BAP260.1 Practical Project

Abstract:

The aim of the present report is to relate the experiment conducted as part of the Bachelor of

Audio Production BAP260 Creative Project II.

The nature of the assignment being a scientific process leading to the achievement of a data

comparison rather than a final and material product, the following report will provide an in-

depth development of the stages, concepts and procedures involved in the establishment of

the project.

The report has been conducted as a scientific paper; as such it concisely relates the process

undertaken to several notions and knowledge associated to the field of study. The subsequent

report that will also be submitted as part of the Creative Project II will therefore be devoted to

the critical analysis and clarification of the science, concepts, theories and models addressed

throughout the present document as well as supporting the choices made through the project.

Table of Content

Clement Bresson - SAE Institute - BAP260.1 Practical Project 1

Introduction

The creative project II is devoted to the science of Architectural Acoustics, especially the

behaviour of the decay of sound in an enclosure.

In the area of study, the decay of sound is commonly referred to the reverberation time or

RT60, and is defined as the time required for a sound to decay by 60dB after interruption of

the emitted signal.

The reverberation time has been widely acknowledged for being a critical factor in the

determination of the overall acoustic quality of an environment, as a result, numerous

researches were conducted in order to investigate and predict its parameters (Kang &

Neubauer 2001).

The science of acoustics was first established by means of various experiments and theories

which then extended to the practical measurements achievable as a result of the development

of the technology.

Nowadays, the reverberation time can be predicted by applying the so-called classic theories

and their numerous revisions or measured in practical situation with the use of measuring

equipment.

However, each method seems to present its own limit of accuracy predominantly set by

environmental factors (Everest, 2001).

In that way, the assumptions on which the theories of reverberation time are based are not

always fulfilled when applied to common environments while the practical measurements

require a specific process in order to achieve results representative of the true behaviour of the

sound field. As a consequence, it appears that a degree of discrepancy persists between the

theoretical and practical approach of measuring the reverberation time.

The aim of the project is therefore to evaluate the degree of divergence observed between the

theory and practice of the reverberation time measurement methods when applied to enclosed

spaces presenting distinct acoustic properties.

In order to do so, three classic theories of reverberation time have been selected for their

relevance in the field and the different assumptions on which they are derived.

The Sabine, Eyring-Norris and Fitzroy theories will thus be applied and evaluated in

conjunction to practical measurements conducted in the same enclosure. The test room will

be subject to different configurations of sound absorption distribution in order to compare the

theory and practice in distinct acoustic conditions commonly observed in diverse listening

environments.


As a result, three room configurations will be assessed. The three selected theories will first be

applied independently for each room configuration for later comparison to the practical

measurements. The reverberation time of the three room configurations will then be measured

by the impulse response method in order to establish a reverberation of reference for each

configuration.

The report is structured such as to firstly present the characteristics of the test room so as to

better define the overall acoustics proprieties of the enclosure with regards to acoustic

parameters susceptible to impact on the prediction and measurement of the reverberation

time.

A second part will be devoted to the theory of reverberation time where the three classic

theories will first be exposed followed by the results obtained for each room configuration

using the different equations. A first comparison will consequently take place between each

theory in order to develop the patterns observed in the three room configurations.

A third part will then proceed to investigate the process of reverberation time measurement by

relating the requirement and procedure necessary to the achievement of the results, as well as

presenting the experiment and the process undertaken to the establishment of a reverberation

time of reference for each room configuration.

The fourth and final part will support the analytical comparison between the results obtained

by the theoretical and practical methods of determining the reverberation time in enclosed

spaces.


Part I/ Data Collection

1/ Room description

The test room is situated at SAE Institute, Sydney, and currently serves as a live/tracking

recording room, linked on both sides to two different control rooms.

The room being situated on the first floor of an educational building located in the city centre,

it appears that the concerns of sound isolation predominated over the acoustical properties

within the room. In fact, partially due to the small size and to the materials construction, the

room is quite “dead” with regards to acoustic parameters such as reverberation, that is, the

average absorption of the room is relatively high and the sound thus tends to decay rather

rapidly compared to spaces qualified as “live” where the sound undergoes numerous

reflections against the room‟s boundaries before dying away. Consequently, the space doesn‟t

match the characteristics of a live room and could rather reflect an average listening

environment such as bedrooms and living rooms of small size, which are often characterised

by a non uniform distribution of sound absorption.

The fact that the room currently set up as a recording space presents several particularities

found in a good number of listening rooms enables to extend the scope of the project to a

larger range of musical related environments rather than only dealing with small recording

spaces.

a) Geometry


In spite of being of rectangular shape, the room presents a few unusual geometric aspects.

First, the enclosure is not a perfect parallelogram as the length, width and height vary

between different spots in the room. As such, the length fluctuates from 5.30 meters on the

left end side to 5.34 meters on the right side. The width also extends from 3.87 meters at the

entrance, to 3.95 at the far end. The room can be separated into two equal zones, with a

height varying from 2.40 meters in the zone 1 to 2.42 meters in the zone two.

These minor changes in dimension cannot be readily incorporated into the calculations; the

length and width were thus averaged while the height of the room remained set to 2.40 meters

which is the standard dimension in numerous Australian constructions.

The pair of opposite walls east and west also presents some divergences, which should be

taken into account. As such, it is necessary to divide each of these walls into three distinct

sections as follow:

These subdivisions of surfaces allow a better representation of the geometry of the room with

regards to the materials‟ areas and the absorption distribution.


b) Materials

The room construction is rather common to such small scale recording space.

The room is composed of an acoustic door made of aluminium and Perspex and two windows

of same dimensions. The door and windows are all mounted with a timber frame.

The ceiling includes two ventilating grills and an air conditioner.

Regarding the other surfaces, the materials remain common to such locations and will be

assessed later on.

The main characteristic of the room is the presence of twenty-one removable acoustic panels

dispersed over the four walls, with the addition of three fixed panels on the ceiling. The panels

are clipped on two timber mountings fixed to the walls which should be taken into account in

the theory computation. The majority of the panels are rectangular, with the exception of two

square panels of 60 centimetres fitted on the south wall and three hexagon panels of 60

centimetres fixed to distinct parts of the ceiling.

The presence of these removable panels enables to alter some acoustic parameters of the

room, especially the reverberation time, and thus provides a good way to investigate the RT60

prediction methods for distinct acoustical conditions.

The following tables list the absorption coefficients as well as describing the configuration of

the materials present in the room:

[ Extract from SAE Institute Perth Handout ]

Walls

All four walls are made of two 16mm plasterboards on 90mm timber studs with mineral wool in the cavity.

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.15 0.10 0.06 0.04 0.04 0.05

Floor

Timber flooring on timber joists with 6mm carpet tile on rubber.

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.01 0.02 0.06 0.15 0.25 0.45

Ceiling

Two 16mm plasterboards in suspended steel grid.

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.15 0.11 0.04 0.04 0.07 0.08


Windows

6mm glass

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.10 0.06 0.04 0.03 0.02 0.02

Door

Acoustic door, aluminium frame, double seals, absorbent in airspace.

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.35 0.39 0.44 0.49 0.54 0.57

Timber frames

Both windows and the door are mounted with a 6.7 cm painted timber frame.

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.15 0.11 0.10 0.07 0.06 0.07

Acoustic panels

All panels are perforated medium density fibreglass filled with glass wool blanket and mounted over a 2cm air space.

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.40 0.80 0.90 0.85 0.75 0.80

Ventilating grills

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz 0.30 0.40 0.50 0.50 0.50 0.40

Note on the absorption coefficients:

It is important to note the limit of the absorption coefficients accuracy and availability.

In fact, most available absorption coefficients are issued from acoustics literature and few

manufacturer specifications. However, it is often difficult to obtain the exact coefficients for

already built materials, and it inevitably involves a compromise between accuracy and

availability.

In that way, the thickness, the size and the configuration of the materials varies dramatically

from one construction to another and it often involves substituting the actual design for the

more available one.

This will inevitably impact on the accuracy of the theories and therefore care must be taken in

order to select the coefficients the most representative of the actual arrangement.


2/ Modal resonances

A valuable stage in the process of predicting a given acoustic parameter of a space is to

evaluate the overall characteristic of the enclosure with regards to other acoustic parameters

which could have an influence on the parameter under study.

