Engineering art: the science of concert
hall acoustics
TREVOR J. COX
Acoustics Research Centre, University of Salford, UK
PETER D’ANTONIO
RPG Diffusor Systems, Upper Marlboro, MD, USA
Modern concert hall design uses science and engineering to make an acoustic which embellishes and enhances the
artistry of the musicians. The modern discipline of concert hall acoustics is a little over a hundred years old, and
over the last century much has been learnt about how to ensure the audience receives high quality sound. During
this period, knowledge from a large number of disciplines has been exploited. It is the intention of this paper to
illustrate how disciplines as diverse as X-ray crystallography, psychology, and mobile telephony have influenced
acoustic design. The paper will concentrate on the design of acoustic diffusers for concert halls, as this is a topic
currently attracting considerable interest within the acoustics industry and academia.
The acoustic of a concert hall contributes an important
part of the sound heard in a classical music performance; the concert hall embellishes the sound. Music
outdoors may be popular when accompanied by fireworks, but the quality of the sound is usually poor.
Move indoors and the sound comes alive, enveloping
and involving the listener in the musicmaking process.
Outdoors, listeners receive sound straight from the
orchestra, there are no reflections from walls, and the
sound appears distant. When music is played in a
room, reflections from the walls, ceiling, and floor
add reverberation and other characteristics to the
sound. As the conductor Sir Adrian Boult said:1 ‘The
ideal concert hall is obviously that into which you
make a not very pleasant sound and the audience
receives something that is quite beautiful. I maintain
that this really can happen in Boston Symphony Hall;
it is our ideal.’ This quote is of great historical
interest, because Boston Symphony Hall was the first
concert hall where the principles of reverberation
were applied. These principles were developed about
a hundred years ago by Wallace Sabine, who was
the first person to apply modern scientific principles
to room design and pioneered modern concert hall
acoustics. Boston Symphony Hall is still seen as one
of the greatest auditoria in the world. Conversely, a
poor hall can have a detrimental effect on the enjoyment of a performance, something that can affect the
musicians as well. As the conductor Sir Simon Rattle
said of one hall (which will remain nameless): ‘The
*** hall is the worst major concert arena in Europe.
The will to live slips away in the first half hour of
rehearsal.’2
Since Wallace Sabine’s work on room acoustics,
much has been learnt about what is important to get
a good acoustic, whether the requirement is to make
© 2003 IoM Communications Ltd DOI 10.1179/030801803225010412
Published by Maney for the Institute of Materials, Minerals and Mining
speech sound intelligible, or to make music sound
beautiful. Modern acoustic science cannot guarantee
a great acoustic every time, but if advice is followed,
technical knowledge should ensure that bad halls are
not built, while significantly increasing the chance of
greatness being achieved. What makes a good concert
hall is a combination of many acoustic and nonacoustic attributes perceived by an audience member
in a complex manner. The acoustic of a concert hall
might be perfect, but if the audience get soaked in
rain walking from the car park, they are unlikely to
rate the experience highly. So a great design is about
accommodating a wide range of requirements, and not
just acoustics. This article, however, will concentrate
on the acoustics. When designing a hall, the acoustic
engineer will look at many acoustic factors: the
background noise level, the amount of reverberation
(the decay of sound after a note has stopped being
played), the amount of sound arriving from the side,
the reflections musicians receive that are necessary
for them to play in time and form a good tone, and
so on. The overall focus of this paper will be on the
role of surface diffusers.
Currently there is much debate about the role
of surface diffusers in concert halls. To take two
examples, one eminent concert hall designer regularly
claims that too much diffusion is detrimental to the
sound quality of the upper strings, while in contrast
other acoustic engineers have blamed the disappointing acoustics of certain major halls on a lack of
surface diffusion. We will return to these contradictory
opinions later, and will try to shed some light on why
they arise, but first it is necessary to describe what a
diffuser is and to outline the current state of the art
of acoustics.
INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 119
1 Spatial and temporal responses of sound reflected
from a plane flat surface (above) and a diffuser
(below)
Treatments
To alter the acoustics of an existing room, treatment
is usually placed on the boundaries, for example if
an office is too reverberant or lively, absorbent ceiling
tiles or carpet might be used to absorb and so remove
some of the acoustic energy. In concert halls, the
sound can be altered by placing treatment on the
walls and ceiling (the floor already has the audience
and seating on it, and so is difficult to alter). There
are three basic forms of treatment, large flat surfaces,
absorbers, and diffusers. Absorbers, such as the ceiling tiles and carpet mentioned already, are not often
used in large concert halls, because they remove sound
energy from the space. In a hall every bit of energy
must be conserved because there is a limit to how
much energy an orchestra can produce. Consequently,
the designer must choose between flat surfaces or
diffusers.
