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Investigation of Open-air theatre design through Geometrical Acoustic
Modelling
Technical Report · June 2017
DOI: 10.13140/RG.2.2.35250.81606
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School of Computing, Science and
Engineering
BEng Audio Acoustics
Investigation of Open-air theatre
design through Geometrical
Acoustic Modelling
Thomas Dimasi
@00368700
t.dimasi@edu.salford.ac.uk
Supervisor: Bruno Fazenda
Reader: Francis Li
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Thomas Dimasi
Investigation of Open-air theatre design
May 2017
Abstract
A review of the evolution and acoustics of ancient open-air theatres is undertaken. The
acoustics of ancient open-air theatres are characterized by remarkably high speech
intelligibility. This is due to the seating area (Cavea) and Orchestra. Thus, the acoustics of the
Cavea is evaluated from the literature and through measurements.
Geometrical acoustics models of open-air theatre are problematic since the lack of
reflection from a roof increases the relevance of the acoustic behaviour of the seating area.
Thus, analysis of how to simulate it in geometrical acoustic software is performed. Different
geometrical representations of the Cavea are tested through validation against measurements
of the Aspendos theatre and scale model reconstructions of the Syracuse theatre. Different
diffractions calculations and algorithms are tested and performance evaluated. A method to
model open-air theatre in geometrical acoustics software is developed. However, the
generalized modelling method has limitations, due to the difficulty in representing the
behaviour of the Cavea. Audience presence in the Aspendos model is evaluated; however, its
effect has not been validated against measurements as none were available.
The acoustic effects of placing reflectors over the stage area and above the seating
area are evaluated through the developed method. Reflectors over the stage area improved
T30 distribution when an audience is present, whereas reflectors over the Cavea generated
undesired T30 predictions around the Cavea due to their interaction between each other. A
model of an open-air theatre with a balcony is also tested. There are discernible improvements
for receivers near the reflective surface of the balcony, however, other positions worsened.
Localized focusing effect is noticed on a specific area of the Cavea.
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Investigation of Open-air theatre design
Acknowledgements
Special thanks to Bruno Fazenda support throughout the project, to Jonathan Hargreaves,
Francis Li, and Paul Kendrick for advice, and to Jens Holger Rindel for kindly sharing the
CAD drawings of the Aspendos theatre and Syracuse theatre Roman reconstruction.
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Table of Contents
BEng Audio Acoustics ........................................................................................................... 1
Thomas Dimasi ..................................................................................................................... 1
@00368700 .......................................................................................................................... 1
Abstract................................................................................................................................. 2
Acknowledgements ............................................................................................................... 3
1. Introduction (Investigation of Open-air theatre design) ...................................................... 6
2. Literature Review .............................................................................................................. 7
2.1 Evolution of open-air theatres ...................................................................................... 7
2.2 The Cavea ................................................................................................................. 11
2.2 Computer Modelling of Open-air Theatres ................................................................. 13
2.3 Open-air Theatre Scale models ................................................................................. 15
3. Theory............................................................................................................................. 18
3.1 Geometrical Acoustics - CATT-Acoustics .................................................................. 18
3.1.1 Algorithms ........................................................................................................... 18
3.1.2 Auto Scattering Function ..................................................................................... 19
3.1.3 Diffraction ............................................................................................................ 20
3.1.4 Limitation of Diffraction calculations .................................................................... 21
4. Methodology ................................................................................................................... 21
4.1 Software used ........................................................................................................... 21
4.2 Validation................................................................................................................... 21
4.2.1 Computer Modelling Prediction Analysis ............................................................. 21
4.2.2 Audience Analysis ............................................................................................... 23
4.2.3 Syracuse theatre GA validation of models ........................................................... 23
4.2.4 Aspendos theatre GA validation of model............................................................ 25
4.3 Design Investigation .................................................................................................. 27
4.3.1 Architectural features investigation ...................................................................... 27
4.3.2 Core Cavea model .............................................................................................. 27
4.3.3 Stage reflectors model ........................................................................................ 28
4.3.4 Clouds model ...................................................................................................... 29
4.3.4 Balcony model .................................................................................................... 29
5. Validation – Results and Analysis ................................................................................... 31
5.1 Modelling the Cavea in GA ........................................................................................ 31
5.1.1 Simple Geometry ................................................................................................ 31
5.1.2 Detailed Geometry .............................................................................................. 33
5.2 Theatre models validation .......................................................................................... 36
5.2.1 Syracuse Theatre Greek model........................................................................... 36
5.2.2 Syracuse Theatre Roman model ......................................................................... 38
5.2.3 Aspendos Theatre model .................................................................................... 41
5.2.3 Aspendos Theatre model with Audience ............................................................. 43
5.3 Discussion of Validation............................................................................................. 44
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Investigation of Open-air theatre design
6. New Designs – Results and Analysis .............................................................................. 45
6.1 Core Cavea model..................................................................................................... 46
6.2 Stage Reflectors model ............................................................................................. 47
6.3 Clouds model ............................................................................................................ 49
6.4 Balcony model ........................................................................................................... 52
6.5 Discussion of New Designs ....................................................................................... 54
7. Conclusions .................................................................................................................... 57
8. Future Work .................................................................................................................... 58
References ......................................................................................................................... 59
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1. Introduction (Investigation of Open-air
theatre design)
The Greeks and Romans were masters of acoustic design more than 2000 years ago.
Since then we have made immense progress in the field of architectural acoustics but most
modern design is protected from issues that our ancestor colleagues had to deal with when
designing for outdoors. This spark the curiosity of wondering what would be a modern design
of an open-air theatre. Theatres evolved from being open-air structures in the Roman Era,
when the Odea, an enclosed structure, was built. Since then theatre design has focussed on
closed structures and the evocative environment offered by the massive Classic and Roman
era open-air theatres has been lost and open air design remained unchanged.
Answering to the question what would be the open-air theatre design which would
represent its evolution in modern times and a beacon of present knowledge of acoustic,
requires an extensive research encompassing many aspects of different fields. However,
addressing the Acoustics aspect, the following questions arise: which are the positive and
negative acoustic aspects of open-air theatres? what acoustic solutions could solve issues?
how should acoustic design be investigated?
This project aims to address these questions by:
•
Reviewing the evolution of Open-Air Theatres acoustics of open-air theatres.
•
Address how to predict performance of open-air theatres using Geometrical
Acoustic (GA) computer modelling.
•
Simulate workable solutions to improve the acoustics of Open-Air Theatres.
To identify how to model open-air theatres in GA a validation against measurements
is conducted. The Syracuse scale models, built by (Farnetani et al. 2006), were chosen for
the validation process since these were measured both with and without specific surfaces.
Hence, through validation of the scale models, the presence or absence of a reflective surface
can be evaluated. The Aspendos theatre was chosen for validation since (M. Lisa, 2004) has
investigated the GA modelling of this theatre and because it is one of the best-preserved
ancient open-air theatres, thus its measurements are good for validation.
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Investigation of Open-air theatre design
During the investigation on how to model open-air theatres the limits of (GA) modelling
failed to simulate the behaviour of the seating area (Cavea) under with the presence of
different architectural features. This is due to the lack of reflections in open-air theatres which
would mask the complex acoustic field of the seating tiers. Thus, much work has been directed
to evaluate the limits of GA predictions and identify a method to predict the acoustic of openair theatres.
Investigation of New Designs carried out here, addresses the influence of reflectors
placed on the rear of the stage area, over the Cavea, and on the back of the Cavea by
investigating a design incorporating a balcony. Since acoustic performance with audience
present is of major interest, New Design are also assessed with the presence of audience.
However, it should be noticed, that validation against measurement with audience present
could not be conducted here due to the lack of comparable data.
2. Literature Review
2.1 Evolution of open-air theatres
Most open-air theatres have undergone modifications over time. Before adopting the
renowned semi-circular shape, we know, Greek theatres were rectangular during the Minoan
period (20th-15th B.C.) and trapezoidal during the Pre-Aeschylean period (15th-6th B.C.).
Minoan theatres suffered from multiple specular reflections and flutter echoes when no
audience was present. When occupied, the Minoan and Pre-Aeschylean theatre acoustic
consisted of the direct sound and reflections, and had low T30 and high STI values. The high
STI was due to the small size of these theatres. When empty, STI drops considerably
(Chourmouziadou and Kang, 2008).
Theatres evolved for centuries, size varied and new architectural features were added
(Barron, 2009). Figure 2.1 shows the shapes of these theatres (image taken from
(Chourmouziadou and Kang, 2008)), while Figure 2.2 illustrates the Classic Greek Theatre
Fig. 2.2a, the Hellenistic Fig. 2.2b, the Roman Fig. 2.2c (image from (Chourmouziadou and
Kang, 2008)).
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Investigation of Open-air theatre design
Figure 2.1: Early shapes of Greek theatres. (a) Minoan, (b) Pre-Aeschylean. (Source: (Chourmouziadou and
(Source: (Chourmouziadou and Kang, 2008))
The Classic Greek theatre semi-circular shape was reached in the 5th century B.C.
Designed to accommodate festivals in honour of Dionysius, it initially hosted musical
performances then Classic Greek drama. The theatre was formed by a performing platform
(called Orchestra) next to a hillside were the audience sat. The semi-circular shape of these
theatres meant that the audience at the side of the orchestra suffered from poor acoustic and
visual. The seating area (called Cavea in Latin) was a structure made of wood or stone
(Barron, 2009). The Cavea increased the reverberance of the theatre proportionally to its size.
