This document is a project report submitted by Eric Ballestero for the Graduate Diploma in Acoustics and Noise Control at the Institute of Acoustics. The report investigates reverberation time as a characterizing criterion for architectural spaces like halls. It aims to show how reverberation time is influenced by factors like room volume and absorption. Measurements were taken in three different halls - a theatre, concert hall, and lecture hall. The results are analyzed and rooms are modeled in software to simulate improvements to reverberation times. The report examines how reverberation time relates to other acoustic criteria and guidelines for optimal reverberation times in different types of spaces.

1. Institute of Acoustics
Project Report
Architectural Acoustics : Reverberation
time as a characterizing criterion of halls
Author:
Eric Ballestero
Supervisor:
Dr. Bob Peters
A project submitted in fulfilment of the requirements
for the Graduate Diploma in Acoustics and Noise Control
October 16, 2015

2. Declaration of Authorship
I, Eric Ballestero, declare that this project titled, ’Architectural Acoustics : Rever-
beration time as a characterizing criterion of halls ’ and the work presented in it are my
own. I confirm that:
Any assistance that I have received and sources of information that I have used
are clearly referenced.
When the work of others is quoted, the source is always given. With the exception
of such quotations, this project is entirely my own work.
I have acknowledged all main sources of help.
Signed:
Date:
i

3. “ If you want to find the secrets of the universe, think in terms of energy, frequency
and vibration.”
- Nikola Tesla
“ We have to remember that what we observe is not nature in itself but nature exposed
to our method of questioning.”
- Werner Heinsenberg
“ Music assists the architect in the use of harmonic and mathematical proportion.”
- Vitruvius

4. INSTITUTE OF ACOUSTICS
Abstract
Graduate Diploma in Acoustics and Noise Control
Architectural Acoustics :
Reverberation time as a characterizing criterion of halls
by Eric Ballestero
The aim of this work was to show that reverberation time is an essential criterion for
characterizing rooms and halls. To understand the importance of reverberation time,
it is necessary to see what factors influence it. In order to demonstrate this statement,
reverberation time has been studied among other objective and subjective criteria of
halls characteristics ; as well as the architectural design techniques involved for those
buildings, e.g. geometric design.
To fully understand reverberation time, this study will allow to see in what ways re-
verberation time of rooms can be critical in understanding or designing acoustic spaces,
or moreover, to see as well what factors can play a role on determining reverberation
time, and to what extent they can influence it, eg. volume and absorption.
Three different halls will be tested : a multi-purpose theatre, a concert hall (audito-
rium) and a university lecture hall.
Before going further on this project, it is necessary to emphazize on the following
hypothesis : to what extent can the reverberation time be a characterizing factor of a
hall ? Is it enough to describe the quality of a hall ; rather than other specific factors ;
or not ? That is to say : to what extent the reverberation time can give us information
about acoustic characteristics of halls. This hypothesis will guide us through all the
topics.

5. Acknowledgements
I would like to thank Dr. Bob Peters for his instructive guidance and my helpful
friend Thibault Guillaume for bearing with all my questions. I am also very grateful in
regards to Emmanuel Merida from Emacoustique for letting me use the material from
his company, and for his support on this project. I also wish to show my gratitude
to Catherine Semidor, Trevor J. Cox and Michael Barron for allowing me to use and
quote some of their sketches, architectural plans or tables into this work, especially to
Catherine Semidor ; from the GRECAU laboratory of the Higher National School of
Architecture and Landscape, University of Bordeaux ; for her support in giving me access
to critical literature references as to her feedback in regards of this study. Of course, a
lot of thanks need to be adressed to the Town Hall of Gradignan, the National Opera of
Bordeaux (ONB) and also to the University of Bordeaux I for giving me access to their
halls.
I am deeply thankfull for all the people who helped improving acoustic techniques
and science all over the world, without whom we will not be at this stage of improve-
ments.
“ No one can appreciate the condition of architectural acoustics - the science of sounds
as applied in buildings - who has not with a pressing case in hand sought through scattered
literature for some safe guidance.”
- W.C. Sabine, Collected Papers on Acoustics, 1922, Introduction.
iv

6. Contents
Declaration of Authorship i
Abstract iii
Acknowledgements iv
Contents v
List of Figures viii
List of Tables xii
Abbreviations xiii
Physical Constants xiv
Symbols xv
Introduction xvi
1 Acoustics and Architecture 1
1.1 A brief history of Architectural Acoustics - Before and after Wallace C.
Sabine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Acoustics in Architectural Design . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1.1 Acoustic Design Criteria . . . . . . . . . . . . . . . . . . 4
1.2.1.2 Speech in Rooms . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1.3 Music in Rooms . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2.1 Shape and Size : Volume . . . . . . . . . . . . . . . . . . 13
1.2.3 Geometrical Design Techniques . . . . . . . . . . . . . . . . . . . . 14
1.2.3.1 Geometry of sound reflection . . . . . . . . . . . . . . . . 14
1.2.3.2 Design for direct sound . . . . . . . . . . . . . . . . . . . 15
1.2.3.3 Design for reflected sound . . . . . . . . . . . . . . . . . . 16
1.2.3.4 Sketches of buildings . . . . . . . . . . . . . . . . . . . . 18
2 Reverberation Time Design 22
v

7. Contents vi
2.1 Reverberation Time : Definition . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Sabine’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 Eyring’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.3 Other methods : Millington-Sette, Fitzroy . . . . . . . . . . . . . . 28
2.1.4 Dictum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Early Decay Time - EDT . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Optimum RT curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Measurements and Analysis 36
3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Preparation : requirements for measurements . . . . . . . . . . . . 37
3.1.2 On site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Room results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2.1 Theatre des Quatre-Saisons . . . . . . . . . . . . . . . . . 44
3.2.2.2 Auditorium of Bordeaux . . . . . . . . . . . . . . . . . . 47
3.2.2.3 University lecture hall . . . . . . . . . . . . . . . . . . . . 50
4 Modeling and Simulations 53
4.1 Modeling of a 3-D space using SketchUp . . . . . . . . . . . . . . . . . . . 54
4.1.1 Brief introduction to SketchUp . . . . . . . . . . . . . . . . . . . . 54
4.1.2 Room Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.3 Exporting model data . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Running simulations using CATT-Acoustic . . . . . . . . . . . . . . . . . 59
4.2.1 Introduction to CATT-Acoustic . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2.1 Highly reverberant room . . . . . . . . . . . . . . . . . . 60
4.2.2.2 Acoustic improvements - middling RT based on adapted
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2.3 Lowered reverberation time and material distribution . . 66
4.2.3 TUCT : SxR & audience mapping simulations - D-50 as a RT
related criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Conclusion 82
A Derivation of Sabine’s formula 84
B Derivation of Eyring’s formula 87
C Absorption coefficients and relative materials 89
D CATT & TUCT specifications 91
E EN ISO 3382 -1 and -2 93

8. Contents vii
Bibliography 94
Other references 95

9. List of Figures
1.1 Examples of acoustical defects (Doelle, 1972). . . . . . . . . . . . . . . . . 9
1.2 Theoretical relationship between STI and T60 (Houtgast et al., 1985). . . 10
1.3 Illustration of the Snell-Descartes Law of Reflection. . . . . . . . . . . . . 15
1.4 Geometrical determination of sight lines ; set out seating rows and deter-
mine eye-(ear-) height above floor at front. Project from sight-(source-)
point through eye line to next row, and add required clearance, which
gives eye height for this row. Repeat until rear of room is reached (A.
Lawrence, 1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Diagrammatic plan of an auditorium designed for speech ; each listener
receives direct sound followed quickly by one or two strong reflections (A.
Lawrence, 1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Diagrammatic plan of an auditorium designed for music ; each listener
receives direct sound followed by many reflected sounds ; note that some
specular reflection by diffusers only is shown - diffraction will also occur
when diffusers are smaller than or similar to λ (A. Lawrence, 1970). . . . 17
1.7 Design method for ceiling (or wall) reflectors. Assume point A is fixed ;
select point P in audience where first ceiling reflection is required, draw
PQ and AX and bisect PAX - this bisector is the normal to the required
plane 1. Find the image of X in this plane, I1 and determine cut-off
point B by selecting last audience position for this reflector (T). repeat
the process by drawing QB, BX and bisecting QBX, etc... (A. Lawrence,
1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Typicall features of a 19th century concert hall ; narrow plan and high
ceiling, shallow side balconies. Poor sight lines (and thus weak direct
sound) to many seats, but modulation of wall and ceiling surfaces together
with narrow plan provides short-path lateral reflections and good diffusion
(A. Lawrence, 1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.9 Typicall features of a recent concert hall ; greater seating capacity and
more generous seating layout leads to an increased volume ; good sight
lines (and strong direct sound) but excessive width and plane surfaces
means lack of short-path reflections and insufficient diffuse reflections (A.
Lawrence, 1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.10 Typicall features of a 19th century opera house ; poor sight lines to many
seats, but feeling of intimacy owing to relative closeness of audience and
stage (A. Lawrence, 1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.11 Typicall features of a recently built opera house ; good sight lines but lack
of short-path lateral reflections ; large volume reduces average loudness
of the sound and loss of intimacy results from remoteness of parts of the
audience (A. Lawrence, 1970). . . . . . . . . . . . . . . . . . . . . . . . . . 20
viii

10. List of Figures ix
1.12 Sectional view of reflections distributed through the Royal Albert Hall in
London (M. Barron, 1993). . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.13 Plan view of reflections distributed through the Berliner Philharmoniker
in Berlin (M. Barron, 1993). . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Reverberation time definition with a sample decay. The slope of the decay
is, in practice, measured between –5 dB and –35 dB of the initial level.
(M.Barron, 1993) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Example of an impulse response and the bouncing paths of sound in a
room. (T.J. Cox, 2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Hypothetical pattern of sound reflections arriving at a listener after a
short sound pulse created at a point in a room. (Institute of Acoustics,
2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Example of Fitzroy reverberation time calculation. (A. Lawrence, 1970) . 30
2.5 Energy conservation when a sound wave hits a partition. . . . . . . . . . . 31
2.6 RT30 and EDT curves compared on a sample of decay. (ref : Synergetic
Audio Concepts : Early Decay Time as a System Performance Benchmark) 34
2.7 Recommended optimum mid-frequency reverberation times for rooms.
(W. T. Grondzik, A. J. Kwok, Mechanical and Electrical Equipment for
Buildings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Pictures of the sound sources and sound level meter used. . . . . . . . . . 40
3.2 Pictures of the Theatre des Quatres-Saisons, Bordeaux . . . . . . . . . . . 41
3.3 Pictures of the Auditorium of Bordeaux . . . . . . . . . . . . . . . . . . . 42
3.4 Example of RT measurement results with the dB-Bati software. . . . . . 43
3.5 Decay curve of an interrupted sound at 1 kHz, with RT = 0.91 s. . . . . . 43
3.6 Diagram of reverberaton times measured at the Theatre des Quatre-
Saisons using the noise interuption (upper) and the impulse response
(lower) methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7 Diagram of averaged reverberaton times measured at the Theatre des
Quatre-Saisons using the noise interuption (INT) and the impulse re-
sponse (IMP) methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8 Diagram of reverberaton times measured at the Auditorium of Bordeaux
using the noise interuption (upper) and the impulse response (lower)
methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.9 Diagram of averaged reverberaton times measured at the Auditorium of
Bordeaux using the noise interuption (INT) and the impulse response
(IMP) methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.10 Diagram of reverberaton times measured at the university lecture hall
using the impulse response method. . . . . . . . . . . . . . . . . . . . . . . 50
3.11 Diagram of averaged reverberaton time measured at the university lecture
hall using the impulse response method. . . . . . . . . . . . . . . . . . . . 51
3.12 Results obtained by dB-Bati showing that the decay curve of 63 Hz has
sufficient energy - not perfect but still adequate to use according to acous-
tic consultants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Example of a quadratic residue diffusor (QRD) created with SketchUp. . 54
4.2 First steps in designing the modeled lecture hall. . . . . . . . . . . . . . . 54
4.3 Final render of the geometry of the lecture hall. . . . . . . . . . . . . . . . 55

11. List of Figures x
4.4 SketchUp material window with different kind of materials manually added
for more conveniance. (Some alpha coefficients are given by A. Lawrence) 56
4.5 MAT.GEO sample file containing relative absorption coefficients (in cen-
ter band frequency) of materials in percents (some materials with 10 value
everywhere are not effective and put to 10 by default). . . . . . . . . . . . 56
4.6 Presentation of different kind of sources and receivers with the Loud-
SpeakersTools pluggin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.7 Example of geometry import from SketchUp to CATT using the SK2GEO
tool. (Euphonia rights reserved) . . . . . . . . . . . . . . . . . . . . . . . 57
4.8 Geometry export from SketchUp to CATT using the SK2GEO tool (up)
and relative .GEO files obtained (down). . . . . . . . . . . . . . . . . . . . 58
4.9 Starting window menu of CATT-Acoustic (left) and General Settings win-
dow (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.10 PL9 and CATT 3-D viewers windows. . . . . . . . . . . . . . . . . . . . . 60
4.11 Image source model function showing the sound paths and some of the
reflections between the source and the receiver. . . . . . . . . . . . . . . . 61
4.12 RT estimations for the first model - highly reverberant. . . . . . . . . . . 62
4.13 ABEC software using the BEM method to observe the modal behaviour
of low frequencies in rooms (Euphonia). . . . . . . . . . . . . . . . . . . . 63
4.14 Second modelling of the lecture hall with different materials incorporated. 64
4.15 RT estimations for the second model - meddly reverberant. . . . . . . . . 65
4.16 Side view for the third simulation, with various materials distributed in
a certain way in order to improve not only the RT but the distribution
of the D-50 criteria (definition) for intelligibility. . . . . . . . . . . . . . . 66
4.17 RT estimations for the third model - low reverberantion. . . . . . . . . . . 67
4.18 Hypothetical illustration of sound distribution, seen as sound rays, within
the room. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.19 TUCT window once a simulation is finished. On the left : a window with
the beam tracing parameters to enter ; on the right : a 3-D viewer with
several display options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.20 RT’ (up) and D-50 (down) mapping of the audience area for the first room
configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.21 STI mapping of the audience area for the first room configuration with
the image source model of the sound rays hiting the room boundaries in
function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.22 RT’ (up) and D-50 (down) mapping of the audience area for the second
room configuration - few acoustic improvements (see Chap 3.4.2.2.2). . . . 75
4.23 STI mapping of the audience area for the second room configuration with
the image source model of the sound rays hiting the room boundaries in
function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.24 RT’ (up) and D-50 (down) mapping of the audience area for the third
room configuration - lowered reverberation time (see Chap 3.4.2.2.2). . . . 78
4.25 STI mapping of the audience area for the third room configuration with
the image source model of the sound rays hiting the room boundaries in
function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.26 Comparison between the second (left column) and third (right column)
configurations in function of RT and D/50 criteria. First line : RT. Second
line : D-50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

12. List of Figures xi
4.27 Comparison between the second (left) and third (right) configurations in
function of STI criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
C.1 Comparison between the second (left) and third (right) configurations in
function of STI criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

13. List of Tables
3.1 Reverberaton times measured at the Theatre des Quatre-Saisons using
the noise interuption method in the frequency range of 63 Hz to 4kHz. . . 44
3.2 Reverberaton times measured at the Theatre des Quatre-Saisons using
the impulse response method in the frequency range of 63 Hz to 4kHz. . . 44
3.3 Reverberaton times measured at the Auditorium of Bordeaux using the
noise interuption method in the frequency range of 63 Hz to 4kHz. . . . . 47
3.4 Reverberaton times measured at the Auditorium of Bordeaux using the
impulse response method in the frequency range of 63 Hz to 4kHz. . . . . 47
3.5 Reverberaton times measured at the university lecture hall using the im-
pulse response method in the frequency range of 63 Hz to 4kHz. . . . . . 50
4.1 RT estimations results for the first model - highly reverberant - in the
frequency range of 125 Hz to 16 kHz. . . . . . . . . . . . . . . . . . . . . . 62
4.2 RT estimations results for the second model - meddly reverberant - in
the frequency range of 125 Hz to 16 kHz. . . . . . . . . . . . . . . . . . . 65
4.3 RT estimations results for the third model - low reverberation - in the
frequency range of 125 Hz to 16 kHz. . . . . . . . . . . . . . . . . . . . . . 67
xii

