SOUND POWER MEASUREMENT IN
A SEMI-CONFINED SPACE
by
WILLIAM R. SCHLATTER
S.B., General Motors Institute
(1971)
Submitted in Partial Fulfillment of
the Requirements for the Degree
of
Master of Science
at the
Massachusetts Institute of Technology
July, 1971
Signature of Author
Department of Mechanical Engineering
July 28, 1971
Certified by
SThesis
Supervisor
I
Accepted by .v
Archives
Chairman, Departmental Committee
on Graduate Students
MASS.
INSL
TECH,
SEP
9 1971
jLR8A
RI
SOUND POWER MEASUREMENT IN
A SEMI-CONFINED SPACE
by
WILLIAM R.
SCHLATTER
Submitted to the Department of Mechanical Engineering on
July 28, 1971, in partial fulfillment of the requirements
for the degree of Master of Science.
ABSTRACT
A sound field can be adequately described if sufficient measurements are made at separate points.
A broad-band noise field requires
only one measurement while a pure-tone or narrow-bandwidth noise field
may require a burdensome number of measurements to get a measured mean
value of squared sound pressure.
If the mean value can be adequately
determined, then the sound power level of a source, in reverberant conditions, can be predicted through a direct relationship between sound
pressure level and sound power level.
This is done for a point source
in a channel.
Thesis Supervisor: Richard I1.Lyon
Title:
Professor of Mechanical Engineering
Table of Contents
ABSTRACT
2
LIST OF FIGURES
4
ACKNOWLEDGMENTS
5
INTRODUCTION
6
CONCLUSIONS
8
DETERMINATION OF SOUND POWER
9
EXPERIMENTAL MEASUREMENT OF SOUND POWER
20
OPTIMAL MEASUREMENT TECHNIQUE
25
CONTINUATION OF THESIS WORK
31
APPENDIX A: DISTRIBUTION OF POINT SOURCES IN INFINITE LINE
32
APPENDIX B: CORRELATION COEFFICIENT BETWEEN TWO POINTS IN
38
A SOUND FIELD
APPENDIX C: TABLES 3 -
REFERENCES
11
39
m
List of Figures
1.
GEOMETRY OF POINT SOURCE AND REFLECTED IMAGES FOR
10
PARALLEL REFLECTING WALLS
2.
L
vs. 6 FOR POINT SOURCE (R = 0)
16
3.
L
vs. 6 FOR MULTIPLE POINT SOURCES (R = 1)
17
4.
L
vs.
19
P
6 FOR DIFFERENT VALUES OF R
5.
(L - L ) vs. 6
p
w
6.
EXPERIMENTAL SETUP
22
7.
SOUND PRESSURE LEVEL AS A FUNCTION OF POSITION
29
-
EXPERIMENTAL RESULTS
21
ACROSS THE CHANNEL
8.
PLOT OF FUNCTION (y
9.
PLOT OF 10 LOG y
=
sin x ci x - cos x si x)
10. PLOT OF BEGINNING OF - 6 dB/dd DROPOFF OF L
P
35
36
37
Acknowledgments
The author wishes to express his appreciation to Professor
Richard H. Lyon for his guidance in the work presented here.
I am
grateful for the encouragement and assistance received from him and
from my fellow graduate students in the acoustics and vibration
laboratory.
My thanks also to the Central Foundry Division of General
Motors Corporation for their financial support during this research.
I. Introduction
The measurement of the sound produced by a machine or other device
is a topic which has been the subject of various papers and the basis of
a standard of the American National Standards Institute (ANSI).
The
American Standard Method for the Physical Measurement of Sound [1] outlines in some detail the method to be used in the determination of sound
power under four conditions: 1) in a free field; 2) in a free field above
a reflecting surface; 3) in a diffuse (reverberant) field; and 4) in a
semireverberant field.
The determination of the sound power of a source requires that an
estimate, p2 , be made of the true, space-averaged value of squared sound
pressure <p2 >_
aged.
where all squared pressures are assumed to be time aver-
The uncertainty in p2 is estimated by its standard deviation a 2*
A limitation of the ANSI standard is the requirement that the sound
output of the source be primarily broad-band noise for the cases involving
measurements in a reverberant or semireverberant field.