Figure 1 contrasts the spatial and temporal responses
of flat and diffusing surfaces. These describe what the
listener would receive if they were close to one of
the surfaces, and if no other surfaces were present.
A flat surface behaves like a mirror reflecting light,
the sound energy being preserved and concentrated
in the specular reflection direction, and with equal
angles of incidence and reflection. The time response
shows the similarity between the direct sound and
the reflection: the flat surface does little to the sound
except change the direction in which it propagates.
Figure 2 shows the resulting frequency response. This
shows how the level (or volume) of the sound will
vary as the pitch (or frequency) of a note changes.
The frequency response of the flat surface is uneven,
with a regularly spaced set of peaks and troughs, and
is known as a comb filter response. This unevenness
means that some frequencies will be emphasised, and
some deemphasised. This leads in turn to coloration
of the sound, where the timbre of notes is altered.
A diffuser, on the other hand, disperses the reflection
both temporally and spatially. The time response
is greatly altered, with reflections arriving over a
longer period, and the frequency response shows less
evidence of comb filtering than the plane surface,
the peaks and troughs being uneven and randomly
spaced. This means that the reflected sound is a more
120 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2
2 Temporal and frequency responses from flat
(above) and diffusing (below) surfaces
faithful rendition of the original sound direct from
the instrument, and less coloration will be heard.
Figure 1 also shows a cross-section through a diffuser,
in this case a reflection phase grating consisting of a
series of wells of different depths but the same width.
There are many types of diffuser, as explained below,
but in principle any non-flat corrugated surface will
have some kind of diffusing ability. Diffusers can also
be constructed from a wide range of materials, such
as wood, gypsum, concrete, metal, and glass, the key
feature being that the material should be hard and
acoustically non-absorbent.
Diffusers are used in a variety of ways. For instance,
they can be used to reduce echoes arising from the
rear walls of auditoria. Sound takes a long time to
travel from the stage to the rear wall of a concert
hall, and if a strong reflection comes back from the
rear wall to the front of the hall, this may be heard
as an echo. In older halls the echo problem would
have been dealt with by placing absorbent on the rear
wall to absorb the acoustic energy, preventing the
reflection occurring and so curing the echo problem
(such a solution was adopted in London’s Royal
Festival Hall ). However absorption removes acoustic
energy and so reduces the loudness of the orchestra.
A modern solution is therefore to use diffusers to
break up and disperse the troublesome reflections,
which can be done without loss of acoustic energy.
An example of the use of reflection phase grating
diffusers, on the rear wall of Carnegie Hall in New
York, can be seen in Fig. 3. In the UK, Glyndebourne
Opera House uses convex curved surfaces on the rear
wall to disperse sound.
Architectural trends
In older, pre-twentieth century halls, such as the
Grosser Musikvereinssaal in Vienna, ornamentation
appeared in a hall because it was the architectural
style of the day. Such walls were therefore naturally
3 Schroeder diffusers (QRDs) applied to the rear
wall of Carnegie Hall to prevent echoes
diffusing; large flat surfaces were very rare. The
Grosser Musikvereinssaal is an interesting example
to acousticians, because it is often cited as one of the
best halls in the world. The hall sound is thought to
have influenced composers including Brahms, Bruckner,
and Mahler. In the Grosser Musikvereinssaal the
influence of the surface diffusion on the sound is very
obvious, with a diffuse sound resulting.
In the twentieth century, however, architectural trends
changed and large expanses of flat areas appeared
in many concert halls. The UK has many post-war
concert halls which have very little ornamentation,
such as the Colston Hall in Bristol. The style of the
day was to produce clean lines following a modernist
style, but these surfaces then had little or no diffusing
capability. The expanse of flat surfaces can lead to
distortion in the sound heard as a result of comb
filtering, echoes, and other mechanisms. It is worth
noting, however, that it is also possible to design
very successful halls with flat surfaces, a good UK
example being Symphony Hall in Birmingham, which
has relatively little surface diffusion.
The key to good diffuser design is to find forms
that complement the architectural trends of the
day. The diffuser must not only meet the acoustic
specification, it must fit in with the visual scheme
required by the architect. As discussed below, modern
diffuser designs have successfully been developed to
complement modern architectural forms.
Schroeder diffusers
The development of the modern diffuser began with
pioneering work by Manfred Schroeder, one of the
twentieth century’s greatest acoustic engineers. In
the 1970s, Schroeder developed the phase grating
diffuser,3 also known as the Schroeder diffuser. An
example of the original design can be seen in Fig. 4
(top). These diffusers offered just what acoustic
designers were looking for, defined acoustic performance based on very simple design equations; for while
it is know that ornamentation produces diffusion, it
does this in an ill defined and haphazard fashion.