Measurements of the theatre of Epidaurus (14000 seats) yield a T30 of over 1 second, while
Ancient Epidaurus (2000 seats) yields a T30 just below 0.6 seconds (Chistian Gade and
Angelakis, 2006). The reflections from the orchestra and the steep slope of the Cavea,
typically 20 to 34 degrees, greatly improve sound transmission across the audience (Barron,
2009). Canacs’ work on image source reflections, shows that the minimum angle between the
image source generated by the orchestra and the sloped Cavea, should be no less than 5
degrees. This is to avoid reflections from the orchestra being blocked by the first audience
rows (Polack,2011). Figure A.3, taken from (Barron, 2009) shows the orchestra-cavea
reflection pattern.
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Investigation of Open-air theatre design
Figure 2.2: Evolution of open-air theatres: (a) Classic Greek Theatre, (b) Hellenistic Theatre, (c) Roman Theatre.
(Source: (Chourmouziadou and Kang, 2008))
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Investigation of Open-air theatre design
Figure 2.3: Reflection from the orchestra. (Source: (Barron, 2009))
As the importance of theatre increased, theatres evolved to meet new requirements.
Developments in Greek Drama required the addition of a raised stage (called Skene), where
the show was moved from the Orchestra (Barron, 2009). Interestingly, during the Greek era,
the Syracuse Theatre stage had a system of carts on tracks which made it possible to change
the stage scenario (Farnetani et al. 2006). The wall behind the Skene (the Proscenium) had
an aesthetic purpose. This wall adds reflections and improves speech intelligibility (Haddad,
2006), however it also increases reverberation by containing reflections from the Cavea
(Chistian Gade and Angelakis, 2006).
Hellenistic theatres were often refurbishments of Classical Greek theatres. The main
modification was the increase in size of the Cavea and of the stage building. Roman theatre
design was substantially different, thanks to advances in structural engineering. Roman
theatres were essentially enclosed spaces with an open ceiling. Evidence shows that a
massive canvas cloth (called Velarium) covered a large part of the opening and was used to
shade the audience from the sun (Barron, 2009), further enclosing the theatre. The stage
building extended over the full height of the cavea and a backwall surrounded the Cavea. Both
the proscenium and the backwall where ornamented, and acted as a diffuser avoiding echoes.
Canac’s study of a scale model of the Roman Theatre of Orange (France) shows that the
proscenium homogenize intensity to the upper part of the Cavea. In Roman theatres, the
Orchestra was occupied by Senators (Barron, 2009), hence the positive reflections from the
orchestra were attenuated. The smaller size and increased inclination of the Cavea minimized
the outcomes of the loss of the Orchestra reflections (Barron, 2009).
(Chourmouziadou and Kang, 2008) outlines the evolution of the theatres and analyses
the acoustic modification at each alteration. Figure 2.4 shows the results obtained by
(Chourmouziadou and Kang, 2008) their analysis. As expected, reverberation time increases
with the evolution of theatres from Classic to Roman. The STI values calculated in
(Chourmouziadou and Kang, 2008) reveal similarity for Classic Greek and Hellenistic Theatres
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Investigation of Open-air theatre design
both when occupied and unoccupied. Roman Theatres have lower STI values due to
reverberation, however, STI values are more consistent between receivers with respect to the
Classic Greek and Hellenistic theatres. Classic Greek and Hellenistic theatres show a wide
variation of both STI and T30 depending on the position of the receiver. Measurement
contours performed by (Van Loenen, et al. 2016) for the Theatre of Epidaurus (Hellenistic
Theatre) and the Odeon of Herodes Atticus (similar to a Roman Theatre) reflect the pattern
described by (Chourmouziadou and Kang, 2008).
Figure 2.4: Ancient open-air theatres T30 variation through eras. (Source: (Barron, 2009))
2.2 The Cavea
Measurements of Epidaurus, performed by (Skarlatos et al. 2011) show that the time
domain of the signal is composed of distinct early reflections from the orchestra and a short
decay of diffracted sound energy. The early reflections decay, of approximately 40dB within
60ms, improves direct signal perception, thus reinforcing speech intelligibility. The frequency
response shows a dip between 170-200 Hz which increases with distance. Resonance peaks
appear between 500Hz to 1000Hz for receivers near the source, while for more distant
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Investigation of Open-air theatre design
positions, additional peaks appear from 300Hz to 1500Hz. This spectral amplification lasts
between 150-200ms. After this amplification, the spectrum becomes more homogenous due
to the dense diffuse reflections.
The spectrum response performed by (Skarlatos et al. 2011) is in line with the numerical
study done by (Declercq, and. Dekeyser, 2007) on the seat rows of the theatre. (Declercq,
and. Dekeyser, 2007) shows how reflections and diffraction effects of the Cavea and the
Orchestra generate a high-pass filter which damps
frequencies between 50Hz to 530Hz and amplifies
frequencies
above
530Hz.
Even
if
the
fundamentals of speech are filtered, the formants
are amplified, thus speech intelligibility increases.
The numerical case presented by (Declercq, and.
Dekeyser, 2007) is an ideal case, hence in reality
the effect is not so strong, however it highlights the
reason behind the great acoustic of Epidaurus.
The high-pass filter revealed by (Declercq, and.
Dekeyser, 2007) depends on the periodicity of the
seat rows. Intuitively, the longer the periodicity (the
row repetition), the lower the frequency band
filtered.
As established, the acoustic of Epidaurus offers
outstanding speech intelligibility. Figure 2.5 shows
the RASTI values respect to receiver distance and
angle from the source (in the centre of the stage)
(Skarlatos
et
al.
2011).
Interestingly,
the
measurements, the dashed lines in Figure 2.5,
show an increase of the Rasti value with distance.
(Skarlatos et al. 2011) suggests this could be due
to variation in the early/late energy reflection ratio.
The computer simulation done by (Skarlatos et al.
2011), the continuous lines in Figure 2.5, predict a
Figure 2.5: Rasti values per receiver distance at different
angles respect to the centre of the Cavea. Value measure(--).
Value simulated (-) (taken from (Skarlatos et al. 2011))
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Investigation of Open-air theatre design
lower value of Rasti especially at 85degrees from the centre of the Cavea. While for 5 and 45
degrees the Rasti value dips for the receiver at the middle rows of the Cavea.
The measurements of Epidaurus performed by (Van Loenen, et al. 2016), see Figure 2.6,
show wide variations in the T20 between the centre of the Cavea and its sides. Inversely, C80
is higher at the centre with respect to the sides of the Cavea. Thus, the seating area on the
side of the Orchestra seems to suffer the worst acoustics in the theatre. The transition point is
between 80 to 90-degree with respect to the centre middle of the Cavea. However, it should
be noted that these measurements were taken with the source right at the centre of the
Orchestra, these could vary if the source is moved onto the stage.
Figure 2.6: Epidaurus measurements and simulations in ODEON. Note scales differ between measurements and
simulations. (Source: (Van Loenen, et al. 2016))
2.2 Computer Modelling of Open-air Theatres
Geometrical Acoustics is a widely used acoustic modelling technique. GA modelling
approximates the behaviour of sound. The concept is developed around ray-acoustics which
is a valid approximation for high frequencies or, more accurately, for the frequency range
whose wavelength is much smaller than the smallest surface dimension of the geometry.
Thus, acoustic models have simplified geometries and details are omitted so as to obtain more
accurate results. At lower frequencies, wave effects such as wave interference, edge
diffractions, finite surface reflections, and interaction between mediums, become more evident
prominent. These effects can be implemented in GA modelling; however, they have their own
limitations (Savioja & Svensson, 2015).
Another problem of GA is the quality of data that can be inputted inside the models. Only
a limited amount of information on absorption and scattering can be inputted, hence a further
approximation. Furthermore, obtaining accurate data for these coefficients is difficult.
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Investigation of Open-air theatre design
Open-air theatres pose a challenge for Geometrical Acoustic computer modelling. The
absence of a ceiling implies a huge loss of energy/rays; hence a high number of rays must be
used to obtain a smooth decay curve. (M. Lisa, 2004) uses 500000 rays in his work. This
number of rays provided a good decay curve for all CATT models run in this work. However,
the real problem for GA modelling of open-air theatres is the absence of a ceiling. The ceiling
provides strong reflections which mask diffractions taking place on the seating area, hence
the Cavea of an open-air theatre cannot just be simulated as a simple diffusing surface.
(Declercq, and. Dekeyser, 2007) provides in depth analysis of the seat row filtering effect,
while (M. Lisa, 2004) highlights the importance of detail modelling the Cavea. Shaping the
seat rows as a plane surface with scattering coefficients typically used for room acoustics will
not reflect enough energy backwards to the stage. Figure 2.7, taken from (M. Lisa, 2004)
shows the reflection paths between the Cavea and the Proscenium. However, the reflection
pattern of the detailed Cavea is valid only for frequencies whose wavelength is smaller than
the height of the seat row.
