Scattering Coefficient Damis Cacavelos

Acoustical Instruments and Measurements May 2015, Argentina

SCATTERING COEFFICIENT

FEDERICO DAMIS1, NAHUEL CACAVELOS

2

Universidad Nacional de Tres de Febrero (UNTREF), Buenos Aires, Argentina.

[email protected], [email protected]

2

Abstract A one-dimensional (1D) diffuser surface was designed, built and measured

according to ISO 17497-1 standard. Basic theory behind diffusion and scattering was analyzed

and applied in the design phase. The design included the use of randomly generated number

series with low autocorrelation in order to improve the diffuser efficiency. The diffuser was

scaled down by a factor of 2, in order to make the measurement process easier and fit the

requirements imposed by the standard. Measurement methodology was fully described and the

scattering coefficient was calculated, as well as its corresponding standard deviation and

measurement uncertainty.

1. INTRODUCTION

Diffusion, in acoustics and architectural

engineering, is the efficacy by which

sound energy is spread evenly in a given

environment. A perfectly diffusive sound

space is the one that has certain key

acoustic properties which are the same

anywhere in space. A non-diffuse sound

space would have considerably different

reverberation time as the listener moved

around the room. Virtually all spaces are

non-diffuse. Spaces which are highly non-

diffuse are ones where the acoustic

absorption is unevenly distributed around

the space, or where two different acoustic

volumes are coupled [1].

Diffusors (or diffusers) are used to treat

sound phenomena in rooms such as

echoes. They are an excellent alternative or

complement to sound absorption because

they do not remove sound energy, but can

be used to effectively reduce distinct

echoes and reflections while still leaving a

live sounding space. Compared to a

reflective surface, which will cause most

of the energy to be reflected off at an angle

equal to the angle of incidence, a diffusor

will cause the sound energy to be radiated

in many directions in time as well as

spatially (as seen in Fig.. 5).

There is a growing trend away from the

traditional use of absorbers on the rear wall

of auditoria towards the use of diffusers.

[2] Fig. 1 shows diffusers applied to the

rear wall of Carnegie Hall in New York.

This form of reflection phase grating was

the starting catalyst for modern diffuser

research. The diffusers were installed in

Carnegie Hall because a long delayed

reflection from the rear wall caused an

echo to be heard on the stage, making it

difficult for musicians to play in time with

each other. Adding diffusers dispersed the

reflection, reducing the reflection level

arriving on the stage and consequently

making the echo inaudible. The diffusers

also improved spaciousness on the main

floor by uniformly diffusing rear wall

reflections and masking echoes from the

boxes in this case. [3]

Figure 1. Schroedder diffusers applied to

the rear wall of Carnegie Hall to prevent

echoes.

The visual and aesthetic aspect of

diffusers is also a relevant matter in


diffuser design. The acoustical treatment

needs to complement the visual appearance

of the room to be acceptable to architects.

If the visual aesthetic is not agreeable, the

treatment will have to be hidden behind

fabric; however, this would turn the side

wall diffusers into absorbers. [2]

In order to show how diffusers disperse

reflections finite difference time domain

(FDTD) models can be used. Fig. 2 shows

a cylindrical wave reflected from a planar

hard surface. An impulse is generated, and

so a single cylindrical wavefront is seen in

frame 0, which is traveling from right to

left towards the surface. The wave simply

changes direction on reflection, traveling

back in the specular reflection direction,

where the angle of incidence equals the

angle of reflection. The reflected

wavefront is spatially unaltered from the

incident sound. Consequently, the sound

from the source reflects straight back, and

is unchanged and not dispersed.

Figure 2. Cylindrical wave reflected from a

flat surface.

Fig. 3 shows the effect of changing a

surface to help disperse reflections. In this

case, part of an ellipse is used. It can be

seen that the reflected wavefront is more

bowed. The change is not great, because

the curve of the ellipse was quite gentle;

even so, in the far field the sound will be

more spatially dispersed. The wavefront

generated is still very ordered, however, so

although single semicylinders or ellipses

are good at spatial dispersion, they are not

the best diffusers, because temporal

dispersion is not achieved.


curved surface.

Fig. 4 shows the effect of using a

Schroeder diffuser; the reflected wavefront

is much more complex than the previous

examples. Inspection of the different

frames shows why this complexity arises.

Sound can be seen taking time to

propagate in and out of wells, causing parts

of the reflected wavefronts to be delayed.

The different depts of the wells cause

different delay times, and the resulting

interference between the reflected waves

forms a complex pattern.



Schroeder diffuser.

Figure 5. Spatial and temporal dispersion

generated by a Schroeder diffuser.

One of the most significant occurrences

in diffuser design, if not the most

important event, was the invention of

phase grating diffusers by Schroeder.

