# Sound Scattering by Gothic Piers and Columns of the Cathédrale Notre-Dame de Paris

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Introduction to the History of the Construction of Notre-Dame de Paris

#### 1.2. General Acoustics of Notre-Dame de Paris

## 2. Materials and Methods

#### 2.1. Columns and Piers of Interest

#### 2.1.1. Compound Piers

**N1**, located at C/D4 in Figure 2, are engaged with four circular colonnettes on a circular core, as shown in Figure 3a. Just besides, the piers

**N2**, located at C/D5, are engaged, with a single one facing the nave (Figure 3b). The central part is the same diameter as the other columns of the nave arcades, except the columns at C/D8 that follow the principle of strong and weak piers, as their diameter is 125 cm. The colonnettes are engaged by less than a quarter of their diameter, penetrating 11 cm and 15 cm, respectively. This type of circular lobe shaft is also found engaged in every responds (Respond: Half-pier or half-column embedded in a wall.). They are piliers cantonnés, a type of pier already used in Romanesque architecture where massive rectangular piers are flanked by semicircular columns, as with Saint-Étienne de Caen Church, Santiago de Compostela or Mainz Cathedrals. Geometries similar to these two couples may be found in other High Gothic cathedrals, as with the naves of Notre-Dame de Chartres and Notre-Dame de Reims, or in the choir of Notre-Dame de Noyon. They are more robust variations of the columns known as “soissonnaises”, which first appeared in Soissons Cathedral, where a single colonnette is engaged in a circular core on the part facing the nave and rises to the vault [57].

**C1**, located at C/D11. They are actually the union of several pilasters, each one receiving a transverse or diagonal rib of the nave or crossing vaults. At the arcade level, this results in an asterisk-shaped section, as shown in Figure 3f. They are the largest piers of the selection, and, with the two previous ones, they directly surround the nave where listeners are located.

**T**, located at C/D2, are selected to study the influence of such shafts. Their cross-section is shown in Figure 3g. At each corner of the diamond shape is engaged a wider column of diameter 34 cm, and on the sides, there are alternately right corners and engaged colonnettes of diameter 19 cm. This pattern is repeated on the wall and outer aisle responds, between each chapels, of the choir [58].

**Ch**and its cross-section is shown in Figure 3c.

#### 2.1.2. Piers with Detached colonnettes en délit

**C2**, located at B/E11, at the entrance of the double aisles at the western wall of the transept. It is represented in Figure 3d. The prismatic piers formed by the union of the arcades and vaults dosserets (Dosseret: Pilaster used as a straight jamb for an arch.) are supplemented by detached colonnettes at each re-entrant corner, separated by a distance of 1 cm.

**N3**, located at B/E5/7/9, is surrounded by 12 detached colonnettes, as shown in Figure 3e. Viollet-le-Duc [60] explained this difference with the single circular cylinder neighbors by considerations of structural strength and stability. In particular, since these piers are in line with the most heavily loaded columns of the nave, they had to take the load of the buttresses, which had existed before 1220, and were allowed to counter the thrust from the sexpartite vaults. However, this has been challenged since [58], and Viollet-le-Duc himself acknowledged that they have a decorative function when installed at the responds after settlement of the building. A couple of them are also included among the coursed shafts that form the piers supporting the towers, located at B/E3. Many examples of such elements can be found in other cathedrals at the time, including, not exhaustively, the cathedrals Notre-Dame de Noyon, Saint-Étienne de Bourges, Notre-Dame de Dijon, and Notre-Dame de Laon [61]. Their use facilitated the way in which the walls are vertically articulated, with the vaults compared to coursed shafts with circular shapes, as in the choir responds. They could be manufactured in mass by standard processes while the walls were built with stones cut in regular rectangular shapes [62]. The piers surrounded by colonnettes are also at the origin of a whole architectural style in England [63]. We can give the examples of the Canterbury Cathedral, the Lincoln Cathedral, or the Salisbury Cathedral, where the colonnettes are in marble of a different color from the central part [64].

#### 2.2. Numerical Methods

#### Setup

^{−1}.

#### 2.3. Experimental Methods

## 3. Results

#### 3.1. Validation of the Experimental Methods with a Rigid Circular Cylinder

^{−1}as the maximum absolute difference. The variations of c between the repeated series are small, so they are compared to the analytic solutions for the plane and spherical incidences calculated with this average value.

#### 3.2. Measurements Compared to Simulations

**N2**and

**N1**(Figure 5a) are also rounded. In addition, for geometries with several elements, a bad straightness, and therefore, the positioning of the small cylinders leads to a different scattering, such as in Figure A2e in comparison to Figure A2f.