Figure 2.7: Ray paths with different geometrical representation of Cavea. (Source: (M. Lisa, 2004))
The conclusions of (Van Loenen, et al. 2016) and (M. Lisa, 2004) agree on the most accurate
method to predict the acoustics of two open-air theatres using ODEON: the theatre of Herodes
(Van Loenen, et al. 2016) and that of Aspendos (M. Lisa, 2004). These two theatres have
similar architectural characteristics: large Proscenium enclosing the Cavea and a backwall
surrounding the Cavea. ODEON is a hybrid GA which combines Image Source method to
predict early reflections and Ray-tracing method to predict the diffuse field. Both (Van Loenen,
et al. 2016) and (M. Lisa, 2004) showed that the most accurate results were obtained when
modelling the stair details of the Cavea and using the image source method up to 2nd order
reflection then switching to ray-tracing. However, (Van Loenen, et al. 2016) shows that for
prediction in Epidaurus the best results are obtained without using the image source method.
The theatre of Epidaurus has only one main reflective surface, the Orchestra, while for
Herodes and Aspendos the Proscenium and backwall are also large reflective surfaces. Due
to the inverse cone shape of the Cavea in Epidaurus there are only two possible image
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Investigation of Open-air theatre design
sources which involve the orchestra: source- Orchestra - receiver or source-Orchestra-Caveareceiver. All other possible image sources will be generated by the Cavea. For the Herodes
and Aspendos theatres there are other possible image sources involving the Proscenium and
the Orchestra. As seen previously, reflections from the Orchestra are very important for the
acoustic of the theatre, likewise other large reflective surfaces are important. Both (Van
Loenen, et al. 2016) and (M. Lisa, 2004) obtained good reverberation time predictions without
using the image source method. The higher 2nd order transition was essential to predict EDT.
(M. Lisa, 2004) also, tried using a higher 5th order transition. Higher transition order leads to
a drastic decrease in the predicted T30, this is not unexpected since the Cavea generates a
highly diffuse field. Thus, the image source method cannot represent the behaviour of the
Cavea.
2.3 Open-air Theatre Scale models
(Farnetani et al. 2006) measured the acoustics of 1:20 scale models of the Theatre of
Syracuse. Scale models were built based on historical reconstructions of the theatre in the
Greek, Hellenistic, and Roman Eras.
The Greek era Syracuse theatre was the smallest with a Cavea diameter of about 85meters, 11-meters high, and included a small stage building. Figure 2.8 shows scale
reconstruction of the Syracuse theatre in Greek Era (Farnetani et al. 2006).
Figure 2.8: Scale model of the Syracuse Theatre Greek Era. (Source: (Farnetani et al. 2006))
The theatre evolved through the Hellenistic Era to the Roman Era. Todays’ ruins are
from the Roman era. The Syracuse theatre in the Roman Era increased in size and
architectural features were added. Figure 2.9 shows the scale model reconstruction of the
Syracuse theatre in Roman times (Farnetani et al. 2006).
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Figure 2.9: Scale model of the Syracuse Theatre Roman Era. (Source: (Farnetani et al. 2006))
As expected, increasing the size of the theatre, the stage, and adding the colonnaded
aisle increased the reverberation time. Figure 2.10 shows the measured T30 of the scale
models of the Syracuse theatre in Greek and Roman times.
2
1.8
1.6
1.4
T30 (s)
1.2
1
Greek
0.8
Roman
0.6
0.4
0.2
0
125
250
500
1000
2000
4000
frequency Octave bands (Hz)
Figure 2.10: T30 octave band measurements Syracuse Theatre scale models in Greek and Roman Era.
(Source:(Farnetani et al. 2006))
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Investigation of Open-air theatre design
(Farnetani et al. 2006) performed interesting measurements adding and removing
architectural features of the Greek and Roman versions. For the Greek version, the theatre
was measured both with and without the stage: Figure 2.11 shows the comparison between
these two measurements.
theatre with and without the stage. For the Roman version,
measurements were taken with and without the colonnaded aisle. Figure 2.12 shows the T30
parameter measured in these two situations.
1.2
1
T30 (s)
0.8
0.6
0.4
0.2
0
125
250
500
1000
2000
4000
frequency Octave bands (Hz)
Greek without Stage
Greek with Stage
Figure 2.11: T30 measurements of the scale models of the Syracuse Theatre (Greek version) without stage building
(Blue) and with stage building (Red). (Source:(Farnetani et al. 2006))
2
1.8
1.6
T30 (s)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
125
250
500
1000
2000
4000
frequency Octave bands (Hz)
Roman without colonnaded aisle
Roman with colonnaded aisle
Figure 2.12: T30 measurements of the scale models of the Syracuse Theatre (Roman version) without
colonnaded aisle and colonnaded aisle. (Source:(Farnetani et al. 2006))
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3. Theory
3.1 Geometrical Acoustics - CATT-Acoustics
3.1.1 Algorithms
CATT-Acoustics is an acoustic prediction software which cone-tracing algorithms and
offers three types of algorithms. However, since CATT allows only Algorithm 2 and Algorithm
3 to be used for an open model, only these two algorithms will be analysed.
For both Algorithm 2 and 3, reflections in CATT are calculated as follows. When the
beam hits a surface: it is attenuated by the absorption coefficient α, and split up into its
specular and diffuse components. The energy split between specular and diffuse reflection
depends on the scattering coefficient. Figure 3.1, taken from (TUCT User’s Manual), shows
the principle of reflection. This process is calculated according to Lambert law (Kuttruff, 2009).
The only energy loss occurring in the reflection process is due to absorption. The repetition of
reflections gradually transforms the energy inside the geometry from specular (early
reflections) to diffuse (diffuse field) as shown in Figure 3.2, taken from (Kuttruff, 2009).
Figure 3.1 Principle of reflection. (Source:(TUCT User’s Manual))
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Investigation of Open-air theatre design
Figure 3.2: Transformation from specular to diffuse energy as reflections inside the system increase. Height of bars
is the total energy reflected; dark bars are the energy of specular reflections; white bars are the energy of the
diffuse reflections. (Source:(Kuttruff, 2009))
In CATT, the split process is performed on each octave band, hence a specular and a
diffuse beam is generated for each band. The difference between Algorithm 2 and 3 is how
the angles of the diffuse reflection component is calculated. In Algorithm 2, specular to
specular and specular to diffuse combinations are deterministic, while the remaining
reflections are random. The non-deterministic reflections are not split into specular and diffuse
components, but randomly reflected either specular or diffuse, depending on the scattering
coefficient. A scattering coefficient of 0.6 (i.e. 60%) means that the chances that the beam will
be diffusely reflected are 6 out of 10.
In Algorithm 3 the number of deterministic combinations is increased. The following
combinations become deterministic: Specular-Diffuse-Specular, Specular-Diffuse-Diffuse,
Diffuse-Specular, and Diffuse-Diffuse.
Due to the highly deterministic calculations, the results of both these algorithms do not vary
relevantly between runs, however this comes at a high computational cost.
3.1.2 Auto Scattering Function
CATT has an Auto-Scat function which makes it possible to assign scattering coefficients
depending on the dimensions of the surface. This is particularly useful to model surfaces such
as reflectors, colonnades, or other surfaces which will not generate specular reflections at
lower frequencies.
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Investigation of Open-air theatre design
3.1.3 Diffraction
Diffraction in CATT is calculated according to the secondary edge-source(SES) method
based on a discrete Huygens interpretation of Biot-Tolstoy (CATT-A, Whitepaper regarding
diffraction, 2016). In this method, the diffracting edges are considered as new sources. The
interpretation is based on the fact that: an impulse (delta function) diffracting from a wedge
will generate a spherical wave composed of the attenuated delta function and the weaker
contributions of the scattered wave (19). This process, illustrated in Figure 3.3 and taken from
(CATT-A, Whitepaper regarding diffraction, 2016), considers the polarity of the diffraction. The
Secondary Edge Source method extends the first order diffraction model to include second
order diffraction (Svensson, Fred and Vanderkooy 1999), hence making it possible to predict
diffraction effects between two separate edges (CATT-A, Whitepaper regarding diffraction,
2016). The Edge Source calculations includes interference effects due to the inherent timesignature and the polarity of diffraction. Hence low-pass filtering performed by a reflector panel
can be simulated. The implementation of diffraction also allows for simulation of the diffuse
effect of a discretize concave or convex geometry (CATT-A, Whitepaper regarding diffraction,
2016).
Figure 3.3: Secondary Edge Source principle. (Source:(CATT-A, Whitepaper regarding diffraction, 2016))
To calculate diffraction effects, CATT traces all possible paths source-edge-receiver. These
can be augmented by also including source-surface-edge-receiver (Sd) and source-edgesurface-receiver (dS) paths. If the SES calculation, edge to edge diffractions, are required, the
number of diffraction paths will increase exponentially. Thus, SES calculation can be
performed only for simple geometries.
Diffraction calculations in CATT do not propagate, but are calculated for a single receiver point,
hence higher order diffractions must be approximated as usual, by using scattering coefficient.
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Investigation of Open-air theatre design
3.1.4 Limitation of Diffraction calculations
The Edge Source method assumes hard edges, hence CATT checks whether the
average 125Hz octave band scattering coefficient between the two intersecting planes is <0.2
(20%), if so the edge will diffract. This check is also performed for the absorption coefficient
(TUCT User’s Manual).
