Engineering Art-The Science of Concert Hall Acoustics

Engineering art: the science of concerthall acoustics

TREVOR J. COXAcoustics Research Centre, University of Salford, UK

PETER D’ANTONIORPG Diffusor Systems, Upper Marlboro, MD, USA

Modern concert hall design uses science and engineering to make an acoustic which embellishes and enhances theartistry of the musicians. The modern discipline of concert hall acoustics is a little over a hundred years old, andover the last century much has been learnt about how to ensure the audience receives high quality sound. Duringthis period, knowledge from a large number of disciplines has been exploited. It is the intention of this paper toillustrate how disciplines as diverse as X-ray crystallography, psychology, and mobile telephony have influencedacoustic design. The paper will concentrate on the design of acoustic diffusers for concert halls, as this is a topiccurrently 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 soundpart of the sound heard in a classical music perform- beautiful. Modern acoustic science cannot guaranteeance; the concert hall embellishes the sound. Music a great acoustic every time, but if advice is followed,outdoors may be popular when accompanied by fire- technical knowledge should ensure that bad halls areworks, but the quality of the sound is usually poor. not built, while significantly increasing the chance ofMove indoors and the sound comes alive, enveloping greatness being achieved. What makes a good concertand 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 memberorchestra, there are no reflections from walls, and the in a complex manner. The acoustic of a concert hallsound appears distant. When music is played in a might be perfect, but if the audience get soaked inroom, reflections from the walls, ceiling, and floor rain walking from the car park, they are unlikely toadd reverberation and other characteristics to the rate the experience highly. So a great design is aboutsound. As the conductor Sir Adrian Boult said:1 ‘The accommodating a wide range of requirements, and notideal concert hall is obviously that into which you just acoustics. This article, however, will concentratemake a not very pleasant sound and the audience on the acoustics. When designing a hall, the acousticreceives something that is quite beautiful. I maintain engineer will look at many acoustic factors: thethat this really can happen in Boston Symphony Hall; background noise level, the amount of reverberationit is our ideal.’ This quote is of great historical

(the decay of sound after a note has stopped beinginterest, because Boston Symphony Hall was the first

played), the amount of sound arriving from the side,concert hall where the principles of reverberationthe reflections musicians receive that are necessarywere applied. These principles were developed aboutfor them to play in time and form a good tone, anda hundred years ago by Wallace Sabine, who wasso on. The overall focus of this paper will be on thethe first person to apply modern scientific principlesrole of surface diffusers.to room design and pioneered modern concert hall

Currently there is much debate about the roleacoustics. Boston Symphony Hall is still seen as oneof surface diffusers in concert halls. To take twoof the greatest auditoria in the world. Conversely, aexamples, one eminent concert hall designer regularlypoor hall can have a detrimental effect on the enjoy-claims that too much diffusion is detrimental to thement of a performance, something that can affect thesound quality of the upper strings, while in contrastmusicians as well. As the conductor Sir Simon Rattleother acoustic engineers have blamed the disappoint-said of one hall (which will remain nameless): ‘Theing acoustics of certain major halls on a lack of*** hall is the worst major concert arena in Europe.surface diffusion. We will return to these contradictoryThe will to live slips away in the first half hour ofopinions later, and will try to shed some light on whyrehearsal.’2they arise, but first it is necessary to describe what aSince Wallace Sabine’s work on room acoustics,diffuser is and to outline the current state of the artmuch 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 DOI 10.1179/030801803225010412

Published by Maney for the Institute of Materials, Minerals and Mining

1 Spatial and temporal responses of sound reflectedfrom a plane flat surface (above) and a diffuser(below)

TreatmentsTo alter the acoustics of an existing room, treatment

2 Temporal and frequency responses from flatis usually placed on the boundaries, for example if(above) and diffusing (below) surfaces

an office is too reverberant or lively, absorbent ceilingtiles or carpet might be used to absorb and so removesome 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 thewalls and ceiling (the floor already has the audience Figure 1 also shows a cross-section through a diffuser,

in this case a reflection phase grating consisting of aand seating on it, and so is difficult to alter). Thereare three basic forms of treatment, large flat surfaces, series of wells of different depths but the same width.

