Line source and site characterizations for defining the sound transmission loss of building facades

Journal of Sound and Vibration (1983) 91(3), 403424

LINE SOURCE AND SITE CHARACTERIZATIONS FOR DEFINING THE SOUND TRANSMISSION LOSS OF BUILDING FACADES F. F. RUDDER, JR Building Acoustics Group, Center for Building Technology, National Bureau of Standards, Washington, D.C. 20234, U.S.A. (Received 28 June 1982, and in revised form 1 March 1983)

An analytical model is presented for defining the sound transmission loss of building facades exposed to noise from line sources. The model describes the non-diffuse sound field incident upon the facade in terms of both source and site parameters. The effects of facade orientation relative to the line source and the sound propagation with distance are introduced as a single term in the definition of the facade sound transmission loss. This term defines a mean angle of incidence for the exterior sound field that is equivalent to a point source location relative to a point on the facade. Numerical results are presented illustrating th, magnitudes of these effects and it is shown that alternative methods for conducting fie‘5hmeasurements of building facade sound transmission loss may be related by using this model. 1. INTRODUCTION

The characterization of the sound transmission loss of building facades is one of the concerns in the design of the building envelope. A difficulty associated with both the prediction and the field measurement of the facade sound transmission loss centers upon the characterization of the exterior sound field incident upon the facade. This paper presents a method for characterizing the sound field incident upon a building facade in which both source and building site parameters are incorporated in the definition of the facade sound transmission loss. This definition is based upon the exterior sound power and the interior sound power on each side of the facade wall. By using this definition, the analysis described in this paper parallels the analytical basis for defining the laboratory measurement of the sound transmission loss of partitions [l, 21. Many important sources of environmental noise may be characterized by a line noise source. For example, highway traffic noise prediction models are based upon line noise sources. Also railway and aircraft noise prediction models may be based upon, respectively, either a linear or a point noise source moving along a straight line path. References [3-61 are only a few of the examples of the vast literature describing such models. Due to this practical importance, the line source model is used, in this paper, to define the sound transmission loss of the building facade. To characterize the exterior sound field incident upon a building facade, it is also necessary to include certain site parameters. In this paper two specific site parameters are considered: site geometry and the possible effect of sound propagation incorporating excess distance attenuation. These idealized building site parameters are examined so that emphasis may be placed upon evaluating overall site conditions such as facade orientation and distance from the line source as they relate to the facade sound transmission loss. 403 0022460X/83/230403+22

$03.00/O

@ 1983 Academic Press Inc. (London) Limited

404

F.

F. RUDl>k.K,

.IK

Keferences [7-l 51 are representative of the literature addressed to the topic of building facade sound transmission loss. Donavan et al. [7] presented a thorough review of this literature and a discussion of field measurement problems. This paper is an extension of the above-referenced research. The present analysis incorporates both site geometry and excess distance attenuation as a single term in the definition of the facade sound transmission loss. The analysis is used to define a point source location that is equivalent to the line source. The present model is also used to interpret alternative field measurement methods for time-varying noise sources. This interpretation relates the time-varying condition and the steady or continuous condition. Hence, the measurements of the facade wall sound transmission loss with a fixed loudspeaker source [ 11, 12, 141, a continuous flow of roadway traffic [9. 10. 131, or a moving point noise source [7, 8, lo] are essentially equivalent measurements insofar as the data from each condition can be interrelated and compared.

2. ASSUMPTIONS 2. I.

AND SITE GEOMETRY

ASSUMPTIONS

The present analysis is based upon simplifying assumptions related to the,source, the sound propagation, and the facade parameters. The environmental noise source is a finite length straight line segment composed of an infinitely large number of incoherent omnidirectional point sources radiating into free space. Each point source is of equal strength so that the line source strength per unit length, W, is constant. All linear dimensions are large relative to the acoustic wavelength. Hence, details concerning the nature of the wave motion may be neglected. The air is characterized by the density p and the speed of sound c and is assumed to be homogeneous. Due to propagation losses, a function E(r) is defined that describes distance attenuation of the acoustic intensity in excess of the geometric spreading (proportional to r-*) of the acoustic radiation from a point source. The facade is a smooth plane surface exposed only to sound radiating directly from the line source. Diffraction of sound around the edges of the facade is ignored. Since the exterior sound field is characterized by the acoustic radiation incident upon the facade, the sound transmission loss is defined in terms of the incident intensity. Reflection or scattering of the sound field from the facade plane, the ground surface, or other surfaces is not included in the analysis. 2.2.

