Page 25 - Summer 2006

Fig. 11. Scattering from a concave arc compared to an optimized curved diffuser. Blue: concave arc, Red: optimized curved diffuser. (After Cox and D’Antonio15).
been possible to add absorption on the wall to remove the reflections, but this would have removed energy from one side of the orchestra, and these reflections are needed so the musicians can hear both themselves and others. Without these the musicians would find it harder to keep in time, form a good tone and blend with the orchestral sound. The solution to the focusing wall is to use diffusers to remove the focusing effect, while preserving the acoustic energy. The polar response after treatment is shown in Fig. 11. It illustrates the effectiveness of the diffuser in remov- ing the “hot-spot.”
Numerical optimization
To design this type of diffuser requires a new method- ology10, and for this it is possible to use numerical opti- mization, a method commonly used in engineering. Numerical optimization may not have the efficiency and elegance of number theory design, but it is extremely effec- tive and the designs it produced are robust. Numerical opti- mization tasks a computer to search to find an optimum solution to a problem. For acoustic diffusers, the computer looks for the surface shape that gives optimal scattering. The procedure follows an iterative scheme. The computer starts by guessing some curved surface shape. The scatter- ing from the surface is predicted in terms of the polar response. The predicted polar response is then rated for its quality in terms of a figure of merit. The computer can then use a process of trial and error, changing the surface shape to try and optimize the figure of merit. The process contin- ues until an optimum design is found. The search process cannot be completely random because this would take too much time. Fortunately, mathematicians have developed many algorithms to allow the search to be done efficiently. Currently, the most popular algorithm is to model the prob- lem as an evolutionary process, using survival of the fittest principles to carry out an efficient search. The technique is called a genetic algorithm.
The genetic algorithm
A genetic algorithm mimics the process of evolution that occurs in nature. A population of individuals is randomly formed with their genes determining the traits of each individ- ual. When designing diffusers, the genes are simply a set of numbers that describe the curved surface shape. Each individ- ual (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. Pairs of breeding parents produce off- spring with genes that are a composite of their parent’s genes. The offspring shape will then have features drawn from the par- ent shapes, in the same way that facial features of a child can often be seen in their parents. A common method for mixing the genes is called multi-point cross over. For each gene, there is a 50% chance of the child’s gene coming from parent A, and a 50% chance of the gene coming from parent B.
If all that happened was a combination of the parent genes, then the system would never look outside the parent popula- tion for better solutions. A fish would never get lungs and walk about on the land. As with biological populations, mutation is needed to enable dramatic changes in the population of shapes. This is accomplished by a random procedure whereby there is a small probability of any gene in the child sequence being ran- domly changed, rather than directly coming 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 the most like- ly to die. This is also true in
the artificial genetic algo-
rithms used in numerical
optimizations. By these
principles, the fitness of
successive populations
should improve. This
process is continued until
the population becomes
sufficiently fit so that the
best shape produced can
be classified as optimum.
Figure 12 (top) shows an example of another optimized curved surface. Because it is now known that periodicity reduces dispersion, this surface has
Fig 12. (top) An optimized curved surface designed to be tiled in a modulated array. (bottom) Ceiling Waveform treatment at KTSU, Houston, TX. (Photo Courtesy of HFP Acoustical Consultants)
A Brief History of Room Acoustic Diffusers 23