The SES method discretises the edge to calculate diffraction, hence there is a sampling
frequency which limits the accuracy at higher frequencies. CATT has the option to increase
the sampling times 4, to obtain better estimations on the higher octaves, however, this is highly
computational expensive and only required for 8kHz and 16kHz estimations (CATT-A,
Whitepaper regarding diffraction, 2016).
The implementation of the surface to edge and the edge to surface diffraction paths have issue
in predicting interference effects if the surface is not rigid. Phase shift, due to an absorptive
surface, is generally unknown (CATT-A, Whitepaper regarding diffraction, 2016).
4. Methodology
4.1 Software used
CATT-Acoustic v9.1a was used to perform prediction and analysis. Theatres’ geometries were
built in SketchUp and exported to CATT-A geometry files using the SUCATT plugin. While,
Matlab was used for data analysis and calculations.
4.2 Validation
4.2.1 Computer Modelling Prediction Analysis
Before investigating open theatre designs, a modelling method is assessed. The aim of this is
not to achieve perfect results, but provide comparable results and to gain more insight into the
open-air condition when modelling. As the largest area of open-air theatres is the Cavea,
defining how to implement it in GA calculations is essential. The dense diffuse field generated
by the Cavea is due to the considerable number of reflections and diffractions occurring there.
Thus, geometry details and the outcomes of applying different calculation settings, including
the type of Algorithm and diffraction, are investigated. The validation focuses on matching the
T30 measurement values. Predictions are also compared to EDT and C50 measurements
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when these are available. Table 4.1 shows specifications of the theatres used for GA
modelling validation.
Table 4.1: Specifications of theatres used for the validation process
Theatre
Syracuse
Greek
Syracuse
Roman
Aspendos
diameter(m)/
height(m)
Orchestra
diameter(m)
Tiers
number/ slope
(°)
37
20°
Stage
wide(m)/
height(m)
23
8.6
85
11
24
143
26
30
64
20°25°
41
20
94
27
27
41
36°
49
27
Source
(Farnetani et al.
2006)
(Farnetani et al.
2006) / (M. Lisa,
2004)
(M. Lisa, 2004)
To assess the impact of different architectural features, the modelling investigation starts with
the scale model of the Greek version of the Syracuse theatre. The architectural features of
this model are: the Cavea, the Orchestra, and a small stage building with columns.
Measurements of this model were taken both with and without the stage building. The absence
of the stage building enables investigation of how the stage building interacts with the Cavea
within a GA model. The analysis proceeds with the Roman version of the Syracuse scale
model. For this theatre measurements were taken with and without the colonnaded aisle
surrounding the Cavea. Thus, making it possible to see how GA is simulating the interaction
between the Cavea and a back wall. The analysis continues with the measurements of the
Aspendos theatre. This is a typical Roman design were the Cavea is enclosed by the stage
building and the colonnaded aisle.
The absence of a roof leads to high energy losses, thus many rays are required to find
all possible paths and to avoid inconsistencies due to the absence of reflections in the decay
curve. Thus, 500000 rays were used for all models, this provided a smooth decay curve at
most receiver positions. Since ancient open-air theatres have a highly diffuse field and the
theatres’ convex shape must be discretized into straight lines, a fixed scattering coefficient of
0.1 was applied to all computer model surfaces unless otherwise stated. This will reduce
possible trapped rays.
The effect of using Algorithm 2 and Algorithm 3 in relation to the diffraction calculations
available on CATT is investigated. Comparative analysis of calculations is performed for every
theatre model, to assess whether increasing calculation detail improves the accuracy of
predictions. Comparison of calculation is performed for the following combinations: Algorithm
2 with 1st order diffraction (Alg2Diff); Algorithm 2 with 1st order diffraction, surface to edge
diffraction, and edge to surface reflection (Alg2DiffSddS); Algorithm 3 with 1st order diffraction
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(Alg3Diff); Algorithm 3 with 1st order diffraction, surface to edge diffraction, and edge to
surface reflection (Alg3DiffSddS).
4.2.2 Audience Analysis
When investigating new designs, predictions in the presence of an audience are of major
interest. Hence, new design models are also run with the presence of audience. Unfortunately,
no measurements of the Aspendos theatre with audience present are available, so this aspect
could not be validated. However, (13) reports simulations of the theatre with an audience
present. Thus, the presence of audience is simulated on the Aspendos theatre and compared
to (13) results. Audience properties are taken from CATT-A material library.
4.2.3 Syracuse theatre GA validation of models
Syracuse theatre models are 1:20 scale. The impulse responses of these models were
measured using a sine sweep from 500Hz to 95kHz, thus the measurements do not fully cover
the 4kHz octave band. The scale models were built out of wood, and painted to increase
reflection. However, the absorption coefficient of wood for the measurement of frequency
range is higher, even if painted, than that of stone. Thus, for the computer models to be
validated an absorption coefficient of 10% is assigned to all surfaces and for each octave
band. (Farnetani et al. 2006) used two source positions during the measurements: one slightly
off centre of the Orchestra and one on the rear part of the Orchestra, near the stage area. For
CATT-A computer simulations, the latter was chosen. Thirty-six receivers were used for
measurements of the Greek version and 42 for the Roman. These were distributed in the
Cavea to cover all possible positions. For the CATT-A models, 10 receiver positions were
used for both versions of the theatre. These receivers were distributed on one half of the
Cavea since the models are symmetrical with respect to the source position.
The CATT-A model of the Greek version of the theatre was built using information in
Table 1 and photos shown in (Farnetani et al. 2006), Fig. 2.4. Two versions of the Greek
theatre are tested, one simulating the Cavea as a flat surface and one modelling the Cavea in
detail. Figure 4.1 shows the simple model, and Figure 4.2 shows the model with detailed
Cavea.
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Figure 4.1: Syracuse Greek model with simple Cavea. Source position red, receivers positions blue.
Figure 4.2: Syracuse Greek model with detailed Cavea. Source position red, receivers positions blue.
The model of the Roman version of the theatre, Figure 4.3, was built from the CAD model
kindly provided by Jens Holger Rindel. The geometries of the CAD model are adapted to
integrate them in CATT-A. Figure 2.5 shows the photo taken from (Farnetani et al. 2006) of
the scale model.
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Figure 4.3: Syracuse Roman model. Source position red, receivers positions blue.
4.2.4 Aspendos theatre GA validation of model
The SketchUp 3D model, from which the geometrical model was derived, is built as a replica
of the 3D CAD model kindly provided by Jens Holger Rindel. The CAD model was designed
based on plans, pictures and information provided by archaeologists and architects involved
in the ERATO project (M. Lisa, 2004). Measurements of the theatre are reported by (M. Lisa,
2004). These were taken in the presence of a wooden stage building used for performances.
Thus, the wooden stage building has been included in the CATT model. The dimensions of
the stage have been inferred from pictures and images. Measurements were performed using
a dodecahedron speaker as source and both an omnidirectional and a Figure of 8 microphone.
Seven receiver positions and one source position were used to perform the measurements.
Measurement receiver positions are shown in Figure 4.4 (image take from (M. Lisa, 2004)).
The source and receiver positions inside the CATT model have been estimated from Figure
4.3. Architectural features in the CATT model have been simplified with respect to the CAD
model provided. Receiver positions are symmetrical with respect to the Odeon model. The
CATT-A model is shown in Figure 4.5, image exported form CATT-A.
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Figure 4.4: Aspendos Theatre Odeon model. Source position red, receivers positions blue. (Source: (M. Lisa,
2004))
Figure 4.5: Aspendos Theatre CATT-A model. Source position red, receivers positions blue.
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4.3 Design Investigation
4.3.1 Architectural features investigation
Once a modelling technique had been assessed, the influence of modern architectural
features could be investigated. The architectural features investigated sought to determine the
effect on the acoustic of reflectors: behind the stage area, overhanging, and behind the
audience. These designs were generated taking into consideration audience sight view. To
assess the influence of reflectors a simple model consisting of the Cavea, Orchestra, and
stage ground is built. This model enables direct comparison between the presence and
absence of reflectors. The absorption coefficient for all surfaces of New designs models is
assumed to be concrete. Values are taken from CATT-A material library.
4.3.2 Core Cavea model
The core Cavea had similar specifications to the theatres used for validation and GA modelling
analysis. The model includes: the Cavea, the Orchestra, and a raised stage. Details of the
geometries used for the core Cavea are shown in Table 4.2. Table 4.2 reports also the
specifications of the balcony design version, since this is the only version for which the
dimensions differ due to the presence of balconies. Twenty receivers are used for all New
models except for the Balcony model. These receivers are placed in four radial lines with
respect to the centre of the Cavea and each has five receivers. In all models the source was
placed at the centre of the raised stage at a height of 1.8 meters and 2 meters from the raised
stage front (11.5 meters from the centre of the Orchestra). Figure 4.6 shows the core Cavea
with receivers and source positions.
Table 4.2: Specifications of Core Cavea model and Balcony
Theatre
Model
Core C.
Balcony
Theatre
dimensions
Diameter Height
(m)
(m)
104.3
18
75.5
23
Orchestra
Tiers
Diameter
(m)
25
25
Num. Slope
(°)
45
25°
33
25°
Raised Stage
dimensions
Width Depth Height
(m)
(m)
(m)
41
12
1.5
41
12
1.5
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Figure 4.6: Core Cavea model. Source position red, receivers positions blue.
