O Robin

A Berry

N Atalla

M Aucejo

B Lossouarn
email: boris.lossouarn@lecnam.net

L Rouleau
email: lucie.rouleau@cnam.fr

J F Deü

C Marchetto

L Maxit

Setting Up Plane And Thin Panels With Representative Simply Supported Boundary Conditions : Comparative Results And Applications In Three Laboratories

Keywords: Rectangular plates, structural response, sound radiation PACS numbers: 43.40.At, 43.40.Dx, 43.40.Rj

Compared to numerous theoretical studies, the problem of experimentally realizing supported conditions on plane panels has received very little attention. A technique to setup a simply supported rectangular plane panel for laboratory vibroacoustic tests is described. Several application cases are reported, including three panels built in Groupe d'Acoustique de l'Université de Sherbrooke (GAUS, Sherbrooke, Canada), three panels built at Laboratoire de Mécanique des Structures et des Systèmes Couplés (LMSSC, Paris, France) and one at Laboratoire Vibrations Acoustique (LVA, Lyon, France). Covered subjects include vibration and sound radiation under deterministic and random excitations (diffuse acoustic field, turbulent boundary layer), similitude laws and passive control of vibrations using piezoelectric patches or viscoelastic treatments. All obtained results highlight the interest of having access to a laboratory tool that behaves closely to an exact mathematical model, and confirm that such a tool can be reproduced and generalized between laboratories.

INTRODUCTION

The book that Arthur Leissa published in 1969 [START_REF] Leissa | Vibration of plates[END_REF] showed the already vast literature that concerned the free vibrations of rectangular plates with '164 pertinent references', even when excluding 'such complicating effects as orthotropy, in-plane forces, variable thickness, the effects of surrounding media, large deflections, shear deformation and rotary inertia, and non homogeneity' [START_REF] Leissa | The free vibration of rectangular plates[END_REF]. Considering all possible combinations of the three classical boundary conditions (i.e. free, clamped, or simply-supported), 21 distinct cases exist for rectangular plates. The specific case of finite simply supported plates is often used as a core problem or statement since such boundary conditions lead to a simple and exact analytical solution to the plate equation of motion.

In [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF] was thoroughly described a technique that allows realizing representative simply supported boundary conditions on a panel and obtaining a functional experimental mean that can easily adapt to typical vibroacoustic laboratory measurements. Three different application cases were reported that concerned mechanical impedance measurement, sound transmission loss and active control applications, and several works from the same laboratory (GAUS) were also cited as possible applications [START_REF] Robin | Experimental vibroacoustic testing of plane panels using synthesized random pressure fields[END_REF][START_REF] Berry | Identification of dynamic loading on a bending plate using the Virtual Fields Method[END_REF][START_REF] Robin | Experimental evidence of modal wavenumber relation to zeros in the wavenumber spectrum of a simply supported plate (L)[END_REF].

This paper reports experimental results obtained using panels built following [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF] in different laboratories. Details (especially theoretical ones) are voluntarily skipped, and the interested reader should pay attention to the indicated references. The three goals of this communication are 1. to briefly recall the methodology presented in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF],

2. to complement previously presented results with other tests conducted at GAUS laboratory since 2016, 3. to include results obtained in other laboratories with panels built following the methodology described in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF].

All the presented results highlight once more the interest of having access to such a laboratory tool. Finally, point 3 confirms that this tool can be effectively reproduced and generalized between laboratories which is a core result.

THEORETICAL FRAMEWORK

The problem under study considers a homogeneous and isotropic rectangular plate of length L x and width L y , and with a thickness h considered small compared to L x and L y (only the transverse deflection w(x, y; ω) is considered, thin panel hypothesis). For such condition, the free vibration equation for small harmonic displacements is

D ∂ 2 ∂x 2 + ∂ 2 ∂y 2 2 w(x, y; ω) -ρhω 2 w(x, y; ω) = 0, (1)
where ρ is the mass density of the plate, ω the angular frequency and D the complex bending stiffness ( D = Ẽh 3 /12(1 -ν 2 ), with Ẽ the complex Young's modulus including the structural damping factor η so that Ẽ = E(1 + jη), and ν the Poisson's ratio). Achieving ideal simply supported conditions implies supporting a plane and thin structure, for which any edge can rotate freely while being restricted from out-of-plane displacements. Bending moments and transverse displacements are thus null on the plate boundaries. In this case, the eigenfrequencies ωmn and eigenfunctions φ mn , solutions of Eq.( 1) take simple closed-form expressions :

ωmn = D ρh 1/2 mπ L x 2 + nπ L y 2 , (2)
and

φ mn (x, y) = sin mπx L x sin nπy L y , (3)
with m, n non-zero, strictly positive integers. The displacement response of the plate at any (x, y) location is given by the following modal expansion

w(x, y; ω) = ∞ m=1 ∞ n=1 q mn (ω)φ mn (x, y), (4)
where q mn (ω) is the complex modal amplitude which for a given modal force F mn (depending on the nature of the plate's excitation) can be written

q mn (ω) = 1 ρhL x L y F mn ω2 mn -ω 2 .
(5)

The ratio of vibration velocity v to an arbitrary force F , Y (ω) = v/F , is the mobility function of the structure. Y (ω) can be derived using Eqs. [START_REF] Leissa | The free vibration of rectangular plates[END_REF][START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF][START_REF] Robin | Experimental vibroacoustic testing of plane panels using synthesized random pressure fields[END_REF][START_REF] Berry | Identification of dynamic loading on a bending plate using the Virtual Fields Method[END_REF], with v calculated as the displacement w multiplied by jω, and finally

Y (ω) = jω 1 ρhL x L y ∞ m=1 ∞ n=1 φ 2 mn (x, y) ω 2 mn -ω 2 . (6)

APPROACHING ACTUAL BOUNDARY CONDITIONS OF A SIMPLY SUPPORTED PANEL

The accurate modeling of the boundary conditions applied along the edges of a structure as their practical implementation share common difficulties. Boundary conditions are generally modeled using a rotational spring of stiffness k r and a linear spring of stiffness k f . Unlike ideal simply supported boundary conditions that imply a null rotational stiffness and an infinite transversal stiffness (i.e. k r = 0 and k f = +∞), realistic simply supported boundary conditions are usually achieved with a small but finite rotational stiffness, and a large but finite transverse stiffness (a small amount of stiffness restrains the edges rotation, and a small but non-zero movement is allowed in the z-axis). The simply supported boundary condition will be achieved here by rigidly attaching the plate's edges to glued thin vertical supports. The supports that will now be called 'blades' throughout the paper, should be flexible enough to provide a small but finite rotational stiffness, but also rigid enough to support the panel's weight and provide sufficient translational restraint. The structure under consideration is a plane panel with dimensions (L x × L y × h) and Young's modulus E. The dimensions of the vertical blades are (L b × l b × h b ), and Young's modulus E b . At a given frequency, the rotation stiffness can be approximated by an equivalent rotational spring constant per unit length for the blades named k b r :

k b r = E b h 3 b 12l b . (7)
A coefficient k p r may also be defined for the panel to be supported :

k p r = Eh 3 12L x , (8)
A simple and necessary condition for approaching simply supported boundary conditions would be k b r < k p r . In order to better predict the error on the plate eigenfrequencies induced by imperfect boundary conditions regarding to perfect simply supported boundary conditions, a criterion based on the ratio of these two coefficients was proposed in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF] and using a similar material for both panel and blades so that their Young's modulus are identical, that is

k b r k p r = h 3 b L x h 3 l b . ( 9
)
This ratio is used to relate frequency shift of eigenfrequencies when the height and thickness of the supporting blades are varied for fixed dimensions of the plate. Only the length L x of the panel is considered for calculations since for a given value of the ratio k b r /k p r calculated with L x and set as an upper bound, the criterion will be implicitly satisfied for L y if L x > L y .

In [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF], it was found that the values of the ratio k b r /k p r nearly vary as the percentage deviation on the frequency of mode (1, 1). Since the percentage change in terms of frequency tended to be smaller for higher modes compared to the fundamental one, the first vibration mode can be used to define a higher bound for frequency precision (that is then reached for all the other vibration modes). The ratio k b r /k p r can be linked to an objective upon frequency precision and be used as a design parameter to define adequate couples of height and thickness ((l b , h b ) values) for the blades supporting a given panel with fixed length, width and thickness. Such couples of values may be defined using a mapping as given in Fig. 4 in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF]. If a maximum frequency deviation %∆f of approximately 5% compared to ideal simply supported conditions is targeted on the first natural frequency of a panel (panel A in this case, for which dimensions are given in Table 1), the following simple empirical condition was finally proposed in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF]

: %∆f ≤ 5% ⇔ k b r k p r ≤ 0.1.

PANEL CONSTRUCTION

This section briefly describes the general method used to construct all the panels that have been used in this paper. Figure 1 schematically describes the main elements and accessories needed. The choices for the panel setup and its constitutive components were guided by simplicity, with only eight main components (parts A to H) and simple building steps (see Figure 2).

Precise descriptions and additional technical details for all needed parts (A to L) are provided in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF]. The frame is made of parts A, B, E and F, all made of steel. The panel (part H) and supporting blades (parts C and D) are made of aluminium. It is precised that steel is chosen for the frame partially for reasons of cost, but mainly to ensure a high 'panel-and-edges to frame' mass ratio so that the frame nearly behaves as a rigid and massive foundation from the panel's point of view. A mass ratio of approximately 0.09 is finally obtained for panel A, described in section 5.1 (with a weight of 20.7 kg for the frame, and 1.8 kg for the panel and edges).

Figure 2 then gives visual instructions for assembly, which are to follow from left to right and top to bottom. It is precised that a main advantage of this setup is that the 'panel-and-blades' part can be easily disassembled and reassembled from the frame.

