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Reflection of noise from a building's facade
Applied Acoustics 43 (1994) 149-157 O 1994ElsevierScienceLimited Primed in Great Britain.All fights reserved 0003-682X/94/$7.00
ELSEVIER
Reflection of Noise from a Building's Facade Rufin Makarewicz & Piotr Kokowski Institute of Acoustics, A. Mickiewicz University, 60-769 Poznari, Poland (Received 8 March 1993; revised version received 23 June 1993; accepted 21 April 1994)
ABSTRA CT
The reflection from the facade of a building is considered in terms of specular reflection and diffuse scattering. Using the concept of the continuous Aweighted sound pressure level, the importance of reflected noise is evaluated by comparing it with the contribution of the direct wave and background noise. The results may be applied directly in the case of buildings flanking one side of a road.
1 INTRODUCTION Urban noise propagation is affected by a number of factors, among them, reflections from buildings' facades. The analysis by Wiener et al., 1 and many others, assumed, that the facades to be perfectly smooth, so that specular reflection occurred. On the other hand, Kuttruff2 based his analysis on surface scattering in all directions. A step forward was made by Davies, 3 Chien and Carroll, 4 who took into account both types of reflections. Similar approaches can be found within the field of ocean acoustics5. Assume that the buildings line only one side of a roadway at a distance D--and we want to predict the equivalent continuous A-weighted sound pressure level (LA,qr) at the opposite side, d away from the roadway (Fig. 1). Do we need to take account of the facade's reflections or not? (The same question in terms of the percentile level, Llo, has been put by Hothersall and Simpson, 6 and by Chew.7) If not, we can simplify considerably the computational procedure, and avoid uncertainty about the value of the reflection coefficient. For such a case, the noise consists of the direct waves and other sounds coming from unidentifiable sources (background noise). 149
R. ~4akarewicz. Piotr Kokowski
15(I
0 d~? I 0 Fig. 1.
Roadway, D m from a row of buildings and d m from the receiver 0.
To answer the above question, we consider the case where reflections would be maximal by making the following assumptions: (i)
The reflection is ideal i.e. no sound energy is absorbed by the facade; (ii) There is no spacing between adjacent buildings so the sound is reflected by a uniform block; (iii) The width, l and the height h, of the block are substantially greater than the distance D (Fig. 2). Let LAeqT and LAeqT denote the continuous A-weighted sound pressure level with and without reflections. The background noise is included in these values. If the permissible error of prediction equals 6L4eqr dB, then the reflections may be neglected when. LAeqT -- L]eq T _< 6LAeqT
(1)
There is the following relationship between LAeqTand the A-weighted sound exposure EA, which characterizes the noise generated by a single vehicle:
LAeqT= IO . lg {T~p2o ~.f Ni . EA, + IO°"£AeqT}
(2)
h
Fig. 2.
A row of buildings (Fig. 1) replaced by a block whose dimensions are larger than the distance to the road: l >> D, h >> D.
Reflection of noise from a building's facade
151
where P0 = 2 x 10-5 N/m 2, the ratio N~/T is the flow rate of the ith class vehicle (veh/s) e.g. automobiles, trucks; and LAeqr expresses the continuous A-weighted sound pressure level of the background noise. The A-weighted sound exposure is defined by the following integral: EA = j_~=p2(t) dt
(3)
Here pZ(t) denotes the time history of A-weighted mean-square sound pressure. 2 DIRECT WAVE To calculate the A-weighted sound exposure for the direct wave, E~d), ((d) denotes direct wave) the function pZ(t) has to be known initially. We assume that the surface of the road and the adjacent terrain is acoustically hard without any excess attenuation. Thus, the A-weighted mean-square sound pressure is
p](t)
:
p2 . lOO.,I3p~A[ro/r(t)]2
(4)
where r(t) denotes the instantaneous source-receiver distance and L~A) is the A-weighted sound pressure level (reference mean emission level) measured at distance r o. Consider the scenario in Fig. 3 where the vehicle and receiver are shown as points S and O, respectively. For a straight road and a steady speed of the vehicle V, the distance SO is r(t) = { ( V t ) 2 + d2} 1/2
(5)
Equations (3-5) yield the A-weighted sound exposure for a direct wave which is generated by a vehicle belonging to the ith class: E~d) = p2. 100,L~p°j . roe rr
V/.d
(6)
Consequently, the continuous A-weighted sound pressure level for the
I
I
S(O,Vt,O)
y
v x O(d,O,O)
Fig. 3.