As it has been thoroughly developed in the literature, modal resonances can have a significant

impact on the diffuseness of a sound field in an enclosure (Knudsen 1932; Mayo 1952;

Louden 1971).

In large numbers, resonant modes reduce the sound field fluctuations because the individual

modal properties combine into a diffuse field with more or less constant properties (Nelson,

1992).

In small numbers however, modal resonances can significantly alter the reverberation time

magnitude on the enhanced frequencies, as well as creating a multiple slope curve due to

different decay rates of the standing waves (Prokofieva, 2009). The decays at the modal

frequencies are decay rates characteristic of individual modes, not the average condition of the

room (Everest, 2001).

In order to get accurate measurements, the room must be large enough so a great number of

modes will overlap within the frequency band in which the decay observation is made (Schultz

& Beranek 1962). Eighteen modes per 1/3-octave band is considered sufficient to define a

diffuse field in a reverberant environment with consistently distributed absorption (Nelson,

1992).

This high modal density, resulting in uniformity of distribution of sound energy and

randomising of directions of propagation, is necessary for reverberation equations to apply.

(Schultz & Beranek 1962)

One of the first steps in order to evaluate the modal resonances of a room is to plot the room

dimensions ratios on the Bolt chart which gives a range of room proportions producing the

smoothest room characteristics at low frequencies in small rectangular rooms (Everest, 2001).


The Bolt‟s ratio is defined as : 𝐻

𝑆𝑚: 𝑊

𝑆𝑚: 𝐿

𝑆𝑚

Where L, W, H : Length, Width, Height of the room (m)

Sm : Smallest dimension of the room (m)

For the room under study, the ratios become:

2.4

2.4∶

3.92

2.4∶

5.32

2.4

1 : 1.63 : 2.22

Extract from Master Handbook of Acoustics,

4th edn, McGraw-Hill.

As revealed on the chart, the room falls into the Bolt area which suggests a relatively smooth

influence of modal resonances, as far as distribution of axial modal frequencies is concerned

(Everest, 2001). In practice however, several other considerations have to be taken into

account regarding the activity of modal resonances.

As such, it is necessary to determine the frequency region dominated by modal resonances,

that is, the region in between the lowest resonant frequency and the critical frequency above

which diffraction and diffusion dominate (Neubauer, 2000).

The lowest resonant frequency of a room is defined as:

𝑓1 = 𝐶

2×𝐿 (1)

Where C : speed of sound (m/s) (Everest, 2001)

L : length of the room (m)

For the room under study, the lowest resonant frequency is:

𝑓1 = 344

2×5.32= 32 𝐻𝑧


The critical (also called Schroeder, crossover or cutoff) frequency is defined as:

𝑓𝑐 = 1.5 × 𝐶

𝑀𝑃𝐹 (2)

Where C : the speed of sound (m/s) (SAE Institute Handout)

MPF : the Mean Free Path (m).

The mean free path defined as the average distance sound travels between reflections is given

by:

𝑀𝐹𝑃 = 4 𝑉

𝑆 (3)

Where V : Volume of the room (𝑚3) (Everest, 2001)

S : Total surface area of the room (𝑚2)

For the room under study, the mean free path becomes:

𝑀𝐹𝑃 = 4 × 50

86= 2.33 𝑚

Consequently, the critical frequency of the room is:

𝑓𝑐 = 1.5 × 344

2.33= 222 𝐻𝑧

As a result, the modal resonances are expected to dominate the frequency region between

32 Hz and 222 Hz.

It is important to evaluate this acoustic characteristic of the room in order to later relate the

reverberation times, especially the decay slope, to the influence of modal resonances.

With that regard, it is then necessary to determine the frequency of the modal resonances

taking place within this region of the audio spectrum.


This is done by the use of the modal calculation given below:

𝑓 = 𝐶

2 𝑝2

𝐿2+

𝑞2

𝑊2+

𝑟2

𝐻2 (4)

Where C : Speed of sound (m/s) (Everest, 2001)

L, W, H : Length, Width, and Height of the room

p, q, r : Integers serving to identify the room modes as Axial, Tangential and

Oblique while also indicating the frequency of a given mode (1 for the

fundamental, 2 for the second harmonics,3,4,...).

This calculation allows to identify each resonant frequency as well as the dimensions in which

these standing waves occur. Being aware of these data will later enable to relate potential

decay slope anomalies to these resonant frequencies.

The modal resonances of the room were thus calculated up to the critical frequency and then

plotted of the following graph with regards to their respective modal strength.

See Appendix for calculations


As it is expressed on the modal resonances plot, the main modal deficiencies susceptible to

affect the measured reverberation time would be the axial mode isolation between 97 Hz and

130 Hz which corresponds to the 10th and 12th mode respectively with only one tangential

mode in between. Another isolation takes place between 195 Hz and 215 Hz corresponding to

the 21st and 22nd mode respectively.

The degeneracy of axial modes occurring around 130 Hz could also have a significant effect on

the decay shape.

Further degeneracy occur in the region between 160Hz and 170Hz and around 215-220Hz but

they can be considered as having a minor effect because involving tangential modes which

only have half the modal strength of axial modes.

It is necessary to be aware of such anomalies caused by modal resonances before undertaking

the comparison between reverberation time theoretical and practical approach as the modal

activities could have a significant effect on the interpretation of potential divergence occurring

at low frequencies.


Part II/ Theory

1/ Sabine Reverberation Time

The reverberation time has for a long time been a purely theoretical approach firstly

introduced and developed by Wallace Clement Sabine around 1900. Based on practical

results, Sabine (1923) derived the empirical formulae that relates the reverberation time

directly to the volume and inversely to the absorption of an enclosure.

The Sabine reverberation time is given by the expression:

𝑅𝑇60 =0.161 𝑉

𝐴 (5)

Where RT60 : Reverberation time (s) (Everest, 2001)

V : Volume of the room (𝑚3)

A : Total absorption of the room (metric sabins).

The total absorption of a room is given by:

𝐴 = 𝑆.ᾱ (6)

Where S : Total surface area of the room (𝑚2) (Long, 2006)

ᾱ : Average absorption coefficient of the room

The average absorption coefficient is expressed as:

ᾱ = 𝑆1 .𝛼1+𝑆2 .𝛼2+⋯+𝑆𝑛 .𝛼𝑛

𝑆 (7)

Where 𝑆1, 𝑆2, 𝑆𝑛 : Surface area of the materials (𝑚2) (Long, 2006)

𝛼1, 𝛼2, 𝛼𝑛 : Absorption coefficient of the respective materials


Sabine derived his equation on the assumption of enclosures having a perfect diffuse sound

field which implies, among other factors, highly uniform distribution of sound energy and

random direction of propagation of the sound (Eyring, 1930).

Joyce (1978) pointed out that diffuse sound field is a function of the enclosure shape as well

as the absorption characteristics of the materials on its surfaces


2/ Eyring-Norris Reverberation Time

In 1930, Carl F. Eyring published a theory of reverberation time based on an idea that was

attributed to R. F. Norris (cited in Long 2006). In his paper, Eyring (1930) pointed out that the

Sabine formula couldn‟t be fulfilled where there is considerable room absorption. In that way,

he described the Sabine reverberation time equation as being essentially applicable to “live”

rooms.

He presented a revised theory of the reverberation time based on the average distance sound

travels between successive reflections and published the so-called Eyring-Norris reverberation

time formula given by the expression:

𝑅𝑇60 =0.161 𝑉

𝑆 ln (1−ᾱ) (8)

Where V : Volume of the room (𝑚3) (Long, 2006)

S : Total surface are of the room (𝑚2)

ᾱ : Average absorption coefficient

Based on Sabine‟s model, the Eyring approach assumes complete diffuseness. As Powell

(1970) pointed out, „Complete diffuseness here is taken in its broadest sense by Eyring,

meaning that after a reflected wave leaves a surface, it has a probability of striking any given

surface equal to the ratio of total surface area to the given surface area, including the one it

just left‟. In that way, The Eyring-Norris equation assumes that all the surfaces are

simultaneously impacted by the initial sound wave, and that successive simultaneous

impacts, each diminished by the average room absorption coefficient, are separated by the

mean free path (Beranek, 2006).