There are many types of diffuser, as explained below,absorbers, and diffusers. Absorbers, such as the ceil-ing tiles and carpet mentioned already, are not often but in principle any non-flat corrugated surface will

have some kind of diffusing ability. Diffusers can alsoused in large concert halls, because they remove soundenergy 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 keymust be conserved because there is a limit to howmuch energy an orchestra can produce. Consequently, feature being that the material should be hard and

acoustically non-absorbent.the designer must choose between flat surfaces ordiffusers. Diffusers are used in a variety of ways. For instance,

they can be used to reduce echoes arising from theFigure 1 contrasts the spatial and temporal responsesof flat and diffusing 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 concertlistener would receive if they were close to one ofthe surfaces, and if no other surfaces were present. hall, and if a strong reflection comes back from the

rear wall to the front of the hall, this may be heardA flat surface behaves like a mirror reflecting 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 rearin the specular reflection direction, and with equalangles of incidence and reflection. The time response wall to absorb the acoustic energy, preventing the

reflection occurring and so curing the echo problemshows the similarity between the direct sound andthe reflection: the flat surface does little to the sound (such a solution was adopted in London’s Royal

Festival Hall ). However absorption removes acousticexcept change the direction in which it propagates.Figure 2 shows the resulting frequency response. This energy and so reduces the loudness of the orchestra.

A modern solution is therefore to use diffusers toshows how the level (or volume) of the sound willvary as the pitch (or frequency) of a note changes. break up and disperse the troublesome reflections,

which can be done without loss of acoustic energy.The frequency response of the flat surface is uneven,with a regularly spaced set of peaks and troughs, and An example of the use of reflection phase grating

diffusers, on the rear wall of Carnegie Hall in Newis known as a comb filter response. This unevennessmeans that some frequencies will be emphasised, and York, can be seen in Fig. 3. In the UK, Glyndebourne

Opera House uses convex curved surfaces on the rearsome deemphasised. This leads in turn to colorationof the sound, where the timbre of notes is altered. wall to disperse sound.

A diffuser, on the other hand, disperses the reflectionboth temporally and spatially. The time response Architectural trendsis greatly altered, with reflections arriving over alonger period, and the frequency response shows less In older, pre-twentieth century halls, such as the

Grosser Musikvereinssaal in Vienna, ornamentationevidence of comb filtering 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 naturallyspaced. This means that the reflected sound is a more

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

3 Schroeder diffusers (QRDs) applied to the rearwall of Carnegie Hall to prevent echoes

diffusing; large flat surfaces were very rare. TheGrosser Musikvereinssaal is an interesting exampleto acousticians, because it is often cited as one of thebest halls in the world. The hall sound is thought tohave influenced composers including Brahms, Bruckner,and Mahler. In the Grosser Musikvereinssaal theinfluence of the surface diffusion on the sound is veryobvious, with a diffuse sound resulting.

In the twentieth century, however, architectural trendschanged and large expanses of flat areas appearedin many concert halls. The UK has many post-warconcert halls which have very little ornamentation,such as the Colston Hall in Bristol. The style of theday was to produce clean lines following a moderniststyle, but these surfaces then had little or no diffusingcapability. The expanse of flat surfaces can lead todistortion in the sound heard as a result of combfiltering, echoes, and other mechanisms. It is worthnoting, however, that it is also possible to designvery successful halls with flat surfaces, a good UKexample being Symphony Hall in Birmingham, whichhas relatively little surface diffusion.

The key to good diffuser design is to find formsthat complement the architectural trends of the

4 Three different Schroeder diffusers: the originalday. The diffuser must not only meet the acoustic design (top), a fractal design (middle), and anspecification, it must fit in with the visual scheme active diffuser (bottom); the diffusers are 0·6m

wide, 0·6m high and about 0·2m deeprequired by the architect. As discussed below, moderndiffuser designs have successfully been developed tocomplement modern architectural forms.

Fowler Centre, New Zealand.4,5 Figure 5 illustrates thisapplication. Marshall and Hyde used large overheadSchroeder diffusersreflectors to provide early reflections to the audiencein the balconies in a revolutionary design. This wasThe development of the modern diffuser began with

pioneering work by Manfred Schroeder, one of the a layout whereby a hall could have good clarity, andyet maintain a large volume for reverberation. Thetwentieth century’s greatest acoustic engineers. In

the 1970s, Schroeder developed the phase grating large volume partly comes from the space behindthe diffusers. Not many years before the design ofdiffuser,3 also known as the Schroeder diffuser. An

example of the original design can be seen in Fig. 4 the hall, it had been established that lateral reflectionswere important in concert halls as they promote a(top). These diffusers offered just what acoustic

designers were looking for, defined acoustic perform- sense of envelopment or spatial impression in rooms.6The evidence for the beneficial effects of lateralance based on very simple design equations; for while

it is know that ornamentation produces diffusion, it reflections came from laboratory measurements onhuman perception, which followed techniques pioneereddoes this in an ill defined and haphazard fashion.