14. Abbreviations
RT Reverberation Time
EDT Early Decay Time
LEV Listener EnVelopment
ASW Apparent Source Width
NC Noise Criterion
AI Articulation Index
SII Sound Intelligibility Index
SLM Sound Level Meter
xiii

15. Physical Constants
Speed of Sound c = 340ms−s at 20 Celsius degrees
xiv

16. Symbols
λ wavelength m
xv

17. Introduction
We all have a certain degree of acoustic awareness - consciously or not. Most are percep-
tive of their acoustic environment. They can tell if a concert hall, theatre, railway station
or an indoor agora may have ‘good’ or ‘bad’ acoustics. As far as speech is concerned
the judgement on the quality of these indoor spaces is relatively simple, rating the ease
with wich the spoken word is understood. However, judging the acoustics of halls with
a particular purpose like music in operas or theatres is a matter of more detailed and
complex observations between objective stimuli and subjective feelings. Indeed, opinions
on acoustics frequently differ as the opinions on the literary and artistic qualities of a
book, a painting or on the architectural design of a new building.
These subjective differencies on evaluating the quality characteristics of rooms or
halls have lead to the general belief that acoustics in these spaces are beyond the scope
of calculations or predictions, and that the study of room acoustics is an art rather than
an exact science[3] - with all the reliability it can demonstrate. This does not mean
that architectural acoustics are not an exact science but due to the constant changing
opinions in regards to a ‘room behaviour’ (how the sound behaves in it) it makes it a
subversive study. As Marshall Long said in the Preface of its Architectural Acoustics,
the latter has been described “as something similar to a black art or perhaps more char-
itably, an arcane science”[6]. It is true that this kind of acoustics depends on these two
quality factors : the science - which is the objective point of view - and the art, or more
precisely people - wich constitute the subjective view.
The hard work of an acoustic consultant in rooms or halls is this ‘two-fold’ problem,
enunciated with clarity by Heinrich Kuttruff in his Introduction of Room Acoustics[3].
The ‘two-fold’ problem can be resumed by taking into account the subjective human
hearing while building the acoustic predictions and calculations of halls on mathemat-
ical models. For indeed, acoustics is what we can call a subjective science. A science
xvi

18. Symbols xvii
based on mathematical models explaining the physical behaviour of the world around
us, but in the end it is reffered to the human ear for a quality examination of the final
product. This allows to clear the point and balance the popular beliefs between art
and science in architectural acoustics. That being said, human behaviour can also be
studied scientifically, but as long as improvements in this area will not be taken further a
significant portion of this field of study will remain subjective, depending on the average
of people’s feelings, as the final quality credit of a room or hall.

19. Chapter 1
Acoustics and Architecture
1.1 A brief history of Architectural Acoustics - Before and
after Wallace C. Sabine
The firsts questions about acoustics began in antiquity with Pythagoras, nearly 6 cen-
turies B.C. It is said that the Greeks of this age had already mastered some acoustic
properties of materials to build amphitheatres, eg. the periodic arrangement of the
rows of seats of the Epidaurus Theatre filters low frequencies (below 500 Hz) of the
background noise – like the rustling of the trees or audience. Later, Aristoxenus and
Vitruvius wrote some works on resonant vases put below the seats of amphitheatres.
More, Vitrivius and Chrysippus stated on those days that sound propagation was be-
having like a droplet into water and then postulate – in an empiric way – the wave
behaviour of sound.
From these days to the dusk of the 19th century, the field of architectural acoustics
wasn’t as developed as it is now. As a matter of fact, this particular field of acoustics
was funded in 1895 by the American physicist Wallace Clement Sabine, carrying out the
first scientific method to this specific acoustic area. The modern architectural acoustics
that we know were born at this relatively late time of scientific history. The change in
understanding the room and building acoustics was that before W.C. Sabine there were
mostly empirical assumptions ; where sensory experience would validate theories ; while
W.C. Sabine brought the first rationalist thoughts in this field of study.
1

20. Chapter 1. Acoustics in Architectural Design 2
“I gave myself pain in mastering this bizarre science [of acoustics] but...nowhere did I
find a positive rule to guide me ; on the contrary, nothing but contradictory statements...
I must explain that I have adopted no principle, that my plan has been based on no theory,
and that I leave success or failure to chance alone... like an acrobat who closes his eyes
and clings to the ropes of an ascending balloon.”
- Garnier, 1880.
Thus Sabine was the one to determinate through experiments that there was a defini-
tive relationship between the quality of acoustics, the size of a hall and the amount of
absorbent surface present. His investigations about acoustics began with the Corpora-
tion of Havard University which proposed him in 1895 to search changes for remedying
the acoustical difficulties in the lecture-room of the Fogg Art Museum, a building that
had just been completed[9]. Rapidly, Sabine encountered some difficulties on explained
literature that wasn’t that helpful, because at that moment of history acoustics were
more an experimental craft rather than a proved construction. He then started to search
the physical relationship between sound energy, absorption, surface and volume of an
indoor space in order to find a formula explaining the sound behaviour in it. And this
way he was finally able to define mathematically the reverberation time of room – a
formula that now every beginner in room acoustics knows on his fingertips.
RT =
0.16V
A
(1.1)
Where RT is the reverberation time,V the room volume and A the total absorption
area. (Units in S.I)
Sabine’s work marked the beginning of the science of room acoustics. Yet, years after
the death of Sabine a return to a simplification of the description of acoustic phenomena
has been noticed. However, even in those years, researchers made significant advances
in the field. On the one hand ; in the description of objective phenomena ; we had the
reformulation of Sabine’s formula by Eyring and the hall’s modal density approach by
H. Kuttruff and M.R. Schroeder. On the other hand, there were the studies relative to
the perceptually dimension : to discover the precedence effect or Haas effect (G. Von
B´ekesy, L.A. Jeffress, H. Haas) and the integration of the energy of the first reflections

21. Chapter 1. Acoustics in Architectural Design 3
(J.P.A. Lochner, H. Haas, O.A. Abdel Alim). Thus from Sabine’s death until now we
came through a long way of improvement in this scientific – but not least human – field.
And even today, there are still a lot of issues to be addressed.
1.2 Acoustics in Architectural Design
This chapter and the following points are intended to give a resumed idea of the gen-
eral aspects of architectural design when dealing with acoustics. Thus the following
subsection will deal with the main acoustical considerations, parameters and techniques
involved for the design of a room, in order to define the important acoustical factors
needed to take into account in its architectural design.
The General Design Considerations will bring two important thoughts to have in mind
: the acoustical design criteria that will allow an objective and subjective evaluation of
an enclosed space ; giving respectively comparative overview elements and qualitative
judgements between rooms ; and the type of the sound source generated within them,
wich can mostly be divided into two different types known as ‘speech’ and ‘music’.
General Design Parameters of architectural acoustics will explain the importance and
the relationship of acoustics in architectural features such as the shape, size and volume
of a room, the early reflections that may arrive at listeners position, and sound diffusion
within the space. This will allow an understanding of the acoustic signature of a room,
as it is a function of these design parameters.
Then, an overview of the Geometrical Design Techniques used in the architecture
of a room for improving and enhancing acoustics will explain the possible solutions to
remedy some acoustical defects or problems using the geometric approach of sound.
It may be usefull to clarify that some quoted criteria in the next subsections will
have a direct impact over this study about reverberation time. Some others will just
complement it for a more broad view context as reverberation time is a fundamental
element in the field of acoustics - having numerous connections with various design and
quality factors of rooms. Therefore it seems appropriate to connect it among those to
see its relative importance as a qualitative endpoint of room acoustics.

22. Chapter 1. Acoustics in Architectural Design 4
1.2.1 Design Considerations
1.2.1.1 Acoustic Design Criteria
Before going any further on the general design parameters of rooms, it is essential to
first set the acoustical criteria used to describe their acoustic quality. To describe the
acoustic quality of a concert hall or theatre, especially through their ability to enhance
the sound of a musical instrument or voice, several objective and subjective dimensions
shall be considered[2][1]. In a concert hall or a conference room a subjective assessment
by the auditors is ultimately important. The objective characteristics are of little inter-
est if they are not completed by subjective variables that measure the perceived sound
quality in the room.
Impulse response, LS/N , absorption, reverberation time (RT), Early Decay Time
(EDT), room radius (Rc), strenght factor G, are all objective characteristics of rooms.
They correspond to the physical phenomenon of sound and are used to compare dif-
ferent rooms. They also help to remedy some defects thus identified : echoes, highly
reverberant levels, standing waves, etc...
On the other hand, the most important subjective characteristics are : reverberance
(or liveness), warmth or brilliance, intimacy, clarity, loudness, and spaciousness (LEV
& ASW).
Also, to understand the difference between objective and subjective criteria, M. Bar-
ron affords a fruitful analogy between room acoustics and wine, where the subjective
nature of wine would be the body, the dryness or sweetness in opposition to the objec-
tive values as the alcohol degree, pH or sugar content. Therefore, in the search for good
acoustics, it is convenient to relate a physical characteristic of the hall to a subjective
quality.
“Experiments have demonstrated that listeners are probably hearing the same subjec-
tive effects, but that they place different individual weightings on these effects for their
judgement of preference. There is thus no ideal concert hall, just as there is no ideal
wine for all.”
- Michael Barron [1].

23. Chapter 1. Acoustics in Architectural Design 5
Thus, the objective acoustic criteria of a room will be defined as characteristics mean-
while subjective criteria will be named as qualities. For example, this project is intended
to study the physical characteristic of reverberation time and differentiate it from its
related subjective quality - reverberance. It is the difference between characteristics
and qualities that have leaded, decades and maybe centuries ago, the general belief of
architectural acoustics being an ‘arcanic science’ rather than an exact one. Some of
the physical characteristics of sound, yet known, will be clearly enunciated again and
treated in Chapter 2 dealing with the sound propagation in rooms, leaving the way clear
to describe the subjective acoustic qualities related to the human perception of sound.
Reverberance represents the sensation of reverberation, which involves the ear. It
depends on reverberation time, but not only. Reverberance is assessed also by the Early
Decay Time (EDT), wich is a measure of the rate of sound decay, expressed in the
same way as a reverberation time, based on the first 10 db portion of the decay. (see
Reverberation Time Design in Chapter 2). Thus RT and EDT could not be the same -
and often, they are not.
The two factors of warmth and brilliance are also related to the reverberation of a
room. Warmth is the adequate sense of bass sound ; as an objective measurement of
the relative level of bass (75Hz - 350Hz) compared with mid-frequency sound (350Hz -
1400Hz). A sense of acoustic brilliance is correspondingly associated with strong high-
frequency sound.
Intimacy refers to the feeling that listeners have of being physically close to the
performing group. A room is generally judged intimate when the first reverberant sound
reaches the listener within about 20 milliseconds of the direct sound. This condition is
met easily in a small room, but it can also be achieved in large halls by the use of
orchestral shells that partially enclose the performers.
Clarity refers to the degree of distinction between two different sounds, eg. different
notes of a piece of music. It is the strength of the initial sound along with early reflections
arriving soon thereafter. Two precision types of clarity are to be defined : the horizontal
; measuring the degree of distinction between blasts ; and vertical ; measuring the degree
of distinction between simultaneous sounds. The desired accuracy value depends on the
type of music or speech.

24. Chapter 1. Acoustics in Architectural Design 6
To paraphrase, loudness refers to the sensation of acoustic strength. Comparing two
rooms with the same reverberation time, the one that has fewer listeners (seats) will
have a louder sound strength. Similarly, a louder sound strength is perceived in a room
very reverberant compared to a less reverberant one, even if they have identical sizes.
In addition, it is possible to differentiate the early loudness level to the reverberant one.
The first is determined by the energy of the direct sound and early reflections while the
second depends on the energy of late reflections (generally after a period of 80ms).
Spaciousness is explained from two key-concepts. The sensation of envelopment (LEV
- Listener EnVelopment) and the apparent source width (ASW - Apparent Source Width)
[2]. LEV describes the feeling of being surrounded by sound. A specific spot where the
reverberant sound is perceived as coming from all directions with the same level is a place
with an optimal LEV index. LEV depends on the direction of incident reflections on the
listener, the separation between the first and late reflections, and the perceived energy
level of reflections. The ASW is described as the sensation that the music comes from
a greater source than the visual width of the actual source. The greater the apparent
width will be, stronger the spaciousness will get. [7].
Thus, for any room design ; whether they should be designed for speech or music ;
it is obviously important to know the physical characteristics of sound that propagates
within it, but also to determinate the different sound qualities perceived by listeners.
Although, it is usefull to notice that in all cases a sufficient sound level is needed, well
above the background noise threshold, resulting in a proper insulation from outside noise
to avoid unwanted sounds coming inside. This opens the way for the two (wanted) types
of sound sources to be considered in rooms : speech and music.
1.2.1.2 Speech in Rooms
Speech is one of the most important sources of sound to man, where its ear’s response is
at its most efficient activity for the frequency range inherent to speech. This is one factor
among others for the total process of speech perception, which also includes linguistic
criteria and context of the enunciated words. These makes it difficult to understand how
must be the acoustic requirements set for speech to be understood.
However, good acoustics for speech to be unerstood can be resumed in one word :
intelligibility. This is the key for speech - to make it intelligible for its listeners. A critical

25. Chapter 1. Acoustics in Architectural Design 7
acoustic requirement for achieving that is having a high signal-to-noise level, in order
that the voice will greatly cover the noise generated within the room. ‘Intelligibility
depends on the masking effects of extraneous sounds on the speech we hear’ [6]. So in
the design of speech rooms as classrooms, conference rooms, and auditoria, the ability to
understand speech is very important. All the architectural components of these rooms
have an influence on intelligibility - size, shape, surface, orientation, materials, as well
as the background noise level. Several fundamental requirements in designing rooms for
speech (Doelle, 1972) contributes in achieving a high signal-to-noise level at the receiver
position :
1. Adequate loudness
Adequate loudness implies a high direct-field level. This requires to set a correct
distance between the source and the receiver, and think about if the speech will be
amplified or not, eg. it could be difficult to understand unreinforced speech beyond
10 to 12 meters, especially in a reverberant space. The latter will determine the
distance needed for speech to be properly understood because there is a region
beyond which the human voice cannot extend without physical or electronic rein-
forcement. This will greatly define the shape of the room, as its walls or ceiling, eg.
making the audience seating area circular minimizes the source-receiver distance.
Also, in addition to the distance between the source and the receiver, the loudness
will be defined by the volume per seat distributed among the audience area. This
one should be low in order to obtain an adequate loudness (2.3 to 4.3 m3) with
an otpimum value of 3.1 m3. The seating capacity plays a role in how big the
volume per seat can be, ie. small seating capacity implies a larger volume per
seat - thus a low loudness, as loudness (or acoustic strenght) represents the gain
in dB in the room with respect to a free field space where the decay is -6dB by
doubling distance. Acoustic strenght is thus inversely proportional to the volume.
The larger the room, the closer it will get to a free field space from the perceived
level (in dB) by a listener.