4.3.3 Stage reflectors model
A 3x3 array of reflectors are suspended over the stage. These are designed to improve
reflections over the Cavea, while avoiding 1st order reflections arriving with a delay greater
than 50ms. The lower reflectors are tilted by 30 degrees, the middle section of reflectors is
tilted by 45 degrees, and the top reflectors are tilted by 60 degrees. All tilts are with respect to
the normal of the stage. Side reflectors are slightly angled to improve specular reflections to
the sides of the Cavea. The Stage Reflectors model is shown in Figure 4.7.
Figure 4.7: Stage Reflector model.
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4.3.4 Clouds model
A model including reflectors overhanging the Cavea is developed. Just as for the stage
reflectors, care was taken to avoid strong specular reflections arriving after 50ms. The
reflectors are placed, in three layers, hovering over the seating area. The first layer consists
of one large reflector at a height of 18-meters. The second layer is a single reflector hovering
at 22 meters. The third layer is formed by 10 reflectors, of different dimensions, 26 meters
high. All heights of the reflectors are relative to the Orchestra level. Each of the layers
addresses a different area of the Cavea. Avoiding obstructing the sight view of the audience,
while minimizing the delay of strong specular reflections, pose a challenge for design. Figure
4.8 show the Clouds model.
Figure 4.8: Clouds model.
4.3.4 Balcony model
The balcony is built at the 19th row of the core Cavea model, which is slightly more than one
fourth of the Core Cavea model diameter. The backwall, formed by the balcony, is 2.4 meters
high, while the balcony extrudes forwards inside the theatre by 4.9 meters. The balcony
reflective surface is 64 degrees tilted with respect to the backwall, redirecting the specular
reflections to the seating area under it. Implementing the balcony splits the Cavea into two
levels, a lower level and higher one. The reflective surfaces composing the balcony are
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replicated on the higher level. Figure 4.9 show Balcony model image exported from SketchUp.
Figure 4.10, exported from CATT-A, shows the positions of receivers and source.
Figure 4.9: Balcony model.
Figure 4.10: Balcony model. Source position red, receivers positions blue.
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5. Validation – Results and Analysis
5.1 Modelling the Cavea in GA
5.1.1 Simple Geometry
Figure 5.1, shows the predictions of T30 octave bands of the Syracuse Greek model
without stage and with the Cavea modelled as a simple flat surface. The predictions are
marked in red, while the scale model measurements, including measurements standard
deviations, are marked in black. Predictions were performed using algorithm 2 and without
calculating diffraction, since there are no geometries present which would generate
diffractions. The more deterministic calculations of algorithm 3 lead to overestimation of
predictions. To reach the measured values, an average scattering coefficient of 0.7 is assigned
to all octave bands except 250Hz and 500Hz of the Cavea surfaces. To reduce the T30 value
of the 250Hz and 500Hz by approximately 20% with respect to other octave bands, a
scattering coefficient of 0.05 is used.
Figure 5.1: T30 octave band predictions of the Syracuse Greek model (no stage) with Cavea modelled has a
simple surface. Predictions are red, measurements black.
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For this geometry, increasing the scattering coefficient increases reverberation time.
Deviations between receiver positions are noticeable. However, these are smaller than the
ones measured. This could be due to the smaller number of receiver positions used in the GA
model.
The scattering coefficients for the Cavea, obtained with the model without the stage building,
are transposed to the version of the theatre with the stage building. Figure 5.2 shows
predictions (red) and predictions including diffraction (blue) against measurements (black).
Calculations are performed using same settings of the model without the stage building except
for 1st order, Sd, and dS diffraction calculations. Predictions are similar to measurements only
for the lower 125Hz octave band. All other octave bands show an underestimation average of
around 40% with respect to measurements. Including 1st order diffraction does not improve
predictions. Diffraction occurs only on the stage building geometries. Comparing Fig. 5.1 and
Fig.5.2 it is noticeable that predictions are mildly affected by the presence of the stage, while
measurements show a 50% increase. The dip at 250Hz and 500Hz has disappeared from
measurements, however, this is not reflected in the predictions. Attempts have been made to
reach measured values. However, modifying scattering coefficients, absorption coefficients,
the algorithm used, and diffraction calculations, did not increase predictions to the T30
measured levels.
Figure 5.2: T30 octave band predictions of the Syracuse Greek model including the stage building with Cavea
modelled as a simple surface. In red are predictions without 1st order diffraction, in blue are predictions including
1st order diffraction, in black are measured values.
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5.1.2 Detailed Geometry
Representing the Cavea in detail drastically increases the number of surfaces of the
computer model. In the Syracuse Greek version without stage, the number of surfaces
increased from 59 to 810 when modelling the Cavea in detail. This, drastically increased
calculation times.
With the detailed modelling of the Cavea, 1st order diffractions occurring on the tiers can be
calculated. Figure 5.3 shows the result of T30 octave band predictions for the Syracuse Greek
version model, without stage and detailed Cavea, against measurements. Predictions using
algorithm 2 with different diffraction calculation are presented: in red are predictions without
diffraction, in green are predictions with 1st order diffraction, in blue are predictions with 1st
order diffraction and Sd, in cyan are predictions using 1st order diffraction, Sd, and dS.
Diffraction calculations reduce average T30 predictions bringing them nearer to measured
values. The effect is most pronounced at lower octave bands. The inclusion of Sd and dS
diffractions increases reduction and reduces variation between receiver positions. However,
predictions are still high since scattering coefficient is still required to represent the diffuse
field.
Figure 5.3: T30 octave band predictions of Syracuse Greek model (no stage) and detailed Cavea using different
diffraction settings.
Predictions reported so far apply a scattering coefficient and an absorption coefficient
of 0.1 for all octave bands. Increasing the scattering or absorption coefficient above 0.15 at
125Hz octave band in the Cavea details will stop CATT-A calculating diffraction for these
edges (TUCT User’s Manual). This can be avoided using the auto scattering function of CATTA (TUCT User’s Manual). Figure 5.4 reports predictions of the same model used for Fig. 5.3,
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with the autoscatt function applied to all the surfaces of the model except ground. The colour
scheme is the same as in Fig 5.3. As expected, the scattering based on surface dimension
has a greater effect at lower frequencies due to the small dimension of the tiers. Predictions
without diffraction lead to higher overestimation. The major difference between diffraction
calculations is between applying 1st order diffraction and 1st order diffraction plus Sd and dS.
The 125Hz octave band predictions, when applying diffraction calculations, have a satisfactory
level of agreement with the measured values. The agreement between predictions and
measured values decreases as the scattering coefficient is reduced. It should be noted that,
at 125Hz, the scattering coefficient applied by autoscatt to the seat tiers, ranges between 0.92
and 0.98. At 250Hz it ranges between 0.46 and 0.98. While at 500Hz the range is between
0.25 and 0.6. The max scattering coefficient reached at 1kHz is 0.25. The fixed 0.1 scattering
coefficient assigned to all surface at all octave bands is to be added to the scattering
coefficients added by autoscatt. Note that if the scattering coefficient increases above 1 it is
truncated to 1.
Figure 5.4: T30 octave band predictions of Syracuse Greek model (no stage) with autoscatt using different
diffraction settings.
Predictions so far have been performed using algorithm 2. Figure 5.5 reports the
comparison between different algorithms, 2 and 3, using different diffraction settings. The
colour scheme is red for Alg2Diff, green for Alg2Diff+Sd+dS, blue for Alg3Diff, and cyan for
Alg3Diff+Sd+dS. The calculations using Algorithm 3, in blue and cyan, lead to strong
overestimations for octave bands where the scattering coefficient assigned by the autoscatt
function is low.
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Figure 5.5: T30 octave band predictions of Syracuse Greek model (no stage) with autoscatt using different
diffraction settings and algorithms.
The seating area is handled in room acoustic GA models by applying a scattering
coefficient that increases from lower to higher octaves. The scattering coefficient in Table 5.1
is applied to the models. Predictions using these values are reported in Figure 5.6. Predictions
using algorithm 3 are still generating a slight overestimation of parameters. Algorithm 2 with
1st order diffraction (red) is the only prediction which, on average, “follows” the dip at 250Hz
and 500Hz octave bands. This is also the calculation setting which has the largest deviation
of T30 between receiver positions. Table 5.2 shows the average absorption and scattering
coefficient of the CATT-A model. The average value for the entire model is very high.
Table 5.1: Scattering coefficient of Cavea
Cavea
Parameters
Abs. Coeff.
Scat.Coeff.
125 Hz
0.1
0.15
Cavea Modelling for Scale models
250 Hz
500 Hz
1k Hz
0.1
0.3
0.1
0.4
0.1
0.6
2k Hz
4k Hz
0.1
0.7
0.1
0.7
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Figure 5.6: T30 octave band predictions of Syracuse Greek model (no stage) with autoscatt and scattering
coefficient.