Notes on Sabine and Eyring-Norris assumptions:

As expressed, Sabine empirical theory is based on the assumption of a diffuse sound field. The

Eyring-Norris theory is a revision of that model which still requires a sufficiently diffuse field

to provide accurate results (Long, 2006).

In many practical cases, the assumption of diffuse field conditions is not in agreement with

the existing absorption distribution. As identified by Neubauer (2000), „The sound field will be,

in general, sufficiently diffuse if there are not large differences in the basic dimensions of the

room, the walls are not parallel, the sound absorbing material is uniformly distributed, and

most internal surfaces are divided into parts‟. In practice, almost all of these requirements are

not fulfilled. As a consequence, numerous researches were conducted in order to revise the

classic theories and to propose a model that can more accurately be applied to common

enclosures characterised by a non uniform distribution of sound absorption.


3/ Fitzroy Reverberation Time

In 1959, Daniel Fitzroy empirically derived an equation through extensive tests in a large

number of rooms, where distribution of sound absorption varies widely in uniformity

(Neubauer & Kostek 2000).

In his paper, Fitzroy (1959) stated that it is possible to take into consideration not only the

physical, but also geometrical aspects of a sound field in an enclosure. In this way the sound

field tends to settle into a pattern of simultaneous oscillation along a rectangular room with

three major axes - vertical, transverse, and longitudinal. The solution appeared to lie in the

relationship between the three possible decay rates occurring along these axes, each being

influenced by the different average absorption normal to these axes, each rate being unique

within its specific axis.

In a rectangular room, one may consider three sets of parallel boundaries. If energy oscillates

simultaneously between each pair of boundaries the average absorption in each pair would

control sound waves travelling between that specific pair during the sound decay period

(Neubauer & Kostek 2000).

Since the sound field encounters the total area of the room‟s boundaries, the three decay

times obtained for each sets of boundaries represent a specific percentage of the total decay of

the room.

Fitzroy used the Eyring-Norris formula to determine the reverberation time peculiar to each

pair of boundaries and controlled by their specific average absorption. Therefore, three

calculations are made by means of the Eyring-Norris formula, each of them being calculated

with different average absorption coefficients. Ratios relating each pair of boundary areas to

the total surface area of the room are then established.

Finally, the reverberation times obtained for each pair of boundary are added together and the

resulting equation yield the reverberation time of the room.

As a result, Fitzroy was the first to derive an empirical reverberation time equation which

considers non-uniform distribution of sound absorption (Neubauer, 2000).

The Fitzroy reverberation time is given by the expression:

𝑅𝑇60 = 0.161 𝑉

𝑆2

𝑥

ln(1−𝛼𝑥 )+

𝑦

ln (1−𝛼𝑦 )+

𝑧

ln(1−𝛼𝑧) (9)

(Neubauer, 2000)

Where x, y, z : Total surface areas of a pair of opposite boundaries (𝑚2)


𝛼𝑥 , 𝛼𝑦 , 𝛼𝑧 : Average absorption coefficient of a pair of opposite boundaries


4/ Predicted reverberation times

The following section will relate the results obtained in each room configuration as well as

providing a first comparison and statement of the recurrent trends observed between the

theories.

The calculations were undertaken using the data included in the first part of the report.

The aim of this project being to compare the degree of accuracy between reverberation time

theory and practical measurement, it was a main concern to take into account as much as

possible the actual configuration of the room at the time the measurement were made.

However, a few materials had to be omitted into the calculation process due to inconsistence

and unavailability of absorption coefficients. As such, the air conditioner, speaker holder,

microphone and stand were neglected in all computations.

All three equations were worked out using an excel spread sheet developed for the project and

provided for each room configuration in the Appendix.


a) Room Configuration #1 – Full

Room #1 - Sabine Reverberation Time

Frequency (Hz) 125 250 500 1000 2000 4000

Sabine Reverberation Times (s) 0.39 0.29 0.29 0.29 0.29 0.23

Room #1 - Eyring-Norris Reverberation Time

Frequency (Hz) 125 250 500 1000 2000 4000

Eyring-Norris Reverberation Times (s) 0.34 0.24 0.24 0.24 0.24 0.18

Room #1 - Fitzroy Reverberation Time

Frequency (Hz) 125 250 500 1000 2000 4000

Fitzroy Reverberation Times (s) 0.46 0.35 0.35 0.30 0.26 0.19

The Fitzroy equation predicts the longest reverberation time with an average of 0.32s while the

Eyring-Norris equation gives the lowest values with an average of 0.25s.

The closest values are observed between the Fitzroy and the Sabine equation with an average

difference of 0.04s.

Reverberation Time Room #1


b) Room Configuration #2 – Empty


Frequency (Hz) 125 250 500 1000 2000 4000



Frequency (Hz) 125 250 500 1000 2000 4000



Frequency (Hz) 125 250 500 1000 2000 4000


The Fitzroy equation predicts again the longest reverberation time with an average of 0.84s.

The Eyring-Norris equations stills provide the lowest values with an average of 0.25s.

The closest values are this time given by the Sabine and Eyring-Norris equation with an

average difference of 0.05s.



c) Room Configuration #3 – Non Uniform


Frequency (Hz) 125 250 500 1000 2000 4000



Frequency (Hz) 125 250 500 1000 2000 4000



Frequency (Hz) 125 250 500 1000 2000 4000


The longest reverberation times are obtained with the Sabine equation with an average of

0.38s, followed closely by the Fitzroy equation with an average of 0.37s.

The largest degree of discrepancy is this time observed between the Sabine and Eyring-Norris

equations with an average of 0.05s.



5/ Theory Comparison

The following tables summarize the reverberation times obtained with each equation as well as

their degree of discrepancy for all three room configurations:

Room #1

Frequency (Hz) 125 250 500 1000 2000 4000 Average(s) Sabine RT60 (s) 0.39 0.29 0.29 0.29 0.29 0.23 0.30

Sab-Ey Discrepancy (s) 0.05 0.05 0.05 0.05 0.05 0.05 0.05

Eyring-Norris RT60 (s) 0.34 0.24 0.24 0.24 0.24 0.18 0.25 Ey-Fitz Discrepancy (s) 0.11 0.11 0.11 0.06 0.02 0.01 0.07 Fitzroy RT60 (s) 0.46 0.35 0.35 0.30 0.26 0.19 0.32 Fitz-Sab Discrepancy (s) 0.06 0.06 0.06 0.01 0.03 0.04 0.04

0.05

Room Average discrepancy (s)

Room #3

Frequency (Hz) 125 250 500 1000 2000 4000 Average(s) Sabine RT60 (s) 0.47 0.38 0.39 0.39 0.37 0.29 0.38 Sab-Ey Discrepancy (s) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Eyring-Norris RT60 (s) 0.42 0.34 0.34 0.34 0.32 0.24 0.33 Ey-Fitz Discrepancy (s) 0.07 0.06 0.06 0.02 0.00 0.00 0.03 Fitzroy RT60 (s) 0.49 0.39 0.40 0.36 0.32 0.25 0.37 Fitz-Sab Discrepancy (s) 0.02 0.01 0.01 0.03 0.05 0.05 0.03

0.04


Smallest Divergence Longest Reverberation Time Shortest Reverberation Time

Room #2

Frequency (Hz) 125 250 500 1000 2000 4000 Average(s) Sabine RT60 (s) 0.65 0.69 0.81 0.74 0.61 0.44 0.66 Sab-Ey Discrepancy (s) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Eyring-Norris RT60 (s) 0.60 0.65 0.76 0.70 0.57 0.39 0.61 Ey-Fitz Discrepancy (s) 0.01 0.02 0.12 0.35 0.44 0.41 0.23 Fitzroy RT60 (s) 0.61 0.67 0.88 1.05 1.01 0.80 0.84 Fitz-Sab Discrepancy (s) 0.04 0.03 0.07 0.31 0.39 0.36 0.20

0.16



According to the data listed above, we can summarise the following:

The Eyring-Norris equation provides the lowest values of reverberation times in all three room

configurations.

In most cases, the Fitzroy equation predicts the longest reverberation time.

As a general rule, the closest values are obtained between the Sabine and the Fitzroy

equations while the largest differences are observed between the Fitzroy and Eyring-Norris

equations.