APPLICATION CASES

LMSSC laboratory

In this section, the pursued application is the study of vibration damping, either provided by viscoelastic patches or by piezoelectric patches (see Figures 3(b-c), respectively). Those experiments follows previous works on (1) inverse characterization of viscoelastic materials using vibration measurements on structures [START_REF] Rouleau | Viscoelastic characterisation of adhesives using inverse techniques[END_REF] and (2) vibration control using piezolectric coupling with a passive electrical network composed of 42 elemental cells [START_REF] Lossouarn | Multimodal vibration damping of a plate by piezoelectric coupling to its analogous electrical network[END_REF]. Three aluminium panels of dimensions 420 × 360 × 3 mm 3 were built at LMSSC laboratory following the indications provided in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF], and are used to evaluate damping performances of different strategies, and perform a crossvalidation between simulation and experimental results. For the bare panel (i.e. without any damping device), operational deflection shapes were obtained using a scanning laser vibrometer (see figure 3(a)). The corresponding results for six vibrations modes [(1,1), (1,2), (3,1), (3,2), (4,1), (4,2)] are reported in Fig. 4, with an excellent agreement with theoretical mode shapes (see Eq. 3). Also, in Table 1, the theoretical and measured eigenfrequencies are reported and confirm the very good agreement between analytical and experimental results. As for the GAUS panel (see next section), the largest deviation is seen for the first vibration mode. For the ten first vibration modes, the mean percentage deviation is 1%.

The theoretical mobility function (vibration velocity upon force) is calculated at the excitation point by using Eq. ( 6) for a unit force input. The experimental mobility function at the excitation point was extracted for the two firstly built panels. Figure 5 shows the comparison of the two experimental and the single theoretical mobility functions. The agreement between analytical and experimental results is very satisfactory on the considered frequency range. The similarity of response between the two built panels is especially impressive. Figure 5: LMSSC laboratory -Measured driving point mobility for two plates of same geometry successively built, compared with an analytical result obtained with Eq.( 6).

Fig. 6 shows preliminary results comparing damping performances of passive viscoelastic and piezoelectric treatments (the case of the bare panel is compared with those two treatments). To achieve damping, the viscoelastic solution consists in a constrained layer damping, while the array of piezoelectric patches is coupled to a multi-resonant passive electrical network. Over the frequency range of interest (0-900 Hz), the amplitude of the first ten resonance peaks is reduced by more than 20 dB. This validates the efficiency of both solutions and allows comparisons with numerical models.

GAUS laboratory

This part of the paper concerns similitudes for the vibration of plates under a random excitation (like the turbulent boundary layer), and both analytical and experimental analyses were conducted in [START_REF] Robin | Exact Geometric Similitude Laws for Flat Plate Vibrations Induced by a Turbulent Boundary Layer[END_REF][START_REF] Robin | Scaling laws for flat plate vibroacoustic response induced by deterministic and random excitations[END_REF]. Additional details can be found in these two references, and results concerning similitude for the dynamic vibration response and radiated acoustic power of flexural plates under a point mechanical excitation are provided in [START_REF] Robin | Scaling laws for flat plate vibroacoustic response induced by deterministic and random excitations[END_REF]. For the case of a turbulent boundary layer excitation, it was shown that under the assumption of proportional sides, perfect similitude in terms of vibration response scaling can theoretically be achieved between plates of variable thicknesses. A scaling procedure was derived using a modal approach and including scaling coefficients for both structure and excitation. Indeed, when the case of a panel under a random excitation is considered, the excitation and the plate vibration response must be both described in terms of auto-spectral and cross-spectral density functions. A modal approach can be considered to calculate the panel response to this excitation, and the auto spectral density function of the displacement at point A(x A , y A ) due to an assigned random distributed

G ww (x A , y A ; ω) = N i=1 M k=1 φ i (x A , y A )φ k (x A , y A ) Z * i (ω)Z k (ω) S p (ω)(L x L y ) 2 γ i γ k A QiQ k (ω), (10)
where for the plate i-th structural mode (i and k indices corresponding to couples of (m, n) values), Z i is the plate modal impedance, γ i = a 0 b 0 φ 2 i (x, y) dy dx, and the terms A QiQ k (ω) are usually known as the acceptances: joint acceptance for i = k, or cross acceptance for i = k. Finally, S p (ω) is the single point spectral density of the forcing pressure field. Theoretical details concerning the calculations for reference panels and derived similitude laws can be found in [START_REF] Robin | Exact Geometric Similitude Laws for Flat Plate Vibrations Induced by a Turbulent Boundary Layer[END_REF][START_REF] Robin | Scaling laws for flat plate vibroacoustic response induced by deterministic and random excitations[END_REF] The interest of setting up a scaled family of simply supported panels was to provide an experimental validation of the theoretical scaling procedure. Three plates are considered and their dimensions are reported in Table 2. In table 1 are also reported the theoretical and measured eigenfrequencies for the panel A, as well as the corresponding percentage frequency deviation (with the largest deviation for the first vibration mode which is a known result, see [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF]). For the ten first vibration modes, the mean percentage deviation is 1.8%. The three panels were all built following [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF]. The material properties are those of a typical aluminum alloy, with a Young's modulus value E = 70 GPa, a mass density ρ = 2700 kg.m -3 , a Poisson's coefficient ν = 0.3 and a structural damping η = 0.02 (considered constant as a function of frequency). The derived similitude ratios (or scaling coefficients) are also reported in Table 2 (denoted r x , r y and r h for length, width and thickness, respectively). The plate A is considered as the reference plate, so that measurement results for panels B and C are scaled to reobtain panel A response.

Measurements were conducted in the anechoic flow facility at Université de Sherbrooke. Each plate was flush-mounted in a wood baffle placed at the convergent nozzle exit, and mechanical decoupling was ensured and verified (the panel was individually supported by its own stand). Seven accelerometers were glued on the panel in order to estimate the vibration autospectrum and cross-spectrum under turbulent boundary layer excitation. The experimental measurements were carried at scaled flow speeds, with speeds of 35, 28 and 21 m/s (for panels A, B and C, respectively). Acceleration autospectral density measured at several discrete points on the three parent panels at different flow speeds are finally scaled following the theoretical procedure defined in [START_REF] Robin | Exact Geometric Similitude Laws for Flat Plate Vibrations Induced by a Turbulent Boundary Layer[END_REF][START_REF] Robin | Scaling laws for flat plate vibroacoustic response induced by deterministic and random excitations[END_REF]. The unscaled and scaled experimental results are presented in figure 7. The agreement between the three structural responses is very satisfactory up to a frequency of 1.5 kHz and confirms the validity of the proposed scaling approach.

LVA laboratory

As a complement to the previous section, another method for calculating the vibration response of a panel to a random excitation considers coupling so-called sensitivity functions and a cross-spectral density function describing the excitation, now both defined in the wave number domain. The one-sided frequency auto spectral density function of the velocity G vv (x, f ) at point x (x, y) can be approximated by a discrete summation [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF]:

G vv (x, y; ω) ≈ 1 4π 2 k∈Ω k |H v (x, y; k; ω)| 2 G pp (k, ω) δk. (11)
with k = (k x , k y ) is the wave vector defined in the plane (x, y), δk represents the wave number resolution and Ω k is the finite wave number domain over which the discrete integration is performed. The function G pp (k, ω) corresponds to the cross-spectral density function of the wall-pressure field (for instance a diffuse acoustic field or a turbulent boundary layer excitation, that can be described using models). The term H v (x, y, k, ω) is called the sensitivity function and characterize the vibration behavior of the panel. A first goal in this work was to show that the sensitivity functions (see Eq. ( 11)) could be determined using the reciprocity principle, which makes it easier to be determined from an experimental point of view (and is a key point into the methods proposed in [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF][START_REF] Marchetto | Experimental prediction of the vibration response of panels under a turbulent boundary layer excitation from sensitivity functions[END_REF]). Theoretically, these functions are defined as the panel's response to wall-pressure plane waves and characterize the panel's vibroacoustic behavior. Practically and thanks to the used ))

SCALED results

Plate A Plate B Plate C simply-supported panel, it has been shown that these functions can be determined from the panel velocity field in the wavenumber domain when the system is excited by a source of unit amplitude at a point of interest. The panel used in this section is similar to panel A in section 5.1, with the difference that the panel and edges part had to be rebuilt at LVA after being damaged during transport between GAUS and LVA laboratories. Compared to original panel A from GAUS, it includes possible variability on construction. The cousin panel at LVA was used in [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF] while the original panel A (see Table 1) was used at GAUS in [START_REF] Marchetto | Experimental prediction of the vibration response of panels under a turbulent boundary layer excitation from sensitivity functions[END_REF]. One great advantage in this study was to be able to conduct distant and parallel tests (between Canada and France) with two panels that behaved closely with differences lower that 1 %. In figure 8 are compared the measured sensitivity functions (using the reciprocity principle) to theoretical ones. The result at a frequency of 178 Hz corresponds to the (2,1) vibration mode, whereas the results at 600 and 1709 Hz are off-resonance cases. The numerical and experimental results are generally in very good agreement, especially within the circle of radius k 0 , the acoustic wave number.

Once the reciprocity approach for determining the sensitivity was validated, the vibration response of the panel to a diffuse acoustic field excitation could be calculated (Eq.( 11)). Figure 9 compares the obtained numerical and experimental results. A good agreement is observed between the two results, which allows validating the proposed methodology for the case of a diffuse acoustic field excitation (prediction of the sound transmission loss was also shown to be feasible [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF]). The methodology was also applied and validated for the case of a turbulent boundary layer excitation [START_REF] Marchetto | Experimental prediction of the vibration response of panels under a turbulent boundary layer excitation from sensitivity functions[END_REF], still using panel A for testing (now in a wind-tunnel) and by performing comparisons with analytical results (obtained for a simply-supported panel).