One of the pioneering applications of Schroeder
diffusers was by Marshall and Hyde in the Michael
4 Three different Schroeder diffusers: the original
design (top), a fractal design (middle), and an
active diffuser (bottom); the diffusers are 0·6 m
wide, 0·6 m high and about 0·2 m deep
Fowler Centre, New Zealand.4,5 Figure 5 illustrates this
application. Marshall and Hyde used large overhead
reflectors to provide early reflections to the audience
in the balconies in a revolutionary design. This was
a layout whereby a hall could have good clarity, and
yet maintain a large volume for reverberation. The
large volume partly comes from the space behind
the diffusers. Not many years before the design of
the hall, it had been established that lateral reflections
were important in concert halls as they promote a
sense of envelopment or spatial impression in rooms.6
The evidence for the beneficial effects of lateral
reflections came from laboratory measurements on
human perception, which followed techniques pioneered
in experimental psychology. These measurements
showed that lateral reflections are important to get a
INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 121
5 Schroeder diffusers in the Michael Fowler
Centre, New Zealand (photo courtesy Dr Harold
Marshall of Marshall Day Acoustics)
sense of involvement with the music. The need for
lateral reflections influenced Marshall and Hyde to
apply diffusers to the large overhead surfaces rather
than using flat reflectors.
Another reason for this new diffuser technology
entering wide use was its commercialisation by an
American company, RPG Diffusor Systems Inc.,
whose interest lay in studio design. Around the time
that Schroeder was developing the new diffuser, a
new design regime for listening and monitoring rooms
was invented. This was the Live End Dead End
(LEDE) layout,7 which was later refined into the
Reflection Free Zone (RFZ) design. Diffusers are
used in small spaces to disperse reflections that would
otherwise arrive early and at a high level and so cause
coloration of the timbre of the sound. This is sometimes referred to as acoustic glare, and is again caused
by comb filtering. Just when studio designers were
looking for diffusers to achieve this design, by happy
coincidence Schroeder diffusers became available.
At that time, one of the founders of RPG, and
also one of the authors of this paper, Peter D’Antonio,
was a diffraction physicist at the Laboratory for the
Structure of Matter at the Naval Research Laboratory
in Washington, DC. Knowing of his interest in music,
a colleague handed him the latest issue of Physics
Today with a cover photo of Manfred Schroeder
seated in an anechoic chamber. The associated article
suggested using Schroeder’s number theoretic diffusers
in concert halls. It became apparent that the reflection
phase gratings suggested by Schroeder were in effect
two-dimensional sonic crystals, which scatter sound
in the same way that three-dimensional crystal lattices
scatter electromagnetic waves. Since the diffraction
theory employed in X-ray crystallographic studies
was also applicable to reflection phase gratings, it was
straightforward to model and design the reflection
phase gratings using techniques first developed in
crystallography.
Figure 6 ( left) shows the scattering from a
Schroeder diffuser in a polar response. A source
illuminates the surface, normal to the surface and
from the right. The polar response shows the energy
scattered from the surface (in decibels) as a receiver
122 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2
6 Scattered levels from a Schroeder diffuser
(left) and a plane surface (right) of the same
dimensions
moves around the surface on a semicircle. A series of
lobes are seen, eleven in this case, which are grating
lobes generated by the periodicity of the surface
structure. Imagine viewing this polar response end
on so that a set of eleven bright spots are seen; it is
this type of image that X-ray crystallographers use
to determine crystal structures. The problem posed in
the acoustic case is however somewhat different from
the crystallographic challenge: in crystallography, the
diffraction patterns of the X-rays are used to determine an unknown structure, whereas in the acoustic
case, the problem to be solved is how to determine
the correct surface structure to achieve a desired
polar response (or diffraction pattern). But before
explaining how Schroeder solved this problem, it is
necessary to explain how diffusers scatter sound.
Huygens
The Huygens construction used in optics is one way
of explaining how diffusing surfaces scatter, though it
is only approximate in many acoustic cases. Consider
a planar surface, the situation illustrated in the upper
half of Fig. 7. When illuminated by a sound source,
a set of secondary sources is generated on the surface,
and these are shown as stars in Fig. 7. Each of these
secondary sources then radiates semicircular waves.
By connecting points on these waves which are in
phase with each other, it is possible to visualise the
waves that are reflected from the surface. (These are
rather like ripples on the surface of water created
when a stone is thrown into the water.) In this
situation, a simple plane wave at right angles to the
surface is generated. The planar surface is acting like
an acoustic mirror, and the wave is unaltered on
reflection (except in its direction). The lower part of
Fig. 7 shows the case for a semicircular surface. In
this case, the reflected waves are now semicircular in
shape. The wave has been altered by the surface, being
dispersed so that the sound reflects in all directions,
a characteristic desirable in a diffuser.
method, Schroeder turned to his favourite subject of
number theory, at first glance a rather obscure form
of abstract mathematics which studies the properties
of natural numbers, but which in practice has proved
to be very useful to scientists and engineers.