One of the pioneering applications of Schroeder in experimental psychology. These measurementsshowed that lateral reflections are important to get adiffusers was by Marshall and Hyde in the Michael

INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2 121

5 Schroeder diffusers in the Michael FowlerCentre, New Zealand (photo courtesy Dr HaroldMarshall of Marshall Day Acoustics)

6 Scattered levels from a Schroeder diffuser(left) and a plane surface (right) of the samesense of involvement with the music. The need fordimensionslateral reflections influenced Marshall and Hyde to

apply diffusers to the large overhead surfaces ratherthan using flat reflectors. moves around the surface on a semicircle. A series of

lobes are seen, eleven in this case, which are gratingAnother reason for this new diffuser technologyentering wide use was its commercialisation by an lobes generated by the periodicity of the surface

structure. Imagine viewing this polar response endAmerican company, RPG Diffusor 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 usethat Schroeder was developing the new diffuser, anew design regime for listening and monitoring rooms to determine crystal structures. The problem posed in

the acoustic case is however somewhat different fromwas invented. This was the Live End Dead End(LEDE) layout,7 which was later refined into the the crystallographic challenge: in crystallography, the

diffraction patterns of the X-rays are used to deter-Reflection Free Zone (RFZ) design. Diffusers areused in small spaces to disperse reflections that would mine an unknown structure, whereas in the acoustic

case, the problem to be solved is how to determineotherwise arrive early and at a high level and so causecoloration of the timbre of the sound. This is some- the correct surface structure to achieve a desired

polar response (or diffraction pattern). But beforetimes referred to as acoustic glare, and is again causedby comb filtering. Just when studio designers were explaining how Schroeder solved this problem, it is

necessary to explain how diffusers scatter sound.looking for diffusers to achieve this design, by happycoincidence Schroeder diffusers became available.

At that time, one of the founders of RPG, and Huygensalso one of the authors of this paper, Peter D’Antonio,was a diffraction physicist at the Laboratory for the The Huygens construction used in optics is one way

of explaining how diffusing surfaces scatter, though itStructure of Matter at the Naval Research Laboratoryin Washington, DC. Knowing of his interest in music, is only approximate in many acoustic cases. Consider

a planar surface, the situation illustrated in the uppera colleague handed him the latest issue of PhysicsToday with a cover photo of Manfred Schroeder half of Fig. 7. When illuminated by a sound source,

a set of secondary sources is generated on the surface,seated in an anechoic chamber. The associated articlesuggested using Schroeder’s number theoretic diffusers and these are shown as stars in Fig. 7. Each of these

secondary sources then radiates semicircular waves.in concert halls. It became apparent that the reflectionphase gratings suggested by Schroeder were in effect By connecting points on these waves which are in

phase with each other, it is possible to visualise thetwo-dimensional sonic crystals, which scatter soundin the same way that three-dimensional crystal lattices waves that are reflected from the surface. (These are

rather like ripples on the surface of water createdscatter electromagnetic waves. Since the diffractiontheory 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 thewas also applicable to reflection phase gratings, it wasstraightforward to model and design the reflection surface is generated. The planar surface is acting like

an acoustic mirror, and the wave is unaltered onphase gratings using techniques first developed incrystallography. reflection (except in its direction). The lower part of

Fig. 7 shows the case for a semicircular surface. InFigure 6 ( left) shows the scattering from aSchroeder diffuser in a polar response. A source this case, the reflected waves are now semicircular in

shape. The wave has been altered by the surface, beingilluminates the surface, normal to the surface andfrom the right. The polar response shows the energy dispersed so that the sound reflects in all directions,

a characteristic desirable in a diffuser.scattered from the surface (in decibels) as a receiver


method, Schroeder turned to his favourite subject ofnumber theory, at first glance a rather obscure formof abstract mathematics which studies the propertiesof natural numbers, but which in practice has provedto be very useful to scientists and engineers.