CONCLUSION

This paper described a technique for realizing representative simply supported boundary conditions on a panel and obtaining a functional experimental mean that can easily adapt to typical vibroacoustic laboratory measurements. In order to ease the spread of this laboratory tool, precise technical guidelines were given with an assembly plan in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF] hoping that it could be acquired, used and improved by the research community. All the obtained experimental results in the three different laboratories and applications cases shown good agreement with analytical solutions and numerical predictions. This confirms the interest and the robustness of the proposed methodology for building panels with such boundary conditions. The results obtained in LMSSC and LVA laboratories with panels built following the instructions provided in [START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF] show that the tool can be effectively reproduced between laboratories.

Figure 1 .

 1 Figure 1. Simply supported panel schematic parts list.

Figure 2 .

 2 Figure 2. Assembly plan (to be read from left to right and top to bottom).

Figure 3 :

 3 Figure 3: Measurement at LMSSC using vibration shaker and scanning laser vibrometer -(a) Bare panel -(b) Viscoelastic patches -(c) Piezoelectric patches.

Figure 4 :

 4 Figure 4: Operational deflection shapes measured at LMSSC laboratory -(a) mode (1,1); (b) mode (1,2); (c) mode (3,1); (d) mode (3,2); (e) mode (4,1); (f) mode (4,2).

Figure 6 :

 6 Figure 6: LMSSC laboratory -Measured mobility for bare panel, panel with viscoelastic treatment and piezoelectric patches.

  Sides, same thickness) 600 525 3.2 1.25 1.25 1.00 C (Proportional Sides, = thickness) 600 525 2.4 1.25 1.25 0.75

Figure 7 :

 7 Figure 7: Measurement at GAUS laboratory -Unscaled (upper part) and scaled (lower part) mean vibration autospectra -flow speeds of 35, 28 and 21 m/s are considered for panels A, B and C, respectively (adapted from [9]).

Figure 8 :

 8 Figure 8: LVA laboratory -Calculated (upper part) and measured (lower part) sensitivity functions at three different frequencies using vibration shaker and scanning laser vibrometer -The indicated circle has a radius k 0 , the acoustic wavenumber -Adapted from [11].

Figure 9 :

 9 Figure 9: LVA laboratory -Velocity ASD functions G vv (dB, ref. 1 m 2 s 2 ) obtained using Eq.(8) with theoretical and measured sensitivity functions (a theoretical model for the diffuse acoustic field is used) -Numerical result (bold gray line) vs experimental result (light black line) -Adapted from[START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF].

Table 1 :

 1 Theoretical and experimental eigenfrequencies for the ten first modes, and corresponding percentage deviation. GAUS laboratory -panel dimensions : 480 × 420 × 3.2 mm 3

	Theoretical frequency
	(Real part, Eq. (2))	77	177.3 207.9 304.1 344.3 426.1 475.2 526.3 578.1 693.3
	(Hz)
	Measured frequency
	at response peak (Hz)	82.1 181.3 212.5 307.8 348.4 431.3 478.1 530.6 584.4 696.9
	Percentage frequency
	deviation %	6.6	2.3	2.2	1.2	1.2	1.2	0.6	0.8	1.1	0.5
	LMSSC laboratory -panel dimensions : 420 × 360 × 3 mm 3
	Theoretical frequency
	(Real part, Eq. (2))	97.5 221.4 266.1 390 427.9 547.2 596.5 671.1 717	877.6
	(Hz)
	Measured frequency
	at response peak (Hz) 102.3 225.4 269.8 391.9 430 547.7 595.2 670.2 718.3 883.1
	Percentage frequency
	deviation %	4.9	1.8	1.4	0.5	0.5	0.1	0.2	0.1	0.2	0.6
	excitation can be found with the following modal expansion:

(m,n) indices of considered mode (1,1) (2,1) (1,2) (2,2) (3,1) (1,3) (3,2) (2,3) (4,1)

[START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF][START_REF] Robin | A plane and thin panel with representative simply supported boundary conditions for laboratory vibroacoustic test[END_REF]

Table 2 .

 2 Plate dimensions (mm) and corresponding scaling coefficients

ACKNOWLEDGMENTS

The authors thanks first their respective laboratory technicians : Patrick Levesque (GAUS), Frédéric Guillerm (LMSSC) and Patrick Blachier (LVA) for their precious technical assistance. Also, Robin Darleux contributed to FRF measurements at LMSSC. Finally, the great contribution of three Italian researchers : Francesco Franco, Sergio De Rosa, Elena Ciappi to the presented similitude results must be especially underlined.


Ihsan El Masri
email: ihsan.elmasri@univ-brest.fr

Pierre-Marie Martin
email: pmartin@univ-brest.fr

Hemanta Kumar Mondal
email: kumar-mondal@univ-ubs.fr

Rozenn Allanic
email: allanic@univ-brest.fr

Cédric Quendo
email: quendo@univ-brest.fr

Christian Roland
email: christian.roland@univ-ubs.fr

Jean-Philippe Diguet
email: jean-philippe.diguet@univ-ubs.fr

Accurate Channel Models for Realistic Design Space Exploration of Future Wireless NoCs Special Session Paper

Keywords: Dipole antennas, electromagnetic propagation, network-on-chip, parallel application, wireless channel modeling

Wireless Networks-on-Chip (WiNoC) are being explored for parallel applications to improve the performances by reducing the long distance/critical path communications. However, WiNoC still require precise propagation models to go beyond proof of concept and to demonstrate it can be considered as a realistic efficient alternative to wired NoC. In this paper, we present accurate 3D models based on measurements in Ka band and Electromagnetic (EM) simulations of transmission on silicon substrate in the V band and the Sub-THz band. Using these EM results, a time-domain simulation is performed using an On-Off Keying (OOK) modulation based transmission with different PA/LNA configurations. Our results highlight the type of performances and tradeoffs to be considered according to different parameters such as power output and amplifier's gain. By improving the knowledge about the signal propagation, one can conduct precise design space exploration for parallel applications. We discuss the realistic channel modeling and we present also hybrid solutions and associated limitations of WiNoC architectures. We conclude the paper with research directions to be explored to make WiNoC a reality.

I. INTRODUCTION

The emergence of the High-Power Computing (HPC) applications, leads to manycore architectures, which necessitates the use of efficient network-on-chip. However, the future manycore architectures will require large NoC that can lead to prohibitive latencies due to long multi-hop paths. Parallel computing relies on cache coherency protocols and synchronization mechanisms (locks, barriers, conditions). The efficiency of cache coherence protocols and synchronization is strongly penalized by multi-hop paths and by sequential messages. There is a great need of alternative solutions to avoid this penalization in parallel computing.

Wireless Networks-on-Chip (WiNoC) is considered as a promising solution in the context of manycore architectures to replace the multi-hop communications [START_REF] Mineo | An adaptive transmitting power technique for energy efficient mm-wave wireless NoCs[END_REF]. The use of wireless links allows single-hop long distance communication between distant nodes. WiNoC also allows taking benefit of real broadcast capabilities. Most existing WiNoC architectures are based on On-Off Keying (OOK) modulation technique as it is power and area efficient. Currently, CMOS-compatible NoC with millimeter-wave (mm-wave) wireless links proves reasonable bandwidth and energy efficiency. With technology advances, mm-wave transceiver can also provide high bandwidth to fulfill the application needs. [START_REF] Deb | Wireless NoC as Interconnection Backbone for Multicore Chips: Promises and Challenges[END_REF] However, WiNoC still suffer from important limitations. The first one is the power consumption of the analog part of the transceivers with its order of magnitude (tens of mW) larger than the power consumption of digital components. Apart from the transceiver component in WiNoC, channel modeling is one of the major concerns. The multiple important radio parameters with strong impacts on communication quality and efficiency are not considered or based on simple and non-meaningful models. Accurate models are complex but signal propagation in air or silicon layers must be taken into account with its physical phenomena implications such as multipath interferences, metal interference structures and timedomain dispersion [START_REF] Zhang | Propagation Mechanisms of Radio Waves Over Intra-Chip Channels With Integrated Antennas: Frequency-Domain Measurements and Time-Domain Analysis[END_REF]. Thus, there is a need of a detailed investigation on channel modeling and antenna models before proceeding to a complete realization of the wireless links at a network level.

The BBC project (on-chip wireless Broadcast-Based parallel Computing) [START_REF]Présentation -CominLabs[END_REF] has as an objective the realization of wireless interconnects for parallel computing application. A complete study is entailed from the radio and physical layer (i.e. the design and study of an integrated antenna /transceiver and of the propagation channel) through the MAC layer to the digital layer (broadcast protocol/architecture).

In this article, we present in the context of BBC an accurate model based on measurements and electromagnetic (EM) simulations of transmission on silicon substrate, in order to facilitate the efficient design of on-chip wireless communication infrastructure. After validating the problems caused by the silicon substrate by measurement, we propose a solution and test its feasibility by EM simulation on 2 different simulation tools and deduce its implications and benefits on the overall system. The retained multilayer architecture for BBC project is illustrated in Fig. 1. The remainder of the paper is organized as follows: a brief overview of existing channel modeling works is presented in Section II. Section III describes the propagation channel measurement and analysis in frequency domain. The time domain analysis of propagation channel is illustrated in Section IV. Overall summary of this work including feasibility, scalability and overheads is discussed in Section V. We conclude our work in Section VI.

II. RELATED WORKS

Emerging interconnect architectures tend to reduce performance limitations of multi-hop communication in conventional wired NoCs. Four emerging interconnects are explored to address the long-range multi-hop communication bottleneck [START_REF] Karkar | A Survey of Emerging Interconnects for On-Chip Efficient Multicast and Broadcast in Many-Cores[END_REF]: three-dimensional (3D) [START_REF] Hwang | 3D nanosystems enable embedded abundant-data computing: special session paper[END_REF], photonic [START_REF] Ohashi | On-Chip Optical Interconnect[END_REF], RF/wireless NoC [START_REF] Boussougou | Electromagnetic analysis of RF interconnect[END_REF]. The mm-wave based WiNoCs have emerged as one of the promising solutions for scalable, energy-efficient CMOS compatible technology. Another alternative to wireless interconnects is an inductive/capacitivecoupled 3D integration technology [START_REF] Kikkawa | Wireless inter-chip interconnects[END_REF], [START_REF] Miura | A 195-Gb/s 1.2-W Inductive Inter-Chip Wireless Superconnect With Transmit Power Control Scheme for 3-D-Stacked System in a Package[END_REF], however it produces electromagnetic interferences through the unwanted coupling. Thus, we choose to use mm-wave interconnects as it is the most adequate for broadcast application.

