basic geometrical room acoustics
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GEOMETRICAL ROOM ACOUSTICS(Approximation at Very High Frequencies)
! At very small wavelengths (compare to
room dimensions), the room surfaces
will be seen by the sound waves as
infinite planes
=> specular reflections
! Waves can then be replaced by energy
rays (see figure), with the propagation
of these rays following that of light
rays. In particular the law of reflection
is that of specular reflection.
!With sound represented by energy rays instead of complex pressure rays,
! diffraction is ignored
! interference is not considered
This assumes the sound rays are
incoherent, which is permissible for a
wide frequency bands (e.g. octave
bands) at high frequencies.
(Note that some modern computer
modelling approaches do use pressurerays with phase information in order to
achieve auralization.)
To adequately represent the actual sound
waves, a large number of rays will be
required. Also the divergence of the
original waves must be accounted for
(hence ray intensity % 1/r2, and ray cross
sectional arc length % r, where r is the propagated distance). To see if a ray is received
or not, the concept of a reception volume around the receiver has to be used. These
complications can be avoided by constructing image sources from the ray reflectionsrather than using only rays for the entire sound field calculation. An example of the
construction of an image source from ray paths is given in the figure. Since rays which
are originated from a single source and reflected at an identical plane surface give rise
to just one single image source, and that the radiation form an image source
automatically includes the spherical divergence, the sound wave propagation can be
adequately represented as long as there are sufficient rays to find all the images. The
sound field in the room can then be calculated from the contributions from the image
sources rather than from the energy carried by individual rays.
Examples of constructing higher order image sources in rooms of arbitrary shapes are
given in the figures. As a simple example, the location of the images of a source
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midway between two parallel walls a distance R apart is given by (with origin taken at
the source)
For a rectangular room of dimensions Lx, Ly, Lz,
and a source at (xo, yo, zo), the images locations
will be
where i, j, k = 0, 1, 2, ....., and the origin of the coordinate system is taken at the
centre of the room.
This can be deduced by first considering the reflection of the source off the -x and +x
walls. The first image generated by the +x wall is given by
The term in the bracket is the mirrored x-distance of the source relative to the reflecting
wall (the +x wall) while the first term in the left hand side is the coordinate of the
reflecting wall relative to the coordinate origin. This corresponds to iLx+(-1)-ixo with
i=1.
Similarly the first image generated by the -x wall is given by (the mirrored distance is
now -ve),
This corresponds to iLx+(-1)-ixo with i=-1. The reflection of this (i=-1) image by the +x
wall is
This corresponds to iLx+(-1)-ixo with i=2.
The reflection of the first +x image (i=1) by the -x wall is given by
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This corresponds to iLx+(-1)-ixo with i=-2.
In general the (i+1) image, with i positive, is generated by the reflection of the -i image
by the +x wall
and the (-i-1) image, with i positive, is generated by the reflection of the +i image by
the -x wall
which give rise to the x-component of our formula for the image location. Similar
consideration can be applied to the y and z components.
If the walls of the room have reflection coefficients of
.x- = (1 - "x-), .x+ = (1 - "x+),
.y- = (1 - "y-), .y+ = (1 - "y+),
.z- = (1 - "z-), .z+ = (1 - "z+),
then the strength of the images is given by
where
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This again can be deduced from the consideration of the image refection. Because in
general the (i+1) image, with i positive, is generated by the reflection of the -i image
by the +x wall, and the (-i-1) image, with i positive, is generated by the reflection of
the +i image by the -x wall, that means that the ith image is generated by alternative
reflections off the -x and +x walls, with each reflection resulted in the strength being
multiplied by the reflection coefficient of the reflecting wall. Hence if i is even then
half of the reflection will be from the -x wall (with .x-) and half from the +x wall (with.x+), and the total strength is modified by the multiplication factor
When i is odd, the last reflection is the odd one out. If i is negative, the last reflection
will be the -x wall the index power of .x- should be increased by 1 from the integer of|i/2| (i=odd), while that of .x+ should be given by the integer of |i/2| . Since i is negative,this corresponds respective to |(i-1)| and |(i+1)|. On the other hand, if i is positive, the
last reflection will be the +x wall the index power of .x+ should be increased by 1 fromthe integer of |i/2| , while that of .x- should be given by the integer of |i/2|. Since i ispositive, this corresponds respective to |(i+1)| and |(i-1)|. Hence in both case the
multiplication factor is given by the same formula,
Similar argument applies to the y and z components.
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VISIBILITY OF GEOMETRICAL IMAGE SOURCES
The use of image sources, though
simplifies the ray representation, gives rise
to the problem of image visibility. Not all
image sources generated by the process are
necessarily visible by a fixed receiver in acomplex room structure.
e.g. S3 - Image blocked from
the receiver by some
room surfaces
S1, S2 - Images generated from
imaginary walls that do
not reflect rays to the
receiver.
Hence S3, S1, and S2 are image sources invisible to the receiver R, but they can
be visible to receivers at other locations.