In the late eighteenth century, Carl Friedrich
Gauss developed the law of quadratic reciprocity,
well known to mathematicians working in number
theory. Dedekind was a doctoral student of Gauss
and wrote a fine description of his supervisor:8
7 Huygens constructions for a plane wave reflected
from a flat surface (above) and a curved surface
(below): normal incidence source with incident
wavefronts excluded for clarity and secondary
sources shown as stars on the surface
Figure 8 shows the case for a simplified reflection
phase grating. In this situation, the reflected waves
are delayed because the waves must travel down
each well and back up again before reflection. The
secondary sources have different delays (phases)
because of the different well depths, and this alters
the reflected wave. This again generates dispersion.
Sequences
In many ways, a reflection phase grating is acting
like an optical diffraction grating. In the acoustic
case, the designer has control over the phases of the
sound waves. To design a reflection phase grating, a
method is required to determine an appropriate well
depth sequence, which then generates a phase distribution on the surface of the diffuser to give the
desired reflected wavefronts. In inventing such a
8 Huygens constructions for a plane wave reflected
from a simplified Schroeder diffuser: the upper
plot shows wavefronts from two wells only
for clarity
… usually he sat in a comfortable attitude, looking down,
slightly stooped, with hands folded above his lap. He
spoke quite freely, very clearly, simply and plainly: but
when he wanted to emphasise a new viewpoint … then
he lifted his head, turned to one of those sitting next to
him, and gazed at him with his beautiful, penetrating
blue eyes during the emphatic speech. … If he proceeded
from an explanation of principles to the development of
mathematical formulae, then he got up, and in a stately
very upright posture he wrote on a blackboard beside
him in his peculiarly beautiful handwriting: he always
succeeded through economy and deliberate arrangement
in making do with a rather small space. For numerical
examples, on whose careful completion he placed special
value, he brought along the requisite data on little slips
of paper.
Although best known to modern physicists for
Gauss’s Law, which explains the properties of the
electric field, it is Gauss’s number theory work
which is of most interest here, because it leads to
the quadratic residue sequence used in the design of
the quadratic residue diffuser (QRD), an example
of which is shown in Fig. 4 (top). The formulation of a
quadratic residue sequence is based on a prime number.
For the diffuser in Fig. 4 (top), the prime number is
7. The depth of the nth well is then proportional to
n2 modulo 7, where modulo indicates the smallest
non-negative remainder. So the third well has a depth
proportional to 32 modulo 7, in other words 2. The
sequence mapped out in this case is 0, 1, 4, 2, 2, 4, 1,
which can be seen in Fig. 4. (The quadratic residue
diffuser in Fig. 4 has zero depth wells on both ends,
but these are half the width of the others, a useful
sleight of hand to make manufacturing and fitting
easier). If this quadratic residue sequence is used to
construct the diffuser, then the diffraction or grating
lobes generated all have the same energy, as shown
in Fig. 6 ( left).
There are many other sequences that can be used.
Another popular one is the primitive root sequence.
The depth of the nth well is then proportional to rn
modulo N, where r is a ‘primitive root’ of N and N
must be a prime. For r to be a primitive root, the
sequence generated must contain every integer from
1 to N−1 without repeat. Thus for N=7, r=3 is a
primitive root, which generates the sequence 1, 3, 2,
6, 4, 5. If this sequence is used to make a diffuser,
the central (specular) lobe will be suppressed, while
the others remain at the same level.
This sequence generation technique is an advanced
form of the number games sometimes played by
children: with some simple generation rules, fantastic
INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 123
answers are produced. But these sequences are
seriously useful, and have been developed for diverse
applications in astronomy, in error checking systems
for computers, and in digital audio data and mobile
telephony. As Schroeder is fond of saying, number
theory is unreasonably useful, considering it was
developed as an abstract mathematical subject.
Improvements
While the basic Schroeder diffuser based on number
theory sequences is an ingenious invention, it has
several Achilles heels. Since the development of the
initial design, several revisions have been suggested
to improve performance. These are used to overcome
key weaknesses related to bandwidth, periodicity, and
appearance and will be discussed in detail below.
In concert hall acoustics, designs have to work
over a wide bandwidth. Human hearing extends over
between ten and eleven octaves (20 Hz to 20 kHz),
and diffuser designs are typically considered over
the seven most important (80 Hz to 5 kHz). This is
one of the key differences between much research in
acoustics and optics, as optics researchers are often
concerned with a single frequency or a narrow bandwidth. In acoustics, the bandwidth is much wider. To
make a wide bandwidth diffuser, the wells need to be
narrow and deep, but this makes the device very
impractical: first of all the structure becomes very
expensive to make, and second it becomes highly
absorbent (air is a viscous fluid, and as with any such
fluid it is difficult to force into narrow wells, acoustic
energy being lost in the process and converted to
heat). A solution to this problem has been developed,
inspired by chaos theory and fractals.
