118 Chapter 5
plots of the diffuse sound field from a diffuser. The less
lobing there is in a diffusion polar plot, the more diffu-
sion and the higher the diffusion coefficients.
Scattering and the scattering coefficients relate to the
amount of energy that is not reflected in a specular
manner. The term specular here denotes the direction of
reflection one would expect if the sound were reflecting
off a hard flat surface. For example, most high frequency
sounds reflect from a hard flat surface at the same angle
as the incident sound, Fig. 5-1. This is referred to as
specular reflection by Cox and D’Antonio. The more
sound that is reflected in a nonspecular manner, the
higher the scattering. Therefore, a simple angled wall
can provide high scattering but low diffusion, since the
reflected sound will still form a lobe, but not in a spec-
ular direction. It should be noted that a significant
amount of absorption makes it difficult to measure scat-
tering. This makes sense since an absorber does not
allow for a high level of specular reflection; absorption
can be mistaken for scattering.
The standardized methods for measuring diffusion
and scattering coefficients are AES-4id-2001 and ISO
15664, respectively.30,31 The AES method provides
guidelines for measuring the performance of diffusive
surfaces and reporting the diffusion coefficients. These
guidelines will allow diffusers of different designs to be
objectively compared. The results cannot, however, be
incorporated into acoustical modeling programs. For
that, the scattering coefficients must be used when
measured in accordance with the ISO method.
Both the AES and ISO methods are relatively new;
the AES method was formalized in 2001 and the ISO
method was published in 2003. Because of this, none of
the independent acoustical test laboratories in North
America are equipped to perform the AES diffusion test
and, as of this writing, only one laboratory in North
America is equipped to perform the ISO scattering test.
Because of this, diffusion and scattering coefficients for
surfaces and treatments (diffusers or otherwise) are diffi-
cult to find. Indeed, because so little testing is being
performed on diffusers, there is some degree of confu-
sion in the industry as to what diffusion and scattering
coefficients actually mean in subjective terms. For
example, what does a diffusion (or scattering) coefficient
of 0.84 at 2500 Hz sound like? There is no denying that
the information is useful and that objective quantifica-
tion of diffusers is necessary. However, comparing diffu-
sion or scattering coefficients for different materials
would be a theoretical exercise at best. The process is
further complicated by the fact that commercial diffusers
vary dramatically in shape and style; each manufacturer
claims some degree of superiority because of some
unique application of some innovative mathematics.
Nonetheless, there are diffusion and scattering coeffi-
cients available in the literature (Cox and D’Antonio
offer a significant amount of laboratory measured diffu-
sion and scattering coefficients^2 ), and some manufac-
turers have begun pursuing independent tests of their
diffusive offerings. Fig. 5-29 pictures examples of
various commercial diffusers. The next few decades will
be a very exciting time for diffusion, particularly if more
independent acoustical laboratories begin to offer AES
and/or ISO testing services.5.3.2 Mathematical (Numerical) DiffusersThe quadratic residue diffuser (QRD) is one form of a
family of diffusers known as reflection phase gratings, or
more generally, mathematical or numerical diffusers.
Numerical diffusers, such as the QRD, are based on the
pioneering work of Manfred Schroeder.^32 Numerical
diffusers consist of a periodic series of slots or wells of
equal width, with the depth determined by a number
theory sequence. The depth sequence is developed via
Eq. 5-9(5-9)
where,
p is a prime number,
n is an integer t 0.The “mod” in Eq. 5-9 refers to modulo, which is a
number theory mathematical process whereby the firstFigure 5-29. Various commercial diffusers.well depth factor=n^2 mod p