#### 3.3. Simulation Results

#### 3.3.1. Time-Frequency Analysis

`cwt`implemented in the Wavelet Toolbox version 5.5 of MATLAB 2020a. It has been computed using Morse wavelets, which can be expressed in the frequency domain

**N1**and

**N2**. In the time domain, the first arrival after the direct sound has a relative peak level of −10.5 dB, which is approximately the value found in the scalogram. A second arrival is visible in the scalogram at low frequency, 30 ms after the direct sound, corresponding to the creeping waves circumventing the cylinder that are strongly attenuated at high frequency. It is not visible in the signal as its relative peak level is −62 dB. For

**N2**with ${\theta}_{0}=90$° (Figure 8b), there are two visible arrivals, corresponding to the reflections on the two circular parts constituting the cross-section, with relative peak levels of −14.5 dB and −18.9 dB. The arrival due to the creeping waves is still visible in the scalogram, and has a level and frequency range similar to the circular cylinder. For

**N1**with ${\theta}_{0}=45$° (Figure 8c), the pressure signal is composed of a three localized pulses between 18 ms and 20 ms after the direct sound with −9.5 ± 1.1 dB relative peak levels. The latter arrivals are due to higher-order reflections between the different part of the cross-section that account for about 4% of the cumulative energy of the backscattered pulse. Its normalized wavelet scalogram also shows a spreading of low frequency, similarly to the previous geometries. For

**C2**with ${\theta}_{0}=90$° (Figure 8e), the wave packet has visible pulses at its onset and offset. They are attributed to the reflections on the plane faces of the cylinder whose normal is colinear with the direction of propagation. In comparison, those of

**N3**(Figure 8d) and

**T**(Figure 8f) look more like diffuse reflections [82].

**N3**,

**C2**, and

**T**, thus favoring multiple interactions during scattering. The column

**N3**(Figure 8d) has two resonances over the considered frequency range. The first one occurs at around the same frequency as

**C2**(Figure 8e), around 400 Hz. The second one is around 850 Hz and seems to decay slightly slower. The resonance of

**T**(Figure 8f) is around 700 Hz. They all have low amplitudes, so they hardly appear on the linear scale of the pressure signals and account for a very small part of the cumulative energy of the backscattered pulses, less than 0.2% for

**T**, for instance.

#### 3.3.2. Reflected-to-Direct Level Differences

**N2**,

**C1**,

**C2**, and

**N3**for two incidence angles for each one. We recall that they are derived from simulations representing the experimental set-ups reported in Table 1, where the distances of the receivers to the center of the section are indicated for each one. The one-third octave band RDLDs are shown on a semicircle only, since the configurations are symmetrical. Moreover, the RDLDs for the positions located in shadow zone are also represented; however, they can not be interpreted as such, because of the interference between the incident and scattered pressures occurring in this region.

**N2**, the two incidence angles considered result in strong spectral and strength differences across the scattering directions, as shown in Figure 9a. For ${\theta}_{0}=90$°, the overall RDLDs represented in Figure 9b are around 3 dB higher in the transverse directions compared to ${\theta}_{0}=45$°. For the piers

**C1**, the RDLDs are very similar, up to 1 kHz, as shown in Figure 9c. Above, the large planar parts of the section favor some directions, according to ray acoustics. These particular directions are therefore highlighted in the overall RDLDs, represented in Figure 9d, where they found their second maximum at around $-4$ dB and $-6$ dB for ${\theta}_{0}=90$° and 0°, respectively. Their maximum of about 2 dB is found in the backscattering direction, as these incidences are normal to the large plane faces of the cylinders. This is also the case for

**C2**, as shown in Figure 9e, where a positive value of nearly 1 dB is found in the backscattering direction for ${\theta}_{0}=0$°. For the two incidences considered, the overall RDLDs, represented in Figure 9f, differ mainly in this region, for ${\theta}_{s}\ge 150$°. Compared to the other section, the RDLDs for

**N3**seems to depend less on the incidence angle, as shown in Figure 9g.

## 4. Discussion

#### 4.1. Gothic Piers as Volumetric Diffusers

**C2**and

**N3**, especially as the small cylinders are close to the central part and to each other. This effect is particularly visible through the existence of resonance frequencies revealed for the latter, as well as for the compound piers with geometric elements of small size, such as

**T**. They are probably the result of coupling between the small cavities formed on the surface of the cylinders creating surface waves, as described in [83,84]. They are, by definition, localized in frequency, and in the cases studied here, count very little for the total energy of the reflections. However, around these frequencies, where the wavelengths are of the order of magnitude of the geometrical elements, i.e., up to about 1 kHz for the geometries considered here, the scattered power is increased without strongly favoring any particular direction. Contrary to beyond, in the limit of small wavelengths, the scattering directions can be determined according to the acoustic ray model, and the overall scattered power is related to the size of the shadow. See [85] for more detailed simulation results of different column geometries analyzed in terms of classical scattering quantities.