To handle many octaves, a diffuser needs to have
roughness on different scales. The use of elements of
different sizes is common in loudspeaker design. In
two way loudspeakers, for example, the large ‘woofer’
is used to radiate bass frequencies, and the smaller
‘tweeter’ generates the treble sound. For diffusers,
some roughness needs to have dimensions metres
in size, and some needs to have dimensions centimetres in size. Fractals are objects which have
scaleable properties, and one of the most famous is
the Mandelbrot set, shown in Fig. 9 at two different
magnifications. When the set is greatly magnified, by
around four thousand times in Fig. 9, a very similar
shape to the original is seen. The same effect can be
achieved for diffusers, as shown in Fig. 4 (middle).
In the surface shown, smaller diffusers are mounted
within larger ones, the smaller scattering the high
frequencies, and the larger the low frequencies. This
type of diffuser is rather fittingly sold under the brand
name Diffractal.9 The example shown has diffusers
with two different scales. Three different sizes of
diffuser are needed to cover bass frequencies, the
largest having a size about fifty times greater than
the smallest.
The issue of periodicity is curious, as in many ways
it is the curse of the structure. Since reflection phase
124 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2
9 The Mandelbrot
magnifications
set
at
two
different
gratings require periodicity to work, many periods
of the diffuser are stacked next to each other. The
diffraction lobes are also a function of periodicity,
and to achieve even energy lobes (Schroeder’s
definition of optimum diffusion) requires the structure
to be periodic. Yet these diffraction lobes represent
energy concentrated into particular directions, with
a lack of reflected energy between. A better diffuser
would be one that distributed the energy more evenly
in all directions without lobes. Consequently there is
a contradiction: to use the original number theory
design, periodicity is required, yet this results in worse
performance. A solution to this paradox was devised
by Angus,10 who showed that techniques devised
in mobile telephony could be adopted for diffusers.
These techniques are also applied to the design of
loudspeaker and microphone arrays.
Code Division Multiple Access (CDMA) systems
are used in mobile telephony to enable multiple users
to use the same transmission bandwidth. Of interest
here are so called spread spectrum techniques. These
techniques take frequency (spectral ) components,
and spread them over a frequency bandwidth. If the
lobes generated by the Schroeder diffuser are viewed
as spatial frequency components, then when spread
spectrum techniques are applied, the lobes will be
spread spatially. This effect is shown in Fig. 10, where
10 Scattered polar distribution from a periodic
arrangement (light line) and a modulated
arrangement (heavy line) of a quadratic residue
diffuser
the spread spectrum process has enabled the scattered
energy to be redistributed from the three lobes in all
directions (all spatial frequencies).
This idea can be applied to diffusers as follows.
Two base Schroeder diffusers are used. The first is
a standard Schroeder diffuser, the second a diffuser
which produces the same polar response only with
opposite phase. This is easy to form by changing the
well depth sequence; in fact the second diffuser is the
reverse of the first. Figure 11 shows two base shapes
arranged in a modulated array, in other words in a
random arrangement. The modulation sequence does
not repeat, and so the diffusers are no longer periodic
and the periodicity lobes disappear. This produces a
much more even polar response.
Whilst in principle it would be possible to flip a
coin to determine the modulation sequence, it is far
better to apply a sequence from number theory (yet
another case of number theory being unreasonably
useful ). This is particularly true if only a few periods
of the diffuser are being considered, as mathematicians
have produced methods for generating the best
possible sequences with the least amount of repetition.
Alternatively it is possible to task a computer to
laboriously search for the best sequences, but this is
rather slow. The first sequence used for modulating
diffusers was the maximum length sequence, also known
as a Galois field because of its basis in mathematics
developed by the nineteenth century mathematician
Evariste Galois. Galois unfortunately met an untimely
death in a duel at the age of twenty-one, but not
before he had sketched out some very important
mathematical concepts. As his director of studies
wrote:11 ‘It is the passion for mathematics which
dominates him, I think it would be best for him if
his parents would allow him to study nothing but
this, he is wasting his time here and does nothing
but torment his teachers and overwhelm himself with
punishments.’ Maximum length sequences are used
widely in digital systems; in acoustics they are best
known for producing efficient measuring systems,
such as listening rooms, filters, and loudspeakers.
The appearance of Schroeder diffusers is an
important impediment to their use, especially given
current taste in architecture and interior design, which
tends to favour curves and more organic shapes.
