- Email: [email protected]
The combination of noise from separate time varying sources
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES P. M. NELSON Transport and Road Research Laboratory, Department of the Environment, Crowthorne, Berkshire (Great Britain) (Received: 10 April, 1972)
SUMMARY Two methods of combining time varying noise sources are proposed and the applications ` of these procedures to practical situations discussed. A design method is suggested that enables both L lo and L90 values to be determined ~hen two or more Gaussian traffic noise distributions are combined. It is indicated that the combination technique may be applied to various noise sources, e.g. the interaction of aircraftfly-past and surface traffic noise.
INTRODUCTION
Environmental noise levels fluctuate through time irrespective of whether they are generated from transport, industry, or domestic sources. The time-scale of the noise level fluctuations may, however, differ appreciably for different emitters. Recent studies of road transport noise have emphasised the importance of the noise level fluctuation in the assessment o f t h e annoyance caused to people, t. 2.3.4 Measurements of noise level made close to the noise source can reasonably be expected to relate to that source alone, although measurements made at greater distances are liable to be affected by noise from other sources and it is then an amalgamation of various source levels that is observed. When the constituent source levels remain constant with time the mechanism of the combination process is well understood. 5 This paper examines the possibilities of providing techniques whereby time varying noise levels can be combined with particular emphasis on the combination of noise from separate road traffic sources. 1 Applied Acoustics (6) (1973)----O Applied Science Publishers Ltd, England, 1973~Printed in Great Britain
2
P.M. NELSON COMBINATION OF NOISE FROM TWO TIME VARYING SOURCES (THEORETICAL CONSIDERATIONS)
General (Method 1) Measurements of noise from time varying sources may conveniently be presented as distributions of probability or frequency of occurrence (per cent) against noise level (decibels). In order to combine two such distributions it is necessary to consider: (1) the number of possible ways that various source levels coincide, and (2) the probability that each of these separate combinations should occur. The first condition is determined by establishing the ranges of the two distributions. For distribution (I) let the range along the x, or noise level axis, be: (L1)~ -< x _< (L2)~ and let the range for distribution (2) be: (L1)2 < x < (L2)2 N.B. The subscripts, 1 and 2, outside the parentheses refer to distribution (1) and distribution (2) respectively. Each distribution may be divided into small finite elements or class intervals of width Ax, forming [(L2)1 - (Lt)l]/Ax = i and [(L2) 2 - (LI)z]/Ax = j classes for distributions (I) and (2) respectively. Taking the noise level associated with each class to be situated at the mid-class point, the location of the nth class interval of distribution (1) may be taken as: x l . = (L1)1 + (n - ½)Ax
(l)
and the probability that x lies within this class interval is given by:
f
Xln+(Ax/2)
(P,)I. = Jxz.-(ax/2) Yxl. dx
(2)
where )'~t. is the frequency of occurrence of the noise level x~,. Similarly. the location of the ruth class interval of distribution (2) is given by: X2m = (L2) 2 + (m - ½)Ax
(3)
and the probability is given by: f x2m+~'/2) Yx2mdx
(4)
The combined noise distribution is obtained by combining each interval contained in distribution (1) with each interval contained in distribution (2) according
3
THE C O M B I N A T I O N OF NOISE FROM SEPARATE TIME VAR.YINO SOURCES
to the 'log law' and determining the corresponding probability. The combined distribution therefore consists of a collection of levels given by: xn,, = lOloglo [10xt"/l° + 10x2"/1°]
(5)
with corresponding probabilities of: (P,)m = ( e , ) , , " (e,)2,,
(6)
w h e r e n = 1 , 2 , 3 . . . . . i a n d m = 1 , 2 , 3 . . . . . j. Method 2 As in method 1, neither the sources nor the combined noise need take any specific distribution of level with time. The basic assumptions are simply (1) that the sources are quite random, (2) that they are incoherent, (3) at any particular short time instant the average level of the combined distribution is equal to the 'log law' source averages over the same period and (4) over long periods of time the probability distributions of level for the sources do not vary. The probability distribution of the combination is derived by considering the probability of various source levels coinciding.
80
< ,.n .J
70
u
//
o z 60
~
e o
Method 1 Eqn. ( 7 } am
50
9g'g
96
go
80
SO
zo
20
5
1
~.1'
C I L ) x 100
Fig. 1. Statistical noise level distributions resulting from (a) the combination of two identical
normal distributions and (b) two identical 'saw tooth' distributions.
4
P.M. NELSON
C o n s i d e r two sources, A a n d B. The p r o b a b i l i t y t h a t the level f r o m A will exceed a noise level, L, is A ( L ) a n d the p r o b a b i l i t y that B will exceed L is B ( L ) . I f the c o m b i n e d noise distribution is C then the p r o b a b i l i t y that the c o m b i n e d level will exceed L is C ( L ) . A a n d B m a y c o m b i n e in several ways to generate a c o m b i n e d level o f L o r greater. C ( L ) is the sum o f the p r o b a b i l i t i e s o f all such A a n d B c o m b i n a t i o n s . It is n o t necessary, however, to consider all c o m b i n a t i o n s in the establishment o f an accurate relationship for C ( L ) p r o v i d e d the m o s t i m p o r t a n t terms which constitute the greatest probabilities are included. I n A p p e n d i x I a list o f c o m b i n a t i o n s o f A and B is given, together with the probabilities. In A p p e n d i x I I an a p p r o x i m a t e relationship for C ( L ) has been derived, utilising the leading terms in A p p e n d i x I. This relationship, eqn. (7) below, is suitable for use with a desk calculator. C ( L ) = 0.7[A(L) + B ( L ) -
A ( L ) B ( L - 3) - B ( L ) A ( L - 3)1 + 0-4A(L - 3)B(L - 3) + 0.3[A(L - 3) +
B(L -
3)]
(7)
A s m o r e terms in the expression for C ( L ) are used, the accuracy is improved. Using all the terms in A p p e n d i x I a n d simplifying, one o b t a i n s :
C(L)
=
A(L) -
3) + A ( L - 3 ) . B ( L -
A(L). B(L -
3) - A(L - 3). B ( L )
+ B ( L ) + [ A ( L - 2) - A(L)][O.3B(L - 5) + 0.7B(L - 4) - B ( L + [ B ( L - 2) - B ( L ) ] [ O . 3 A ( L - 5) + 0.7A(L - 4) - A ( L - 3)] + [ A ( L - 2) - A ( L - 1)]
3)]
x [0.18B(L - 7) + 0.42B(L - 6) + 0.1B(L - 5) - 0.7B(L - 4)] + [B(L
-
2) -
B(L
-
l)]
× [0.18A(L - 7) + 0.42A(L - 6) + 0.1A(L - 5) - 0.7A(L - 4)] + [ A ( L - 3) - A ( L - 2)][0.15B(L - 5) + 0-35B(L - 4) - 0.5B(L - 3)] + [ B ( L - 3) - B ( L - 2)][0.15B(L - 5) + 0.35A(L - 4) - 0-5A(L - 3)] + [ A ( L -- 1) -- A(L)] x [0.1 + 0.2B(L - 13) + 0-2B(L - 10) + 0.2B(L - 9) + 0.2B(L - 8) + 0.09B(L - 7) + 0.01B(L - 6) - 0-3B(L - 5) - 0.7B(L - 4)] + [ B ( L - 1) - B(L)][0.1 + 0.2A(L - 13) + 0.2A(L - 10) + 0"2A(L - 9) + 0.2A(L - 8) + 0.09A(L - 7) + 0.01A(L - 6) - 0.3A(L - 5) - 0.7A(L - 4)] (8) This expression is p r o b a b l y n o t suitable for use with desk calculators b u t can be readily i n c o r p o r a t e d into c o m p u t e r p r o g r a m s . It provides a much faster c o m b i n a t i o n calculation t h a n m e t h o d 1. R e l a t i v e accuracy o f the combination m e t h o d s
T h e a c c u r a c y o f m e t h o d 1 for c o m b i n i n g noise level d i s t r i b u t i o n s is d e p e n d e n t on the m a g n i t u d e o f the t r u n c a t i o n errors f o r m e d by classifying each d i s t r i b u t i o n a n d the c o m b i n e d d i s t r i b u t i o n into class intervals o f width Ax. Clearly, as A x is m a d e
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
5
smaller, such errors are reduced. However, by reducing Ax the number of times eqns. (5) and (6) have to be evaluated to calculate the combined distribution increases in proportion to Ax-z. A compromise between accuracy and computational efficiency is, therefore, required. In this paper all calculations using method 1 have been carried out with x equal to 0.1 dB(A). The resulting combined distributions are then also accurate to 0.1 dB(A). The accuracy of eqn. (7) can be estimated by comparing its performance with the combination procedure of method 1. Figure 1 shows the results of combining two identical normal distributions and two identical 'saw tooth' (skew) distributions using both method 1 and eqn. (7). In both cases, the distributions differ by less than 0-5 dB(A) for approximately 70 per cent of their total range and by less than 1.0 dB(A) for 99 per cent of the total range. It is expected that this order of precision is adequate provided only two distributions are to be combined. The accuracy of eqn. (8) was also estimated by applying it to the combination of two normal and two 'saw tooth' distributions. The differences between these combined distributions and those obtained using method I were less than 0.2 dB(A).
THE COMBINATION OF NOISE FROM TRAFFIC ON SEPARATE ROADS
Traffic noise constitutes the major proportion of all environmental noise sources and, in recent years, much work has been carried out and a considerable amount of traffic noise distribution data collected. It is now possible to predict, with some confidence, the noise distribution for an area adjacent to major noise sources provided it is assumed that the traffic is freely flowing. 6"7 The predictions are restricted, however, to noise from one road only. This section describes how the noise from traffic on two or more roads may be combined. The aim is to provide a design guide suitable for inclusion in current prediction methods. It was shown in the previous section that, in order to combine two or more time varying noise sources, either the equation describing the probability distribution of noise level with time, or the probability distribution in 1 dB intervals, is required for each of the composite sources. In order to fulfil this requirement in this section two assumptions are made: (1) That the fundamental traffic noise, L 1o, the level exceeded for 10 per cent of the stated time, and L g o , the level exceeded for 90 per cent of the stated time, are known or can be estimated for each of the composite traffic sources. (2) That the noise levels for each traffic source are distributed normally. The second assumption has been found to apply in practice to freely flowing dense traffic, s. 9 These assumptions allow the calculation of the normal distribution
P.M. NELSON
6
parameters from the following equations:
1 [ - - ( X - - /-/)2] y~ - a(2n)~ exp ~-i
(9)
where the standard deviation, a, and mean, It*, are expressed in terms of L ~o and L9o according to: a = ~
1
[L,o - L9o]
(lO)
and IL = ½[Llo + L9o]
(11)
99"99 per cent of a normal distribution is contained between the limits: -4.0a
+ lt-< x < 4 . 0 a
+it
This range is adequate for the combination of noise levels. Using method 1, the x-axes of the traffic noise distributions were each divided into elements of class width Ax, forming a total of 8 a j A x = i and 8~r2/Ax = j classes for distributions (1) and (2) respectively. The location of the nth class interval of distribution (1) is then given by eqn. (1) where (LI)~ = I~ - 4o-~, and the probability that x lies within this class interval is given by eqn. (2), where: l
-
Yx~. = a,(2rc)~ exp ~
2a12
,,)
Similarly, the location of the ruth class interval of distribution (2) is given by eqn. (3) where (L,)2 = Itz - 4a2 and the probability is given by eqn. (4) where: v
=
1
exp
[-x,m \
2a2 2
/
The combined distribution is determined using eqns. (5) and (6). The increase in L~o and L9o when two traffic noise level distributions are combined
In order to calculate the increase in Llo and L9o for the range of traffic noise distributions likely to be encountered in practice it is necessary to determine the probable ranges of the variables involved. In practice it is rare to find traffic noise distributions with standard deviations greater than 8 or less than 2. For example, a's of the order of 3-5 dB(A) are usual for motorways and roads where the flow of vehicles is heavy. The difference between the means of the distributions to be combined (lt t - #2), may vary from a maximum of the order of 20 dB(A) when the noisiest source, i.e. the distribution possessing the greater mean level, would be expected to dominate the quiet one, to a minimum of zero dB(A) when the two distributions have the same mean. Consequently, the working range for (p~ - S'2)
* p = Ls0, the level exceeded for 50 per cent of the stated time.
