basic geometrical room acoustics

8/2/2019 Basic Geometrical Room Acoustics

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AEOF3/AEOF4 Acoustics of Enclosed Spaces Dr. Y.W. Lam 1995

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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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basic geometrical room acoustics

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