In the late eighteenth century, Carl FriedrichGauss developed the law of quadratic reciprocity,well known to mathematicians working in numbertheory. Dedekind was a doctoral student of Gaussand wrote a fine description of his supervisor:8

… usually he sat in a comfortable attitude, looking down,slightly stooped, with hands folded above his lap. Hespoke quite freely, very clearly, simply and plainly: butwhen he wanted to emphasise a new viewpoint … thenhe lifted his head, turned to one of those sitting next tohim, and gazed at him with his beautiful, penetratingblue eyes during the emphatic speech. … If he proceeded7 Huygens constructions for a plane wave reflectedfrom an explanation of principles to the development offrom a flat surface (above) and a curved surfacemathematical 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 besidesources shown as stars on the surface him in his peculiarly beautiful handwriting: he always

succeeded through economy and deliberate arrangementin making do with a rather small space. For numericalFigure 8 shows the case for a simplified reflectionexamples, on whose careful completion he placed specialphase grating. In this situation, the reflected wavesvalue, he brought along the requisite data on little slipsare delayed because the waves must travel downof paper.each well and back up again before reflection. The

Although best known to modern physicists forsecondary sources have different delays (phases)Gauss’s Law, which explains the properties of thebecause of the different well depths, and this alterselectric field, it is Gauss’s number theory workthe reflected wave. This again generates dispersion.which is of most interest here, because it leads tothe quadratic residue sequence used in the design of

Sequences the quadratic residue diffuser (QRD), an exampleof which is shown in Fig. 4 (top). The formulation of aIn many ways, a reflection phase grating is actingquadratic residue sequence is based on a prime number.like an optical diffraction grating. In the acousticFor the diffuser in Fig. 4 (top), the prime number iscase, the designer has control over the phases of the7. The depth of the nth well is then proportional tosound waves. To design a reflection phase grating, an2 modulo 7, where modulo indicates the smallestmethod is required to determine an appropriate wellnon-negative remainder. So the third well has a depthdepth sequence, which then generates a phase distri-proportional to 32 modulo 7, in other words 2. Thebution on the surface of the diffuser to give thesequence mapped out in this case is 0, 1, 4, 2, 2, 4, 1,desired reflected wavefronts. In inventing such awhich can be seen in Fig. 4. (The quadratic residuediffuser in Fig. 4 has zero depth wells on both ends,but these are half the width of the others, a usefulsleight of hand to make manufacturing and fittingeasier). If this quadratic residue sequence is used toconstruct the diffuser, then the diffraction or gratinglobes generated all have the same energy, as shownin 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 rnmodulo N, where r is a ‘primitive root’ of N and Nmust be a prime. For r to be a primitive root, thesequence generated must contain every integer from1 to N−1 without repeat. Thus for N=7, r=3 is aprimitive 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, whilethe others remain at the same level.

8 Huygens constructions for a plane wave reflectedThis sequence generation technique is an advancedfrom a simplified Schroeder diffuser: the upper

form of the number games sometimes played byplot shows wavefronts from two wells onlyfor clarity children: with some simple generation rules, fantastic


answers are produced. But these sequences areseriously useful, and have been developed for diverseapplications in astronomy, in error checking systemsfor computers, and in digital audio data and mobiletelephony. As Schroeder is fond of saying, numbertheory is unreasonably useful, considering it wasdeveloped as an abstract mathematical subject.

ImprovementsWhile the basic Schroeder diffuser based on numbertheory sequences is an ingenious invention, it hasseveral Achilles heels. Since the development of theinitial design, several revisions have been suggestedto improve performance. These are used to overcomekey weaknesses related to bandwidth, periodicity, andappearance and will be discussed in detail below.

In concert hall acoustics, designs have to workover a wide bandwidth. Human hearing extends overbetween ten and eleven octaves (20 Hz to 20 kHz),and diffuser designs are typically considered overthe seven most important (80 Hz to 5 kHz). This isone of the key differences between much research inacoustics and optics, as optics researchers are oftenconcerned with a single frequency or a narrow band-width. In acoustics, the bandwidth is much wider. Tomake a wide bandwidth diffuser, the wells need to benarrow and deep, but this makes the device veryimpractical: first of all the structure becomes veryexpensive to make, and second it becomes highlyabsorbent (air is a viscous fluid, and as with any such