SITE GEOMETRY

Alternative descriptions of the site geometry are illustrated in Figure 1. These alternative descriptions are used to define the angle of incidence, 0, at a point on the facade relative to a point on the line source. As indicated in Figure 1, the facade may be considered to be a vertical plane with the line source located in an imaginary horizontal plane. The facade plane is oriented at an angle 0 (measured in the horizontal plane) relative to the line source. For the facade perpendicular to the line source, the orientation angle is @ = 0. For the facade parallel to the line source, the orientation angle is @ = 7r/2. A point on the line source is denoted by the co-ordinate 5 with the origin as indicated in Figure 1. The distance between a point on the line source and a point on the facade is denoted by r. Finally, the point on the line source, &, that is, the intersection of the facade plane with the line source (or the extension of the line source), represents the physical limit that is used to calculate the acoustic radiation from the line source. This

A DEFINITION

OF BUILDING

FACADE

(01

Figure 1. Nomenclature angular co-ordinates I$.

405

TL

(bl

for site geometry

and facade

orientation:

(a) for rectilinear

co-ordinates,

.$; (b) for

limit is specified since diffraction of sound around the physical edges of the facade is ignored. 2.3. ANGLE OF INCIDENCE To calculate the incident acoustic intensity at a point on the facade, it is required to express cos 0 in terms of the line source co-ordinate 5 and the other site parameters. For the nomenclature of Figure l(a), the relationship is c0s@=(~*-~)c0s@/r,

(C&,

(la)

where & = D tan @ and r* = R2 + [*. For the nomenclature of Figure 1(b), the corresponding relationship is cosO=cosysin@cos~-cos@sin$~, where 4 = tan-’ (t/R),

f$s4*,

(lb)

C#J* = tan-’ ( t2/ R) and cos y = D/R.

3. ACOUSTIC INTENSITY AND EXTERIOR SOUND POWER LEVELS In order to determine the sound transmission loss of an element of the wall structure in a facade, the exterior sound power on the surface area of the wall must be estimated. 3. I. ACOUSTIC INTENSITY LEVEL The detailed derivation of the relationships among the acoustic intensity level, the incident sound pressure level and the site parameters is presented in Appendix A. Since it is assumed that all linear dimensions are large relative to the acoustic wavelength, the acoustic pressure and particle velocity are assumed to be in phase and the incident sound field may be characterized by plane waves. The form of this relationship is (see Figure 1 and equation (A13)) (a list of nomenclature is given in Appendix B) L,(R)=L,(R)+lOlog[cos

01

dB reI,,,.

(2)

In equation (2), the term cos is physically interpreted as the mean value of the cosine of the angle of incidence [9]. The rationale for this interpretation is evident from the analysis contained in Appendix A. The functional form for cos is given in equation (A14) and represents a spatial mean or average value defined by an integration over the length of the line source exposed to the facade.

406

I-. F. RUDDtK.

Figure

3.2. EXTERIOR

SOUND

2. Facade

POWER

wall area,

.iI<

S,.. and site geometry

LEVEL

Figure 2 illustrates a facade exposed to a line source. The facade wall area, S,, represents the exterior wall area of a room and is indicated by the shading in Figure 2. The exterior sound power at the facade wall is obtained by integrating the acoustic intensity over the wall area, S,.. Due to the complicated functional form for the incident intensity, it is necessary to conduct this integration numerically. However, from the mean value theorem for surface integrations it is always possible to determine a point within the area S, so that the following relationship holds: 10 L,(K’)/r() dS, ~ SwIOL,(RVIO,

(3)

