Engineering art: the science of concert

hall acoustics

TREVOR J. COX

Acoustics Research Centre, University of Salford, UK

PETER D’ANTONIO

RPG DiVusor 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 in uenced
acoustic design. The paper will concentrate on the design of acoustic diVusers 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

speech sound intelligible, or to make music sound

part of the sound heard in a classical music perform-

beautiful. Modern acoustic science cannot guarantee

ance; the concert hall embellishes the sound. Music

a great acoustic every time, but if advice is followed,

outdoors may be popular when accompanied by Ž re-

technical knowledge should ensure that bad halls are

works, but the quality of the sound is usually poor.

not built, while signiŽ cantly increasing the chance of

Move indoors and the sound comes alive, enveloping

greatness being achieved. What makes a good concert

and involving the listener in the musicmaking process.

hall is a combination of many acoustic and non-

Outdoors, listeners receive sound straight from the

acoustic attributes perceived by an audience member

orchestra, there are no re ections from walls, and the

in a complex manner. The acoustic of a concert hall

sound appears distant. When music is played in a

might be perfect, but if the audience get soaked in

room, re ections from the walls, ceiling, and  oor

rain walking from the car park, they are unlikely to

add reverberation and other characteristics to the

rate the experience highly. So a great design is about

sound. As the conductor Sir Adrian Boult said:1 ‘The

accommodating a wide range of requirements, and not

ideal concert hall is obviously that into which you

just acoustics. This article, however, will concentrate

make a not very pleasant sound and the audience

on the acoustics. When designing a hall, the acoustic

receives something that is quite beautiful. I maintain

engineer will look at many acoustic factors: the

that this really can happen in Boston Symphony Hall;

background noise level, the amount of reverberation

it is our ideal.’ This quote is of great historical

(the decay of sound after a note has stopped being

interest, because Boston Symphony Hall was the Ž rst

played ), the amount of sound arriving from the side,

concert hall where the principles of reverberation

the re ections musicians receive that are necessary

were applied. These principles were developed about

for them to play in time and form a good tone, and

a hundred years ago by Wallace Sabine, who was

so on. The overall focus of this paper will be on the

the Ž rst person to apply modern scientiŽ c principles

role of surface diVusers.

to room design and pioneered modern concert hall

Currently there is much debate about the role

acoustics. Boston Symphony Hall is still seen as one

of surface diVusers in concert halls. To take two

of the greatest auditoria in the world. Conversely, a

examples, one eminent concert hall designer regularly

poor hall can have a detrimental eVect on the enjoy-

claims that too much diVusion is detrimental to the

ment of a performance, something that can aVect the

sound quality of the upper strings, while in contrast

musicians as well. As the conductor Sir Simon R attle

other acoustic engineers have blamed the disappoint-

said of one hall (which will remain nameless): ‘The

ing acoustics of certain major halls on a lack of

*** hall is the worst major concert arena in Europe.

surface diVusion. We will return to these contradictory

The will to live slips away in the Ž rst half hour of

opinions later, and will try to shed some light on why

rehearsal.’2

they arise, but Ž rst it is necessary to describe what a

Since Wallace Sabine’s work on room acoustics,

diVuser is and to outline the current state of the art

much has been learnt about what is important to get
a good acoustic, whether the requirement is to make

of acoustics.

INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 119

© 2003 IoM Communications Ltd D OI 10.1179/030801803225010412
Published by Maney for the Institute of Materials, Minerals and Mining

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

2 Temporal and frequency responses from flat

is usually placed on the boundaries, for example if

(above) and diffusing (below) surfaces

an oYce 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

faithful rendition of the original sound direct from
the instrument, and less coloration will be heard.

sound can be altered by placing treatment on the
walls and ceiling (the  oor already has the audience

Figure 1 also shows a cross-section through a diVuser,
in this case a re ection phase grating consisting of a

and seating on it, and so is diYcult to alter). There
are three basic forms of treatment, large  at surfaces,

series of wells of diVerent depths but the same width.
There are many types of diVuser, as explained below,

absorbers, and diVusers. Absorbers, such as the ceil-
ing tiles and carpet mentioned already, are not often

but in principle any non- at corrugated surface will
have some kind of diVusing ability. DiVusers can also

used in large concert halls, because they remove sound
energy from the space. In a hall every bit of energy

be constructed from a wide range of materials, such
as wood, gypsum, concrete, metal, and glass, the key

must be conserved because there is a limit to how
much energy an orchestra can produce. Consequently,

feature being that the material should be hard and
acoustically non-absorbent.

the designer must choose between  at surfaces or
diVusers.