9 The Mandelbrot set at two differentfluid it is difficult to force into narrow wells, acousticmagnifications

energy being lost in the process and converted toheat). A solution to this problem has been developed,inspired by chaos theory and fractals. gratings require periodicity to work, many periods

of the diffuser are stacked next to each other. TheTo handle many octaves, a diffuser needs to haveroughness on different scales. The use of elements of diffraction lobes are also a function of periodicity,

and to achieve even energy lobes (Schroeder’sdifferent sizes is common in loudspeaker design. Intwo way loudspeakers, for example, the large ‘woofer’ definition of optimum diffusion) requires the structure

to be periodic. Yet these diffraction lobes representis used to radiate bass frequencies, and the smaller‘tweeter’ generates the treble sound. For diffusers, energy concentrated into particular directions, with

a lack of reflected energy between. A better diffusersome roughness needs to have dimensions metresin size, and some needs to have dimensions centi- would be one that distributed the energy more evenly

in all directions without lobes. Consequently there ismetres in size. Fractals are objects which havescaleable properties, and one of the most famous is a contradiction: to use the original number theory

design, periodicity is required, yet this results in worsethe Mandelbrot set, shown in Fig. 9 at two differentmagnifications. When the set is greatly magnified, by performance. A solution to this paradox was devised

by Angus,10 who showed that techniques devisedaround four thousand times in Fig. 9, a very similarshape to the original is seen. The same effect can be in mobile telephony could be adopted for diffusers.

These techniques are also applied to the design ofachieved for diffusers, as shown in Fig. 4 (middle).In the surface shown, smaller diffusers are mounted loudspeaker and microphone arrays.

Code Division Multiple Access (CDMA) systemswithin larger ones, the smaller scattering the highfrequencies, and the larger the low frequencies. This are used in mobile telephony to enable multiple users

to use the same transmission bandwidth. Of interesttype of diffuser is rather fittingly sold under the brandname Diffractal.9 The example shown has diffusers here are so called spread spectrum techniques. These

techniques take frequency (spectral ) components,with two different scales. Three different sizes ofdiffuser are needed to cover bass frequencies, the and spread them over a frequency bandwidth. If the

lobes generated by the Schroeder diffuser are viewedlargest having a size about fifty times greater thanthe smallest. as spatial frequency components, then when spread

spectrum techniques are applied, the lobes will beThe issue of periodicity is curious, as in many waysit is the curse of the structure. Since reflection phase spread spatially. This effect is shown in Fig. 10, where


better to apply a sequence from number theory (yetanother case of number theory being unreasonablyuseful ). This is particularly true if only a few periodsof the diffuser are being considered, as mathematicianshave produced methods for generating the bestpossible sequences with the least amount of repetition.Alternatively it is possible to task a computer tolaboriously search for the best sequences, but this israther slow. The first sequence used for modulatingdiffusers was the maximum length sequence, also knownas a Galois field because of its basis in mathematicsdeveloped by the nineteenth century mathematicianEvariste Galois. Galois unfortunately met an untimelydeath in a duel at the age of twenty-one, but notbefore he had sketched out some very importantmathematical concepts. As his director of studieswrote:11 ‘It is the passion for mathematics whichdominates him, I think it would be best for him ifhis parents would allow him to study nothing butthis, he is wasting his time here and does nothingbut torment his teachers and overwhelm himself withpunishments.’ Maximum length sequences are usedwidely in digital systems; in acoustics they are bestknown for producing efficient measuring systems,such as listening rooms, filters, and loudspeakers.

The appearance of Schroeder diffusers is animportant impediment to their use, especially givencurrent taste in architecture and interior design, whichtends to favour curves and more organic shapes.With Schroeder diffusers the acoustic treatmentimposes a distinctive visual aesthetic, and while thereare architects who favour the argument that formshould follow function, most prefer to determinetheir own aesthetic. If an architect thinks a diffuser