Apart from very simple constructions,

previous diffusers had not dispersed sound

in a predictable manner. The Schroeder

diffuser offered the possibility of

producing optimum diffusion, and also

required only a small number of simple

design equations. Single plane or 1D

Schroeder diffusers consist of a series of

wells of the same width and different

depths. The wells are separated by thin

fins. The depths of the wells are

determined by a mathematical number

sequence, such as the quadratic residue

sequence. Single plane diffusers cause

scattering in one plane, in the other

direction, the extruded nature of the

surface makes it behave like a plane

surface. Because of this, it is normal to just

consider the plane of maximum dispersion.

Plane waves are reflected from the bottom

of the wells and eventually re-radiate into

the space. All these waves have the same

magnitude but a different phase because of

the phase change due to the time it takes

the sound to go down and up each well.

Consequently, the polar distribution of the

reflected pressure from the whole surface

is determined by the choice of well depths.

Schroeder showed that by choosing a

quadratic residue, the energy reflected into

each diffraction lobe direction is the same.

Nevertheless, it is important to note that

lobes can be generated in the energy polar

patterns due to surface periodicity.

Figure 6. QRD diffuser cross-section.

For the design theory to be correct,

plane wave propagation within the wells

must dominate. Consequently, an upper

frequency for the diffusion to follow the

simple design principles can be found

from:

(1)

where is the minimum wavelength

before cross-modes in the wells appear,

and w is the well width. Above this


limit dispersion will continue to occur

because these are complicated

structures. Consequently, this is just a

limit of applicability of a theory, and

not necessarily an upper limit for the

diffusion quality. Schroeder diffusers

work at integer multiples of a design

frequency, f0. The design frequency is

normally set as the lower frequency

limit, given by:

(2)

where is the maximum number of

the sequence used, c is the speed of

sound, N is the amount of wells and

is the maximum well depth.

Similarly to fmax, this design frequency

is not the lowest frequency at which the

surface produces more dispersion than

a plane surface, it is just the first

frequency at which even energy

diffraction lobes can be achieved. It has

been shown that Schroeder diffusers

reflect differently from a plane hard

surface an octave or two below the

design frequency [4].

2. DIFFUSER DESIGN

An optimized diffuser was designed

and built, taking into account the principles

of diffusion and scattering.

(Mathematically) optimized diffusers

differ from non-optimized diffusers

because the latter offer only spatial energy

dispersion whereas optimized ones offer

dispersion both in space and time.

Figure 7. 1D optimized diffuser.

Figure 8. Examples of non-optimized

diffusers.

Due to costs, transportation and

constructive limitations, a 1D (one

dimensional Fig. 7) diffuser surface of 1

m2 was proposed (1m x 1m). The material

used was expanded polystyrene (EPS) of

20 kg/m3 density. As mentioned earlier, in

order to achieve a great amount of

diffusion, we need to avoid surface

periodicity. If we consider a numerical

series, in direct relation with the surface

shape, we need to achieve maximum

randomness of the series, randomizing the

surface accordingly. This numerical series

will be then considered as a discrete signal,

and its Fourier Transform will give

information about the amount of pressure

in the far field, as a function of the output

angle of the sound wave propagated,

according to Equation 3[5]:

| ( )| | ( ) ( ) ( ) | (3)


The Autocorrelation function is a great

mathematical tool to analyze the

periodicity of a signal; it shows the

similarity between observations of a signal

as a function of the time lag between them.

It is defined in the discrete domain as:

( )

(4)

The WienerKhinchin theorem states

that the autocorrelation function of wide-

sense-stationary random process has a

spectral decomposition given by the power

spectrum of that process [6]. By applying

this theorem, it can be deduced that in

order to obtain a plain power spectrum

(which equals to uniform distribution of

energy in space), the autocorrelation of a

given signal must be as similar as possible

to a Delta Dirac function (Fig. 9).

Figure 9. Fourier Transform of a Delta Dirac

function.

Consequently, if the numerical series

autocorrelation approaches a Delta Dirac

function, the diffusion will be higher for

the given surface. For that matter, we

define an Autocorrelation Quality Factor

(ACQF) by means of the following:

(5)

where is the autocorrelation function

of the numerical series used. As ACQF

reaches 0 greater decorrelation is achieved

in the signal, causing greater pseudo-

randomness of the diffuser shape and

better diffusion of the sound energy, since

its autocorrelation approaches a Delta

Dirac function (Fig. 10).

Figure 10. Autocorrelation for different

sequences.