The theories tend to better agree in the first half of the frequency band.

The biggest change in reverberation time from room configuration #1 to #2 is found with the

Fitzroy equation with an average increase of 0.52s.

The biggest change in reverberation time from room configuration #1 to #3 is equally found by

using the Sabine and Eyring-Norris equation with an average increase of 0.09s.

The biggest change in reverberation time from room configuration #2 to #3 is found by using

the Fitzroy equation with an average decrease of 0.47s.

The higher level of agreement between theories is found in room configuration #3 with an

average discrepancy of 0.4s followed by room #1 with 0.5s.

The higher degree of discrepancy between theories occurs in the room configuration #2 where

the Fitzroy equation predicts on average 30% higher values than the Sabine and Eyring-Norris

equations.

A first global comparison between theories is important at this stage of the process in order to

detect potential incoherence in the results wether due to incertitude in the collected data or

the particularity of the space under test.

In that sense, the large discrepancy observed in the room configuration #2 between the

Fitzroy equation and the other two seems to be a peculiarity of this trial as no such trend has

been observed in the other two configurations. Care must be consequently taken when

comparing the results with the purpose of conclude on the overall degree of discrepancy

between theories as the situation was peculiar to this configuration only and doesn‟t represent

the overall trends observed.

Taken as a whole however the theories seem to point in the same direction despite an average

discrepancy of 0.08s between the reverberation times predicted.


PART III/ Practical measurements

1/ The theory behind the practice

A common practice for reverberation time measurement is to “excite” the enclosure with a

sound source which is then switched off. A microphone captures the resultant decay which is

recorded for later analysis.

The decay is then plotted on a logarithm scale of amplitude versus time and the curve is

approximated by a straight line using linear regression. The slope of the recorded decay curve

is subsequently estimated, yielding the reverberation time (Guzina, 1984).

By definition, the evaluation of RT60 requires to be made over a 60 dB decay range. In

practice however, this condition is rarely fulfilled due to background noise which requires

using a high level sound source sufficiently above the noise floor (Schultz & Beranek 1962).

The ISO standard 3382 (1997) requires to set a noise margin of 10 dB in order to minimise

noise residues in the tail of the decay to be measured.

Consequently, the reverberation time has to be measured over smaller windows than 60dB,

with the extrapolated RT60 then being calculated, rather than actually measured.

The type of the definition from which the slope of the decay curve is calculated depends on the

difference between the background noise level and the sound source level. The RT 60

parameter can then be calculated using three definitions: EDT, RT20 and RT30. The ISO 3382

standard (1997) prescribes EDT (Early Time Decay) measurement to be made over the first 5

dB decrease, RT 20 measurement over the regression range of -5 to -25 dB and RT30 over the

-5 to -35 dB range.

Once the EDT, RT20 or RT30 is calculated, it is a simple mater or extrapolation in order to

determine RT 60. Since 20dB is one third of 60dB, the decay time over this 20dB window

would be 1\3 of the total RT60 (RT60= 12 × EDT or 2 × RT30).

Another aspect of the acoustic measurements is the need to repeat the measurements several

times in order to obtain a better statistical picture of the behaviour of the sound field in the

room (Everest, 2001).

The repetition of the experiment under the same conditions resulting in different decay curves

is due to the randomness of the excitation signal, not any changes in the characteristics of the

enclosure (Shroeder, 1964). The uncertainty arising from these fluctuations in the decay

curves is minimized by recording several decays and averaging the reverberation times of the

individual curves. Five or more decays are sufficient to provide satisfying statistical accuracy

(D'Antonio & Eger 1986).


Another source of fluctuation in the decay shape also arises for different sound source and

microphone positions, as such, it is requires to record several decays for different combination

of sound source and microphone positions (Guzina, 1982).

For each combination of microphone and loudspeaker positions, an ensemble averaging

procedure should be used, involving the superposition of several repeated excitations of the

room to obtain a single decay curve from which the reverberation time can be evaluated

(Nordtest, 1985). Such an ensemble averaging should be made by averaging the instantaneous

mean-square pressure values. Preferably, the ensemble averaging should also be extended to

include all the source and microphone positions (ISO 3382 standard, 1997).

If the difference between the reverberation times or noise levels measured for two positions in

the same room exceeds 10%, they cannot be averaged (Prokofieva, 2009).

If an ensemble averaging is made in each combination of microphone and loudspeaker

positions, the overall reverberation time of the room is given by the arithmetic mean of the

total number of reverberation time measurements (Nordest, 1985).

By averaging over a sufficient number of measurements made in the same frequency band

and with the same microphone and loudspeaker position, the problem of insufficient accuracy

and reproducibility of the results, especially at low frequencies, can be overcome (Guzina,

1984).

Measurements are usually made in narrow bands (octave or 1/3 octave), rather than

broadband (20Hz to 20KHz) in order to obtain a steady and dependable indication of the

average acoustical effects taking place within that particular slice of the spectrum (Everest,

2001). Octave and 1/3 octave filters are thus used partly to form the necessary selectivity

when broadband excitation is used and partly to improve the signal-to-noise ratio (Schultz &

Beranek 1962).


2/ Sound source

One of the main requirements is that the sound sources used to excite the enclosure must

have enough energy throughout the spectrum to ensure decays sufficiently above the

background noise to give the required accuracy (Everest, 2001).

The sound sources used for RT60 measurement fall into two categories: Steady state and

impulse sound sources.

Steady state sources use random noise (white or pink noise) reproduced through a

loudspeaker while impulse sound source consists of exciting and capturing the impulse

response of a room.

Only impulse source will be used in the project, thus the steady state measurement method

won‟t be elaborated in this report.

Impulse sound sources were and sometimes still are under the form of electrical spark

discharges, pistols firing blanks, pricked balloons and even small canons (Everest, 2001).

Nowadays, The most commonly used excitation signals are deterministic, wide-band signals

known as:

• MLS (maximum-length sequence) and IRS (inverse repeated sequence), which use

pseudorandom white noise

• Time-stretched pulses and SineSweep, which use time varying frequency signals.

These sound sources are defined as non-impulsive excitation signals (Nelson, 1992). Thus, a

deconvolution process is required in order to extract the impulse response from the recorded

signal. In order to do so, a known input signal is generated and the system output is

measured. The recorded signal is then mathematically compared with the generated signal

and the impulse response of the room can be extracted (Stan, Embrechts & Archambeau

2002).

This category of impulse sources offers some advantages over the mechanically produced

impulses (Cann & Lyon1979):

Significantly more energy can be fed into the room, improving the signal to noise ratio.

The monitoring and recording levels can be more easily and precisely set.

It is less likely to overload the recording medium as these sources don‟t fluctuate as much

as impulsive noise.


Sine sweeps

The sine sweeps are sinusoids that have their instantaneous frequency varying in the time.

Sweeps can be linear or logarithmic. Logarithmic sweep exhibits a pink spectrum, that is, its

amplitude decays at a rate of 3dB/octave. It means that the signal has the same energy per

octave. The frequency values double in time at a fixed rate (Ueda, Kon & Iazzetta 2005).

In practice a silence of sufficient duration is added at the end of the SineSweep signal in order

to recover the tail of the impulse response (Guzina, 1982).

To minimize the influence of the transients introduced by the measurement system and

appearing at the beginning and end of the emission of the excitation signal, the ends of the

SineSweep signal are exponentially attenuated (exponential growth at the beginning and

exponential decrease at the end) (Stan, Embrechts & Archambeau 2002).

The sine sweep technique allows to set the initial and final frequency (usually from 20Hz to

20KHz), the sweep duration and the duration of silences inserted after each sweep.

Sound source requirements

In order to measure room acoustical parameters in compliance with the ISO 3382 standard

(1997), an omnidirectional sound source should be used.

Three different source positions shall be used. One of these positions shall be in a corner. By

aiming the loudspeaker into a corner of the room (especially in smaller rooms), all resonant

modes are excited, because all modes terminate in the corners (Everest, 2001).

The sound source positions should be at least 2 meters apart (Nordtest, 1985).

The power of the electrical noise signal fed to the loudspeaker shall remain the same during

all the excitations (Nordtest, 1985).