With Schroeder diffusers the acoustic treatment
imposes a distinctive visual aesthetic, and while there
are architects who favour the argument that form
should follow function, most prefer to determine
their own aesthetic. If an architect thinks a diffuser
looks ugly, it will not be used, however important
the treatment is to the acoustic design. Consequently,
there is a need for designs that complement modern
architectural trends. Figure 12 shows a modern diffuser
design on the ceiling of a cinema in Seattle. This is a
curved diffuser designed to offer a visual complement
to the curtaining on the stage, while providing the
required acoustic performance. The diffuser disperses
reflections from the ceiling which would otherwise
colour the sound. To design this sort of diffuser requires
a new methodology, and for this it is possible to use
numerical optimisation. This is a method commonly
used in engineering, for example to design parts of
the space shuttle. Numerical optimisation does not
have the efficiency and elegance of number theory
design, but it is extremely effective and robust.
Numerical optimisation tasks a computer to search
for an optimum solution to a problem, for instance
it could look to optimise a car engine component
to minimise weight while ensuring sufficient strength
and durability. In the case of diffusers, the computer
11 Cross-section through a modulation scheme using an N=7 quadratic residue diffuser and its inverse
INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 125
12 The Cinerama in Seattle, WA with a diffusing
ceiling (photo courtesy University of Salford
and Harris-Grant Associates)
looks for the surface shape which gives optimal scattering. The procedure works by iteration. The computer
starts by guessing some curved surface shape, and
the scattering from the surface is then predicted in
terms of the polar response. The predicted polar
response is rated for its quality in terms of a ‘figure
of merit’, which by a process of trial and error the
computer can try and minimise by changing the surface shape. The process continues until an optimum
design is found, which occurs when a minimum in
the figure of merit is determined. The search process
is not completely random because this would be too
slow – fortunately mathematical algorithms have been
developed to allow the search to be done efficiently.
Currently the most popular approach, using a so
called genetic algorithm, models the problem as
an evolutionary process, using survival of the fittest
principles to carry out an efficient search.
A genetic algorithm mimics the process of evolution
that occurs in biology. A population of individuals
is randomly formed. Each individual is determined
by its ‘genes’, which in this case are simply a set of
numbers which describe the curved surface shape.
Each individual (or shape) has a fitness value (figure
of merit) that indicates how good it is at diffusing
sound. Over time new populations are produced by
combining (breeding) previous shapes, and the old
population dies off. Offspring are produced by pairs
of parents breeding, and the offspring have genes that
are a composite of the parents’. The offspring shape
will then have features drawn from the parent shapes,
in the same way that facial features of a child can
often be recognised in the parents. A common method
of mixing genes is called multipoint crossover. For
each gene, there is a fifty per cent chance of the
child’s gene coming from parent A, and a fifty per
cent chance of it being from parent B.
If all that happened was combination of the
parents’ genes, then the system would never look
outside the parent population for better solutions
(the ‘fish’ diffuser would never get lungs and walk
126 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2
13 Optimised curved surface in the Edwina
Palmer Hall, Hitchin, UK (photo courtesy Arup
Acoustics)
about on land). As with biological populations, to
enable dramatic changes in the population of shapes,
mutation is required. This is a random procedure
whereby a small probability is defined of any gene in
the child sequence being randomly changed, rather
than coming direct from the parents.
Selecting shapes to die off can be done randomly,
with the least fit (the poorest diffusers) being most
likely to be selected. In biological evolution, the fittest
are most likely to breed and pass on their genes, and
the least fit are the most likely to die; this is also true
with an artificial genetic algorithm used to design
diffusers. Using these principles, the fitness of successive
populations should improve, and the process is continued until the population becomes sufficiently fit
for the shape produced to be classified as optimum.
Figure 13 shows an example of another optimised
curved surface. In this case a concave wall in a music
practice room would have caused problems of focusing if untreated. Concave walls focus sound to a
point in the same way that concave mirrors focus
light. In acoustics, focusing effects are obvious in
whispering galleries, such as the dome in St Paul’s
Cathedral in London. Figure 14 shows the polar
response for a concave wall, revealing that the
scattered energy level is much greater for the receiver
at the focal point. In treating this focusing problem,
it would have been possible to add absorption on the
wall to remove the reflections from the curved surface.
But this would have removed energy from one side of
the orchestra, and these reflections are needed so the
musicians can hear themselves and their colleagues.
Reflections are needed for the musicians to keep in
14 Scattering from a concave arc (light line)
compared with an optimised curved diffuser
(heavy line)
time, form a good tone, and blend the overall ensemble
sound. The solution is therefore to use diffusers, as
these remove the focusing effect from the sound while
preserving the acoustic energy. Figure 14 illustrates the
effectiveness of the diffuser in dispersing the focused
sound.
Absorption for diffusion
In recent years, interest has been returning to number
theory to generate a different kind of diffuser, the
hybrid surface. Construction of a hybrid surface is
shown schematically in Fig. 15. It consists of a piece
of acoustic absorbent covered with a perforated mask,
the mask then being hidden from view by a thin piece
of acoustically transparent cloth. Where there are
holes in the mask, absorption is generated; where the
sound strikes a solid part of the mask, reflection occurs.