26. Chapter 1. Acoustics in Architectural Design 8
2. Sufficient diffusion
For good intelligibility the sound level needs to be relatively uniform within the
space, that is to say that a sufficient diffusion is required in order to avoid ‘dead
spots’ in the room as to enhance the speech transmission with reflective surfaces.
The reflectors must be close enough so that the reflection delay time is less than
30 to 50 msec. This is to avoid intelligibility issues due to a to high delay time
between the source and the receiver.
3. Appropriate Reverberation Time
As Marshall Long said, ‘the reverberation time can be the boon or the bane of the
acoustic performance of a room’. Generally, reverberation times for speech are to
be low (< 0.8 or 1 sec) as longer ones are preferred for music. Long reverberation
times can cause serious damage to the understanding of speech, as it is difficult
then to differentiate the beginning from the ending between consecutive words, eg.
a university lecture room with a high RT value can harm the learning skills of
students so that they will no longer follow the professor’s speech.
4. Good Signal-to-Noise ratio & low background noise
As said previously, a sufficient sound level from the source is needed to be correctly
understood, well above the background noise level. Some authors (Peutz and Klein,
1974) recommend that the received level must be at least 25 dB higher that the
background noise level for adequate intelligibility, while others (Bradley, 1986)
hold that a 10 to 15 dB margin is sufficient and a more reasonable choice. This
means that the background noise level must be relatively low - small rooms or
lecture halls are designed to an NC* 30 value (35 dBA) and larger auditoria to
an NC 25 value (30 dBA). Signal-to-noise ratio and background noise are thus
important features to take care of in the design of a room - especially when speech
is involved, due to its low sound level compared to musical instruments.
5. Acoustical defects
As they seem to indicate, acoustical defects are not a goal to reach in a room.
Conversely, it is generally understood to aim to a room free of any particular
defect. Acoustical defects can be defined by multiple or long-delayed reflections,
focusing, low frequency phenomena (room resonances), locally high-amplitude lev-
els, and echoes. These are the main acoustical defects that can contribute to poor

27. Chapter 1. Acoustics in Architectural Design 9
intelligibility and general discomfort in rooms. For example, a small lecture hall
at the University of Bordeaux I has two parallel walls that lead to the rise of a
flutter echoe between these, detracting speech signal in a considerable way - and
this should be avoided.
Figure 1.1: Examples of acoustical defects (Doelle, 1972).
So generally, a good design for speech intelligibility lies in a good proportion of
direct sound over reflected sound. The solutions and techniques involved are dis-
cussed in subsection 1.3 Geometrical Design Techniques.
There are some ways to know the efficiency of how speech from a source could be
transmitted to the receiver in a defined space, which will allow to measure the intelligibil-
ity of the spoken word. The measurements could be subjective, predictive, or objective.
The objective and predictive measurements of speech intelligibility allow the intelligibil-
ity evaluation of speech without resorting to people playing the role of auditors - as it
is the case for subjective evaluations. The main objective and predictive measurements
are the Articulation Index (AI), the Speech Intelligibility Index (SII), Speech Transmis-
sion Index (STI) and Speech Interference Level (SIL), performed using artificial signals
that will allow to run acoustical simulations to make approximate predictions of speech
behaviour. Signal-to-noise ratios are thus commonly used for this kind of calculations.
For what concerns the relation between reverberation time and speech, we will just
talk about the Speech Transmission Index as a relevant index for speech intelligibility.
This criterion is based on the fact that speech is a modulated signal. Intelligibility is
particularly important especially when modulations are well perceived. In other words,
good clarity corresponds to a signal when the ‘peaks’ stand many ‘gaps’. However, the
presence of reverberation reduces the correct signal modulation - masking the ‘gaps’ -
and thus impairing intelligibility.

28. Chapter 1. Acoustics in Architectural Design 10
Moreover, Bistafa and Bradley (2000) plotted in a study the STI values versus rever-
beration times for unamplified speech in classrooms (Figure X.X). We can see that for
a given signal-to-noise ratio the intelligibilty can be maximized as a function of rever-
beration. Lower the reverberation time is, the better is the speech transmission. But
care is needed, as some reverberant field in the room is required to make it ‘natural’
for its listeners, eg. a lecture hall cannot have an RT value of 0.1 or 0.2 secs, where
listeners will feel uncomfortable from the unnatural silence (deaf feeling). It will also
have an impact on the acoustic strenght needed for the source, making it more louder
to be correctly understood (sound absorption).
Figure 1.2: Theoretical relationship between STI and T60 (Houtgast et al., 1985).
Then, it is good to say that reverberation time plays a critical role on determining
how intelligible speech is in rooms, as optimum RT curves for defined rooms restrict the
possible STI values for speech, setting a narrow band scale for speech transmission.
1.2.1.3 Music in Rooms
Where the speech is used to inform, music is used to marvel. When it comes to music in
rooms, the purpose of sound communication in these places shall be reconsidered. Be-
cause unlike speech, the goal here is no more informative but artistic, supposed to bring
listeners to a cathartic state - to feel relievied with pleasure by ‘melodies that carry the
soul out of itself’ (Aristotle). Therefore, the acoustic criteria will no longer be the same
in function of the type of sound source used. Where for speech the importance is given
to speech transmission (STI), the music would appeal to a sense of ‘musical transmis-
sion’, where there would be no more a single predominant factor as intelligibility is for
speech but a consortium of volatile and arduous criteria, as it is hard to describe clearly

29. Chapter 1. Acoustics in Architectural Design 11
something that urges our sensitive ; and sometimes unconscious ; feelings. This ‘musical
transmission’ could be defined as the capacity to transmit the music the way it should
be - and the tricky point is there. What is the best way music should be transmitted ?
How can it be defined ?
There isn’t one unique method to enhance general music on the best way for a par-
ticular hall. There are so many kinds of music with their own specifities, that could be
amplified or unamplified, that it is really restrictive to build an auditoria just for one
kind of music. But there are some commom elements where there is a general agreement,
and it is the ability that a room has to give sufficient clarity (or definition) to musical
sound as enhancing the listeners envelopment too. That could be the equivalent of intel-
ligibility for music : clarity and envelopment. And, as it will be seen, the reverberation
time of a hall plays a major role in those. Following there is a resumed list of the general
agreements for an ideal listening environment (Marshall Long [6]) :
1. Audience should feel enveloped or surrounded by sound. This requires a significant
portion of the energy arriving as lateral reflections, from the side particularly, after
80 msec from the arrival of the first sound.
2. A reverberant field whose duration depends on the type of music being played,
thus supporting instrumental sound. For a good sense of warmth, a reverberation
time that rises with decreasing frequency below 500 Hz would be beneficial.
3. The room must provide enough clarity and definition so that the rapid musical
passages can be appreciated in detail. It requires reflective surfaces located close
to the source or the receiver so that the initial time delay gap is short.
4. The sound coming from the source must have an adequate loudness, evenly dis-
tributed throughout the hall. In big halls (above 2600 seats) loudness and def-
inition are reduced whereas in small auditoria care must be needed so that the
loudness does not become overbearing.
5. The room needs to support a wide bandwidth as musical instruments generate
sounds from 30 Hz to 12,000 Hz - which is much broader than speech. However
the room must not color the natural frequency balance of music.
6. An excellent noise control is needed. Exterior sources and mechanical equipment
mustbe controlled so that the quietest instrumental sound can be heard. Thus

30. Chapter 1. Acoustics in Architectural Design 12
background noise levels should not exceed NC 20 in small halls and NC 15 in large
halls.
7. The detailed reverberation characteristics of the space should be well controlled
with a smooth reverberant tail and no echoes, shadowing, coloration, or other
defects.
8. The ability that performers should hear each other clearly (support factor) and re-
ceive from the space a reverberant return close to that experienced by the audience
is very important.
As it can be seen, the reverberation time takes here a prominent place in regard to
the establishment of basic parameters of a listening environment. It is a fundamental
pillar of the design of musical rooms such as concert or symphony halls and auditorium.
For music, reverberation must be well controlled not only to avoid the clarity of mu-
sical sound being detracted but also to optimize the listening experience. Because it
will indeed play on other factors such as warmth (or brilliance), loudness, spaciousness
(LEV/ASW), intimacy, without mentionning clarity again, as some acoustical defects
too (coloration, echoes, ...). Thus reverberation times of these places shall be considered
with great care, and normally they are the ones that will set the acoustical parameters
around a cornerstone (as reverberation is intrinsically connected with those, and con-
versely).
This subsection helped to shed light on the acoustical considerations to have in mind
when dealing with acoustics in enclosed spaces, defining the many quality criteria that
exist (objective & subjective) in literature and to enlighten the differences upon the
characterizing criteria between the two types of sound sources that are respectively
speech and music.
This will allow to see now how the architectural parameters of a room can affect its
acoustics, and hence its reverberation time.

31. Chapter 1. Acoustics in Architectural Design 13
1.2.2 Design Parameters
1.2.2.1 Shape and Size : Volume
Shape and size of indoor spaces are two significant parameters of room acoustics.
Shape and Size
The shape of rooms dedicated to listening is a critical factor as to how they will
respond to sound stimuli. Whether that partitions (walls, floor, ceiling, or accessories)
are parallel-faced or not, that they focus sound or diffuse it, any of these components
play an important role in the acoustics of a room. The shape of a room does not
necessarily have a big impact on its reverberation time (even if it affects it), but indeed
on its reverberation process and hypothetically on its ensuing defects (long reflections,
echoes, resonnances, etc...). Of course, the criteria will not be the same depending on
the type of generated sound source - speech or music.
For music, shoebox halls with strong lateral reflections and relatively low sized (<
1800 seats) will be preferred[5], whereas speech rooms will tend to be hemi-circular with
strong first reflections and small size for a better distance with the speaker.
As it was mentionned above, size is another parameter with heavy consequences on
reverberation time. This is a factor, which paired with shape, that brings the idea of
volume in a room. Because size is not necessarily the inner volume of the room, and that
shape thereof influence in a complementary way the real volume of a space. However,
the more sized is a room, the more the chances to get a bigger volume are high.
Thus, it is the combination of these two architectural parameters (size and shape)
that will define a room volume, the latter is one of the greatest parameters involved
in reverberation times, along with absorption, that will allow thereafter an acoustic
characterizion of a room.
Volume
Being present in all reverberation equations, volume is one of two major fundamental
factors related to reverberation time, along with absorption. Indeed, RT is stricly
proportional to volume, and inversely proportional to absorption. Room volume will
influence both reverberation time and room gain (loudness or strenght factor).

32. Chapter 1. Acoustics in Architectural Design 14
RT ∝ V olume ; RT ∝ 1
A
Thus, according to Sabine’s formula, the more the room will have a big volume/ab-
sorption ratio (0, 16.V > A), the higher the reverberation time will be, and conversely if
the volume/absoprtion ratio is small (0, 16.V < A), the reverberation time will be low
too.
There is a trinity for assessing some acoustic criteria in rooms involving room volume,
room purpose and reverberation time. For example, the volume and purpose of a room
(studio, opera, workplace, etc...) can allow to define the best suited reverberation time
for this kind of room. Hence, the volume of a room is a prominent parameter, based
on both size and shape, characterizing the acoustic behaviour of a room through its
reverberation time.
1.2.3 Geometrical Design Techniques
As A. Lawrence said, it is useful to employ geometrical techniques of sound at the
sketch design step, based on optical theory, to analyse the distribution of sound in a
room. There are, however, serious limitations to the use of such techniques, and should
be seen as an approximate point of view.
1.2.3.1 Geometry of sound reflection
In the geometric model of sound propagation, applicable for high-frequencies, sound
travels in a given geometry as light rays do, according to Fermat’s principle : ‘light (or
sound to some extent) travels from one point to another on given trajectories such as
the duration of the travel is locally minimal’.
The optical theory based on the laws of Snell-Descartes describes the behaviour of
light at the interface of two media. These laws are two in number, one for reflection and
another one for refraction. The law of reflection is stated as follows :
• the reflected light ray is in the plane of incidence
• the incident and reflected angles are equal in absolute values ; θ2 = −θ1

33. Chapter 1. Acoustics in Architectural Design 15
Thus, the angle made between the reflected ray and the surface is equal to the angle
made by the incident ray. This is the basis of sound reflection as well. Nevertheless, the
approach based on optical theory is valid only when the reflecting surface is significantly
larger than the wavelength of the incident sound. This is the reason why this method
only works for high frequencies and not low ones, as the lower audible frequency has a
wavelength of 17 m, which is larger or equivalent in size to some room surfaces. To this
extent, diffraction theory suits in a better way for this kind of acoustic behaviour.
Only considering mid- and high-frequencies, there can be some geometrical design
techniques used in rooms for enhancing their acoustics.
Figure 1.3: Illustration of the Snell-Descartes Law of Reflection.
1.2.3.2 Design for direct sound
First, it is necessary to establish the layout of audience in regard to the sound emission
location (eg. proscenium, stage, etc...). Whether for sight or hearing, it is always good
to reduce the distance between the source and the receiver as much as possible. However,
care is needed as the logical fan-shape or circular auditorium which results from this
principle may have serious acoustic limitations (eg. focusing phenomenon resulting from
concave surfaces should be avoided).
A design that provides good sight lines in a room will also produce good direct sound
paths. In order to get a good direct sound, it is essential to think about this aspect of
the sound propagation. That is the important paramater for direct sound in rooms.
Furthermore, when sight lines are not important and that other criteria may be
suggested to sound paths, different compromises may be established depending on the
room type, eg. in a concert hall, the source point may be taken higher than the sight
point.

34. Chapter 1. Acoustics in Architectural Design 16
Figure 1.4: Geometrical determination of sight lines ; set out seating rows and deter-
mine eye-(ear-) height above floor at front. Project from sight-(source-) point through
eye line to next row, and add required clearance, which gives eye height for this row.
Repeat until rear of room is reached (A. Lawrence, 1970).
1.2.3.3 Design for reflected sound
In most rooms (except anechoic rooms), sound will be reflected - to some extent - from
all surfaces. It is then possible to shape the room surfaces so that reflected sound (at
least in the medium and high frequencies) will be directed in a suitable way. Figures
below show some examples of reflected sound paths in different rooms.
Figure 1.5: Diagrammatic plan of an auditorium designed for speech ; each listener
receives direct sound followed quickly by one or two strong reflections (A. Lawrence,
1970).
According to A.Lawrence, it can be good to note that ‘reflected components may add
considerably some loudness to the sound received, and since those parts farther from the
source receive weaker direct sound (attenuated by spherical divergence, air absorption
and absorption by intermediate surfaces), it is usual to attempt to reflect more sound to
those parts than to those close to the source’. Thereby, reflected sounds may be more

35. Chapter 1. Acoustics in Architectural Design 17
Figure 1.6: Diagrammatic plan of an auditorium designed for music ; each listener
receives direct sound followed by many reflected sounds ; note that some specular
reflection by diffusers only is shown - diffraction will also occur when diffusers are
smaller than or similar to λ (A. Lawrence, 1970).
suitable for farther audience than the closer one, enhancing the received sound whose
energy has been lowered by many factors. But care is needed, as echoes may be per-
ceived if sounds are received at sufficient loudness more than 50 msec after the arrival
of the direct sound.
Another example of useful geometrical design of reflections is illustrated in the figure
below, showing ceiling reflections to cover a larger audience.
Figure 1.7: Design method for ceiling (or wall) reflectors. Assume point A is fixed ;
select point P in audience where first ceiling reflection is required, draw PQ and AX
and bisect PAX - this bisector is the normal to the required plane 1. Find the image
of X in this plane, I1 and determine cut-off point B by selecting last audience position
for this reflector (T). repeat the process by drawing QB, BX and bisecting QBX, etc...
(A. Lawrence, 1970).

36. Chapter 1. Acoustics in Architectural Design 18
These design techniques ; for speech, music, with direct and reflected paths, coming
directly from the source, walls, or ceiling ; should be taken into consideration when
establishing an adequate reverberated field in the room. As reverberation time is an
important criterion when listening, choice of reflection paths must be wisely thought.
1.2.3.4 Sketches of buildings
In this subsection, it will be shown briefly different types of buildings, old and new, with
their acoustic qualities and defects.
Concert Halls
Figure 1.8: Typicall features of a 19th century concert hall ; narrow plan and high
ceiling, shallow side balconies. Poor sight lines (and thus weak direct sound) to many
seats, but modulation of wall and ceiling surfaces together with narrow plan provides
short-path lateral reflections and good diffusion (A. Lawrence, 1970).

37. Chapter 1. Acoustics in Architectural Design 19
Figure 1.9: Typicall features of a recent concert hall ; greater seating capacity and
more generous seating layout leads to an increased volume ; good sight lines (and
strong direct sound) but excessive width and plane surfaces means lack of short-path
reflections and insufficient diffuse reflections (A. Lawrence, 1970).
Opera Houses
Figure 1.10: Typicall features of a 19th century opera house ; poor sight lines to
many seats, but feeling of intimacy owing to relative closeness of audience and stage
(A. Lawrence, 1970).