For a source
emitting line or narrow bandwidth spectra, the ANSI standard recommends
against determining sound power under diffuse field or semireverberant
field conditions.
There are cases, however, when it is necessary to make
measurements under these conditions when a source has pure tone components and still be reasonably confident of the results.
The purpose of this thesis is to present the work done in attempting
to determine the sound power of a source located between two parallel
walls, a case similar to a car in a city street which is flanked by tall,
closely packed buildings.
The problem involves both broad band noise and
superimposed pure tone components where phase interference in reflected
portions of the sound field cause large fluctuations in sound pressure
level with small changes in position of source or observer.
The general format of this thesis is a) a general conclusion of the
results of this work, b) an analysis of sound power determination for
broad-band noise from a source in a channel, c) a description of the
experimental program for measuring the mean and variance of the squared
sound pressure, d) the experimental results giving an optimal measurement technique for minimizing the variance in p 2 , and e) a brief outline
for continuing the work begun in this thesis.
II. Conclusions
It has been shown that for the case of an omnidirectional point
source of sound in a channel, there is a direct relationship between the
space-averaged sound pressure level around a given location and the sound
power level of the source.
A single measurement position is sufficient
to define the sound field when the sound is primarily broad-band noise.
When a pure tone predominates in the sound field, several discrete
sample points will give an average mean value of sound pressure that is
as accurate as the average obtained by a continuous line average.
The
discrete points should be chosen at one-half wavelength spacings to get
uncorrelated samples.
A total of 6 sample points spaced :at X/2 apart will
give a mean value of sound pressure that has no more variance in the mean
value than a continuous line average over 3 wavelengths (or 1/4 street
width) distance.
III.
Determination of Sound Power
The total acoustical power radiated from a sound source operating
in a given acoustic environment is frequently a more useful quantity than
the sound pressure measured at a certain position.
If the environment of
the sound source should change, the sound pressure measured at a given
position could change greatly whereas the radiated acoustical power is
generally not changed appreciably.
However, if the source is near one or
more reflecting surfaces, the radiation impedance may differ appreciably
from that of free space [2].
Under the condition that the reflecting
surfaces remain fixed with respect to the source location, these surfaces
may be considered a part of the source and the sound power determined as
such.
A car on a city street is an example of this since the street re-
mains a fixed surface with respect to the car even though the car may be
moving.
This problem is further discussed in Section IV where direction-
ality of the source is considered.
The radiated power of a sound source can be indirectly measured
since the acoustic power is proportional to the mean-square sound pressure.
The sound pressure level obtained by a single measurement in a reverberant sound field depends greatly upon the position of the microphone if
the sound field contains significant pure tone or narrow bandwidth components.
This section deals only with those cases of wide band noise and a
later section covers the cases of pure tone sources.
The computation of sound power can be made using the following relationship:
L
w
= L
p
- 10 log X - C
(1)
10
I-
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(i
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\
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SoUi.c.
PAIRALL.EI
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,I7
/
s
W.W"..EWc
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where
10 logl 0 (W/10- 12 watts), (dB)
Lw
W
Lp
E
sound power of the source in watts
20 loglo (p/0.0002 microbar), (dB)
p = sound pressure in microbars
C =10 log 1 0
(
B
528
'
½
) ]I60+
+ .5dB
[1]
6 = the air temperature in degrees Fahrenheit
B = the barometric pressure in inches of mercury
X = a factor determined by the geometry of the source, the source
environment, and the directivity factor.
The main part of the following discussion centers around the determination and evaluation of the factor X for the semireverberant case considered in this paper.
At this point we describe the environment and type of source being
considered.
We consider the situation of a car being driven down a city
street flanked by tall buildings spaced very close together.
shows a graphic representation of the propagation model.
Figure 1
The ray theory
of acoustics is employed and multiple images replace the actual single
source and reflecting surfaces.
We shall also ignore source direction-
ality at first and treat the car as an onmidirectional point source.
For a single point source, the sound pressure is related to the
sound power by the formula:
p2 = pcW/47rr 2
(2)
where
p 2 /pocW = X = 1/47r
2
(3)
W = The acoustical power of the omnidirectional source in watts
r = the distance from the source in meters.