5, When R is assumed to be known or can be approximated, the exterior sound the facade wall .is expressed in terms of the incident sound pressure level as W, = lrcfSlv cos 0 lOr-~~(~”I” If the facade is exposed to several calculated using the expression

line noise sources,

W, = lrefSw C (cos 0 10LJx)“(‘)

power on

watts. the exterior

watts,

(4) sound

power

may be

(5)

where the summation is over all line sources. The approximation indicated in equation (5) is a result of the assumed value of R for estimating the integration of the acoustic intensity over the area S,. Since L,(R’) does not vary rapidly over the surface, this approximation will be used in the remainder of the discussion. However, the results will be presented as equalities rather than with the = symbol. The utility of equation (5) is evident in Figure 3. Although equation (5) is functionally equivalent to Lewis’ result [9], the present analysis allows the user to consider line sources with oblique orientation relative to the plane of the building facade. Lewis’ results are limited to the cases of facades either parallel or perpendicular to the line source [9]. Figure 3(a) illustrates a plan view of a building site with a curved highway alignment approximated by a series of straight line segments. Figures 3(b) and (c) illustrate that each facade will be exposed to a different set of line source segments. Inspection of Figure 3 shows that in some instances subsegments of a finite length line segment must be considered in the computation of the acoustic intensity contribution using equation (5). This requirement is a result, when using the present model, of ignoring diffraction of sound at the edge of the facade.

A DEFINITION

Figure 3. Definition of facade A; (c) line sources

line sources exposed exposed to facade B.

OF

BUILDING

to a facade:

FACADE

407

2-L

(a) plan view of site; (b) line sources

exposed

to

For a single line source, the exterior sound power level on the building facade wall is obtained by using equation (4) as L,(R)

= L,(R)+10

log[cos O]+lO log (S,&,/

W,,,)

dB re Wrer.

(6)

Equation (6) expresses the exterior sound power level in terms of the incident sound pressure level, the facade wall area, S,, and the building site conditions incorporated in the cos 0 term. This expression and the functional expressions for cos 0 described below are the central results of the analysis. 4. FACADE SOUND TRANSMISSION LOSS The facade sound transmission loss is defined in terms of the incident sound power on the facade exterior and the transmitted sound power on the facade interior. By assuming a diffuse interior sound field, the transmitted sound power may be expressed in terms of the average interior sound pressure level, J$,, and the total sound absorption of the receiving room, A. The functional relationship among the parameters defines the facade sound transmission loss: TL= lOlog[W,/AI,,,]-L,+6

dB.

(7)

This definition is used with the appropriate expression for the facade exterior sound power, W,, to obtain explicit functions for the facade sound transmission loss in terms of the incident sound pressure level, the site parameters included in cos 8, and the receiving room parameters. 4.1. FACADE EXPOSED TO A SINGLE LINE SOURCE For a facade exposed to a single line source, the sound transmission loss is obtained by using equations (4) and (7), and is defined as TL=L#)-L,+lOlog[cosO]+lOlog(S,/A)+6

dB.

(8)

408

F-. I‘.

KUDDt.K.

IK

The incident exterior sound pressure level, L,,t K 1. and the mean value of the angle of incidence, cos 0, depend upon the site conditions as discussed in section 3.2. Expression (8) for TL is commonly stated in terms of the exterior sound pressure level measured in the presence of the facade surface. This is accomplished bv redefining the exterior sound pressure level and deleting or modifying the 6 dB constant-in equation (8) to approximate surface reflections and scattering of the incident sound field. Such adjustments are not discussed here but may be found in references [7. 9. 1 1. 121. 4.2. t ACADE

EXPOSED

1‘0 \kVERAL

LINE

SOUK(‘hS

For a facade exposed to several line sources, the sound transmission using equations (5) and (7) to obtain the definition TL-1010g[~(cos010’~~~‘K”“‘)]-~,,+1010g(S,,/A)+6

loss is derived

dB,

by

(9)

where the summation is over all line sources. This result applies to a single point on the facade characterized by a wall of area S,. For composite facade walls such as window-wall systems, this result may also be used provided that the wall areas of the individual components and the average transmitted intensity for each component are defined. Further, since in the present analysis diffraction, scattering, and reflection at the edge of the facade are ignored, the predictions thus obtained may not be accurate for corner rooms.

5. MEANING

OF THE TERM

10 log [cos 01

In the previous sections functional relationships among the exterior sound power, the incident sound pressure level, and the building site parameters have been described. The building site parameters are implicitly incorporated into the term cos 0. In this section the potential significance of the building site parameters as related to both the measurement and the prediction of the facade sound transmission loss is discussed. Results developed in Appendix A are used. 5.1.

