the tapered extremity. It allows to define a correction length for the tapered termination that appears to be quasi-constant with respect to frequency. More detailed study could be done to investigate in depth the links between geometry, material and correction length. A review of the existing models is made in Sec. 4; a comparison of the models between them shows that a beam model and plate model yield the same result but that a geometrical acoustics approach is not adapted to the studied geometry, which may not have a sufficient smoothness. A comparison between the model and the experiment shows a good agreement even if the experimental result is less regular than the result given by the model. It can be concluded that the estimation method proves to be usable and that the available models are satisfying to describe the overall behaviour of R. Besides, modelling the imperfections at the extremity may improve the adequacy to the experiment.

Figure 1 :

 1 Figure 1: (a) Scheme of the experiment. (b) Photo of the experimental setup.

Figure 2 :

 2 Figure 2: Scheme of the ABH profile at beam extremity for (a) aluminium and (b) polymer cases.

120 2 . 1 . 3 .

 213 Measurement conditionsTransfer mobility measurements are performed at 21 abscissas 5 mm apart from each other. In the case of the aluminium ABH beam, the closest abscissa is located 0.1 m way from the tapered region. Let A 0 be the point on the neutral axis of the beam at absissa x i (see Fig.4a). Let A l and A r be two

Figure 3 :

 3 Figure 3: Complex variations of (a) the real part and (b) the imaginary part of the Young modulus E of the polymer material.

Figure 4 :

 4 Figure 4: (a) Left, neutral axis and right measurement points A l , A 0 and A r at abscissa x i . (b) Transfer mobility measurement Y (A i , E) (full gray line) , Y (A r , E) (dashed gray line) and Y (A 0 , E) (thick full black line) on the uniform aluminium beam (dB, ref. 1 m.s -1 .N -1 ). Force is applied at x f = 0.58 m, velocity is measured at x i = 0.3 m. (c) Transfer mobility measurement Y (A 0 , E) for uniform (gray line) and ABH (black line) beams made of aluminium (dB, ref. 1 m.s -1 .N -1 ). Force is applied at x f = 0.58 m, velocity is measured at x i = 0.3 m.

Figure 5 :

 5 Figure 5: Scheme of the beam showing the excitation point x f , the N + 1 measurement points x 0 to x N , the attenuating and propagating waves and the boundary condition at x=0.

Figure 6 :

 6 Figure 6: Estimated reflection coefficient R for the free end of uniform beams made of aluminium (full black line), polymer with fitted wavenumber (dashed dark gray line) and polymer with non-fitted wavenumber (full light gray line), in (a) modulus and (b) phase.

2 )

 2 from the edge and plotted on Fig. 7 (full light gray line). It is compared with the phase the reflection coefficient of the aluminium ABH termination (named R ABH ). The phase discrepancies between R Ref and R ABH are studied in order to investigate the role of the ABH thickness profile. The phase of R ABH decreases more than the phase of R Ref , for the same actual termination length. It means that an incident wave spends more time in the tapered

Figure 7 :

 7 Figure 7: Estimated reflection coefficient R for the termination of the aluminium (full black line) and polymer (dashed gray line) ABH beams and reflection coefficient for the reference beam estimated at distance l ABH of the free boundary condition (full light gray line), in (a) modulus and (b) phase.

Figure 8 :

 8 Figure 8: Measured dimensionless correction length ∆l/l ABH due to ABH termination for aluminium (full black line) and polymer (dashed gray line) ABH beams.

Figure 9 :

 9 Figure 9: Reflection coefficient calculated with geometrical model (dash-dotted line), beam waveguide model (dashed line) and plate model (full line) in (a) modulus and (b) phase.

model for an ABH termination with parameters mentioned in Tab. 1. On this figure, R is clearly monotonously decreasing.

Acknowledgements

The authors are grateful to the Centre de Transfert de Technologies du Mans (CTTM) for the manufacture of the polymer beams used in this study and to Stanislas Renard for the manufacturing of the aluminium ABH terminations.

of the ABH effect. Such a phenomenon can contribute to the differences between model and experiment.

Conclusions

405

This paper is dedicated to the experimental measurement of the reflection coefficient of an Acoustic Black Hole termination. In Sec. 2, an experimental method for estimating the reflection coefficient of a beam extremity is presented and applied to beams with Acoustic Black Hole termination. In Sec. [START_REF] Vemula | Attenuation of waves in plates and bars using a graded impedance interface at edges[END_REF]


V Denis

A Pelat

F Gautier

Scattering effects induced by imperfections on an Acoustic Black Hole placed at a structural waveguide termination

Keywords: reflection coefficient, wave method, flexural vibration, damping, acoustic black hole, inhomogeneous structural waveguide, imperfections, scattering

The so-called "acoustic black hole" effect (ABH) is a passive vibration control technique based on the flexural waves properties in thin structure of varying thickness. A usual implementation consists in using a plate with tapered extremity with a power-law profile, covered with a thin damping layer. The inhomogeneity of the structure leads to a decrease of flexural wave speed and an increase of their amplitude, therefore resulting in an efficient energy dissipation if damping layer is placed where the thickness is minimal. The manufacture of an efficient extremity is difficult because of the small thickness, and often generates imperfections and tearing. Moreover, previous works suggest multiple that flexural modes are propagating across the width of the ABH tip. A model of an ABH multimodal waveguide taking into account an imperfect termination is developed. It shows that an elementary imperfection can affect the reflection coefficient of the extremity and reduce it. Scattering and propagation properties of the extremity

Introduction

The control of unwanted vibrations is important for reliability, stability and comfort in many industrial applications [START_REF] Rao | Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes[END_REF]. Indeed, vibrations can generate radiated noise [START_REF] Cremer | Structure-Borne Sound: structural vibrations and sound radiation at audio frequencies[END_REF] but are also a source of structural damage [START_REF] Jones | Influence of additive damping on resonance fatigue of structures[END_REF]. Most of the classical vibration control methods involve surface damping treatment.