10 Scattered polar distribution from a periodiclooks ugly, it will not be used, however importantarrangement (light line) and a modulatedthe treatment is to the acoustic design. Consequently,arrangement (heavy line) of a quadratic residue

diffuser there is a need for designs that complement modernarchitectural trends. Figure 12 shows a modern diffuserdesign on the ceiling of a cinema in Seattle. This is athe spread spectrum process has enabled the scatteredcurved diffuser designed to offer a visual complementenergy to be redistributed from the three lobes in allto the curtaining on the stage, while providing thedirections (all spatial frequencies).required acoustic performance. The diffuser dispersesThis idea can be applied to diffusers as follows.reflections from the ceiling which would otherwiseTwo base Schroeder diffusers are used. The first iscolour the sound. To design this sort of diffuser requiresa standard Schroeder diffuser, the second a diffusera new methodology, and for this it is possible to usewhich produces the same polar response only withnumerical optimisation. This is a method commonlyopposite phase. This is easy to form by changing theused in engineering, for example to design parts ofwell depth sequence; in fact the second diffuser is thethe space shuttle. Numerical optimisation does notreverse of the first. Figure 11 shows two base shapeshave the efficiency and elegance of number theoryarranged in a modulated array, in other words in adesign, but it is extremely effective and robust.random arrangement. The modulation sequence does

Numerical optimisation tasks a computer to searchnot repeat, and so the diffusers are no longer periodicfor an optimum solution to a problem, for instanceand the periodicity lobes disappear. This produces ait could look to optimise a car engine componentmuch more even polar response.to minimise weight while ensuring sufficient strengthWhilst in principle it would be possible to flip a

coin to determine the modulation sequence, it is far 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


12 The Cinerama in Seattle, WA with a diffusingceiling (photo courtesy University of Salfordand Harris-Grant Associates)

looks for the surface shape which gives optimal scatter-ing. The procedure works by iteration. The computerstarts by guessing some curved surface shape, andthe scattering from the surface is then predicted in

13 Optimised curved surface in the Edwinaterms of the polar response. The predicted polarPalmer Hall, Hitchin, UK (photo courtesy Arup

response is rated for its quality in terms of a ‘figure Acoustics)of merit’, which by a process of trial and error thecomputer 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 inthe figure of merit is determined. The search process mutation is required. This is a random procedure

whereby a small probability is defined of any gene inis not completely random because this would be tooslow – 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 efficiently.Currently the most popular approach, using a so Selecting shapes to die off can be done randomly,

with the least fit (the poorest diffusers) being mostcalled genetic algorithm, models the problem asan evolutionary process, using survival of the fittest likely to be selected. In biological evolution, the fittest

are most likely to breed and pass on their genes, andprinciples to carry out an efficient search.A genetic algorithm mimics the process of evolution the least fit are the most likely to die; this is also true

with an artificial genetic algorithm used to designthat occurs in biology. A population of individualsis randomly formed. Each individual is determined diffusers. Using these principles, the fitness of successive

populations should improve, and the process is con-by its ‘genes’, which in this case are simply a set ofnumbers which describe the curved surface shape. tinued until the population becomes sufficiently fit

for the shape produced to be classified as optimum.Each individual (or shape) has a fitness value (figureof merit) that indicates how good it is at diffusing Figure 13 shows an example of another optimised

curved surface. In this case a concave wall in a musicsound. Over time new populations are produced bycombining (breeding) previous shapes, and the old practice room would have caused problems of focus-

ing if untreated. Concave walls focus sound to apopulation dies off. Offspring are produced by pairsof parents breeding, and the offspring have genes that point in the same way that concave mirrors focus

light. In acoustics, focusing effects are obvious inare a composite of the parents’. The offspring shapewill then have features drawn from the parent shapes, whispering galleries, such as the dome in St Paul’s

Cathedral in London. Figure 14 shows the polarin the same way that facial features of a child canoften be recognised in the parents. A common method response for a concave wall, revealing that the

scattered energy level is much greater for the receiverof mixing genes is called multipoint crossover. Foreach gene, there is a fifty per cent chance of the at the focal point. In treating this focusing problem,

it would have been possible to add absorption on thechild’s gene coming from parent A, and a fifty percent chance of it being from parent B. wall to remove the reflections from the curved surface.

But this would have removed energy from one side ofIf all that happened was combination of theparents’ genes, then the system would never look the orchestra, and these reflections are needed so the

musicians can hear themselves and their colleagues.outside the parent population for better solutions(the ‘fish’ diffuser would never get lungs and walk Reflections are needed for the musicians to keep in


technique exists for folding a sequence into a two-dimensional array, a technique commonly referred toas the Chinese Remainder Theorem.