T H E C O M B I N A T I O N OF NOISE FROM SEPARATE TIME V A R Y I N G SOURCES
CHANGE IN
TABLE 1 L I 0 BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS (ltt -- IZ2) dB(A)
o t dB(A)
o2 dB(A)
0
5
10
15
20
ALto dB(A); Value added to L l o o f distribution (1) 2-0 4.0 6.0 8-0 2-0 4.0 6.0 8.0 2-0 4.0 6.0 8.0 2-0 4-0 6.0 8.0
2.0
4.0
6.0
8.0
2.4 1-4 0.8 0.5 4-0 2.5 1.5 0-8 5-9 4.1 2.8 1-8 8.2 5.9 4-4 3.3
0-9 0-4 0-2 0.1 1-5 0.8 0.4 0.2 2-6 1.6 1.0 0.6 4.1 2-7 2.0 1-4
0.3 0-1 0 0 0.4 0-2 0.1 0.1 0-9 0-5 0-2 0.1 1.6 1.1 0.7 0-4
0.1 0 0 0 0.1 0 0 0 0.3 0.1 0 0 0.5 0.3 0.! 0
0 0 0 0 0 0 0 0 0.1 0 0 0 0.2 0.I 0 0
h a s b e e n c h o s e n t o b e 0 < 1~I - / ~ 2 < 2 0 d B ( A ) a n d t h e r a n g e f o r e l a n d a2 h a s b e e n c h o s e n t o b e 2 . 0 _< a < 8.0 d B ( A ) . D i s t r i b u t i o n s w h o s e p a r a m e t e r s fall w i t h i n t h e s e r a n g e s h a v e b e e n c o n s t r u c t e d a n d c o m b i n e d i n p a i r s t o d e r i v e c o m b i n e d L 1 o a n d L 9 o v a l u e s a c c u r a t e t o 0.1 d B ( A ) . T h e d i f f e r e n c e s b e t w e e n t h e s e d e r i v e d v a l u e s a n d L Io a n d L 9 o o f d i s t r i b u t i o n (1) a r e g i v e n as A L t o a n d A L g o v a l u e s i n T a b l e s 1 a n d 2 r e s p e c t i v e l y . TABLE 2 CHANGE IN L 9 0 BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS
(P 1 --/t2) dB(A) 01 dn(.4)
(72 dB(A)
0
5
10
15
20
AL90 dB(A); Value added to Lgo o f distribution (1) 2"0 4-0 6'0 8'0 2"0 4"0 6'0 8"0 2 "0 4'0 6"0 8"0 2"0 4"0 6-0 8"0
2"0
4-0
6 "0
8"0
3-9 5-9 7'9 10.1 3.2 5.2 7-1 9.2 2"8 4"5 6-4 8"4 2.4 4.0 4'8 7'8
1"9 3"1 4"7 6-4 1-8 3.0 4.4 6"1 1"6 2-8 4-2 5-8 1"5 2"6 4-0 5-5
0"8 1.3 2.2 3-4 0-8 1-5 2"3 3'6 0"8 1"6 2'4 3.6 0"8 1"6 2-5 3"6
0.2 0"5 0"9 1"4 0'3 0"6 1'0 1"5 0.4 0"3 1.2 1'6 0"5 0.8 1"4 1"8
0-1 0.2 0"3 0-5 0'1 0.2 0.4 0"7 0-2 0-3 0-6 1-1 0"3 0-4 0-7 1-2
P. M. NELSON dBlA)
dSlA)
dRIA) O"t
%
dS(A] or"I
dS{A)
%.
o"~8(A) ( P l " PZ )=15
(I
Fig. 2.
dS(A)
A l i g n m e n t charts for AL ) o ( n u m b e r s on the curves are values of AL ] o).
dB(A)
dB(A)
dS(A)
~'~
dB(A)
7
6
5
,L
3
2
3"
2
4
6
dB(A
k) [ Idl "~JZ ) = 10
Fig. 3.
8
2
~
G
8
cr'Z (pl-pZ)=
dS(A] 15
A l i g n m e n t charts for AL9o ( n u m b e r s on the curves are values o f AL9o).
2~
9
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
To use these tables it is necessary to estimate the standard deviations and the means of the two distributions to be combined. Provided L~o and L9o can be estimated for both noise sources, then, by assuming that the distributions are Gaussian, the mean and standard deviations can be obtained using eqns. (10) and
01). Accurate estimates of ALto and ALgo may be obtained either by interpolating between the tabulated data or by using the alignment charts shown in Figs. 2 and 3. A worked example showing the procedure to adopt in using these charts is given in Appendix III.
I
I o
•
t.
I
I
I
I
I
07.2 6 8
-----
Log. taw
.....
Quadratic
AL~o= 10 tog. l l o l L l ° h l l * Q-IO ( L 1 ° l z / l O l - l k l o h equation
ALlo ,, 2.71 -0./,.7 [(,'~L~o h -(ALIo)z I • 0.02 I(AL~d~ - (ALlo,)I 12
"--3 o ..a <1
!
~.
5
6
7
8
9
(Lmh - (t.lo)2 dB (A) Fig. 4.
Comparison of the values of ALl0 listed in Table 1 with the log law and quadratic expressions.
In Fig. 4 the values of AL~ o listed in Table 1 are plotted against (L 1o) 1 - (LI 0)2. It can be seen that by using the log law expression for the summation of noise levels to sum values of L t o, i.e. by making the approximation that: ALIo = 101Ogle (10(Lie)l/l° + 10 (LI0)2/10) - - ( L I o ) I
(12)
a curve is obtained that passes close to the tabulated data although it overestimates the majority of the data points. The maximum error is of the order of 0-6 dB(A). Alternatively, the quadratic equation: AL~o = 2.71 -- 0"47[(L1o)1 - (L~o)21 + 0"02[(L1o)1 - (L~o)212
(13)
10
P. M. NELSON
passes through the data points with a maximum error of only 0.4 dB(A). If errors of this order of magnitude are acceptable in the estimation of A L t o, eqn. (13) provides a simple alternative to the interpolation procedures implicit in the use of Table 1 or the alignment charts. It is important to note that these expressions only apply to ALto. No similar relationship could be found to fit the A L 9 o values. Combination of noise distributions from more than two traffic sources The combined noise exposure from three or more separate traffic sources can be obtained by a process of successive summation using Tables 1 and 2 or the alignment charts, provided the mean and the standard deviation can be estimated for each source.
For example, if the combined noise exposure from three traffic sources is to be determined, the combined values of Lto and L 9 o a r e found first for two of the distributions for the appropriate values of a t, cr2 and/~t - It2 and then the final values of Lto and L 9 o a r e determined by combining this resultant distribution with the remaining distribution. This process can be repeated if a larger number of separate traffic noise distributions is to be combined.
Effect on the noise variability of combining two traffic noise sources It seems likely that in the future a limit incorporating a noise variability factor will be used for planning. At present two such units have been postulated, the Traffic Noise Index ( T N I ) t ' 2 : T N I = (Ll0 - L 9 0 ) + L 9 0 - 30, and the Noise TABLE
3
CHANGE IN TNI BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS
It I -- l+2 d B ( A ) a t dB
tr2 dB(A)
0
5
10
15
20
A T N I ; Value to be added to T N I o f distribution (1) 2.0 4"0 6'0 8"0 2"0 4"0 6.0 8"0 2.0 4-0 6"0 8"0 2'0 4"0 6"0 8"0
2"0
4"0
6"0
8"0
--2 --12 --21 --29 6 --6 --15 --27 14 3 --7
--2 --7 --13 --19 I --6 - - 12 - - 18 6 --2 --9
--18
--15
26 12 0 --10
12 3 -4 --ll
--1 --4 --7 --10 --1 --4 --7 --10 0 --5 --6 --10 4 0 -5 -9
0 --2 --3 --4
0 --1 -- 1 -- 1
--1
0
--~ --3 --4 0 --2 --4
-- 1 -- 1 --2 0 -- 1 --2
--6
--3
l --1 -4 - 7
0 --I -2 -3
11
T H E C O M B I N A T I O N OF NOISE F R O M S E P A R A T E TIME V A R Y I N G SOURCES
TABLE 4 CHANGE IN L N P BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS
Oat -- ltz) dB(A) at dB(A)
az dB(A)
0
5
10
15
20
ALNI,; Value to be added to L N P o f distribution (1) 2"0 4"0 6"0 8"0 2"0 4-0 6"0 8'0 2-0 4-0 6-0 8"0 2"0 4'0 6"0 8"0
2"0
4"0
6"0
8'0
1-4 --2"1 --5"7 --9"8 4'6 0"3 --3'8 --8"3 7'8 3"8 --0"7 --5-4 12'8 7'6 3-0 2'0
0"2 --1"7 --4"2 --6"8 1'3 --l'O --3'5 --6"4 3'3 0'6 --2-4 --3"8 6-0 2'8 0 --1-2
0 --0"9 --2-2 --4'0 O-1 --0"9 --2-1 --4"0 1-O --0"4 --2'0 --3"0 2'2 0"7 --1"1 --2"2
0 --0"4 --0"9 --1-5 0 --0"5 --1-0 --1"5 0"2 --0"5 --1-2 --1.0 0"5 --0-1 --1-2 --1.8
0 --0"2 --0"3 --0-4 --0"1 --0-2 --0'4 --0-7 0 --0'3 --0-6 --0-9 0-1 --0-1 --0"1 --1"3
P o l l u t i o n Level ( L N p ) 3 : : LNp = Lea + 2"56a where Leq is the level o f mean noise energy. I t seemed useful, therefore, to p r o v i d e a simple way o f determining the c h a n g e s in T N I a n d LNp when noise distributions are c o m b i n e d . T a b l e s 3 and 4 give values of: A T N I = 4 A L t o - 3AL9 o and ALNp =
A L t o + AL9o ( A L t o - ALgo 5"120"t~ 2 + ( A L t o - AL9o) 1 + 56 + 56 /
respectively, where A T N I is the change in T N I (i.e. the value to be a d d e d to the T N I o f d i s t r i b u t i o n (1) to o b t a i n the T N I o f the c o m b i n e d distribution) and ALNp the c o r r e s p o n d i n g change in LNp*. I t can be seen from T a b l e 1 that, in a l m o s t every case AL9o is greater t h a n A L t o . Consequently, the noise variability m a y be reduced when two distributions a r e c o m b i n e d . The effect o f this r e d u c t i o n is a p p a r e n t in T a b l e 3 where A T N I is o f t e n negative, implying a reduction o f T N I . R e d u c t i o n s o f Lsp, T a b l e 4, are n o t so c o m m o n o r so m a r k e d since this unit puts less e m p h a s i s on the noise variability. * The assumption is made ia the derivation of these equations that the resultant combined distribution is Gaussian. This is not rigorously true, but, in all cases studied, no significant differences from truly Gaussian curves were obtained even when the standard deviations of the separate distributions were large.