DiVusers are used in a variety of ways. For instance,

they can be used to reduce echoes arising from the

Figure 1 contrasts the spatial and temporal responses

of  at and diVusing surfaces. These describe what the

rear walls of auditoria. Sound takes a long time to
travel from the stage to the rear wall of a concert

listener would receive if they were close to one of
the surfaces, and if no other surfaces were present.

hall, and if a strong re ection comes back from the
rear wall to the front of the hall, this may be heard

A  at surface behaves like a mirror re ecting light,
the sound energy being preserved and concentrated

as an echo. In older halls the echo problem would
have been dealt with by placing absorbent on the rear

in the specular re ection direction, and with equal
angles of incidence and re ection. The time response

wall to absorb the acoustic energy, preventing the
re ection occurring and so curing the echo problem

shows the similarity between the direct sound and
the re ection: the  at surface does little to the sound

(such a solution was adopted in London’s R oyal
Festival Hall ). However absorption removes acoustic

except change the direction in which it propagates.
F igure 2 shows the resulting frequency response. This

energy and so reduces the loudness of the orchestra.
A modern solution is therefore to use diVusers to

shows how the level (or volume) of the sound will
vary as the pitch (or frequency) of a note changes.

break up and disperse the troublesome re ections,
which can be done without loss of acoustic energy.

The frequency response of the  at surface is uneven,
with a regularly spaced set of peaks and troughs, and

An example of the use of re ection phase grating
diVusers, on the rear wall of Carnegie Hall in New

is known as a comb Ž lter response. This unevenness
means that some frequencies will be emphasised, and

York, can be seen in F ig. 3. In the UK, Glyndebourne
Opera House uses convex curved surfaces on the rear

some deemphasised. This leads in turn to coloration
of the sound, where the timbre of notes is altered.

wall to disperse sound.

A diVuser, on the other hand, disperses the re ection

both temporally and spatially. The time response

Architectural trends

is greatly altered, with re ections arriving over a
longer period, and the frequency response shows less

In older, pre-twentieth century halls, such as the
Grosser Musikvereinssaal in Vienna, ornamentation

evidence of comb Ž ltering than the plane surface,
the peaks and troughs being uneven and randomly

appeared in a hall because it was the architectural
style of the day. Such walls were therefore naturally

spaced. This means that the re ected sound is a more

120 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2

3 Schroeder diffusers (QRDs) applied to the rear

wall of Carnegie Hall to prevent echoes

diVusing; large  at 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 in uenced composers including Brahms, Bruckner,
and Mahler. In the Grosser Musikvereinssaal the
in uence of the surface diVusion on the sound is very
obvious, with a diVuse sound resulting.

In the twentieth century, however, architectural trends

changed and large expanses of  at areas appeared
in many concert halls. The U K 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 diVusing
capability. The expanse of  at surfaces can lead to
distortion in the sound heard as a result of comb
Ž ltering, echoes, and other mechanisms. It is worth
noting, however, that it is also possible to design
very successful halls with  at surfaces, a good UK
example being Symphony Hall in Birmingham, which
has relatively little surface diVusion.

The key to good diVuser design is to Ž nd forms

that complement the architectural trends of the

4 Three different Schroeder diffusers: the original

day. The diVuser must not only meet the acoustic

design (top), a fractal design (middle), and an

speciŽ cation, it must Ž t in with the visual scheme

active diffuser (bottom); the diffusers are 0·6 m

wide, 0·6 m high and about 0·2 m deep

required by the architect. As discussed below, modern
diVuser designs have successfully been developed to
complement modern architectural forms.

