PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Acoustics inside a gypsum sphere with 7 m of diameter
Ernesto ACCOLTI 1; Fernando DI SCIASCIO; Daniel GANDOLFOInstituto de Automática, National University of San Juan and National Scientific and Technical Research Council,
Argentina

ABSTRACT
Concave surfaces are known to focalize or diffuse sound depending on the curvature and the source position
regarding the surface. Spherical rooms are frequently found as theoretical cases in the literature. In this work,
results from models and measurements of the acoustics inside a 7-m-diameter gypsum sphere are compared.
Keywords: concave surfaces, spherical rooms, architectural acoustics

1. INTRODUCTION
That concave surfaces are likely to focus sound energy is a well-known problem in room acoustics(1, 2, 3). Whisperings gallery is frequently presented as a theorical case (3).
In this article the acoustics of a 360° cinema inside a gypsum truncated sphere is studied and a
project for the acoustic conditioning is proposed. The study has two goals: giving support to theory andidentifying acoustic conditioning needs.
The 360° cinema was constructed at the same time the present study was being carried out. Thus,
the main conditioning is not yet implemented, but acoustical measurements of the unconditioned
cinema were taken in order to improve the proposals.
Schemes of the cinema are shown in Figure 1 and Figure 2. Figure 1 shows an elevation crosssection joint to three proposals of loudspeakers configuration. Figure 2 shows the floor plan joint tomeasurements points.

0,65 m

90º
86º

90º
100º

100º

86º

Figure 1 – Scheme with directive loudspeakers from above16074

eaccolti@inaut.unsj.edu.arN

R2
2,7S
0,5R3

1,0

1,00,5R1Figure 2 – Scheme with source S and measurements points R1, R2, and R3.

2. PREDICTIONS
The main method for predictions is the Kirchhoff integral applied to spherical surfaces as inVercammen pp. 50 (2).
𝜃𝑚𝑝i 𝑅𝑝(𝐴, 𝜛) =∫4𝜋

2𝜋

1 + 𝑘𝑢1 + 𝑗𝑘𝑠𝑒 −𝑗𝑘(𝑢+𝑠)∫ sin 𝜃 (cos 𝜑 +cos 𝛼)
d𝜃d𝜙
𝑢𝑠𝑠𝑢

(1)

𝜃=𝜃𝑀 𝜙=0

where p(A,ω): is the sound pressure at a receiver point A, for the angular frequency ω=2πf, linear
frequency f, wave number k, 𝑝i is the source sound pressure measured at 1 m in free field, s is the surface
differential distance to source, u is the surface differential distance to receiver, R is the surface
differential distance to the center of the sphere, θ is the elevation angle of the surface differential, 𝜙 isthe azimuth angle of the surface differential, α is the local angle from the receiver to the center of thesphere at surface differential, and φ is the local angle from the source to the center of the sphere atsurface differential.
In order to include all the first order reflections, the reflection in the floor was added using theimage source method (3)𝑑𝑒 𝑗𝜛 ⁄𝑐(2)+ 𝑝(𝐴, 𝜛)𝑑where p is the sound pressure including all first order reflections,Equation (2) was implemented in a mathematical software to calculate the sound pressure. The
surface was discretized in 100 steps of elevation and in 900 steps of azimuth. Receivers were modeledin two categories: surface receivers and punctual receivers.
Surface receivers were located in a plane located 1.4 m above the floor. The surface was sampledusing a squared grid of 40 cm side. For this category of receivers, the sound pressure was estimatedwith equation (2) at the center frequency of the 1/1 octave bands.
Figure 3 shows the sound pressure due to first order reflections for the 125 Hz frequency. The figure
shows the source position with a red truncated pyramid. The floor is the gray surface . Every 10elevation elements a black circle is shown. The blue pyramids are other loudspeakers.
𝑝f (𝐴, 𝜛) = 𝑝i6075

Figure 3 – Sound pressure distribution. Surface receivers and one source located at 4 m highPunctual receivers were located at positions R1, R2 and R3 at 1.4 m high. These receivers wereused to estimate the impulse response corresponding to one reflection in the conca ve surface. For thiscategory of receivers, the sound pressure was estimated with equation (2) at 4 096 linearly spaced
frequencies in the range from 0 Hz to 6 kHz. Then, using the inverse Fourier Transform, the impulse
response was calculated.
Figure 4 shows the impulse responses of the first order reflections calculated with this method.