12
P.M. NELSON
An example of the esthnation of total noise exposure from separate traffic sources Figure 5 shows a plan of a hypothetical urban roadway layout. It is assumed that traffic flows freely on both roads and that road I is a major road and road 2 a minor one. The traffic flow specifications assumed for this example are included on the plan. The problem is to assess the values of L t o and possibly Lgo at sites 1-6 when traffic noise is propagated from both roads. Road 2
Flow Speed
= 200 vehicte/h. = /,0 k m / h .
Composition = 0 % heavy vehicle.
t 6!
\\', \ \
Scale
I
\
\
\
i
~
\
\\'1
"
"
?
'P
m
,.\M
Two storey building.
\
\
\
\
\
\
\
\
\
5~t i
! I
,\",1 ,\M \
\xl
\ ~' \
\
\ \ ,
\
\
\
Two storey
building.
\ \
\
\
\
\
\
\
M
\\M ,.\M
, \ \
3T
\
\
NI
\\"1 ,\",1 \
\
",1
i
1,
I t i
Road 1
Flow :2000 vehicle/h; Speed : 48 km/h;Composition : 1 0 % heavy vehicles.
Fig. 5. Plan of a hypothetical roadway layout.
It was shown earlier that the combined L I o and L9o levels may be determined at any site provided that, for each source, either the values o f L ~o can be estimated or the mean,/+, and standard deviation, a, of the normal noise level-time distribution can be determined. If the first condition holds eqn. (13) may be applied to calculate the total L~o exposures from the separate estimates of L to. If the second condition holds/a and a may be calculated for each source and Tables 1 and 2 used to obtain the total L~o and Lgo exposures.
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
13
Predicted L to Values may be obtained at the prescribed sites from design rules published by Scholes and Sargent 6 and equations published by Delany et aL to giving the attenuation down side roads. Table 5 gives the predicted values of L ~o at the six sites due to the two different traffic sources, the increment, AL,o, obtained using eqn. (13) and the combined L~o value. The second method of calculation requires an estimate of the statistical parameters, ~r and p. These may be determined from estimates of L~o and L 9 o and by using eqns. (10) and (I1). For the road system under consideration L l o values TABLE 5 ESTIMATIONOF THE TOTAL L l 0 EXPOSUREUSING EQN. (13)
Road 1 Site number 1 2 3 4 5 6
Road 2
Predicted L i e dB(A) 77"0 73"0 67"0 63'0 60"0 58"0
59"0 59"0 59"0 59"0 59-0 59.0
( L l o ) l -- (Lio)2 ALlodB(A) frorn dB(A) eqn. (13) 18'0 14"0 8"0 4"0 1 '0 1 "0
0 0 0"4 1'2 2"3 2"3
Total Llo dB(A) 77"0 73"0 67"4 64'2 62"3 61 "3
have been predicted above. At present there are no design rules for the prediction of L9o. However, it is possible to estimate L9o at sites 1-6 by making comparisons with experimental values obtained at similar positions and for similar traffic flows, speeds and compositions. The values of L9o obtained in this way are noted in Table 6, together with the k lo values taken from Table 5 and a and p calculated from eqns. (10) and (11). The values of AL~o and AL9o shown in the table were estimated using Tables 1 and 2 for the appropriate values of ~rt and a , and p t - P 2 given in Table 6. The total values of L t o and L9o at each site are included in Table 6. It can be seen from Tables 5 and 6 that both methods give approximately the same estimates of ALl o, the differences result mainly from interpolation errors in the use of Tables 1 and 2. The second method has the advantage of enabling estimates of both ALto and AL9o to be made, thereby taking into account changes in the variability of the noise. Since noise units incorporating variability factors are subjectively important both TNI and LNp have been calculated from the L~0 and L9o estimates at each site. These values are included in Table 6. It will be noticed that at points close to the main road, noise from traffic on the minor road has little effect on the Lto values. Increases in L9o are, however, not insignificant as increases of 1.0 dB(A) and 2.0 dB(A) were found for sites (1) and (2) respectively. At points further from the main road the noise from the minor road has a greater effect. At site 6, for example, increases amounting to 2.3 dB(A) of Lto and 4.0 dB(A) of L9o were found.
I 2 3 4 5 6
Site No.
LIo
77.0 73.0 67.0 63-0 60"0 58.0
62"0 59-0 54.0 52.0 50-0 48"0
Lgo
Road I
a
5"9 5-5 5" I 4.3 3-9 3-9
69"5 66-0 60.5 57'5 55-0 53"0
59.0 59'0 59.0 59.0 59.0 59.0
LIO
e,.,.ai,.,.,I,,,I,,,..,.,18(A) It
TABLE 6
50.0 50.0 50"0 50.0 50-0 50.0
Lgo
Road 2
o" 3.5 3.5 3-5 3'5 3'5 3'5
It 54.5 54.5 54.5 54-5 54.5 54.5
15"0 11"5 6.0 3.0 1.0 1 '5
0 0 0"5 1.2 2-0 2-3
",~B(J;; dB(A) ^L,o
ESTIMATION OF VARIOUS NOISE UNITS USING 'FAILLES 1 AND 2
1 '0 2"0 3'8 4"3 4.5 4.0
aL,,)
dB(A)
77.0 73.0 67'5 64.2 62.0 61.3
Total Lio dB(A)
63.0 61 '0 57-8 56"3 54"5 54-0
Total Lgo dB(A)
89 79 65 58 53 52
TNI
88 82 73 69 66 65
LNP
Z
7: rrl g,,,.
~r
7-.
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
15
The variation of Llo along the side road is shown in Fig. 6. If noise from the minor road traffic is excluded, curve (b) is obtained. When the noise from road 1 is combined with curve (b) the sigmoid curve (c) is obtained. This curve follows closely curve (b) at distances up to 20 metres from road 1 but as the distance is increased further it rapidly departs from curve (b) as, theoretically, it asymptotically approaches the L1 o level predicted for road 1. 80~ 5ite~ {b) j 70 ~ite 3 r $it,~ 5
v
m
6Q
S~ -J
50
10
20
/.0
60
8o
100
Distance from the nearside kerb (m)
Fig. 6. Values of L10 obtained when (a) traffic flows along road 2 only, (b) traffic flows along road 1 only, and (c) traffic flows along both roads. The methods used in predicting the noise exposure in this simple example can be readily applied to more complex roadway networks provided the appropriate design rules are available for predicting the noise exposure from the separate road sources involved. It is expected that as more research is carried out on such features as the shielding effects of housing and roadway configurations and the effects of interrupted traffic flow on noise propagation, the combination techniques will find increased application in predicting traffic noise exposures in urban areas.
APPLICATION OF THE COMBINATION PROCEDURES TO THE SUMMATION OF NOISE FROM SEPARATE ENVIRONMENTAL SOURCES
The method proposed earlier for combining road traffic noise from multiple sources is important since this type of traffic noise constitutes the major proportion of all
16
v.M. NELSON
environmental noise. However, there is an increasing proportion of noise from other sources, particularly from aircraft. In regions close to major airports or in areas subjected to a large number of aircraft movements, aircraft noise may well be the principal cause of annoyance. Other transport modes such as trains and future modes such as hovercraft and helicopters may also influence the noise climate at some sites. The proposition of units such as LNp implies that annoyance is a function of total noise and variation of level and is independent of the nature of the transport
"B z
99.9
99 98
90
80
60
/,0
20
5
1
0.1
PercentQge of time CL level is exceeded
Fig. 7. Statistical noise level distribution for (a) typical aircraft movement measured 250 m from runway centre line, (b) for typical motorway traffic measured 30 m from the nearside lane and (c) for combined (a) and (b). generator. It is important, therefore, for both prediction and planning purposes to be able to combine noise not only from separate transport sources of the same type but also from various transport modes. A typical example would be to determine the additive effects of aircraft fly-past noise and surface traffic noise. Such calculations may be simply carried out provided the noise level distribution of each composite source is known or can be estimated. As an example, typical aircraft and traffic noise distributions are given in Fig. 7. Since the aircraft noise distribution is obviously not Gaussian, the probabilities were taken from the curves at intervals
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
17
of 1 dB(A) and combined using eqn. (7) and, to obtain a more accurate result, eqn. (8). The combined distributions obtained are also shown in Fig. 7. Unfortunately, distributions of noise level from transport sources, other than from road traffic, are either seldom measured or are not generally available and so it is not possible at present to study comprehensively the effects of various noise source interactions. It is expected that future measurements of transport noise will incorporate distribution data and therefore enable combination calculations to be carried out.
ACKNOWLEDGEMENTS
This paper was prepared in the Environment Section (Head, Mr L. H. Watkins) of the Transportation Division. It is contributed by permission of the Director of Road Research and is reproduced by permission of the Controller of HM Stationery Office. The author also wishes to acknowledge the guidance and encouragement given by D. G. Harland.
REFERENCES 1. I. O. GRIFFITHSand F. J. LANGDON,Subjective response to road traffic noise, Jotu'nal of Sound and Vibration, 8(16) (1968). 2. F. J. LANGDON and W. E. SCHOLES,The traffic noise index--a method of controlling noise nuisance, Building Research Station Current Paper CP 38/68. 3. D. W. ROBINSON, The concept of noise pollution level, Aero Report 38, 1969, National Physical Laboratory. 4. D. W. ROmNSON, An outline guide to criteria for the limitation of urban noise, Aero Report 39, 1968, National Physical Laboratory. 5. A. P. G. PETERSONand E. E. GROSS,Handbook of noise measurement, General Radio Company, West Concord, 1967. 6. W. E. SCHOLES and J. W. SARGENT, Designing against noise from road traffic, Building Research Station Current Paper CP 20/71. Applied Acoustics, 4 (1971) pp. 203-234. 7. D. R. JOHNSONand C. G. SAUNDERS,The evaluation of noise from freely flowing road traffic, Aero Report 29, 1967, National Physical Laboratory. 8. D. R. JOHNSON, A note on the relationship between noise exposure and noise probability distribution, Aero Report 40, 1969, National Physical Laboratory. 9. W. E. SCHOLESand G. H. VULKAN,A note on the objective measurement of road traffic noise, Applied Acoustics, 2(3) (1969) pp. 185-197. 10. M. E. DELANY, W. C. COPELANOand R. C. PAYNE, Propagation of traffic noise in typical urban situations, Acoustics Report Ac 54, 1971. National Physical Laboratory.