38. Chapter 1. Acoustics in Architectural Design 20
Figure 1.11: Typicall features of a recently built opera house ; good sight lines but lack
of short-path lateral reflections ; large volume reduces average loudness of the sound
and loss of intimacy results from remoteness of parts of the audience (A. Lawrence,
1970).
Royal Albert Hall & Berliner Philharmoniker
Figure 1.12: Sectional view of reflections distributed through the Royal Albert Hall
in London (M. Barron, 1993).

39. Chapter 1. Acoustics in Architectural Design 21
Figure 1.13: Plan view of reflections distributed through the Berliner Philharmoniker
in Berlin (M. Barron, 1993).

40. Chapter 2
Reverberation Time Design
“ Hearing is the effect of a stroke which is transmitted through the ears by means of the
air, brain, and blood to the soul, beginning at the head and extending to the liver. The
sound which moves swiftly is acute ; that which moves slowly is grave ; that which is
uniform is smooth, and the opposite is harsh. Loudness depends on the quantity of the
sound. Of the harmony of sounds I will hereafter speak.”
- Plato, Timaeus, Section 1.
For centuries, this was the grasp human beings had on sound nature and behaviour.
In those ages, there were only empiric assumptions. With time it can be said that these
theories were well-funded or not, as nowadays humans have been able to pierce most of
secrets of this invisible phenomenon – sound. At best, we not only gained knowledge
with well suited explanations, but thanks to the rising of the arithmetic language from
Middle- and Far-East countries, we have been able to translate word-based concepts into
numbers and algebraic formulations. And today, complex acoustic phenomena such as
reverberation time can be described clearly by words and equations.
22

41. Chapter 2. Sound and Rooms 23
2.1 Reverberation Time : Definition
The reverberation time of a room can be defined as the time it takes for sound to drop
by -60 dB after switching the source off. When measuring, there is no need to measure
the actual time taken by the reverberant energy to decay by 60 dB, but due to the
linearity of the decay, we can just observe the slope of 30 dB decay and then extrapolate
the corresponding time for a decrease of 60 dB (see RT results in Chapter 3). This is
because the ‘tail’ of the reverberation time decay often falls below the background noise
in the room, which may distort calculations.
Figure 2.1: Reverberation time definition with a sample decay. The slope of the decay
is, in practice, measured between –5 dB and –35 dB of the initial level. (M.Barron,
1993)
Statistical model of reverberation
Reverberation time is the most common example that can be made to illustrate
the statistical approach for sound in rooms. This approach limits itself to predicting
statistical average values of acoustic data which are assumed to be constant throughout
a room. It assumes that sound propagation whithin a room is analogous to a snooker
ball bouncing around a billiard table, ie. where sound rays will hit the room boundaries
and be reflected in that way (Figure 2.2). Thus, in any room, boundaries will dominate
the behaviour of sound. With time, as the direct sound is shutted off the reflection

42. Chapter 2. Sound and Rooms 24
density will increase, and when the reflections become very dense this is termed the
reverberant field.
Figure 2.2: Example of an impulse response and the bouncing paths of sound in a
room. (T.J. Cox, 2009)
Also, it is right to say that reverberation time is somehow the acoustic signature of an
enclosed space, as it is a derivation (backward integration) from the impulse response,
which characterizes the acoustic behaviour of a room (Figure 2.2 and 2.3). The impulse
response is a pressure versus time graph showing a response at a receiver position when
somewhere else in the room a short impulse is created.
Thereby, reverberation time formulations are statistical models of acoustic room be-
haviour, and are only applicable where there are a large number of reflections and the
sound field is diffuse. In the case of low frequency content, a modal behaviour of the
room will make the sound field non-diffuse. Thus there is a lower frequency bound on
the applicability of statistical absorption formulations, called the Schroeder frequency,
given by :
f ≥ 2000
T60
V
(2.1)
where T60 is the reverberation time and V the room volume.

43. Chapter 2. Sound and Rooms 25
Figure 2.3: Hypothetical pattern of sound reflections arriving at a listener after a
short sound pulse created at a point in a room. (Institute of Acoustics, 2015)
2.1.1 Sabine’s formula
As said in the beginning of Chapter 1, pioneer work in the determination of acoustical
requirements for rooms was carried out by W.C. Sabine at the edge between the 19th
and 20th century. He concluded that the rate of decay of sound energy, once the source
has ceased, was one of the most important acoustical properties in rooms (Sabine, 1922
[9]). ‘While greater theoretical minds in the nineteenth century probably imagined that
the problem was not amenable to simple mathematical analysis’ (Barron, 1993 [1]),
Sabine discovered around the year 1900 [9] that only two quantities determined the
reverberation time in a room : the room volume (V ) and the total acoustic absorption
(A). For him, the time taken for a sound to decay to an inaudible level after the source
was stopped (an organ pipe), was measured and called reverberation time, RT.
The concept of reverberation time was later defined in objective terms as previously
(decay rate of -60dB). The Sabine’s relation is as follows :
RT =
0.16V
A
(2.2)
Where RT is the reverberation time in seconds,V the room volume m3 and A the total
absorption area in m2 (Units in S.I). It may seem that the formula is not homogeneous
at first sight (m3 as numerator and m2 as denominator) but it is still correct, for the
0.16 factor is appropriate. (See derivation in Appendix A)

44. Chapter 2. Sound and Rooms 26
This equation is the basis of virtually all reverberation time predictions in rooms.
The calculation is performed at each octave band frequency of interest (A is frequency
dependent, so is RT), at least from 125 Hz to 2000 Hz but generally extended from 63
Hz to 4000 Hz.
Limitations of Sabine’s formula
To investigate the validity of the Sabine’s formula, it is needed to consider extreme
values of absorption. If the absoption coefficient is very low (α → 0), the absorption
of the room will then approach a zero value (A → 0). Thus RT tends to a great value
(RT → ∞), which is actually the case in a very reverberant room.
Conversely, if the room is very absorbent, the absorption coefficient would tends to 1
(α → 1). In this case, A approaches S (A → S) and RT would tends towards the value
: RT = 0.16V
A . This is not very logical because, as the room absorbs all sound energy,
RT should be null.
These results reveal the imperfections of Sabine’s theory. In fact, the latter is based
on the assumption - incorrectly assumed - of a reverberant field evenly distributed in the
room - or that sound energy is being absorbed continuously with time. But for that to
be strictly accurate, it is necessary that the room has to be sufficiently reflective. This
is why Sabine’s theory gives even less valid results that the room is absorbent.
In practice, Sabine’s formula is used when the mean absorption coefficient of the
room is less than 0.2 (some surfaces may have an absorption coefficient greater than
that value, while the average coefficient remains below 0.2). For more absorbent rooms,
other calculation methods are used to define the reverberation time.
2.1.2 Eyring’s formula
It has been seen that Sabine’s theory does not suit all situations and gives inconsistent
results when the room is too absorbent. In this case it is better to use another formula
proposed by Eyring in 1930. This formula is also based on a number of assumptions,
and therefore does not apply to all situations ; there are other formulas for calculating
RT though, even if they are less commonly used (Millington/Sette and Fitzroy).
Thereby, the Sabine’s method is a statistical approach that assumes that the reverber-
ant energy is being absorbed continuously with time. Now, for the energy to have enough

45. Chapter 2. Sound and Rooms 27
time to distribute in this way within the room it is necessary that the absorption must
be relatively low. However Eyring’s model is based on a ‘microscopic’ approach. Rather
than considering the reverberant energy being absorbed in the same way throughout the
room, Eyring’s method consists in following the path of a sound ray through it (mean
free path between reflections), and calculate the energy absorbed during each reflection.
That is to say that the absorption occurs discretely at each reflection with the room
boundaries, which is a more correct assumption than Sabine’s theory. (see derivation in
Appendix B)
The formula of Eyring’s reverberation time is then :
RTE = −
0.16V
Sln(1 − α)
(2.3)
Where RTE is the reverberation time in seconds,V the room volume m3 and α the
absorption coefficient. (Units in S.I)
As regards to other formulas (reverberated intensity, reverberated sound pressure
level, room radius, etc...), they still remain valid, only replacing the absorption area A
by a new quantity, Rc, called the room constant in m2.
If α is very small, (1 − α) ≈ 1, hence Rc → A.
Limitations of Eyring’s formula
If the absorption of the room is very small, α → 0. As α is positive, ln(1 − α) → 0
(while being negative). This implies RTe → +∞, as in Sabine’s formula, which is
consistent.
If the absorption is very strong, α → 1. As α < 1, ln(1 − α) → −∞ and thus
RTE → 0. Again, the result makes sense, since if the room absorbs all the reverberated
energy, RT is necessarily null (while Sabine’s formula gave outlier results).
It is interesting to note that when ¯α (mean value) is very small, the formulae may be
simplified and the result may be the same as the original Sabine’s equation (for ‘lively’
rooms). It is important to note as well that reverberation time equation of both Sabine
and Eyring assume a uniform absorption for all surfaces of the room - that means having
an averaged equal absorption for all surfaces. And in some cases, this is no more a real
statement.

46. Chapter 2. Sound and Rooms 28
2.1.3 Other methods : Millington-Sette, Fitzroy
Millington-Sette’s Formula
Eyring assumes that sound coming from a source in a room is successively reflected by
boundaries having a similar averaged α coefficient. Each time a wave strikes one of the
boundaries, a fraction (α) of the energy is absorbed, and a fraction (1 − α) is reflected.
However, this method has a weak point that appears to be when the various components
of the room no longer have similar absorption coefficients, but rather different ones. That
is why Millington and Sette (1932-1933) derived an equation to predict the reverberation
time that is based on similar assumptions as Eyring’s formula, but differs in the way
in which absorption coefficients of the various portions of boundaries are averaged, ie.
containing different absorption coefficients. This leads to the Millington-Sette’s formula
:
RTMS =
0.16V
−
n
i=n
Siln(1 − αi)
(2.4)
where Si is the surface area of the ith material, and Ai is its actual absorption
coefficient. In the limit of all αi << 1, it reduces to Sabine’s formula with αi = α(Eyring)i.
Fitzroy’s Formula
The reason why the method of Fitzroy takes its legitimate place, being really different
from previously quoted works, is that it is no longer based on the assumption of an
evenly distributed sound field, ie. allowing a better prediction of the reverberation time
in the case of non-uniformly distributed sound absorption. This method makes sense as
the reverberation time frequency characteristics (in this scenario) cannot be predicted
accurately using Sabine’s or other classical reverberation theories. The reason is that
these theories are based on the assumption that the sound field considered is completely
diffuse, and it will be sufficiently diffuse if there are no large differences in the basic
dimensions of the room, that walls are not parallel, and that sound absorbing material
is uniformly distributed throughout the space. But in practice, almost none of these
requirements is fulfilled[8].
That is why Fitzroy introduced in 1959, after a further 25 years since Millington-Sette

47. Chapter 2. Sound and Rooms 29
formulation, an empirically derived equation that considers non-uniform distribution of
absorption. This supports the theory that if some surfaces are reflective while others are
absorbent it is likely that some of the sound energy will persist much longer than the
remainder. Fitzroy considers then that the sound field may tend to develop reflection
patterns involving the three major axis of a rectangular room, and that each of these
patterns will undergo decay at different rates depending only on the average absorption
of surfaces involved in each case. The decay is thus considered as the sum of the decays
between each pair of surfaces, each contributing in proportion to the ratio of the area
of each pair to the whole surface area. The empirically Fitzroy derived equation is
ammended as follows [8][4] :
RTFtz = 0.16 ·
V
S2
−x
ln(1 − αx)
+
−y
ln(1 − αy)
+
−z
ln(1 − αz)
(2.5)
where : x, y, z - total areas of two opposite parallel walls in m2,
αx, αy, αz - average absorption coefficients of a pair of opposite walls,
S - total surface area of the room in m2,
V - total volume of the room in m3.
A comparative example between Fitzroy and Eyring methods is explained by Anita
Lawrence in its book ‘Architectural Acoustics, 1970’ just below, with room dimensions,
surfaces, and relative absorption coefficients. Calculation of Eyring’s reverberation time
is spared to get directly to the result of 1.2 sec of reverberation. As it is shown in
the example below, Fitzroy’s calculation shows a reverberation time of approximatively
4 sec. This represents a significant difference and although the effect of non-uniform
distribution of absorption modified by reflections. This example does illustrate that
caution should be exercised when using Eyring’s or Sabine’s formula if the assumptions
made are not reasonably accurate in a particular room.

48. Chapter 2. Sound and Rooms 30
Figure 2.4: Example of Fitzroy reverberation time calculation. (A. Lawrence, 1970)
2.1.4 Dictum
It is worth noting that acousticians are not altogether satisfied with existing formulae
on reverberation time, thus - according to some literature - the European standard con-
cerning this issue is still an open question.
The methods mentioned above are not the only methods of calculation for the rever-
beration time of rooms. Indeed, there is a variety of modifications about reverberation
time made by several scientists/researchers based on classic formulae, eg. Tohyama
and Suzuki (1995) presented an ‘almost-two-dimensional’ diffuse field theory which may
occur in a room with hard walls perpendicular to an absorbent floor. But there were
also others like Franklin (1903), Schroeder (1965), Cremer and M¨uller (1978), Kuttruff

49. Chapter 2. Sound and Rooms 31
(1975), Nilsson (1992), as Bistafa and Bradley (2000) - just to mention some of the best
known.
However, it may be noted that in most situations the so-called ‘conventional’ or
‘classic’ methods are still a good way to approximate ; without too much deviation ;
the practical reverberation time of a room - knowing the assumptions that need to be
handled, of course.
2.2 Absorption
When a sound wave encounters an obstacle, some of its energy is absorbed : this is the
sound absorption phenomenon - a certain quantity of sound energy is transformed into
heat. For this it is necessary that the acoustic wave energy causes enough vibration of
matter.
Energy balance
When an incident sound wave Ei (charged with energy) meets a partition, an Er
part is reflected in the incident plane, a portion Et is transmitted directly through the
partition, a portion Ea is absorbed and transformed into heat inside the wall, and finally,
one last Ef part consists of ‘flanks’ and parasites transmitting in building side structures.
Figure 2.5: Energy conservation when a sound wave hits a partition.
Thus, and according to the laws of physics, the total energy is conserved, which gives
:
Ei = Ea + Er + Et + Ef (2.6)

50. Chapter 2. Sound and Rooms 32
Absorption coefficient
What is of interest here is the absorbed part of the energy, relative to its incident
part. The ratio of absorbed energy over incident energy is what is called the absorption
coefficient of a material.
α =
Ea
Ei
(2.7)
where α is a unitless number (a ratio) that ranges from 0 (for a totally reflecting
partition) to 1 (for a totally absorbent partition) :
0 < α < 1
The absorption coefficient of a material depends on the frequency of the sound wave
and its angle of incidence. However, the absorption coefficient is usually given as a
function of frequency without specifying the angle of incidence : this corresponds to the
average absorption coefficient for all angles of incidence, ie. to a diffuse field. Appendix
C provides a sample table of absorption coefficient values for different materials.
Absorption area
Lets consider a room whose partitions consist of n surfaces coated with various mate-
rials (wood walls, carpet flooring, etc ...). Lets call then S1, S2, ... , Sn the relative areas
of these materials, and α1, α2, ... , αn their respective absorption coefficients. What is
called absorption area (in m2) of a said surface Si with an absorption coefficient αi is
the following quantity :
Ai = Siαi (2.8)
For example, a 2 m2 curtain with an absorption coefficient α = 0.35 (at 250Hz) has
an equivalent absorption area A = 0.7 m2.
This implies that the absorption of a material with an absorption coefficient α and
surface S is equal to the surface of a perfectly absorbing material (α = 1), which would
absorb the same energy as the real one.
By extension, the absorption of a room may be defined by the sum of all absorption
surfaces that compose it. Which gives :