The efficiency of these methods has been widely proven, but a major drawback is that they involve an increased mass of the treated structure. This is a concern in many engineering domains for economical or ecological cost.

The development of passive vibration control techniques without added mass is interesting in this matter.

The Acoustic Black Hole (ABH) effect [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF][START_REF] Krylov | Experimental investigation of the acoustic black hole effect for flexural waves in tapered plates[END_REF][START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF] is a passive vibration control technique taking advantage of bending wave properties in structures of decreasing thickness in order to attenuate the reflections at the edge, and consequently decreasing the resonant behaviour. An example of such boundary damping is given in the pioneer work of Vemula et al. [START_REF] Vemula | Attenuation of waves in plates and bars using a graded impedance interface at edges[END_REF] which proposes to use a graded impedance interface at the edges of a beam, consisting in the association of several pieces of beams made with different materials. The results show that lower reflectivity is caused by energy dissipation within the composite material at the free end coupled with relatively large amplitude vibrations caused by the impedance gradation. In the ABH effect, as first described by Mironov [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF], the thickness of a beam edge decreasing smoothly to zero cause flexural waves to slow down and stop without being reflected. The condition of sufficient smoothness can be fulfilled by a power-law thickness profile h(x) in the form:

h(x) = x m , ( 1
)
where x is the spatial coordinate and m ≥ 2 (see Fig. 1(a)). If the thickness is strictly zero at the edge [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF], it can be shown that the time taken by a wave to reach the edge becomes infinite. For a practical structure with a finite thickness, the reflection coefficient tends to zero with the decrease of the residual thickness at the tip of the tapered profile. It is however shown that manufacturing processes are such that this residual thickness can never be small enough for the effect to be attractive.

As it is shown by Krylov et al. [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF][START_REF] Krylov | Experimental investigation of the acoustic black hole effect for flexural waves in tapered plates[END_REF] in the framework of geometrical acoustics, the negative effect of the finite thickness at the edge can be compensated by covering the profile with a thin damping layer. A low reflection coefficient can be obtained for the ABH termination. This model has been refined by Georgiev et al. [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF] by using a Euler-Bernoulli beam modelling. In this work, reflection and impedance matrices along the tapered beam are computed solving a Riccati equation. Practical rules for determining the op-timal geometrical and material properties of the damping layer [START_REF] Georgiev | Numerical and experimental investigation of the acoutic black hole effect for vibration damping in beams and elliptical plates[END_REF] are found using this model. Further investigations from Denis et al. [START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF] show that the small thickness of the extremity induces a local plate behaviour in a beam with an ABH extremity, and that a two-dimensional behaviour has to be taken into account to model the tapered zone; local transverse eigenmodes can be found in the beam structure. Note that two-dimensional ABH have also been described in the literature: circular ABH used as plate vibration damper has been firstly proposed by Gautier et al. [START_REF] Gautier | Experimental investigation of the acoustic black hole effect for vibration damping in elliptical plates[END_REF] and studied both experimentally [START_REF] Gautier | Experimental investigation of the acoustic black hole effect for vibration damping in elliptical plates[END_REF][START_REF] Bowyer | Effect of geometrical and material imperfections on damping flexural vibrations in plates with attached wedges of power law profile[END_REF][START_REF] Bowyer | Experimental investigations of damping flexural vibrations in plates containing tapered indentations of power-law profile[END_REF] and theoretically [START_REF] O'boy | Damping of flexural vibrations in circular plates with tapered central holes[END_REF][START_REF] Aklouche | Scattering of flexural waves from an acoustic black hole in an infinite thin plate[END_REF][START_REF] Aklouche | Model of the scattering of flexural waves from a two-dimensional acoustic black hole[END_REF].

Most of the literature [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF][START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF][START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF] considers a perfect tapered extremity: the free edge is considered straight and normal in the direction of propagation

x. The consequence is that an incident plane wave propagating along the

x-axis remains plane and that the reflected wave is also stricly plane. Thus there should be no excitation of the trapped eigenmodes mentioned in [START_REF] Denis | Modal overlap factor of a beam with an acoustic black hole termination[END_REF].

However the practical realisation fails the assumption of perfect edge because the manufacture of such small thicknesses is difficult and leads to irregular and teared extremities (see Figs. 1(b) and (c)). Moreover, it is observed that the wave field is not unidimensional for some frequencies [START_REF] Denis | Vibration damping in beams using the acoustic black hole effect[END_REF][START_REF] Denis | Characteristics of the vibration field inside an acoustic black hole placed on a beam[END_REF]. Bowyer

Model of an imperfect ABH termination

Problem statement

The ABH extremity (see Fig. 2) is considered as an inhomogeneous structural waveguide. In order to describe coupling phenomena between guided modes at the extremity from imperfect edge conditions and to extend the study of the reflection coefficient made in [START_REF] Denis | Measurement and modelling of the reflection coefficient of an acoustic black hole termination[END_REF], the aim is to obtain the reflection matrix of the extremity or of a region of the waveguide. This reflection matrix can be obtained by either the knowledge of the impedance matrix of the waveguide when it is ended with a free boundary condition, or as part of the scattering matrix of a region of the waveguide. We propose in this section to compute numerically the scattering of a waveguide region.