APPENDIX I: PROBABILITY CONSTITUENTS OF C(L) T h e f o l l o w i n g listing gives s o m e o f t h e possible w a y s t h a t A a n d B c a n c o m b i n e t o g e n e r a t e a level o f L o r greater. T h e c o r r e s p o n d i n g p r o b a b i l i t i e s are given in e a c h
18
v . M . NELSON
case. T h e p r o b a b i l i t y t h a t C exceeds L, C(L), is t h e s u m o f all t h e c o n s t i t u e n t probabilities. Listing of the constituents of C(L)
R e a s o n t h a t C s h o u l d exceed L
Probability
1. I f A o r B exceeds L
A(L)
2. I f A a n d B in t h e i n t e r v a l L to (L - 3)
[A(L -
3) - A ( L ) ] [ B ( L
3. I f o n e lies in the i n t e r v a l L t o (L - 2) a n d t h e o t h e r in t h e i n t e r v a l (L - 3) to (L - 4-3)
[A(L -
2) -
-- A ( L ) B ( L )
+ B(L)
3) -
-
A(L)][0.38(L
+ 0 . 7 B ( L - 4) + [B(L
-
2) -
x [0.3A(L -
-
B(L
B(L)]
5)
3)]
-
S(L)]
5) + 0 . 7 A ( L -
4)
A(L
-
3)1
4. I f o n e lies in the i n t e r v a l L t o (L - 1) a n d t h e o t h e r in the i n t e r v a l (L - 43) to (L - 6-9)
[A(L -
5. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s o n e lies in the i n t e r v a l (L - 2) t o (L - 3) a n d the o t h e r lies in t h e i n t e r v a l (L - 3) to (L - 4-3)
0.5{[A(L - 3) - A ( L - 2)] × [0.3B(L - 5) + 0.TB(L - 4) - B ( L - 3)] + [ B ( L - 3) - B ( L - 2)][0.3A(L - 5) + 0 . 7 A ( L - 4) - A ( L - 3)]',.
6. I f o n e lies in t h e i n t e r v a l L - 1 to L - 1.6 a n d the o t h e r in the i n t e r v a l
0.42{[A(L - 2) - A ( L - 1)] × [ B ( L - 5) - B (L - 4)]
1) - A ( L ) I [ O . 9 B ( L - 7) + 0 . 1 B ( L - 6) - 0 . 3 B f L - 5) - 0 . 7 B ( L - 4)] + [ B ( L - 1) - B ( L ) ] [ O . 9 A ( L - 7) + 0 . 1 A ( L - 6) - 0"3A(L - 5) - 0-7A(L - 4)]
+ [ B ( L -- 2)
L - 4-3 t o L -
5
-
B(L
-
I)][A(L -
5) -
7. I f o n e lies in t h e i n t e r v a l L - 1 t o L - 1.2 a n d t h e o t h e r in the i n t e r v a l L5toL6
A(L
-
0.2{[A(L - 2) - A ( L - 1)][B(L - B ( L - 5)] + [ B ( L - 2) - B(LI ) ] [ A ( L - - 6) --
A(L
8. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s 0-18{[A(L - 2) - A ( L - 1)] o n e is in the i n t e r v a l L - 1 t o L - 1.2 x [ B ( L - 7) - B ( L - 6)] a n d the o t h e r is in the i n t e r v a l + [ B ( L - 2) - B ( L - 1)] L6toL6.9 x [A(L7)- A(L-
--
4)]} 6)
5)]',
6)]}
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
19
9. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s 0.4{[A(L - 2) - A ( L - 1)] o n e is in t h e i n t e r v a l L - 1.2 to L - 1.6 × [ B ( L - 6) - B ( L - 5)] a n d the o t h e r is in t h e i n t e r v a l L - 6 + [ B ( L - 2) - B ( L - 1)] toL5 x [A(L-6)-A(L-
5)]}
10. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s 0.28{[A(L - 2) - A ( L - 1)] o n e is in the i n t e r v a l L - 1.6 to L - 2 x [ B ( L - 5) - B ( L - 4)] a n d t h e o t h e r is in the i n t e r v a l L - 5 + [ B ( L - 2) - B ( L - 1)1 t o L - 4.3 × [ A ( L - 5) - A ( L
-
4)1}
11. I f o n e lies in the i n t e r v a l L t o L - 0.8 a n d t h e o t h e r in t h e i n t e r v a l L6.9toL8
0 . 8 ( [ A ( L - l ) - A ( L ) ] [ a ( L - 8) - 0 . 9 B f L - 7) - 0 . 1 B ( L - 6)] + [ B ( L - 1) - B ( L ) I [ A ( L 8) - 0 . 9 A ( L - 7) - 0 - 1 A ( L - 6)1}
12. I f o n e lies in the i n t e r v a l L to L - 0.6 a n d the o t h e r in t h e interval L - 8 to L - 9
0.6{[A(L
13. I f o n e lies in t h e i n t e r v a l L to L - 0.4 a n d t h e o t h e r in t h e i n t e r v a l L-9toL10
0.4{[A(L
14. I f o n e lies in the i n t e r v a l L t o L - 0.2 a n d t h e o t h e r in t h e i n t e r v a l L.10toL13
0.2{[A(L-
15. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s o n e lies in t h e i n t e r v a l L - 0-8 t o L - 1 a n d t h e o t h e r in t h e i n t e r v a l L - 8 t o L - 6-9
0.1 {[A(L - 1) - A ( L ) ] [ B ( L - 8) - 0 . 9 B ( L - 7) - 0 . 1 B ( L - 6)] + [ B ( L - 1) - ( B L ) ] [ A ( L - 8) - 0 . 9 A ( L - 7) - 0 - 1 A ( L - 6)]}
16. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s o n e lies in t h e i n t e r v a l L - 0.6 t o L.0.8 a n d t h e o t h e r in the i n t e r v a l L-9toL-8
0.1{[A(L -
17. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s o n e lies in the i n t e r v a l L - 0-4 t o L - 0.6 a n d t h e o t h e r in t h e i n t e r v a l L - 10toL - 9
0.1{[A(L -
-
-
-
[a(L
9)
-
1)
-
-
8)]}
1) - A ( L ) ] [ B ( L - 10) 9)] + [ B ( L - 1) ( B L ) ] [ A ( L - 10) - A ( L -
9)]}
-
9) -
A(L
-
-
B(L -
-
1)-
B(L
-
13) 1)
A(L)][B(L-
10)] +
B(L)][A(L
-
-
A(L)][B(L
8)1 +
B(L)I[A(L
-
-
[B(L
13) -
1) - A ( L ) ] [ B ( L - 8)] + [ B ( L B(L)[A(L - 9) -
-
A(L
B(L
1) - A ( L ) ] [ B ( L 9)] + [ B ( L [ A ( L - 10) -
18. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s 0-1 {[A(L - I) - A ( L ) ] [ B ( L o n e lies in t h e i n t e r v a l L - 0-2 to - B ( L - 10)] + [ B ( L L -- 0.4 a n d t h e o t h e r in t h e i n t e r v a l -B(L)][A(L - 13) L-- 13toL-10
9)]}
1) A(L
-
+
-
9)
-
B(L -
-
1) -
~(L
-
10) 1) -
A(L
-
-
13) I)
A(L
-
-
8)]}
B(L)
9)]}
10)]}
20
P . M . NELSON
19. For approximately half the occasions 0.1 {[A(L - 1) - A(L)][I - B ( L - 13) one lies in the interval L to L - 0.2 + [ B ( L - 1) - B(L)] and the other below L - 13 x [1 - A ( L - 13)]}-
APPENDIX lI:
A SIMPLE EQUATION FOR THE COMBINATION OF NOISE FROM TWO TIME VARYING SOURCES A AND B
Each distribution is considered to comprise three regions, shown diagrammatically below. (L t)A L
(L 2)n L
L-2
L-3
(L2)A
(L2)B
A and B may combine to produce a level o f L or greater when, (1) either A or B exceeds L (2) if both A and B are in the interval L to (L - 3) (3) for X o f the occasions when A is in the interval L to (L - 3) and B is in the interval (L - 3) to (L2)B and for Y o f the occasions when B is in the interval L to (L - 3) and A is in the interval (L - 3) to (L2)A. The probabilities for (1) and (2) are given in Appendix I. The probability that C will exceed L for (3) is X[I - A ( L - 3)][B(L - 3) - B(L)] + Y[1 - B ( L - 3)][A(L - 3) - A(L)] The total probability that C will exceed L , C ( L ) , is given by the total probabilities o f (I), (2) and (3). Therefore: + B ( L ) + A ( L - 3)B(L - 3) - A ( L ) B ( L - 3) - A ( L - 3)B(L) + X [ I - A ( L - 3)][B(L - 3) - B(L)] + Y[1 - B ( L - 3)][A(L - 3) - A(L)]
C(L) = A(L)
04) In order to complete the above equation, suitable values are required for the correction factors X and Y. A simple analysis revealed that X and Y may be most suitably assigned the value: X=
Y = 0-3
(15)
Substituting (15) into (14) and simplifying, one obtains: C(L)
= 0-7[A(L)
+ B(L) -
A(L)B(L
-
3) -- B ( L ) A ( L
-
3)]
+ 0 . 4 A ( L - 3)B(L - 3) + 0.3[A(L - 3) + B ( L - 3)]
THE COMBINATIONOF NOISE FROM SEPARATE TIME VARYING SOURCES
21
APPENDIX III: ESTIMATIONOF A L t o AND AL9o USING THE ALIGNMENT CHARTS
The following example of the estimation of ALto and ALgo will illustrate the use of the alignment charts (Figs. 2 and 3). Figures 2 and 3 are each divided into four panels, each panel representing a particular value of le t-It 2 within the range 0-15 dB(A). The panels are bounded by the vertical axis graduated on the left side in a 2 dB(A). Between the boundaries and at the bottom of each panel a horizontal axis has been constructed which is graduated in a2 dB(A). As an example of the use ofthe charts, consider a typical case where a t = 4 dB(A), 0"2 = 6 dB(A) and (/tt - It2) = 0. According to Tables 1 and 2, ALto is 4.1 dB(A) and AL9o is 4"5 dB(A). To obtain the corresponding estimates using the alignment charts a line is drawn joining a t = 4 dB(A) and a z = 6 dB(A) on the vertical scales. A perpendicular is then constructed from the horizontal scale at a2 = 6 dBA to intersect this line. This procedure is illustrated in Figs. 2 and 3 by the broken lines. The position of the point of intersection, labelled A on the ALto chart and B on the ALgo chart, then identifies the values of ALto and ALgo. It can be seen that point A lies very close to the AL t o = 4 dB(A) curve and B lies approximately midway between the AL9o = 5 dB(A) and AL9o = 4 dB(A) curves. Using the charts therefore AL~o and ALgo are estimated to be 4.0 dB(A) and 4.5 dB(A) respectively. These estimates are in good agreement with the tabulated results. For most practical cases ltt - #2 will not be equal to either 0, 5, 10 or 15 dB(A) and so it is necessary to interpolate between the panels. This can be most conveniently achieved by constructing a graph of either AL~ o or AL9o against (/z 1 - I~2) where/~t - It2 takes the values 0, 5, 10, 15 dB(A), using the alignment charts in the manner described above, so that the values of ALto and ALgo can then be identified for any value of ( I t t - IL2).