Table 5.2: Average Abs. and Scat. Coefficients of the Syracuse Greek model without stage
Theatre
Parameters
Avg.Abs. %
Avg.Scat. %
Syracuse Greek Theatre without Stage Building
125 Hz
250 Hz
500 Hz
1k Hz
2k Hz
0.10
0.75
0.10
0.70
0.10
0.64
0.10
0.62
0.10
0.63
4k Hz
0.10
0.6
5.2 Theatre models validation
5.2.1 Syracuse Theatre Greek model
The stage building is added to the previous model and auto-scattering function is applied to
stage building surfaces. Columns, stage front border and the small wall in front of the stage
are the surfaces mainly affected by the auto-scattering function. All other surfaces do not vary
much. The average parameters of the theatre are reported in Table 5.3. The average
scattering coefficient has decreased by around 0.1 for most octave bands with respect to the
model without a stage building. Prediction results using different calculation settings are
reported in Figure 5.8, together with measured values from (Farnetani et al. 2006). Predictions
using only 1st order diffraction (red and blue) exhibit lower reduction of reverberance with
respect to the more complex diffraction calculations (green and cyan), and, on average, are
more similar to the measured values.
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Table 5.3: Average Abs. and Scat. Coefficients of the Syracuse Greek model with stage
Theatre
Parameters
Avg.Abs. %
Avg.Scat. %
125 Hz
0.10
0.67
Syracuse Greek Theatre
250 Hz
500 Hz
1k Hz
0.10
0.61
0.10
0.54
0.10
0.52
2k Hz
4k Hz
0.10
0.53
0.10
0.51
Figure 5.8: T30 octave band predictions of Syracuse Greek model.
For this model C50 measurements are reported by (Farnetani et al. 2006). Figure 5.9
shows the comparison between C50 measurements and predictions at different octave bands
for different calculation settings: Alg2Diff (blue), Alg2Diff+Sd+dS (cyan), Alg3Diff (green),
Alg3Diff+Sd+dS (orange), measurements (yellow). Using algorithm 2 and 1st order diffraction
offers more accurate predictions of C50 at most octave bands. The noticeable dip at 250Hz
on measurements is due to a comb filtering effect visible on the impulse response of
measurements (Farnetani et al. 2006). Predictions do not show this effect; thus, the reflection
interference pattern is not correctly represented. Lower predictions of more complex
diffraction, Sd and dS, are occurring although T30 predictions are lower. Thus, the interference
pattern of generated by Sd and dS is likely to be exaggerated. Improvements in predictions
may be achieved if it was possible to calculate 2nd order diffractions.
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Figure 5.9: C50 octave band predictions of Syracuse Greek model. Alg2Diff (blue), Alg2Diff+Sd+dS (cyan), Alg3Diff
(green), Alg3Diff+Sd+dS (orange), measurements (yellow).
5.2.2 Syracuse Theatre Roman model
The Syracuse Theatre Roman version of the scale model, was measured by (Farnetani et al.
2006) with and without the colonnaded aisle surround the Cavea. Table 5.4 shows the average
scattering coefficient of the Syracuse Roman version model without and with the colonnaded
aisle. T30 measurements and predictions using different calculation settings are reported in
Figure 5.10. Algorithm 3, as in the Greek version of the model, leads to overestimation of T30.
However, for this model, the deviation between receivers’ positions using algorithm 3 is
noticeably larger than in the Greek models. Algorithm 2 predictions are similar to
measurements, however, for this model, deviation from measurements is larger than in the
Greek model. This is likely due to the stage building. Scattering coefficient is assigned to the
stage building surfaces using the autoscat function, thus also preserving diffraction from the
stage details. Scattering of the proscenium is increased by subdividing it into smaller surfaces
to increasing the scattering coefficient at 125Hz and 250Hz, however, this also increases the
scattering coefficient to 0.25 at 500Hz. The model was tested using different conditions and
the 0.25 at 500Hz increase is not sufficient, in itself, to justify the underestimation when using
algorithm 2. The overestimation of algorithm 2 at 4kHz cannot be brought down within 100ms
even when increasing the scattering coefficient of the entire proscenium to 0.7. This suggests
that the absorption coefficient of the material used to build the scale model should be higher
than the assumed 0.1.
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Table 5.3: Average Scat. Coefficients of the Syracuse Roman model.
Syracuse Roman Theatre model with and without colonnaded aisle
Theatre
125 Hz
250 Hz
500 Hz
1k Hz
2k Hz
Parameters
Without c. aisle
0.72
0.65
0.57
0.53
0.55
Avg. Scat.
With c. aisle
0.68
0.6
0.52
0.48
0.49
Avg. Scat.
4k Hz
0.52
0.48
Figure 5.10: T30 octave band predictions of Syracuse Roman model (no colonnade aisle).
Figure 5.11 reports the T30 measurement and predictions using different calculation
settings of the Syracuse Theatre Roman version. Predictions are more similar to the measured
value with respect to the model without the colonnaded aisle. Algorithm 2 is still showing
greater similarity to measured values on average. Overestimation 2kHz and 4kHz is as high,
if not higher, with respect to the version without the colonnaded aisle. Measurements between
the two versions show a substantial increase of T30 at the lower 125Hz and 250Hz octave
bands. The computer model does not show such a substantial increase at the lower octave
bands. Attempts have been made to simulate this effect by altering the scattering coefficient
of the wall surrounding the Cavea, however this could not be achieved. The effect is probably
linked to the interaction between the Cavea and the surrounding wall. C50 values are reported
by (Farnetani et al. 2006) for this version of the theatre. Figure 5.12 shows the comparison
between C50 measurements and predictions using different calculation settings. C50
measured values are considerably lower than in the Syracuse Greek version, except for the
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4kHz octave band. There is a small dip at 250Hz in measurements. Predictions do not show
the sharp increase in the 2kHz and 4kHz, however this is probably linked to the T30
overprediction for these octave bands. Predictions using algorithm 2 are more similar to
measured values.
Figure 5.11: T30 octave band predictions of Syracuse Roman model.
Figure 5.12: C50 octave band predictions of Syracuse Roman model.
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5.2.3 Aspendos Theatre model
The Aspendos Theatre model, built from CAD drawings, uses the absorption coefficients
and scattering coefficients reported in (M. Lisa, 2004) except for the Cavea scattering
coefficients which use those reported in Table 5.1. These measurements are reported in Table
5.4. T30 measurements and prediction results are reported in Figure 5.13. T30 predictions are
overestimated when using algorithm 3 while they show a good match when using algorithm 2.
Differences in the diffraction calculations are small, the main one being the deviation of
predicted values between receivers’ positions. (M. Lisa, 2004) also reports EDT
measurements, these are compared against predictions in Figure 5.14. EDT values are more
similar using algorithm 2. Differences in diffraction calculation settings do not seem to play a
major role. EDT between receivers’ positions varies more than in T30 measurements.
Algorithm 2 provides good agreement with measured results in this model. As a further
comparison, Table 5, shows the average JND error of T30, G, C80, and absolute error of STI
between the 500Hz and 1kHz octave bands. These errors are calculated against values
obtained by the computer model performed by (13). JND errors are calculated for the mid
frequencies, 500Hz and 1kHz, as for (M. Lisa, 2004) and using (BS EN ISO 3382-1:2009)
guidelines. For algorithm 2 errors are below one unit error, while algorithm 3 exhibits
perceivable differences. The CATT-A model and the Odeon model performed by (13) possess
a satisfactory level of agreement when using similar absorption and scattering coefficients.
Table 5.4: Average Scat. Coefficients of the Aspendos model.
Theatre
Parameters
Avg.Abs. C. %
Avg.Scat.C. %
125 Hz
0.078
0.66
Aspendos Theatre
250 Hz
500 Hz
0.13
0.56
0.13
0.48
1k Hz
2k Hz
4k Hz
0.13
0.44
0.13
0.44
0.13
0.44
Table 5.5: Average Scat. Coefficients of the Aspendos model.
Calculation
Alg2_Diff
Alg2_Diff_SddS
Alg3_Diff
Alg3_Diff_SddS
T30 JND
error
0.7
0.4
2.5
2.8
G JND
error
0.5
0.4
0.7
0.6
C80 JND
error
0
0
0.9
1
STI abs.
error
0
0
0.1
0.1
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Figure 5.13: T30 octave band predictions of Aspendos model.
Figure 5.14: EDT octave band predictions of Aspendos model.
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5.2.3 Aspendos Theatre model with Audience
For the Aspendos model, simulation is performed with an audience present, even
though are no measurements currently available for the Aspendos theatre with an audience
present. The model is run using absorption coefficients found in the CATT-A library. Average
coefficients of the theatre are reported in Table 5.6. The absorption coefficient has increased,
giving the T30 results presented in Figure 5.15. These were calculated using algorithm 2 with
1st order diffraction. As expected, predictions show reduction starting from 250Hz increasing
for higher octave bands. Parameters are compared to results reported by (13) in Table 5.7.
These are compared as JND error for T30, G, C80, and absolute error for STI. Only the 500Hz
and 1kHz are used to compare results as specified by (BS EN ISO 3382-1:2009). The only
noticeable error between the CATT-A computer model and (13) the Odeon computer model
is for C80, where the CATT-A model underestimates of 1dB respect to (13) the Odeon model.
Table 5.6: JND and absolute error between CATT-A model and (M. Lisa, 2004) Odeon model.
Theatre
Parameters
Avg.Abs. Coeff.
%
Avg.Scat.Coeff.