x z Bending stiffness D(x) and thickness h(x) describe the waveguide and their expressions are given by (see Fig. 2)

-x 0 0 L x l x ABH h 0 h t h l h(x) (a)
D(x) = Eh(x) 3 12ρ(1 -ν 2 ) (1 + jη(x)), (2)
where E, ρ and ν are the Young's modulus, mass density and Poisson ratio of the plate material, respectively, and η(x) is the loss factor, and the thickness is

h(x) =      h 0 (x+x 0 ) m (x 0 +x ABH ) m if x ≤ x ABH , h 0 if x > x ABH . (3)
The effect of the viscoelastic layer can be modelled by an equivalent loss factor [START_REF] Ross | Damping of plate flexural vibrations by means of viscoelastic laminae[END_REF] which depends on x via [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF]

η(x) =      η p , if x ≤ x l , η p + η l 3E l h l Eh(x) , if x > x l , (4)
where η p is the loss factor of the plate, and E l , η l , h l the Young's modulus, the loss factor and the thickness of the viscoelastic layer, respectively.

Governing equations

The governing equations for the flexural motion of plate with inhomogeneities along the longitudinal x-direction (plate parameters remains invariant along transverse y-direction), and in harmonic regime at pulsation ω (with the e jωt time convention) give the relations between variables of displacement w, slope θ x , bending moment M x and effort V x and their firstorder partial derivatives with respect to x [START_REF] Leissa | Vibration of plates[END_REF]. After several derivations, the following relations are obtained:

∂w ∂x = θ x , (5)
∂θ x ∂x = - 1 D M x -ν ∂ 2 w ∂y 2 , (6)
∂M x ∂x = V x -2(1 -ν)D ∂ 2 θ x ∂y 2 , (7)
∂V x ∂x = -ρhω 2 w + D(1 -ν 2 ) ∂ 4 w ∂y 4 -ν ∂ 2 w ∂y 2 . ( 8
)
For simplicity, the two edges along x (y=0 and y=b) are simply supported, allowing to write a Levy-type analytical solution [START_REF] Gorman | A general solution for the free vibration of rectangular plates resting on uniform elastic edge supports[END_REF][START_REF] Lee | Levy-type solution for the analysis of nonuniform plates[END_REF][START_REF] Xiang | Levy solutions for vibration of multispan rectangular plates[END_REF] for Eqs. ( 5)- [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF] as a product of beam functions; this cannot properly be done assuming free conditions on the four edges [START_REF] Leissa | Vibration of plates[END_REF]. Such lateral boundary conditions define the waveguide which will be studied. This waveguide is represented on Fig. 2(c)

and is called a Levy-waveguide. The boundary condition is not realistic for general insight on beam ABH applications at low frequencies, but is a reasonable condition for high frequencies. Therefore, this choice rules out a direct comparison with the experiment made with beams but allows a multimodal analysis of the effects of the imperfection. The two edges along y (x=0 and x=L) are free. The boundary condition of the edge x = 0 is modified in Sec. 2.5 for taking into account the imperfect extremity.

Each variable g(x, y) (standing for w, θ x , M x and V x ) is then written as the multimodal expansion

g(x, y) = ∞ q=1 g q (x)Ψ q (y), (9)
where

Ψ q (y) = √ 2 sin qπ b y (10)
are the orthogonal modes of a simply supported beam [START_REF] Guyader | Vibration in Continuous Media[END_REF] also called the transverse modes of the waveguide.

The projection of Eqs. ( 5)-( 8) on the transverse modes yields a set of projected equations that can be rewritten as the following state equation

d dx W(x) = H(x)W(x), (11)
where the state vector W(x)

W(x) =         w(x) θ x (x) M x (x) V x (x)         , (12)
is the concatenation of displacement, slope, bending moment and force modal vectors. For example, if we consider that the series expansion in Eq. ( 9) is truncated with K terms, we have:

w(x) = [w 1 (x), w 2 (x), ..., w K (x)] T . ( 13
)
Matrix H embeds the plate equations and is expressed as

125 H(x) =   H 1 (x) H 2 (x) H 3 (x) H 4 (x)   , (14)
with

H 1 (x) =   O I -νI 2 O   , H 2 (x) =   O O -I/D O   , H 3 (x) =   O -2(ν -1)DI 2 -ρhω 2 I + (1 -ν 2 )DI 4 O   , H 4 (x) =   O I -νI 2 O   , ( 15
)
where I is the K × K identity matrix, O is the K × K zero matrix and I 2qj

and I 4qj are defined as

I 2qj = - qπ b 2 δ qj , I 4qj = qπ b 4 δ qj . (16)
Moreover, the local impedance matrix Z(x) can be defined as

  M x (x) V x (x)   = jωZ(x)   w(x) θ x (x)   . (17)
Substituting Eq. ( 17) in Eq. ( 11), a Riccati non-linear equation is obtained for the impedance matrix [START_REF] Felix | Multimodal analysis of acoustic propagation in three-dimensional bends[END_REF][START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF]:

130 ∂Z(x) ∂x = -Z(x)H 1 (x) -jωZ(x)H 2 (x)Z(x) + H 3 (x) jω + H 4 (x)Z(x). ( 18
)
The boundary value problem is transformed into an initial value problem.

The free boundary condition translates into the initial condition Z(x = 0) = O.