SUMMARY Two methods of combining time varying noise sources are proposed and the applications ` of these procedures to practical situations discussed. A design method is suggested that enables both L lo and L90 values to be determined ~hen two or more Gaussian traffic noise distributions are combined. It is indicated that the combination technique may be applied to various noise sources, e.g. the interaction of aircraftfly-past and surface traffic noise.
INTRODUCTION
Environmental noise levels fluctuate through time irrespective of whether they are generated from transport, industry, or domestic sources. The time-scale of the noise level fluctuations may, however, differ appreciably for different emitters. Recent studies of road transport noise have emphasised the importance of the noise level fluctuation in the assessment o f t h e annoyance caused to people, t. 2.3.4 Measurements of noise level made close to the noise source can reasonably be expected to relate to that source alone, although measurements made at greater distances are liable to be affected by noise from other sources and it is then an amalgamation of various source levels that is observed. When the constituent source levels remain constant with time the mechanism of the combination process is well understood. 5 This paper examines the possibilities of providing techniques whereby time varying noise levels can be combined with particular emphasis on the combination of noise from separate road traffic sources. 1 Applied Acoustics (6) (1973)----O Applied Science Publishers Ltd, England, 1973~Printed in Great Britain
2
P.M. NELSON COMBINATION OF NOISE FROM TWO TIME VARYING SOURCES (THEORETICAL CONSIDERATIONS)
General (Method 1) Measurements of noise from time varying sources may conveniently be presented as distributions of probability or frequency of occurrence (per cent) against noise level (decibels). In order to combine two such distributions it is necessary to consider: (1) the number of possible ways that various source levels coincide, and (2) the probability that each of these separate combinations should occur. The first condition is determined by establishing the ranges of the two distributions. For distribution (I) let the range along the x, or noise level axis, be: (L1)~ -< x _< (L2)~ and let the range for distribution (2) be: (L1)2 < x < (L2)2 N.B. The subscripts, 1 and 2, outside the parentheses refer to distribution (1) and distribution (2) respectively. Each distribution may be divided into small finite elements or class intervals of width Ax, forming [(L2)1 - (Lt)l]/Ax = i and [(L2) 2 - (LI)z]/Ax = j classes for distributions (I) and (2) respectively. Taking the noise level associated with each class to be situated at the mid-class point, the location of the nth class interval of distribution (1) may be taken as: x l . = (L1)1 + (n - ½)Ax
(l)
and the probability that x lies within this class interval is given by:
f
Xln+(Ax/2)
(P,)I. = Jxz.-(ax/2) Yxl. dx
(2)
where )'~t. is the frequency of occurrence of the noise level x~,. Similarly. the location of the ruth class interval of distribution (2) is given by: X2m = (L2) 2 + (m - ½)Ax
(3)
and the probability is given by: f x2m+~'/2) Yx2mdx
(4)
The combined noise distribution is obtained by combining each interval contained in distribution (1) with each interval contained in distribution (2) according
3
THE C O M B I N A T I O N OF NOISE FROM SEPARATE TIME VAR.YINO SOURCES
to the 'log law' and determining the corresponding probability. The combined distribution therefore consists of a collection of levels given by: xn,, = lOloglo [10xt"/l° + 10x2"/1°]
(5)
with corresponding probabilities of: (P,)m = ( e , ) , , " (e,)2,,
(6)
w h e r e n = 1 , 2 , 3 . . . . . i a n d m = 1 , 2 , 3 . . . . . j. Method 2 As in method 1, neither the sources nor the combined noise need take any specific distribution of level with time. The basic assumptions are simply (1) that the sources are quite random, (2) that they are incoherent, (3) at any particular short time instant the average level of the combined distribution is equal to the 'log law' source averages over the same period and (4) over long periods of time the probability distributions of level for the sources do not vary. The probability distribution of the combination is derived by considering the probability of various source levels coinciding.
80
< ,.n .J
70
u
//
o z 60
~
e o
Method 1 Eqn. ( 7 } am
50
9g'g
96
go
80
SO
zo
20
5
1
~.1'
C I L ) x 100
Fig. 1. Statistical noise level distributions resulting from (a) the combination of two identical
normal distributions and (b) two identical 'saw tooth' distributions.
4
P.M. NELSON
C o n s i d e r two sources, A a n d B. The p r o b a b i l i t y t h a t the level f r o m A will exceed a noise level, L, is A ( L ) a n d the p r o b a b i l i t y that B will exceed L is B ( L ) . I f the c o m b i n e d noise distribution is C then the p r o b a b i l i t y that the c o m b i n e d level will exceed L is C ( L ) . A a n d B m a y c o m b i n e in several ways to generate a c o m b i n e d level o f L o r greater. C ( L ) is the sum o f the p r o b a b i l i t i e s o f all such A a n d B c o m b i n a t i o n s . It is n o t necessary, however, to consider all c o m b i n a t i o n s in the establishment o f an accurate relationship for C ( L ) p r o v i d e d the m o s t i m p o r t a n t terms which constitute the greatest probabilities are included. I n A p p e n d i x I a list o f c o m b i n a t i o n s o f A and B is given, together with the probabilities. In A p p e n d i x I I an a p p r o x i m a t e relationship for C ( L ) has been derived, utilising the leading terms in A p p e n d i x I. This relationship, eqn. (7) below, is suitable for use with a desk calculator. C ( L ) = 0.7[A(L) + B ( L ) -
A ( L ) B ( L - 3) - B ( L ) A ( L - 3)1 + 0-4A(L - 3)B(L - 3) + 0.3[A(L - 3) +
B(L -
3)]
(7)
A s m o r e terms in the expression for C ( L ) are used, the accuracy is improved. Using all the terms in A p p e n d i x I a n d simplifying, one o b t a i n s :
C(L)
=
A(L) -
3) + A ( L - 3 ) . B ( L -
A(L). B(L -
3) - A(L - 3). B ( L )
+ B ( L ) + [ A ( L - 2) - A(L)][O.3B(L - 5) + 0.7B(L - 4) - B ( L + [ B ( L - 2) - B ( L ) ] [ O . 3 A ( L - 5) + 0.7A(L - 4) - A ( L - 3)] + [ A ( L - 2) - A ( L - 1)]
3)]
x [0.18B(L - 7) + 0.42B(L - 6) + 0.1B(L - 5) - 0.7B(L - 4)] + [B(L
-
2) -
B(L
-
l)]
× [0.18A(L - 7) + 0.42A(L - 6) + 0.1A(L - 5) - 0.7A(L - 4)] + [ A ( L - 3) - A ( L - 2)][0.15B(L - 5) + 0-35B(L - 4) - 0.5B(L - 3)] + [ B ( L - 3) - B ( L - 2)][0.15B(L - 5) + 0.35A(L - 4) - 0-5A(L - 3)] + [ A ( L -- 1) -- A(L)] x [0.1 + 0.2B(L - 13) + 0-2B(L - 10) + 0.2B(L - 9) + 0.2B(L - 8) + 0.09B(L - 7) + 0.01B(L - 6) - 0-3B(L - 5) - 0.7B(L - 4)] + [ B ( L - 1) - B(L)][0.1 + 0.2A(L - 13) + 0.2A(L - 10) + 0"2A(L - 9) + 0.2A(L - 8) + 0.09A(L - 7) + 0.01A(L - 6) - 0.3A(L - 5) - 0.7A(L - 4)] (8) This expression is p r o b a b l y n o t suitable for use with desk calculators b u t can be readily i n c o r p o r a t e d into c o m p u t e r p r o g r a m s . It provides a much faster c o m b i n a t i o n calculation t h a n m e t h o d 1. R e l a t i v e accuracy o f the combination m e t h o d s
T h e a c c u r a c y o f m e t h o d 1 for c o m b i n i n g noise level d i s t r i b u t i o n s is d e p e n d e n t on the m a g n i t u d e o f the t r u n c a t i o n errors f o r m e d by classifying each d i s t r i b u t i o n a n d the c o m b i n e d d i s t r i b u t i o n into class intervals o f width Ax. Clearly, as A x is m a d e
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
5
smaller, such errors are reduced. However, by reducing Ax the number of times eqns. (5) and (6) have to be evaluated to calculate the combined distribution increases in proportion to Ax-z. A compromise between accuracy and computational efficiency is, therefore, required. In this paper all calculations using method 1 have been carried out with x equal to 0.1 dB(A). The resulting combined distributions are then also accurate to 0.1 dB(A). The accuracy of eqn. (7) can be estimated by comparing its performance with the combination procedure of method 1. Figure 1 shows the results of combining two identical normal distributions and two identical 'saw tooth' (skew) distributions using both method 1 and eqn. (7). In both cases, the distributions differ by less than 0-5 dB(A) for approximately 70 per cent of their total range and by less than 1.0 dB(A) for 99 per cent of the total range. It is expected that this order of precision is adequate provided only two distributions are to be combined. The accuracy of eqn. (8) was also estimated by applying it to the combination of two normal and two 'saw tooth' distributions. The differences between these combined distributions and those obtained using method I were less than 0.2 dB(A).
THE COMBINATION OF NOISE FROM TRAFFIC ON SEPARATE ROADS
Traffic noise constitutes the major proportion of all environmental noise sources and, in recent years, much work has been carried out and a considerable amount of traffic noise distribution data collected. It is now possible to predict, with some confidence, the noise distribution for an area adjacent to major noise sources provided it is assumed that the traffic is freely flowing. 6"7 The predictions are restricted, however, to noise from one road only. This section describes how the noise from traffic on two or more roads may be combined. The aim is to provide a design guide suitable for inclusion in current prediction methods. It was shown in the previous section that, in order to combine two or more time varying noise sources, either the equation describing the probability distribution of noise level with time, or the probability distribution in 1 dB intervals, is required for each of the composite sources. In order to fulfil this requirement in this section two assumptions are made: (1) That the fundamental traffic noise, L 1o, the level exceeded for 10 per cent of the stated time, and L g o , the level exceeded for 90 per cent of the stated time, are known or can be estimated for each of the composite traffic sources. (2) That the noise levels for each traffic source are distributed normally. The second assumption has been found to apply in practice to freely flowing dense traffic, s. 9 These assumptions allow the calculation of the normal distribution
P.M. NELSON
6
parameters from the following equations:
1 [ - - ( X - - /-/)2] y~ - a(2n)~ exp ~-i
(9)
where the standard deviation, a, and mean, It*, are expressed in terms of L ~o and L9o according to: a = ~
1
[L,o - L9o]
(lO)
and IL = ½[Llo + L9o]
(11)
99"99 per cent of a normal distribution is contained between the limits: -4.0a
+ lt-< x < 4 . 0 a
+it
This range is adequate for the combination of noise levels. Using method 1, the x-axes of the traffic noise distributions were each divided into elements of class width Ax, forming a total of 8 a j A x = i and 8~r2/Ax = j classes for distributions (1) and (2) respectively. The location of the nth class interval of distribution (1) is then given by eqn. (1) where (LI)~ = I~ - 4o-~, and the probability that x lies within this class interval is given by eqn. (2), where: l
-
Yx~. = a,(2rc)~ exp ~
2a12
,,)
Similarly, the location of the ruth class interval of distribution (2) is given by eqn. (3) where (L,)2 = Itz - 4a2 and the probability is given by eqn. (4) where: v
=
1
exp
[-x,m \
2a2 2
/
The combined distribution is determined using eqns. (5) and (6). The increase in L~o and L9o when two traffic noise level distributions are combined
In order to calculate the increase in Llo and L9o for the range of traffic noise distributions likely to be encountered in practice it is necessary to determine the probable ranges of the variables involved. In practice it is rare to find traffic noise distributions with standard deviations greater than 8 or less than 2. For example, a's of the order of 3-5 dB(A) are usual for motorways and roads where the flow of vehicles is heavy. The difference between the means of the distributions to be combined (lt t - #2), may vary from a maximum of the order of 20 dB(A) when the noisiest source, i.e. the distribution possessing the greater mean level, would be expected to dominate the quiet one, to a minimum of zero dB(A) when the two distributions have the same mean. Consequently, the working range for (p~ - S'2)
* p = Ls0, the level exceeded for 50 per cent of the stated time.