The WiNoC system has two main components on the physical layer: an antenna and a transceiver. The OOK modulation is generally envisaged for WiNoC as associated low power transceivers are easily implementable in CMOS technology [START_REF] Lee | A Low-Power Low-Cost Fully-Integrated 60-GHz Transceiver System With OOK Modulation and On-Board Antenna Assembly[END_REF], [START_REF] Zhu | A Low-Power Low-Cost 45-GHz OOK Transceiver System in 90-nm CMOS for Multi-Gb/s Transmission[END_REF], [START_REF] Moghadami | A 210 GHz Fully-Integrated OOK Transceiver for Short-Range Wireless Chip-to-Chip Communication in 40 nm CMOS Technology[END_REF], [START_REF] Byeon | /s[END_REF]. The OOK modulation is the simplest amplitude-shift keying modulation with a data rate up to half of the available bandwidth. It is convenient for many high-speed applications and short-distance systems as in WiNoC case.

The on-chip integrated antennas for WiNoC operating in the GHz-range have been implemented in [START_REF] Triantafyllou | High transmission performance integrated antennas on SOI substrate for VLSI wireless interconnects[END_REF], [START_REF] Kim | Characteristics of integrated dipole antennas on bulk, SOI, and SOS substrates for wireless communication[END_REF], [START_REF] Tavakoli | An optimized phased-array antenna for intra-chip communications[END_REF], [START_REF] Sun | Performance of Intra-Chip Wireless Interconnect Using On-Chip Antennas and UWB Radios[END_REF], [START_REF] Benech | Design and performance of integrated antennas for wireless intra chip interconnections[END_REF], [START_REF] Bialkowski | Investigations into intrachip wireless interconnection for ultra large scale integration technology[END_REF]. Though several works highlight the advantages of WiNoC architectures, however, it is also important to note that some existing analyses are not based on physical implementations of on-chip wireless links and some analyses make the fundamental assumption that signal propagation is through the free space. Apart from these, few works have analyzed the behavior of signal propagation in intra-chip wireless channel [START_REF] Liu | Innovative propagation mechanism for inter-chip and intra-chip communication[END_REF]. In [START_REF] Zhang | Propagation Mechanisms of Radio Waves Over Intra-Chip Channels With Integrated Antennas: Frequency-Domain Measurements and Time-Domain Analysis[END_REF], propagation mechanisms in intra-chip channels is explored and shows the variation of path loss factor and propagation delay in free space transmission. In [START_REF] Cheema | The last barrier: on-chip antennas[END_REF] the WiNoC propagation problems involved with channel model on Silicon is addressed. However, no effective solutions were provided without modifying the CMOS structure.

None of the existing works thoroughly examine the effects of the complex intra chip geometries on wireless channel and signal propagation characteristics that can have significant impact on circuit design, network and system performance notably when the carrier frequencies increase. Thus, we provide an initial deep EM analysis of the channel, to be followed later by the effects of the metal interferences structures.

III. PROPAGATION CHANNEL MEASUREMENT AND ANALYSIS IN FREQUENCY DOMAIN

A first study concerns the design and the electromagnetic analysis of the antenna and of the propagation channel. Usually, in frequency domain, the channel is characterized through the scattering parameters (S-parameters) representation (S 12 and S 21 are the transmission coefficients and S 11 and S 22 are the reflection coefficients). Such parameters are computed by using EM field solver simulation tools or by direct measurement from realized prototypes. In this paper, Sparameters are used to investigate and estimate the channel behavior in several configurations of carrier frequencies. First of all, we validate the numerical electromagnetic model by comparison with measurements in Ka band and then we highlight the problem of multipath reflection of the EM wave on the boundary between air and silicon.

A. Propagation Channel Measurements in Ka Band

To be able to satisfy high data rate and small dimension antennas, the use of Extremely High Frequency (EHF) band (30-300 GHz) as the frequency range for propagation is investigated in this work. This choice leads to high data rates due to the high bandwidths expected. It is also interesting for the reduction of the size of the antenna due to the wavelength decrease. In order to validate the EM-Simulation, we have realized antennas in Ka band [START_REF] Mondal | Interference-Aware Wireless Network-on-Chip Architecture Using Directional Antennas[END_REF][START_REF] Mineo | Exploiting antenna directivity in wireless NoC architectures[END_REF][START_REF] Rayess | Antennas and Channel Characteristics for Wireless Networks on Chips[END_REF][START_REF] Demoustier | Review of two microwave applications of carbon nanotubes: nanoantennas and nano-switches[END_REF][START_REF] Huang | Performance Prediction of Carbon Nanotube Bundle Dipole Antennas[END_REF][START_REF] Abadal | Graphene-enabled wireless communication for massive multicore architectures[END_REF][START_REF] Pano | Wireless Network-on-Chip analysis of propagation technique for on-chip communication[END_REF][START_REF] He | Improvement of integrated dipole antenna performance using diamond for intra-chip wireless interconnection[END_REF][START_REF] Shinde | A 0.24pJ/bit, 16Gbps OOK Transmitter Circuit in 45nm[END_REF][START_REF] Subramaniam | A 0.36pJ/bit, 17Gbps OOK receiver in 45-nm CMOS for inter and intra-chip wireless interconnects[END_REF][START_REF] Sosa | A Diversity Scheme to Enhance the Reliability of Wireless NoC in Multi-Path Channel Environment[END_REF][START_REF] Ganguly | A Unified Error Control Coding Scheme to Enhance the Reliability of a Hybrid Wireless Network-on-Chip[END_REF](38)(39)(40).

Based on the architecture retained (Fig. 2) and to limit the complexity of the environment, we have studied the simplest propagation channel model on a simplified CMOS structure. It consists of printed dipole antennas as they are adequate to the WiNoC in broadcast applications (omnidirectional radiation pattern and simple design). The antennas were designed based on CMOS-like Structure with grounded High Resistivity silicon (HR-Si) with: a relative permittivity ε r = 11.9, a resistivity of ρ = 2.5 kΩ.cm and a thickness: Hsub = 655 μm. A small layer of 330 nm of SiO 2 (ε r = 4) is placed on the top level as an isolation. The dipole's length is close to L d = λ g /2, with λ g is the guided wavelength. The layer and the design of dipole antenna are presented in Fig. 2. In order to facilitate the probe measurements a transition between GSG (Ground Signal Ground) pads and the dipole antenna has been designed. The test structures have been realized at the Laboratory of GREMAN in Tours -France, based on the procedure explained in [START_REF] Allanic | Three-state Microwave Tunable Resonator Integrating Several Active Elements on Silicon Technology in a Global Design[END_REF]. The measurements have been done at the Lab-STICC laboratory in Brest -France using a probe station.

The results presented in Fig. 3 and Fig. 4 show a good agreement between the simulations (on the EM simulation Software HFSS TM ) and the measurement results which validate the EM-simulation tools and modeling.

The matching of the antennas is presented Fig. 3. In both cases measurement and simulation, a minimum of S 11 = -37 dB is obtained at 37 GHz. The presence of multiple resonance frequencies on the reflection parameter S 11 is observed. The difference between the -10 dB matching bandwidth in simulation F h1 -F l1 = 6.3 GHz (17.4%) and in measurement F h2 -F l2 = 11.1 GHz (30.8%) is due to the uncertainty on silicon losses, the antenna environment and the measurement conditions. [START_REF] Masri | Integrated dipole antennas and propagation channel on silicon in Ka band for WiNoC applications[END_REF] A good agreement between the simulated and measured S 12 transmission parameter is shown in Fig. 4. With a separation between the 2 antennas of 7.5 mm, the maximum of S 12 is -12.5 dB which is similar to other measurements found in literature [START_REF] Deb | Wireless NoC as Interconnection Backbone for Multicore Chips: Promises and Challenges[END_REF]. If we consider a variation of -3 dB in comparison of the maximum of S 12 parameter, a 4 GHz bandwidth is obtained. Notice that this band is clearly smaller than the -10 dB matching criterion previously presented. Similarly to the resonances observed in adaptation case, we notice the presence of the transmission dips in the transmission parameter S 12 . The dips limit the -3 dB bandwidth. They are due to a cavity effect caused by the difference of relative permittivity between the silicon and the surrounding air (11.9 vs. 1). This interface silicon/air behaves like magnetic walls around the silicon; associated with finite dimensions, it implies destructive interferences [START_REF] Cheema | The last barrier: on-chip antennas[END_REF]. The difference between rays traveling through different paths, mainly the path that goes directly in the air and the path that goes in the silicon also implies transmission dips.

B. EM Simulations in V band

To really highlight this phenomenon of cavity effect and multiple paths which induced dips in transmission we have conducted several EM simulations with different scenarios as depicted in Fig. 7. To clearly highlight the cavity effect, two simulation cases are compared in this section in the V band (40-75 GHz). At the end of this paragraph, a solution to overcome the negative behavior of the cavity is proposed.

As already was seen in measurement, the problem resides in the cavity formed along the silicon substrate and due to the reflections on the dielectric/air interface, the ideal case would be to consider the lateral dimensions of the dielectric substrate as infinite (i.e. radiation boundaries on the air box surrounding the substrate). In reality, the substrate will have finite dimensions and surrounded by air.

So, the 2 cases simulated and compared are:

1. Ideal scenario: Antenna on HR-Si with infinite lateral dimensions simulated using radiation boundary on the silicon substrate edges Fig. 7 1).

2. Real scenario: Antenna on HR-Si with finite lateral dimensions simulated using an air box surrounding the silicon substrate Fig. 7 2).