The noise reaches the receiver 0 directly from radiation of the source S as it moves with steady speed V along a block.
152
R. Makarewicz, Piotr Kokowski
sum of the direct wave and the background noise, without reflections from the facade, is given by (Eqns (2) and (3)): LAeqr = 10. lg
N i . EA(~) + 100'lLAmar
(7)
The value of L~enr may be calculated using the above equations or may be directly measured in an open area at a distance d from a road. We will consider it as a known quantity.
3 R E F L E C T E D WAVES The exterior facade of a typical building does not reflect noise in a purely specular manner. Because the dimensions of the facade's irregularities (decorative elements, recesses of windows, etc.) are comparable to the wavelengths of traffic noise, the specular reflection is disturbed by surface scattering (Fig. 4). Following Davies 3, Chien and Carroll 4 Urick 5, we will use the mixed reflection law which allows for both types of reflection. To enhance the effects of specular reflection and scattering we take into consideration the block described in the Section 1 (Fig.2). The analysis of reflected waves will be carried out for a receiver O(d,0,0) located at a distance (d + D) from the facade, i.e. d m away from the vehicle stream (Figs. 2 and 3). 3.1 Specularly reflected wave
For the geometry shown in Fig. 3, the A-weighted mean-square sound pressure of the specularly reflected wave is: p~q : p2o . 100IL(p °j . ~ . [ro/rq] 2
(8)
where the path length of the reflected wave is: r(t) = {(Vt) 2 + (d + 2D)2} v2
(9)
Fig° 4° The incident wave (solid line) is specularly reflected (long dashed line) and
scattered (short dashed lines).
Reflection of noise from a building's facade
153
The reflection coefficient/3 is assumed to be constant, i.e. it is the same at all points of the facade and for all angles of incidence (when/3 = O, only scattered waves occur). Under the assumption 1 >> D (Fig. 2), the combination of eqns (3),(6),(8),(9) yields the A-weighted sound exposure for the specularly reflected wave: EA(q) = EA(d}. /3.
1
(10)
1 + 2D/d
where EA(d) is the A-weighted sound exposure for direct wave of sound which is emitted by a vehicle of the ,ith class at the distance d from the vehicle stream (Eqn. (6)). 3.2 Scattered waves
If a noise characterized by A-weighted intensity, IA, impinges on the surface dS at an angle O (Fig. 4), then some fraction of the sound energy may be scattered, dPA = y . ira COS O dS (11) As for the reflection coefficient, the scattering coefficient y is assumed to be constant. (For a perfectly smooth surface, we have 3' = 0 and the undisturbed specular reflection occurs.) The area dS can be treated as a secondary noise source with the Aweighted power dPa given above. Accordingly, at a distance R the A-weighted intensity of scattered wave due to the surface dS is d l A - a ( 6 ) dPA R2
(12)
where Q($) plays the role of a directivity factor (Fig.4). In the far field we can write: pJ = 1A • pc, where pc is the characteristic impedance of air. Thus, making use of Lambert's law s, Q($) = (1/,r) cos$, we obtain the resultant A-weighted mean square sound pressure of noise scattered by the whole facade of area S (Eqns. (11) and (12)): P~s = ~
II c°s O
COS
¢
zrR2
pZA dS •
(13)
s
Here p2 is the A-weighted mean-square sound pressure of the wave incident on the facade (see Eqn. 4). For the geometry shown in Fig. (5), the distance between the moving source S and the scattered element dS, and the angle of incidence are given by: r = { D 2 + (y -