T H E C O M B I N A T I O N OF NOISE FROM SEPARATE TIME V A R Y I N G SOURCES
CHANGE IN
TABLE 1 L I 0 BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS (ltt -- IZ2) dB(A)
o t dB(A)
o2 dB(A)
0
5
10
15
20
ALto dB(A); Value added to L l o o f distribution (1) 2-0 4.0 6.0 8-0 2-0 4.0 6.0 8.0 2-0 4.0 6.0 8.0 2-0 4-0 6.0 8.0
2.0
4.0
6.0
8.0
2.4 1-4 0.8 0.5 4-0 2.5 1.5 0-8 5-9 4.1 2.8 1-8 8.2 5.9 4-4 3.3
0-9 0-4 0-2 0.1 1-5 0.8 0.4 0.2 2-6 1.6 1.0 0.6 4.1 2-7 2.0 1-4
0.3 0-1 0 0 0.4 0-2 0.1 0.1 0-9 0-5 0-2 0.1 1.6 1.1 0.7 0-4
0.1 0 0 0 0.1 0 0 0 0.3 0.1 0 0 0.5 0.3 0.! 0
0 0 0 0 0 0 0 0 0.1 0 0 0 0.2 0.I 0 0
h a s b e e n c h o s e n t o b e 0 < 1~I - / ~ 2 < 2 0 d B ( A ) a n d t h e r a n g e f o r e l a n d a2 h a s b e e n c h o s e n t o b e 2 . 0 _< a < 8.0 d B ( A ) . D i s t r i b u t i o n s w h o s e p a r a m e t e r s fall w i t h i n t h e s e r a n g e s h a v e b e e n c o n s t r u c t e d a n d c o m b i n e d i n p a i r s t o d e r i v e c o m b i n e d L 1 o a n d L 9 o v a l u e s a c c u r a t e t o 0.1 d B ( A ) . T h e d i f f e r e n c e s b e t w e e n t h e s e d e r i v e d v a l u e s a n d L Io a n d L 9 o o f d i s t r i b u t i o n (1) a r e g i v e n as A L t o a n d A L g o v a l u e s i n T a b l e s 1 a n d 2 r e s p e c t i v e l y . TABLE 2 CHANGE IN L 9 0 BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS
(P 1 --/t2) dB(A) 01 dn(.4)
(72 dB(A)
0
5
10
15
20
AL90 dB(A); Value added to Lgo o f distribution (1) 2"0 4-0 6'0 8'0 2"0 4"0 6'0 8"0 2 "0 4'0 6"0 8"0 2"0 4"0 6-0 8"0
2"0
4-0
6 "0
8"0
3-9 5-9 7'9 10.1 3.2 5.2 7-1 9.2 2"8 4"5 6-4 8"4 2.4 4.0 4'8 7'8
1"9 3"1 4"7 6-4 1-8 3.0 4.4 6"1 1"6 2-8 4-2 5-8 1"5 2"6 4-0 5-5
0"8 1.3 2.2 3-4 0-8 1-5 2"3 3'6 0"8 1"6 2'4 3.6 0"8 1"6 2-5 3"6
0.2 0"5 0"9 1"4 0'3 0"6 1'0 1"5 0.4 0"3 1.2 1'6 0"5 0.8 1"4 1"8
0-1 0.2 0"3 0-5 0'1 0.2 0.4 0"7 0-2 0-3 0-6 1-1 0"3 0-4 0-7 1-2
P. M. NELSON dBlA)
dSlA)
dRIA) O"t
%
dS(A] or"I
dS{A)
%.
o"~8(A) ( P l " PZ )=15
(I
Fig. 2.
dS(A)
A l i g n m e n t charts for AL ) o ( n u m b e r s on the curves are values of AL ] o).
dB(A)
dB(A)
dS(A)
~'~
dB(A)
7
6
5
,L
3
2
3"
2
4
6
dB(A
k) [ Idl "~JZ ) = 10
Fig. 3.
8
2
~
G
8
cr'Z (pl-pZ)=
dS(A] 15
A l i g n m e n t charts for AL9o ( n u m b e r s on the curves are values o f AL9o).
2~
9
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
To use these tables it is necessary to estimate the standard deviations and the means of the two distributions to be combined. Provided L~o and L9o can be estimated for both noise sources, then, by assuming that the distributions are Gaussian, the mean and standard deviations can be obtained using eqns. (10) and
01). Accurate estimates of ALto and ALgo may be obtained either by interpolating between the tabulated data or by using the alignment charts shown in Figs. 2 and 3. A worked example showing the procedure to adopt in using these charts is given in Appendix III.
I
I o
•
t.
I
I
I
I
I
07.2 6 8
-----
Log. taw
.....
Quadratic
AL~o= 10 tog. l l o l L l ° h l l * Q-IO ( L 1 ° l z / l O l - l k l o h equation
ALlo ,, 2.71 -0./,.7 [(,'~L~o h -(ALIo)z I • 0.02 I(AL~d~ - (ALlo,)I 12
"--3 o ..a <1
!
~.
5
6
7
8
9
(Lmh - (t.lo)2 dB (A) Fig. 4.
Comparison of the values of ALl0 listed in Table 1 with the log law and quadratic expressions.
In Fig. 4 the values of AL~ o listed in Table 1 are plotted against (L 1o) 1 - (LI 0)2. It can be seen that by using the log law expression for the summation of noise levels to sum values of L t o, i.e. by making the approximation that: ALIo = 101Ogle (10(Lie)l/l° + 10 (LI0)2/10) - - ( L I o ) I
(12)
a curve is obtained that passes close to the tabulated data although it overestimates the majority of the data points. The maximum error is of the order of 0-6 dB(A). Alternatively, the quadratic equation: AL~o = 2.71 -- 0"47[(L1o)1 - (L~o)21 + 0"02[(L1o)1 - (L~o)212
(13)
10
P. M. NELSON
passes through the data points with a maximum error of only 0.4 dB(A). If errors of this order of magnitude are acceptable in the estimation of A L t o, eqn. (13) provides a simple alternative to the interpolation procedures implicit in the use of Table 1 or the alignment charts. It is important to note that these expressions only apply to ALto. No similar relationship could be found to fit the A L 9 o values. Combination of noise distributions from more than two traffic sources The combined noise exposure from three or more separate traffic sources can be obtained by a process of successive summation using Tables 1 and 2 or the alignment charts, provided the mean and the standard deviation can be estimated for each source.
For example, if the combined noise exposure from three traffic sources is to be determined, the combined values of Lto and L 9 o a r e found first for two of the distributions for the appropriate values of a t, cr2 and/~t - It2 and then the final values of Lto and L 9 o a r e determined by combining this resultant distribution with the remaining distribution. This process can be repeated if a larger number of separate traffic noise distributions is to be combined.
Effect on the noise variability of combining two traffic noise sources It seems likely that in the future a limit incorporating a noise variability factor will be used for planning. At present two such units have been postulated, the Traffic Noise Index ( T N I ) t ' 2 : T N I = (Ll0 - L 9 0 ) + L 9 0 - 30, and the Noise TABLE
3
CHANGE IN TNI BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS
It I -- l+2 d B ( A ) a t dB
tr2 dB(A)
0
5
10
15
20
A T N I ; Value to be added to T N I o f distribution (1) 2.0 4"0 6'0 8"0 2"0 4"0 6.0 8"0 2.0 4-0 6"0 8"0 2'0 4"0 6"0 8"0
2"0
4"0
6"0
8"0
--2 --12 --21 --29 6 --6 --15 --27 14 3 --7
--2 --7 --13 --19 I --6 - - 12 - - 18 6 --2 --9
--18
--15
26 12 0 --10
12 3 -4 --ll
--1 --4 --7 --10 --1 --4 --7 --10 0 --5 --6 --10 4 0 -5 -9
0 --2 --3 --4
0 --1 -- 1 -- 1
--1
0
--~ --3 --4 0 --2 --4
-- 1 -- 1 --2 0 -- 1 --2
--6
--3
l --1 -4 - 7
0 --I -2 -3
11
T H E C O M B I N A T I O N OF NOISE F R O M S E P A R A T E TIME V A R Y I N G SOURCES
TABLE 4 CHANGE IN L N P BY COMBINING TWO GAUSSIAN NOISE LEVEL DISTRIBUTIONS
Oat -- ltz) dB(A) at dB(A)
az dB(A)
0
5
10
15
20
ALNI,; Value to be added to L N P o f distribution (1) 2"0 4"0 6"0 8"0 2"0 4-0 6"0 8'0 2-0 4-0 6-0 8"0 2"0 4'0 6"0 8"0
2"0
4"0
6"0
8'0
1-4 --2"1 --5"7 --9"8 4'6 0"3 --3'8 --8"3 7'8 3"8 --0"7 --5-4 12'8 7'6 3-0 2'0
0"2 --1"7 --4"2 --6"8 1'3 --l'O --3'5 --6"4 3'3 0'6 --2-4 --3"8 6-0 2'8 0 --1-2
0 --0"9 --2-2 --4'0 O-1 --0"9 --2-1 --4"0 1-O --0"4 --2'0 --3"0 2'2 0"7 --1"1 --2"2
0 --0"4 --0"9 --1-5 0 --0"5 --1-0 --1"5 0"2 --0"5 --1-2 --1.0 0"5 --0-1 --1-2 --1.8
0 --0"2 --0"3 --0-4 --0"1 --0-2 --0'4 --0-7 0 --0'3 --0-6 --0-9 0-1 --0-1 --0"1 --1"3
P o l l u t i o n Level ( L N p ) 3 : : LNp = Lea + 2"56a where Leq is the level o f mean noise energy. I t seemed useful, therefore, to p r o v i d e a simple way o f determining the c h a n g e s in T N I a n d LNp when noise distributions are c o m b i n e d . T a b l e s 3 and 4 give values of: A T N I = 4 A L t o - 3AL9 o and ALNp =
A L t o + AL9o ( A L t o - ALgo 5"120"t~ 2 + ( A L t o - AL9o) 1 + 56 + 56 /
respectively, where A T N I is the change in T N I (i.e. the value to be a d d e d to the T N I o f d i s t r i b u t i o n (1) to o b t a i n the T N I o f the c o m b i n e d distribution) and ALNp the c o r r e s p o n d i n g change in LNp*. I t can be seen from T a b l e 1 that, in a l m o s t every case AL9o is greater t h a n A L t o . Consequently, the noise variability m a y be reduced when two distributions a r e c o m b i n e d . The effect o f this r e d u c t i o n is a p p a r e n t in T a b l e 3 where A T N I is o f t e n negative, implying a reduction o f T N I . R e d u c t i o n s o f Lsp, T a b l e 4, are n o t so c o m m o n o r so m a r k e d since this unit puts less e m p h a s i s on the noise variability. * The assumption is made ia the derivation of these equations that the resultant combined distribution is Gaussian. This is not rigorously true, but, in all cases studied, no significant differences from truly Gaussian curves were obtained even when the standard deviations of the separate distributions were large.
