Figure 1. Simulation of QRD polar responses for a) N=17 2 periods, b) N=17 25 periods, c) N=89 2 periods, d) uniform diffusion paradigm

In 1983, RPG® introduced the first commercial reflection phase grating (RPG) diffusors based on Schroeder’s ingenious suggestion of using periodic sequences with flat power spectra. Scattering from any periodic sequence is characterized by scattering into well-defined diffraction orders or directions (light radial lines in Fig. 1). The diffraction directions are determined solely by the size of the repeat unit, which is equal to the product of the number of wells, N, and the width of a single well, w. To insure that the scattering in these diffraction directions is uniform, Schroeder suggested sequences, such as quadradic residues, that exhibit a flat power spectrum. The polar distribution of the scattered energy about the diffraction directions (width of the lobes) depends on the number of periods. It can be seen in Fig. 1a that scattering from 2 periods of a QRD with 17 wells at the design frequency f0 is distributed into 5 broad lobes at 00, +/-240, and +/- 590. As the number of periods increases from 2 to 25 (Fig. 1b) the scattered energy in the far field is concentrated into narrower diffraction orders. To approach uniform diffusion, shown in Fig. 1d, a number theory diffusor must be based on a large Nw product to produce many non-evanescent diffraction directions. Thus, in Fig. 1c, an increase in the number of diffraction orders by a factor of 5 is seen when the number of wells is increased from 17 to 89. The number theoretic diffusor has proven to be an extremely powerful tool in architectural acoustics, however, there is room for improvement.

Figure 2. Comparison between the diffusion parameter (zero ideal) for normal incidence of a QRD with dividers and a POD with no dividers

In Vol. 3, Issue 1, a new shape optimization technique was introduced. RPG® is now proposing a new generation of phase optimized diffusors (POD) based on the paradigm of uniform scattering, as shown Fig. 1d, instead of the flat power spectrum of the number theory diffusors. These new diffusors are designed by combining multi-dimensional optimization techniques with boundary element predictions, which are more accurate than the far-field Fraunhofer theory of the number theoretic diffusors. In addition, these new diffusors can be designed with or without dividers between wells. The uniformity of a polar response (shown in Fig. 1), can be quantified by its standard deviation. The uniform diffusion response in Fig. 1d would have a standard deviation of zero. In Fig. 2 we illustrate how phase optimization can be used to lower the standard deviation of a POD with no dividers and provide superior diffusion compared to a traditional QRD® with dividers. In this example, a POD without dividers was optimized for normal incidence up to 1,250 Hz.

Figure 3. Right half of a symmetrical 125 step phase optimized diffusor

RPG® provides this optimization service to acoustical consultants as part of our collaborative CHAOS™ program. In Fig. 3 we show the right half of a 21' (6.4 m), 125 step, symmetrical, phase-optimized diffusor designed for the rear wall of a performing arts facility. It was optimized for random incidence between 100 - 4,000 Hz. To provide this service, RPG® requires the source and receiver positions, desired coverage, and allowable width and depth.

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