Fowler Centre, New Zealand.4,5 Figure 5 illustrates this
application. Marshall and Hyde used large overhead

Schroeder diffusers

re ectors to provide early re ections to the audience
in the balconies in a revolutionary design. This was

The development of the modern diVuser began with
pioneering work by Manfred Schroeder, one of the

a layout whereby a hall could have good clarity, and
yet maintain a large volume for reverberation. The

twentieth century’s greatest acoustic engineers. In
the 1970s, Schroeder developed the phase grating

large volume partly comes from the space behind
the diVusers. Not many years before the design of

diVuser,3 also known as the Schroeder diVuser. An
example of the original design can be seen in F ig. 4

the hall, it had been established that lateral re ections
were important in concert halls as they promote a

(top). These diVusers oVered just what acoustic
designers were looking for, deŽ ned acoustic perform-

sense of envelopment or spatial impression in rooms.6
The evidence for the beneŽ cial eVects of lateral

ance based on very simple design equations; for while
it is know that ornamentation produces diVusion, it

re ections came from laboratory measurements on
human perception, which followed techniques pioneered

does this in an ill deŽ ned and haphazard fashion.

One of the pioneering applications of Schroeder

in experimental psychology. These measurements
showed that lateral re ections are important to get a

diVusers was by Marshall and Hyde in the Michael

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)

6 Scattered levels from a Schroeder diffuser

(left) and a plane surface (right) of the same

sense of involvement with the music. The need for

dimensions

lateral re ections in uenced Marshall and Hyde to
apply diVusers to the large overhead surfaces rather
than using  at re ectors.

moves around the surface on a semicircle. A series of
lobes are seen, eleven in this case, which are grating

Another reason for this new diVuser technology

entering wide use was its commercialisation by an

lobes generated by the periodicity of the surface
structure. Imagine viewing this polar response end

American company, R PG DiVusor Systems Inc.,
whose interest lay in studio design. Around the time

on so that a set of eleven bright spots are seen; it is
this type of image that X-ray crystallographers use

that Schroeder was developing the new diVuser, a
new design regime for listening and monitoring rooms

to determine crystal structures. The problem posed in
the acoustic case is however somewhat diVerent from

was invented. This was the Live End Dead End
(LEDE ) layout,7 which was later reŽ ned into the

the crystallographic challenge: in crystallography, the
diVraction patterns of the X-rays are used to deter-

R e ection Free Zone (R FZ ) design. DiVusers are
used in small spaces to disperse re ections that would

mine an unknown structure, whereas in the acoustic
case, the problem to be solved is how to determine

otherwise arrive early and at a high level and so cause
coloration of the timbre of the sound. This is some-

the correct surface structure to achieve a desired
polar response (or diVraction pattern). But before

times referred to as acoustic glare, and is again caused
by comb Ž ltering. Just when studio designers were

explaining how Schroeder solved this problem, it is
necessary to explain how diVusers scatter sound.

looking for diVusers to achieve this design, by happy
coincidence Schroeder diVusers became available.

At that time, one of the founders of R PG, and

Huygens

also one of the authors of this paper, Peter D’Antonio,
was a diVraction physicist at the Laboratory for the

The Huygens construction used in optics is one way
of explaining how diVusing surfaces scatter, though it

Structure of Matter at the Naval Research Laboratory
in Washington, DC. Knowing of his interest in music,

is only approximate in many acoustic cases. Consider
a planar surface, the situation illustrated in the upper

a colleague handed him the latest issue of Physics
Today with a cover photo of Manfred Schroeder

half of F ig. 7. When illuminated by a sound source,
a set of secondary sources is generated on the surface,

seated in an anechoic chamber. The associated article
suggested using Schroeder’s number theoretic diVusers

and these are shown as stars in F ig. 7. Each of these
secondary sources then radiates semicircular waves.

in concert halls. It became apparent that the re ection
phase gratings suggested by Schroeder were in eVect

By connecting points on these waves which are in
phase with each other, it is possible to visualise the

two-dimensional sonic crystals, which scatter sound
in the same way that three-dimensional crystal lattices

waves that are re ected from the surface. (These are
rather like ripples on the surface of water created

scatter electromagnetic waves. Since the diVraction
theory employed in X-ray crystallographic studies

when a stone is thrown into the water.) In this
situation, a simple plane wave at right angles to the

was also applicable to re ection phase gratings, it was
straightforward to model and design the re ection

surface is generated. The planar surface is acting like
an acoustic mirror, and the wave is unaltered on

phase gratings using techniques Ž rst developed in
crystallography.