This then forms a surface which partially absorbs,
and any reflected energy is diffused because of the
random arrangement of holes. The key to good dispersion is the arrangement of the holes, which is best
done following a pseudo random binary sequence with
optimal autocorrelation properties ( least repetition).
When the sequence has a 1, a hole is drilled in the
mask, when it has a 0, no hole is drilled. Any repetition
in the sequence will lead to lobes, so sequences are
needed which are dissimilar from shifted versions
of themselves. Again, number theory can provide a
whole range of sequences.
Angus12 started by looking at maximum length
sequences, the same sequences which were used originally to modulate Schroeder diffusers. The problem is
that maximum length sequences are just strings of 1s
and 0s, and what is needed for hybrid surfaces is
a two-dimensional array of numbers. Fortunately, a
15 Construction of a hybrid surface: porous
absorber (left), mask (middle), and cloth (right)
technique exists for folding a sequence into a twodimensional array, a technique commonly referred to
as the Chinese Remainder Theorem.
An example of a Chinese remainder problem
was posed by Sun Tsu Suan-Ching in the fourth
century :13 ‘There are certain things whose number
is unknown. … Divided by 3, the remainder is 2; by 5
the remainder is 3; and by 7 the remainder is 2. What
will be the number?’ From this rather unpromising
start, a method of sequence folding can be generated
which has been used in coding systems, cryptology,
and X-ray astronomy. Incidentally, one answer to
the above problem is 23. The mask shown in Fig. 15
is a length 1023 maximum length sequence which has
been folded into a 31 by 33 array using the Chinese
Remainder Theorem.
The problem with maximum length sequences is
that they are devised for systems that are bipolar,
consisting of +1s and −1s. The hybrid surface on
the other hand produces areas of no reflection (0s)
and reflection (1s), and so is inherently unipolar. This
can be a problem when designing these diffusers.
Most electronic systems have bipolar capabilities, and
can produce signals of the opposite sign, but this
is not true of fibre optic systems, and hence optical
sequences have been developed. Optical sequences
were developed for use in fibre optical CDMA processes. Fibre optic CDMA sequences, where the light
intensity is either on or off, cannot have cancellation,
and hence use unipolar sequences. These sequences
can be used to design hybrid diffusers.
Figure 16 compares the scattering from a hybrid
absorber–diffuser with that from a plane surface. The
hybrid surface provides greater dispersion. This dispersion can be improved even further if the hybrid
surface is bent and made corrugated, as this breaks
up the specular reflection component further. This
type of design is proving to be very popular in studio
control and listening rooms.
Active technology
The final technology to be discussed here is active
control technology. Active noise control has caused
much interest in the last twenty years or so, but the
application of this technology is not widespread
because the practical implementation is costly and
problematic. It has, however, been used successfully
in controlling ventilation, car, and aircraft noise.
Attention has now been turned to whether an active
diffuser can be generated. In active noise control,
the noise source is cancelled by generating a signal
from a secondary source of the same magnitude
but opposite phase. The waves from the noise and
secondary source then cancel each other out. Active
diffusers do something slightly different: they start by
cancelling out a reflection, but then the secondary
source adds in an artificial reflection which mimics
the characteristics of a diffuser.
Figure 4 (bottom) shows an artist’s impression of
an active diffuser. Loudspeakers are placed at the
INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 127
most successful halls are usually those where there
is a good rapport between the acoustician and
the architect, where the engineering facilitates and
embellishes the art.
As concert hall designs have been improved over
the century, attention has been focused on different
aspects of acoustic design. Currently, there is much
interest in understanding the role of surface diffusers.
As this paper has shown, much is now known about
the design of diffusers, but many questions remain
unanswered. The most important question to answer
is how much diffusion should be applied, and where
diffusers should be used. While acoustic designers
have produced many innovative new designs, the
understanding of how and why to apply diffusers lags
behind and is still largely based on precedence.
In 2001, Manfred Schroeder attended the International Congress on Acoustics in Rome. He commented on some photos of optimised curved surfaces,
saying we had produced diffusers which were beautiful.
We were flattered by such a tribute from the pioneer
of modern acoustic diffuser design. Using optimisation
it is now possible to design simultaneously to given
visual and acoustic requirements – perhaps a case of
‘engineering art’?
16 Hemispherical polar balloons showing scattering from surfaces: hybrid surface (above),
plane hard surface (below)
bottom of some of the wells. By changing how these
controlled loudspeakers respond to incident sound, it
is possible to change the characteristics of the wells.