Visibility can be checked by 'backtracing': rays re-constructed from R to see if the
image can be hit or not. For an image to be visible, the path between image and receiver
must cross all surfaces involved in the generation of the image but no others, and the
last wall must be a physical wall.
1. Only walls that generate the image should have their imaginary part considered.
2. All walls must be plane in the generation of images.
e.g. rays can be constructed from R to the visible images easily, but not from R to hit
S3, S1, and S2.
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Computer Models
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REFLECTOGRAM AND NUMBER OF IMAGES
An example of the network of image
sources of a source in a rectangular room is
shown in the figure.
Consider the case of an impulse source, i.e.excitation given by
s(t) = So*(t) , Delta function *(t)=1at t=0, otherwise=0.
where So is the impulse strength and
the impulse occurs at t=0.
Sound from the image sources will each arrive at a receiver after a certain time delay,
given by
ti = ri/c
where ri is the distance between the ith
image source and the receiver.
Each of these arrivals represents a refection
arrival. It will be attenuated at each wall
reflection it experienced according to the
walls it crosses in the way from the image
source to the receiver. If we record these
energy arrivals as a function of time, with
time t relative to the arrival time of the
direct sound, we obtain a reflection
diagram (a reflectogram) as shown in the
figure.
One major usefulness of the geometrical approach is that reflectograms, especially the
early part, can be constructed easily. As will be seen later, the early part of a
reflectogram is subjectively very important in room acoustics. Wave theory approach
requires very difficult calculations before a reflectrogram can be produced. Diffuse
field approach is too approximate to generate detailed reflectorgrams. The geometricalapproach is therefore the most useful approach in room acoustics to obtain energy-time
relationship.
As shown by the figure, the early part of the reflectogram has distinct reflection
arrivals, due to the relatively small number of image sources generated in the short time
duration. As time goes on, the density of the reflection arrivals increase rapidly and
very soon individual reflections will no longer be separable from each other.
The number of image sources at any time t (which is equal to the number of reflection
arrivals from 0 to t) can be estimated by referring to the figure. Assuming that time t is
sufficiently long that we can take
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ri. distance from the ith image source to the centre of the room
We can then use the radius R=ct from the centre of the room to construct a sphere
which will enclose all image sources that have contributed reflections to the receiver.
The volume of this sphere is
Vt = 4B (ct)3
/ 3
As seen from the figure, each image of the source occupies, just as the source does, a
volume of the room. Hence the total number of image sources enclosed by the sphere
is
Nt = Vt/V = 4B(ct)3/3V
The temporal density of the image sources, and hence the temporal density of reflection
arrivals, is
The density increases with t2.
It is obvious that the number of image sources even at a modest delay time will soon
become too numerous to be constructed by hand, and computer modelling have to be
used (e.g. ray tracing, hybrid ray tracing/image source method).
AVERAGE NUMBER OF REFLECTIONS PER SECOND &&&&n AND MEAN FREE PATH&&&&RRRR
The above equation gives the number of
image reflections arriving at the receiver
per second, not the number of reflections
actually happening per second in the room.
To estimate &n , we need to consider the
propagation of sound rays.
Consider a rectangular room of dimensions
Lx, Ly, Lz. Geometrical room acoustics
models the sound field in the room by a
large number of sound rays which areassumed to be distributed uniformly over
all possible directions of propagation. Consider a sound ray whose angle with respect
to the x-axis is 2x. Let N2x be the number of crossings of mirror walls perpendicular tothe x-axis it undergoes in a time t. The total distance travelled in this time is ct. The
projection of this distance onto the x-axis is therefore ctcos2x. Since the room isrectangular, there is a mirror wall perpendicular to the x-axis for every Lx distance.
Ignoring the dependence of the first (real) wall crossing on the source position, then
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(Note that |cos2x| is required to account for rays in the 2x>B/2 directions)
The number of crossings per second is then
Averaging |cos2x| over all possible directions of propagation (o to B), using the x-axisas the polar axis in the formula for the solid angle, yields
Hence nx, the average of n2x over 2x, is
nx = c/2Lx
Similarly for the crossings of walls perpendicular to the y- and z-axes
ny = c/2Ly , nz = c/2Lz
The total number of wall reflections per second, averaged over all rays, is
where S is the total surface area 2(LxLy+LxLz+LyLz), and V is the volume LxLyLz.
Note that eqn.(2) is not the average number of wall reflections per second for one ray
averaged over time, but is rather the average number of wall reflections per sound
averaged over all possible ray directions. However, if we assume that somehow eachray will change its direction, e.g. by non-specular wall reflections or obstacles,
sufficiently many times in a given time duration during the reverberation process, then
the time average will be equivalent to the direction average that is carried out in the
derivation of eqn.(2). In such cases
The above assumption will be achieved if the sound field in the room is diffuse, forexample, as a result of diffuse wall reflections, scattering by obstacles, or irregular
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room shapes.
Assuming that eqn.(3) is correct, then the mean free path of a ray, i.e. the average
distance travelled by a ray between successive reflections, is
Note that in the limit of very high frequencies, the above equations are all valid for
rooms of arbitrary shapes.