12
P.M. NELSON
An example of the esthnation of total noise exposure from separate traffic sources Figure 5 shows a plan of a hypothetical urban roadway layout. It is assumed that traffic flows freely on both roads and that road I is a major road and road 2 a minor one. The traffic flow specifications assumed for this example are included on the plan. The problem is to assess the values of L t o and possibly Lgo at sites 1-6 when traffic noise is propagated from both roads. Road 2
Flow Speed
= 200 vehicte/h. = /,0 k m / h .
Composition = 0 % heavy vehicle.
t 6!
\\', \ \
Scale
I
\
\
\
i
~
\
\\'1
"
"
?
'P
m
,.\M
Two storey building.
\
\
\
\
\
\
\
\
\
5~t i
! I
,\",1 ,\M \
\xl
\ ~' \
\
\ \ ,
\
\
\
Two storey
building.
\ \
\
\
\
\
\
\
M
\\M ,.\M
, \ \
3T
\
\
NI
\\"1 ,\",1 \
\
",1
i
1,
I t i
Road 1
Flow :2000 vehicle/h; Speed : 48 km/h;Composition : 1 0 % heavy vehicles.
Fig. 5. Plan of a hypothetical roadway layout.
It was shown earlier that the combined L I o and L9o levels may be determined at any site provided that, for each source, either the values o f L ~o can be estimated or the mean,/+, and standard deviation, a, of the normal noise level-time distribution can be determined. If the first condition holds eqn. (13) may be applied to calculate the total L~o exposures from the separate estimates of L to. If the second condition holds/a and a may be calculated for each source and Tables 1 and 2 used to obtain the total L~o and Lgo exposures.
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
13
Predicted L to Values may be obtained at the prescribed sites from design rules published by Scholes and Sargent 6 and equations published by Delany et aL to giving the attenuation down side roads. Table 5 gives the predicted values of L ~o at the six sites due to the two different traffic sources, the increment, AL,o, obtained using eqn. (13) and the combined L~o value. The second method of calculation requires an estimate of the statistical parameters, ~r and p. These may be determined from estimates of L~o and L 9 o and by using eqns. (10) and (I1). For the road system under consideration L l o values TABLE 5 ESTIMATIONOF THE TOTAL L l 0 EXPOSUREUSING EQN. (13)
Road 1 Site number 1 2 3 4 5 6
Road 2
Predicted L i e dB(A) 77"0 73"0 67"0 63'0 60"0 58"0
59"0 59"0 59"0 59"0 59-0 59.0
( L l o ) l -- (Lio)2 ALlodB(A) frorn dB(A) eqn. (13) 18'0 14"0 8"0 4"0 1 '0 1 "0
0 0 0"4 1'2 2"3 2"3
Total Llo dB(A) 77"0 73"0 67"4 64'2 62"3 61 "3
have been predicted above. At present there are no design rules for the prediction of L9o. However, it is possible to estimate L9o at sites 1-6 by making comparisons with experimental values obtained at similar positions and for similar traffic flows, speeds and compositions. The values of L9o obtained in this way are noted in Table 6, together with the k lo values taken from Table 5 and a and p calculated from eqns. (10) and (11). The values of AL~o and AL9o shown in the table were estimated using Tables 1 and 2 for the appropriate values of ~rt and a , and p t - P 2 given in Table 6. The total values of L t o and L9o at each site are included in Table 6. It can be seen from Tables 5 and 6 that both methods give approximately the same estimates of ALl o, the differences result mainly from interpolation errors in the use of Tables 1 and 2. The second method has the advantage of enabling estimates of both ALto and AL9o to be made, thereby taking into account changes in the variability of the noise. Since noise units incorporating variability factors are subjectively important both TNI and LNp have been calculated from the L~0 and L9o estimates at each site. These values are included in Table 6. It will be noticed that at points close to the main road, noise from traffic on the minor road has little effect on the Lto values. Increases in L9o are, however, not insignificant as increases of 1.0 dB(A) and 2.0 dB(A) were found for sites (1) and (2) respectively. At points further from the main road the noise from the minor road has a greater effect. At site 6, for example, increases amounting to 2.3 dB(A) of Lto and 4.0 dB(A) of L9o were found.
I 2 3 4 5 6
Site No.
LIo
77.0 73.0 67.0 63-0 60"0 58.0
62"0 59-0 54.0 52.0 50-0 48"0
Lgo
Road I
a
5"9 5-5 5" I 4.3 3-9 3-9
69"5 66-0 60.5 57'5 55-0 53"0
59.0 59'0 59.0 59.0 59.0 59.0
LIO
e,.,.ai,.,.,I,,,I,,,..,.,18(A) It
TABLE 6
50.0 50.0 50"0 50.0 50-0 50.0
Lgo
Road 2
o" 3.5 3.5 3-5 3'5 3'5 3'5
It 54.5 54.5 54.5 54-5 54.5 54.5
15"0 11"5 6.0 3.0 1.0 1 '5
0 0 0"5 1.2 2-0 2-3
",~B(J;; dB(A) ^L,o
ESTIMATION OF VARIOUS NOISE UNITS USING 'FAILLES 1 AND 2
1 '0 2"0 3'8 4"3 4.5 4.0
aL,,)
dB(A)
77.0 73.0 67'5 64.2 62.0 61.3
Total Lio dB(A)
63.0 61 '0 57-8 56"3 54"5 54-0
Total Lgo dB(A)
89 79 65 58 53 52
TNI
88 82 73 69 66 65
LNP
Z
7: rrl g,,,.
~r
7-.
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
15
The variation of Llo along the side road is shown in Fig. 6. If noise from the minor road traffic is excluded, curve (b) is obtained. When the noise from road 1 is combined with curve (b) the sigmoid curve (c) is obtained. This curve follows closely curve (b) at distances up to 20 metres from road 1 but as the distance is increased further it rapidly departs from curve (b) as, theoretically, it asymptotically approaches the L1 o level predicted for road 1. 80~ 5ite~ {b) j 70 ~ite 3 r $it,~ 5
v
m
6Q
S~ -J
50
10
20
/.0
60
8o
100
Distance from the nearside kerb (m)
Fig. 6. Values of L10 obtained when (a) traffic flows along road 2 only, (b) traffic flows along road 1 only, and (c) traffic flows along both roads. The methods used in predicting the noise exposure in this simple example can be readily applied to more complex roadway networks provided the appropriate design rules are available for predicting the noise exposure from the separate road sources involved. It is expected that as more research is carried out on such features as the shielding effects of housing and roadway configurations and the effects of interrupted traffic flow on noise propagation, the combination techniques will find increased application in predicting traffic noise exposures in urban areas.
APPLICATION OF THE COMBINATION PROCEDURES TO THE SUMMATION OF NOISE FROM SEPARATE ENVIRONMENTAL SOURCES
The method proposed earlier for combining road traffic noise from multiple sources is important since this type of traffic noise constitutes the major proportion of all
16
v.M. NELSON
environmental noise. However, there is an increasing proportion of noise from other sources, particularly from aircraft. In regions close to major airports or in areas subjected to a large number of aircraft movements, aircraft noise may well be the principal cause of annoyance. Other transport modes such as trains and future modes such as hovercraft and helicopters may also influence the noise climate at some sites. The proposition of units such as LNp implies that annoyance is a function of total noise and variation of level and is independent of the nature of the transport
"B z
99.9
99 98
90
80
60
/,0
20
5
1
0.1
PercentQge of time CL level is exceeded
Fig. 7. Statistical noise level distribution for (a) typical aircraft movement measured 250 m from runway centre line, (b) for typical motorway traffic measured 30 m from the nearside lane and (c) for combined (a) and (b). generator. It is important, therefore, for both prediction and planning purposes to be able to combine noise not only from separate transport sources of the same type but also from various transport modes. A typical example would be to determine the additive effects of aircraft fly-past noise and surface traffic noise. Such calculations may be simply carried out provided the noise level distribution of each composite source is known or can be estimated. As an example, typical aircraft and traffic noise distributions are given in Fig. 7. Since the aircraft noise distribution is obviously not Gaussian, the probabilities were taken from the curves at intervals
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
17
of 1 dB(A) and combined using eqn. (7) and, to obtain a more accurate result, eqn. (8). The combined distributions obtained are also shown in Fig. 7. Unfortunately, distributions of noise level from transport sources, other than from road traffic, are either seldom measured or are not generally available and so it is not possible at present to study comprehensively the effects of various noise source interactions. It is expected that future measurements of transport noise will incorporate distribution data and therefore enable combination calculations to be carried out.
ACKNOWLEDGEMENTS
This paper was prepared in the Environment Section (Head, Mr L. H. Watkins) of the Transportation Division. It is contributed by permission of the Director of Road Research and is reproduced by permission of the Controller of HM Stationery Office. The author also wishes to acknowledge the guidance and encouragement given by D. G. Harland.
REFERENCES 1. I. O. GRIFFITHSand F. J. LANGDON,Subjective response to road traffic noise, Jotu'nal of Sound and Vibration, 8(16) (1968). 2. F. J. LANGDON and W. E. SCHOLES,The traffic noise index--a method of controlling noise nuisance, Building Research Station Current Paper CP 38/68. 3. D. W. ROBINSON, The concept of noise pollution level, Aero Report 38, 1969, National Physical Laboratory. 4. D. W. ROmNSON, An outline guide to criteria for the limitation of urban noise, Aero Report 39, 1968, National Physical Laboratory. 5. A. P. G. PETERSONand E. E. GROSS,Handbook of noise measurement, General Radio Company, West Concord, 1967. 6. W. E. SCHOLES and J. W. SARGENT, Designing against noise from road traffic, Building Research Station Current Paper CP 20/71. Applied Acoustics, 4 (1971) pp. 203-234. 7. D. R. JOHNSONand C. G. SAUNDERS,The evaluation of noise from freely flowing road traffic, Aero Report 29, 1967, National Physical Laboratory. 8. D. R. JOHNSON, A note on the relationship between noise exposure and noise probability distribution, Aero Report 40, 1969, National Physical Laboratory. 9. W. E. SCHOLESand G. H. VULKAN,A note on the objective measurement of road traffic noise, Applied Acoustics, 2(3) (1969) pp. 185-197. 10. M. E. DELANY, W. C. COPELANOand R. C. PAYNE, Propagation of traffic noise in typical urban situations, Acoustics Report Ac 54, 1971. National Physical Laboratory.
