In our continuing effort to expand the acoustical palette of aesthetic and quantifiable scattering shapes available to the architect and acoustician, we now turn to fractals.

Fractals are self-similar: as a surface is magnified a similar looking surface is found. Consequently, each surface revealed by different magnifications should be capable of scattering different frequency ranges. We have previously taken advantage of the self-similar property of fractals by combining it with the uniform scattering of the QRDŽ. This has yielded the amplitude modulated diffusing fractal called the DiffractalŽ. In this study we investigate methods to optimize the actual shape of fractal surfaces that are not necessarily self-similar, but statistically self-similar. This is mathematically referred to as self-affine.

Optimization techniques are much more efficient when the number of parameters required to represent a surface are reduced. Fractal construction techniques can be adapted to form systems that represent complex shapes with few defining parameters, so they should be capable of being optimized. We examine two techniques for constructing fractal or fractal-inspired diffusors. The Fourier Synthesis method has been tested in an effort to relate scattering quality to surface profile. The Step Function Addition method has been developed to allow complex surfaces to be represented by only a few shape parameters. This has enabled these surfaces to be optimized for the best diffusing performance.

Figure 1. Schematic diagram showing Fourier synthesis construction technique

Fourier Synthesis
Fractal surfaces can be constructed from spectral shaping of a Gaussian white noise source. Figure 1 is a schematic showing how the surfaces can be generated. To acousticians, such a scheme is likely to be more familiar as a time signal filtering process. The Gaussian white noise is passed through a filter, which is implemented using simple Fourier techniques. The decrease in spectral content per octave is characterized by the gain of the filter at each frequency. The filter gain A (f) is given by:

Where beta =1 for 1/f or pink noise, beta =2 for Brownian motion or brown noise . If there are M points determining the Gaussian white noise, then there is a need for M+1 parameters to define the shape. For the surface shapes considered here there will be a few hundred parameters. A problem arises when trying to optimize such surfaces as the number of parameters to define the shape will be too large.

Figure 2. Fractal generation by step function addition. Bottom line: 1 step function, middle line: 10 step functions, top line: 20 step functions

Step Function Addition
To overcome this obstacle, another fractal generation technique has been used which enables the reduction of the number of shape determining parameters. Brownian motion can be simulated by a series of randomly displaced step functions. It is not always as mathematically pure as a Fourier Synthesis technique, but more simply enables the reduction of the number of parameters required to represent the surface shape. To get proper Brownian motion requires the addition of an infinite number of step functions. Each step function has random amplitude and the position of the step is a random position somewhere along the width of the diffusor. The displacement of the diffusor from a flat surface y at a distance x along the diffusor is given by:

This has reduced the number of independent parameters to N+2, the set of displacements xi and the amplitude decay rates a and e.

Figure 3. Diffusion parameter for four surfaces at a 60° angle of incidence. Zero is ideal. The surfaces are as follows:

Figure 3 illustrates that the diffusion performance of the optimized fractal is comparable to the optimized curve and better than an arc or planar surface. Therefore, the optimized fractal may be considered as a new shape in the acoustical palette.