#### 4.2. Audibility of Scattering by Cylindrical Obstacles

**C1**, where values exist up to −6 dB and −4 dB, for ${\theta}_{0}=0$° and 90°, and ${\theta}_{s}=95$° and 113°, respectively.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FD | Finite difference |

FV | Finite volume |

RDLD | Reflected-to-Direct Level Difference |

SNR | Signal-to-noise ratio |

## Appendix A. Measurements vs. Simulations: Additional Examples

**Figure A1.**$|{p}_{s}/{p}_{i}|$ for the compound piers

**N1**with (

**a**,

**b**) ${\theta}_{0}=90$° and (

**c**,

**d**) ${\theta}_{0}=45$°, and

**C1**with (

**e**,

**f**) ${\theta}_{0}=0$° and (

**g**,

**h**) ${\theta}_{0}=90$°. Scale model measurements (

**left**) compared to simulations (

**right**). The frequency axis of the measurements is scaled according to the factors given in Table 1.

**Figure A2.**$|{p}_{s}/{p}_{i}|$ for the piers with colonnettes

**C2**, with (

**a**,

**b**) ${\theta}_{0}=0$° and (

**c**,

**d**) ${\theta}_{0}=90$°, and

**N3**with (

**e**,

**f**) ${\theta}_{0}=0$° and (

**g**,

**h**) ${\theta}_{0}=15$°. Scale model measurements (

**left**) compared to simulations (

**right**). The frequency axis of the measurements is scaled according to the factors given in Table 1.

## References

- Cox, T.; D’Antonio, P. Acoustic Absorbers and Diffusers: Theory, Design and Application; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar] [CrossRef]
- Rindel, J.H. Design of new ceiling reflectors for improved ensemble in a concert hall. Appl. Acoust.
**1991**, 34, 7–17. [Google Scholar] [CrossRef] - Bradley, D.T.; Müller-Trapet, M.; Adelgren, J.; Vorländer, M. Comparison of Hanging Panels and Boundary Diffusers in a Reverberation Chamber. Build. Acoust.
**2014**, 21, 145–152. [Google Scholar] [CrossRef] - Rindel, J.H. Attenuation of Sound Reflections due to Diffraction. In Proceedings of the Nordic Acoustical Meeting, Aalborg, Denmark, 20–22 August 1986. [Google Scholar]
- Szeląg, A.; Kamisiński, T.; Lewińska, M.; Rubacha, J.; Pilch, A. The Characteristic of Sound Reflections from Curved Reflective Panels. Arch. Acoust.
**2014**, 39, 549–558. [Google Scholar] [CrossRef] - Rathsam, J.; Wang, L.M. Planar Reflector Panels with Convex Edges. Acta Acust. United Acust.
**2010**, 96, 905–913. [Google Scholar] [CrossRef] - Medwin, H.; Clay, C.S. Fundamentals of Acoustical Oceanography; Applications of Modern Acoustics, Academic Press: San Diego, CA, USA, 1998. [Google Scholar] [CrossRef]
- Standard ISO 17497-1:2004; Acoustics—Sound-Scattering Properties of Surfaces—Part 1: Measurement of the Random-Incidence Scattering Coefficient in a Reverberation Room. International Organization for Standardization: Geneva, Switzerland, 2004.
- Standard ISO 17497-2:2012; Acoustics—Sound-Scattering Properties of Surfaces—Part 2: Measurement of the Directional Diffusion Coefficient in a Free Field. International Organization for Standardization: Geneva, Switzerland, 2012.
- Ryu, J.K.; Jeon, J.Y. Subjective and objective evaluations of a scattered sound field in a scale model opera house. J. Acoust. Soc. Am.
**2008**, 124, 1538–1549. [Google Scholar] [CrossRef] - Jeon, J.Y.; Jo, H.I.; Seo, R.; Kwak, K.H. Objective and subjective assessment of sound diffuseness in musical venues via computer simulations and a scale model. Build. Environ.
**2020**, 173, 106740. [Google Scholar] [CrossRef] - Kim, Y.H.; Kim, J.H.; Jeon, J.Y. Scale Model Investigations of Diffuser Application Strategies for Acoustical Design of Performance Venues. Acta Acust. United Acust.
**2011**, 97, 791–799. [Google Scholar] [CrossRef] - Shtrepi, L.; Astolfi, A.; Pelzer, S.; Vitale, R.; Rychtáriková, M. Objective and perceptual assessment of the scattered sound field in a simulated concert hall. J. Acoust. Soc. Am.
**2015**, 138, 1485–1497. [Google Scholar] [CrossRef] - Shtrepi, L.; Astolfi, A.; Puglisi, G.E.; Masoero, M.C. Effects of the Distance from a Diffusive Surface on the Objective and Perceptual Evaluation of the Sound Field in a Small Simulated Variable-Acoustics Hall. Appl. Sci.
**2017**, 7, 224. [Google Scholar] [CrossRef] - Suzumura, Y.; Sakurai, M.; Ando, Y.; Yamamoto, I.; Iizuka, T.; Oowaki, M. An evaluation of the effects of scattered reflections in a sound field. J. Sound Vib.
**2000**, 232, 303–308. [Google Scholar] [CrossRef] - Strutt, H.J.X. On the light from the sky, its polarization and colour. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1871**, 41, 107–120. [Google Scholar] [CrossRef] - Morse, P.; Ingard, K. Theoretical Acoustics; International Series in Pure and Applied Physics; Princeton University Press: Princeton, NJ, USA, 1986. [Google Scholar]
- Bowman, J.; Senior, T.; Uslenghi, P.; Asvestas, J. Electromagnetic and Acoustic Scattering by Simple Shapes; Taylor & Francis: Abingdon, UK, 1988. [Google Scholar]