51. Chapter 2. Sound and Rooms 33
Atot =
n
i=0
Siαi (2.9)
Atot = S1α1 + S2α2 + ... + Snαn (2.10)
Atot = A1 + A2 + ... + An (2.11)
If a room has an absorption A = 100 m2, this means that all room partitions absorb
as much as 100 m2 of perfectly absorbent material. That is why A can be called an
‘equivalent absorption area’.
Air absorption
Vibrating under the action of a sound wave, air molecules (oxygen, nitrogen, carbon
dioxide, etc...) undergo friction with each other. They thus produce heat. The result is
a loss of energy because the energy transformed into heat is deduced from the acoustic
energy emitted by the source. The attenuation of sound that results is called attenuation
by atmospheric dissipation or divergence.
Although frictional processes do occur as part of the mechanism of air absorption
there are other mechanisms as well involving the rotation and vibration of the molecules
in the air.
This attenuation increases with frequency (more precisely proportional to the square
of the frequency). This is why when being at long distance of a source, the bass sound
is more audible than acute sounds (eg. background noise of a city). The low frequency
sounds can therefore spread far beyond high frequency ones. Due to the low dissipation
of bass sounds, some animals like elephants or whales use infra-sounds to communicate.
The absorption of sound by air is strongly dependent on relative humidity, and af-
fected to a lesser extent by air temperature - which is a strong factor in air velocity
though. So normally, air absorption of sound is usually given in function of the relative
humidity of the location.
2.3 Early Decay Time - EDT
As said in Chapter 1.2.1, the Early Decay Time (EDT) is a measure of the sound decay,
expressed in the same way as reverberation time, based on the first 10 dB portion of

52. Chapter 2. Sound and Rooms 34
decay.
For speech and music, it is impossible for a listener to perceive a decay of 60 dB, since
the latter portion of decay is masked by the following sounds or the background noise.
Thus the subjective impression of reverberation ; reverberance ; is found to depend on
something more than RT60. It depends mostly of the initial decay rates. This is because
classical theory predicts an exponential decay (which is represented as a straight line on
a logarithmic scale), but in pratice deviations from this decay are found (Figure 2.6).
Figure 2.6: RT30 and EDT curves compared on a sample of decay. (ref : Synergetic
Audio Concepts : Early Decay Time as a System Performance Benchmark)
In highly diffuse spaces where the decay is completely linear, the two quantities, RT
and EDT, would be identical. In less diffuse spaces, different values would be found, as
in Figure 2.6, and the impression of reverberance would be more based on EDT rather
than RT. [1][4]
Early Decay Time is thus ultimately important to enhance clarity or intelligibility,
as it allows to distinguish more or less the beginning or ending of spontaneous sounds.
Hence, a good control on early reflections is a leading parameter in the design of a room.
2.4 Optimum RT curves
Opinions regarding the ideal reverberation time for a given room or hall vary depending
on people’s subjective weightings. But in general, some guidelines can be determined
for different kind of rooms.
Rooms to be used for speech should have a fairly short reverberation time and rooms
used for music should have longer ones. The volume also is of importance. For example,
longer reverberation times are more acceptable in large auditoriums than in small ones.

53. Chapter 2. Sound and Rooms 35
Figure 2.7 shows recommended mid-frequency reverberation times for different pur-
poses in different sized rooms.
Figure 2.7: Recommended optimum mid-frequency reverberation times for rooms.
(W. T. Grondzik, A. J. Kwok, Mechanical and Electrical Equipment for Buildings.
Sometimes, a room or hall is to be used for several purposes and some compromise
must be made between requirements for speech and music. In many cases this will be
acceptable to most of the audience, but in major multi-purpose auditoriums it may be
necessary to incorporate some method of varying reverberation to suit more to the kind
of occasion.
It is clear that optimum reverberation times are a strong factor for characterizing an
adequate reverberation time for a room depending on its purpose and volume. They are
a helpfull guideline in order to determinate the suitability of a particular reverberation
time of a room.

54. Chapter 3
Measurements and Analysis
3.1 Methodology
The measurements carried out under this project will serve to illustrate the different
ways to measure reverberation times of rooms. The procedure for measurements is
based on standard EN ISO 3382-1 and 3382-2 : 2009 ‘Acoustics – Measurement of room
acoustic parameters – Part 1 : Performance spaces – Part 2 : Reverberation times in
rooms’. This standard will however, serve as a guideline for acoustic measurements in
rooms, as the measuring equipment available does not fully meet the standard method.
The purpose of these measurements is to see from a practical point of view how
the reverberation time of a space can be measured. On this occasion, several rooms
with different characteristics will be available : a university lecture hall (for speech),
an auditorium (for music), and a multi-purpose theatre (for a combination of both
speech and music). Thus, this will allow to see if their reverberation times match the
characterization criteria issued by their use, i.e. if they correspond to the optimum RT
values of those rooms.
36

55. Chapter 3. Measurements and Analysis 37
3.1.1 Preparation : requirements for measurements
Measurement conditions
As stated in the standard in section 4 ‘measurement conditions’, it is necessary to
complete and to be aware of some important criteria when measuring RT values, such
as the potential absorption added by people or audience, the characteristics of the mea-
suring equipment, as the procedure to measure with good accuracy the reverberation
time.
Criteria can be as follows :
• It is necessary to ensure that the sound source is as omnidirectional as possible.
For expertise or precision measurements, it must meet the requirements of ISO
3382. However, it is acceptable to use other sources as long as they can produce
a sufficient sound pressure level in order to generate acceptable decay curves in
the minimum dynamic range required, without contamination of the background
noise.
• The source positions can be the usual positions of sound generation in the room,
depending on its use.
• The microphone must not exceed a certain diameter. It must be as small as possible
to avoid any diffraction phenomenon in the high frequency content of the sound
generated. Moreover, its integration time, i.e. the time constant of an exponential
averaging device, must be less than T/30. Thus, it is commonly accepted that the
sound level meter should be at least of Class 2, or Class 1 in the best instance.
• In a multi-microphone measuring framework, it is essential that these should not
be too close to each other, in order to obtain an adequate acoustic coverage of the
room. In that event, they should be placed at least half a wavelength away, i.e. a
minimum distance of 2 m for the usual frequency range.
• It is appropriate to place the microphone at least a quarter wavelength away (about
1 m) of any reflecting surface, floor included.
• No microphone position must be too close to the source position in order to avoid
excessive influence of the direct field. The minimum distance dmin in meters, can
be determined using the following equation :

56. Chapter 3. Measurements and Analysis 38
dmin = 2 V
cTe
where V is the room volume, c is the sound velocity in m.s−1and Te the estimated
reverberation time in seconds.
• It is agreed that the measured frequency range must be at least from 250 Hz to
2000 Hz. For more precise measurements, it is good to measure between 125 Hz
to 4000 Hz.
• It is customary to have at least 2 to 3 measurement positions to get a good average
of the reverberation time of the room. This average is simply the arithmetic
average of all the RT values for all positions.
The previous list allows to establish most of the methodological criteria to be taken
into account before proceeding to any measurement.
Measurement methods
Moreover, measurements of reverberation time can be conducted by various methods,
namely the methods by noise interruption and by impulse response. Both methods have
the same theoretical average.
On one side, the noise interruption method works as follows :
- A speaker must be used and the signal it receives must be a random broadband
electrical noise, ie. white or pink noise. The source must be able to produce
a sufficient sound pressure level in order to ensure a decay curve beginning at
least 35 dB above the background noise in the corresponding frequency range. If,
however, a T30 is to be measured, it is then necessary to create a level at least
45 dB above the background noise, for more accuracy. The method then consists
in generating the broadband noise signal to pull the room in a steady state, and
suddenly turning off the source to observe the immediate decay of sound, ie. its
reverberation time.

57. Chapter 3. Measurements and Analysis 39
On the other side, the impulse response method is based on different criteria :
- It is required to generate a pulse type signal just like a gunshot, an air balloon
explosion, or even sinusoidal sweeping and maximum length sequences (MLS). It
is really necessary that the source is not itself reverberant and of short duration,
with a wideband spectrum. As for the noise interruption method, the impulse
must be able to produce a sufficient peak sound pressure level to ensure a decay
curve beginning at least 35 dB above the background noise. MLS and sweep
sine methods do particularly allow to generate the impulse response only after
appropriate treatment of the recorded signal on the sound level meter, improving
the signal-to-noise ratio. From these measurements, the reverberation time is
deduced by the reverse integration method.
Concerning our measurements, both methods will be used to see and compare the
results when possible. This will put into practice what was said in the previous para-
graphs.
3.1.2 On site
Equipment
The measurements of this project were made possible through the provision of mea-
surement equipment from the company Emacoustique in Bordeaux.
Equipment specifications :
Sound Level Meter 01dB-Stell Solo Type 7121
calibrated at 93.6 dB for 1kHz
Microphone Sensor 01dB-Stell 21S Type 10673
Active sound source Bruel & Kjaer Type 4224
Inflatable balloons
Access to computers with softwares 01dB Suite (dB-Bati & dB-Trait)

58. Chapter 3. Measurements and Analysis 40
Figure 3.1: Pictures of the sound sources and sound level meter used.
Here is the equipment that will be used for measurements, allowing to use the noise
interruption method and to try the impulse response method when it is possible - or see
where it cannot work well enough (air balloons have less energy than gunshots).
As said previously, this equipment is validated as the minimum required for RT
measurements. The sound source used is not perfectly omidirectionnal in all its frequency
response, but can be assumed adequate without the need to put it to a corner to stimulate
the room. The sensor of the sound level meter is categorified as Class 1 (50 mV/Pa)
and thus meets the relative requirements of the standard.
Halls
In each room, the methodology will remain basically the same. First, the source will
be installed in the usual place of sound production (stage or proscenium, most of the
time). Normally, a certain distance Dmin is to be respected, eg. for an approximate
volume of 17000 m3 the Auditorium of Bordeaux will have a minimal distance of mea-
surement Dmin equal to 10 m. However as the measurements will almost always take
place in the audience area, the distance Dmin can be reduced to the distance between
the source and audience as it is the defined listening area. Sometimes, it may be that the
first rows of seats will be at the limit distance of an excessive direct field. The relative
measurement positions in the different rooms will be the same as said in their respective
description paragraphs.
For measurements using the noise interruption method, the sound source will be ac-
tivated and locked to a sufficient sound power level of 110 or 120 dB, in order to give
enough energy to the room. Correct settings for this procedure shall be checked in the
sound level meter (SLM) menu (noise interruption). Hence the SLM will start to mea-
sure the sound pressure level. The sound source shall be then shutted off, in order to
allow the recording of the decay of reverberant sound. Finally, the recorded sound tracks

59. Chapter 3. Measurements and Analysis 41
will be written in a sheet of paper to ensure a better post-treatment of measurements.
Once this is done, the impulse response method will be tried (using air balloons) to see
if the results will remain the same or if some discrepancies will appear. For this, correct
settings for this procedure must be renewed and wait for an impulse signal. Then, the
air balloons will be inflated and exploded in the same locations as previoulsy. After the
explosion, a few seconds of processing will allow to see the decay curve of reverberation
time. The number of the recorded tracks will be written down on a sheet of paper.
The three rooms available for measurements are as follows :
Theatre des Quatres-Saisons
This theatre of 404 seats (volume ≈ 2, 000m3 ) is a multi-purpose room. It host
musical events as well as lyrical performances or other shows. The RT requirements for
this room are quite difficult to define as it may enhance sound in a good way for many
kinds of sound sources (music/speech). RT shall be in the range between music and
speech optimum values.
Figure 3.2: Pictures of the Theatre des Quatres-Saisons, Bordeaux
Measurements will be made at the front, middle, rear and lateral zone of the audience
area.
Auditorium of Bordeaux
The name of the performance room is Henri Dutilleux. It has 1449 seats (volume
≈ 12, 000m3), a stage with multiple configurations (10, 12 or 14 m deep and 20.5 meters
width), and has 17 adjustable shelves to let the ’chef’ the free disposal of musicians
according to their kind of performance. A special arrangment also allows a totally
horizontal platform for dance. Under the proscenium, a mobile fossa of 160 m2 can
accommodate 120 musicians.

60. Chapter 3. Measurements and Analysis 42
Figure 3.3: Pictures of the Auditorium of Bordeaux
This room was designed by Eckard Khale from Khale Acoustics a few years ago (2013-
2014). It is definitively a good room for music, and its RT should be a good example
to measure.
Measurements will be done at the front, middle, rear zones of ground audience, at
the rear upper balcony, behind the stage area and at a lateral balcony.
University lecture hall
This lecture hall (volume ≈ 3, 000m3) was of interest because of rumors about its
bad acoustics. Students and teachers are nearly all agree on this phenomenon - needless
to say that most of people are not happy with this room. When going inside the room,
from an acoustic consultant point view, the reason is obvious : the reverberation time.
This room shall allow to see how reverberation time can be bad for intelligibility.
Measurements will be made at the front, middle, rear and lateral zone of the audience
area - like the theatre.
3.2 Results
3.2.1 Data processing
Once measurements are done in these places it is fundamental to process the data thus
obtained by the sound level meter. To do this, an access to softwares specialized for
these kind tasks is helpfull. Here, the 01dB software ’dB-Bati’ will help processing raw
data and visualize the decay curves recorded.

61. Chapter 3. Measurements and Analysis 43
Figure 3.4: Example of RT measurement results with the dB-Bati software.
It is important to check if the RT value is correctly determined. To define a reverber-
ation time, it is essential to ensure that the decay curves roughly follow a straight line.
In the example above, one can observe that the decrease slope at 63 Hz is not really
right. This may indicate a lack of sound energy at low frequencies, or some resonances
and echoes phenomena that may interfere with the sound decay in this frequency range.
The picture below shows the decay of sound in a room at 1 kHz. The decay curve is
more like a straight line now, which justifies the determined reverberation time.
Figure 3.5: Decay curve of an interrupted sound at 1 kHz, with RT = 0.91 s.

62. Chapter 3. Measurements and Analysis 44
3.2.2 Room results
After processing the recorded data, results of both methods will be presented for each
hall as follows : resumed values of RT values into a table, a time/frequency diagram,
and discussion about some measured details as the corresponding optimum RT values
for these spaces.
3.2.2.1 Theatre des Quatre-Saisons
Reverberation time measurements (noise interruption and impulse response methods)
carried within the Theatre des Quatre-Saisons at the indicated positions have given the
following results :
Reverberation Times (s)
Frequency (Hz) Proscenium Front Middle Rear Lateral Avge
63 2.08 0.68 (10.73) (0.21) 1.41 1.4
125 1.08 1.12 0.76 1.09 1.4 1.1
250 0.81 1.11 0.88 1.08 1.1 1
500 0.76 0.89 0.92 0.98 1.13 0.93
1k 0.71 0.83 0.91 0.9 0.91 0.85
2k 0.58 0.82 0.85 0.86 0.92 0.80
4k 0.65 0.67 0.83 0.81 0.81 0.75
Table 3.1: Reverberaton times measured at the Theatre des Quatre-Saisons using the
noise interuption method in the frequency range of 63 Hz to 4kHz.
Reverberation Times (s)
Frequency (Hz) Proscenium Front Middle Rear Lateral Avge
63 1 1.61 1.18 1.21 1.76 1.35
125 0.92 1.06 1.14 1.33 1.16 1.12
250 0.88 1.09 1.06 1.05 1.02 1.02
500 0.99 0.99 0.9 0.97 0.94 0.96
1k 0.93 0.92 0.84 0.93 0.86 0.9
2k 0.84 0.97 0.82 0.9 0.88 0.88
4k 0.75 0.91 0.8 0.8 0.73 0.8
Table 3.2: Reverberaton times measured at the Theatre des Quatre-Saisons using the
impulse response method in the frequency range of 63 Hz to 4kHz.