In view of the fact that a 1D diffuser

was proposed, the most important thing to

be determined is the well widths and

depths. By using a fixed well width of 5

cm, a series of 20 well depths were

required. For the sake of simplicity, only 6

different well depths were considered, each

one differing 2 cm from each other. The

maximum depth was then 10 cm. In order

to adapt to measurement requirements, the

diffuser built was scaled by a factor of 2

(section 3.2 Scaling) so bandwidth was

calculated according to full scale

dimensions (Eq. 6-7). The numerical series

to establish the diffuser surface shape was

then required to contain a series of 20

numbers from 0 to 5. An algorithm was

coded in MATLAB, so as to find the

optimum numerical series with the lowest

ACQF. After several hours of continuously

running the algorithm, it found that the

optimum sequence was: [2 2 2 2 1 2 3 0 1

0 5 0 1 1 0 3 4 3 3 2] with an ACQF value

of 0,1784.

Once set the sequence values,

maximum and minimum frequencies of

operation were determined according to

Eq. 1 and 2 (even though the equations

apply to Schroeder diffusers only):

(6)


(7)

The theoretical bandwidth of the

diffuser was 3200 Hz approximately.

However, it must be noted that due to its

low amount of mass, the diffuser will

hardly work at such low frequencies, since

the acoustical impedance presented will be

very low. Diffusing behavior will only be

expected at mid and mid-high frequencies.

Figure 11. 3D model of the designed diffuser.

The sample was first modeled in

Google Sketchup and then constructed

(Fig. 11-13). The sample consisted of a flat

base of 4 cm thick of EPS, and several bars

of 2 cm thick according to the calculated

sequence. Materials were glued with

Unipox glue and then reinforced with duct

tape. The footprint surface was 1 m2, total

volume was 0,077 m3 and weight was 1,54

kg.

Figure 12. Final construction.

Figure 13. Final construction.

3. MEASUREMENT PROCEDURE

3.1 PRINCIPLE

Measurements were made to the

diffuser sample according to ISO 17497-

1[7]. This International Standard is used to

determine the sound-scattering properties

of a surface. The degree of acoustic

scattering from surfaces is very important

in all aspects of room acoustics, and is

used mostly by acoustical simulation

software. Insufficient scattering may cause

strong deviations from exponential sound

pressure decay. On the other hand, an

approximately diffuse sound field may be

obtained with highly scattering surfaces in

a room. It is of great relevance to

differentiate between scattering and

diffusion coefficients. While the scattering

coefficient is a rough measure that

describes the degree of scattered sound, the

diffusion coefficient describes the

directional uniformity of the scattering; i.e.

the quality of the diffusing surface.

Therefore, there is a need for both concepts

and they have different applications. To

sum up, results obtained with this standard

can be used to describe how much the

sound reflection from a surface deviates

from a specular reflection.

The general principle of the method can

best be explained by looking at the effect

of reflection and scattering in the time

domain. Fig. 14 shows three bandpass-

filtered pulses which were reflected from a

corrugated surface for different


orientations of the test sample in the free

field.

Fig 14. Examples of band-pass filtered impulse

responses measured at three different positions

of the test sample.

The initial parts of the reflections are

highly correlated. This coherent part is

identical with the specular component of

the reflection. In contrast, the later parts

are not in phase and depend strongly on the

specific orientation. The energy in the

tail of the reflected pulse contains the

scattered part. The principle of the

measurement method is to extract the

specular energy from the reflected pulses.

This is done by means of averaging the

impulse responses obtained for different

sample orientations. In addition to

conventional measurements of absorption

coefficient (based on ISO 354), the sample

is placed on a turntable and impulse

responses are obtained for different sample

orientations. By synchronized averaging of

the pressure impulse responses, the

specular components add up in phase,

whereas the scattered sound interferes

destructively. Assuming statistical

independence between scattered

components, it can be shown (Fig. 14) that

after synchronized addition of n room

impulse responses, the initial decay is

related to the combined effects of

absorption and an apparent energy loss due

to sound scattered from the sample.

3.2 SCALING

In the interest of ease of measurement,

scaling can be applied to both the test

sample and the reverberation room used. A

physical scale ratio of 1:N can be used,

that is, the ratio of any linear dimension in

a physical scale model to the same linear

dimension in full scale. However, it must

be noted that the wavelength of the sound

used in a scale model for acoustic

measurements obeys the same physical

scale ratio. So, if the speed of sound is the

same in the model as in full scale, the

frequencies used for the model

measurements will be a factor of N times

than those in full scale. A scale factor of 2

was used in this case. Since the frequency

range should be measured from 100 Hz to

5000 Hz, in third-octave bands (full scale),

taking into account the scale factor, the

scaled frequency range will be from 200

Hz to 10000 Hz, in third octave bands also.