re ection (except in its direction). The lower part of
Fig. 7 shows the case for a semicircular surface. In

F igure 6 ( left) shows the scattering from a

Schroeder diVuser in a polar response. A source

this case, the re ected waves are now semicircular in
shape. The wave has been altered by the surface, being

illuminates the surface, normal to the surface and
from the right. The polar response shows the energy

dispersed so that the sound re ects in all directions,
a characteristic desirable in a diVuser.

scattered from the surface (in decibels) as a receiver

122 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2

method, Schroeder turned to his favourite subject of
number theory, at Ž rst 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 Ž ne description of his supervisor:8

… 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

7 Huygens constructions for a plane wave reflected

from an explanation of principles to the development of

from a flat surface (above) and a curved surface

mathematical formulae, then he got up, and in a stately

(below): normal incidence source with incident

wavefronts excluded for clarity and secondary

very upright posture he wrote on a blackboard beside

sources shown as stars on the surface

him in his peculiarly beautiful handwriting: he always
succeeded through economy and deliberate arrangement
in making do with a rather small space. For numerical

Figure 8 shows the case for a simpliŽ ed re ection

examples, on whose careful completion he placed special

phase grating. In this situation, the re ected waves

value, he brought along the requisite data on little slips

are delayed because the waves must travel down

of paper.

each well and back up again before re ection. The

Although best known to modern physicists for

secondary sources have diVerent delays (phases)

G auss’s Law, which explains the properties of the

because of the diVerent well depths, and this alters

electric Ž eld, it is Gauss’s number theory work

the re ected wave. This again generates dispersion.

which is of most interest here, because it leads to
the quadratic residue sequence used in the design of

Sequences

the quadratic residue diVuser (QR D), an example
of which is shown in Fig. 4 (top). The formulation of a

In many ways, a re ection phase grating is acting

quadratic residue sequence is based on a prime number.

like an optical diVraction grating. In the acoustic

For the diVuser in F ig. 4 (top), the prime number is

case, the designer has control over the phases of the

7. The depth of the nth well is then proportional to

sound waves. To design a re ection phase grating, a

n2 modulo 7, where modulo indicates the smallest

method is required to determine an appropriate well

non-negative remainder. So the third well has a depth

depth sequence, which then generates a phase distri-

proportional to 32 modulo 7, in other words 2. The

bution on the surface of the diVuser to give the

sequence mapped out in this case is 0, 1, 4, 2, 2, 4, 1,

desired re ected wavefronts. In inventing such a

which can be seen in Fig. 4. (The quadratic residue
diVuser in F ig. 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 Ž tting
easier). If this quadratic residue sequence is used to
construct the diVuser, then the diVraction 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 diVuser,
the central (specular) lobe will be suppressed, while
the others remain at the same level.

8 Huygens constructions for a plane wave reflected

This sequence generation technique is an advanced

from a simplified Schroeder diffuser: the upper

form of the number games sometimes played by

plot shows wavefronts from two wells only

for clarity

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 diVuser 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 diVuser designs are typically considered over
the seven most important (80 Hz to 5 kHz). This is
one of the key diVerences between much research in
acoustics and optics, as optics researchers are often
concerned with a single frequency or a narrow band-
width. In acoustics, the bandwidth is much wider. To
make a wide bandwidth diVuser, the wells need to be
narrow and deep, but this makes the device very
impractical: Ž rst of all the structure becomes very
expensive to make, and second it becomes highly
absorbent (air is a viscous  uid, and as with any such

9 The

Mandelbrot

set

at

two

different

 uid it is diYcult to force into narrow wells, acoustic

magnifications

energy being lost in the process and converted to
heat). A solution to this problem has been developed,
inspired by chaos theory and fractals.

gratings require periodicity to work, many periods
of the diVuser are stacked next to each other. The