%
Aspendos Theatre with Audience average coefficients
125 Hz
250 Hz
500 Hz
1k Hz
2k Hz
4k Hz
0.1
0.19
0.25
0.33
0.37
0.35
0.66
0.56
0.48
0.44
0.44
0.44
Figure 5.15: EDT octave band predictions of Aspendos model.
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Table 5.7: JND and absolute error between CATT-A model and (M. Lisa, 2004) Odeon model with audience
present.
Calculation
Alg2_Diff
T30 JND
error
0.25
G JND
error
0.3
C80 JND
error
1.1
STI abs.
error
0
5.3 Discussion of Validation
The importance of how to represent the Cavea in GA models has been explored by (Van
Loenen, et al. 2016) and (M. Lisa, 2004). (Van Loenen, et al. 2016) concludes that for the
theatre of Epidaurus, which is similar in shape to the Syracuse Greek without stage scale
model version, representing the Cavea as a flat surface leads to more accurate predictions
closer to measurements. T30 predictions for the Syracuse Greek version without stage are
more accurate when the Cavea is simulated as a simple flat surface. However, once the stage
building is inserted into the model, modelling the Cavea as a simple flat surface, makes it
impossible to bring results closer to measured values: only modelling the Cavea in detail can
do so. Thus, if a reflective surface is added in front of the Cavea, tier details must be added to
increase the interaction between the two architectural features.
The issue is that modelling the Cavea in detail, breaks the validity of GA for frequencies
whose wavelength is comparable to surface dimensions. Thus, GA modelling requirements
are not satisfied for octave bands lower than 2kHz. However, good estimations of T30 and
C50 have been achieved in the Syracuse Greek model even at lower octave bands.
Modelling the Cavea in detail, to increase the interaction with the stage building, is likely
to undermine interactions between the Cavea and the backwall surrounding the Cavea.
Predictions of the Syracuse Roman version with and without the colonnaded aisle surrounding
the Cavea show that the increase at lower frequencies from the model without and the model
with the colonnaded aisle could not be fully achieved. The reason lies in the fact that lower
frequencies would see the Cavea as a flat surface rather than as a series of reflective surfaces.
Thus, more low frequency energy would be reflected to the colonnaded aisle and reflected
back inside the theatre, rather than be dispersed by the extremely high scattering required.
However, this problem is reduced by the presence of the large stage building which interacts
with the colonnaded aisle. Unfortunately, no scale model was measured with only the
colonnaded aisle. Thus, the magnitude of this issue could not be assessed.
The practice of applying a scattering coefficient based on surface dimensions improves
predictions at lower frequencies. Treating all surfaces, except the ground, as barriers, hence
reflections from these are weaker at low frequencies, improves the energy balance inside
model.
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Syracuse Greek and Syracuse Roman show a decrease in T30 energy when Sd and dS
are calculated, while for Aspendos just a neglectable increase occurs. The reason for this
difference lies in the larger amount of reflection generated in the more enclosed design of the
Aspendos theatre which mask diffraction effects.
The C50 dip in the 250Hz octave band in the Syracuse Greek model, noticeable in the
measurements and due to comb filtering (Farnetani et al. 2006) is likely related to the comb
filtering below 530Hz showed by (Declercq, and. Dekeyser, 2007). Hence originates also from
reflections occurring on the Orchestra. It would seem using Sd calculations would be
appropriate. However. The predicted dip in C50 at 250Hz is too small to be considered
relevant. And. The use of more complex diffraction, Sd and dS calculations when detail
modelling the Cavea generates excessive destructive interference effects affecting C50
predictions. Thus, partially accurate diffraction of the Cavea reduces prediction accuracy. 2nd
order diffraction calculations, if possible, may improve predictions.
The presence of an audience could not be validated due to the lack of measurements
with audience present. However, if absorption and scattering coefficients developed for room
acoustic GA models are valid also for open-air theatre conditions, it is likely that simulating the
audience in an open-air theatre would lead to more accurate predictions. Since. The core
issue in open-air theatre GA modelling remains how to simulate the complex acoustic field
generated by the Cavea and by its interaction with other architectural features in the theatre.
Thus, by adding an audience, these effects would be partially masked. However, there are still
likely to be problems at the lower 125Hz octave band because the audience will have a smaller
impact at these frequencies.
6. New Designs – Results and Analysis
From the validation process it was observed that:
1)
1st order diffraction plays a significant role in predicting the acoustics of open
air theatres. Extending diffraction calculations to Sd and dS does not improve
predictions and could have an adverse effect on T30 and C50 predictions.
2)
Applying scattering coefficients based on the dimension of surfaces improves
energy balance of predictions at lower frequencies.
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Investigation of Open-air theatre design
Correct response at higher frequency is achieved by applying to the Cavea a
scattering coefficient similar to the scattering coefficient of the seating area.
4)
Using the more deterministic algorithm, algorithm 3, leads to overestimation of
T30 and underestimation of C50.
Thus, New Design models are investigated using algorithm 2, 1st order diffraction,
applying autoscatt function to all surfaces except ground, and using the scattering coefficients
reported in Table 5.1 for the Cavea.
6.1 Core Cavea model
The Core Cavea is built as specified in Methodology. T30 octave bands and C50
values per receiver predictions are reported respectively in Figure 6.1a and Figure 6.1b.
Receivers’ positions can be observed on Fig. 4.6. The predictions without audience are in
blue, those with audience, in red. Predictions of the empty theatre show an average T30 of
0.8 seconds for the higher octave bands which gradually decrease, in the lower octave bands,
to 0.7 seconds. The audience T30 reduction effect increases gradually at frequencies from
250Hz, as noticed for the Aspendos model. C50 when empty is very similar to the Syracuse
Greek model using Alg2Diff calculation settings. The presence of the audience noticeably
increases C50 for most receivers except for receivers 8, 12, and 14. The C50 value of receiver
13
is
negative,
around
-10dB.
Figure 6.1a: T30 octave band Core Cavea model predictions with (red) and without (blue) audience.
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Figure 6.1b: C50 predictions at receivers’ position of Core Cavea model. With audience (red) and without audience
(blue).
6.2 Stage Reflectors model
This model is designed to assess reflections from the stage area. The model consists of
the Core Cavea model with an 3x3 array of reflectors over the stage. Reflectors are tilted to
improve distribution over the seating area. Figure 6.2a shows T30 predictions with (red) and
without (blue) the reflectors in place. Reverberation time has increased at all octave bands,
deviation between receiver’s positions has not improved. Similarly, C50, not reported, does
not show improvements except for a few positions. Figure 6.2b shows T30 octave band
predictions with (red) and without (blue) stage reflectors with audience present. T30 deviation
from median value has improved at octaves above 250Hz, however, there are still some
positions which are suffering from variations in the T30 spectrum. Figure 6.2c shows T30
parameters against distance of receivers positioned at different angles respect to the centre
of the Cavea. Prediction of the Core Cavea model with audience are in blue, while predictions
of the Stage Reflector model with audience are in red. Overall, T30 values have improved in
the Cavea, except for receiver position 13 as can be seen on the 67.7-degree angle.
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Figure 6.2a: T30 octave band predictions of Core Cavea model (blue) and Stage Reflectors model (red).
Figure 6.2b: T30 octave band predictions with audience present of Core Cavea model (blue) and Stage Reflectors
model (red).
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Figure 6.2c: T30 predictions per distance at different angles with audience present of Core Cavea model (blue)
and Stage Reflectors model (red).
6.3 Clouds model
The scope of this model is to assess reflections from above the seating area. This model
consists of 12 reflectors distributed, in layers, at three different heights. Each layer addresses
different areas of the Cavea. Reflectors are placed to avoid obstructing audience view and to
avoid strong specular reflections occurring after 50ms. Figure 6.3a shows T30 octave band
values predicted for the Core Cavea model (blue) and the Clouds model (red). Cloud reflectors
increase T30. This increase is greater for the lower octaves. T30 distribution around the Cavea
has not improved. Deviations of T30 octave band values are present at certain locations.
These deviations shift to different octave bands and receiver positions between runs. Figure
6.3b shows the T30 value with distance for receivers at different angle respect to the centre
of the Orchestra. T30 increased over the entire Cavea, but with a more pronounced effect at
receiver 6 at 36.7-degree. This localized T30 increase at a single receiver varies between
runs. Predictions simulating the audience are shown in Figure 6.3c, in blue are the predictions
without the clouds, in red predictions with clouds. Adding reflective surfaces has increased
T30 values at all positions. These are more stable with distance for the central receivers of
the Cavea, while more variable for the side ones. Receiver 15, at 90-degrees, shows a T30
increase even larger than in the absence of audience. This localized increase varies between
runs.
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Figure 6.3a: T30 octave band predictions of Core Cavea model (blue) and Clouds model (red).
Figure 6.3b: T30 predictions per distance at different angles of Core Cavea model (blue) and Clouds model (red).
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Figure 6.3c: T30 predictions per distance at different angles with audience present of Core Cavea model (blue)
and Clouds model (red).
Figure 6.3d shows the T30 octave band predictions of the Cloud model (red) without
audience removing the two lower layers. Variations between receiver positions has not varied
substantially. Excursions in T30 octave bands predictions have reduced. This remains
consistent between runs. T30 values increase more at lower octave bands than previously.
Figure 6.3d: T30 octave band predictions of Core Cavea model (blue) and Clouds model (red) without lower clouds
layer.