- Hughes, R.J.; Angus, J.A.S.; Cox, T.J.; Umnova, O.; Gehring, G.A.; Pogson, M.; Whittaker, D.M. Volumetric diffusers: Pseudorandom cylinder arrays on a periodic lattice. J. Acoust. Soc. Am.
**2010**, 128, 2847–2856. [Google Scholar] [CrossRef] - Yashiro, K.; Ohkawa, S. Boundary element method for electromagnetic scattering from cylinders. IEEE Trans. Antennas Propag.
**1985**, 33, 383–389. [Google Scholar] [CrossRef] - Liu, Y. On the BEM for acoustic wave problems. Eng. Anal. Bound. Elem.
**2019**, 107, 53–62. [Google Scholar] [CrossRef] - Ihlenburg, F. (Ed.) Finite Element Analysis of Acoustic Scattering; Springer: Berlin/Heidelberg, Germany, 1998. [Google Scholar] [CrossRef]
- Taflove, A.; Hagness, S. Computational Electrodynamics: The Finite-Difference Time-Domain Method; Artech House Antennas and Propagation Library, Artech House: Norwood, MA, USA, 2005. [Google Scholar]
- Redondo, J.; Picó, R.; Roig, B.; Avis, M.R. Time Domain Simulation of Sound Diffusers Using Finite-Difference Schemes. Acta Acust. United Acust.
**2007**, 93, 611–622. [Google Scholar] - Hoare, S.; Murphy, D. Prediction of scattering effects by sonic crystal noise barriers in 2d and 3d finite difference simulations. In Proceedings of the Acoustics 2012, Nantes, France, 23–27 April 2012; Société Française d’Acoustique, Paris, France, 2012. [Google Scholar]
- Redondo, J.; Picó, R.; Sánchez-Morcillo, V.J.; Woszczyk, W. Sound diffusers based on sonic crystals. J. Acoust. Soc. Am.
**2013**, 134, 4412–4417. [Google Scholar] [CrossRef] - Tornberg, A.K.; Engquist, B. Consistent boundary conditions for the Yee scheme. J. Comput. Phys.
**2008**, 227, 6922–6943. [Google Scholar] [CrossRef] - Bilbao, S.; Hamilton, B.; Botts, J.; Savioja, L. Finite Volume Time Domain Room Acoustics Simulation under General Impedance Boundary Conditions. IEEE/ACM Trans. Audio Speech Lang. Process.
**2016**, 24, 161–173. [Google Scholar] [CrossRef] [Green Version] - Hamilton, B.; Bilbao, S. Hexagonal vs. rectilinear grids for explicit finite difference schemes for the two-dimensional wave equation. J. Acoust. Soc. Am.
**2013**, 133, 3532. [Google Scholar] [CrossRef] - Olive, S.E.; Toole, F.E. The Detection of Reflections in Typical Rooms. J. Audio Eng. Soc.
**1988**, 37, 539–553. [Google Scholar] - Buchholz, J.M. A quantitative analysis of spectral mechanisms involved in auditory detection of coloration by a single wall reflection. Hear. Res.
**2011**, 277, 192–203. [Google Scholar] [CrossRef] [PubMed] - Burgtorf, W.; Oehlschlägel, H.K. Untersuchungen über die richtungsabhängige Wahrnehmbarkeit verzögerter Schallsignale. Acta Acust. United Acust.
**1964**, 14, 254–266. [Google Scholar] - Seraphim, H. Über die Wahrnehmbarkeit mehrerer Rückwürfe von Sprachschall. Acta Acust. United Acust.
**1961**, 11, 80–91. [Google Scholar] - Brunner, S.; Maempel, H.J.; Weinzierl, S. On the Audibility of Comb Filter Distortions. In Proceedings of the 122nd AES Convention, Vienna, Austria, 5–8 May 2007; Audio Engineering Society: New York, NY, USA, 2007. [Google Scholar]
- Begault, D.R.; McClain, B.U.; Anderson, M.R. Early Reflection Thresholds for Anechoic and Reverberant Stimuli within a 3-D Sound Display. In Proceedings of the 18th International Congress on Acoustics (ICA04), Kyoto, Japan, 4–9 April 2004. [Google Scholar]
- Zhong, X.; Guo, W.; Wang, J. Audible Threshold of Early Reflections with Different Orientations and Delays. Sound Vib.
**2018**, 52, 18–22. [Google Scholar] [CrossRef] - Walther, A.; Robinson, P.W.; Santala, O. Effect of spectral overlap on the echo suppression threshold for single reflection conditions. J. Acoust. Soc. Am.
**2013**, 134, EL158–EL164. [Google Scholar] [CrossRef] - Pelegrín-García, D.; Rychtáriková, M. Audibility Thresholds of a Sound Reflection in a Classical Human Echolocation Experiment. Acta Acust. United Acust.
**2016**, 102, 530–539. [Google Scholar] [CrossRef] - Pelegrín-García, D.; Rychtáriková, M.; Glorieux, C. Single Simulated Reflection Audibility Thresholds for Oral Sounds in Untrained Sighted People. Acta Acust. United Acust.