Vt) 2 + z2} v2,
cosO = D / r
(14)
Reflection of noise from a building's facade
155
The above expression can be rewritten as (Eqns (10), (17) and (18)):
L~,qr = 10"lg { F(D/d) . T . 1p2 ~i Ni " Etd! + 100.1LAoqr}
(19)
where
F(D/d)=I+(/3+2y)_ 1 (20) 7r 1 + (D/d) describes the combined effect of both types of reflection: specular reflection and surface scattering. N o t e that/3 = O and 7 = O (no reflection) implies F = 1 and consequently LAeqT = LAeqT, i.e. Eqn (19) yields the continuous A-weighted sound pressure level of noise without reflections which consists of the direct wave and b a c k g r o u n d noise (see Eqn (7)). Assume that/~Acqr, i.e. the continuous A-weighted sound pressure level of the b a c k g r o u n d noise, and the value of LAeqT a r e known. T h e n the contribution of the direct waves is given by (Eqn (7)): 1 r.p
~ U i . E~dA~= 100~L~eqT -- 100~LA°qr ,
(21)
T o overestimate the role of reflections we write:/3 = 1 and y = 1. Hence, eqns (1) and (19)-(21) determine the region where the direct waves play the major role:
0 < d < 2D/(A - 1)
(22)
where D is the distance between the vehicle stream and the perfectly reflecting block (Fig. 2). The quantity A is given by:
( A =
2_~1 -- lo0"ltLAeqT--L~eqT] 1 +
100'16LAeq T -
1
(23)
where L**qr is the continuous A-weighted sound pressure level without reflections at the observation point (Eqn. 7). It would be unreasonable to set 6LAcqr = 0, i.e., only an accuracy of 100% is acceptable. In such a case A ~ oo and inequalities (22) yield 0 < d < 0. This means that there is no place where reflections are negligible. Conversely, if we d o n ' t care a b o u t accuracy at all, 6LA~qT >> ldB, then A ---> 1 and inequalities (22) take the form, 0 < d < 0% i.e.; one can omit the reflections at any distance from the vehicle stream. H o w large m u s t the permissible error be, to create such a situation? Let's note that d = ,o for A = 1 (Eqn 22). Consequently, eqn (23) can be rearranged to:
F o r example, assume that m e a s u r e m e n t s give L * ~ r = 50 dB and/~A~qr =
156
R. Makarewic=, Piotr Kokowski
45 dB. The above equation yields 3.3 dB. This implies that with that accuracy or less (~LAeqZ->>3.3 dB) we may neglect reflections, i.e., calculate L]eqr (Eqn. 7) for all distances d instead of LAeqr (Eqn. 18). The same conclusion is true when
6L4*qr-
L~eqr= - 1 0 " l g
1 - (10°~6L~qr -
1)
+
For example, let us take 6LA~qr = 2 dB. In such a case, reflections can be neglected for all distances d when the difference LAeqr * - LAeqr equals 1.9 dB or less. Let's consider a practical situation as illustrated in Fig. I. Do we need to take into account the facade's reflections at the receiver O? The first step is the calculation of continous A-weighted sound pressure level without reflections at the receiver, i.e., the value of LAeqTdefined by Eqn (7). In general, traffic noise is annoying when it clearly exceeds the background noise, LAeqr. Otherwise the former is masked by the latter. Thus * -- £4eqT 2>> 1 dB. Consequently, eqn (23) can be we may write, LAeqT rewritten as A ~- ( 1 + 2)/(lO016LAcqI'--l) (26) For the prediction e r r o r ~tAeq.1 = 2 dB, which is small compared to other uncertainties, one gets A -- 2.8. Finally, inequalities (22) take the approximate form: 0 < d < D (27) where d and D are distances shown in Fig. 1.