3/ Microphone

The microphone should be as small as possible and preferably have the maximum dimension

of 13 mm. The smaller the microphone, the less its directional effects, using a small

microphone is considered best for essentially uniform sensitivity to sound arriving from all

angles (Everest, 2001).

The microphone should be positioned on a tripod, usually at ear height for a listening room, or

microphone height for a room used for recording.

When the room volume is less than 250 m3, three microphone positions are required for each

source position. In larger rooms three more positions shall be added. The positions shall be

evenly distributed in the room (Schultz & Beranek 1963). The ISO 3382 standard (1997)

prescribes three or four source positions in most cases.

The microphone positions should be located at least λ/2 apart, where λ is the wavelength of

sound for the centre frequency of the frequency band of interest.

Only one microphone should be used at a time. The microphones should be at least 0,5 m

from any absorber or diffuser and at least 2 m from the sound source (Nordtest, 1985).

Variations of the decay slope can sometimes be observed and is often caused by sources of

absorption or reflection in close proximity to the measurement point. This is a major source of

variation when patches of reflective or absorptive material are greater than a wavelength in

size and situated less than a wavelength away.

Because of the relatively high sound absorption coefficients typical of common listening

spaces, it is often the case that the direct field of a loudspeaker dominates throughout nearly

the entire enclosure. As Nelson (1992) expressed, the standard equation for the critical

distance Dc may be used as an estimation of the extent of the direct field. For studies of the

reflected sound field the direct field is essentially "noise" which obscures the reflected sound

field. In order to avoid such interference, classical methods for measuring reverberation time

instruct that the measurement points should be situated several times the critical distance

away from the sound source (Guzina, 1982). If a standoff of at least twice the critical distance

cannot be maintained between source and measurement point, special precautions and/or

analysis will probably be necessary to remove the influence of the direct field from the

reflected wave signature (Schultz & Beranek 1963).


The critical distance is given by the expression:

𝐷𝐶 = 2 × 𝑉

𝐶×𝑅𝑇60 (10)

Where V : Volume of the room (𝑚3) (Nelson, 1992)

C : Speed of sound (m/s)

4/ Requirements

The measurements were undertaken with the use of two different acoustic measurement

softwares offering roughly the same capabilities.

The fact of conducting every measurement with both software for each room configurations,

microphone and sound source positions allows a first range of comparison in order to avoid

any aberrant figures. As such, it is possible to verify if potential fluctuations occurring

between several decays within a measuring set and between each configuration of microphone

and sound source positions are of the same nature. As a result of such process, it is possible

to evaluate whether these fluctuations are due to variations of the acoustic properties of the

room at different locations or result from difference in computational processing of the two

softwares.

The first software is AcMus, an open source software based on a java plug-in framework. The

software was developed in cooperation with different departments of the University of Sao

Paulo, Brazil with for aim to provide an open-source extensible software for estimation,

measurement, analysis, and simulation of rooms especially designed for musical performance

(Ueda, Kon & Iazzetta, 2005).

The second software is ETF5, developed by Acoustisoft in 1996 as a “Loudspeaker and Room

Acoustics Analysis Program”. The program was one of the first software-based measurement

programs intended to turn an ordinary multimedia PC into a proquality measurement too l

(Everest, 2001).

Both softwares make use of the impulse response in order to extract the acoustic parameters

of the room, as such they enable the use of MLS and Sine sweep as excitation signals. They

also both undertake the same process in order to extract the impulse response of the room.

The operator connects a test microphone and microphone preamplifier to one of the

computer‟s sound card input. One of the interface output channels is then connected to the

sound system. The output channel sends the signal back to the second interface input. This

signal is taken as a reference to sync the signal that is recorded by the microphones.


In this way, the program can subtract any response anomalies caused by the sound card so

that what remains is actually caused by the system under test (Everest, 2001).

A test signal is then played from the sound card‟s first output through the sound system and

is simultaneously recorded through the sound card‟s inputs and stored as a wave file on the

computer for analysis. The impulse response is finally obtained by comparing both recording

signals (the actual room recording and the reference signal) and applying the deconvolution

process (Ueda, Kon & Iazzetta, 2005).

The software automatically processes the recording and then generates an impulse response

of the system being measured, which is stored in the computer for post-processing. Therefore,

all acoustic parameters can be analysed in post-processing after the measurements take

place.

Once the impulse response is obtained, the softwares generate the Energy Time Curve (or

Energy Delay Curve) by using the Schroeder Integration method according to the ISO 3392

standard (D'Antonio, 1986).

The reverberation time is finally obtained by applying a linear least square fit to the Schroeder

integration over a specified time interval (Everest, 2001).

In this way, both softwares allow to determine the reverberation time from the definition of

RT20, RT30 and EDT (AcMus only), depending on the available dynamic range for analysis.


5/ Measurement Process

The measurements were undertaken using a logarithm sine sweep as excitation signal. The

initial frequency was set to 20 Hz and the final frequency to 20 KHz with a total duration of 6

seconds. The level of the excitation signal remained the same for both softwares and for every

measurement. The final value of reverberation time was obtained from the extrapolation of

RT20.

Three sound source positions were used.

The position 1 was located in the corner of walls east and north.

The position 2 was placed centred and 0.80m away from the north wall.

The position 3 was located in the corner of walls west and north.

The positions were spaced 1.90m away from each other.

For all three positions, the speaker was elevated at 1.30 m from the floor.

Four microphone positions were used.

The position A, B, C were separated by 1m from each other and between 0.85 and 1 meter

from the nearest reflective and absorptive surfaces.

The position D was the closest to the sound source while still being more than the critical

distance away (the critical distance was estimated from the lowest reverberation time obtained

in the theoretical part of the project).

The microphone, positioned on a stand, was elevated 1,58m from the floor.

Five decays were captured for each combination of sound source/microphone positions.

Equipment

MacBook Pro laptop

Digidesign MBox2 Mini audio interface

Phonic MU 1722X mixer (served as microphone preamplifier and monitoring system

for output and input levels)

Genelec 1030A active monitor (directional speaker)

Behringer ECM8000 omnidirectional microphone


The following picture represents the set up of the measurement process:

The measurement process undertaken for each room configuration is summarized below:

Room #

Sound Source 1

Sound Source 2

Sound Source 3

AcMus Microphone A 5 Decays 5 Decays 5 Decays

Microphone B 5 Decays 5 Decays 5 Decays

Microphone C 5 Decays 5 Decays 5 Decays

Microphone D 5 Decays 5 Decays 5 Decays

ETF5 Microphone A 5 Decays 5 Decays 5 Decays

Microphone B 5 Decays 5 Decays 5 Decays

Microphone C 5 Decays 5 Decays 5 Decays

Microphone D 5 Decays 5 Decays 5 Decays

As expressed above, the number of decays required amount to 120 for one room configuration.

With the same process occurring in three room configurations, the total number of decays

rises to 360.

The experiment was conducted from 9pm to 2am in order to avoid as much background noise

level as practically possible. The room temperature was 21.4 °C.


Averaging Process:

The averaging process required to establish a final reverberation time for each room

configuration took place as follow:

For both softwares:

-Averaging of the reverberation times obtained for all five decays in each combination of sound

source/microphone positions.

- Averaging of the reverberation time obtained for all four microphone positions in each sound

source positions.

- Averaging of the reverberation time obtained for all three sound source positions.

The final reverberation time of the room was obtained by averaging the results obtained with

both softwares.

The following section relates the results obtained for each room configuration.


Room Configuration #1 - Full

The room was left in its usual arrangement with all 21 acoustic panels mounted.


1/ Room #1 - AcMus Software

2/ Room #1 - ETF5 Software

AcMus Reverberation Time Room #1

ETF 5 Reverberation Time Room #1

AcMus RT 60 (s)

Frequency(Hz) 125 250 500 1000 2000 4000

0.85 0.62 0.36 0.26 0.22 0.21

ETF 5 RT 60 (s)

Frequency(Hz) 125 250 500 1000 2000 4000

0.83 0.66 0.35 0.33 0.29 0.27


3/ Room #1 - Reverberation Time

Reverberation Time Room #1 (s)

Frequency(Hz) 125 250 500 1000 2000 4000

0.84 0.64 0.36 0.29 0.25 0.24

As expressed, the reverberation times obtained with both softwares show good conformity in

the first half of the frequency bands with an average difference of 0.02s while the second half

presents a larger divergence of approximately 0.07s.