Wave expansion of the state vector

Eigenspace of matrix H(x) is described by 4K eigenvalues and their as-135 sociated eigenvectors. The relation between them is

E(x)H(x) = Λ(x)E(x), (19)
where Λ(x) is the diagonal matrix containing the eigenvalues of H(x) on its diagonal and

E(x) =   E 1 E 2 E 3 E 4   , (20)
with E 1 to E 4 being 2K × 2K matrices, is a matrix whose columns are the associated eigenvectors; E is also the transition matrix between eigenspace and state space. The change of basis is applied using the relation

W(x) = E(x)V(x), (21)
where V(x) is the wave vector, describing for each mode q the propagating and attenuating waves travelling towards x > 0 and x < 0.

For the sake of simplicity, let us first assume a single transverse mode q verifying the dispersion equation

k 4 q (x) = -k 4 f (x) + qπ b 4 , (22)
where k f = 4 ω 2 ρh/D is the flexural wave number and k q is the guide wave number. Then the eigenvalue matrix Λ of H writes:

Λ = diag (jk 1q , k 2q , -jk 1q , -k 2q ) , (23)
where

k 1q = k 2 f -(qπ/b) 2 , (24)
k 2q = k 2 f + (qπ/b) 2 . ( 25
)
It can easily be shown that the eigenvalues verify the dispersion equation [START_REF] Gorman | A general solution for the free vibration of rectangular plates resting on uniform elastic edge supports[END_REF]. In the absence of dissipation, second and fourth eigenvalues are real and are related to attenuating waves. Depending on the sign of

k 2 f -(qπ/b) 2 ,
first and third eigenvalues are real or imaginary and are related to effectively

propagating waves (k 2 f -(qπ/b) 2 > 0) or evanescent waves (k 2 f -(qπ/b) 2 < 0)
. The sign of the eigenvalue indicates the travelling direction of the associated wave: from the chosen convention first and second eigenvalues are related to travelling waves towards x < 0 and third and fourth eigenvalues are related to travelling waves towards x > 0. When several transverse modes are considered, a similar classification is chosen, thus:

Λ = diag(p-, a-, p+, a+), (26)
where p and a indicate a vector of eigenvalues related to propagating or attenuating waves, respectively, and the sign indicates the direction of travel of the associated waves. The wave vector is thus noted

V =         V - p V - a V + p V + a         . (27)

Scattering matrices in the ABH waveguide

Let Ω be a region of the waveguide delimited by two abscissas t -and t + (see Fig. 3). V out and V in are the outcoming wave vector from Ω and incoming wave vector to Ω, respectively. They write

V out =   V -(t -) V + (t + )   and V in =   V -(t + ) V + (t -)   . (28)
The scattering matrix S (size 4K × 4K) of Ω is defined by

V out = SV in , (29)
and writes

S =   T +-R - R + T -+   , (30)
where matrix R + (size 2K × 2K) represents the reflection of the incident waves at side t + (denoted V -(t + )) on the reflected waves at side t + (denoted

V + (t + )). Matrix T +-(size 2K × 2K
) represents the transmission of the incident waves at side t + on the transmitted waves at side t -. Moreover, following the form of the wave vector (see Eq. 27), the term (R + pa ) qj quantify the reflection from the incident propagating mode q on the reflected attenuating mode j. For any region of the waveguide defined by the abscissa x including the end of the waveguide at x=0, the scattering matrix S [0, x] representing the waveguide free extremity is defined in agreement with the definition of Ref. [START_REF] Vemula | Attenuation of waves in plates and bars using a graded impedance interface at edges[END_REF]: transmission towards the surrounding medium T +-, transmission towards the inside T -+ and reflection of the outside R -are zero. Only R + is non-zero. Hence:

x y t - t + V -(t+) V + (t+) V + (t-) V -(t-) Ω
S [0, x] =   O O R + (x) O   . (31)
Matrix R + (x) is then called the reflection matrix of the guide extremity (i.e.

the region [0, x]). Eqs. ( 17), ( 20) and [START_REF] Denis | Measurement and modelling of the reflection coefficient of an acoustic black hole termination[END_REF] yields the relation between R + (x)

and Z(x): condition is (see Fig. 4)

R + (x) = (-jωZ(x)E 2 + E 4 ) -1 (-E 3 + jωZ(x)E 1 ). ( 32
)
S x=0 S ]0, x ABH ]
S [0, x ABH ] = S x=0 S ]0, x ABH ] , (33)
where the assembly operator is defined in Appendix A. It is therefore important to distinguish R + (x) and R + ]0, x] . Practically, Eq. ( 33) will be used to determine S [0, x ABH ] . Details about S x=0 and S ]0, x] are given in Sec. 2.5 which proposes a model for the imperfect boundary condition, and 190 Sec. 2.6 which proposes to find numerically the scattering in the ABH profile.

Imperfect boundary condition

Identifying and modelling imperfections or defaults in a structure is closely related to structural health monitoring [START_REF] Doebling | A summary review of vibration-based damage identification methods[END_REF] where the main interest is usually cracks. Those are usually modelled by local stiffness changes [START_REF] Joshi | A unified approach to free vibration of damaged beams having various homogeneous boundary conditions[END_REF]. A simpler model is used in this study: the imperfection is described as a material extension of small variable length a(y) along direction x. It is constituted of infinitesimal rigid bars of width dy, that are not coupled between each other (see Fig. 5). These bars are supposed to follow the kinematics of the last cross section of the waveguide: this leads to the fact that the displacement w(0, y) and slope θ x (0, y) of the last cross section are such as the ones of the bars. These bars have a mass density ρ identical to that of the waveguide and their thickness h is assumed constant and equal to that of the extremity of the waveguide. The only actual parameter is then the length a(y) which controls, for each infinitesimal element, the force V x (0, y) and bending moment M x (0, y) applied on the guide extremity. Finally, this simple model is well adapted to represent any arbitrary imperfection by a distribution of mechanical load at the extremity.