5 CONCLUSION The initial assumptions were chosen so as to increase the role of specular reflection and surface scattering as compared to real life conditions: instead of a large and non absorbing block, there is usually a row of buildings which absorb some fraction of the sound energy (/3 < 1, y < 1). Therefore one can ignore reflections from the building facade and in this way simplify considerably the computational procedure when the distance from the road d (Fig. 1) fulfills eqn (27).
ACKNOWLEDGEMENTS The authors thank reviewers for their comments and Dr. B. C. J. Moore (University o f Cambridge) for corrections to the English text.
Reflection of noise from a building'sfacade
157
REFERENCES AND BIBLIOGRAPHY 1. Wiener, F. M., Malme, C. I. & Gogos, C. M., Sound propagation in urban areas, J. Acoust. Soc. Am., 37 (1965), 738-747. 2. Kuttruff, H., Zur Berechnung yon Pegelmittelwerten und Schwankungsgr6ssen bei Strassenl~rm Acustica, 32 (1975), 57-69. 3. Davies, H. G., Multiple-reflection diffuse-scattering model for noise propagation in streets. J. Acoust. Soc. Am., 64 (1978), 517-521. 4. Chien, C. F. & Carroll, M. M., Sound above a rough absorbent plane, J. Acoust. Soc. Am., 67 (1980), 827-829. 5. Urick, R. J., Principles of Underwater Sound, McGraw-Hill, New York, 1975. 6. Hothersall, D. C. & Simpson S., The reflection of road traffic noise; J. Sound Vibr., 90 (1983), 399-405. 7. Chew, C. H., Prediction of traffic noise from expressway Appl. Acoust., 28 (1989), 203-212. 8. Kuttruff, H., Room acoustics. Applied Science Publishers, second edition, London, 1979. 9. Gradshteyn, I. S. & Ryzik, I. M., Table of Integrals, Series and Products, Academic Press, New York, 1975.
ELSEVIER
Reflection of Noise from a Building's Facade Rufin Makarewicz & Piotr Kokowski Institute of Acoustics, A. Mickiewicz University, 60-769 Poznari, Poland (Received 8 March 1993; revised version received 23 June 1993; accepted 21 April 1994)
ABSTRA CT
The reflection from the facade of a building is considered in terms of specular reflection and diffuse scattering. Using the concept of the continuous Aweighted sound pressure level, the importance of reflected noise is evaluated by comparing it with the contribution of the direct wave and background noise. The results may be applied directly in the case of buildings flanking one side of a road.
1 INTRODUCTION Urban noise propagation is affected by a number of factors, among them, reflections from buildings' facades. The analysis by Wiener et al., 1 and many others, assumed, that the facades to be perfectly smooth, so that specular reflection occurred. On the other hand, Kuttruff2 based his analysis on surface scattering in all directions. A step forward was made by Davies, 3 Chien and Carroll, 4 who took into account both types of reflections. Similar approaches can be found within the field of ocean acoustics5. Assume that the buildings line only one side of a roadway at a distance D--and we want to predict the equivalent continuous A-weighted sound pressure level (LA,qr) at the opposite side, d away from the roadway (Fig. 1). Do we need to take account of the facade's reflections or not? (The same question in terms of the percentile level, Llo, has been put by Hothersall and Simpson, 6 and by Chew.7) If not, we can simplify considerably the computational procedure, and avoid uncertainty about the value of the reflection coefficient. For such a case, the noise consists of the direct waves and other sounds coming from unidentifiable sources (background noise). 149
R. ~4akarewicz. Piotr Kokowski
15(I
0 d~? I 0 Fig. 1.
Roadway, D m from a row of buildings and d m from the receiver 0.