We have studied these scenarios considering 4 clusters of cores on a chip, and therefore 4 antennas are placed on the silicon substrate as per the Fig. 5. The distance of separation between 2 adjacent antennas is set to 10 mm (the distance between diagonal antennas is of 14.1 mm). The characteristics of the substrate were kept the same as those used in measurements in Ka band. The comparison between the 2 cases has been conducted on the EM SW CST Microwave Studio © (Similar results were validated on HFSS as well) and the Scattering S-parameters are presented in Fig. 6.

The reflection parameters S 11 (≈S ii ) (Fig. 6-a) shows a very different behavior between the two cases. In the case (1) with infinite dimensions a single resonance corresponding to the dipole resonance is observed as well as a large matching bandwidth (BW) that extends from 42 to 74 GHz (>50%). In case of finite dimensions, multiple resonances are present and it is difficult to identify the resonance due to the dipole.

The transmission parameter between diagonal antennas S 21 looks stable on a large BW in the case (1) and we can note a -3 dB BW over 30 GHz. While the chaotic parameter obtained in the case of a finite substrate dimension (2) will make the transmission impossible. For the transmission parameter between adjacent antennas S 31 (and its equivalents S 41 ), the same remarks can be said as for the diagonal transmission. Nevertheless, in the case (1), more fluctuations are observed which may limit the transmission bandwidth (multiple bands of 4 GHz around multiple frequencies are discernible). In the next section, we present a solution to avoid the cavity behavior.

C. Proposed Solution Simulations in V band

As the ideal case (1) is practically not realizable because of the limited dimensions of the chips, and to get back to this case we suggest surrounding the Silicon layer by an absorbing material layer. In this absorbing layer, the waves will be evanescent before reaching the air and thus the reflections would be reduced significantly. This proposed solution is called thereafter case (3) and is illustrated in Fig. 7-3. We can see in the literature propositions for absorbent materials like graphene as an example [START_REF] Wu | Experimental demonstration of a transparent graphene millimetre wave absorber with 28% fractional bandwidth at 140 GHz[END_REF].

A comparison between the cases (1) and ( 3) have been made using EM-simulation and the S-parameters results are presented in Fig. 8. The use of an absorbing layer permits to limit the number of resonances and to obtain a smooth variation like the infinite dimensions case. The contribution of the EM-absorbent can also be shown on the electric field distribution as depicted in Fig. 9. The fields' cartography in the case (2) seems chaotic, a huge number of modes interfere which is typical of an oversized cavity while in the case (3) fields distribution can be compared to the propagation in an infinite environment.

Lower levels of transmission are observed between adjacent antennas than between the diagonal antennas despite that the separation distance is higher for the diagonal antennas than for the adjacent ones. This is due to the radiation pattern of the grounded dipole antenna which presents lower gain in the adjacent antenna's direction than in the diagonal antenna's direction. Nevertheless, the proposed solution helps to bypass the silicon substrate problems. To avoid this directivity problem, a solution is to use monopole antennas placed inside the silicon substrate.

D. EM Simulations for Monopole Antennas in Sub-THz band

As can be seen in the above paragraph, the antenna directivity and placement plays an important role in improving the transmission for the WiNoC channel. This hypothesis was studied in [START_REF] Tavakoli | An optimized phased-array antenna for intra-chip communications[END_REF], [START_REF] Mondal | Interference-Aware Wireless Network-on-Chip Architecture Using Directional Antennas[END_REF], [START_REF] Mineo | Exploiting antenna directivity in wireless NoC architectures[END_REF]. But in each of these publications no deep EM analysis was provided. As broadcast applications are addressed, omnidirectional vertical monopoles are good candidates [START_REF] Rayess | Antennas and Channel Characteristics for Wireless Networks on Chips[END_REF]. In order to have a reasonable height compared to the substrate height, these monopoles have been designed in the Sub-THz band (at 200 GHz on silicon the wavelength is of ≈435μm, so a quarterwavelength monopole would be after optimization of ≈143μm height excited by a coaxial cable conform with their dimensions) as it is presented Fig. 10. These monopoles could be metallic if the technology permits, otherwise carbon nanotubes could be integrated [START_REF] Demoustier | Review of two microwave applications of carbon nanotubes: nanoantennas and nano-switches[END_REF], [START_REF] Huang | Performance Prediction of Carbon Nanotube Bundle Dipole Antennas[END_REF], [START_REF] Abadal | Graphene-enabled wireless communication for massive multicore architectures[END_REF]. EM simulation has been done considering a Silicon substrate surrounded by an absorbing layer as explained previously. The distance of separation between 2 adjacent antennas is set to 10 mm and between the diagonal antennas would be 14.1 mm.

The simulations have been conducted on HFSS TM and the results of S-parameter are shown in Fig. 11 for the reflection parameter and in Fig. 12 for the transmission parameters.

A good matching (S 11 <-20db) is obtained and the -10 dB bandwidth is extended to 62 GHz which represents a relative band of 31%.

As for the transmission parameters S 21 & S 31 , we can see fluctuations that stay in a level between -25 and -35 dB. This behavior may be explained by the multiple reflections vertically through the layers [32] [33] than by the horizontal reflections on the edges and borders of the substrate interfaces. The use of omnidirectional antennas implies higher transmission between the adjacent antennas than the diagonal antennas as the adjacent antennas are closer.

However, using high carrier frequency implies lower transmission levels for a same distance. This is due to the FRIIS equation who expresses the ratio of received power P r to transmitted power P t in 3D space as:

(1)

Where G t and G r represent the antenna gains, D is the separating distance between the two elements and  is the wavelength and  is the channel attenuation. So, the greater the frequency, the lower is this ratio. To evaluate the channel behavior in time domain, the S parameters obtained by measurements or EM simulations have been used in a very simple circuit modeling to obtain the eye diagram characteristics. This part is presented in the next section.

IV. TIME DOMAIN ANALYSIS OF THE PROPAGATION CHANNEL

A. Transceiver Design and Time Domain Simulations in Ka Band

To assess the channel performances, we have used the block diagram of a simple OOK transceiver shown in Fig. 13. The transmitter consists of a binary sequence generator and a sine wave generator multiplexed in an OOK modulator. The channel is formed by the touchstone file (.sNp) extracted from the measurements or the EM simulation between 2 or more antennas. The receiver includes a Low Noise Amplifier (LNA) of variable gain G and an envelope detector to recover the signal. Eye diagrams help us to check the state of the signal after each step and precisely at the receiver side.

The specific advantage of this transceiver is besides its simple design, the absence of a power amplifier at the transmitter side which will help us later to decrease the power consumption.

Based on the above design, the performances of the measured Ka band channel have been evaluated using ADS© circuit simulator. The aim of this simple study is to estimate the Signal to Noise Ratio (SNR) in several configurations of binary rate and amplifier gain. The noise figure of the LNA have been set arbitrarily at 10 dB and the filter of the envelop detector is simply modeled by RC cells. The simulation results are summarized in the Table 1. The representation of the 1V level: L 1 , the 0V level: L 0 , the height H and the width W are presented in Fig. 14.

The SNR is estimated using:

(2)

Where d 1 and d 0 are the noise levels for L 1 and L 0 respectively (also illustrated on Fig. 14).

The SNR depends both on the binary rate F bin and on the global gain G of the transmission chain. The higher the data rate F bin is, the higher the global gain G should be and thus the higher the power consumption. In Ka band, based on the channel measurements results, we set the carrier frequency to 35 GHz, for a data rate of 2 Gbps, no global gain implies an SNR of around 17 dB which seems sufficient for WiNoC applications [START_REF] Karkar | A Survey of Emerging Interconnects for On-Chip Efficient Multicast and Broadcast in Many-Cores[END_REF].For a data rate of 4 Gbps, the global gain needed to obtain an SNR greater than 12 dB is around 5 dB. In both binary rate cases, the use of 10 dB global gain ensures a good level of SNR implying a bit error rate lower than 10 -12 which is adequate for WiNoC applications.

B. Time Domain Simulations in the mm Wave Band

Based on the EM simulations, we use here the same transceiver design used earlier for the measurement in Ka band, to realize the time domain simulations for the case (3) in mm Wave band for the monopole antennas designed with a surrounding absorbing layer and a carrier frequency of 200 GHz.

The results of the eye diagram simulations are resumed in the Table 2 and Table 3 corresponding respectively to the adjacent antennas transmission case (10 mm distance) and the diagonal antennas transmission case (14.1 mm distance). The noise figure of the different amplifiers is set to 10 dB.

Due to the Friis relation, using higher frequencies implies the need of a greater amplifier gain. This is why to achieve a data rate greater than 4 Gbps, it is necessary to use at least a global gain of 15 dB; the greater the data rate, the higher the gain needed. As an example, if 8 Gbps is needed, a global gain of 25 dB minimum is required to achieve a SNR > 9dB.

The ripple of the S-parameters limits the band, while if the response is flat, higher data rates are possible. This drawback can be compensated by using a numerical equalizer as it has a limited power consumption compared to the analog part. The lower the transmission parameters level, the higher the global gain in the chain must be.

V. DISCUSSIONS

In this study we first observe that reflections and cavity effects must be considered to build a realistic WiNoC. We show that an absorption layer is required to get a workable solution and to avoid intractable placement constraints. Moreover considering technology constraints and cost, it is crucial that this layer is CMOS compliant. Another well identified issue is the channel uniformity with horizontal dipole antenna. We show that vertical quarter-wavelength monopole is a possible solution in the Sub-THz Band that can be integrated in a 200μm silicon layer.