**2017**, 103, 492–505. [Google Scholar] [CrossRef] - Robinson, P.W.; Walther, A.; Faller, C.; Braasch, J. Echo thresholds for reflections from acoustically diffusive architectural surfaces. J. Acoust. Soc. Am.
**2013**, 134, 2755–2764. [Google Scholar] [CrossRef] - Wendt, F.; Höldrich, R. Precedence effect for specular and diffuse reflections. Acta Acust.
**2021**, 5, 1. [Google Scholar] [CrossRef] - Kleiner, M.; Svensson, P.; Dalenbäck, B.I. Auralization of qrd and other diffusing surfaces using scale modelling. In Proceedings of the 93rd AES Convention, San Francisco, CA, USA, 1–4 October 1992; Audio Engineering Society: New York, NY, USA. [Google Scholar]
- Meyer, J.; Savioja, L.; Lokki, T. A case study on the perceptual differences in finite-difference time-domain-simulated diffuser designs. In Proceedings of the 146th AES Convention, Dublin, Ireland, 20–23 March 2019; Audio Engineering Society: New York, NY, USA. [Google Scholar]
- Kritly, L.; Sluyts, Y.; Pelegrín-García, D.; Glorieux, C.; Rychtáriková, M. Discrimination of 2D wall textures by passive echolocation for different reflected-to-direct level difference configurations. PLoS ONE
**2021**, 16, e0251397. [Google Scholar] [CrossRef] - Aubert, M.; Vitry, P. La Cathédrale Notre-Dame de Paris: Notice Historique et Archéologique; Notices Historiques et Archéologiques sur les Grands Monuments; D.-A. Longuet: Paris, France, 1919. [Google Scholar]
- Bruzelius, C. The Construction of Notre-Dame in Paris. Art Bull.
**1987**, 69, 540–569. [Google Scholar] [CrossRef] - Sandron, D.; Tallon, A.; Cook, L. Notre Dame Cathedral: Nine Centuries of History; Penn State University Press: University Park, PA, USA, 2020. [Google Scholar] [CrossRef]
- Celtibère, M.; Lemercier, M.; Hibon, A.; Ribaud, A.; Normand, A.; Bisson, B. Monographie de Notre-Dame de Paris et de la Nouvelle Sacristie de MM. Lassus et Viollet-Le-Duc; A. Morel: Paris, France, 1853. [Google Scholar]
- Mullins, S.S.; Canfield-Dafilou, E.K.; Katz, B.F.G. The development of the early acoustics of the chancel in Notre-Dame de Paris: 1160–1230. In Proceedings of the 2nd Symposium: The Acoustics of Ancient Theatres, Verona, Italy, 6–8 July 2022. [Google Scholar]
- Lassus, J.B.A.; Viollet-le-Duc, E.E. Projet de Restauration de Notre-Dame de Paris par MM. Lassus et Viollet-Leduc: Rapport…Annexé au Projet de Restauration, Remis le 31 Janvier 1843; Imprimerie de Mme de Lacombe: Paris, France, 1843. [Google Scholar]
- Travaux de Notre-Dame de Paris—Journal des Travaux 1844–1865. Available online: https://mediatheque-patrimoine.culture.gouv.fr/travaux-de-notre-dame-de-paris-1844-1865 (accessed on 1 June 2022).
- Postma, B.N.; Katz, B.F.G. Acoustics of Notre-Dame Cathedral de Paris. In Proceedings of the 22nd International Congress on Acoustics (ICA), Buenos Aires, Argentina, 5–9 September 2016; pp. 5–9. [Google Scholar]
- Katz, B.F.G.; Weber, A. An acoustic survey of the Cathédrale Notre-Dame de Paris before and after the fire of 2019. Acoustics
**2020**, 2, 791–802. [Google Scholar] [CrossRef] - Hoey, L.R. Pier Form and Vertical Wall Articulation in English Romanesque Architecture. J. Soc. Archit. Hist.
**1989**, 48, 258–283. [Google Scholar] [CrossRef] - Thurlby, M. Aspects of the architectural history of Kirkwall Cathedral. Proc. Soc. Antiqu. Scotl.
**1998**, 127, 855–888. [Google Scholar] - CNRS/MC. Chantier Scientifique Notre-Dame de Paris. Available online: https://www.notre-dame.science/ (accessed on 1 June 2022).
- Klein, B. Naissance et formation de l’architecture gothique en France et dans les pays limitrophes. In L’art Gothique, Architecture-Sculpture-Peinture; Toman, R., Ed.; Könemann: Köln, Germany, 1999; Chapter 3; pp. 28–114. [Google Scholar]
- Murray, S. Notre-Dame of Paris and the Anticipation of Gothic. Art Bull.
**1998**, 80, 229–253. [Google Scholar] [CrossRef] - Davis, M.T. Splendor and Peril: The Cathedral of Paris, 1290–1350. Art Bull.
**1998**, 80, 34–66. [Google Scholar] [CrossRef] - Viollet-le-Duc, E.E. Dictionnaire Raisonné de l’Architecture Française du XIe au XVIe Siècle; B. Bance: Paris, France, 1859. [Google Scholar]
- Fernie, E. La fonction liturgique des piliers cantonnés dans la nef de la cathédrale de Laon. Bull. Monum.