3.3 REVERBERATION ROOM

The reverberation room must comply

with the ISO 354 [8] standard, but with

certain differences. For example, diffusing

elements within the room must be fixed;

moving diffusers like rotating vanes shall

not be used. The room and its contents

should be invariant, as far as possible. The

temperature and humidity have a very

significant effect. Any devices such as

circulation systems that cause movement

or change the properties of the air in the

room should not be operated. For the

current test room, this was not an issue,

since the room was empty during

measurements, except for the equipment

used and its operators.

The dimensions of the test room are

shown in Fig. 15. This gives a total volume

of 62,2 m3 (for the extended volume

analysis section 3.4 of the present work

must be checked) and a total surface of

98,38 m2. The total volume, in cubic

meters, shall be at least:


(8)

(9)

Figure 15. Test room dimensions

Reverberation time (T20) was measured

in the room to verify maximum absorption

area. Results are shown in Table 1.

Absorption area was calculated according

to 8.1.2 section of ISO 354 as:

(10)

where V is the room volume, c is the speed

of sound and T20 is the reverberation time.

Frequency T20 (s) A1 (m2)

100 1,53 6,55

125 1,74 5,75

160 1,97 5,10

200 1,60 6,26

250 1,70 5,91

315 1,56 6,42

400 1,70 5,91

500 1,70 5,90

630 1,69 5,92

800 1,57 6,39

1000 1,60 6,29

1250 1,53 6,54

1600 1,47 6,82

2000 1,44 6,96

2500 1,42 7,05

3150 1,31 7,65

4000 1,20 8,35

5000 1,12 8,95

6300 1,03 9,75

8000 0,91 11,04

10000 0,77 13,00 Table 1. Reverberation time and absorption

area for the test room.

The equivalent absorption area of the

empty room, A1, including the air

attenuation should not exceed:

(11)

(12)

This was not achieved due to the

limitations of the test room dimensions and

room volume, which lowered the

reverberation time, even though the

surfaces were very reflective.

As regards shape of the room, if the

room is not rectangular, none of its

surfaces should be parallel. If the room is

rectangular, its proportions should be

selected so that the ratio of any two

dimensions does not equal or closely

approximate an integer. The proportions

1:21/3

:41/3

are frequently used. Other room

dimension ratios that have been found to

be satisfactory are given in Table 2 [9].

ly/lx lz/lx

0,83 0,47

0,83 0,65

0,79 0,63

0,68 0,42

0,7 0,59

NOTE The symbols lx, ly and lz represent the room dimensions.

Table 2. Recommended test room

proportions.

Room proportion determined was 1:

1.1: 2 since ly/lx=1,81 and lz/lx=0,9

(considering lx=3,36m; ly= 6,09m; lz=

3,04m). Consequently the test room will

not guarantee an even modal distribution in

the low frequencies. However, since the

lowest frequency of analysis was 200 Hz,

modal distribution issues were not that

significant. Also, according to ISO 354 the

length of the longest straight line (Imax)

which fits within the boundary of the room

should obey the following equation:


(13)

(14)

This conditions could not be met either, for

the same reasons already established.

3.4 DIFFERENCES IN ROOM

VOLUME

The reverberation room had a

suspended ceiling built in PVC (6mm

thick). Above this surface, there is a

distance of 0,3 m up to a concrete ceiling.

It should be clarified then, that the actual

volume of the room will be a function of

frequency. For low frequencies, the roof

has a very high rate of transmissibility, so

all the sound radiated on its surface is

easily transmitted to the air gap between

ceilings, thereby increasing the volume to

be considered. But at high frequency, the

transmissibility rate is sufficient to prevent

sound propagation. By means of

calculating Transmission Loss (TL) of the

ceiling material this effect can be

considered. TL simulation was performed

with a MATLAB software algorithm to

estimate this parameter.

Figure 16: Tranmission loss calculated for the

suspended ceiling.

Two relevant ranges of frequency can

be noted (Fig. 16). Above 1 kHz, the TL is

high enough to consider a constant room

volume. Below 1 kHz the TL is less than

30 dB, so volumetric increase must be

considered in this range. Fig 17 shows the

volume as a function of frequency, where

volume linearly decreases up to 1 kHz, and

then it remains constant.

Figure 17: Corrected volume of the room.

3.5 TEST SAMPLE

The area of the test sample should be as

large as possible in order to obtain good

measurement accuracy. The test sample

should be ideally circular, but the standard

allows the use of square samples (flush-

mounting the sample in the turntable). The

sample shall have a minimum edge size of

. In this case this gives an amount

of 1,5 m, which was not achieved. Flush

mounting was not done either. Structural

depth of the test sample should be

(Fig. 18).

Figure 18. Structural depth of test sample.

where d is the diameter of the test sample

and h is the structural depth. In this case,

this condition could not be achieved either,

since h = 0.1 m and d=1m. As a result,

scattering coefficients obtained with this

sample could be above 1 and below 0 for

particular third-octave frequency bands,

because edge effects can occur due to

variations in the height of the sample along

the edge of the test sample. The surface of

the perimeter of the test sample should be

as smooth as possible and rigid as possible.