An example of a Chinese remainder problemwas posed by Sun Tsu Suan-Ching in the fourthcentury :13 ‘There are certain things whose numberis unknown. … Divided by 3, the remainder is 2; by 5the remainder is 3; and by 7 the remainder is 2. Whatwill be the number?’ From this rather unpromisingstart, a method of sequence folding can be generatedwhich has been used in coding systems, cryptology,

14 Scattering from a concave arc (light line) and X-ray astronomy. Incidentally, one answer tocompared with an optimised curved diffuser

the above problem is 23. The mask shown in Fig. 15(heavy line)is a length 1023 maximum length sequence which hasbeen folded into a 31 by 33 array using the Chinese

time, form a good tone, and blend the overall ensemble Remainder Theorem.sound. The solution is therefore to use diffusers, as The problem with maximum length sequences isthese remove the focusing effect 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 oneffectiveness of the diffuser in dispersing the focused the other hand produces areas of no reflection (0s)sound. and reflection (1s), and so is inherently unipolar. This

can be a problem when designing these diffusers.Most electronic systems have bipolar capabilities, andAbsorption for diffusioncan produce signals of the opposite sign, but this

In recent years, interest has been returning to number is not true of fibre optic systems, and hence opticaltheory to generate a different kind of diffuser, the sequences have been developed. Optical sequenceshybrid surface. Construction of a hybrid surface is were developed for use in fibre optical CDMA pro-shown schematically in Fig. 15. It consists of a piece cesses. Fibre optic CDMA sequences, where the lightof acoustic absorbent covered with a perforated mask, intensity is either on or off, cannot have cancellation,the mask then being hidden from view by a thin piece and hence use unipolar sequences. These sequencesof acoustically transparent cloth. Where there are can be used to design hybrid diffusers.holes in the mask, absorption is generated; where the Figure 16 compares the scattering from a hybridsound strikes a solid part of the mask, reflection occurs. absorber–diffuser with that from a plane surface. TheThis then forms a surface which partially absorbs, hybrid surface provides greater dispersion. This dis-and any reflected energy is diffused because of the persion can be improved even further if the hybridrandom arrangement of holes. The key to good dis- surface is bent and made corrugated, as this breakspersion is the arrangement of the holes, which is best up the specular reflection component further. Thisdone following a pseudo random binary sequence with type of design is proving to be very popular in studiooptimal autocorrelation properties ( least repetition). control and listening rooms.When the sequence has a 1, a hole is drilled in themask, when it has a 0, no hole is drilled. Any repetition

Active technologyin the sequence will lead to lobes, so sequences areneeded which are dissimilar from shifted versions The final technology to be discussed here is activeof themselves. Again, number theory can provide a control technology. Active noise control has causedwhole 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 widespreadsequences, the same sequences which were used origin- because the practical implementation is costly andally to modulate Schroeder diffusers. The problem is problematic. It has, however, been used successfullythat 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 activea two-dimensional array of numbers. Fortunately, a diffuser can be generated. In active noise control,

the noise source is cancelled by generating a signalfrom a secondary source of the same magnitudebut opposite phase. The waves from the noise andsecondary source then cancel each other out. Activediffusers do something slightly different: they start bycancelling out a reflection, but then the secondarysource adds in an artificial reflection which mimicsthe characteristics of a diffuser.

Figure 4 (bottom) shows an artist’s impression of15 Construction of a hybrid surface: porousabsorber (left), mask (middle), and cloth (right) an active diffuser. Loudspeakers are placed at the


most successful halls are usually those where thereis a good rapport between the acoustician andthe architect, where the engineering facilitates andembellishes the art.

As concert hall designs have been improved overthe century, attention has been focused on differentaspects of acoustic design. Currently, there is muchinterest in understanding the role of surface diffusers.As this paper has shown, much is now known aboutthe design of diffusers, but many questions remainunanswered. The most important question to answeris how much diffusion should be applied, and wherediffusers should be used. While acoustic designershave produced many innovative new designs, theunderstanding of how and why to apply diffusers lagsbehind and is still largely based on precedence.

In 2001, Manfred Schroeder attended the Inter-national Congress on Acoustics in Rome. He com-mented on some photos of optimised curved surfaces,saying we had produced diffusers which were beautiful.We were flattered by such a tribute from the pioneerof modern acoustic diffuser design. Using optimisationit is now possible to design simultaneously to givenvisual and acoustic requirements – perhaps a case of‘engineering art’?