**1987**, 145, 257–266. [Google Scholar] [CrossRef] - Olson, V. Colonnette Production and the Advent of the Gothic Aesthetic. Gesta
**2004**, 43, 17–29. [Google Scholar] [CrossRef] - Bony, J. French Influences on the Origins of English Gothic Architecture. J. Warbg. Court. Inst.
**1949**, 12, 1–15. [Google Scholar] [CrossRef] - Hoey, L. Piers versus Vault Shafts in Early English Gothic Architecture. J. Soc. Archit. Hist.
**1987**, 46, 241–264. [Google Scholar] [CrossRef] - Fabero, J.; Bautista, A.; Casasús, L. An explicit finite differences scheme over hexagonal tessellation. Appl. Math. Lett.
**2001**, 14, 593–598. [Google Scholar] [CrossRef] [Green Version] - Yamashita, O.; Tsuchiya, T.; Iwaya, Y.; Otani, M.; Inoguchi, Y. Reflective boundary condition with arbitrary boundary shape for compact-explicit finite-difference time-domain method. Jpn. J. Appl. Phys.
**2015**, 54, 07HC02. [Google Scholar] [CrossRef] - Tolan, J.G.; Schneider, J.B. Locally conformal method for acoustic finite-difference time-domain modeling of rigid surfaces. J. Acoust. Soc. Am.
**2003**, 114, 2575–2581. [Google Scholar] [CrossRef] [PubMed] - Bilbao, S.; Hamilton, B. Passive volumetric time domain simulation for room acoustics applications. J. Acoust. Soc. Am.
**2019**, 145, 2613–2624. [Google Scholar] [CrossRef] [PubMed] - Vázquez, P.; Menéndez, B.; Denecker, M.F.C.; Thomachot-Schneider, C. Comparison between petrophysical properties, durability and use of two limestones of the Paris region. In Sustainable Use of Traditional Geomaterials in Construction Practice; Geological Society of London: London, UK, 2016. [Google Scholar] [CrossRef]
- Botteldooren, D. Acoustical finite-difference time-domain simulation in a quasi-Cartesian grid. J. Acoust. Soc. Am.
**1994**, 95, 2313–2319. [Google Scholar] [CrossRef] - Hamilton, B. Finite Volume Perspectives on Finite Difference Schemes and Boundary Formulations for Wave Simulation. In Proceedings of the 17th International Conference on Digital Audio Effects (DAFx-14), Erlangen, Germany, 1–5 September 2014. [Google Scholar]
- Du, Q.; Faber, V.; Gunzburger, M. Centroidal Voronoi Tessellations: Applications and Algorithms. SIAM Rev.
**1999**, 41, 637–676. [Google Scholar] [CrossRef] - Geuzaine, C.; Remacle, J.F. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Rasmussen, K. Calculation Methods for the Physical Properties of Air Used in the Calibration of Microphones; Report PL-11b; Department of Acoustical Technology, Technical University of Denmark: Kongens Lyngby, Denmark, 1997. [Google Scholar]
- Sheaffer, J.; van Walstijn, M.; Fazenda, B. Physical and numerical constraints in source modeling for finite difference simulation of room acoustics. J. Acoust. Soc. Am.
**2014**, 135, 251–261. [Google Scholar] [CrossRef] [PubMed] - Müller, S.; Massarani, P. Transfer-Function Measurement with Sweeps. J. Audio Eng. Soc.
**2001**, 49, 443–471. [Google Scholar] - Robinson, P.; Xiang, N. On the subtraction method for in-situ reflection and diffusion coefficient measurements. J. Acoust. Soc. Am.
**2010**, 127, EL99–EL104. [Google Scholar] [CrossRef] - Zhang, L.; Wu, X. On cross correlation based-discrete time delay estimation. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’05), Philadelphia, PA, USA, 23 March 2005; Volume 4, pp. iv/981–iv/984. [Google Scholar] [CrossRef] [Green Version]
- Krynkin, A.; Umnova, O.; Vicente Sánchez-Pérez, J.; Yung Boon Chong, A.; Taherzadeh, S.; Attenborough, K. Acoustic insertion loss due to two dimensional periodic arrays of circular cylinders parallel to a nearby surface. J. Acoust. Soc. Am.
**2011**, 130, 3736–3745. [Google Scholar] [CrossRef] - Markowskei, A.J.; Smith, P.D. Measuring the effect of rounding the corners of scattering structures. Radio Sci.
**2017**, 52, 693–708. [Google Scholar] [CrossRef] - Lilly, J.; Olhede, S. Higher-Order Properties of Analytic Wavelets. IEEE Trans. Signal Process.
**2009**, 57, 146–160. [Google Scholar] [CrossRef] - Robinson, P.W.; Xiang, N.; Braasch, J. Understanding the perceptual effects of diffuser application in rooms. Proc. Meet. Acoust.
**2011**, 12, 015002. [Google Scholar] [CrossRef] - Berry, D.L.; Taherzadeh, S.; Attenborough, K. Acoustic surface wave generation over rigid cylinder arrays on a rigid plane. J. Acoust. Soc. Am.
**2019**, 146, 2137–2144. [Google Scholar] [CrossRef] - Farhat, M.; Chen, P.Y.; Bağcı, H. Localized acoustic surface modes. Appl. Phys. A
**2016**, 122, 1–8. [Google Scholar] [CrossRef] - Weber, A.; Katz, B.F.G. An investigation of sound scattering by ancient and Gothic piers and columns. Acta Acust. 2022, submitted.
- Zitron, N.R.; Davis, J. A note on scattering of cylindrical waves by a circular cylinder. Can. J. Phys.
**1966**, 44, 2941–2944. [Google Scholar] [CrossRef]