To handle many octaves, a diVuser needs to have

roughness on diVerent scales. The use of elements of

diVraction lobes are also a function of periodicity,
and to achieve even energy lobes (Schroeder’s

diVerent sizes is common in loudspeaker design. In
two way loudspeakers, for example, the large ‘woofer’

deŽ nition of optimum diVusion) requires the structure
to be periodic. Yet these diVraction lobes represent

is used to radiate bass frequencies, and the smaller
‘tweeter’ generates the treble sound. For diVusers,

energy concentrated into particular directions, with
a lack of re ected energy between. A better diVuser

some roughness needs to have dimensions metres
in size, and some needs to have dimensions centi-

would be one that distributed the energy more evenly
in all directions without lobes. Consequently there is

metres in size. Fractals are objects which have
scaleable properties, and one of the most famous is

a contradiction: to use the original number theory
design, periodicity is required, yet this results in worse

the Mandelbrot set, shown in Fig. 9 at two diVerent
magniŽ cations. When the set is greatly magniŽ ed, by

performance. A solution to this paradox was devised
by Angus,10 who showed that techniques devised

around four thousand times in F ig. 9, a very similar
shape to the original is seen. The same eVect can be

in mobile telephony could be adopted for diVusers.
These techniques are also applied to the design of

achieved for diVusers, as shown in F ig. 4 (middle).
In the surface shown, smaller diVusers are mounted

loudspeaker and microphone arrays.

Code D ivision Multiple Access (CDMA ) systems

within larger ones, the smaller scattering the high
frequencies, and the larger the low frequencies. This

are used in mobile telephony to enable multiple users
to use the same transmission bandwidth. Of interest

type of diVuser is rather Ž ttingly sold under the brand
name D iVractal.9 The example shown has diVusers

here are so called spread spectrum techniques. These
techniques take frequency (spectral ) components,

with two diVerent scales. Three diVerent sizes of
diVuser are needed to cover bass frequencies, the

and spread them over a frequency bandwidth. If the
lobes generated by the Schroeder diVuser are viewed

largest having a size about Ž fty times greater than
the smallest.

as spatial frequency components, then when spread
spectrum techniques are applied, the lobes will be

The issue of periodicity is curious, as in many ways

it is the curse of the structure. Since re ection phase

spread spatially. This eVect is shown in Fig. 10, where

124 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2

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 diVuser 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 Ž rst sequence used for modulating
diVusers was the maximum length sequence, also known
as a Galois Ž eld 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 eYcient measuring systems,
such as listening rooms, Ž lters, and loudspeakers.

The appearance of Schroeder diVusers 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 diVusers 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 diVuser

10 Scattered polar distribution from a periodic

looks ugly, it will not be used, however important

arrangement (light line) and a modulated

the treatment is to the acoustic design. Consequently,

arrangement (heavy line) of a quadratic residue

diffuser

there is a need for designs that complement modern
architectural trends. Figure 12 shows a modern diVuser
design on the ceiling of a cinema in Seattle. This is a

the spread spectrum process has enabled the scattered

curved diVuser designed to oVer a visual complement

energy to be redistributed from the three lobes in all

to the curtaining on the stage, while providing the

directions (all spatial frequencies).

required acoustic performance. The diVuser disperses

This idea can be applied to diVusers as follows.

re ections from the ceiling which would otherwise

Two base Schroeder diVusers are used. The Ž rst is

colour the sound. To design this sort of diVuser requires

a standard Schroeder diVuser, the second a diVuser

a new methodology, and for this it is possible to use

which produces the same polar response only with

numerical optimisation. This is a method commonly

opposite phase. This is easy to form by changing the

used in engineering, for example to design parts of

well depth sequence; in fact the second diVuser is the

the space shuttle. Numerical optimisation does not

reverse of the Ž rst. Figure 11 shows two base shapes

have the eYciency and elegance of number theory

arranged in a modulated array, in other words in a

design, but it is extremely eVective and robust.

random arrangement. The modulation sequence does

Numerical optimisation tasks a computer to search

not repeat, and so the diVusers are no longer periodic

for an optimum solution to a problem, for instance

and the periodicity lobes disappear. This produces a

it could look to optimise a car engine component

much more even polar response.

to minimise weight while ensuring suYcient strength

Whilst in principle it would be possible to  ip a

coin to determine the modulation sequence, it is far

and durability. In the case of diVusers, 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 scatter-
ing. The procedure works by iteration. The computer
starts by guessing some curved surface shape, and
the scattering from the surface is then predicted in

