Civil Engineering Reference
In-Depth Information
planes. We shall confine ourselves to the first type, as the extension to two dimensions is
reasonably straightforward, conceptually at least.
Schroeder (1975) began his work on what we may term mathematical diffusers by
investigating the scattering from surfaces shaped in the form of a maximum length
sequence (MLS). We showed in section 1.5.2 the particular Fourier properties of these
sequences giving a completely flat power spectrum. Then, quoting Schroeder: “Thus,
because of the relation between the Fourier transform and the directivity pattern, a wall
with reflection coefficients alternating between +1 and -1, would scatter an incident
plane wave evenly (except for a dip in the specular direction which corresponds to the
DC component in the spectrum).” The “MLS wall” was realized as a hard wall with
“grooves” or wells a quarter of a wavelength deep in the area where a reflection factor of
-1 was called for. In practice, such diffusers work, however, over a rather limited
frequency range, approximately one octave. There are means of increasing the workable
bandwidth, as recent research shows, but this implies adding active components to the
diffuser (see Cox et al. (2006)).
However, there are other periodic sequences having useful Fourier properties,
which make them excellently suited for modelling diffusing elements having a much
broader bandwidth than the MLS. These are the
quadratic residue
sequences and the
primitive root
sequences (see e.g. Schroeder (1999), Cox and Antonio (2004)). The
sequence forming the base for making a quadratic residue diffuser (QRD) is given by
2
MOD
n
sm
=
N
where
m
=
1, 2, 3,...
(4.65)
This means that
s
n
is the reminder when
m
2
is divided by the prime number
N
. Taking
N
=7 as an example, we get the following sequence: 0, 1, 4, 2, 2, 4, 1. In a similar way as
for the MLS diffuser the numbers are transformed into the corresponding depths
d
n
of the
grooves or wells of the surface, but these are now not constant:
d
max
max
ds
=
.
(4.66)
n
n
()
s
n
So how do we choose the maximum depth
d
max
and also the width of each well?
Certainly, to make the diffuser work properly there should be plane wave propagation in
each well and there must be a significant phase change for the waves reflected from the
bottom. The design rule normally used for the latter, which determines the maximum
workable wavelength or the equivalent minimum frequency, is expressed as:
s
λ
sc
n
max
n
0
d
=
or
f
=
,
(4.67)
n
min
2
N
2
Nd
n
where
c
0
is the speed of sound. This design rule implies that the mean depth of the wells
at this frequency is of the order of a quarter of a wavelength. As for the width
w
of each
well, we should ensure plane wave propagation, which implies being below the cut-off
frequency giving
λ
c
min
0
w
=
or
f
=
.
(4.68)
max
2
2
w