**Figure 2.**Floor plan with piers, columns, and responds locations. The shafts of identical cross-sections from the selected groups under study are highlighted with a common color. The columns of the nave with circular shafts are indicated with a circle.

**Figure 3.**Cross-sections of the selected shafts. Piers of the western bay of the nave, (

**a**)

**N1**, (

**b**)

**N2**. Pier (

**c**)

**Ch**in the southern ambulatory. Piers with detached colonnettes (

**d**)

**C1**, western wall of the transept, and (

**e**)

**N3**, nave aisles. Western crossing piers (

**f**)

**C1**at the arcade level. Piers (

**g**)

**T**, supporting the tribune between the towers. Dimensions are given in cm. A cylindrical coordinate system is assigned to each, centered on the blue point, and the directions of propagation of the incident waves are indicated with respect to the abscissa, as shown in (

**b**) in the following.

**Figure 4.**Examples of hybrid meshes in proximity to a curved boundary. (

**a**) Voronoi diagram of grid points enclosing the boundary. (

**b**) Centroidal Voronoi diagram after 10 iterations of Lloyd’s method. Voronoi cells are shown with blue edges, and their corresponding generating sites with blue dots and square. They are clipped by the boundary of the object shown in cyan, and bounded in the outer direction by the regular grid points enclosing them, shown with red dots. Regular hexagonal grid is shown in black.