63. Chapter 3. Measurements and Analysis 45
Figure 3.6: Diagram of reverberaton times measured at the Theatre des Quatre-
Saisons using the noise interuption (upper) and the impulse response (lower) methods.
It is clear that both methods differ in the results obtained - mostly in particular RT
measurements. But it is surprising to see that the average of the multiple reverberation
times taken to different locations provides an averaged curve relatively similar between
the two methods - 1.4 sec at 63 Hz and 0.8 sec at 4 kHz in a same decrease pattern.
Figure 3.7 shows the two averaged reverberation times and the total relative average
between them, which can be assumed to be a close approximation of the real mean
value of the reverberation time of the theatre. It can be said that, in spite of the differ-
ence between some measurements, the two methods of noise interruption and impulse
response converge on an approximate value of the reverberation time of the room.

64. Chapter 3. Measurements and Analysis 46
N.B : In previous tables, it can be seen that there is a proscenium value recorded
despite the fact that it was said that a distance Dmin should be respected. I though that
the proscenium should be recorded too as it is the location where perfomers generate
sound and thus receive a feedback from the room. Reverberation time in this location
should be of interest to see how perfomers hear reverberated sound when they talk or
play on a music instrument. Moreover, as they are located in the sound source area
they might benefit of some more intelligibility or clarity to hear each other due to the
proximity with the direct field. Also, given the many measurements, the integration of
the reverberation time taken on the proscenium within other measurements affects the
final curve in small proportions.
It can also be seen that some values may seem unusual, eg. RT values at 63 Hz for
middle and rear locations using the noise interruption method. This may be because
of some sort of room modes or wave behaviour of sound, or maybe a lack of energy
although this would correspond more to the use of balloons. In all cases, the averaging
helps stabilizing these measurements around an adequate value.
Figure 3.7: Diagram of averaged reverberaton times measured at the Theatre des
Quatre-Saisons using the noise interuption (INT) and the impulse response (IMP)
methods.
As said previously, Figure 3.7 shows the two averaged reverberation times and the
total relative average between them. The two methods of noise interruption and impulse
response thus converge on an approximate value of the reverberation time of the room.
This value can be considered adequate as it is the result of many averages between single
measurements and their relative mean value.

65. Chapter 3. Measurements and Analysis 47
Concerning the validity of obtained values, it can be seen that RT values for this
theatre fall around 1 sec of reverberation. They extend to 1.4 secondes in low frequencies,
which may give a bit sense of warmth and enhance music, to nearly 0.8 secondes in higher
frequencies (1 to 0.8 sec from 250 Hz to 4kHz), allowing a short reverberation to enhance
speech transmibility when it is required. As it has been said in Chapter 1, optimum
reverberation times for music shall fall above 1.5 to 2 sec and less or equal to 1 sec for
speech. As it is a multi-purpose theatre, it is good to find that RT tends to a compromise
between these two types of sources that are speech and music.
3.2.2.2 Auditorium of Bordeaux
Thus, as for the previous room, reverberation time measurements concering the two
methods of noise interruption and impulse response will be presented as follow :
Reverberation Times (s)
Frequency (Hz) Proscenium Front Middle Rear Stage Lateral Upper Rear Avge
63 3.18 2.19 2.63 4.03 3.95 4.12 3.35
125 1.55 0.65 2.57 2.77 2.23 2.7 2.08
250 2.15 2.02 1.93 2.21 2.31 2.4 2.17
500 1.82 1.8 2.08 1.99 2.14 1.83 1.95
1k 1.83 1.45 2.01 1.94 2 1.96 1.86
2k 1.49 1.59 1.92 1.95 1.93 1.87 1.8
4k 1.49 1.72 1.73 1.72 1.8 1.76 1.7
Table 3.3: Reverberaton times measured at the Auditorium of Bordeaux using the
noise interuption method in the frequency range of 63 Hz to 4kHz.
Reverberation Times (s)
Frequency (Hz) Proscenium Front Middle Rear Stage Lateral Upper Rear Avge
63 1.51 1.39 4.64 4.03 1.76 1.21 2.42
125 2.49 2.57 2.53 2.77 1.16 1.33 2.14
250 2.08 1.8 2.2 2.2 1.02 1.05 1.72
500 2.14 1.95 2 2 0.94 0.97 1.66
1k 1.94 1.83 2 1.94 1.86 0.93 1.6
2k 1.87 1.94 1.94 1.95 0.88 0.9 1.6
4k 1.77 1.77 1.77 1.72 0.73 0.8 1.42
Table 3.4: Reverberaton times measured at the Auditorium of Bordeaux using the
impulse response method in the frequency range of 63 Hz to 4kHz.

66. Chapter 3. Measurements and Analysis 48
Figure 3.8: Diagram of reverberaton times measured at the Auditorium of Bordeaux
using the noise interuption (upper) and the impulse response (lower) methods.
Here, unlike the theatre, one can see a significant difference between the two kind of
reverberation times obtained by measurements. For example, it can be seen that some
of the data obtained by the impulse response method are quite disparate. The measure-
ments made close to the sound source shows a high RT value (lateral balcony and rear
stage locations) while slightly more remote measurements demonstrate a lack of accu-
racy - mostly in low frequency due the low sound energy dissipated by the air balloon. In
terms of the noise interruption method, the same curve pattern can be observed, more or
less, streched on different scales and intensities in function of the microphone location.
These differences show that RT values are not the same everywhere. However, it may
also come from other uncontrolled factors when measuring, ie. reflexions from upper

67. Chapter 3. Measurements and Analysis 49
balconies or the back of the room, or some diffraction phenomenon around structures.
Thereby, this is why several measurement points had been taken in order to average
the whole distinct values into one single approximate curve.
As it was done for the Theatre des Quatre-Saisons, it is still a good thing to get
the mean value of the two averaged curves from each method in order to obtain a
better approximation of the reverberation time of the hall. The average of these two
reverberaton time curves is shown in Figure 3.9.
Figure 3.9: Diagram of averaged reverberaton times measured at the Auditorium of
Bordeaux using the noise interuption (INT) and the impulse response (IMP) methods.
The Auditorium of Bordeaux is a large room, specialized in music production. There-
fore, the main goal here is to see if the reverberation time of the Auditorium is sufficiently
appropriate for music.
First of all, one can observe that the reverberation time in the low frequency range
is quite high, reaching nearly 3 secs at 63 Hz. This may indicate the need to establish a
warmth sense to sound, as warmth gives a extra quality to perfomed music. Then it can
be seen in the rest of the curve that the reverberation time remains between 2 and 1.5
seconds, which is acceptable (and even desired for music in rooms) and can correpond
to a wish of giving more clarity for medium and acute sounds - easily starting from 1
kHz up to 12 kHz. Thus, this reverberation time can provide warmth and bold coating
for low frequency sounds, as clarity and brilliance for acute ones.

68. Chapter 3. Measurements and Analysis 50
As the optimal reverberation time for an auditorium of approximately 17000 m3 is be-
tween 1.5 and 2.5 secs for mid-frequencies, one can say that the Auditorium of Bordeaux
correctly fulfills its function of musical hall - pretty much even according rumors.
3.2.2.3 University lecture hall
Due to some difficulties in arranging the schedule between the University of Bordeaux
and Emacoustique, equipment to make measurements of the reverberation time with the
noise interruption method could not be available. That said, as the volume of the room
was relatively small (150 seats) with high RT values, the measurements were conducted
with the impulse response method (using air balloons).
Reverberation Times (s)
Frequency (Hz) Front Middle Rear Lateral Avge
63 2.55 2.4 2.28 2.89 2.53
125 1.7 1.54 1.39 1.86 1.62
250 1.55 2 1.88 1.94 1.84
500 1.96 2.11 1.96 2.14 2
1k 2.03 2.05 1.9 2.14 2
2k 1.88 1.91 1.9 1.9 1.9
4k 1.63 1.76 1.65 1.68 1.68
Table 3.5: Reverberaton times measured at the university lecture hall using the im-
pulse response method in the frequency range of 63 Hz to 4kHz.
Figure 3.10: Diagram of reverberaton times measured at the university lecture hall
using the impulse response method.

69. Chapter 3. Measurements and Analysis 51
At first sight, the RT curves seem to be ordered around a same mean value, repre-
sented as the average curve (shown alone in Figure 3.11). One can also distinguish high
reverberation time values in the low frequency range. It should however be said that
these high values can be over-evaluated, not because of a lack of energy produced by
the explosion of the balloon (Figure 3.12), but possibly due to the presence of strong
resonnances and also to the fact that a modal behaviour of the room makes it non-
diffuse at these frequencies - possibility of flutter echoe in low frequencies. The Shroeder
Frequency (Chap 1.1) has been calculated with an approximate volume to find that RT
values under 90-100 Hz are no longer perfectly accurate. Thus, according to Figure 3.11
(gap area), and assuming that the volume was approximate, it might be correct to say
that innacurate values really stand below 150-250 Hz - which is still high.
Figure 3.11: Diagram of averaged reverberaton time measured at the university lec-
ture hall using the impulse response method.
Figure 3.12: Results obtained by dB-Bati showing that the decay curve of 63 Hz has
sufficient energy - not perfect but still adequate to use according to acoustic consultants.

70. Chapter 3. Measurements and Analysis 52
There was some rumors about this room on how it had poor acoustics for lec-
tures. Now, seeing the reverberation time graphs, one can understand why. Nor-
mally, reverberation times for lecture halls (or speech rooms) must not exceed 1 second
(RTspeech ≤ 0.8 ∼ 1). But in this room, reverberation time is always much higher than
1, with a lower limit at 1.6 secs. As said in the subsection Speech in Rooms (Chapter
1.2.1.2), excessive reverberation harms speech transmission, and thus intelligibility. This
is a clear example where the reverberation time has not been taken into account when
designing the room, which has plenty of reverberant surfaces and low absorption.
This opens the way to Chapter 4, where a solution to a similar scenario will be tried
with computer modelling and simulation softwares. It will consist in creating a room
(based on the previous unversity lecture hall) with high reverberant values and try to
lower them using different materials with other absorption coefficients.

71. Chapter 4
Modeling and Simulations
As stated at the end of the last chapter, this chapter will show the 3-dimensional model-
ing of a room and simulate the reverberation time that should probably exist within it.
Two key softwares will be used for achieving this purpose : a modeling software named
SketchUp and a renowned acoustic simulation software named CATT-Acoustic. These
simulations will aim to show the importance of acoustic absorption of rooms.
The methodology will be as follows : a lecture hall with a high reverberation time,
based on the architecture of the University Lecture Hall measured in Chapter 3, will
be created, and its reverberation time estimated and calculated through different ways.
The first simulation will consist of integrating highly reverberant materials, i.e. with
low acoustic absorption, to push to the limits the case of a very high reverberation time
in a room of that size. Secondly, more absorptive materials will be integrated to roughly
meet the same RT value found with the previous measurements in Chapter 3 - adapting
the model in a simply way. And thirdly, the last simulation will consist in the good dis-
tribution of acoustic panels and reflective surfaces in order to obtain not only a good RT
value but a good sound distribution within the room. Auralizations for each simulations
will be accessible on the attached CD-ROM. Of course, some assumptions will be made
to simplify this exercice that could be very detailed and time consumming otherwise.
However, this will illustrate the impact of absorption in rooms, and how materials shall
be carefully selected and distributed in the design step of any room.
53

72. Chapter 4. Modeling and Simulations 54
4.1 Modeling of a 3-D space using SketchUp
4.1.1 Brief introduction to SketchUp
SketchUp is a modeling software distributed by Google, allowing to build a virtual model
of a 3-D item or architectural space - in our concernings, a lecture hall. This software
will help designing the geometry of the room in a simple way in order to import it into
a specialized simulation software to see how sound will behave in it.
Figure 4.1: Example of a quadratic residue diffusor (QRD) created with SketchUp.
4.1.2 Room Modelling
The modeled room (University Lecture Hall) is based on a hexagonal 2-D shape, then
extruded on the vertical axis in order to obtain a 3-D model (see Figure 4.2 below).
Figure 4.2: First steps in designing the modeled lecture hall.

73. Chapter 4. Modeling and Simulations 55
Thereafter, the room is gradually built inside and surfaces that are no longer in
contact with the inside volume are removed to simplify the obtained model, which gives
the final render presented in Figure 4.3.
Figure 4.3: Final render of the geometry of the lecture hall.
It is worth noting that here the geometric model is approximated in its design in order
to lighten the amount of superficial data that will then be processed by CATT-Acoustic.
In addition, the more number of surfaces and corners existing in the model is high,
the more is the probability of having geometric errors in CATT. So normally, models
are simplified and various data such as scattering coefficients are added as parameters
related to specific surfaces.
In this situation, it will be assumed that the room is empty and that the desks are not
installed*. What will be of interest will be in evaluating a fairly high reverberation time
in the room, in order to lower it with more absorbent materials in the next simulation.
* Nota Bene : Desks could be modelized as 2-D boards without any thickness and given
them a related scattering coefficient in CATT, which will normally be the thing to do
for RT simulations. However, a higher scattering coefficient will be integrated to the
ground in order to compensate. More details will be written over the CATT appendix.
Materials / Source and receiver
Once the geometric model is complete, there are two things that must be settled
before sending the geometry data to CATT.
First, it is important to grant specific materials to model surfaces. This process is
made very easy with SketchUp, where a bank of materials can be defined and used in
correlation with CATT - including the ability to configure relative absorption coefficients.

74. Chapter 4. Modeling and Simulations 56
Figure 4.4 shows the SketchUp material window, allowing to color surfaces with desired
materials. Figure 4.5 shows the text file where data of absorption coefficients is defined.
Figure 4.4: SketchUp material window with different kind of materials manually
added for more conveniance. (Some alpha coefficients are given by A. Lawrence)
Figure 4.5: MAT.GEO sample file containing relative absorption coefficients (in center
band frequency) of materials in percents (some materials with 10 value everywhere are
not effective and put to 10 by default).
For our first model, walls will be built with rendered brickwork or concrete, the floor
will be made of plastered concrete, the platform will be a hollow wood podium, and a
13mm plaster board on 100mm studs will be built for the ceiling.
Secondly, and ultimately important for the afterpart, it is necessary to place one or
more sources in the geometric model as also one or more receivers. In this model, there
will be only a pair Source/Receiver to keep the CATT parameters simple. (The source
will be omnidirectional)

75. Chapter 4. Modeling and Simulations 57
Intregration of sources and receivers is made possible by the LoudSpeakersTools plug-
gin for SketchUp from Euphonia, which is part from a greater pluggin named SK2GEO
(SketchUp to .Geo files). It is an electro-acoustic source positioning module. It will
allow to put sources into SketchUp and export them with the global model geometry.
Figure 4.6: Presentation of different kind of sources and receivers with the Loud-
SpeakersTools pluggin.
4.1.3 Exporting model data
The two previous tools (LoudSpeakerTool and SK2GEO) are necessary to export data
into CATT software. They will allow to export geometry, material specifications, and
the relative positions of sources and receiveirs. A quick menu allows to do so in a few
moments, creating different kinds of ’.GEO’ and ’.LOC’ files in the selected folder. Fig-
ure 4.8 shows the quick menu tool and the different files thus created.
Figure 4.7: Example of geometry import from SketchUp to CATT using the SK2GEO
tool. (Euphonia rights reserved)

76. Chapter 4. Modeling and Simulations 58
Figure 4.8: Geometry export from SketchUp to CATT using the SK2GEO tool (up)
and relative .GEO files obtained (down).
Therefore, a few new files are thus created after the data export which are as follows
:
• MASTER.GEO : It is the master file that will first be used to open CATT-Acoustic.
It will link together all the other files to allow the good running for CATT simu-
lations.
• MAT.GEO : MAT stands for ’materials’. This is the material file containing all
the relative data concerning the selected materials in the model, eg. absorption
and scattering coefficients (Figure 4.5).
• SRC.LOC : It is the source file location into the model. Basically, it has the
carthesian coordinates of its location in SketchUp to be able then to replace it
correctly into CATT.
• REC.LOC : This is the same kind of file as the SRC.LOC but aiming to keep the
location coordinates of the receiver.
• Others files containing extra data.
Once these files are obtained after the export session in SketchUp, it is possible to
go then through CATT-Acoustic and begin the simulation.