The second point is related to power consumption, which is one of the main impediments to WiNoC implementation. PA and LNA are of the most significant sources of power consumption. PA efficiency is extremely low (17% in [START_REF] Shinde | A 0.24pJ/bit, 16Gbps OOK Transmitter Circuit in 45nm[END_REF]) and worse than LNA. Recent work in 45nm show very impressive power gain for PA [START_REF] Shinde | A 0.24pJ/bit, 16Gbps OOK Transmitter Circuit in 45nm[END_REF] and LNA [START_REF] Subramaniam | A 0.36pJ/bit, 17Gbps OOK receiver in 45-nm CMOS for inter and intra-chip wireless interconnects[END_REF], namely 3.9 and 1.3mw respectively, and the LNA efficiency is three times better (0.08 vs 0.24pJ/bit). So, we explore the possibility to get rid of the PA. Our simulations show that a 25dB gain provides 8 Gbps data rate for a 9.5dB SNR, which is enough to retrieve the information with a simple transmission scheme. If we assume a classic linear power model with respect to the gain, we can perform rough estimations. Let's consider a solution without a PA and with an LNA only from [START_REF] Shinde | A 0.24pJ/bit, 16Gbps OOK Transmitter Circuit in 45nm[END_REF]. We must increase the gain of the LNA from 16 to 25dB, which means a power consumption of about 2mW. This estimation must be compared to 5.2mW that is required for the LNA and PA case and a global gain of 32dB. On the contrary, note that if we consider a solution with a single PA from [START_REF] Shinde | A 0.24pJ/bit, 16Gbps OOK Transmitter Circuit in 45nm[END_REF], we must increase the gain of the PA from 16 to 25dB and obtain a power consumption of about 6mW. So, it seems to be possible to use an LNA only, this is a promising perspective since significantly lower power consumption means possible implementation and scalability. In other words, we can expect WiNoC with a significant number of routers.

Thirdly, let's consider the frequency Band. 200GHz allows designing vertical antennas but the drawback is the energy efficiency that decreases proportionally with the frequency. Compared to 60GHz, we get a loss of about 6dB, however it is already included in our simulation scheme and in our results. Moreover digital communications methods can additionally be applied to reduce power with lower SNR. This option is motivated by the power consumption of the digital part, which is an order of magnitude lower than the analog part. Basically, we can expect to introduce Direct Sequence Spread-Spectrum (DSSS) [START_REF] Sosa | A Diversity Scheme to Enhance the Reliability of Wireless NoC in Multi-Path Channel Environment[END_REF] with low spread factor and Error-Correcting Codes (ECC) [START_REF] Ganguly | A Unified Error Control Coding Scheme to Enhance the Reliability of a Hybrid Wireless Network-on-Chip[END_REF] techniques with high code ratio in order to maintain high data rates. Finally, digital equalization [START_REF] Sosa | A Diversity Scheme to Enhance the Reliability of Wireless NoC in Multi-Path Channel Environment[END_REF] can also be used to improve the bandwidth and so the data rate.

VI. CONCLUSION

Our study highlights the crucial need for realistic channel models that consider cavity effects and multi-paths. However, it also shows that WiNoC remains a promising alternative to prohibitive wired multi-hops paths and to provide efficient broadcast capabilities if appropriate design, frequency band, absorbing layer, antenna design and digital processing techniques are applied.

The immediate perspectives are firstly an analytical accurate channel model, thus we are currently running timeconsuming simulations. Then we intend to provide the details and the experimental results of the CMOS-compliant absorbing layer that was briefly introduced in this paper. Finally, we intend to combine accurate RF models and digital techniques to evaluate the consistency of the complete proposed solution.

Fig. 1 :

 1 Fig. 1 : The different layers of a Wireless network on Chip

Fig. 2 :

 2 Fig. 2: Design of dipole antenna: (a) stack layer; (b) 3D view of dipole antenna with feeding transition

Fig. 3 :

 3 Fig. 3: Comparison between measurement and simulation of the reflection parameter S11 of the structure

Fig. 4 :

 4 Fig. 4: Comparison between measurement and simulation of the transmission parameter S12 of the structure

Fig. 5 :

 5 Fig. 5: Scenario used for the 4 antennas

Fig. 6 :

 6 Fig. 6: S-parameters comparison for the ideal (1) and real (2) cases: (a) S11; (b) S21; (c) S31

Fig. 7 :Fig. 9 :

 79 Fig. 7: The 3 scenarios under comparison

Fig. 10 :

 10 Fig. 10: Monopole geometry for 200 GHz

Fig. 11 :

 11 Fig. 11: S11 parameter for the monopole antenna

Fig. 13 :

 13 Fig. 13: Electric circuit for OOK simulation and eye diagram modeling

Fig. 14 :

 14 Fig. 14: Basic parameters obtained on ADS for the channel evaluationTable 1: Main parameters of the Eye Diagram of the measured channel in Ka band F bin. (Gbps) Global (PA + LNA) Gain (dB)

Table 2 :

 2 Eye diagram characteristics for adjacent antenna transmissions in the mm Wave band

	F bin. (Gbps)	Global (PA + Gain (dB) LNA)	L1 (mV)	L0 (mV)	H (mV)	W(ps)	SNR (dB)
		15	18	3	-	-	7.9
	6	20	70	7	33	138	11
		30	405	53	195	145	12
		20	62	3	33	105	11
	8	25	167	12	93	105	13
		30	363	33	204	105	13.8
	Table 3: Eye diagram characteristics for diagonal antenna transmissions in the
	mm Wave band
	F bin. (Gbps)	Global (PA + LNA) Gain (dB)	L1 (mV)	L0 (mV)	H (mV)	W(ps)	SNR (dB)
		15	7	1	-	-	7
	6	20	50	5	7	110	8.6
		30	330	37	140	130	11,5
		20		Not significant
	8	25	124	10	36	100	9.5
		30	283	32	108	101	9,1

ACKNOWLEDGMENT

The authors would like to thank the "Laboratoires d'Excellence" COMINLABS and the TECHYP (the High Performance Computing Cluster of the Lab-STICC).


Jean-Christophe Valière

Towards a history of architectural acoustics using archaeological evidence: Recent research contributions to understanding the use of acoustic pots in the quest for sound quality in 11 th -17 th -century churches in France

Introduction

The history of Western acoustics has yet to be written, let alone the history of acoustic practices. For one thing, current knowledge is based on the great philosophical texts of antiquity, their transmission during the Medieval and Modern periods (respectively, the fifth to fifteenth centuries and the sixteenth to seventeenth), and the works of the first scientists (eighteenth and nineteenth centuries). For another thing, experiments in sound obviously do not leave archaeological traces, and we know of only a few historical objects whose deliberately acoustic functions are certain. Among these objects, acoustic pots are a rare example whose use is confirmed by texts, although the 'theoretical' bases underlying their installation still elude our full comprehension. An improved understanding of the intentions underlying the choice of pots and their insertion into religious buildings' walls and vaults will help us to discern the foundations of acoustic practices of the time.

In this article, we will first present our findings from fifteen years of measurements and observations conducted in some fifty churches in France and adjacent countries (Switzerland Germany). We will compare these results with the major historical texts on the subject and previous studies in France and neighbouring countries. We will then analyse how such pots are distributed in the liturgical space according to the function of the building, focusing particularly on the practice of frequency-tuning the pots, their placement in the building, and their association with decorative paintings related to angelic singing and the harmony of spheres.

Prior knowledge

Scholars and scientists rediscovered the technique of acoustic pots in the nineteenth and twentieth centuries, and were quick to offer interpretations of both their hypothetical function and historical origin. In our earlier publications (Palazzo-Bertholon andValière 2012, Valiere et al. 2013) we commented extensively on most authors' assertion that the practice spread as a result of the circulation of a text by the Roman architect Vitruvius ([1 st c. BC] 2009) 1 . Although his writings influenced the spread of pots as a technical concept during the Middle Ages, there is thus far no concrete evidence that medieval pots were directly modelled on Vitruvius' description. There are indeed examples of pots dating from the first centuries of the Christian era in the Mediterranean basin [START_REF] Zakinthinos | The effect of ceramic vases on the acoustics of old Greek orthodox churches[END_REF]Skarlatos 2007, Barone 2015), but recent studies show that pots only seem to have emerged in Western Europe (notably in France) at the end of the ninth century (Carolingian period) 2 , with a revival between the fourteenth and sixteenth centuries before their use was gradually abandoned. According to recent studies in Italy (Barone 2015), there is no proven temporal continuity between the Byzantine-influenced Sicilian churches with pots and those in the north of Italy, which may be more influenced by Western European building practices. Similarly, the remaining examples in Serbia are related to construction techniques of the Dalmatian coast (Croatia) influenced by northern Europe, with no association with ancient traditions (Đorđević Z. 2016) 3 . Proving a direct relationship between building practices in south-eastern and north-western Europe that progressively spread across Western Europe may not seem to be the most pertinent approach today, but this line of enquiry is still being actively pursued and could prove the source of new results in years to come.

For the purposes of this article, we will concentrate only on French and European examples that have been studied in the last sixty years (Palazzo-Bertholon and Valière, 2012) 4 . Our recent research, whose early findings have already been published, will serve as the basis for the analysis that we offer here. In sum, we have identified more than 400 buildings in Europe, and in France we have measured over 1000 pots in 41 churches (out of 50 visited). We conducted an archaeological survey of each building and collected any related available documents.

The guiding principle of our research is uncovering the technical knowledge that led contractors, builders, architects, or religious authorities (monks, canons, bishops) to insert pots in buildings. This knowledge was probably rooted in classical texts by Aristotle and Vitruvius that were read and discussed in the Middle Ages and in oral traditions (Anonymous 1432). The first architectural treatises [START_REF] Alberti | L'Art d'Edifier [De re aedificatoria[END_REF]) and translations of classical texts (Vitruvius 1547, Alberti 1553) are likely to have fostered the final resurgence of the technique in the Renaissance and Modern times, when the emergence of polyphonic singing was likely another important factor in their revival.

Analysis of these sources and comparison with the data we collected in the studied churches leads us to interpret acoustic pots according to three dimensions that we will refer to using the Latin terms Vox, Locus, Transitus.