To answer the above question, we consider the case where reflections would be maximal by making the following assumptions: (i)
The reflection is ideal i.e. no sound energy is absorbed by the facade; (ii) There is no spacing between adjacent buildings so the sound is reflected by a uniform block; (iii) The width, l and the height h, of the block are substantially greater than the distance D (Fig. 2). Let LAeqT and LAeqT denote the continuous A-weighted sound pressure level with and without reflections. The background noise is included in these values. If the permissible error of prediction equals 6L4eqr dB, then the reflections may be neglected when. LAeqT -- L]eq T _< 6LAeqT
(1)
There is the following relationship between LAeqTand the A-weighted sound exposure EA, which characterizes the noise generated by a single vehicle:
LAeqT= IO . lg {T~p2o ~.f Ni . EA, + IO°"£AeqT}
(2)
h
Fig. 2.
A row of buildings (Fig. 1) replaced by a block whose dimensions are larger than the distance to the road: l >> D, h >> D.
Reflection of noise from a building's facade
151
where P0 = 2 x 10-5 N/m 2, the ratio N~/T is the flow rate of the ith class vehicle (veh/s) e.g. automobiles, trucks; and LAeqr expresses the continuous A-weighted sound pressure level of the background noise. The A-weighted sound exposure is defined by the following integral: EA = j_~=p2(t) dt
(3)
Here pZ(t) denotes the time history of A-weighted mean-square sound pressure. 2 DIRECT WAVE To calculate the A-weighted sound exposure for the direct wave, E~d), ((d) denotes direct wave) the function pZ(t) has to be known initially. We assume that the surface of the road and the adjacent terrain is acoustically hard without any excess attenuation. Thus, the A-weighted mean-square sound pressure is
p](t)
:
p2 . lOO.,I3p~A[ro/r(t)]2
(4)
where r(t) denotes the instantaneous source-receiver distance and L~A) is the A-weighted sound pressure level (reference mean emission level) measured at distance r o. Consider the scenario in Fig. 3 where the vehicle and receiver are shown as points S and O, respectively. For a straight road and a steady speed of the vehicle V, the distance SO is r(t) = { ( V t ) 2 + d2} 1/2
(5)
Equations (3-5) yield the A-weighted sound exposure for a direct wave which is generated by a vehicle belonging to the ith class: E~d) = p2. 100,L~p°j . roe rr
V/.d
(6)
Consequently, the continuous A-weighted sound pressure level for the
I
I
S(O,Vt,O)
y
v x O(d,O,O)
Fig. 3.
The noise reaches the receiver 0 directly from radiation of the source S as it moves with steady speed V along a block.
152
R. Makarewicz, Piotr Kokowski
sum of the direct wave and the background noise, without reflections from the facade, is given by (Eqns (2) and (3)): LAeqr = 10. lg
N i . EA(~) + 100'lLAmar
(7)
The value of L~enr may be calculated using the above equations or may be directly measured in an open area at a distance d from a road. We will consider it as a known quantity.
3 R E F L E C T E D WAVES The exterior facade of a typical building does not reflect noise in a purely specular manner. Because the dimensions of the facade's irregularities (decorative elements, recesses of windows, etc.) are comparable to the wavelengths of traffic noise, the specular reflection is disturbed by surface scattering (Fig. 4). Following Davies 3, Chien and Carroll 4 Urick 5, we will use the mixed reflection law which allows for both types of reflection. To enhance the effects of specular reflection and scattering we take into consideration the block described in the Section 1 (Fig.2). The analysis of reflected waves will be carried out for a receiver O(d,0,0) located at a distance (d + D) from the facade, i.e. d m away from the vehicle stream (Figs. 2 and 3). 3.1 Specularly reflected wave
For the geometry shown in Fig. 3, the A-weighted mean-square sound pressure of the specularly reflected wave is: p~q : p2o . 100IL(p °j . ~ . [ro/rq] 2
(8)
where the path length of the reflected wave is: r(t) = {(Vt) 2 + (d + 2D)2} v2
(9)
Fig° 4° The incident wave (solid line) is specularly reflected (long dashed line) and
scattered (short dashed lines).