77. Chapter 4. Modeling and Simulations 59
4.2 Running simulations using CATT-Acoustic
4.2.1 Introduction to CATT-Acoustic
CATT-Acoustic is an advanced acoustic simulation software in the field of virtual acous-
tics widely used in building acoustics engineering and sound reinforcement systems.
Plenty of calculations can be done using adequately CATT tools.
The software can be opened in different ways, either by creating a new project or open-
ning a CATT file such as the MASTER.GEO file previously exported from SketchUp.
It will be considered here that CATT is directly opened with the MASTER.GEO file.
When opening the software, a window appears offering various settings (Figure 4.9). In
order to start viewing the model, source files that CATT will use must first be defined
in a repertory ; they will be the files previously exported from SketchUp (GEO, SRC,
REC files).
Figure 4.9: Starting window menu of CATT-Acoustic (left) and General Settings
window (right).
Once settings are correctly defined, the project must be saved and a CATT.MD9 file
will be created, which will be the next opening file containing all the previously specified
settings. Thereafter, the display option for viewing the imported model can be started
by clicking on the ’Save and Run’ option.

78. Chapter 4. Modeling and Simulations 60
If any problem related to the geometry model exists, CATT will send a message say-
ing that the room is not closed and that it will affect estimations - which will be no
longer accurate. If the geometry is correct, two windows will appear allowing the 3-D
visualization of the model (Figure 4.10).
Figure 4.10: PL9 and CATT 3-D viewers windows.
As shown in the figure above, the imported room model is correct and the source
and receiver are in their respective locations. Moreover, several surfaces are colored in
function of their kind of material.
4.2.2 Simulation results
4.2.2.1 Highly reverberant room
Room specifications are as follows (see Appendice C for alpha coefficients) :
• Walls : rendered brickwork or concrete
• Floor : plastered concrete
• Platform : wooden podium
• Ceiling : 13mm plaster on 100mm studs
• Room volume : 10253 m3 and Total approximative inner area : 3820 m2

79. Chapter 4. Modeling and Simulations 61
The room is intended to have a high reverberation time, which may be excessively
high though - but its value does not really matters for now as it remains relatively high.
Once the model is correctly runned, RT estimations can be made.
There are different ways for proceeding to RT calculations with CATT : RT estima-
tions, SxR predictions (source/receiver) and Audience mapping. Estimations are based
in a really simple approach of geometrical acoustics, beam tracing a few acoustic rays
(some decades) that will estimate the reverberation time values for different frequencies.
Also, they are given with Sabine and Eyring theories to see the relative deviation of the
said values. On the other side SxR predictions are more precise calculations of acoustic
behaviour in rooms. They can be made following 3 kinds of algorithms (short, longer,
and much longer calculations) depending on how much degrees of reflections are wanted
(reflection order from 0 to 3). The CATT module sofwate named TUCT will allow such
calculations.
Figure 4.11: Image source model function showing the sound paths and some of the
reflections between the source and the receiver.
As it can be seen RT estimations in CATT-Acoustic have given the following results
presented in Figure 4.12.

80. Chapter 4. Modeling and Simulations 62
Figure 4.12: RT estimations for the first model - highly reverberant.
The results above show several kind of information. There is a time versus frequency
graph showing the different RT estimations, a table of values corresponding to the pre-
vious graph, and two important specifications that are the Schr¨oder frequency and the
cut-off limit of the geometric acoustics approach.
Frequency (Hz)
RT (s) 125 250 500 1k 2k 4k 8k 16k
T-30 4.37 6.09 6.66 6.16 4.67 2.85 1.22 0.45
TS 3.34 5.54 6.27 5.98 4.51 2.81 1.24 0.43
TE 3.12 5.31 6.10 5.83 4.39 2.75 1.22 0.43
Table 4.1: RT estimations results for the first model - highly reverberant - in the
frequency range of 125 Hz to 16 kHz.

81. Chapter 4. Modeling and Simulations 63
It can be deducted by these two lastest informations that results beyond 250 Hz are
no longer accurate for a geometric approach of sound, due to a high modal density in
low frequencies calculated from the Shroeder frequency, which is at nearly 45 Hz. Other
softwares can however determine the acoustic behaviour beyond this limit as it is the
case of ABEC softwares using the BEM method (Boundary Element Method) that espe-
cially allows to observe the modal behaviour of low frequencies in rooms (Figure 4.13).
Furthermore, reverberation time results could be a bit over-evaluated, as scattering
coefficient has been set as an estimation, e.g. for the ground a value ‘L’ of 1.2 meters
has been added to take into account the gap between the different stairs that should be
present in reality, which can modify - in a non-negligible way sometimes - the results
thus obtained. Also, no audience has been set and no furnitures are present.
Figure 4.13: ABEC software using the BEM method to observe the modal behaviour
of low frequencies in rooms (Euphonia).
Hence, the reverberation time estimated by CATT-Acoustic is of approximatively
5 seconds, neglecting data below 250 Hz. This is a high value, comparable to that of
a church or cathedral. But this is quite normal, as there is no furniture or audience
in the room allowing additional scattering and absorption, with only hard reflecting
surfaces, and a big volume. These are all the required parameters to set an extremely
long reverberant field.
In the following subsection, the same room will be modelized with different materials.
This shall bring out a slighty different reverberation time.

82. Chapter 4. Modeling and Simulations 64
4.2.2.2 Acoustic improvements - middling RT based on adapted model
Just as the first simulation, we must repeat the procedure back from the design step
on SketchUp. The room remains the same. The dimensions do not change, only the
materials incorporated into the surfaces. This time the room specifications will be as
follow (see appendix C) :
• Front, rear, and lateral walls : 4mm crossband plywood, d=4mm holes, cc 10mm,
75mm distance.
• Floor : plastered concrete.
• Platform : wooden podium.
• Ceiling : strongly vibrating ceiling (wooden panel over air space).
• Room volume : 10253 m3 and Total approximative inner area : 3820 m2
Figure 4.14: Second modelling of the lecture hall with different materials incorpo-
rated.
Then, once the SketchUp model is adjusted, the export of files to CATT can be
started. Repeating the same procedure as previously, CATT will present the results for
its RT estimation.
The new reverberation time estimated by CATT-Acoustic (Figure 4.15) with more
absorbent materials is of between 1.5 and 2 seconds, neglecting data below 125 Hz
(Schr¨oder frequency = 27 Hz). This is a much lower value, with at least 4 seconds of

83. Chapter 4. Modeling and Simulations 65
Figure 4.15: RT estimations for the second model - meddly reverberant.
difference, which is a huge gap. Also, this value is a bit over-evaluated for the same
reasons as for the first estimation : no furniture or audience in the room allowing ad-
ditional scattering and absorption, and a big volume. Some surfaces are however softer
or with air gaps behind, which makes the difference between the two estimations. As
these parameters are neglected in the same way for the two estimations, this lighten the
difference that materials can make in the acoustics of a room.
Once furnitures will be installed and people seated in the room, adding some more
absorption and taking into account scattering phenomenon, the reverberation time shall
lower to a much correct value around 1 or 1.2 secs (RTOptSpeech ≈ 0.8 ∼ 1).
Frequency (Hz)
RT (s) 125 250 500 1k 2k 4k 8k 16k
T-30 2.02 1.67 1.33 1.50 1.69 1.56 0.98 0.43
TS 1.98 1.94 1.66 1.77 1.93 1.70 1.02 0.41
TE 1.76 1.72 1.44 1.56 1.75 1.59 0.99 0.41
Table 4.2: RT estimations results for the second model - meddly reverberant - in the
frequency range of 125 Hz to 16 kHz.

84. Chapter 4. Modeling and Simulations 66
However, the auralizations made in the virtual room (see TUCT section) still demon-
strate that the distribution of sound within the room is not homogenuous and thus
create a low reverberant field which has strong consequencies over speech intelligibility.
4.2.2.3 Lowered reverberation time and material distribution
This time, it will be interresting to see how the distribution of materials can affect
reverberation time and the sound propagation within the room. Indeed, it will be
shown in the next section (audience mapping) that good RT values don’t always mean
having a good sound distribution throughout space. Back to RT estimations, it will be
considered that the same room does not change, only the materials incorporated into the
surfaces and mostly their location. This time the room specifications will be as follow
(see appendix C) :
Figure 4.16: Side view for the third simulation, with various materials distributed
in a certain way in order to improve not only the RT but the distribution of the D-50
criteria (definition) for intelligibility.
• Walls per color :
– blue : 4 mm perforated crossband plywood, d=4 mm holes, cc 10 mm, 75
mm distance, 25 mm mineral wool.
– green : panel w. slot = 8 mm, on 25 mm studs with mineral wool.
– turquoise : thick wood-chip board in from of air cushion.
– occer red : perforated metal tiles with mineral pad backing over 410 mm air
space.
• Floor : plastered concrete.
• Platform : wooden podium.

85. Chapter 4. Modeling and Simulations 67
• Ceiling per color :
– purple : 25 mm thick, 200 mm from ceiling.
– brown beige : strongly vibrating ceiling (wooden panel over air space).
– green : panel w. slot = 8 mm, on 25 mm studs with mineral wool.
• Room volume : 10253 m3 and Total approximative inner area : 3820 m2.
Figure 4.17: RT estimations for the third model - low reverberantion.
Once again, CATT results from its estimation calculus are plotted in the previous
Figure 4.17 and Table 4.3.
It can be seen that RT values are now much more suitable for speech, with a flat
curve of approximatively 1 sec between 500 Hz and 4 kHz. Also the Sch¨oder frequency
remains at a low value (25 Hz) hence allowing relatively good predictions between the
range of 125 Hz to 8 or 16 kHz.
Frequency (Hz)
RT (s) 125 250 500 1k 2k 4k 8k 16k
T-30 1.93 1.21 0.86 0.89 1.04 1.01 0.73 0.35
TS 1.96 1.43 1.06 1.13 1.13 1.06 0.75 0.36
TE 1.73 1.21 0.84 0.90 0.93 0.90 0.68 0.35
Table 4.3: RT estimations results for the third model - low reverberation - in the
frequency range of 125 Hz to 16 kHz.

86. Chapter 4. Modeling and Simulations 68
The big difference between this simulation and the previous one (with all the surfaces
covered with the same material) is that the material distribution on the various partitions
of the room is heterogeneous - unevenly distributed, allowing 2 or 3 different materials
to be present on one surface, e.g. the ceiling (see figure 4.16). In this way, sound will be
more absorbed in certain critic spots, and reflected where it needs to be. This will allow
not only a good direct sound from the source but distribute the first reflections where
they are more needed and to prevent these ones to interfere with the poeple enjoying
mostly direct sound.
Therefore, the front wall (behind the lecturer) has been made of thick wood boards
that will mainly reflect the sound to the audience. On the other side of the room (end
of audience), many sound absorbers had been installated in order to respect two rules :
1) avoid late echoes to get back to the audience area (perf. metal tiles), and 2) absord
some of the incident sound but allowing a few strong first reflections to be heard by the
end part of the audience (wood panels on 25 mm studs with mineral wool inside).
On the same logic, the first part of the ceiling (near the lecturer), which is high, will
be made of blocks of mineral wool with a 200 mm gap to absorb some low frequencies.
This will try to avoid some late reflections to be heard from the people being close to
the source and thus benefiting a strong direct sound. The other half of the ceiling will
be again divided in two. The middle part will have the same kind of air gap to absorb
low frequencies but without mineral wool. This technique should give some vertical
reflections to the end part of the audience. The last ceiling area will be made of the
same absorbing material as the rear walls in order to shut any kind of echoes.
Lately, lateral walls will participate to lower the overall reverberation time and will
focus some reflections to the main audience (4mm plywood with 4mm perforation and
25 mm mineral wool).
Figure 4.18: Hypothetical illustration of sound distribution, seen as sound rays,
within the room.

87. Chapter 4. Modeling and Simulations 69
Thus, as the CATT data has demonstrated (and will continue to do so in the next
subsection) we can see that the sum of these different materials has leaded to a good
estimate of reverberation time - way much better than the two previous ones.
Hence, the material characteristics are thus a really important factor to take into
account, as they can greatly change reverberation time values. Replacing the hard con-
crete walls by perforated wood boards is a good choice for adding some extra absorption.
Also, placing some more absorbent materials over the ceilling is a great idea, as it is
a really large area with no restrictions for its use, eg. the floor must be designed for
people to step on it, reducing the choices of available materials.
Nevertheless, the distribution of these materials is also a important parameter to deal
with as it can change not only the overall RT values, but also the sound propagation in
the room. This last parameter is of great value as the source, here it will be the speech
of a lecturer, must be well understood through all the audience area.
This is why the next and last subsection will deal with an RT derived criterion ; the
Definition, called D-50 ; and hence STI values for speech transmission.
4.2.3 TUCT : SxR & audience mapping simulations - D-50 as a RT
related criterion
In this subsection, the aim will be to run acoustic simulations for Source and Receiver
echograms (SxR) and to map the results in the audience area. For achieving this objec-
tive, the CATT related program named TUCT will be used. As said in the CATT manual
‘“CATT-Acoustic” means the software as a whole, including TUCT, while “CATT-A”
means the main program CATT9.exe. Although with its own name TUCT is an integral
part of CATT-Acoustic.’
TUCT stands for The Universal Cone-Tracer and offers three different cone-tracing
algorithms (short, long, and very long, depending on the level on complexity that is de-
sired), but also many room acoustics measurements and analysis functions for echograms,
impulse responses, colored maps, as well as direct relative calibrated impulse response
playing, and convolution for auralization.
Now, it would be interesting to see how sound will behave in the different room
configurations. To do so, the mapping option will be of great interest. As we will
obtain the audience mapping of the area according to several acoustic criteria such as

88. Chapter 4. Modeling and Simulations 70
RT, STI, G (strenght), C-50, D-50, the SxR simulation will just be used to obtain a
basic auralization of the said configurations (which will be available on the CD-ROM
attached to this project). This will allow to hear the difference between each of them.
(See Appendice D)
Therefore, we will just focus on the audience mapping. For the mapping to work cor-
rectly, it requires to set the various acoustic related parameters in order to run correctly
through the simulation. They are : the height and steps of audience, the estimated
number of rays or cones (100.000 or 200.000 constitute good values), and the echogram
length to set in fonction of the estimated reverberation time of the room. Once these
parameters are correctly set, the simulation can run - which can take from a few seconds
to a decade of minutes or hours in function on the complexity.
Figure 4.19: TUCT window once a simulation is finished. On the left : a window
with the beam tracing parameters to enter ; on the right : a 3-D viewer with several
display options.
Highly reverberant configuration
The mapping of the first configuration has given the following results shown in Fig-
ures 4.20 and 4.21. (Appendice C for alpha coefficients and materials)

89. Chapter 4. Modeling and Simulations 71
Figure 4.20: RT’ (up) and D-50 (down) mapping of the audience area for the first
room configuration.