MAGNITUDE

OF THE

TERM

10 log [COS @]

The general functional form for cos is given by equation (A14). Explicit functional forms are obtained by substituting the appropriate excess distance attenuation function, as defined by equations (A.5) through (A7), and performing the necessary integrations. Three attenuation functions are considered: geometric attenuation, equation (A5); power law attenuation, equation (A6); and exponential attenuation, equation (A7). In the case of geometric attenuation the intensity varies inversely as the square of the distance from a point source (proportional to rP2). For the power law attenuation model the intensity is assumed to be proportional to rC(2+a), where (Y is a dimensionless number. In the U.S.A., a = 0.5 is a typical value used to model excess distance attenuation for highway traffic noise studies [3, 4, 161. For the exponential attenuation model the intensity is assumed to be proportional to exp(-qr)/r*, where r) is a constant with dimensions (length)-‘. The corresponding attenuation constant, expressed in dB per unit length, is 10 log(e)T. The exponential model may be used to simulate, for example, scattering of the sound waves as they propagate away from the source [17]. The parameters CYand n are, of course, frequency dependent. In Appendix A expressions are presented for the sound pressure level attenuation with distance away from both a point source, equation (A9); and a line source, equation (Al 1). Since numerical examples will be used to indicate the magnitude of the term 10 log [cos 01, it is worthwhile to illustrate the sound pressure level decrease with distance for the values (Y= 0.5 and n =0.001/m (0.0043 dB/m or 4.3 dB/ 1000 m) and to compare

A DEFINITION

OF BUILDING

FACADE

409

TL

these with the geometric attenuation model results. (The values for (Y and 77 are only representative values used for the purpose of illustration.) Figure 4(a) shows the decrease in sound pressure level with distance, AL,(r), for a point source. A reference distance, Z? = 15 m, is selected for illustration. As expected, the geometric model predicts attenuation at a constant rate of 6 dB per doubling of distance and the power law model predicts attenuation at a constant rate of 7.5 dB per doubling of distance for (Y= 0.5. The exponential model predicts attenuation between these two curves for the value 77=0*001/m. Figure 4(b) shows similar results for an infinite length straight line source. As expected, the geometric attenuation model predicts attenuation at a constant rate of 3 dB per doubling of distance.

G 2 T

-20

I

-

’ ‘I”“1 (b)

-

90

a” -30

-

-40

-

I IO

100

Distance

from

1000

point source, r

I

iI,,,,

IO

Cm)

I

1

I

I

I I I 1I \\.

100 Dtstonce

from

1000 line

source,

R (ITI)

Figure 4. Relativesound pressure level versus distance for (a) a point source, R = 15 m and (b) an infinite length line source, R = 1.5 m. -, Geometric attenuation; ---, power law attenuation, a =0.5; ---, exponential attenuation; 7 =0.001/m.

5.1.1. Geometric attenuation For the case of geometr’ic attenuation, the explicit functional form for cos 0 is obtained by using equations (A14) and (A15). The result is cos 0 = [sin (&/$][cos

y sin @ cos &--cos @ sin $1.

(10)

This result applies to a finite length line source with the geometry illustrated in Figure 2. For an infinite length line source, the result may be simplified by substituting the values d;,= -r/2and qS2=tan-’ (cos y tan @) into equation (10). These substitutions yield the result cos@=(G+co~-ysin@)/(&+~/2).

(11)

The geometry defining the parameters Q, and y is illustrated in Figures 1 and 2. This result is presented in Figure 5(a) as a plot of 10 log [cos 01 as a function of the facade orientation angle @ with the elevation angle y as a parameter. For @ = O”, the facade is perpendicular to the infinite length line source and equation (11) reduces to cos 0=2/rr.

(12)

For @ = 90”, the facade is parallel to the infinite length line source and equation (11) reduces to cos 0 = (2/ 7T)cos y.

(13)

410

F. F. RUDDER,

-5

.JK

-

-10 0 -

-5

(bi

-

-10 0 -

y:150

y=oo

(cl

/-!

1

y’75” -5 -

/ 7130’

r=60° \ H

+I50

K//

A.