Room Configuration #2 - Empty

All the 21 acoustic panels were removed, only the three fixed panels remained on the ceiling.


1/ Room #2 – AcMus Software

AcMus RT 60 (s)

Frequency(Hz) 125 250 500 1000 2000 4000

1.14 1.42 1.23 0.81 0.65 0.63

2/ Room #2 – ETF5 Software

ETF 5 RT 60 (s)

Frequency(Hz) 125 250 500 1000 2000 4000

0.97 1.38 1.04 0.73 0.62 0.63




Room #2 – Reverberation Time


Frequency(Hz) 125 250 500 1000 2000 4000

1.05 1.40 1.13 0.77 0.63 0.63

This time the results show better agreement in the second half of the frequency bands with an

average 0.04s divergence against 0.13s for the first half. The AcMus software provides the

highest values for all the frequency bands.

The reverberation time presents the particularity of a sudden rise at 250Hz which hasn‟t been

observed in the previous room configuration.




Room Configuration #3 – Non Uniform

The third room configurations consists of producing a non uniform distribution of sound

absorption as much as possible by placing the acoustic panels opposite to the reflective

surfaces and removing the panels present on these surfaces. As a result, the South and East

walls were uncovered while the North and West walls were left with the acoustic panels on.


Room #3 - AcMus Software

AcMus RT 60 (s)

Frequency(Hz) 125 250 500 1000 2000 4000

0.93 0.72 0.50 0.33 0.25 0.25

Room #3 - ETF5 Software

ETF 5 RT 60 (s)

Frequency(Hz) 125 250 500 1000 2000 4000

0.89 0.74 0.45 0.36 0.28 0.27




Room #3 - Reverberation Time


Frequency(Hz) 125 250 500 1000 2000 4000

0.91 0.73 0.48 0.34 0.26 0.26

This third room configuration offers the best conformity between the results obtained with

both softwares with an average difference of 0.03s throughout the frequency band.




6/ Practice Comparison

The first step in the analysis and comparison of the practical results was to evaluate the

degree of difference between values obtained with different sound source and microphone

positions. It is then important to investigate if these discrepancies are of same nature in all

room configurations and for both softwares.

Another important aspect is to trace any pattern that could be observed in the impulse

response recorded and being potentially related to a specific sound source and microphone

position, or a combination of both. This sort of phenomenon could easily translate the relation

of the sound source/microphone position to a specific acoustic aspect of the room such as

modal resonance or comb filtering and could consequently affect the true estimation of the

decay rate.

A first evaluation was thus undertaken over the impulse responses recorded at various sound

source and microphones positions in the three room configurations. After inspection, a

persisting difference was noticed in the model of the impulse response captured by the

microphone position D for every sound source positions and room configurations. In this

microphone position, all impulses present a distinct slope following the direct sound which

could appear to be caused by the ringing of the loudspeaker usually associated with its

volume or port resonance (Nelson, 1992). The microphone position D being the closest to the

sound source, it may be the most prone to capture the ringing of the speaker. This

phenomenon however didn‟t affect the reading of the decay rate and the microphone position

was consequently included in the averaging process.

After further examination, the average difference between the values of reverberation time

obtained with the three sound source positions was found to be 0.04 seconds for both

softwares and in all three room configurations.

The difference between the four microphone positions observed in all three room

configurations was larger with the use of ETF5 which provided an average discrepancy of

0.08s compared to the results obtained with AcMus which gave an average of 0.05s of

difference between the positions.

This first evaluation of the results is required in order to evaluate the degree of accuracy of the

acoustic measurement device. The rather close results obtained between sound source and

microphone positions as well as the repeatability of the divergence of same nature observed in

the three room configurations and by using two different softwares provide a strong basis for

the establishment of a reverberation time of reference for each room configuration.

As a consequence, it is fairly accurate to take the average value of the combination of sound

source and microphone positions independently for each software used.


Once the reverberation times of reference are established for each software, it is then requires

to evaluate the degree of discrepancy between each ones in order to determine is the average

of the two will provide enough accuracy so as to establish reverberation times of reference

representative of the overall behaviour of the sound field in each room configuration.

The results obtained with the two software show fairly good conformity with an average

difference of 7% and 9% in room configuration #3 and #2 respectively despite a bigger

divergence observed in the configuration #1 with 13% on average.

A similarity in the pattern of the results were observed in room configuration #1 and #3 where

the software ETF5 provides the highest value for the second half of the frequency bands as

well as at 250Hz with an average of 0.04s longer reverberation times measured. This trend

doesn‟t occur in room configuration #2 where the AcMus software gives the longest

reverberation times throughout the entire frequency range.

The several divergences observed between the results obtained with the different softwares

and to a lesser extent between the microphone and sound source positions may be due to

several independent causes.

First the randomness of the time at which the excitation signal is switched on and off implies

a constant variation of room resonances between each decay captured (Schultz & Beranek,

1963). This agitation of room modes will tends to alter the measured reverberation times in

the lowest frequency bands to a greater extent than the upper bands as modal resonances will

increase with frequency and thus provide better distribution and modal smoothing (Louden,

1971). This rationalization is in agreement with the results obtained as the discrepancies

between sound source and microphone positions appear larger in the first half of the

frequency band.

Another possibility lies in the difference between processing and averaging methods used by

the softwares. Although both softwares uses the Fast Hadamard Transform in order to extract

the impulse response and the Schroeder Integration method for calculating the decay rate, a

few parameters differ from one another such as the gate time which sets the interval on which

the linear least squares fit takes place. As a result, slight divergence in the determination of

reverberation time can arise from the same impulse response analysed through different

softwares.


As a general rule however, the low average divergence between values obtained with the

different softwares and the persisting nature of these differences observed in two room

configurations allow to average the results between softwares while still preserving the main

characteristic observed between the reverberation times measured in each set. In that way,

the decay slope and reverberation curve obtained after averaging the results obtained for both

softwares still provide the same information given by each set of measurement while at the

same time presenting an overall picture of the sound field in the room rather than only a

characteristic attributed to on a given position and/or software property.

As a result, the final reverberation times obtained in each room configuration can serve as

strong basis for later comparison to the theory.


PART IV/ Analytical Comparison

Once all the reference values of reverberation times have been established and that all three

theories have been applied in every room configurations, the main objective of the project can

be assessed, that is determining the degree of discrepancy between the reverberation time

predicted by the selected theories and measured in-situ.

The comparison will first take place in each room configuration in order to relate the degree of

accuracy of the theories to the acoustic conditions peculiar to a given environment.

1/ Room Configuration #1 – Full

As easily perceived, the biggest difference between the theories and practice arises for the first

two frequency bands where the Sabine, Eyring-Norris and Fitzroy equations produce an

average error percentage of 54%, 61% and 46% respectively.

Nevertheless, the theories appear to fairly agree with the practice for the second half of the

frequency band with an average error percentage of 3%, 16% and 6% for the Sabine,

Eyring-Norris and Fitzroy equations respectively.

Reverberation Times


The degrees of error between theories and practical measurement are summarized below:

Frequency (Hz) 125 250 500 1000 2000 4000

Sabine 53% 54% 19% 2% 13% 2%

Eyring-Norris 59% 62% 33% 19% 7% 24%

Fitzroy 46% 46% 1% 2% 2% 21%

Highest degree of error

Lowest degree of error

As expressed, the Fitzroy equation provides the closest value throughout most of the

frequency bands even though the 46% of error for the first two bands remain a too large

amount to statistical comply with the measurements.

It is noteworthy that the Eyring-Norris equation provides the highest degree of error

throughout most of the frequency bands with an average of 34% compared to 24% and 20%

for the Sabine and Fitzroy equations respectively.

The average absorption coefficient of the room being estimated to 0.32 on average, the Eyring-

Norris equation according to the literature would tend to provide more accurate results than

the Sabine equation which requires a maximum average absorption of 0.2 on average in order

to better suit the assumptions on which the theory is based (Long, 2006).

The large difference in average absorption between the more reflective surfaces containing the

windows and door and their respective opposite surface recovered with acoustic panel

generates a highly non uniform sound field which seems to better be taken into account by

the Fitzroy equation. As a result, the Fitzroy equation provides the closest results to the

practical measurement, especially in the frequency bands between 500Hz and 2000Hz with an

average discrepancy of 2%.