3. MEASUREMENTSIn order to both validate the proposed model and carry out a diagnosis of the room a number of
acoustical measurements were taken. For that purpose, the impulse responses were measured using
both balloon bursts and sweeps.
The source position S as well as the receivers’ positions R1, R2 and R3 are shown in Figure 2. The
source was located at a height of 4 m and receivers were located at a height of 1.4 m. Three bursts wereused for each receiver.
Figure 5 shows the absolute value of one of the impulse responses for each receiver estimated fromthe sweeps eliminating the effect of the signal as in ISO 18233 (4). The impulse responses are limited
up to about the octave frequency band of to 4 kHz (0 Hz to 6 kHz). Each row is scaled by the amplitudeof the direct sound and time shifted by the time the direct sound requires to reach the corresponding
receiver.
Finally, using the balloon bursts, the reverberation time was estimated for each burst. The
estimators are EDT, T10, T20, and T30 as defined in ISO 3382 (5). Figure 6 shows the average of theestimations for the 9 bursts corresponding to the 3 repetitions for each of the 3 receivers.

6076

Figure 4 – Calculated impulse responses at punctual receivers located at R1, R2, and R3

Figure 5 – Measured impulse responses for receivers at R1, R2, and R3.

6077

Reverberation time (s)

6,0
5,0
4,0
3,0
2,0
1,00,0
125

250

500

1000

2000

4000

Frecuencia central de banda de frecuencia de 1/1 octava (Hz)
EDT

T10

T20

T30

Figure 6 – Reverberation time estimations.

4. DISCUSSION
This article shows an application of numerical calculation of reflected sound pressure due toconcave surfaces models. The predicted results are in accordance with measurements and expectedbehavior for concave surfaces.
The time of the calculated first order reflections is very similar to that of measured reflections.
However, the amplitude is not exactly the same possibly due to directivity of loudspeakers or
propagation of the discretization errors. In the measured impulse responses several additionalreflections are present. These additional reflections are likely to appear due to higher order reflections.
The focalization of sound energy shown in Figure 3 is one of the expected results. Differences ofabout 20 dB are present in the receiver surface. This effect was not measured with electronicequipment but was perceived aurally by the authors.
A consequence of such a focalization is that the sound field is not diffuse. This conclusion can also
be drawn by analyzing the reverberation time estimations in figure 6. In case of diffuse so und fields,
the estimations of EDT, T10, T20 and T30 tend to be similar.
Even the focalizations are relatively small problems compared with the main problem: perceivableechoes. These echoes can be distinguished from data shown in Figure 4 and Figure 5 as thosereflections comparable in amplitude to the direct sound and delayed about 20 ms or more. A change inthe direction of arrival of the echoes was also perceived by the authors as they walked throw the room.
The red line in Figure 1 shows the screen of the cinema. The intention of the commissioner is to
avoid acoustics materials in the screen zone. With the model used in this work it is possible to evaluate
several configurations such as those shown in Figure 1 affecting the coverage angle of loudspeake rs oracoustic treatment in several surfaces.

ACKNOWLEDGEMENTS
The authors acknowledge the support from the following Argentine institutions: the Secretaría deEstado de Ciencia Tecnología e Innovación de la Provincia de San Juan (Secretary of State for Science
Technology and Innovation of the Province of San Juan), the National Scientific and TechnicalResearch Council, and the National University of San Juan.

REFERENCES
1. Vercammen M. Sound concentration caused by curved surfaces. Proceedings of Meetings on Acoustics,
2013; 19:015053.
2. Vercammen M. Sound concentration caused by curved surfaces. Doctoral thesis, Eindhoven Universityof Technology, 2012.
3. Kuttruff H. Room acoustics. Elsevier Science Publ.,1973, (fourth ed., Taylor & Francis, London, 1999)

6078

4. ISO 18233:2006. Acoustics – Application of new measurement methods in building and room acoustics.
InternationalOrganization for Standardization, Geneva, 2006.
5. ISO, International Standard Organization. ISO 3382-1: Acoustics. Measurement of room acoustic
parameters. Part 1: Performance spaces, Geneva, 2009

6079