13 Optimised curved surface in the Edwina

terms of the polar response. The predicted polar

Palmer Hall, Hitchin, UK (photo courtesy Arup

response is rated for its quality in terms of a ‘Ž gure

Acoustics)

of merit’, which by a process of trial and error the
computer can try and minimise by changing the sur-
face shape. The process continues until an optimum

about on land ). As with biological populations, to
enable dramatic changes in the population of shapes,

design is found, which occurs when a minimum in
the Ž gure of merit is determined. The search process

mutation is required. This is a random procedure
whereby a small probability is deŽ ned of any gene in

is not completely random because this would be too
slow – fortunately mathematical algorithms have been

the child sequence being randomly changed, rather
than coming direct from the parents.

developed to allow the search to be done eYciently.
Currently the most popular approach, using a so

Selecting shapes to die oV can be done randomly,

with the least Ž t (the poorest diVusers) being most

called genetic algorithm, models the problem as
an evolutionary process, using survival of the Ž ttest

likely to be selected. In biological evolution, the Ž ttest
are most likely to breed and pass on their genes, and

principles to carry out an eYcient search.

A genetic algorithm mimics the process of evolution

the least Ž t are the most likely to die; this is also true
with an artiŽ cial genetic algorithm used to design

that occurs in biology. A population of individuals
is randomly formed. Each individual is determined

diVusers. Using these principles, the Ž tness of successive
populations should improve, and the process is con-

by its ‘genes’, which in this case are simply a set of
numbers which describe the curved surface shape.

tinued until the population becomes suYciently Ž t
for the shape produced to be classiŽ ed as optimum.

Each individual (or shape) has a Ž tness value (Ž gure
of merit) that indicates how good it is at diVusing

F igure 13 shows an example of another optimised

curved surface. In this case a concave wall in a music

sound. Over time new populations are produced by
combining (breeding) previous shapes, and the old

practice room would have caused problems of focus-
ing if untreated. Concave walls focus sound to a

population dies oV. OVspring are produced by pairs
of parents breeding, and the oVspring have genes that

point in the same way that concave mirrors focus
light. In acoustics, focusing eVects are obvious in

are a composite of the parents’. The oVspring shape
will then have features drawn from the parent shapes,

whispering galleries, such as the dome in St Paul’s
Cathedral in London. F igure 14 shows the polar

in the same way that facial features of a child can
often be recognised in the parents. A common method

response for a concave wall, revealing that the
scattered energy level is much greater for the receiver

of mixing genes is called multipoint crossover. For
each gene, there is a Ž fty per cent chance of the

at the focal point. In treating this focusing problem,
it would have been possible to add absorption on the

child’s gene coming from parent A, and a Ž fty per
cent chance of it being from parent B.

wall to remove the re ections from the curved surface.
But this would have removed energy from one side of

If all that happened was combination of the

parents’ genes, then the system would never look

the orchestra, and these re ections are needed so the
musicians can hear themselves and their colleagues.

outside the parent population for better solutions
(the ‘Ž sh’ diVuser would never get lungs and walk

R e ections are needed for the musicians to keep in

126 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2

technique exists for folding a sequence into a two-
dimensional array, a technique commonly referred to
as the Chinese R emainder Theorem.

An example of a Chinese remainder problem

was posed by Sun Tsu Suan-Ching in the fourth
century ad: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,

14 Scattering from a concave arc (light line)

and X-ray astronomy. Incidentally, one answer to

compared with an optimised curved diffuser

the above problem is 23. The mask shown in F ig. 15

(heavy line)

is a length 1023 maximum length sequence which has
been folded into a 31 by 33 array using the Chinese

time, form a good tone, and blend the overall ensemble

R emainder Theorem.

sound. The solution is therefore to use diVusers, as

The problem with maximum length sequences is

these remove the focusing eVect from the sound while

that they are devised for systems that are bipolar,

preserving the acoustic energy. Figure 14 illustrates the

consisting of +1s and ­ 1s. The hybrid surface on

eVectiveness of the diVuser in dispersing the focused

the other hand produces areas of no re ection (0s)

sound.

and re ection (1s), and so is inherently unipolar. This
can be a problem when designing these diVusers.
Most electronic systems have bipolar capabilities, and