2/ Room Configuration #2 – Empty

The discrepancy at low frequencies persists with both Sabine and Fitzroy equations providing

a 39% average error for the first half of the frequency bands and Eyring-Norris giving an

average of 43%.

The second half of the frequency band presents good conformity of the results between the

Sabine equation and the measurement up to 2000Hz with 0.02s of difference but the

difference rises to 0.19s in the last band.

Error Percentage

Frequency (Hz) 125 250 500 1000 2000 4000

Sabine 38% 50% 29% 3% 3% 30%

Eyring-Norris 43% 54% 33% 9% 11% 38%

Fitzroy 42% 52% 23% 37% 59% 27%



Reverberation Times


The Sabine equation appears to provide the closest values to the measurement with an

average error percentage of 26% compared to 31% for the Eyring-Norris and 40% for the

Fitzroy equation. Despite the fact that the Fitzroy equation presents the lowest percentages of

error throughout most of the frequency bands compared to the Eyring-Norris equation, the

average of the error percentage appears to be larger for the Fitzroy equation which is due to

the large difference observed for the 1000Hz and 2000Hz bands compared to the

measurement. This point out the limit of accuracy when averaging the degree of error

throughout the entire frequency bands in order to deduce the statistical divergence of the

theory compared to the practice. As a result, it appears more correct to divide the frequency

bands in two half and assess the degree of discrepancy occurring between theory and practice

within a particular slice of the spectrum. The overall pictures will then give better statistical

results representative of a specific segment of the audio spectrum rather than a broadband

generalisation.

As such, the results obtained with the Fitzroy equation appear closer to the measurement

than the Eyring-Norris equation in the first three bands despite the fact that the later equation

provides an overall better agreement to the practice than the former.

These results however still provides considerable divergence in the first half of the frequency

bands compared to the measurement and as a consequence do not constitute an accurate

basis for predicting the reverberation times at low frequencies.

It is nevertheless interesting to observe that the Sabine equation provides the better

agreement with the practical measurement, especially in the 1000Hz and 2000Hz octave

bands. The room average absorption being estimated to 0.15 on average, this situation suits

the assumptions on which the theory is verified.

Moreover, the configuration #2 is the one that presents the better agreement to the

assumption of uniform distribution of sound absorption as much as it was practically

possible. In this sense, the ceiling was the only surface mounted with acoustic panels and as

such offered a better symmetrical distribution of absorption with regards to the large

absorption provided by the carpet flooring. The walls were left bare which may have diminish

the non homogenous reflection and absorption of sound between the surfaces containing

reflective materials (such as windows and door) and their opposite surfaces which are

mounted with acoustic panels in the other two room configuration.

In that way, this room configuration appeared the one suiting the best the assumptions of

uniform distribution of sound absorption made by all three theories; all this related to the

scale of the project.


3/ Room Configuration #3 – Non Uniform

The theories persist to provide notable divergence in the low frequency range notably in the

first two bands with an average error of 47% for the Sabine and Fitzroy equations and 54% for

the Eyring-Norris.

The last three frequency bands present much less divergence with an average error of 21% for

the Sabine, 14% for the Eyring-Norris, and 11% for the Fitzroy equation.

As displayed in the table below, the Fitzroy equation provides the least degree of error

throughout most of the frequency bands.

Error percentage

Frequency (Hz) 125 250 500 1000 2000 4000

Sabine 48% 47% 17% 13% 39% 13%

Eyring-Norris 53% 54% 28% 1% 21% 7%

Fitzroy 46% 46% 16% 3% 21% 5%



Reverberation Times


It is interesting to put in relation the establishment of these results to the assumption on

which each theory is derived. As expressed earlier, all three theories have for common basis

the assumption of a relatively diffuse sound field and homogenous propagation of sound

energy. The aim of the room configuration #3 was to establish an as non uniform distribution

of sound absorption as practically possible so as to diverge from the acoustic properties

required by the theories in order to provide accurate results.

This configuration involves significant difference in the average absorption between surfaces

and this consequently impacts on the validity of the theory. In fact, the three classic theories

under study differentiate themselves from one another by the way the average absorption of a

room is taken into account.

The Sabine and Eyring-Norris theories assume an equal average absorption for all surfaces

(Beranek, 2006) while the Fitzroy equations takes into account the geometry of the room with

regards to sound distribution; thus the divergences in average absorptions between the

surfaces along the lateral, transversal and longitudinal axes of the room are accounted for in

the Fitzroy equation.

As a result, and as verified in this room configuration, the Fitzroy theory tends to better

approximate the reverberation time in environment characterised by non uniform distribution

of sound absorption. Despite the overall high degree of discrepancy between theory and

practice, the Fitzroy equation provides on average of 10% less error than the other two

equations. The Sabine equation presents severe divergence in the upper frequency bands with

average error of 20% compared to 10% obtained with the other two theories. This observation

in a way translates the limit of the Sabine theory when applied to environment with

significantly non uniform distribution of sound absorption. The Fitzroy equation due to its

assumption of different average absorption for each pair of parallel boundaries nevertheless

provides the less error.

The divergences observed between the theoretical values and practical measurements of

reverberation time have been assessed for the three room configurations representing different

acoustic environments. However, the alteration of the acoustic properties of one given space

by the only change in the amount of absorption and its distribution is unfortunately

somewhat unrepresentative of the acoustical divergence that could undergo a sound field from

a large living space to a small absorbent one.

As a consequence the tendencies observed between theory and practice cannot establish a

general deduction verified in diverse acoustic environments, but rather are a peculiarity of the

room under investigation and of the project. The room however being rather common to such

locations, the assumptions concluded upon the completion of the project may nonetheless be

applicable to environments of the same scale and function, and presenting similar acoustic

properties.


Overall Analysis

After repetition of the investigation of reverberation time prediction methods in an enclosed

space presenting different configuration of absorption distribution, the overall evaluation of

the degree of discrepancy between theory and practice can finally take place.

As it has been observed in each room configuration, the major source of divergence between

the values predicted with the theory and the practical measurement arises in the low

frequency bands. For every room configurations, an average discrepancy of 40% occur in the

first three frequency bands where the theory predicts on average 0.45s lower reverberation

times than practically measured.

These large divergences observed at low frequencies for each equation compared to the

practice may be an indication of the limit of the accuracy of the theory when applied to small

spaces. In this sense, the assumption of all three theories regarding a diffuse sound field is

often unfounded in such environment where the influence of modal resonances dominates the

low frequency spectrum (Knudsen, 1932). As it has been evaluated earlier, the modal

resonances of the room are expected to dominate from 32Hz up to the Schroeder frequency of

222Hz. According to the modal resonance chart, only twelves modes reside between the

fundamental resonant frequency and the first octave band of 125Hz and only thirteen more

modes up to the critical frequency. These rather small numbers of modal frequencies between

octave bands imply a quite uneven distribution of modal energy throughout the first two

octave bands and as a result may impact the homogeneity of the sound field in the lower

frequency range. The various isolations and degeneracy detected in this interval are indication

of modal anomalies susceptible to affect the decay trace of the measured reverberation time.

The practice requires taking multiple averaging of the behaviour of sound at different points in

the room in order to overcome the limit of strong dependence of the decay rate to modal

resonance activity (Schultz & Beranek 1963). The results obtained with the theory however

remain in the domain of the prediction and it is partially the reason why the estimation of the

room resonances is required in order to evaluate the point from which the nature of the

results obtained begins to depart from the influence of modal resonance. Although the effect of

modal resonance is expected to diminish around the second octave band, the theory tends to

better agree with the practical measurement from the third frequency band, especially in the

room configuration #1 and #3. A reasonably fair conformity of the results is then maintained

throughout the rest of the spectrum.


The lowest discrepancy between theory and practice is equally observed in room configuration

#1 and #3 with 27% while the second configuration provides an average error of 32%. The

configuration #2 also surprisingly presents the highest level of discrepancy observed between

theories and between measurements made with the different softwares. No particular reason

can be advanced to justify this trend.