The spoken or sung voice (Vox) is undeniably the builders' primary focus in pot placement. All studies, historical or scientific, agree on this fact, and there is no lack of evidence. In the first place, the texts are very explicit, as evidenced by one dating from 1432 found in Metz by scholars in the nineteenth century (Anonymous 1432) and others found in Switzerland [START_REF] Desarnaulds | De l'acoustique des églises en Suisseune approche pluridisciplinaire[END_REF]. From an acoustic point of view, the pots' resonance frequencies are always between 100 and 500 Hz (average 209 Hz), and for the most part between 100 and 200 Hz, meaning the lower frequencies of the human speaking voice or the most solemn notes of the singing voice. Pots were likely selected according to their vocal resonance frequency because every text on the subject since antiquity has attempted to explain the amplification of sound as a result of cavities. The explanation given by Saint Thomas Aquinas, picking up on Aristotle (2001 [4th c. BCE), is very explicit: "Hollow bodies give out sound very well, when struck, because they hold the air in, and, as this air first set in motion cannot at once come out, it disturbs the rest of the air, and thus by repercussion the impact and the sounding are multiplied" [START_REF] Aquinas | Commentary on Aristotle's De Anima[END_REF] 1994, p. 141).

Our second line of interpretation suggests that the pots were inserted to have a potential impact on the place (Locus). It is first of all important to distinguish between the two dimensionsthe acoustic source (the voice, discussed above) and the placebecause, given the understanding of acoustics at the time, it is conceivable that the pots were intended to improve perception of the voice, independent of placement. The hypothesis that pots were installed to act on buildings' acoustics is subject to debate, with solid arguments for and against 5 . One prominent claim in favour came from some twentieth-century acousticians that argued that the pots were an early technique for making what came to be known as a Helmholtz resonator, a low-frequency absorber used to improve the acoustics of small rooms such as recording studios. This is a point of strong contention. Regardless, today we know that the amplifying capacity of pots has been known since Aristotle, as seen in the quotation from Saint Thomas Aquinas. On the other hand, no text mentions the role of an absorber of acoustic energy.

Another point of contention lies in the fact that ancient texts rarely document the sound qualities of specific places, especially with reference to inserted pots (they only mention the resonance of the voice). In more recent analyses, reverberation (wrongly referred to as 'resonance') is often seen as beneficial to singing, rather than an impediment. This perspective is probably greatly influenced by recent enthusiasm for plain-song and the practice of religious singing outside of the liturgical sphere or by specialized monastic chorales (such as Ligugé or Solesme in France).

References in classical and medieval literature nevertheless point to a search for improving the sound quality of places. Vitruvius, for example, classifies places into four categories according to their sonority: resonant (with echoes), dissonant (damping), consonant (amplifying the sound) and circumsonant (reverberant). He takes an especially negative view of the latter type, saying that circumsonant places prevent a good understanding of speech [START_REF] Valière | Acoustic Pots in Ancient and Medieval buildings: Literary analysis of ancient texts and comparison with recent observations in French churches[END_REF]. Later, in the period when pots were installed in many churches in Europe, the Italian architect [START_REF] Alberti | L'Art d'Edifier [De re aedificatoria[END_REF] indicated in his treatise that wooden ceilings make it easier to hear the voice of the priest than the stone vaults of a church. Understanding speech in voluminous (and thus highly reverberant) places was a widespread problem that many tried to address, but the question remains as to whether pots can be understood as being such an attempt. No text sheds any light on this point, and it is reasonable to wonder about the builders' intent.

Our measurements produced some particularly compelling elements for consideration. The most important point is that the number of pots increases overall with the size of the building. It is also interesting to note that sometimes two groups of pots with two different central frequencies were chosen. These frequencies are often separated by a fourth or fifth interval, which is consistent with medieval and renaissance musical theories. The reason for using two (or more) tuned pot frequencies is subject to speculation. One explanation is the considerable influence of Vitruvius in the Middle Ages, who claimed that the acoustic resonators in Greek theatres (called echea) were tuned by fourth, fifth, octave, and double octave. The development of polyphonic singing during these periods could be a second explanation. If we apply the medieval conception of acoustics, it is plausible that the pots were inserted to amplify the voice, and that each group of pots was to stand for a group of voices. Obviously, as we indicated previously, this would be a misunderstanding of the real function of such resonators.

Nevertheless, our surveys show that this frequency-related technique is predominantly present in the larger churches of our sample, which can be interpreted as the pursuit of a more dramatic impact in larger buildings. In addition, a text on the Abbey of Montvilliers in Normandy indicates that the vault was lowered to 'relieve the voices' of the nuns 6 . Pots were inserted in this lowered vault, showing that an architectural action was associated with the acoustic intent of the pots.

The last dimension, which we refer to as 'Transitus', represents the symbolism of pot placement relative to the connection between secular and divine in medieval thought. They are sometimes arranged in the form of a cross or a triangle manifesting patterns of Christian symbolism, as seen in the church of Oberwinterthur in Switzerland (Palazzo-Bertholon, 2012).

Moreover, acoustic pots are iconographically associated with cosmic representations of the celestial vault: the stars and planets painted in Asbeck and Baindt churches (Germany) are scattered with acoustic pots. In the choir of San Severin of Cologne, the trumpets of angels painted above the stalls have acoustic pots inserted in each bell centre. Moreover, it is probably no coincidence that the acoustic pots placed at the horns' mouths are located above the stalls, since throughout the Middle Ages the voice was a metaphor for the spirit. The Church Fathers, and especially Saint Augustine, established that the relationship between humans and the heavens was built on song. Later, many medieval authors, including Amalarius of Metz (ninth century) or Regino of Prüm (tenth century), emphasized the idea that human music imitates angelic choirs. According to Guillaume Durand, bishop of Mende in the thirteen century. 'the two choirs of singers reflect the angels and the spirits of elected men who praise God with a reciprocal and mutual desire to do good'. In his theological view, God is the composer of all music and Christ is the conductor. The Holy Spirit provides the Divine breath that keeps the musical order in the worlds of Heaven and Earth. For Ugolino of Orvieto (1380-1457), 'all earthly music is a symbol of heavenly harmony'.

In addition, the study of the symbolism of acoustic pots follows a path signalled by one the great theologians of the twelfth century, Hildegarde von Bingen. In a letter to Elisabeth von Schönau, a Benedictine nun with whom she was friendly and corresponded regularly, Hildegarde makes a comparison between humans and clay pots, saying that although both are fragile, the resonance of each is nonetheless associated with divine inspiration: "I am a miserable little clay vase. This, I do not say of myself but in thanks to the true light: Man is a vessel which God created for himself and filled with his inspiration in order to accomplish his work" 7 . As composer and author of "The Symphony of the Harmony of Celestial Revelations", Hildegarde probably knew the ancient theories of the harmony of the spheres (Palazzo-Bertholon, 2016).

Blending medieval concepts, the acoustic pot is thus simultaneously an object useful for strengthening the voice in places whose sound quality is deemed unsatisfactory and a medium symbolizing the link between the temporal and the spiritual. With these three lines of general interpretation in mind, we will now consider the relationship between pots and liturgical space.

In order to make the discussion clearer, Vox will be used to refer to voice-related interpretations, Locus for those related to place, and Transitus for those associated with the symbolic.

Spacing and arrangement of pots

Variations according to church function

Let us emphasize first of all that acoustic pots are found in both parish and monastic churches. There are, however, statistical differences between location types. For one thing, we found that the trends described in the previous section are stronger in monastic or canonical churches than in parish churches. For example, pots with frequencies in the 100-200 Hz range are statistically more common in monastic churches. Similarly, the technique of associating pots of two frequencies, observed in half of our sample, is more frequent in monastic churches.

Thus far, findings from our current sample indicate that no monastic order predominates in the use of pots, which were found in churches of the Benedictines (Pommiers-en-Forez, Auvergne), Cordeliers (Notre-Dame des Anges, Brittany), and the Cistercians (Abbey de l'Epau, Loire Region). Similarly, churches of women's convents (Montivilliers, Normandy) are as concerned as monasteries for men. As a result, at this stage our analysis will only distinguish between monastic churches and parish churches, because, despite the magnitude of our study, findings do not permit us to identify specific patterns relating to particular monastic orders.

Heights

As a rule, pots are inserted as high as possible in the buildings: preferably at the top of walls, the base of vaults, or in vaults themselves [START_REF] René | Contribution à l'étude des Vases Acoustiques du Moyen Âge[END_REF]. This observation corresponds perfectly to a Transitus interpretation, in the sense that the amplification of sound (in intention, rather than reality) is essentially localized in the upper parts of the building. But it is not incompatible with an intention to act on the place (the Locus interpretation), since the overall effect of the pots is related to their number. It is for this reason that twentieth-century acousticians hypothesised that the pots placed in church masonry could be Helmholtz resonators, a more recently discovered acoustic device, and this conviction is strengthened by the fact that the number of pots increases with the volume of the church.

On the other hand, although the pots are tuned to the voice, it would be surprising if they were designed to amplify the audience's perception of the priest's voice like a sounding board, although period texts seem to credit this hypothesis.

Positioning with reference to the liturgical space

The placement of pots relative to the liturgical floor plan presents some interesting characteristics. We identified two distinct groups, the first (A) concerning churches whose pots are installed in the liturgical space (including the choir, the transept and the first bay of the nave), and the second (B) comprising churches in which the pots are dispersed throughout the building, especially in the nave.

Although pots are distributed almost everywhere in most churches (25 of the 36 churches surveyed) (group B), group A has a higher concentration of monastic churches (7/11) than parish churches (4/11). In contrast, group B contains a large majority of parish churches, meaning that they have more widely dispersed pots (15/25) relative to monastic churches (10/25). This situation is compatible with the Locus interpretation in the sense that the pots were thought to amplify the voice, so they act on the space where the faithful or monastics assemble.

For monastic churches, in many cases the pots are unsurprisingly located in the liturgical space (group A) (7/11). The most significant examples in France are in Pommiers-en-Forez and Montivilliers, where the pots are located just above the places where monks or nuns would have been positioned. In many monastic churches, they are placed behind the chancel separating monks from the faithful. It is not systematic in monastic churches, however: in Lavaudieu (Auvergne) and in the Abbaye de Mercoire (Auvergne), for example, the pots are distributed along the length of the nave (group B). It is likely that such churches were used exclusively by the congregation.