Newton's Second Law is applied to each element and leads to ρha(y)dyz(y) = -V x (0, y),

I 0 θx (0, y) x = -M x (0, y), (34)
with ρha(y) the mass of the element, assumed concentrated in its centre of mass, whose vertical displacement is z(y) = w(0, y) + a(y) 2 θ x (0, y). The moment of inertia I 0 writes: Eqs. ( 34) and ( 35) then write

I 0 = a(y) 0 s 2 ρhdy ds = ρha(y) 3 dy 3 . (36)
V x (0, y) = ρha(y)dyω 2 (w + aθ x 2 ), (37)
M x (0, y) = ρha(y) 3 dy 3 ω 2 θ x . (38)
From the impedance matrix definition of Eq. ( 17), the boundary condition at x=0 is expressed as an initial impedance condition:

Z 0 = 1 jω   O ρhω 2 3b b 0 a(y) 3 Ψ q Ψ j dy ρhω 2 b b 0 a(y)Ψ q Ψ j dy ρhω 2 2b b 0 a(y) 2 Ψ q Ψ j dy   . (39)
In the general case, matrix Z 0 does not have a purely analytical expression. The reality of imperfections is complex but this model gives a simple way to represent it with a single parameter a(y). This parameter has to be small with regard to the wavelength at x=0 for the model to be valid and the bars to respect the assumption of inflexibility. Following this hypothesis, the imperfect termination adds inertial effects but no elastic effects.

The elementary academic imperfection presented here induces wave conversion mechanisms similar to those induced by a complex imperfection. The couplings due to elementary imperfections can then be studied.

Because of a(y) is supposed to be small, terms ρhω 2 2b b 0 a(y) 2 Ψ q Ψ j dy and ρhω 2 3b b 0 a(y) 3 Ψ q Ψ j dy can actually be neglected in regards of ρhω 2 b b 0 a(y)Ψ q Ψ j dy. The model of Eq. ( 39) is then simplified in the followings: the effects of rigid bars are restricted to their localised masses, therefore

Z 0 = 1 jω   O O ω 2 b b 0 m(y)Ψ q Ψ j dy O   , (40)
with m(y) = ρha(y). Computing S x=0 is immediate using Eqs. ( 31), [START_REF] Iserles | On the implementation of the method of magnus series for linear differential equations[END_REF] and (40).

2.6. Numerical resolution for the ABH profile 2.6.1. Magnus method

230

A Magnus scheme [START_REF] Pagneux | Multimodal admittance method in waveguides and singularity behavior at high frequencies[END_REF][START_REF] Schiff | A natural approach to the numerical integration of Ricatti differential equations[END_REF][START_REF] Lu | A fourth-order magnus scheme for helmholtz equation[END_REF] can be used to solve the Riccati equation ( 18) in a similar way as in [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF]. It is used in this paper for computing the scattering matrix of a section. A fourth-order Magnus scheme is used. The waveguide is discretized along the x-direction with a constant step. The Magnus scheme is applied to Eq. ( 11) and yields:

235 W(x n+1 ) = e Ωn W(x n ), ( 41
)
where xn is the longitudinal discrete coordinate and Ω n is the fourth-order

Magnus matrix [START_REF] Iserles | On the implementation of the method of magnus series for linear differential equations[END_REF]:

Ω n = ∆ 2 (H 1 + H 2 ) + √ 3 12 ∆ 2 [H 2 , H 1 ], (42)
where ∆ = xn+1xn is the constant spatial step,

H 1 = H(x n + 1 2 - √ 3 6 ∆), (43)
H 2 = H(x n + 1 2 + √ 3 6 ∆), (44)
and [H 2 , H 1 ] is the commutator between H 2 and H 1 .

Computation of the elementary scattering matrix from Magnus scheme

The scattering matrix S elem of an elementary section of length ∆ is obtained from Magnus scheme. Combining Eq. (41) and Eq. ( 21) gives

V xn+1 = E -1 xn+1 e Ωn E xn V xn . (45)
The local transfer matrix Q [START_REF] Wang | Waves in cylindrical shells with circumferential submembers : a matrix approach[END_REF] is identified in Eq. ( 45):

Q = E -1 xn+1 e Ωn E xn =   Q 1 Q 2 Q 3 Q 4   . (46)
Rewriting V xn+1 and V xn in V out and V in (referring to the elementary section) allows to reorganize matrix Q in order to obtain the scattering matrix S elem of the elementary section:

S elem =   Q 1 -1 -Q 1 -1 Q 2 Q 3 Q 1 -1 Q 4 -Q 3 Q 1 -1 Q 2   . (47)
In the following, the computation is done with ∆ = 10 -4 m and 30 modes have been taken into account. A convergence study (not presented in this paper) shows that using 30 modes for this configuration in the frequency range of interest is sufficient to accurately model the effect of the imperfection.

The number of modes used should be in any case greater than the number of propagating modes at the ABH tip at the highest frequency.