90. Chapter 4. Modeling and Simulations 72
Figure 4.21: STI mapping of the audience area for the first room configuration with
the image source model of the sound rays hiting the room boundaries in function of
time.
First, lets have a quick look over this new criterion that is D-50, also called the
Definition. The Definition works in a similar way as for Clarity. Where Clarity (C-50) is
an index that defines the clarity of sound - especially suited for music, but adaptable for
speech - giving the ratio in dB of the energy arriving in the first 50 ms over the energy

91. Chapter 4. Modeling and Simulations 73
arriving after 50 ms (for a short pulse), the Definition, for instance, is better suited for
speech and gives the ratio in % of the energy arriving in the first 50 ms (t0 → 50ms)
over the total energy (t0 → ∞). Indeed, the equations for both are relatively similar.
D50 =




t=50ms
t=0
p2
(t) dt
t=∞
t=0
p2
(t) dt



 × 100(%) (4.1)
C50 = 10log10




t=50ms
t=0
p2
(t) dt
t=∞
t=0
p2
(t) dt −
t=50ms
t=0
p2
(t) dt



 (dB) (4.2)
Thus, D50 is a measure of the per-cent total sound energy arriving within 50 ms
after an initial pulse of sound. If most of the energy of the sound impulse is within
this 50 ms window, then it will be much easier for people in this room to understand
speech rather than the opposite. On theory, a “good” listening room from a speech-
intelligibility perspective has a D50 > 50%, even if on practice all values below 60%
are considered as “mediocre”. Nevertheless, D50 ≥ 50% is the minimum recommended
value.
Another criteria shown in figure 4.21 which is explained more in depth in Chapter
1.2.1.2 is the Speech Transmission Index (STI), assessing the quality of voice transmission
in a room. It is the relationship between the emergence of items (sentences, words or
syllables) correcty identified as reliable and the noise of the room studied in seven octave
bands. A 0.6 value is considered as a start of good intelligibility. Here, it will be combined
with RT’ and D-50 criteria to judge to correct sound distribution within the modelized
room - i.e. its intelligibility all over the audience area.
Concerning the mapping figures 4.20 and 4.21, several criteria are shown for the same
room modelization, i.e. RT’, D-50 and STI values. It can be seen that RT’ isn’t really
homogenuous throughout the space, and even more, that its values are very high - which
comfirms the estimated data of RT ≈ 5secs previously calculated in the CATT main
program.
We can see that the minimum RT’ overall value is about 4 secs near the source and
that it reaches an average of 6.8 secs in all the remaining area - including some red spots

92. Chapter 4. Modeling and Simulations 74
with an even higher value.
Then, the map below RT’ values shows the D-50 criterion in the same configuration.
An obvious reading indicates that the Definition in the room - some meters away from
the source - don’t overpass the 40%, with values under 30% or 20% all over the room.
This means that speech would be awfully difficult to understand - as it can be heard in
the audio files for the higly reverberant configuration (see CD-Rom).
The same conclusion can be adressed by the STI mapping in Figure 4.21 where it
can be clearly seen that the overall STI values (which are measured between 0 and 1)
stand at a maximum of 0.5 near the source and below 0.4 and less with more distance.
STI values comfirm once again the poor intelligibility of the room.
Finally, the image source model of sound rays in Figure 4.21 shows some of the sound
reflections in function of time. It can be seen that many of them arrive after 50 ms until
150 ms, which will make many late sound reflections to disturb with the direct sound at
the listener position.
Regarding the RT design, this room makes a clear example of how high RT values
(i.e. lots of long reflections) can impair hearing and intelligibility. Moreover, the ability
of seing the reverberation time distribution through space allows not only to see the
latter as a unique value - a number - but a fluctuating component with a decreasing
energy that can be non-linear. The next configuration will make an even more difference
between a relatively good reverberation time and intelligibily criteria.
Reverberant configuration
The mapping of the second configuration has given the following results shown in
Figures 4.22 and 4.23 on the next page. (Appendice C for alpha coefficients and mate-
rials)

93. Chapter 4. Modeling and Simulations 75
Figure 4.22: RT’ (up) and D-50 (down) mapping of the audience area for the second
room configuration - few acoustic improvements (see Chap 3.4.2.2.2).

94. Chapter 4. Modeling and Simulations 76
Figure 4.23: STI mapping of the audience area for the second room configuration
with the image source model of the sound rays hiting the room boundaries in function
of time.
Now, the same criteria results are shown for the second configuration room - with
some absorbent materials homogenuously applied to the room surfaces, i.e. one material
per surface.

95. Chapter 4. Modeling and Simulations 77
We know that the previous RT estimation made by CATT had given approximately
1.5 to 2 secs of reverberation at a central spot in the audience. Figure 4.22 shows slightly
lower values, but remaining around 1.4 to 1.6 secs for the main area. These are more
appreciable RT values, however, we can observe that the D-50 and STI values aren’t
good enough to qualify the room as ‘intelligible’.
In fact, if we map only the lower limit values allowed for a good intelligibility, we
would see that only 1/6 or 1/4 of the audience would hear in a intelligibile way the
sound source. Because it can be seen that D-50 values are mostly equal or under 50%
and STI values are just at 0.6 for some listeners but under 0.5 for all the others. We can
see as well that the time distribution of strong sound reflections is still over 125 ms.
Changing the surface materials of the room was good enough to lower its reverbera-
tion time, but good hearing - intelligibility - is still an issue in the room. Auralizations
for the ‘reverberant room’ on the attached CD-ROM can emphasis over this conclusion.
Reverberation time has been lowered by more than a half, but it is still a reverberant
place to hear the speech of a lecturer.
Low reverberant configuration
This last configuration will be made with the unevenly distributed acoustic materials
for which CATT gave an RT value of 0.8 to 1 sec of reverberation. The objective will be
to see how the material distribution can not only affect RT values but also that sound
is properly distributed through the audience in order to improve intelligibility.
On the next pages will be shown the mapping results for this kind of configuration -
third.

96. Chapter 4. Modeling and Simulations 78
Figure 4.24: RT’ (up) and D-50 (down) mapping of the audience area for the third
room configuration - lowered reverberation time (see Chap 3.4.2.2.2).

97. Chapter 4. Modeling and Simulations 79
Figure 4.25: STI mapping of the audience area for the third room configuration with
the image source model of the sound rays hiting the room boundaries in function of
time.
As we can see, not only RT results are strongly adequate but that D-50 and STI
values are very good as well. Therefore the room has an average RT value of 0.5 to 0.8
secs, which is a bit too much maybe as there is no audience or furnitures already installed
in the room, but it is a good value for this example. On the other side, D-50 values are
at least at 50% in the weakest spots and over 70% for the remaining audience, which are

98. Chapter 4. Modeling and Simulations 80
good results. The same happens for STI values with minimum values of 0.5/0.6 at the
weakest spots (4 spots according to TUCT) and a good average of 0.7/0.8 for the rest.
Certainly, there are still a few weak spots in the room, but their values match the
minimum required for the speech to be intelligible. We can thus say that RT plays a
major role on determining the shape and materials of the room, but those are to be
carefully choosen as not only one RT value is of interest, but it is its general shape -
energy decreasing curve - that needs to be wrought in order to set correctly other RT
related parameter such as C-50 and STI. Because indeed, C-50 depends greatly on the
number of reflections arriving before 50 ms over the overall number of them. If the RT
is short, the chances to have a good C-50 value are strong, an even more if the Early
Decay Time is short.
In order to finish this subsection, a comparison can be seen in the next Figure 4.26
which will illustrate the difference between the two last configurations : the second one
with a relatively good reverberation time but low intelligibility, and the third one with
a good RT and a good intelligibility as well.
Figure 4.26: Comparison between the second (left column) and third (right column)
configurations in function of RT and D/50 criteria. First line : RT. Second line : D-50.

99. Chapter 4. Modeling and Simulations 81
Figure 4.27: Comparison between the second (left) and third (right) configurations
in function of STI criterion.
The aim of this chapter was to illustrate the impact of absorption in rooms, and how
materials shall be carefully selected in the design step. Thus, it can be said that the
impact of absorption is a non-negligible factor of room acoustics - if not vital, just as
for volume (see Chapter 1.2.2). If the room volume is large, the longer will be the rever-
beration time. In the same way, less absorption in a room leads to a long reverberation
time. A hall with less absorption and large volume will thus be an acoustic nightmare,
and conversely, an intimate room volume with good (not too much) absorption will be
comfortable and smooth.

100. Conclusion
The aim of this work was to show that reverberation time is an essential criterion for
characterizing indoor spaces, and to see to what extent it is a fundamental factor of
room acoustics.
To fully understand this phenomenon several aspects of reverberation time have been
studied, determining in what ways it may be critical in designing acoustic spaces, as to
what factors may have an influence on it.
It has been seen that :
• Long reverberation times can cause serious damage to the understanding of speech,
that is, intelligibility - how intelligible speech will be in rooms.
• Reverberaton time is a fundamental pillar of the design of musical rooms, suited
for the kind of music played, supporting instrumental sound and providing enough
clarity and definition.
• Various acoustic criteria such as warmth, brilliance, loudness, spaciousness, inti-
macy, clarity, definition have strong connections, and are thus defined in function
of the said reverberation time.
• The volume of a room is a critical parameter, based on the shape and size of the
room, characterizing the acoustic behaviour of the said room through its reverber-
ation time.
• Reverberation time is a statistical model of acoustics, with plenty of theories,
giving a correct average of possible values.
82

101. Chapter 4. Modeling and Simulations 83
• As a derivation from the impulse response of a room, reverberation time can be
considered somehow as the acoustic signature of an indoor space.
• Acoustic absorption is the basis of reverberation time, and this parameter shall
be carefully designed as it’s has consequences not only on RT values, but on other
related criteria as D-50 and SRI.
It has been shown that reverberation time is a prominent element in the field of acous-
tics ; having numerous connections with various design and quality factors of rooms.
Thus optimum values for reverberation times appear to be a strong factor characteriz-
ing suitable acoustics for indoor spaces.
Therefore, it can be said that architecture and acoustics of rooms are intrinsically
connected and that the connection of both allows the establishment of basic ; but im-
portant ; parameters of a listening environment - whether it is for speech or music.
Since many parameters are well related to subjective assessments of room acoustics,
it is most important for the acoustical design process to provide an objective general
design tool, RT, which enables prediction of relevant sensations in the stage of planning
a certain type of room, with good values of uncertainty.
Hence, reverberation time defines the acoustic behaviour of a room, and to some
extent, it is the only determining factor in the acoustic design of a room, in its entirety.
It is not the only characterizing factor of a room, yet it is a vital criterion to give a first
diagnosis of a room (in a building or designing stage) and obtain its most important
acoustic characteristics. It allows to class the room in a defined context and verify that
it matches, acoustically speaking, its predefined purpose.
“Reveberation time can be the boon or the bane of the acoustic performace of a room”
- Marshall Long, Architectural Acoustics

102. Appendix A
Derivation of Sabine’s formula
The following derivation is a written copy from Appendix 2, Unit 5 (Sound Absorbers
and Room Acoustics) of the General Principles of Acoustics module coursework of the
Institute of Acoustics (IOA). All copyrights belong to the IOA.
Assumptions :
• sound density is uniform throughout the room
• sound energy is transmitted equally in all directions
• sound source maintains constant supply of sound energy
• sound energy loss occurs only at room walls (no air absorption)
• sound absorption distributed evenly over room surfaces
Decay of sound in room (after sound source switched off)
Rate at which sound is being absorbed by room surfaces = αIS
Total amount of sound energy in room = εV
Rate of change of sound energy :
−
d
dt
(εV ) = −V
ε
dt
(Note - sign because ε is decreasing with time)
84

103. Appendix A. Derivation of Sabine’s formula 85
But
ε =
4I
c
Therefore rate of decay of energy :
−V
d
dt
4I
c
= −
4V
c
dI
dt
Rate of decay = rate of absorption
−
4V
c
dI
dt
= αSI
Therefore
dI
dt
= −
αSc
4V
I = −
Ac
4V
I
(since αS = A)
This is a standard differential equation of an exponential decay with time of a quantity
(in this case I ). It has a standard solution :
I = I0e−(Ac
4V
t) ←→
I
I0
= e−(Ac
4V
t)
ln
I
I0
= −
Ac
4V
t
Using the relation between ln and log10 : logN = lnN
2.3
10log
I
I0
= −
Ac
4V
10
2.3
t
By definition, when t = T (reverberation time), 10log(I/I0) = 60dB
Therefore :
60 = −
10c
2.3 × 4

104. Appendix A. Derivation of Sabine’s formula 86
Taking c = 340m.s−1 it gives :
T =
0.16V
A
where T was the estimated reverberation time according to Sabine.

105. Appendix B
Derivation of Eyring’s formula
The following derivation is a written copy from Appendix 4, Unit 5 (Sound Absorbers
and Room Acoustics) of the General Principles of Acoustics module coursework of the
Institute of Acoustics (IOA). All copyrights belong to the IOA.
Let I0 be the sound intensity at time t = 0 when the source is disconnected and the
average absorption coefficient of the surfaces of the enclosures be αavge. Then :
• the intensity after one reflection : I1 = (1 − αavge)I0
• the intensity after two reflections : I2 = (1 − αavge)2I0
• the intesnsity after n reflections : In = (1 − αavge)nI0
The reverberation time is the time for the sound intensity to fall to 1/1, 000, 000 of
the original, so n is such that :
In
I0
= (1 − αavge)n
= 10−6
By taking log10 of each side :
log10(1 − αavge)n
= −6
87

106. Appendix B. Derivation of Eyring’s formula 88
n × log10(1 − αavge) = −6
n =
6
log10(1 − αavge)
How long n reflections take on average ? If sound waves are treated like billiard balls
particules, it can be shown that the mean distance travelled between collisions with walls
in a rectangular room, volume V in m3 and total surface area S given by 4V/S.
So time taken for n reflections (reverberation time) is RT = 4V n/(cS) seconds.
After substituting for n, putting a numerical value for c, and replacing log10 by
loge = ln = 2.303 × log10, the Reverberation Time is given by :
RT =
0.16V
−Sln(1 − αavge)
which is the Eyring’s formula.

107. Appendix C
Absorption coefficients and
relative materials
The following absorption coefficients in Figure C.1 (see next page) are mostly listed into
the CATT surface properties windows, giving values between 125 Hz to 16 kHz. The
source of data are the following : Petersen, Karlen, Bies Hansen, Cremer and Muller,
and Lawrence.
89

108. AppendixC.Absorptioncoefficientsandrelativematerials90
Figure C.1: Comparison between the second (left) and third (right) configurations in function of STI criterion.

109. Appendix D
CATT & TUCT specifications
Here are some specifications regarding the CATT-acoustic parameters that have been
used.
CATT-A programm
Floor :
The floor of the modelized room is in reality more like stairs but had been modelized
as a flat area. In order to reduce the accuracy errors in CATT, the absorption coefficient
of the floor have been multiplied by two due to the stair shape that increases nearly by
two the actual floor area. If the surface is actually 2 times bigger than the one built in
the model, on solution into multiply by 2 the actual absorption coefficients.
Another point concerning the floor is its scattering coefficient. The gap between each
stair is of approximately 1 or 1.2 meters. At this scale, some low frequencies - around
280 Hz, as λ ≈ 1.2m - could be reflected in random patterns. This is why an estimated
scattering coefficient L has been given to the floor, to try to take into account as best
as it can be the scattering phenomenon of the ground.
TUCT programm
Auralization :
Algorithm : Short algorithm split order 2 (number of reflections)
Calculation parameters : 100,000 cones/rays ; air absorption
91

110. Appendix D. CATT & TUCT specifications 92
Files in the CD-ROM are classified following this order :
• Type of Sound Source (Man, Woman, Guitar)
– Selection of the audio track in function of Higly Reverberant, Reverberant,
or Low Reverberation files.
Highly Reverberant : 1st Configuration. Reverberant : 2nd Configuration. Low
Reverberation : 3rd Configuration.
Mapping :
Map data : step = 0.5 m, height = 0.7 m. Calculation parameters : 100,000
cones/rays ; air absorption

111. Appendix E
EN ISO 3382 -1 and -2
Find attached to this project the said standard.
93

112. Bibliography
[1] Michael Barron. Auditorium Acoustics and Architectural Design. Taylor & Francis,
second edition, 2010.
[2] Leo L. Beranek. Concert and Opera Hall. How they sound. Acoustical Society of
America, 1996.
[3] Heinrich Kuttruff. Room Acoustics. Spon Press, fifth edition, 2009.
[4] Anita Lawrence. Architectural Acoustics. Applied Science Publishers Limited of
London, Elsevier Inc., first edition, 1970.
[5] Marshall Long. What is so special about shoebox halls ? Acoustics Today, 2009.
[6] Marshall Long. Architectural Acoustics. Academic Press, second edition, 2014.
[7] Y. Furue M. Morimoto, K. Iida. Relation between Auditory Source Width in Various
Sound Fields and Degree of Interaural Cross-Correlation. Applied Acoustic, 38:291
– 301, 1993.
[8] B. Kostek R. Neubauer. Prediction of the Reverberation Time in Rectangular Rooms
with Non-Uniformly Distributed Sound Absorption.
[9] Wallace C. Sabine. Collected Papers on Acoustics. Oxford University Press, first
edition, 1922.
94

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