SOUND DECAY IN A ROOM
The contribution from an image source which arrives at time t will have its energy
attenuated by
a) Distance according to inverse square law, i.e. 1/(ct)2
b) Reflections at walls. The number of reflections experienced by a ray is on average
&nt. Assuming the absorption on the walls can be accounted for by an average
absorption coefficient &", then at each reflection the ray will loss &" fraction ofenergy, with (1-&") fraction of energy remaining. After &nt reflections the energy willthen be attenuated by (1-&")t.
c) Air absorption. Let m be the absorption constant normally used in room acoustics,
then the air absorption attenuation factor is e-mct.
Accounting for all three attenuations, the energy of the ray arriving is then
where Ao is a constant representing the source strength. Note that &", i.e. the means ofaveraging, has not yet been defined, but if all walls are identical then &" is theabsorption coefficient.
Multiplying the equation with the density of reflection arrivals
then the total energy arriving is in the form of
The reverberation time, i.e. the time in which the energy falls to 10-6 of its initial value,
is then
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&n is given by eqn.(3), hence,
Hence by geometrical considerations we can arrive at a formula of reverberation time,which is still the most important objective parameter in room acoustics.
The average absorption coefficient may be derived by either of the following two
assumptions:
Sabine's Assumption:
Total absorption = Simple sum of absorption of individual surfaces
The absorption coefficients "i of individual surfaces are usually determined bymeasurements (see e.g. BS3638) based on Sabine's RT formula.
This assumption will only hold if the reflected direction at each reflection is totally
diffuse, so that the probability of a ray hitting a surface will be totally independent of
the previous angle of incidence, and depends only on the area ratio S i/S. The T60formula then becomes
This assumption can only be realised by total diffusion at each reflection, which is
difficult to realize in practice.
Geometrical Distribution
Consider each reflection as an event. In each reflection/event, the energy is reduced by
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" so that the result of the event is to multiply the incident energy by a factor of (1-").If" is constant then after m events the result is a multiplication of a factor (1-")m.
Let N be the total number of reflections/events. The average absorption coefficient is
defined such that after N events the result is a multiplication of (1-&")N.
Let "i and Si respectively be the absorption coefficient and surface area of the ithsurface in the room. The result on a reflection/event on this surface is a factor (1-"i).If, on average, the number of reflections fallen on to the ith surface is proportional to
the surface area Si, such that it is equal to the ratio Si/S multiplied by the total number
of reflections N, then the actual result of all N reflections is given by
By definition this must be equal to (1-&")N.
Hence the T60 formula becomes
This eqn. suffers a problem in that if any of the surface absorption coefficients "i, e.g."j, has a value of unity, then T60=0 regardless of the size of the surface Sj. This isobviously a physically incorrect result. This is due to the over-simplification of the
decay process by the assumption of eqn.(9), that all surfaces, no matter how small it is,will participate in the decay process of a ray at all time. Hence the factor (1-"i)Si/S is
present in the eqn. for all surfaces, which will give a value of zero in the eqn. if "i=1as long as Si>0. This may not be a serious problem in that if "i's are determined bycalculations of the random incidence integration, then they will have a theoretical upper
limit of approximately 0.95.
WALL DIFFUSION
Geometrical room acoustics is useful in that it
a) allows simple constructions of reflectograms (a picture of the temporal distributionof energy arrivals) especially in the early part of the room response - which is very
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important in auditorium acoustics as we shall see later,
b) allows us to deal with complex room shapes and absorption distributions by means
of geometric ray constructions,
c) gives estimates of reflection density, mean free path and T60 which are all important
parameters in room acoustics.
However as the reflection density increases with t2, the number of higher order image
sources will soon become too large to be handled. In addition to that, real wall surfaces
are inevitably non-smooth (surface irregularities) and perfect specular reflections
cannot be achieved. Furthermore the finite sizes of walls, decorations on walls etc. all
diffract energy away from the geometrical specular directions. The errors associated
with these latter two problems increase rapidly with the order of reflections (image
sources). It is therefore inevitable that the law of specular reflection will become
increasingly incorrect while the sound field becomes more diffuse as time goes on. In
practice the geometrical specular reflection law generally will only be followed
reasonably well by the first few order of image sources, say up to the 6th order. Giventhe difficulty of handling large numbers of reflections and the errors with diffusion at
higher order image sources, the simple geometrical approach is only good for the early
part of a room's impulse response/reflectogram. Indeed most acoustic consultants will
only use such an approach to investigate in detail reflections up to the 5th order. For the
later part of the impulse response diffuse field approach is more appropriate.
In fact in most performance spaces, a diffuse field is desirable to obtain a uniform
sound field and to avoid unexpected acoustic peculiarities. Often purposely built
scattering surfaces are introduced to increase the diffuseness. General scatters (such as
rough surfaces and or grids of randomly arranged plane or curved reflectors) can be
used. An effective diffusing surface is the Schroeder diffuser which is based on the
quadratic residue number sequence
where m is a prime number, n is the height of the kth well in the diffuser (see figure for
the example of m=19). According to theory, uniform scattering (maximum diffusion)
can be achieved for a infinitely extended diffuser using such sequences.
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