In the great majority of parish churches, distribution is quite regular: either in two lines in the vaults, or at the top of the walls and along the nave in particular. In addition to the symbolic interpretation already mentioned (Transitus), this regular distribution can also be associated with a better distribution of sound throughout the building (Locus).

Positioning frequency and tuning of pots

We have already mentioned the fact that pot frequencies were chosen deliberately and pots were attuned in many churches where two pot frequencies were chosen. But beyond the choice of frequencies, is their distribution in the church space consistent with a rationale of voluntary implantation?

In general, pots are aligned along the walls or in the vault, and each row may consist of pots of a particular frequency. At Saint-Blaise d'Arles, in Provence, the low frequency (220 Hz) is on the upper line, while the high frequency (330 Hz, a fifth higher) is in the lower line, in a relatively counter-intuitive placement. In Pommiers-en-Forez, the pots are arranged roughly along three lines, the low frequency being in the median position. In Ploaré (Brittany), the vast majority of low-frequency pots (141 Hz) are set in the lower line in the nave, while the upper line contains only higher-frequency pots (184 Hz, i.e. the top quartet). Although this is not a general rule, there are other cases where the high frequencies are placed at a height, which can be explained by a link between musical and pot frequency. This characteristic is found in various configurations. In Saint-Martin d'Angers, the low-frequency pots (100 Hz) are set in the lower portion of the church, while the high-frequency pots (150 Hz) are placed in the vaults. In the subterranean chamber of the cathedral of Noyon (also called a "caveau phonocamptique"), the higher frequencies are also mostly placed in the upper part of the walls.

The distribution of pots also follows a logical orientation within churches. Thus, in the Romanesque vault of Montivilliers 8 , the low-frequency pots (143 Hz) are placed to the south and west, while the high-frequency pots (199 Hz) are placed in the north and east walls. In Saint-Victor de Marseille, the low frequencies (143 Hz) are also in the north, and the high frequencies (199 and 280 Hz) more to the south. A distribution according to the points of the compass is found in the Noyon vault, where the low frequencies are arranged in the north and west walls, while more high frequency pots are placed to the south. During the emergence of polyphonic religious music, it is likely that pot distribution was related to the direction of the singing voices, so the low frequencies were located in a northerly direction, facing the choir. Note, however, that the chapel of Mercoire (seventeenth century) does not follow this rationale. Indeed, the pots are placed at the same level in the vault and tuned in fifth: the low-frequency (181 Hz) pots are located to the south and the east, while the high-frequency (279 Hz) ones are placed to the north and west. But the orientation of the chapel suggests that the altar was in fact located to the north, which would ultimately mean that the distribution of frequencies in the liturgical space fits the same pattern as the other examples.

Geometric patterns and association with painted scenes

In a previous article, Palazzo-Bertholon (2012) associates the geometric organization of pots with Christian symbolism. They are sometimes arranged in the shape of a cross or a triangle, as can be seen in Oberwinterthur, Switzerland, where the pots form a complex pattern on the wall that not only takes the forms of crosses and triangles, but also lines of numbers (7,10,11) whose Christian symbolism could be expanded upon. This arrangement, which to our knowledge is unique, is found in the church choir.

Acoustic pots were moreover associated with wall decorations (paintings and stuccoes) in Switzerland and Germany. In the churches of Asbeck and Baindt, pots are at the centres of painted stars representing the celestial vault. In these churches, the stars and heavenly bodies make celestial music 'too perfect to be perceived by human ears, according to Plato, or too deafening to be understood by men, according to Aristotle and Macrobe' (Palazzo-Bertholon, 2012).

Elsewhere pots are associated with angelic music. In the churches of Villette and Gelterkinden in Switzerland, the pot-associated murals represent angels. In Oetenbach monastery in Zürich and that of Gelterkinden (Switzerland), the pots are arranged in lines surrounded by a crown of painted points that seem to express the divine sound that escapes. At Saint-Severin in Cologne, two paintings face each other over the stalls in the liturgical space, and four angels blowing horns are depicted at each corner, framing the crowning and ascension of the Virgin Mary. The bell end of each angel's trumpet contains an acoustic pot that opens into the church space.

Synthesis and conclusion

The research presented in this article comes from the largest survey of acoustic pots in Europe, based on the work of previous researchers and including those from neighbouring countries. Archaeological, historical and acoustic data were systematically recorded in nearly fifty churches throughout France, and the included buildings covered a long historical time span, from the eleventh to the seventeenth centuries.

Survey results for France revealed that during the studied period, understandings of acoustics included several elements associated with geometry and music, two disciplines of the quadrivium. Medieval authors established a constant relationship between architecture and music, following the classical tradition inherited from Vitruvius. Architectural beauty depended on symmetry, eurythmy, and decoration. Symmetry corresponded theoretically and objectively to the musical symphony: all parts of a building should harmonize according to the law of exact proportionality. This definition of 'purely rational and rigid' beauty can also be applied to music that is created on the basis of mathematical and geometrical relationships. Interpretation of these results is a delicate business, as the study of acoustic pots has suffered (and still does) from overly hasty analyses that began with the earliest observations in the nineteenth and early twentieth centuries and continued in the first acoustical approaches in the mid-twentieth century.

The interpretation that pots were used as absorbers, in a kind of intuitive medieval pre-science, is not supported by texts on the subject, but neither does it rule out the possibility that builders intended to act on the acoustics of the place.

Prior knowledge and our own research first of all made it possible to confirm that pots were clearly implanted to act on the voice, and the sung voice in particular (Vox). We then demonstrated that the pots carried multiple strong spiritual and symbolic messages (Transitus) through their spatial arrangement and association with paintings and decoration. Finally, the improvement of the sonority of the place was probably another objective (Locus). We focused analysis in this article on the location of the pots within the building, taking other parameters such as height, frequency, orientation, and symbolism into account.

Our research has led to several significant findings. The pots are mostly arranged according to the place's use and function. In many monastic churches, the pots are placed above the officiants or in a horizontal line all along the nave, linking the assembly of the faithful with the liturgical space. The fact that pots are arranged according to frequency suggests a connection between their position (high / low) and the voice (high / low), referring to the usual representation of music (Vox), but also to symbolism (Transitus). Further study is needed to interpret this arrangement, especially with regard to the musical symbolism in images [START_REF] Charles-Dominique | Musiques savantes, musiques populaires : les symboliques du sonore en France (1200-1750[END_REF]. Analysis of the positioning of pots according to compass points reveals that most lowfrequency pots are positioned to the north, and high-frequency pots to the south. With the emergence of polyphony, there could be a link between the musical use of the place and the arrangement of singers within the space; this possibility requires further research, based on particular buildings with surviving documentation of musical and liturgical practices. Lastly, the spatial distribution and ornamental arrangement of the pots relative to wall decorations requires further study in order to understand the variants and their symbolism.

To conclude, the three dimensions that we propose for interpreting acoustic pots (Vox, Locus Transitus) provide a general framework for improving our understanding of what guided builders of the time, whilst avoiding the pitfalls of oversimplification or anachronism. Further research into the musical and scientific knowledge of the concerned periods is essential for accurate and complete interpretation of the data. Our research irrefutably shows that the intention was to act on the voice in a way that is dependent on the form and function of the place, together with the symbolism characteristic of the Medieval and the Renaissance periods.

  figure 1

Notes

1 Among the chapters of the book edited by Palazzo-Bertholon and Valière (2012), focused mainly on Germany, Italy (around Genoa), Spain, Croatia, Switzerland, and France, is a broad survey of acoustic pots in Europe and an exhaustive bibliography of old and recent publications. For an earlier summary in English, [START_REF] Arns | Resonant cavities in the history of architectural acoustics[END_REF] is a good starting point. See also [START_REF] Desarnaulds | Efficiency of 13 th century acoustic ceramic pots in two churches[END_REF], [START_REF] Mijic | Analysis of Contribution of Acoustic Resonators Found in Serbian Orthodox Churches[END_REF], [START_REF] Zakinthinos | The effect of ceramic vases on the acoustics of old Greek orthodox churches[END_REF]. 2 The oldest identified in Western Europe is the Abbey of Meschede (Germany) according to [START_REF] Kottmann | St. Walburga in Meschede: der karolingische Bau und das Schalltopfensemble[END_REF].

3 Zorana Đorđević's article on this specific point is in publication. The original source was published by [START_REF] Djuric | Dubrovački graditelji u Srbiji srednjeg veka[END_REF]. 4 In our study we also used the published data of Floriot (1964), [START_REF] Fontaine | Contribution à l'étude des vases acoustiques disposés dans les églises (XI e -XVIII e s.)[END_REF] and [START_REF] Desarnaulds | De l'acoustique des églises en Suisseune approche pluridisciplinaire[END_REF]. 5 See for instance [START_REF] Tallon | Acoustics at the Intersection of Architecture and Music: The Caveau Phonocamptique of Noyon Cathedral[END_REF], for whom a deliberate attempt to 'improve' the sound of the church is anachronistic. Clearly, even if some pots have a limited impact on sound quality in large churches, it does not mean that builders were not trying to improve the building's acoustics. Indeed, there is evidence that this was their intention, but based on Medieval and Renaissance conceptions of acoustics. 6 Recently discovered in the chronicle of the abbey of Montvilliers uncovered by Pauline Carvalho, who gives a complete description of the abbey (2012). 7 Ed. L. van Acker, CCCM, 91°, p. 456. In G. Iversen, Chanter avec les Anges. Poésie dans la messe médiévale, interprétations et commentaires, Paris, 2001, p. 250. 8 There are also two sets of pots at Montvilliers: one two-frequency set in the Romanesque vault, which were probably retroactively inserted (probably in the fifteenth or sixteenth century, the dating is in progress), and another set of pots of identical frequency in the seventeenth-century vault.