**Figure 5.**Photographs of scale models and experimental set-up. Piers and columns: (

**a**)

**N2**and

**N1**, (

**b**)

**C1**, (

**c**)

**C2**, (

**d**)

**N3**, and (

**e**)

**Ch**and

**T**. Experimental set-up: (

**f**) overview with the sound source on the right, (

**g**) view of the platform, turntable, and microphone mounted on an articulated arm.

**Figure 6.**Measured scattered fields (3 repetitions) for a rigid circular cylinder compared to analytic solutions in frequency domain. (

**a**) Polar diagrams of relative scattered pressure level, $20{log}_{10}|{p}_{s}/{p}_{i}|$, as functions of the scattering angle ${\theta}_{s}$ at different central frequencies of octave bands with the corresponding Helmholtz numbers $ka$ indicated, where k is the wavenumber and a the radius. Magnitude ratio, $|{p}_{s}/{p}_{i}|$, from (

**b**) measurements (Meas. 3) with scaled frequency, and (

**c**) analytic solutions for plane wave incidence.

**Figure 7.**$|{p}_{s}/{p}_{i}|$ for

**N1**with ${\theta}_{0}=90$° (

**a**,

**b**),

**T**with ${\theta}_{0}=90$° (

**c**,

**d**), and

**Ch**with ${\theta}_{0}=0$° (

**e**,

**f**). Scale model measurements (

**left**) compared to simulations (

**right**). The frequency axis of the measurements is scaled according to the factors given in Table 1.

**Figure 8.**Pressure signal (

**top**) and corresponding wavelet scalogram (

**bottom**) normalized by the free-field maximum for (

**a**) a circular cylinder of diameter 133 cm, (

**b**)

**N2**with ${\theta}_{0}=90$°, (

**c**)

**N1**with ${\theta}_{0}=45$°, (

**d**)

**N3**with ${\theta}_{0}=0$°, (

**e**)

**C2**with ${\theta}_{0}=90$°, and (

**f**)

**T**with ${\theta}_{0}=0$° at 4 m from their center in the backscattering direction.

**Figure 9.**RDLDs for

**N2**(

**a**,

**b**),

**C1**(

**c**,

**d**),

**C2**(

**e**,

**f**), and

**N3**(

**g,h**) for different plane wave incidence angles ${\theta}_{0}$ as functions of the scattering angle ${\theta}_{s}$: One-third octave bands (

**a**,

**c**,

**e**,

**g**) and overall (

**b**,

**d**,

**f**,

**h**) results. The receiver positions are reported in Table 1.

**Table 1.**Experimental set-ups and parameters for the measurements on scale models with the Courant number $\lambda $ used in each corresponding simulation.

Label (Fig.) | Scale Factor | Incidence Angle ${\mathit{\theta}}_{0}$ | Distance from Center ^{1} | c [m s^{−1}] | $\mathit{\lambda}$ | |
---|---|---|---|---|---|---|

Source [cm] | Receiver [cm] | |||||

N1 (Figure 3b) | 1:12 | 90° | 31 | 307 | 346.2 | 0.755 |

N2 (Figure 3a) | 1:12 | 90° | 32 | 307 | 345.7 | 0.751 |

45° | 32 | 307 | 345.8 | 0.731 | ||

C1 (Figure 3f) | 1:12 | 0° | 32 | 306 | 345.7 | 0.753 |

90° | 32 | 306 | 346.2 | 0.742 | ||

T (Figure 3g) | 1:10 | 90° | 31 | 319 | 346.1 | 0.744 |

Ch (Figure 3c) | 1:10 | 0° | 31 | 319 | 346.5 | 0.743 |

C2 (Figure 3d) | 1:8.5 | 90° | 33 | 307 | 346.1 | 0.753 |

0° | 33 | 307 | 345.9 | 0.739 | ||

N3 (Figure 3e) | 1:8.5 | 0° | 33 | 338 | 346.6 | 0.745 |

15° | 37 | 317 | 346.6 | 0.762 |

^{1}Represented by a blue dot in Figure 3a–e.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Weber, A.; Katz, B.F.G.
Sound Scattering by Gothic Piers and Columns of the Cathédrale Notre-Dame de Paris. *Acoustics* **2022**, *4*, 679-703.
https://doi.org/10.3390/acoustics4030041

**AMA Style**

Weber A, Katz BFG.
Sound Scattering by Gothic Piers and Columns of the Cathédrale Notre-Dame de Paris. *Acoustics*. 2022; 4(3):679-703.
https://doi.org/10.3390/acoustics4030041

**Chicago/Turabian Style**

Weber, Antoine, and Brian F. G. Katz.
2022. "Sound Scattering by Gothic Piers and Columns of the Cathédrale Notre-Dame de Paris" *Acoustics* 4, no. 3: 679-703.
https://doi.org/10.3390/acoustics4030041