Reflection of noise from a building's facade
153
The reflection coefficient/3 is assumed to be constant, i.e. it is the same at all points of the facade and for all angles of incidence (when/3 = O, only scattered waves occur). Under the assumption 1 >> D (Fig. 2), the combination of eqns (3),(6),(8),(9) yields the A-weighted sound exposure for the specularly reflected wave: EA(q) = EA(d}. /3.
1
(10)
1 + 2D/d
where EA(d) is the A-weighted sound exposure for direct wave of sound which is emitted by a vehicle of the ,ith class at the distance d from the vehicle stream (Eqn. (6)). 3.2 Scattered waves
If a noise characterized by A-weighted intensity, IA, impinges on the surface dS at an angle O (Fig. 4), then some fraction of the sound energy may be scattered, dPA = y . ira COS O dS (11) As for the reflection coefficient, the scattering coefficient y is assumed to be constant. (For a perfectly smooth surface, we have 3' = 0 and the undisturbed specular reflection occurs.) The area dS can be treated as a secondary noise source with the Aweighted power dPa given above. Accordingly, at a distance R the A-weighted intensity of scattered wave due to the surface dS is d l A - a ( 6 ) dPA R2
(12)
where Q($) plays the role of a directivity factor (Fig.4). In the far field we can write: pJ = 1A • pc, where pc is the characteristic impedance of air. Thus, making use of Lambert's law s, Q($) = (1/,r) cos$, we obtain the resultant A-weighted mean square sound pressure of noise scattered by the whole facade of area S (Eqns. (11) and (12)): P~s = ~
II c°s O
COS
¢
zrR2
pZA dS •
(13)
s
Here p2 is the A-weighted mean-square sound pressure of the wave incident on the facade (see Eqn. 4). For the geometry shown in Fig. (5), the distance between the moving source S and the scattered element dS, and the angle of incidence are given by: r = { D 2 + (y -
Vt) 2 + z2} v2,
cosO = D / r
(14)
Reflection of noise from a building's facade
155
The above expression can be rewritten as (Eqns (10), (17) and (18)):
L~,qr = 10"lg { F(D/d) . T . 1p2 ~i Ni " Etd! + 100.1LAoqr}
(19)
where
F(D/d)=I+(/3+2y)_ 1 (20) 7r 1 + (D/d) describes the combined effect of both types of reflection: specular reflection and surface scattering. N o t e that/3 = O and 7 = O (no reflection) implies F = 1 and consequently LAeqT = LAeqT, i.e. Eqn (19) yields the continuous A-weighted sound pressure level of noise without reflections which consists of the direct wave and b a c k g r o u n d noise (see Eqn (7)). Assume that/~Acqr, i.e. the continuous A-weighted sound pressure level of the b a c k g r o u n d noise, and the value of LAeqT a r e known. T h e n the contribution of the direct waves is given by (Eqn (7)): 1 r.p
~ U i . E~dA~= 100~L~eqT -- 100~LA°qr ,
(21)
T o overestimate the role of reflections we write:/3 = 1 and y = 1. Hence, eqns (1) and (19)-(21) determine the region where the direct waves play the major role:
0 < d < 2D/(A - 1)
(22)
where D is the distance between the vehicle stream and the perfectly reflecting block (Fig. 2). The quantity A is given by:
( A =
2_~1 -- lo0"ltLAeqT--L~eqT] 1 +
100'16LAeq T -
1
(23)