Absorption for diffusion

can produce signals of the opposite sign, but this

In recent years, interest has been returning to number

is not true of Ž bre optic systems, and hence optical

theory to generate a diVerent kind of diVuser, the

sequences have been developed. Optical sequences

hybrid surface. Construction of a hybrid surface is

were developed for use in Ž bre optical CDMA pro-

shown schematically in F ig. 15. It consists of a piece

cesses. Fibre optic CDMA sequences, where the light

of acoustic absorbent covered with a perforated mask,

intensity is either on or oV, cannot have cancellation,

the mask then being hidden from view by a thin piece

and hence use unipolar sequences. These sequences

of acoustically transparent cloth. Where there are

can be used to design hybrid diVusers.

holes in the mask, absorption is generated; where the

F igure 16 compares the scattering from a hybrid

sound strikes a solid part of the mask, re ection occurs.

absorber–diVuser with that from a plane surface. The

This then forms a surface which partially absorbs,

hybrid surface provides greater dispersion. This dis-

and any re ected energy is diVused because of the

persion can be improved even further if the hybrid

random arrangement of holes. The key to good dis-

surface is bent and made corrugated, as this breaks

persion is the arrangement of the holes, which is best

up the specular re ection component further. This

done following a pseudo random binary sequence with

type of design is proving to be very popular in studio

optimal autocorrelation properties ( least repetition).

control and listening rooms.

When the sequence has a 1, a hole is drilled in the
mask, when it has a 0, no hole is drilled. Any repetition

Active technology

in the sequence will lead to lobes, so sequences are
needed which are dissimilar from shifted versions

The Ž nal technology to be discussed here is active

of themselves. Again, number theory can provide a

control technology. Active noise control has caused

whole range of sequences.

much interest in the last twenty years or so, but the

Angus12 started by looking at maximum length

application of this technology is not widespread

sequences, the same sequences which were used origin-

because the practical implementation is costly and

ally to modulate Schroeder diVusers. The problem is

problematic. It has, however, been used successfully

that maximum length sequences are just strings of 1s

in controlling ventilation, car, and aircraft noise.

and 0s, and what is needed for hybrid surfaces is

Attention has now been turned to whether an active

a two-dimensional array of numbers. Fortunately, a

diVuser 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
diVusers do something slightly diVerent: they start by
cancelling out a re ection, but then the secondary
source adds in an artiŽ cial re ection which mimics
the characteristics of a diVuser.

F igure 4 (bottom) shows an artist’s impression of

15 Construction of a hybrid surface: porous

absorber (left), mask (middle), and cloth (right)

an active diVuser. 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 diVerent
aspects of acoustic design. Currently, there is much
interest in understanding the role of surface diVusers.
As this paper has shown, much is now known about
the design of diVusers, but many questions remain
unanswered. The most important question to answer
is how much diVusion should be applied, and where
diVusers should be used. While acoustic designers
have produced many innovative new designs, the
understanding of how and why to apply diVusers lags
behind and is still largely based on precedence.

In 2001, Manfred Schroeder attended the Inter-

national Congress on Acoustics in R ome. H e com-
mented on some photos of optimised curved surfaces,
saying we had produced diVusers which were beautiful.
We were  attered by such a tribute from the pioneer
of modern acoustic diVuser 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 scatter-

Acknowledgements

ing from surfaces: hybrid surface (above),

plane hard surface (below)

Figure 1 is taken from t. j. cox and p. d’antonio:
A coustic A bsorbers and DiVusers; 2003, London,
Spon; Figs. 2, 6–8, 10, 14, and 16 from p. d’antonio

bottom of some of the wells. By changing how these

and t. j. cox: ‘D iVusor application in rooms’,

controlled loudspeakers respond to incident sound, it

A pplied A coustics, 2000, 60, 113–142; and Fig. 3 from

is possible to change the characteristics of the wells.

p. d

’antonio and t. j. cox: ‘Two decades of sound

For example, it would be possible to make a well

diVusor design and development part 1: applications

appear longer than it really is. But why would one

and design’, Journal of the A udio Engineering Society,

go to such lengths, as the active system with its

1998, 46, 955–976.

electronics and control structure is diYcult to develop
and expensive to implement? Active systems are
being investigated to gain additional diVusion at
low frequency, in a range where it is very diYcult to

Notes and literature cited

gain diVusion by normal (passive) means. To obtain

1. l. l. beranek: Concert and Opera H alls: H ow they

diVusion at low frequency is very diYcult because

Sound, 69–74; 1996, Woodbury, N Y, Acoustical Society

the size of acoustic waves (wavelength) becomes many

of America.

metres in size, which means that a diVuser should be

2. s. moss: ‘Things can only get better’, Guardian, 1999,

extremely large as the diVuser’s dimensions need to

23 July.

be comparable with the wavelength. Active diVusion

3. m. r. schroeder: ‘Binaural dissimilarity and optimum

is a method whereby the acoustic Ž eld can be dis-

ceilings for concert halls: more lateral sound diVusion’,

turbed at a low frequency using a shallower device.