Another aspect of the analytical comparison between theory and practice could be to evaluate

the degree of similarity between the changes observed for different room configurations. In this

sense, a difference of 0.50s is observed between the reverberation time measured in room #1

and #2; that is the full and empty configuration. It is interesting to note that the Fitzroy

equation predicts an average increase of 0.52s while both Sabine and Eyring-Norris predict a

0.36s increase only. The Fitzroy equation therefore follows closely the trend observed with the

measurements. The change in reverberation time from room configuration #1 to #3 is

evaluated to be 0.06s by measurement and again the Fitzroy equation falls very closely with a

prediction of 0.05s compared to 0.09s found with the Sabine and Eyring-Norris equation.

From room configuration #2 to #3 the change is expected to be 0.44s according to the

measured reverberation times, Fitzroy this time predicting 0.47s and the Sabine and Eyring-

Norris equations 0.28s. This closer look to the degree of discrepancy observed between the

anticipating changes in reverberation time from one room configuration to another predicted

by the theory and by the practical measurements enables to perceive another limit of the

accuracy of the prediction methods. It has consequently been expressed than the results

obtained with the Fitzroy equation are once more the most in agreement with the practical

measurements.

An important point has to be made regarding the accuracy of the absorption coefficients used

in the equations. As expressed earlier, a compromise between accuracy and availability

inevitably took place in order to establish the average absorption for each material. The

divergence in size and thickness of the materials as well as their disposition in the room is an

important factor of disparity in the determination of the true level of absorption provided by a

given material. In this sense the absorption will increase with air gaps and perforated

materials depending on the space and dimensions and these discrepancies are not often

specified in the available absorption coefficients publication. As a result, various degree of

error can rapidly be introduced in the calculations and the final results can be severely altered

from the true value of reverberation time. This appears to be one of the main limits of the

theory.


Summary

Three of the classic theories of reverberation time have been applied to the same room for different

configurations of sound absorption distribution. The reverberation times have been measured in-situ

for each room configuration by the capture of the impulse response using a sine sweep test signal and

by recording several decays at different microphones and sound source positions with the use of two

different acoustic measurement softwares.

The practical measurements were first compared at different scales in order to evaluate the degree of

discrepancy introduced between different measuring positions in the room and for different acoustic

measurement software. A good conformity of the results and the repeatability of the trends observed in

each room configuration enabled a strong basis for establishing a reverberation time of reference in

each room configuration in an attempt to later compare these values to the results predicted with the

theory.

The main aim of the project could finally be assessed, that is determining the level of accuracy of the

classical reverberation time theories when applied to enclosed spaces presenting diverse sound field

properties governed by a change the distribution of sound absorption. The largest divergence between

predicted and measured values was commonly observed in the first half of the octave bands with

theories providing an average of 40% discrepancy compared to practical measurements. The theories

provided nevertheless truthful results in the second half of the octave bands with an average

discrepancy of 15% with the practice.

After evaluation of the modal activity of the room, it appears that the activity of modal resonance

dominate in the first two frequency bands and thus the condition of homogeneous distribution of sound

energy required by the theories is not fulfilled within that particular range of the spectrum.

As a consequence, the estimation of reverberation time at low frequencies remains affected by the

influence of modal resonances which is not taken into account in the theories‟ assumption of diffuse

sound field.

Another aspect impacting on the assumption of diffuse sound field is the variation of absorption

between the room surfaces which implies that the decay rate will be partially governed by the average

absorption encountered during the sound propagation. In this case, the Fitzroy equation was found to

give the closest statistical estimation of the reverberation time compared to the practical measurement

and as such, the recognition of the Fitzroy theory to better take into account the absorption distribution

of a room was verified.

The Sabine equation provided the less discrepancy in the room configuration having the lowest average

absorption and thus complies with the assumption of which the theory‟s accuracy is acknowledged. The

Eyring-Norris theory however didn‟t provide any recurrent figure and as a result no statistical deduction

can be carried out regarding its accuracy in distinct acoustic environments. As a conclusive statement,

the theory of reverberation time appears to be limited to an approximate method of predicting the

overall trend that the decay of sound is susceptible to undertake according to the acoustic propriety of

the space under test.


References

Beranek, LL 2006, Analysis of Sabine and Eyring equations and their application to concert hall audience and chair absorption, 2 June 2006, Cambridge.

Beranek, LL 2008, „Concert hall acoustics‟, Journal of the Audio Engineering Society, Vol. 56, no. 8, August 2008, Cambridge. Cann, RG & Lyon RH 1979, „Acoustical impulse response of interior spaces‟, Journal of

the Audio Engineering Society, Vol. 27, No. 12, December.

D'Antonio , P 1986, „T60- how do I measure thee, let me count the ways‟, in the 81st Convention, Audio Engineering Society, 12-16 November, Los Angeles. Everest, FA 2001, The Master Handbook of Acoustics, 4th edn, McGraw-Hill, New York.

Eyring , CF 1930, „Reverberation time in „dead‟ rooms‟, Journal of the Acoustic Society of America, Vol. 1, January, Melville. Fitzroy, D 1959, „Reverberation formulae which seems to be more accurate with non-uniform distribution of absorption‟, Journal of the Audio Engineering Society, Vol. 31. Guzina, B 1982, „Measurement of reverberation time using different signal sources‟, in the 71th Convention, 2–5 March, Audio Engineering Society, Montreux. Guzina, B 1984, „Measurement of reverberation time using a microproceesor audio analyzer‟, in the 75th Convention, Audio Engineering Society, 27-30 March, Paris. Joyce, WB 1978 „Exact effect of surface roughness on the reverberation time of a uniformly

absorbing spherical enclosure‟, Journal of the Acoustic Society of America, Vol 65, Melville.

Kanga, J & Neubauer, RO 2001, „Predicting reverberation time: comparison between analytic formulae and computer simulation‟, in the 17th International Congress on Acoustics, Audio Engineering Society, Rome. Knudsen, VO 1932, „Resonances in small rooms‟, Journal of the Acoustic Society of America,

July.

Long, M 2006, Architectural acoustics (applications of modern acoustics), Elsevier, Oxford. Louden, MM 1971, „Dimension ratios of rectangular rooms with good distribution of eigentones‟, Acustica, Vol. 24.

Mayo, CG 1952, „Standing-wave patterns in studio acoustics', Acustica, Vol.2, no.2.

Nelson, DA 1992, „Impulse response of very small rooms‟, in the 93rd Convention, 1-4 October, Audio Engineering Society, San Francisco. Neubauer, RO 1999, „Prediction of reverberation time in rectangular room using a modified-fitzroy equation‟, in the 8th International Symposium on Sound Engineering and Mastering‟, 9-11 September, Gdansk, Poland.


Neubauer, RO 2000, „Estimation of reverberation time in rectangular rooms with non-uniformly distributed absorption using a modified fitzroy equation‟, in the Seventh International Congress on sound and vibration, 4-7 July 2000, Germany. Nordtest 1985, Rooms: reverberation time, NT Acou 053, approved 1985-05.

Powell, JG 1970, „Choosing a formula for calculating the absorption coefficient from reverberation chamber measurements‟, Journal of the Audio Engineering Society, Vol. 18, No. 6, California. Prokofieva, E 2009, „Selection of loudspeaker positions for reverberation time and sound field measurements‟, in the 126th Convention, 7–10 May 2009, Audio Engineering Society, Munich, Germany. Sabine, WC 1923, Collected papers on acoustics, Harvard University Press, New York. Schultz, TJ & Beranek, B 1962, „How would you measure the reverberation time‟, in the 14th annual meeting of the Audio Engineering Society, 15-19 October 1962, Cambridge. Schultz, TJ & Beranek, B 1963, „Problems in the measurement of reverberation time‟, Journal of the Audio Engineering Society, Vol. 11, No. 4, Cambridge. Stan, G, Embrechts, J & Archambeau D 2002, „Comparison of different impulse response measurement techniques‟, Journal of the Audio Engineering Society, Vol. 50, No. 4, April, 2002. Ueda, LK, Kon,F & Iazzetta, F 2005, „An open-source platform for musical room acoustics research‟, in the 2005 Congress and Exposition on Noise Control Engineering, 7-10 August 2005, Rio de Janeiro.

BAP260.1 Practical Project

Documents

Transcript of BAP260.1 Practical Project