16 Hemispherical polar balloons showing scatter- Acknowledgementsing from surfaces: hybrid surface (above),plane hard surface (below) Figure 1 is taken from . . and . ’:

Acoustic Absorbers and Diffusers; 2003, London,Spon; Figs. 2, 6–8, 10, 14, and 16 from . ’bottom of some of the wells. By changing how theseand . . : ‘Diffusor application in rooms’,controlled loudspeakers respond to incident sound, itApplied Acoustics, 2000, 60, 113–142; and Fig. 3 fromis possible to change the characteristics of the wells.. ’ and . . : ‘Two decades of soundFor example, it would be possible to make a welldiffusor design and development part 1: applicationsappear longer than it really is. But why would oneand design’, Journal of the Audio Engineering Society,go to such lengths, as the active system with its1998, 46, 955–976.electronics and control structure is difficult to develop

and expensive to implement? Active systems arebeing investigated to gain additional diffusion atlow frequency, in a range where it is very difficult to Notes and literature citedgain diffusion by normal (passive) means. To obtain

1. . . : Concert and Opera Halls: How theydiffusion at low frequency is very difficult becauseSound, 69–74; 1996, Woodbury, NY, Acoustical Society

the size of acoustic waves (wavelength) becomes many of America.metres in size, which means that a diffuser should be 2. . : ‘Things can only get better’, Guardian, 1999,extremely large as the diffuser’s dimensions need to 23 July.be comparable with the wavelength. Active diffusion 3. . . : ‘Binaural dissimilarity and optimumis a method whereby the acoustic field can be dis- ceilings for concert halls: more lateral sound diffusion’,turbed at a low frequency using a shallower device. Journal of the Acoustical Society of America, 1979,

65, 958–963.These devices, however, are very much in their infancy4. . . and . . : ‘Some practical con-and it remains to be seen whether a practical device

siderations in the use of quadratic residue diffusingcan be constructed.surfaces’, Proceedings of the 10th InternationalCongress on Acoustics, Sydney, 1980, paper E7.3.

Summary 5. . . , . . and . . . : ‘Theacoustical design of Wellington Town Hall: 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. . : ‘The subjective effects of first reflections inscience means that concert hall design involves a concert halls – the need for lateral reflections’, Journal

of Sound and Vibration, 1971, 15, 475–494.combination of precedence, engineering, and art. The


7. . and . : ‘The LEDE concept for the 10. . . . : ‘Using grating modulation to achievewideband large area diffusers’, Applied 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 Society, 1980, 28, 585–595.

8. See www-gap.dcs.st-and.ac.uk/~history/Mathematicians/ Galois.html.12. . . . : ‘Sound diffusers using reactive absorptionGauss.html.

9. . ’ and . : ‘The QRD diffractal: a grating’, Proceedings of the 98th Convention of theAudio Engineering Society, 1995, preprint 3953.new one- or two-dimensional fractal sound diffusor’,

Journal of the Audio Engineering Society, 1992, 40, 13. . : The Penguin Book of Curious and InterestingPuzzles; 1992, London, Penguin.113–129.

Trevor Cox is Reader in Acoustics at Salford University, UK.He graduated with a degree in physics from BirminghamUniversity, UK in 1988 and went on to study auditoriumacoustics at Salford University, where he was awarded hisdoctorate in 1992. Between 1993 and 1995, Dr Cox workedat South Bank University, London. A large proportion of

Dr Trevor J. Cox his research concerns diffusing surfaces: his use of numericalAcoustics Research Centre optimisation techniques has led to the worldwide appli-University of Salford cation of innovative diffusing surfaces. This paper is basedSalford M5 4WT on his Isambard Kingdom Brunel Award lecture given toUK the British Association for the Advancement of Science’[email protected] Festival of Science held in Leicester in 2002.

Peter D’Antonio received his BS degree from St John’sUniversity in 1963 and his PhD from the PolytechnicInstitute of Brooklyn in 1967. Dr D’Antonio has specialisedin a wide variety of scientific disciplines including spectro-scopy, X-ray and electron diffraction, electron microscopy,software development, and architectural acoustics. Hecarried out research in diffraction physics at the Naval

Dr Peter D’Antonio Research Laboratory, Washington, DC for thirty years.RPG Diffusor Systems Inc. He is now president of RPG Diffusor Systems Inc., which651-C Commerce Drive was founded in 1983 to carry out basic research in roomUpper Marlboro acoustics and to develop designs and innovative numberMD 20774 theoretic architectural surfaces, to enhance the acoustics ofUSA critical listening and performance environments.


Engineering Art-The Science of Concert Hall Acoustics

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