APPENDIX I: PROBABILITY CONSTITUENTS OF C(L) T h e f o l l o w i n g listing gives s o m e o f t h e possible w a y s t h a t A a n d B c a n c o m b i n e t o g e n e r a t e a level o f L o r greater. T h e c o r r e s p o n d i n g p r o b a b i l i t i e s are given in e a c h
18
v . M . NELSON
case. T h e p r o b a b i l i t y t h a t C exceeds L, C(L), is t h e s u m o f all t h e c o n s t i t u e n t probabilities. Listing of the constituents of C(L)
R e a s o n t h a t C s h o u l d exceed L
Probability
1. I f A o r B exceeds L
A(L)
2. I f A a n d B in t h e i n t e r v a l L to (L - 3)
[A(L -
3) - A ( L ) ] [ B ( L
3. I f o n e lies in the i n t e r v a l L t o (L - 2) a n d t h e o t h e r in t h e i n t e r v a l (L - 3) to (L - 4-3)
[A(L -
2) -
-- A ( L ) B ( L )
+ B(L)
3) -
-
A(L)][0.38(L
+ 0 . 7 B ( L - 4) + [B(L
-
2) -
x [0.3A(L -
-
B(L
B(L)]
5)
3)]
-
S(L)]
5) + 0 . 7 A ( L -
4)
A(L
-
3)1
4. I f o n e lies in the i n t e r v a l L t o (L - 1) a n d t h e o t h e r in the i n t e r v a l (L - 43) to (L - 6-9)
[A(L -
5. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s o n e lies in the i n t e r v a l (L - 2) t o (L - 3) a n d the o t h e r lies in t h e i n t e r v a l (L - 3) to (L - 4-3)
0.5{[A(L - 3) - A ( L - 2)] × [0.3B(L - 5) + 0.TB(L - 4) - B ( L - 3)] + [ B ( L - 3) - B ( L - 2)][0.3A(L - 5) + 0 . 7 A ( L - 4) - A ( L - 3)]',.
6. I f o n e lies in t h e i n t e r v a l L - 1 to L - 1.6 a n d the o t h e r in the i n t e r v a l
0.42{[A(L - 2) - A ( L - 1)] × [ B ( L - 5) - B (L - 4)]
1) - A ( L ) I [ O . 9 B ( L - 7) + 0 . 1 B ( L - 6) - 0 . 3 B f L - 5) - 0 . 7 B ( L - 4)] + [ B ( L - 1) - B ( L ) ] [ O . 9 A ( L - 7) + 0 . 1 A ( L - 6) - 0"3A(L - 5) - 0-7A(L - 4)]
+ [ B ( L -- 2)
L - 4-3 t o L -
5
-
B(L
-
I)][A(L -
5) -
7. I f o n e lies in t h e i n t e r v a l L - 1 t o L - 1.2 a n d t h e o t h e r in the i n t e r v a l L5toL6
A(L
-
0.2{[A(L - 2) - A ( L - 1)][B(L - B ( L - 5)] + [ B ( L - 2) - B(LI ) ] [ A ( L - - 6) --
A(L
8. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s 0-18{[A(L - 2) - A ( L - 1)] o n e is in the i n t e r v a l L - 1 t o L - 1.2 x [ B ( L - 7) - B ( L - 6)] a n d the o t h e r is in the i n t e r v a l + [ B ( L - 2) - B ( L - 1)] L6toL6.9 x [A(L7)- A(L-
--
4)]} 6)
5)]',
6)]}
THE COMBINATION OF NOISE FROM SEPARATE TIME VARYING SOURCES
19
9. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s 0.4{[A(L - 2) - A ( L - 1)] o n e is in t h e i n t e r v a l L - 1.2 to L - 1.6 × [ B ( L - 6) - B ( L - 5)] a n d the o t h e r is in t h e i n t e r v a l L - 6 + [ B ( L - 2) - B ( L - 1)] toL5 x [A(L-6)-A(L-
5)]}
10. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s 0.28{[A(L - 2) - A ( L - 1)] o n e is in the i n t e r v a l L - 1.6 to L - 2 x [ B ( L - 5) - B ( L - 4)] a n d t h e o t h e r is in the i n t e r v a l L - 5 + [ B ( L - 2) - B ( L - 1)1 t o L - 4.3 × [ A ( L - 5) - A ( L
-
4)1}
11. I f o n e lies in the i n t e r v a l L t o L - 0.8 a n d t h e o t h e r in t h e i n t e r v a l L6.9toL8
0 . 8 ( [ A ( L - l ) - A ( L ) ] [ a ( L - 8) - 0 . 9 B f L - 7) - 0 . 1 B ( L - 6)] + [ B ( L - 1) - B ( L ) I [ A ( L 8) - 0 . 9 A ( L - 7) - 0 - 1 A ( L - 6)1}
12. I f o n e lies in the i n t e r v a l L to L - 0.6 a n d the o t h e r in t h e interval L - 8 to L - 9
0.6{[A(L
13. I f o n e lies in t h e i n t e r v a l L to L - 0.4 a n d t h e o t h e r in t h e i n t e r v a l L-9toL10
0.4{[A(L
14. I f o n e lies in the i n t e r v a l L t o L - 0.2 a n d t h e o t h e r in t h e i n t e r v a l L.10toL13
0.2{[A(L-
15. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s o n e lies in t h e i n t e r v a l L - 0-8 t o L - 1 a n d t h e o t h e r in t h e i n t e r v a l L - 8 t o L - 6-9
0.1 {[A(L - 1) - A ( L ) ] [ B ( L - 8) - 0 . 9 B ( L - 7) - 0 . 1 B ( L - 6)] + [ B ( L - 1) - ( B L ) ] [ A ( L - 8) - 0 . 9 A ( L - 7) - 0 - 1 A ( L - 6)]}
16. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s o n e lies in t h e i n t e r v a l L - 0.6 t o L.0.8 a n d t h e o t h e r in the i n t e r v a l L-9toL-8
0.1{[A(L -
17. F o r a p p r o x i m a t e l y h a l f the o c c a s i o n s o n e lies in the i n t e r v a l L - 0-4 t o L - 0.6 a n d t h e o t h e r in t h e i n t e r v a l L - 10toL - 9
0.1{[A(L -
-
-
-
[a(L
9)
-
1)
-
-
8)]}
1) - A ( L ) ] [ B ( L - 10) 9)] + [ B ( L - 1) ( B L ) ] [ A ( L - 10) - A ( L -
9)]}
-
9) -
A(L
-
-
B(L -
-
1)-
B(L
-
13) 1)
A(L)][B(L-
10)] +
B(L)][A(L
-
-
A(L)][B(L
8)1 +
B(L)I[A(L
-
-
[B(L
13) -
1) - A ( L ) ] [ B ( L - 8)] + [ B ( L B(L)[A(L - 9) -
-
A(L
B(L
1) - A ( L ) ] [ B ( L 9)] + [ B ( L [ A ( L - 10) -
18. F o r a p p r o x i m a t e l y h a l f t h e o c c a s i o n s 0-1 {[A(L - I) - A ( L ) ] [ B ( L o n e lies in t h e i n t e r v a l L - 0-2 to - B ( L - 10)] + [ B ( L L -- 0.4 a n d t h e o t h e r in t h e i n t e r v a l -B(L)][A(L - 13) L-- 13toL-10
9)]}
1) A(L
-
+
-
9)
-
B(L -
-
1) -
~(L
-
10) 1) -
A(L
-
-
13) I)
A(L
-
-
8)]}
B(L)
9)]}
10)]}
20
P . M . NELSON
19. For approximately half the occasions 0.1 {[A(L - 1) - A(L)][I - B ( L - 13) one lies in the interval L to L - 0.2 + [ B ( L - 1) - B(L)] and the other below L - 13 x [1 - A ( L - 13)]}-
APPENDIX lI:
A SIMPLE EQUATION FOR THE COMBINATION OF NOISE FROM TWO TIME VARYING SOURCES A AND B
Each distribution is considered to comprise three regions, shown diagrammatically below. (L t)A L
(L 2)n L
L-2
L-3
(L2)A
(L2)B
A and B may combine to produce a level o f L or greater when, (1) either A or B exceeds L (2) if both A and B are in the interval L to (L - 3) (3) for X o f the occasions when A is in the interval L to (L - 3) and B is in the interval (L - 3) to (L2)B and for Y o f the occasions when B is in the interval L to (L - 3) and A is in the interval (L - 3) to (L2)A. The probabilities for (1) and (2) are given in Appendix I. The probability that C will exceed L for (3) is X[I - A ( L - 3)][B(L - 3) - B(L)] + Y[1 - B ( L - 3)][A(L - 3) - A(L)] The total probability that C will exceed L , C ( L ) , is given by the total probabilities o f (I), (2) and (3). Therefore: + B ( L ) + A ( L - 3)B(L - 3) - A ( L ) B ( L - 3) - A ( L - 3)B(L) + X [ I - A ( L - 3)][B(L - 3) - B(L)] + Y[1 - B ( L - 3)][A(L - 3) - A(L)]
C(L) = A(L)
04) In order to complete the above equation, suitable values are required for the correction factors X and Y. A simple analysis revealed that X and Y may be most suitably assigned the value: X=
Y = 0-3
(15)
Substituting (15) into (14) and simplifying, one obtains: C(L)
= 0-7[A(L)
+ B(L) -
A(L)B(L
-
3) -- B ( L ) A ( L
-
3)]
+ 0 . 4 A ( L - 3)B(L - 3) + 0.3[A(L - 3) + B ( L - 3)]
THE COMBINATIONOF NOISE FROM SEPARATE TIME VARYING SOURCES
21
APPENDIX III: ESTIMATIONOF A L t o AND AL9o USING THE ALIGNMENT CHARTS
The following example of the estimation of ALto and ALgo will illustrate the use of the alignment charts (Figs. 2 and 3). Figures 2 and 3 are each divided into four panels, each panel representing a particular value of le t-It 2 within the range 0-15 dB(A). The panels are bounded by the vertical axis graduated on the left side in a 2 dB(A). Between the boundaries and at the bottom of each panel a horizontal axis has been constructed which is graduated in a2 dB(A). As an example of the use ofthe charts, consider a typical case where a t = 4 dB(A), 0"2 = 6 dB(A) and (/tt - It2) = 0. According to Tables 1 and 2, ALto is 4.1 dB(A) and AL9o is 4"5 dB(A). To obtain the corresponding estimates using the alignment charts a line is drawn joining a t = 4 dB(A) and a z = 6 dB(A) on the vertical scales. A perpendicular is then constructed from the horizontal scale at a2 = 6 dBA to intersect this line. This procedure is illustrated in Figs. 2 and 3 by the broken lines. The position of the point of intersection, labelled A on the ALto chart and B on the ALgo chart, then identifies the values of ALto and ALgo. It can be seen that point A lies very close to the AL t o = 4 dB(A) curve and B lies approximately midway between the AL9o = 5 dB(A) and AL9o = 4 dB(A) curves. Using the charts therefore AL~o and ALgo are estimated to be 4.0 dB(A) and 4.5 dB(A) respectively. These estimates are in good agreement with the tabulated results. For most practical cases ltt - #2 will not be equal to either 0, 5, 10 or 15 dB(A) and so it is necessary to interpolate between the panels. This can be most conveniently achieved by constructing a graph of either AL~ o or AL9o against (/z 1 - I~2) where/~t - It2 takes the values 0, 5, 10, 15 dB(A), using the alignment charts in the manner described above, so that the values of ALto and ALgo can then be identified for any value of ( I t t - IL2).