The perimeter was not covered during

measurements. The absorption coefficient

of the test sample should not exceed a

value of s = 0,50 because the

measurement method will not produce


reliable results for samples with high

absorption coefficient.

3.6 BACKGROUND NOISE

When measuring impulse responses in a

room it is important to have a good amount

of signal-to-noise ratio (S/N) for each

frequency band in order to obtain accurate

values of reverberation times. When

measuring T20 for instance, a S/N ratio of

25 dB is the minimum required in order to

linearize the decay recorded. The reason

for this is described in the requirements of

decay signals, found at section 3.5 of the

present work. In the interest of reaching

higher values of S/N ratio, the source

sound pressure level was set notably high

and perceptually above the background

noise. To complement this, all impulse

responses measured were checked for

appropriate S/N ratio using Aurora

software (Fig. 19). It is worth noting that

the log sine sweep method was used,

which improves the actual S/N ratio

performance of the measurement.

Figure 19. S/N ratio evaluation for an

impulse response measured.

3.7 MEASUREMENT OF IMPULSE

RESPONSES

Impulse responses were measured

without and with the test sample following

ISO 354. Two source positions and three

microphone positions were used, giving a

total of six measurements for each

reverberation time. The reverberation time

was then estimated as the arithmetic

average of the individual reverberation

times determined in each position.

Measurement overview can be seen in

Table 3.

Reverberation Time

Test sample

Turntable

T1 not

present not

rotating

T2 present not

rotating

T3 not

present rotating

T4 present rotating

Table 3. ISO 17497-1 Measurement overview.

A turntable was required in order to

rotate the sample. The turntable shall be

provided with a rigid base. The base plate

shall be symmetrical with respect to the

axis of rotation. The size of the base plate

shall correspond to the maximum

dimension of the test sample. In this case,

an Outline ET250-3D electronic turntable

was used. It has a circular base with a

diameter of 350 mm. Dimension of the

turntable can be seen in Fig. 20.

Figure 20. Dimensions of Outline ET250-3D

turntable.

No part of the turntable may be closer

than N-1

. 1,0 m to the walls of the room (in

this case 0,5 m). The scattering coefficient

for the base plate itself shall be measured

to check the quality of the arrangement.

For each combination of source and

receiver positions, (in some cases


according to Table 3) the test signal was

continuously radiated and received while

the turntable was rotating. The total

measurement duration was equal to the

time of one revolution of the turntable. The

test signal used to measure impulse

responses shall be deterministic since the

evaluation requires a coherent averaging.

The integrated impulse response method

shall be applied. In this case, the signal

was a sine sweep from 50 Hz to 16000 Hz

which lasted 180 seconds. In order to avoid

measurement error due to air movements

or other unstable conditions in the room,

the measurements were not started until 15

minutes after closing the door.

Figure 21. Experimental setup

Audio samples were recorded using a

Tascam US-1641 usb audio interface, with

16 bit resolution and 44.1 kHz sampling

frequency. Measurements were made using

three simultaneous measurement

microphones: two Earthworks M50 and

one DPA 4007; all of them being

omnidirectional with great impulse and

frequency response. Test signal was

generated in Aurora 4.4 software and then

reproduced in an Outline Globe Source

Radiator and Subwoofer, as seen in Figs.

21 and 23. Audio samples were then

deconvolved with inverse filters using

Aurora to obtain impulse responses.

Impulse response decay was analyzed in

third-octave bands using Easera 1.0

software. According to ISO 17497-1, the

backward integration shall be restricted to

the linear slope of the impulse response

level. The decays for T1, T2 and T3 should

be linear down to the background noise

level, whereas the decay for T4 consists of

two superposed decay curves, and only the

first decay should be evaluated. This can

clearly be seen in Fig. 22.

Figure 22. Example of reverberation times

slopes for the different setups.

Integration limit must be set at -30 dB

and the reverberation time must be

evaluated in the range between -5 dB and

-20 dB provided that the first decay is

within the range. In order to comply with

the standard as much as possible, T20 were

used for these particular reasons.

Figure 23. Experimental setup.

For the specification and positioning of

microphones and sources ISO 354 was

referred, but taking into consideration the

scale factor applied. As a result, the

minimum distance between microphones

was 0,75 m; the minimum distance from

microphones to source was 1m and the


minimum distance to any room surface

was 0,5 m. Both sound source positions

had to be at least 1,5 m apart.

Figure 24. Microphone and source positions for

the first measurement setup.

