Engineering art: the science of concert hall acoustics

Transcription

1 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 enjoy- ment 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 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 disappoint- ing 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 IoM Communications Ltd DOI / Published by Maney for the Institute of Materials, Minerals and Mining INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO

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

3 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

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

5 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 8 Huygens constructions for a plane wave reflected from a simplified Schroeder diffuser: the upper plot shows wavefronts from two wells only for clarity 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 distri- bution on the surface of the diffuser to give the desired reflected wavefronts. In inventing such a 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 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 1toN 1 without repeat. Thus for N=7, r=3 isa 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

6 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 9 The Mandelbrot set at two different magnifications 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 124 INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2

7 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

8 12 The Cinerama in Seattle, WA with a diffusing ceiling (photo courtesy University of Salford and Harris-Grant Associates) 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 con- tinued 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 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

9 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 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 (heavy line) 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 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 is these 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 on effectiveness 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. Absorption for diffusion Most electronic systems have bipolar capabilities, and can 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 optical theory to generate a different kind of diffuser, the sequences have been developed. Optical sequences hybrid surface. Construction of a hybrid surface is were developed for use in fibre optical CDMA proshown schematically in Fig. 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 off, 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 diffusers. holes in the mask, absorption is generated; where the Figure 16 compares the scattering from a hybrid sound strikes a solid part of the mask, reflection occurs. absorber diffuser with that from a plane surface. The This then forms a surface which partially absorbs, hybrid surface provides greater dispersion. This disand any reflected energy is diffused because of the persion can be improved even further if the hybrid random arrangement of holes. The key to good dissurface is bent and made corrugated, as this breaks persion is the arrangement of the holes, which is best up the specular reflection 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 in the sequence will lead to lobes, so sequences are Active technology needed which are dissimilar from shifted versions The final 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 originbecause the practical implementation is costly and ally to modulate Schroeder diffusers. 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 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. 15 Construction of a hybrid surface: porous Figure 4 (bottom) shows an artist s impression of absorber (left), mask (middle), and cloth (right) an active diffuser. Loudspeakers are placed at the INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO

10 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), Acknowledgements plane hard surface (below) Figure 1 is taken from T. J. COX and P. D ANTONIO: Acoustic Absorbers and Diffusers; 2003, London, bottom of some of the wells. By changing how these Spon; Figs. 2, 6 8, 10, 14, and 16 from P. D ANTONIO controlled loudspeakers respond to incident sound, it and T. J. COX: Diffusor application in rooms, is possible to change the characteristics of the wells. Applied Acoustics, 2000, 60, ; and Fig. 3 from For example, it would be possible to make a well P. D ANTONIO and T. J. COX: Two decades of sound appear longer than it really is. But why would one diffusor design and development part 1: applications go to such lengths, as the active system with its and design, Journal of the Audio Engineering Society, electronics and control structure is difficult to develop 1998, 46, 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 Notes and literature cited gain diffusion by normal (passive) means. To obtain 1. L. L. BERANEK: Concert and Opera Halls: How they diffusion at low frequency is very difficult because Sound, 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. S. MOSS: 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. M. R. SCHROEDER: Binaural dissimilarity and optimum is 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, These devices, however, are very much in their infancy 65, and it remains to be seen whether a practical device 4. A. H. MARSHALL and J. R. HYDE: Some practical con- can be constructed. siderations in the use of quadratic residue diffusing surfaces, Proceedings of the 10th International Summary Congress on Acoustics, Sydney, 1980, paper E A. H. MARSHALL, J. R. HYDE and M. F. E. BARRON: The acoustical 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 bination of precedence and luck. Nowadays, acoustic 6. M. BARRON: The subjective effects of first reflections in science means that concert hall design involves a concert halls the need for lateral reflections, Journal combination of precedence, engineering, and art. The of Sound and Vibration, 1971, 15, INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO. 2

11 7. D. DAVID and C. DAVIS: The LEDE concept for the 10. J. A. S. ANGUS: Using grating modulation to achieve control of acoustic and psychoacoustic parameters wideband large area diffusers, Applied Acoustics, 2000, in recording control rooms, Journal of the Audio 60, Engineering Society, 1980, 28, See www-gap.dcs.st-and.ac.uk/~history/mathematicians/ 8. See www-gap.dcs.st-and.ac.uk/~history/mathematicians/ Galois.html. Gauss.html. 12. J. A. S. ANGUS: Sound diffusers using reactive absorption 9. P. D ANTONIO and J. KONNERT: The QRD diffractal: a grating, Proceedings of the 98th Convention of the new one- or two-dimensional fractal sound diffusor, Audio Engineering Society, 1995, preprint Journal of the Audio Engineering Society, 1992, 40, D. WELLS: 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 Between 1993 and 1995, Dr Cox worked at South Bank University, London. A large proportion of Dr Trevor J. Cox his research concerns diffusing surfaces: his use of numerical Acoustics Research Centre optimisation techniques has led to the worldwide appli- University of Salford cation of innovative diffusing surfaces. This paper is based Salford M5 4WT on his Isambard Kingdom Brunel Award lecture given to UK the British Association for the Advancement of Science s [email protected] Festival of Science held in Leicester in Peter D Antonio received his BS degree from St John s University in 1963 and his PhD from the Polytechnic Institute of Brooklyn in 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 Dr Peter D Antonio Research Laboratory, Washington, DC for thirty years. RPG Diffusor Systems Inc. He is now president of RPG Diffusor Systems Inc., which 651-C Commerce Drive was founded in 1983 to carry out basic research in room Upper Marlboro acoustics and to develop designs and innovative number MD theoretic architectural surfaces, to enhance the acoustics of USA critical listening and performance environments. INTERDISCIPLINARY SCIENCE REVIEWS, 2003, VOL. 28, NO