where L**qr is the continuous A-weighted sound pressure level without reflections at the observation point (Eqn. 7). It would be unreasonable to set 6LAcqr = 0, i.e., only an accuracy of 100% is acceptable. In such a case A ~ oo and inequalities (22) yield 0 < d < 0. This means that there is no place where reflections are negligible. Conversely, if we d o n ' t care a b o u t accuracy at all, 6LA~qT >> ldB, then A ---> 1 and inequalities (22) take the form, 0 < d < 0% i.e.; one can omit the reflections at any distance from the vehicle stream. H o w large m u s t the permissible error be, to create such a situation? Let's note that d = ,o for A = 1 (Eqn 22). Consequently, eqn (23) can be rearranged to:
F o r example, assume that m e a s u r e m e n t s give L * ~ r = 50 dB and/~A~qr =
156
R. Makarewic=, Piotr Kokowski
45 dB. The above equation yields 3.3 dB. This implies that with that accuracy or less (~LAeqZ->>3.3 dB) we may neglect reflections, i.e., calculate L]eqr (Eqn. 7) for all distances d instead of LAeqr (Eqn. 18). The same conclusion is true when
6L4*qr-
L~eqr= - 1 0 " l g
1 - (10°~6L~qr -
1)
+
For example, let us take 6LA~qr = 2 dB. In such a case, reflections can be neglected for all distances d when the difference LAeqr * - LAeqr equals 1.9 dB or less. Let's consider a practical situation as illustrated in Fig. I. Do we need to take into account the facade's reflections at the receiver O? The first step is the calculation of continous A-weighted sound pressure level without reflections at the receiver, i.e., the value of LAeqTdefined by Eqn (7). In general, traffic noise is annoying when it clearly exceeds the background noise, LAeqr. Otherwise the former is masked by the latter. Thus * -- £4eqT 2>> 1 dB. Consequently, eqn (23) can be we may write, LAeqT rewritten as A ~- ( 1 + 2)/(lO016LAcqI'--l) (26) For the prediction e r r o r ~tAeq.1 = 2 dB, which is small compared to other uncertainties, one gets A -- 2.8. Finally, inequalities (22) take the approximate form: 0 < d < D (27) where d and D are distances shown in Fig. 1.
5 CONCLUSION The initial assumptions were chosen so as to increase the role of specular reflection and surface scattering as compared to real life conditions: instead of a large and non absorbing block, there is usually a row of buildings which absorb some fraction of the sound energy (/3 < 1, y < 1). Therefore one can ignore reflections from the building facade and in this way simplify considerably the computational procedure when the distance from the road d (Fig. 1) fulfills eqn (27).
ACKNOWLEDGEMENTS The authors thank reviewers for their comments and Dr. B. C. J. Moore (University o f Cambridge) for corrections to the English text.
Reflection of noise from a building'sfacade
157
REFERENCES AND BIBLIOGRAPHY 1. Wiener, F. M., Malme, C. I. & Gogos, C. M., Sound propagation in urban areas, J. Acoust. Soc. Am., 37 (1965), 738-747. 2. Kuttruff, H., Zur Berechnung yon Pegelmittelwerten und Schwankungsgr6ssen bei Strassenl~rm Acustica, 32 (1975), 57-69. 3. Davies, H. G., Multiple-reflection diffuse-scattering model for noise propagation in streets. J. Acoust. Soc. Am., 64 (1978), 517-521. 4. Chien, C. F. & Carroll, M. M., Sound above a rough absorbent plane, J. Acoust. Soc. Am., 67 (1980), 827-829. 5. Urick, R. J., Principles of Underwater Sound, McGraw-Hill, New York, 1975. 6. Hothersall, D. C. & Simpson S., The reflection of road traffic noise; J. Sound Vibr., 90 (1983), 399-405. 7. Chew, C. H., Prediction of traffic noise from expressway Appl. Acoust., 28 (1989), 203-212. 8. Kuttruff, H., Room acoustics. Applied Science Publishers, second edition, London, 1979. 9. Gradshteyn, I. S. & Ryzik, I. M., Table of Integrals, Series and Products, Academic Press, New York, 1975.