140 Building acoustics
The width w is normally chosen in the range of 5–10 cm. Making the wells too narrow
may increase the surface area too much giving unwanted surface sound absorption,
especially when the wells have separating walls (see Figure 4.17 a).
The other type of sequence having Fourier properties that makes them useful in the
construction of broadband diffusers, giving little specular reflections, is the primitive
root sequences. These are calculated in a slightly different way than the quadratic residue
ones, given by:
spn==nMODNwhere n 1, 2, 3,... (4.69)
The number p is denoted a primitive root modulo N, also called a generating element
because it generates a complete residue system in some permutation. As an example,
choosing N equal to 7 there are two primitive roots, being 3 and 5. We shall use a higher
number N in our example below, choosing N equal to 13 where the lowest primitive root
is 2. Using (4.69) to calculate this sequence gives the values shown in Table 4.1.
Table 4.1 Primitive root sequence for N equal 13 and primitive root p equal 2. Well depths in mm for design
frequency 1000 Hz.
n 1 2 3 4 5 6 7 8 9 10 11 12
sn 2 4 8 3 6 12 11 9 5 10 7 1
dn
(mm)28 57 113 43 85 170 156 128 71 142 99 14
In the last row the corresponding depths of the wells are given, calculated by
equation (1.3) choosing a design frequency (fmin) of 1000 Hz. It should be noted that
there is only N –1 cells in the sequence. As is apparent from the table and also from
Figure 4.17, where we have put three such periods on a row, diffusers based on a
primitive root sequence (PRD) are unsymmetrical.
a)
b)
Figure 4.17 Sketch of a ceiling having three periods of a primitive root diffuser (PRD) with N equal 13.
a) With dividing walls between the wells (grooves), b) Without dividing walls.
Prediction methods for the acoustic pressure field, i.e. the sum of the direct field
from a source and the scattered field, is normally based on the Helmholtz-Kirchhoff
integral equation (see e.g. Cox and Lam (1994)). This means using Equation (3.44) in
Chapter 3 with an added term representing the direct field. If only the far field is of
interest, a computational method based on the analogue Fraunhofer diffraction method in