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6.4 Balcony model
To accommodate balconies the dimensions of the theatre must change. For comparison,
the height of the Cavea excluding the reflectors surrounding the Cavea is identical to the Core
Cavea. The Balcony model Cavea diameter is 75.5 meters, while the Core Cavea model is
104.3 meters. Number of tiers of the Cavea has been reduced from 45 in the Core Cavea to
33 in the Balcony model.
Figure 6.4a showed the T30 octave band predictions of the Core Cavea (blue) and the
Balcony model (red). The T30 values of the Balcony model have increased at all octave bands.
The increase is more pronounced at lower frequencies. There is no noticeable improvement
in T30 octave band distribution, and few values present large deviations. Figure 6.4b shows
the T30 against distance for each radial line. The curve generated by the polynomial curve
fitting suggests exaggerated trends. This visualization error is due the different size of the
models, thus different distances source to receivers. Figure 6.1b shows that T30 is not varying
excessively between the two levels of the Cavea. Receivers positioned in the higher part of
the Cavea are the two most distant on each radial line. Also, the T30 level for positions located
under the balcony, the third receiver on each radial line, does not suffer from lower T30 levels.
Figure 6.4c shows C50 values predicted at each receiver, the Core Cavea values are in blue,
the Balcony model, in red. Clarity has strongly decreased at specific positions. Clarity
reduction occurs strongly at the receivers positioned in the higher part of the Cavea: receivers
3,4, 8, and 18, but not at receivers 14 and 19 at the more distant point of the 67.7 and 90degree radial line. C50 has increased for positions 14 and 19. C50 reduction also occurs
strongly at positions on the 8th tier of the Cavea, these correspond to receivers 1, 6, 11, and
16. Receivers under the balcony (receivers 2, 7, 12, 17) do not suffer from their location
compared to the Core Cavea model and still show a C50 value lower than 0.
Figure 6.4a: T30 octave band predictions of Core Cavea model (blue) and Balcony model (red).
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Figure 6.4b: T30 predictions per distance at different angles with audience present of Core Cavea model (blue)
and Balcony model (red).
Figure 6.4c: C50 predictions at receivers’ position of Core Cavea model (blue) and Balcony model (red).
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Figure 6.4d shows C50 values at each receiver position in the presence of an audience.
The Core Cavea model predictions are in blue, the Balcony model, in red. Large drops in C50
have reduced, and most receivers in proximity of large reflective surfaces are benefiting from
higher C50 values. These are receivers 7, 9, 12, 14, 17, and 19.
Figure 6.4d: C50 predictions at receivers’ position with audience present of Core Cavea model (blue) and Balcony
model (red).
6.5 Discussion of New Designs
T30 predictions of the Core Cavea model are likely to be overestimates as the geometry
of the Cavea is detailed. T30 values without audience are 0.4 seconds lower than the
measurements of the theatre of Epidaurus as reported by (Chistian Gade and Angelakis,
2006). This is consistent with the smaller diameter, around 12% smaller than Epidaurus, and
reduced slope of the cavea, 1.6 degrees less than in Epidaurus. However, even considering
a possible overestimation of the Core Cavea model, the difference seems exaggerated.
Epidaurus exhibits a more stable spectral balance, with respect to the Core Cavea, with
a small dip at 250Hz. The Core Cavea model T30 reduction in the lower octaves is similar to
measurements of the Syracuse Greek scale model without stage, except for the 125Hz octave
band. Source positions are likely to be the reason for inconsistencies, since the measurements
of both Epidaurus and the scale model differ from the Core Cavea model. T30 octave band
variation depending on source position can been seen in measurements reported in (Chistian
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Gade and Angelakis, 2006). This opens a new requirement in the investigation of new designs
for open air theatres: uniformity of acoustic which depends on source position.
At receiver position 13, T30 is low when compared with receivers 12 and 14. Receiver
13’s parameters did not vary between runs and, removing diffraction calculations, did not
change this prediction. Even when the position of receiver 13 is changed in the Balcony model,
both C50 and T30 exhibit large deviation of values with respect to similar positions. This
suggests that the effect is occurring at a specific area along the 67.7degree radial line. Looking
at the echogram at position 13 Figure 6.5, it clear that a vast amount of energy is arriving at
the receiver positioned within 150ms, suggesting this is subject to a focusing effect, due to the
concave shape of the Cavea. This effect occurs at all frequency bands despite the high nonuniform scattering coefficient of the Cavea. Focusing effects in the seating area have not been
reported in the literature analysed, suggesting that this is just a GA modelling error.
The Stage Reflectors model octave band T30 values show higher values in the 500Hz
and 1kHz octave bands than do the measurements of the Syracuse Roman scale model
without the colonnade aisle. The implementation of stage reflectors improves T30 distribution
when an audience is present, however, spectral distribution is still non-uniform for some
positions.
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The impact of stage reflectors is likely to differ when a source such as a speaker is
present, and, considering that open-air theatres possess fewer reflective surfaces, it is likely
that directivity of the source will play a major role in defining the design of an open-air theatre.
The Clouds model exhibits run to run variations as high T30 predictions at different
locations. This is probably connected to rays being trapped between the layers of reflectors,
since removing the two central layers eliminates high, localized T30 predictions. These
reflectors are modelled as highly reflective concrete surfaces, an unlikely condition. The use
of cloud reflectors, made from other weather resistant materials, such as plastic, should be
investigated.
The Clouds model, with only the top layer, and the balcony model show noticeable
increases in the T30 value in the 125Hz and 250Hz octave bands. Large increases in T30
values in the lower octave bands can also be noticed among the measurements of the
Syracuse Roman scale model with and without the colonnaded aisle. Thus, reflective surfaces
placed in proximity to the higher part of the Cavea increase T30 in the lower octave bands.
The Balcony model shows C50 improvements for receivers placed in proximity to
reflecting surfaces. Thus, a design which includes reflective surfaces distributed inside the
seating area, like the Vineyard shape, is likely to improve the acoustics of the theatre when
an audience is present.
Receivers on the higher section of the Cavea suffer from a noticeable reduction in C50
with respect to the model. Looking at the echogram these positions do not benefit from the
Orchestra reflections. The presence of two strong specular reflections occurring approximately
at a 100ms are visible at position 3. These are due to specular reflections generated by the
balcony. Similar reflections, at approximately 55ms, are affecting receiver 1 on the 8th tier of
the Cavea. These strong specular reflections can be seen at most receivers. These effects
were not noticeable in the Syracuse Roman and Aspendos models. Thus, any reflective
surface surrounding the Cavea are required to be highly diffusive.
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7. Conclusions
From the validation process it was observed that the Cavea geometry must be detailed
if a reflective surface is added on the stage. However, modelling the Cavea in detail will
detrimentally affect the interaction between the Cavea and reflective surfaces placed at the
back of the Cavea. Approximated predictions can be obtained modelling the Cavea in detail
even though, doing so, breaks the GA assumption that surface size must be larger than the
lowest frequency calculated. Predictions, however, are less accurate at lower frequency
bands, especially at 250Hz. This band is affected by interference effects which cannot be
simulated even if using beam tracing software and simulating 1st order diffraction. Simulation
of 1st order diffraction does, however, improve predictions. Adding more advance diffraction
calculation such as including diffraction of 1st order reflections and reflections of diffractions
does not improve predictions and may have an adverse impact on temporal acoustic
distribution. More deterministic and “accurate” calculations generate overestimation of T30
and underestimation of C50 at higher frequencies. Thus, a certain degree of randomness is
beneficial. Energy balance inside the model is improved by assigning a scattering coefficient
based on surface dimensions. The discretization of the circular shape of the Cavea generates
focusing effects in localized areas.
The inability of reproducing the interaction between the Cavea and other surfaces,
limits the validity of New designs investigation. This is likely to generate overestimation or
underestimation depending on the position of the reflective surfaces.
Investigation of consequences of applying reflectors to different areas of the Cavea,
revealed that reflectors placed over the stage improve reverberation throughout the seating
area, this is valid for an omnidirectional source. When modelling clouds reflectors over the
Cavea ensuring specular reflections arrive within 50ms over the seating area, thus force
locating them at different heights, undesired interaction between different reflectors layers
occurred. The introduction of large reflective surfaces inside the Cavea improves clarity for
the audience near the surface, however, it could generate adverse effects for other receivers,
those distant from the surface. The introduction of a balcony design undermines the positive
influence of reflections from the Orchestra. From measurement and predictions, it is noticed
that reverberation at lower frequencies increases when reflective surfaces are placed near the
highest rows of the Cavea.
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8. Future Work
Open-air theatre design would greatly benefit if a solution to address the complex
acoustic interference effect generated by the Cavea in GA is developed. This should also
address the Cavea interaction with other surfaces.
Open-air theatre design could benefit if investigate through hybrid deterministic /
statistical acoustic prediction software.
There are many possible design to be investigated including Vineyard shape, variations
of the Cavea shape, different patterns of cloud reflectors, and different version of balconies
integrations.
Gaining greater insight in the Cavea behaviour depending on source position would
help the design of new Cavea shapes.
Modern theatre uses amplification not only to amplify performance but also to enchanted
it. Thus, investigation open-air theatre acoustic design integrated with electro-acoustic
amplification would satisfy modern requirements.
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