All this distance conditions were met

accordingly; distance was measured with a

Bosch GLR225 laser distance meter. In

Fig. 24-25 the two measurement setups

used can be seen.

Figure 25. Microphone and source positions for

the second measurement setup.

As regards temperature and relative

humidity, changes during the course of

measurement can have a large effect on the

measurement results, especially at high

frequencies. Reducing the air attenuation

improves the measuring accuracy.

Therefore, temperature and relative

humidity shall be measured in the room

before and after each of the four

measurement situations. This was partially

accomplished, since only temperature

measurements were made before and after

all measurements. Overall changes were

minimal, a difference of 0,7C. The initial

temperature was 18,6 C whereas the final

temperature was 19,3 C. Measurements

were made using a Luft digital

thermometer with a precision of 0,1C. No

relative humidity measurements were

made due to lack of measurement

equipment for that matter.

4. RESULTS

In pursuit of calculating the scattering

coefficient of the diffuser, others

coefficients had to be calculated first. One

of them was the random-incidence

absorption coefficient s. According to ISO

17497-1, it shall be calculated for third-

octave bands using the formula:

(

)

( ) (15)

where V is the volume of the reverberation

room in cubic meters (m3); S is the area of

the test sample in square meters (m2); T1 is

the reverberation time obtained without

sample but with the plate present, in

seconds (s); T2 is the reverberation time

obtained for the test sample in seconds (s);

c1 is the speed of sound in air, in meters

per second (m/s), during measurements of

T1; c2 is the speed of sound in air, in meters

per second (m/s) during the measurement

of T2; m1 is the energy coefficient of air, in

reciprocal meters (m-1

), calculated

according to ISO 9613-1, using the

temperature and relative humidity during

the measurement of T1; m2 is the energy

attenuation coefficient of air, in reciprocal

meters (m-1

), during the measurement of

T2. The reverberation times T1 and T2 are

measured without rotation of the turntable.


The speed of sound in atmospheric air

can be calculated according to ISO 9613-1

as:

(16)

In this case, since the test room volume

was not high enough to consider air

absorption, m coefficients were zeroed.

Also, the specular absorption coefficient

spec was calculated similarly to s:

(

)

( ) (17)

where V is the volume of the reverberation

room in cubic meters (m3); S is the area of

the test sample in square meters (m2); T3 is

the reverberation time obtained for the

rotating base plate without sample, in

seconds (s); T4 is the reverberation time

obtained for the test sample on a rotating

turntable in seconds (s); c3 is the speed of

sound in air, in meters per second (m/s),

during measurements of T3; c4 is the speed

of sound in air, in meters per second (m/s)

during the measurement of T4; m3 is the

energy coefficient of air, in reciprocal

meters (m-1

), during the measurement of

T3; m4 is the energy attenuation coefficient

of air, in reciprocal meters (m-1

), during the

measurement of T4. Again, m coefficients

were zeroed in this equation.

Finally, the determination of s and spec leads to the calculation of the random-

incidence scattering coefficient using the

following formula:

(18)

Spatially averaged results obtained for

reverberation time (T20) in all cases can be

seen in Table 4 and Fig. 26. It must be

noted that all values were transposed to

full scale measurement: for example,

values for 100 Hz were measured at 200

Hz, and so on. From this data, it can be

observed that in low frequency there are

some variations due to uneven modal

density, but then the reverberation time

tends to decrease, as expected. Also, T2

and T4 values show a determined amount

of decrease respect to T1 and T3, given the

fact that in these cases the test sample was

present and provided some absorption.

Frequency T1 T2 T3 T4

100 1,60 1,44 1,54 1,39

125 1,70 1,52 1,72 1,48

160 1,56 1,46 1,51 1,46

200 1,70 1,53 1,72 1,55

250 1,70 1,52 1,67 1,56

315 1,69 1,54 1,71 1,59

400 1,57 1,50 1,63 1,48

500 1,60 1,48 1,60 1,51

630 1,53 1,38 1,51 1,41

800 1,47 1,34 1,49 1,32

1000 1,44 1,29 1,45 1,30

1250 1,42 1,28 1,38 1,28

1600 1,31 1,19 1,32 1,19

2000 1,21 1,10 1,20 1,09

2500 1,10 1,02 1,13 1,04

3150 1,05 0,95 1,04 0,94

4000 0,92 0,85 0,92 0,85

5000 0,78 0,74 0,78 0,73 Table 4. Results obtained for reverberation

time (T20).

Figure 26. Results obtained for reverberation

time (T20).


Standard deviation of the reverberation

time was estimated according to Annex A

of the ISO 17497-1 as:

( )

( ) (19)

where N is the number of measurements of

the reverberation time, and the spatial

average of the reverberation chamber is:

(20)

The 95 % confidence limit for T20

values was estimated as two times the

standard deviation in all cases. Table 5

shows measurements as well as their

corresponding expanded uncertainty.