The case of a single point source can readily be extended to that
for many point sources by summing the contributions produced by each of
the sources at the observer's position.
When each of the point sources
is producing incoherent or wide band noise so that phase interference between sources can be ignored, we can simply add the mean square pressures
due to each source to obtain the total mean square pressure.
For the
multiple image case shown in Figure 1, the contribution of the n-th image
to the mean square pressure is
p n = pcWRIn/4,r2
n
(4)
where rn = the distance from the n-th image
R = the energy reflection coefficient of the walls;
W(reflected)/W(incident)
WRInI
= the effective sound power of the n-th image.
The total contribution, at a point, of all the point sound sources
would be the summation of all p2 from Equation 4:
n
p
22=
n=-w
n
E
P cWR I n /4,r
n=-n
o
2
n
(5)
and
X =
E RIn / 4 ,r2
n
n=-_
.
(6)
13
X is proportional to the sum of the inverse square of the distance to
each source for the case of R = 1:
X
E
2
1/r
n
n=-=
l/[d 2 + (nL - a) 2 ]
=
(7)
n=-C
where (a) is as shown in Figure 1. The summation in Equation 7 has been
evaluated by Johnson and Saunders [3] for the case of equal strength
noise sources (R = 1) equally spaced:
S
/[
2
2
+a)
]
=
SdL
sinh (27d/L)
cosh (2id/L) - cos (2wa/L)
(8)
and the quantity X for the case R = 1 becomes
1
X = 41,
sinh (2r6)
cosh (2w6)
- cos (2'a)
(9)
where 6 and a are distances normalized to the street width (or source
separation distance).
Equation 9 can be made even more general for the case shown in
Figure 1 in which the sources are not evenly spaced.
In this case, the
series of points are broken up into two separate series of odd and even
numbered locations.
These two new series each have evenly spaced p6ints
2L apart and a new value for X is given by
= 1
X
sinhw6
coshw6 - costr (a -b)
L
sinhr6
r(a +
coshw6 + cos w+
L b)
L
(10)
which reduces to Equation 9 for the case when b - 0. Equations 9 and 10
are listed in Table 1 along with a few other simple cases for comparison
purposes.
'TABLE
FOR
TypE oFsouge
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15
A plot of the simplest case, a single point source (R = 0), shown
in Figure 2 shows some interesting results.
Beyond two street widths
from the source in the direction along the street, the mean-square pressure
level, L , is within a 1 dB range regardless of the positions of the
source and observer across the width of the street.
of L
The limiting value
is determined by the separation w as the distance d, or 6 = d/L,
decreases.
The result in Figure 2 has only reference value.
As the
reflection coefficient, R, increases from 0 to 1i,the mean-square pressure
level, Lp, for the same source will always be equal to or higher than
the level shown by Figure 2.
When R = 1, a plot of Equation 10 (the relation graphed in Figure 3)
shows the maximum values one can obtain for L
as a function of source
p
and observer positions in the street.
Some important points shown in
Figure 3 are that when the sources are evenly spaced (i.e., b = 0) and
the observer is also centered in the street (a = 0), the level L
p
approaches that of a single point source for values of 6 less than 0.2.
When the source is against one wall, the first image corresponds with the
original source and the levels along that wall approach a limit of 3 dB
above that of the single source in the same position.
Likewise, having
the source and observer on opposite sides of the street will produce a
minimum level of greater than 6 dB above that of a single source due to
two double images spaced at one street width from the observer at the
closest position.
Again, in the case of multiple points (R > 0), the levels all
approach a limit where source and observer positions across the street
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are unimportant.
In this case the distance where the levels are less
than 1 dB from the limit is reached at 6 = 1.0 (or d = L) whereas this
limit occurs at 6 = 2.0 (or d = 2L) for a single point source when R = 0.
The limit for the case of R = 1 turns out to be the -3 dB per doubling of
distance line one would get if an infinite line source had a strength of
W watts per unit length where the unit length is equal to one street
width.
This is a useful result since it shows that beyond a distance
proportional to the street width, the mean-square sound pressure level is
independent of observer position across the street and of whether the
source and its images are discrete points of arbitrary spacing or distributed in a continuous line.