1

y=600

;:I

\ y=O”

y =75”

Y

-10

.I

///

,90

I//

,

-60

I

,

,

-30 Facade

0

onentotlon

1

30

1

I

60

I

1 I 90

angle,0 (degrees)

Figure 5. 10 log [cos 81 as a function of facade orientation angle @ and elevation angle y: (a) geometric attenuation; (b) power law attenuation, a = 0.5; (c) exponential attenuation, ?R = 0.50.

The results presented in equations (12) and (13) are identical to Lewis’ results [9]. The effect of the elevation angle, y, is most pronounced when the facade is parallel to the line source. For moderate elevation angles (y G 45”), the magnitude of 10 log [cos O] varies between - 1 and - 4 dB over the rather wide range of orientation angles, - 45” s @ =z90”.

A DEFINITION

OF BUILDING

FACADE

411

TL

51.2. Power law and exponential attenuation For the case of an infinite length line source, equation (A14) was numerically evaluated for both the power law attenuation model and the exponential attenuation model. Figure 5(b) shows the result for the power law model for the case a = 0.5. The general shape of the curves shown in Figure 5(b) is similar to the shape of the curves presented in Figure 5(a). For any fixed values of @ and 7, the values of 10 log [cos 61 are within 1 dB for these two attenuation models. Figure 5(c) shows the numerical evaluation of equation (A14) for the exponential model with qR = 0.5. As an example with the value q= 0*001/m, Figure 5(c) corresponds to a facade located at a distance of R = 500 m from the line source (see Figure 4(b)). The curves of constant elevation angle in Figure 5(c) are similar in shape to the curves shown in Figure 5(a) and (b). For any fixed values of @ and 7, the values of 10 log [cos 01 for the exponential attenuation model are within l-2 dB of the value estimated by using the geometric model. As discussed in Appendix A, the integrations required to evaluate cos 0 for the power law and the exponential attenuation model may be expressed in terms of tabulated functions. When the building facades are either perpendicular or parallel to an infinite length line source, the expressions for cos 0 are explicit functions of the attenuation parameter. For facades perpendicular to an infinite length line source (@ = 0’) and the power law attenuation model, one uses equations (A14) and (A16) to obtain

&-&

(2/T) J-r iid r (??/Wl.

For the exponential attenuation equations (A14) and (A17):

model the corresponding

(14)

result is obtained by using

cos 0 = E2( nR)/Ki,( TR).

(15)

These results may be used to estimate the effect of excess distance attenuation for facades perpendicular to an infinite length line source. Figure 6(a) shows a plot of 10 log [cos 01 versus a for the range 0 =Z(Ys 1 as obtained by using the result of equation (14). For CY = 0, equation (14) reduces to equation (12) as one would expect. Figure 6(b) shows

1

!

1

(b) -10

Figure 6. 10 log [cos 81 as a function of excess distance to an infinite length line source: (a) power law attenuation;

log (2/r)

attenuation parameters for facades (b) exponential attenuation.

perpendicular

t;. k.

412

RUDDER.

.IK

the corresponding result obtained by using equation ( 15) for the range of values 0 s VR 5 2.0. For 71=O, equation (15) also reduces to equation (12). The general trend indicated by Figure 6 is that consideration of excess distance attenuation decreases the value of the facade sound transmission loss by l-2 dB relative to the estimate from equation (12). Similar results are obtained when the building facade is parallel to an infinite length line source. Using the results in Appendix A and the power law attenuation model, one obtains the expression

For the exponential

attenuation

model,

the corresponding

result

is

cosO=~Rcosy[{K,(~R)/Ki,(~R)}-11.

(17)

Figure 7 shows plots of 10 log [cos O/cos r] as a function of CYand qR, respectively. These were obtained by using equations (16) and ( 17). In this case, the consideration of excess distance attenuation increases the predicted facade sound transmission loss by l-1.5 dB relative to the corresponding estimate from equation (13).

Figure 7. 10 log [cos @/cos y] as a function of excess distance attenuation parameters to an infinite length line source: (a) power law attenuation: (h) exponential attenuation.

for facades

parallel

Based upon the numerical results presented in Figures 4(b)-7, it appears that for design purposes the magnitude of cos 0 can be obtained by using the simple expression given by equation (10) to estimate facade orientation effects. However, one may still use the excess distance attenuation models as discussed above and in Appendix A to estimate the incident sound pressure level at the exterior surface of the building facade. Equations (AlO) and (All) of Appendix A provide the basis for these estimates.