Frequency T1 T2 T3 T4

100 1,6 0,1 1,44 0,15 1,54 0,12 1,39 0,14

125 1,7 0,13 1,52 0,12 1,72 0,18 1,48 0,14

160 1,56 0,08 1,46 0,12 1,51 0,1 1,46 0,09

200 1,7 0,13 1,53 0,07 1,72 0,09 1,55 0,14

250 1,7 0,09 1,52 0,08 1,67 0,1 1,56 0,11

315 1,69 0,09 1,54 0,06 1,71 0,15 1,59 0,09

400 1,57 0,09 1,5 0,06 1,63 0,04 1,48 0,06

500 1,6 0,06 1,48 0,03 1,6 0,02 1,51 0,04

630 1,53 0,04 1,38 0,04 1,51 0,04 1,41 0,05

800 1,47 0,04 1,34 0,03 1,49 0,05 1,32 0,06

1000 1,44 0,05 1,29 0,03 1,45 0,04 1,3 0,08

1250 1,42 0,04 1,28 0,02 1,38 0,04 1,28 0,06

1600 1,31 0,03 1,2 0,01 1,31 0,02 1,18 0,03

2000 1,2 0,02 1,09 0,04 1,22 0,03 1,09 0,02

2500 1,12 0,04 1,03 0,01 1,12 0,02 1,04 0,02

3150 1,03 0,03 0,95 0,03 1,04 0,02 0,94 0,03

4000 0,91 0,02 0,86 0,02 0,93 0,01 0,85 0,03

5000 0,77 0,01 0,73 0,02 0,78 0,01 0,73 0,02

Table 5. T20 results with expanded uncertainty.

Figs. 27-28 show graphically the expanded

uncertainty for T1 and T2 in order to

compare both measurements. T3 and T4 are

not displayed because the values are very

similar to the latter. The data dispersion is

greater for low frequencies due to room

resonances, and it decreases as frequency

increases.

Figure 27. Measurement uncertainty for T1.

Figure 28. Measurement uncertainty for T2.

Final values obtained for scattering are

showed in Table 6, and obtained according

to Eq. 15-18. The first thing to analyze is

the fact that scattering values calculated

are not trustworthy below 1250 Hz for

various reasons. The most crucial factor is

assumed to be the test sample surface,

which was not enough to fit the standard.

Other factors could have been the low

amount of diffusion of the test room (the

second most relevant factor), design

limitations and considerations such as

weight and spatial dimension of diffusion

(1D diffusers offer lower scattering than

2D diffussers), the edge absorption of the


test sample due to the fact that the sample

was not flush mounted, low amount of

decay samples (source and microphone

positions), instrumental error and other

secondary requirements of the ISO 17497-

1 which were not met and specified during

this product specification. However, for

the frequencies above 1600 Hz, results

obtained were much more reliable, partly

because diffraction at higher frequencies

(due to the short wavelengths) is a

phenomenon with a higher probability to

occur.

Frequency alfa s alfa spec s

100 0,72 0,71 -0,02

125 0,69 0,93 0,78

160 0,46 0,22 -0,45

200 0,66 0,67 0,03

250 0,69 0,46 -0,76

315 0,59 0,45 -0,33

400 0,30 0,63 0,48

500 0,51 0,39 -0,25

630 0,74 0,50 -0,97

800 0,69 0,87 0,59

1000 0,82 0,75 -0,35

1250 0,77 0,59 -0,81

1600 0,78 0,83 0,22

2000 0,83 0,87 0,25

2500 0,72 0,81 0,35

3150 1,02 1,01 0,39

4000 0,90 0,94 0,44

5000 0,70 0,88 0,61 Table 6. Results calculated for s, spec and

scattering coefficient (s).

The uncertainties in the absorption

coefficients are obtained by means of the

following equations:

(

) (

) (21)

(

) (

) (22)

Finally, the standard deviation in the

scattering coefficient is:

|

| (

) (

) (23)

Frequency s spec s

100 0,84 0,90 4,43

125 0,69 0,86 2,79

160 0,68 0,61 2,17

200 0,53 0,65 2,43

250 0,46 0,58 3,22

315 0,43 0,64 2,08

400 0,47 0,32 0,57

500 0,28 0,22 0,84

630 0,29 0,29 2,52

800 0,27 0,43 1,43

1000 0,31 0,51 3,63

1250 0,24 0,41 2,59

1600 0,21 0,27 1,46

2000 0,33 0,26 2,12

2500 0,31 0,26 1,17

3150 0,38 0,35 1,76

4000 0,34 0,37 1,58

5000 0,36 0,36 1,28 Table 7. Standard deviation for s, spec and s

coefficients

Standard deviations for all coefficients

are shown in Table 7. High values of

standard deviation were introduced, due to

the uncertainties already mentioned.