This result enables one to evaluate the
intermediate cases where R is between 0 and 1 by distributing the point
sources across the street in a continuous line.
This is done in Appendix
A to get the results shown in Figure 4.
The noise models used have resulted in three graphs showing the
relationship between mean-square sound pressure and sound power of a
source for measurements taken at various positions.
Because of approxi-
mations for the intermediate values of R, that plot is only valid for
6 Z 1.5 or 2.0.
The next step is to see if the results found can be
verified experimentally.
results.
The following section presents experimental
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IV.
Experimental Measurement of Sound Power
In the preceding section, the relationship between sound pressure
level and sound power level was given for the case of a sound source
placed between two parallel reflecting surfaces of various reflection
coefficients.
The relationship
L
w
= L
p
- 10 log X - c
(1)
with R = 1.0 giving the result
sinhi6
1
X=
+
coshL6 - cos
L
(a - b)
sinhnd
coshw6 + cos
L
(a + b)
(10)
has been experimentally verified for both broad-band noise and pure tone
sources.
Figure 5 shows the theoretical curve for Equations 1 and 10
with the experimental results plotted on the same graph (c = 0.5 dB).
The experimental setup used to get these results was a 32:1 scale
model using a point sound source located midway between two parallel walls
(Figure 6).
As shown, the sound source used was a dynamic microphone
driven by a voltage equivalent to the type of sound desired (e.g., pure
tone-sine wave, white noise, filtered octave band or 1/3 octave band of
white noise, etc.).
The sound source was placed at the floor level pointed
into the edge between the end and floor to get an omnidirectional source
in the field of interest.
The sound power and the directionality of the
source were easily determined by making sound pressure level readings on
the surface of an imaginary quarter sphere centered on the source and with
a radius small enough that "free" field measurements were made in the region where the -6 dB/dd law applies, and was observed.
The distance used
a.
4
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in this experiment was a radius of 3 inches.
The problem of source directivity enters when source power is determined.
In this problem, the source has three mirrored images located at
the same point as the source.
Therefore, the sound power observed is that
of a single point source in free space having four times the sound power
or equivalently, a single source having a directionality of four in the
region of space in the channel and zero elsewhere.
Thus, in measuring
the sound pressure level in the free-field part of the far field,
Equation 9.1 from L.L. Beranek's Noise Reduction book
L
where L
p
= L + DI 8 - 20 log r - 1 dB
w
e
(11)
= sound-power level of-source, dB re 10- 1 2 watt
r = distance of receiver from source in feet
DIO = directivity index of source in direction 0, dB
shows that if the true power level of the source is used with a directivity of four, the same result could be found using a directivity of one
and a source strength of four times the true strength.
In the experi-
mental determination of sound power in this work, directionality was
ignored, and the sound pressure level observed on one-quarter of a sphere
centered on the source was implicitly assumed to be the same on the other
three quarters.
This assumption can be made for the work in this thesis
but in stating the sound power of a source in field work, care must be
taken to use this directionality so as not to overrate the sound power
output of a particular source.
In the experimental work done, measurements of L
were made with
p
the top and far end of the channel open.
The measuring microphone was
protected from floor vibrations by a pad of fiberglass.
In all parts of
the experimental work, the source was kept in the center of the channel
and the receiver was either in the center of the channel or at a distance
down the channel where position across the channel was unimportant.
Broad Band Noise - In the first part of the experimental work, one octave
of white noise centered at 4000 Hz was used.
This broad band of noise
results in a frequency average which is equivalent to a spatial average
for discrete frequencies [4].
Measurements of L as a function of disp
tance along the center of the channel were made and compared to the sound
power level of the source.
The result plotted in Figure 5 shows fairly
close correlation between theory and experiment.
Measured values are
listed in Table 3 in the appendix.
Pure Tone Sound - To compare theory to experiment for pure tone sound, a
tone of 4000 Hz was used and a spatial average was required to arrive at
a single number level L as a function of distance down the channel.
To
do this, a sweep across the channel was made at distances where average
level was predicted to be independent of receiver position across the
channel.
A continuous average was made of this sweep and a level L was
p
determined for each of several locations 6 and the results are listed in
Table 4 and plotted in Figure 5.