For example, it would be possible to make a well
appear longer than it really is. But why would one
go to such lengths, as the active system with its
electronics and control structure is difficult to develop
and expensive to implement? Active systems are
being investigated to gain additional diffusion at
low frequency, in a range where it is very difficult to
gain diffusion by normal (passive) means. To obtain
diffusion at low frequency is very difficult because
the size of acoustic waves (wavelength) becomes many
metres in size, which means that a diffuser should be
extremely large as the diffuser’s dimensions need to
be comparable with the wavelength. Active diffusion
is a method whereby the acoustic field can be disturbed at a low frequency using a shallower device.
These devices, however, are very much in their infancy
and it remains to be seen whether a practical device
can be constructed.
Summary
Much has been learnt about the design of concert
halls over the last hundred years. A little over a
century ago, the design of a hall was based on a combination of precedence and luck. Nowadays, acoustic
science means that concert hall design involves a
combination of precedence, engineering, and art. The
128 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2
Acknowledgements
Figure 1 is taken from . .  and . ’:
Acoustic Absorbers and Diffusers; 2003, London,
Spon; Figs. 2, 6–8, 10, 14, and 16 from . ’
and . . : ‘Diffusor application in rooms’,
Applied Acoustics, 2000, 60, 113–142; and Fig. 3 from
. ’ and . . : ‘Two decades of sound
diffusor design and development part 1: applications
and design’, Journal of the Audio Engineering Society,
1998, 46, 955–976.
Notes and literature cited
1. . . : Concert and Opera Halls: How they
Sound, 69–74; 1996, Woodbury, NY, Acoustical Society
of America.
2. . : ‘Things can only get better’, Guardian, 1999,
23 July.
3. . . : ‘Binaural dissimilarity and optimum
ceilings for concert halls: more lateral sound diffusion’,
Journal of the Acoustical Society of America, 1979,
65, 958–963.
4. . .  and . . : ‘Some practical considerations in the use of quadratic residue diffusing
surfaces’, Proceedings of the 10th International
Congress on Acoustics, Sydney, 1980, paper E7.3.
5. . . , . .  and . . . : ‘The
acoustical design of Wellington Town Hall: design
development, implementation and modelling results’,
Proceedings of the Institute of Acoustics, Edinburgh,
1982.
6. . : ‘The subjective effects of first reflections in
concert halls – the need for lateral reflections’, Journal
of Sound and Vibration, 1971, 15, 475–494.
7. .  and . : ‘The LEDE concept for the
control of acoustic and psychoacoustic parameters
in recording control rooms’, Journal of the Audio
Engineering Society, 1980, 28, 585–595.
8. See www-gap.dcs.st-and.ac.uk/~history/Mathematicians/
Gauss.html.
9. . ’ and . : ‘The QRD diffractal: a
new one- or two-dimensional fractal sound diffusor’,
Journal of the Audio Engineering Society, 1992, 40,
113–129.
Dr Trevor J. Cox
Acoustics Research Centre
University of Salford
Salford M5 4WT
UK
[email protected]
Dr Peter D’Antonio
RPG Diffusor Systems Inc.
651-C Commerce Drive
Upper Marlboro
MD 20774
USA
10. . . . : ‘Using grating modulation to achieve
wideband large area diffusers’, Applied Acoustics, 2000,
60, 143–165.
11. See www-gap.dcs.st-and.ac.uk/~history/Mathematicians/
Galois.html.
12. . . . : ‘Sound diffusers using reactive absorption
grating’, Proceedings of the 98th Convention of the
Audio Engineering Society, 1995, preprint 3953.
13. . : The Penguin Book of Curious and Interesting
Puzzles; 1992, London, Penguin.
Trevor Cox is Reader in Acoustics at Salford University, UK.
He graduated with a degree in physics from Birmingham
University, UK in 1988 and went on to study auditorium
acoustics at Salford University, where he was awarded his
doctorate in 1992. Between 1993 and 1995, Dr Cox worked
at South Bank University, London. A large proportion of
his research concerns diffusing surfaces: his use of numerical
optimisation techniques has led to the worldwide application of innovative diffusing surfaces. This paper is based
on his Isambard Kingdom Brunel Award lecture given to
the British Association for the Advancement of Science’s
Festival of Science held in Leicester in 2002.
Peter D’Antonio received his BS degree from St John’s
University in 1963 and his PhD from the Polytechnic
Institute of Brooklyn in 1967. Dr D’Antonio has specialised
in a wide variety of scientific disciplines including spectroscopy, X-ray and electron diffraction, electron microscopy,
software development, and architectural acoustics. He
carried out research in diffraction physics at the Naval
Research Laboratory, Washington, DC for thirty years.
He is now president of RPG Diffusor Systems Inc., which
was founded in 1983 to carry out basic research in room
acoustics and to develop designs and innovative number
theoretic architectural surfaces, to enhance the acoustics of
critical listening and performance environments.
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