Z 0 = - jm 0 ω b   O O Ψ j (y 0 )Ψ k (y 0 ) O   . (49)
In this configuration, mode 1 is propagating above 360 Hz in the homogeneous region (this is assessed further in Sec. 3.3) and the next odd mode is mode 3 which is propagating above 2000 Hz. Therefore, numerical results concerning mode 1 are shown in the 400-2000 Hz frequency range.

Typical results for the reflection coefficient

Fig. 6 presents variations of reflection coefficient (R + pp ) 11 (of incident mode 1 on reflected mode 1) with frequency, for terminations described in Tab. 1

Geometrical characteristics Characteristics of material

Aluminium plate

L=0.8 m, b=0.1 m, E 1 =70 GPa, η 1 =0.2 %,
x ABH =0.06 m, x 0 =0.01 m, ρ 1 =2700 kg.m -3 , ν=0.3.

h 0 =1.5 mm, m=2.
Damping layer

h l =0.1 mm, E 2 =7 GPa, η 2 =20 %, x l =0.05 m. ρ 2 =1000 kg.m -3 .
Imperfection

y 0 =0.05 m m 0 =1.5× 10 -5 kg
Table 1: Geometrical and material characteristics of the simulated ABH waveguide.

with or without imperfection. The reflection on the ABH extremity without imperfection and without damping layer (full gray line) is close to unity. This is not surprising since the non covered ABH is known to be inefficient [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF].

Covered ABH without imperfection (dashed gray line) gives an interesting result as its reflection coefficient can be as low as 0.5 with oscillations; this results can be compared to earlier results [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF].

The case of the non covered but imperfect ABH termination (single mass at the centre of the cross-section, full black line) is mostly identical to the perfect termination except that it displays several very deep minima localised in frequency. With a damping layer the imperfect termination (dashed black line), |(R + pp ) 11 | clearly leads to a smaller reflection than the perfect covered case; the difference can be as much as 0.1. Notice that local minima seen on the non-covered imperfect extremity can still be observed in the covered case but are much less localised in frequency.

Cut-off frequency

In an homogeneous structure of a given thickness, whether a mode q is propagating or not only depends on the frequency according to the dispersion relation for mode q (see Eq. ( 22)). In an inhomogeneous structure such as the tapered profile studied here (see Fig. 7(a)), the propagating behaviour 295 depends on both the frequency and the thickness, therefore on the axial coordinate.

At a given frequency, it is shown on Fig. 7(b) that the eigenvalues associated with propagating waves of modes q=1, 3, 5, 7 (for example) are functions of abscissa (see Eq.( 22)). A mode becomes propagating at its cut-300 off abscissa, which is indicated by the change of nature of the associated eigenvalue (real or imaginary). These cut-off abscissas allow for each mode q to define coloured region on Fig. 7(a). Fig. 7(c) shows that the cut-off frequency of mode q depends on the abscissa, i.e. on the local thickness of the waveguide. In this case, in the uniform region of the waveguide mode 1 is propagating above 360 Hz while modes 3, 5 and 7 are not. They are thus confined in the tapered extremity.

Reflection induced by the tapered profile itself

The geometrical acoustics analysis [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF], which is based on the WKB approximation and only considers propagating waves, suggests that an incident wave reflects only at the boundary and that there is no reflection inside the tapered profile. The model presented in Sec. 2 gives the scattering properties in a region ]0, x ABH ] of the waveguide that does not include the free boundary condition. Zero reflection implies that the scattering matrix is diagonal: it contains only transmission terms and reflection terms are zero. Scattering matrices S ]0, x] and S [0, x] are computed with respect to frequency and allow to observe the reflection of the ABH profile with or without the free condition, respectively. Fig. 8 shows indeed that for mode 1, there is a non zero reflection along the tapered profile that does not depend on the free boundary condition: the tapered section reflects a low part of incident waves (full gray line).

Note that covering the profile with a damping layer does not affect this behaviour consistently (full black line). The observed reflection coefficient displays oscillations that can directly be compared to oscillations in the full ABH termination (i.e. including the boundary condition, dashed black line).

Oscillations of the reflection coefficient have been observed first in [START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF], where the authors hypothesize that oscillations are due to sharpness and length of the profile. Note that the geometrical acoustics approach and the associated WKB solution [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF][START_REF] Krylov | Acoustic 'black holes' for flexural waves as effective vibration dampers[END_REF] does not yield oscillations; therefore, it may indicate that the approximations made in this approach are not valid for the studied 330 geometry, especially the approximation of sufficient smoothness. This is also recently suggested in a study of Feurtado et al. [START_REF] Feurtado | A normalized wave number variation parameter for acoustic black hole design[END_REF] who study the parameters yielding a smooth profile, but was also briefly discussed in Ref. [START_REF] Krylov | Acoustic 'black holes' for flexural waves as effective vibration dampers[END_REF]. ]0, xABH] with damping layer (full black), without damping layer (full grey) and of the reflection matrix R + (x) with damping layer (dashed black).

Coupling mechanisms and mode trapping

Let us consider a propagating incident wave q arriving at the extremity.

The reflection matrix R + pp (x = 0) is displayed on Fig. 9(a). It shows the mode coupling at the imperfect free end: the diagonal terms of |R + pp | represent the reflection of a mode on itself, and are less that unity in this case. The out-ofdiagonal terms represent the reflection of an incident mode on one or several scattered modes; some of these terms are non-zero. The incident energy is thus redistributed on modes that are propagating at the extremity. This phenomenon is present at any frequency. The case of a single mass located in the centre of the extremity induces that only odd modes are concerned with couplings, as expected. Note that while an incident mode q is partially reflected on mode j, the incident mode j is partially reflected on mode q.