The following section is an analysis in more detail of how the spatial average can be most easily made when a pure tone source is used.
V. Optimal Measurement Technique
In the preceding section, some values of sound pressure level were
given from measurements made in a pure tone sound field.
The accuracy
of these values is dependent upon how the measurements were made.
This
section contains an analysis of the previous work done by various authors
to determine an optimum measurement technique for arriving at a reasonably precise value for the average squared sound pressure without making
the 'measurement work too tedious.
An experimental analysis is also made
of the particular sound field being described in this thesis.(a pure
tone point source located in a channe3) to determine an acceptable means
of measuring the average sound pressure level.
Theory - The time-averaged value of squared sound pressure (called
sound intensity here, for convenience) is taken as a random variable
whose statistical properties can be determined.
The variance of sound
intensity is the best single index of the size of the sound level fluctuations.
The variance is generally emphasized for this reason.
Further-
more, a normalized variance is used so that results are independent of the
magnitude of sound intensity.
Lubman [5] makes an analysis of the nor-
malized variance (V2 ) in sound intensity in a reverberant room and arrives
at some results showing that it has unit value for a pure tone:
V2 = 1
(12)
I
and when a pair of tones are introduced in the reverberant room, V 2 has
2
a range of values bounded by:
< V2 <
2
2 -
.
(13)
The limits on V 2 , where the subscript stands for the number of tones, are
2
reached under several conditions. In order to achieve minimum variance
(V2 - 1/2), the amplitude of the two tones must be equal and the frequency
2
separation of the tones must be larger than the bandwidth of a typical
room resonance.
The limit of 1 is approached by equating the frequencies
or making the amplitudes unequal by an order of magnitude or more.
Extending the variance of intensity for a multitone field of M tones,
Lubman gives the bounds as:
1/M < V2 < 1.
M-
(14)
Finally, the result is extended for a narrow-band noise field to the
approximate equation,
V2
n
[ + BT6 0 /6.9] 1-
(15)
which is valid for the product BT6 0 > 20
where T 6 0 = the 60 dB reverberation time of the room
B = the modal bandwidth of the room.
These results will be useful for comparison to the experimental values
of normalized variance.
The normalized variances described so far have been for single
measurement positions.
In a later paper by Lubman, he derived the nor-
malized variance for spatial averaging in a diffuse sound field [6]. When
a spatial average is made over a continuous straight-line traverse, a
normalized variance in a pure-tone or extremely narrow-band noise field
is found to be:
V2 - 1/(1 + 2L/)
(16)
where L = the length of traverse of the averaging path
A = the wavelength of the tone.
Researchers at Carrier Corporation [7] have arrived at a slightly modified form of Equation 16 through experimental means for an experimental
normalized variance in a reverberant room of:
V 2 = 1/1.56(1 + .55L/X) 2
(17)
Continuous-line averaging in a reverberant sound field will give
reasonable results for the mean sound intensity with a low variance in
intensity.
Averaging at discrete, well-separated points will give re-
sults very nearly the same as the continuous line averaging.
Waterhouse
and Lubman [8] published a paper in which they argue three points:
1) continuous averaging is not always better than discrete averaging at
a fixed interval; 2) a discrete average taken in a certain way is always
better than a continuous average; and 3) the sample variance decreases
with the size of the discretely sampled region in one, two, or three
dimensions.
They conclude that discrete averaging is generally to be
preferred to continuous averaging, as the former is simpler to perform
experimentally and, in addition, yields better results.
The key to the accuracy in discrete point averaging is in choosing
points in the sound field where the cross-correlation function (also
called the covariance function) for the mean-square sound pressure is
zero or as small as possible.
If all points are chosen so that zero
correlation exists between points, then the result will be the most accurate value of the mean.
Cook et. al. [9] in 1953 showed theoretically and confirmed experi-
mentally that the cross-correlation function between the sound pressures
in a diffuse sound field of narrow bandwidth was equal to:
R(r) = (sin kr)/kr
(18)
where k = the wavenumber
r = the distance between the points at which the pressure
measurements are taken.
Thus, for readings taken at points spaced X/2 apart, the two values of
sound pressure (and mean-square pressure) have zero correlation.