Journal of the Acoustical S ociety of America, 1979,
65, 958–963.

These devices, however, are very much in their infancy

4. a. h. marshall and j. r. hyde: ‘Some practical con-

and it remains to be seen whether a practical device

siderations in the use of quadratic residue diVusing

can be constructed.

surfaces’, Proceedings of the 10th International
Congress on Acoustics, Sydney, 1980, paper E7.3.

Summary

5. a. h. marshall, j. r. hyde and m. f. e. barron: ‘The

acoustical design of Wellington Town H all: design

Much has been learnt about the design of concert

development, implementation and modelling results’,

halls over the last hundred years. A little over a

Proceedings of the Institute of Acoustics, Edinburgh,

century ago, the design of a hall was based on a com-

1982.

bination of precedence and luck. Nowadays, acoustic

6. m. barron: ‘The subjective eVects of Ž rst re ections in

science means that concert hall design involves a

concert halls – the need for lateral re ections’, Journal
of S ound and V ibration, 1971, 15, 475–494.

combination of precedence, engineering, and art. The

128 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2

7. d. david and c. davis: ‘The LED E concept for the

10. j. a. s. angus: ‘U sing grating modulation to achieve

wideband large area diVusers’, A pplied Acoustics, 2000,

control of acoustic and psychoacoustic parameters
in recording control rooms’, Journal of the Audio

60, 143–165.

11. See www-gap.dcs.st-and.ac.uk/~history/Mathematicians/

Engineering S ociety, 1980, 28, 585–595.

8. See www-gap.dcs.st-and.ac.uk/~history/Mathematicians/

G alois.html.

12. j. a. s. angus: ‘Sound diVusers using reactive absorption

Gauss.html.

9. p. d’antonio and j. konnert: ‘The QRD diVractal: a

grating’, Proceedings of the 98th Convention of the
Audio Engineering Society, 1995, preprint 3953.

new one- or two-dimensional fractal sound diVusor’,
Journal of the A udio Engineering S ociety, 1992, 40,

13. d. wells: The Penguin Book of Curious and Interesting

Puzzles; 1992, London, Penguin.

113–129.

Trevor Cox is Reader in Acoustics at Salford University, UK.
H e graduated with a degree in physics from Birmingham

U niversity, U K in 1988 and went on to study auditorium
acoustics at Salford U niversity, where he was awarded his

doctorate in 1992. Between 1993 and 1995, D r Cox worked
at South Bank University, London. A large proportion of

Dr Trevor J. Cox

his research concerns diVusing surfaces: his use of numerical

Acoustics R esearch Centre

optimisation techniques has led to the worldwide appli-

U niversity of Salford

cation of innovative diVusing surfaces. This paper is based

Salford M5 4WT

on his Isambard K ingdom Brunel Award lecture given to

U K

the British Association for the Advancement of Science’s

[e-mail]

Festival of Science held in Leicester in 2002.

Peter D ’Antonio received his BS degree from St John’s
U niversity in 1963 and his PhD from the Polytechnic

Institute of Brooklyn in 1967. D r D ’Antonio has specialised
in a wide variety of scientiŽ c disciplines including spectro-

scopy, X-ray and electron diVraction, electron microscopy,
software development, and architectural acoustics. H e

carried out research in diVraction physics at the N aval

Dr Peter D’Antonio

R esearch Laboratory, Washington, D C for thirty years.

R PG D iVusor Systems Inc.

He is now president of RPG D iVusor Systems Inc., which

651-C Commerce D rive

was founded in 1983 to carry out basic research in room

U pper Marlboro

acoustics and to develop designs and innovative number

MD 20774

theoretic architectural surfaces, to enhance the acoustics of

U SA

critical listening and performance environments.

INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 129