Let us now consider a reflected wave j resulting from the scattering described above. The wave is coming from the edge and its interaction with the propagating medium is described by the reflection matrix R - pp associated to scattering matrix S ]0, x] . The reflection matrix R - pp is diagonal. x = 0

x j c q q j j q j (1) (2)
(3) Figure 10: Scheme of the excitation and reflection mechanisms in the imperfect ABH termination. x j c indicates the cut-off abscissa for mode j. In step (1) the incident mode q is scattered on mode j, in step (2) mode j is reflected towards the extremity and in step (3) mode j is scattered on mode q.

3.6. Effect of the imperfection parameters 3.6.1. Effect of the position of the imperfection Let us consider the same mass figuring the imperfection at two more positions: y 0 =b/3 and y 0 =4b/5. Reflection matrices for these cases are plotted on Fig. 11 and show the couplings between modes; note the slight differences compared to Fig. 9(a). Fig. 12 shows the results for |(R + pp ) 11 | in these configurations. In the case of the non covered tapered profile, reflection coefficients for b/2 and b/3 are different as they do not display the same minima: this can be expected since they do not couple the same modes. When the termination is covered, the three cases are rather similar (see Fig. 13). It is however noticeable, that the best performance is reached when y 0 =b/2, which is the position of maximum displacement for mode 1. of |(R + pp ) 11 |, assuming that there is no cancellation effects. This fact suggests that naturally obtained imperfections are not necessarily detrimental and that controlled irregularities of the ABH extremity can be used in order to enhance its damping performance for a given minimum thickness; it is reminded that a thinner extremity provides better results.

Effect of the number of imperfections

Experimental investigations on an imperfect ABH termination

Experimental setup

This experiment uses the setup and the method detailed by the authors in 395 [START_REF] Denis | Measurement and modelling of the reflection coefficient of an acoustic black hole termination[END_REF] for measuring the reflection coefficient of a beam. The method is similar to Kundt-method in acoustics. A wave model and a least-square technique are used to get the reflection coefficient from frequency response functions at several locations on the beam. The tested beam is vertically suspended and is excited by a swept-sine from 100 to 8000 Hz using a shaker. Beam velocity 400 is measured using a laser vibrometer (Polytec OFV-400) at 21 abscissas in a 0.1 m long measurement zone.

The unique sample consists in an aluminium beam with an ABH extremity (see Fig. 14(a)), whose parameters are described in Tab. 2. The tip is as clean and undamaged as possible but displays a small permanent deformation 405 (less than a millimeter) and a small notch. A thin damping layer (striped tape) is stuck on the flat side of the tapered profile (see Fig. 14(b)). This configuration is referred as "non-damaged". With the help of a FEM model (COMSOL) of this beam, the first transverse ABH mode can be found at 1050 Hz. After a first measurement of the reflection coefficient of the beam, an point mass (tin drop) is stuck at the tip of the tapered profile (see Fig. 14(c)). This second configuration is referred as "damaged". A second measurement of the reflection coefficient is then made on the damaged beam.

Reflection coefficient of an imperfect ABH extremity

The resulting reflection coefficient for the two configurations can be seen in Fig. 15. On the one hand the reflection coefficient for the non damaged configuration (gray curve) decreases with oscillations below 4000 Hz to reach very low values (0.1 in magnitude) above 4000 Hz. This behaviour, including the oscillatory phenomenon, is conform to the ABH models developed in the literature [START_REF] Krylov | New type of vibration dampers utilising the effect of acoustic 'black holes[END_REF][START_REF] Georgiev | Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect[END_REF][START_REF] Denis | Measurement and modelling of the reflection coefficient of an acoustic black hole termination[END_REF]. On the other hand, the damaged configuration (black curve) displays a similar behaviour but presents lower values in the 0-4000 Hz region. Above 4000 Hz, there is no sensible difference between the two configurations. The differences in magnitude can reach 0.

Conclusions

This paper investigates the role of imperfections that are usually found at the tip of the ABH tapered profile when it is made very thin, due to the limitations of the manufacturing techniques.

A multimodal model of inhomogeneous waveguide with an ABH termination with a simple imperfection is developed in order to obtain the scattering matrix of a region of the waveguide as well as the reflection matrix of the ABH extremity. It is numerically solved using a Magnus scheme. It is shown that an elementary imperfection on the free extremity of the tapered profile affects the reflection coefficient and reduces it. Effects are spectrally localised when the damping is weak, but are broadly extended when the damping is increased due to the viscoelastic layer. At a given frequency, guided modes can

Figure 1 :

 1 Figure 1: (a) Nominal shape of the ABH termination (perspective view), (b) scheme of imperfect ABH termination and (c) picture of imperfections taking place at the tip of an ABH extremity (top view). Deformations come from stress relaxation during machining.

Figure 2 :

 2 Figure 2: Model of plate with ABH extremity: edges along x are simply supported, edges along y are free. (a) Side view and (b) front view.

Figure 3 :

 3 Figure 3: Region Ω of the waveguide delimited by x = [t -, t + ]. Incoming and outgoing waves are represented.

Figure 4 :

 4 Figure 4: Boundary condition at x = 0 and region ]0, x ABH ]. The assembly of the two regions defines region [0, x ABH ] (see Eq. (33)).

Figure 5 :

 5 Figure 5: Model of imperfection as an extension constituted of infinitesimal rigid bars: (a) side view and (b) top view.