In considering the case of a point source in a channel, a crosscorrelation function between sound pressures was derived (see Appendix B):
0
R(r) =
Rn
R1nj
---r
cos k(r
- r
/
rn r n
n1
n2 n=-m r r
n=-=
n 1 n2
1
2
where the parameters have been defined at Equation 4.
(19)
This expression
has not yet been evaluated and the optimal point spacing has not been
determined.
Experiment - Experimental measurements in both continuous-line averaging
and discrete point averaging have been made.
For the experimental case
of this thesis (described in Section III), a slow sweep of the microphone
across the channel produced a graphic level recorder output of the following form in Figure 7.
Each chart was broken up into approximately 50 intervals of equal
spacing.
Each interval represented a microphone sweep distance of approxi-
mately 0.70 inches.
A quasi-continuous line average was made by reading
the average sound pressure level (not a dB average) of each interval and
~~
I-- um .4ooH..
ltn
-OM
.
FIGURE 7
Sound Pressure Level as a Function of Position Across the Channel
recording it in Table 4 in the appendix.
Each level was then converted
to a squared sound pressure, in microbars squared, in Table 6.
The
figures in Table 6 were manipulated to get the average levels plotted in
Figure 5 and the variances discussed below.
For the discrete point readings used, the same charts were read at
points representing the ends of the adjoining continuous intervals and
the levels recorded in Table 5 and converted to squared sound pressures
in Table 7.
Variances normalized to the mean squared pressures predicted by
theory are calculated and listed in Table 2 below for both continuousline averaging and discrete point averaging.
ance come from Equation 16.
Values of theoretical vari-
TABLE 2: AVERAGE VARIANCES
CONTINUOUS LINE:
DISCRETE POINT:
% CIANNEL WIDTH
EXP. VAR.
THEORY
100%
.0384
.0457
50%
.0495
.0873
25%
.1594
.1605
POINT SPACING
.22M
.44X
NO. OF POINTS
VARIANCE
48
.0309
24
.0488
12
.1388
24
.0365
12
.0495
6
.1663
.87X
.1444
.1963
.4764
A conclusion from the above results is that discrete point readings
taken at spacings of X/2 will provide uncorrelated samples.
The number
of readings needed will have to be determined by the accuracy required
(i.e., more readings mean better accuracy).
VI.
Continuation of Thesis Work
The conclusion stated in the preceding section is based on a limited
amount of data.
Further work must be done to get enough values of vari-
ance so that curves can be plotted showing the amount of variance as a
function of the number of sample points and the spacing of the points.
Enough data is available in Tables 4 - 7 in the appendix for this to be
done.
A computer program for evaluating the correlation function between
points in the sound field (Appendix B) is needed for an analytic result
for optimal spacing of the sample points.
The number of sample points
needed is based on the accuracy required.
Waterhouse and Lubman [8,10]
have attacked the problem of determining the probability of a measured
value of mean-square pressure lying within ±1 dB, or some other range, of
the mean value as a function of the number of sample points used for each
measurement.
This work will have to be extended for the purpose of this
thesis to determine the minimum acceptable number of sample points required.
A further step in the continuation of this work is to investigate
the effects of surface irregularities on the channel walls to find what
effect sound diffusion will have on measurement accuracy.
A moving
source should also be incorporated in future studies to simulate real
life conditions.
-~--
·1
Appendix A
Distribution of Point Sources in Infinite Line
For all cases of R between 0 and 1i,the expression for X is given by
Equation 6:
X =
RInl /4 ,r2
E
n
n=-m
(6)
which has not been evaluated for any case other than R = 0 and R = 1.
An approximation can be made by replacing the series of points by a continuous line source having a decaying acoustical strength so that the net
acoustic power in each interval of the line source is the same as the net
power of the discrete point source which would have occupied that interval of space.
An expression of the form
w(x) = a
where i = -
-
L
e
(A-l)
In R will be used.
The value of a can be found by equating the sound power in the n-th
interval
n+ )L
w(x) dx = WR n
W = J
(A-2)
(n-')L
or
n+" ))L
( -- e
L
(n-½)L
x
)
dx = IWRn
(A-3)
yields
a
ZR In
a
R
ln R
R-i
(A-4)
33
In this simplified model, the infinite line of decaying strength
is symmetric about the center of the street.