Current results for scattering were

compared and contrasted to four diffuser

solutions offered by the company RPG,

only in the frequencies above 1600 Hz.

The diffusers compared were 1D and can

be seen in Fig. 29. Fig. 30 shows data

comparison. As it can be seen, scattering

of the designed diffusor at these

frequencies is not as good as other

products scattering, but it shows similar

performance as the FlutterFree diffuser.

However, it must be noted that the

comparison is only made at high

frequencies; most of the RPG diffusers


shown offer scattering at lower frequencies

also.

Figure 29. 1D commercial diffusers:

FlutterFree, QRD 734, FlutterFree-T,

Formedffusor (from top left to bottom right).

Figure 30. Comparison between the designed

diffuser and commercially available products.

5. CONCLUSION

A diffuser design was proposed and

constructed, taking into consideration the

background theory associated with diffuse

sound fields and acoustics. Measurements

of the diffuser were carried out according

to ISO 17497-1 and it was determined that

a very important factor was the sample

size; since we are calculating the scattering

coefficient indirectly, by means of

reverberation time, if the sample footprint

has a low amount of surface, the

reverberation time considered will not vary

accordingly and error will be introduced.

As stated, diffusion is a very desirable

effect on room acoustics and even though

the quantity of diffusion surface is hard to

define, it is empirically demonstrated and

well known that in order to obtain good

sounding rooms there are some obligatory

locations for diffusers. For example, in

control rooms for recording/mixing

studios, a preferable place for diffusers is

the rear wall, because sound incidence is

critical and highly probable for that

surface. The election of 1D or 2D diffusers

will depend on the overall geometry of the

room; 2D will tend to distribute the energy

along the horizontal and vertical axis of

incidence whereas 1D will only sparse the

energy in a unique axis depending on the

diffuser mounting. In the case of control

rooms, if sound reflections from the rear

wall to the ceiling are to be avoided, 1D

diffusers mounted vertically will be

preferable, and they will only scatter the

sound energy laterally; this could be the

case when the height of the control room is

not very large and would avoid strong

second order reflections at the sweet spot.

In the case of concert halls, the

situation will be different since these

rooms have greater volume and different

geometry in general. Here, the compulsory

location for diffusers will be the front side

of the balcony surface; scattering on this

surface can be accomplished by means of

optimized diffusers or non-optimized ones

such as ornaments which are very common

in concert halls. The scattering on the

balcony will avoid reflections from the

stage causing unwanted echoes which

could perceptually affect the performers or

the audience located at the front rows.

Also, diffusion can be introduced on the

rear walls for the same reasons as in

control rooms, but considering the distance

between the rear walls and the last row of

seats. Diffusers tend to behave like

resonators in the short distance, and its


frequency of resonance will be determined

by well depth and width. Therefore,

listeners at the last rows will perceive

variations in the frequency response that

will not be applied to the rest of the

audience.

In the end, diffusion and scattering are

very complex phenomena in the room

acoustics field and even though objective

influence of these parameters in actual

enclosures and sound fields are not well

established, they can improve acoustical

preference of rooms when used properly,

and the design of surfaces which introduce

diffusion and scattering behavior such as

the present in this specification is of great

relevance.

6. REFERENCES

[1] Strube, H. W. (1981). More on the

diffraction theory of Schroeder

diffusors. The Journal of the Acoustical

Society of America, 70(2), 633-635.

[2] Cox, T. J., & D'antonio, P. (2009).

Acoustic absorbers and diffusers: theory,

design and application. CRC Press.

[3] L. L. Beranek, Concert and Opera

Halls: How They Sound. AIP Press, 6974 (1996).

[4] T. J. Cox and Y. W. Lam. Prediction

and evaluation of the scattering from

quadratic residue Diffusors. J. Acoust.

Soc. Am., 95(1), 297305 (1994).

[5] Schroeder, M. (1997) Number Theory in science and Communication. Chapters 13 & 15.Third edition. Springer. 1997.

[6] C. Chatfield (1989). The Analysis of

Time Series An Introduction (Fourth Ed.). Chapman and Hall, London. pp. 9495.ISBN 0-412-31820-2.

[7] ISO 17497-1. Acoustics - Sound-

scattering properties of surfaces - Part 1:

Measurement of the random-incidence

scattering coefficient in a reverberation

room.

[8] ISO 354. Acoustics - Measurement of

sound absorption in a reverberation room.

[9] ISO 3741. Acoustics - Determination of

sound power levels of noise sources using

sound pressure - Precision methods for

reverberation rooms.

Scattering Coefficient Damis Cacavelos

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