3. Scattering induced by imperfections 3 . 1 .

 31 Characteristics of the simulated waveguideThe case of an ABH waveguide made of aluminium is considered. Parameters used in the numerical applications of the model are summarized in Tab. 1; parameters for the damping layer are realistic but do not result from characterization tests. Moreover, as a first step the imperfection is modelled by a single mass localised at (x = 0, y 0 = b/2). The mass represents about 0.2% of the total mass of the ABH termination. This mass localisation induces that only waveguide modes with odd numbers are concerned with possible coupling. The initial condition (40) then simplifies:

11 |Figure 6 :

 116 Figure 6: Modulus of the reflection coefficient (R + pp ) 11 for perfect non covered (full gray line), perfect covered (dashed gray line), imperfect non covered (full black line) and imperfect covered (full dashed line) ABH terminations (parameters of Tab. 1).

Figure 7 :

 7 Figure 7: (a) Thickness profile of the extremity with zone of propagation of modes k < (white), k < 5 (light gray), k < 7 (dark gray) and k > 7 (black) at 1000 Hz. (b) Real (grey) and imaginary (black) parts of λ 1 (thick full), λ 3 (thin full), λ 5 (dashed) and λ (dashdotted) at 1000 Hz. (c) Cut-off frequency for mode 1 (thick full), 3 (thin full), (dashed), 7 (dashdotted).

Figure 8 :

 8 Figure 8: Modulus of the term (1, 1) of the reflection matrix R + ]0, xABH] with damping layer (full black), without damping layer (full grey) and of the reflection matrix R + (x) with

Fig. 9

 9 Fig. 7(b). A mode j excited by the imperfection at x=0 can then be reflected toward the extremity x=0 at some point in the tapered profile. A scheme for the proposed excitation and reflection mechanisms is displayed on Fig. 10.

Figure 9 :

 9 Figure 9: (a) Reflection matrix |R + pp | at point x=0 and for f =1000 Hz and (b) Third (full), fifth (dashed) and seventh (dashdotted) diagonal terms of the reflection matrix |R - pp | of the region ]0, x] at f =1000 Hz.

Fig. 13 presents 11 |Figure 12 :

 131112 Fig. 13 presents results on |(R + pp ) 11 | for combined imperfections. Combination b/2+b/3 and b/2+b/3+4b/5 are studied. Note that theses configurations combines the minima of cases studied in Sec. 3.6.1. Cancellation of mutualeffect may appear at some frequencies when the termination is not covered.Broad drops of |(R +pp ) 11 | appears (at 1600 Hz) that are the consequence of two close consecutive minima. In the covered case, combining imperfections seems beneficial for the reflection coefficient which is reduced in the 500-700, 900-1100 and 1400-2000 Hz ranges when masses are added to the model.The gain reaches 0.2 at some frequencies. Compared to Sec. 3.6.1, it appears that multiplying the imperfections helps obtaining a significant reduction

11 |Figure 13 :

 1113 Figure 13: Modulus of the reflection coefficient (R + pp ) 11 for imperfect cases of ABH termination with mass at y=b/2 (black), y=[b/2;b/3] (light grey) and y=[b/2;b/3;4b/5] (dark grey), non covered (full lines) and covered (dashed lines).

  Geometrical characteristics Characteristics of materialAluminium beamL=0.8 m, b=0.02 m, E 1 =70 GPa, η 1 =0.2 %, x ABH =0.06 m, x 0 =0.01 m, ρ 1 =2700 kg.m -3 . h 0 =1.5 mm, m=2.Imperfection (tin drop)y 0 =0.01 m m 0 =3.6× 10 -4 kgTable2: Geometrical and material characteristics of the sample ABH.

Figure 14 :Figure 15 :

 1415 Figure 14: (a) Machined side of the tapered profile, (b) flat side of the tapered profile, covered with viscoelastic layer (striped tape) and (c) extremity with stuck tin drop.

be propagating in the tapered region while they are attenuating in the thicker uniform region of the waveguide. They are thus confined or localised in the tapered region. Due to couplings at the extremity, an incident waveguide mode is partially reflected on multiple modes. Because of the imperfection, guided modes that are confined in the ABH are excited, leaving the possibility of local resonances that can explain the drop of the reflection coefficient for the incident mode. Furthermore, the results suggest the use of controlled imperfection of the tip of the ABH profile in order to enhance its damping performance. A key point for researchers and designers of ABH systems is that imperfect extremities are not detrimental and can at best be beneficial.

Moreover, it is also shown that waves may be reflected inside the tapered profile itself and not only by the extremity. It suggests that many experimental ABH profiles do not match the sufficient smoothness conditions described in [START_REF] Mironov | Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval[END_REF] and can be improved practically to avoid internal reflections.

Finally, a measurement of the reflection coefficient of an artificially damaged ABH extremity is realised and shows that it differs from the reflection coefficient of a non damaged extremity. It is used to qualitatively confirm the results of the model.

Appendix A. Assembly rule for scattering matrices

In order to compute the scattering matrix of a whole inhomogeneous guide from scattering matrices of elementary regions, it is useful to define an operator that combine the scattering matrices S A and S B of two consecutive regions A and B. If S AB is the scattering matrix of the concatenation of the two regions, it can be shown that S AB can be computed from S A and S B by using a operator [START_REF] Vemula | Attenuation of waves in plates and bars using a graded impedance interface at edges[END_REF]:

with

Eq. (A.1) is also called assembly rule for scattering matrices. The four sub- .6)