In equating the sound power
in the O-th, or street interval, an additional factor must be added in
the form of a point source of strength OW.
L
Ja E e
SW + 2
dx = W
(A-5)
L
o
implies
= 1-
(A-6)
1+ /r
and gives a factor of X for this model of:
X = 4rd+
2
4 L(d 2 + x2 )
dx
(A-7)
o
which can be evaluated using Equation 3.354 on page 312 of I.S. Gradshteyn and I.M. Ryzhik's book of mathematical functions to get a result
of
+ 2aL [sin(pd) ci(pd) - cos(pd) si(pd)]
X= 4-
where ci (cosine integral) =
si (sin integral) =
-dt
cos t
sin t
sin t dt
-
(A-8)
St
- -
t
sin
t
2
x
dt
o
Figure 8 is a plot of the magnitude vs. x for the function
[sin x ci x - cos x si x] and Figure 9 is a plot of the logl0 of the
same function which approaches a limiting asymptote of -3 dB per doubling
4C dA4T
o
stance.
i
is
is
an
in
lt
f-
iP
i4
4
f1
r
34
any value of the reflection coefficient R, with the exception of R
there exists some distance beyond which the level L
1,
-
drops off at 6 dB
per doubling of distance since p2 is also inversely proportional to distance.
The distance beyond which L
drops off at 6 dB per doubling of
distance is found from Figure 8 to be:
6
-7
-n
In
R
(A-9)
and this asymptote will be parallel to the R = 0 line and above it by
an amount given by
10 log
fl + R
, dB
-10)
(A-10)
which can be found by integrating Equation A-I over all of x to find
the total acoustic power for any particular value of R (the amount of
BW must be added to the total, too).
Figure 10 is a plot showing the
limiting curve below which all curves for any value of R run parallel
to the R = 0 line.
The points marked on this curve are points where
the various plots for R enter this region and were determined by
Equations A-9 and A-10.
_
__
__
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~I·
_II·_
_···
_~_·_
2
~· I
mL
M'
i
...
j
dl-·;',4-
SEMI -r.OGAPRIHMIC
46 5050
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J
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-
APPENDIX
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References
[1] American Standard Method for the Physical Measurement of Sound,
Amer. Standards Assoc., New York, 51.2 - 1962.
[2] Waterhouse, R.V. "Output of a Sound Source in a Reverberant Chamber
and Other Reflecting Environments." J. Acoust. Soc. Am., 30, pp. 4 - 13,
1958.
[3] Johnson, D.R., and Saunders, E.G., "The Evaluation of Noise from
Freely Flowing Road Traffic," J. Sound and Vibr., 7, pp. 287 - 309, 1968.
[4] Schroeder, M.R., "Effect of Frequency and Space Averaging on the
Transmission Responses of Multimodal Media," J. Acoust. Soc. Am., 46,
pp. 277 - 283, 1969.
[5] Lubman, D., "Fluctuation of Sound with Position in a Reverberant
Room," J. Acous. Soc. Am., 44, pp. 1491 - 1502, 1968.
[6] Lubman, D., "Spatial Averaging in a Diffuse Sound Field," J. Acoust.
Soc. Am., 46, pp. 532 - 534, (L), 1969.
[7] Ebbing, C.E. and Ingalls, D.V., "Experimental Evaluation of Microphone Traverses in a Reverberant Room," Carrier Corporation report,
November 7, 1969.
[8] Waterhouse, R.V. and Lubman, D., "Discrete versus Continuous
Averaging in a Reverberant Sound Field," J. Acoust. Soc. Am., 48,
pp.
1 -
5, 1970.
[9] Cook R.K., Waterhouse, R.V., Berendt, R.D., Edelman, S. and
Thompson, M.C., Jr., "Measurement of Correlation Coefficients in Reverberant Sound Fields," J. Acoust. Soc. Am., 27, pp. 1072 - 1077, 1955.
(10] Waterhouse, R.V., "Statistical Properties of Reverberant Sound
Fields," J. Acoust. Soc. Am., 43,
pp.
1436 - 1444, 1968.