6. EQUIVALENT

POINT

SOURCE

LOCATION

The discussion in section 5 was concerned with the magnitude of the term cos 0 and the potential significance of facade orientation and excess distance attenuation on the magnitude. The physical interpretation of cos 0 is that it represents a mean or average angle of incidence for the exterior sound field. This section presents one result of this physical interpretation of cos 0. The result is based upon the analysis contained in Appendix A and defines a point source location equivalent to a line source. The

A DEFINITION

OF BUILDING

FACADE

413

7Z

equivalency is based on the difference between the two exterior intensity levels such that the difference is independent of the site geometry. The difference between the line source and the point source intensity levels is expressed in terms of the incident sound pressure levels by using equations (A12) and (A13). The result is LI (R) - L,(r)

= Lp( R) - L,( r,,) + ld log [cos O/cos O,]

dB,

(18)

where the subscript 0 denotes a point source. By substituting into this equation expressions from equations (A8), (AlO) and (A14), the intensity level difference can be expressed in terms of the source sound power levels and the site parameters as L,(R)-L,,(r,)=Lm-L,,+lOlog

ri j cos OE(R

set 4) d+

RRE( r,)cos OO

’

(19)

where all terms are defined in Appendix A and the limits on the integration are defined by the length of the line source as illustrated in Figure 1. From equation (19), it is clear that the difference in the intensity levels at the facade for the line source and the point source will be constant provided that the expression in braces equals unity. For the case of geometric attenuation, a very simple physical interpretation results. Substituting equations (A15b) into equation (19) and setting the terms in braces equal to unity, one obtains r$cos G/cos O,] = RR/(2

sin 6).

(20)

By selecting the point source location (r,, 0,) so that this equality is preserved, one obtains r~=I?R/(2sintj),cosOo=cosO=cosysin~cos~-cos@sin~.

(21a, b)

The angle of incidence of the equivalent point source, Oo, is established by the facade orientation angle, @, the elevation angle, y, and the angle 6 as indicated in equation (21b). Physically, 4 is the angle between the reference line of length R (see Figures 1, 2 and 8) and the line connecting the midpoint of the line source to the point on the facade. The application of equation (21) to the field measurement of the facade sound transmission loss is evident-the site geometry relating the facade orientation relative to the line source is directly related to the equivalent point source location. The equivalent point source location, defined by equation (21), is illustrated in Figure 8. This location may be related to the point source co-ordinates defined for the standardized field measurement of facade sound transmission loss [ll, 121. By using the present method, however, one can relate the facade sound transmission loss to a line source, such as a highway, to that of an equivalent point source at a location based on the site characteristics.

__--

___a

_--. __--

-.__ --._ ___--- _-: t I

I I ! I

I I

_J __--___--Figure 8. Equivalent point source location (rO, 0,)

(comparewith Figures

1 and 2).

414

F.

F. RUDDER,

JK

If the facade is exposed to several line sources, then the above results may be used to define a point source location equivalent to each line source. For field measurement of the facade sound transmission loss, the sound power level of each point source must be established relative to the sound power level of the corresponding line source. The field measurement would then be conducted with all point sources operated simultaneously. For the case of an infinite length line source, equation (21) may be further simplified to express the equivalent point source location (r,,, O,,) in terms of the site parameters R, y, CDand the reference distance R. In this case, the integration limits are 6, = - 7r/2 and & = tan-’ (cos y tan @) and equation (21) reduces to r~=RR/[Jl+co~~~*Gl-COSTS], cos0,=~[(cosysin@+cos@)Jl+cos~

(22a)

Z* (cos y sin @ - cos @)J’l - cos &I,

(22b) where the + sign is used if C&> 0 and the - sign is used if & < 0. For a facade perpendicular to th_e infinite length line source (@ = O’), the equivalent point source location is r, = (RR/J2)“’ and 0, = 45”. For a facade parallel to the infinite length line source ( @ = 90”), the eavalent point source location is r. = (RR/2) “* and O. = y. Figure 9 is a plot of r,,/Jl?R as a function of the facade orientation angle, 0, with the elevation angle, y, as a parameter. This result is based upon equation (22a). Figure 10 is a plot of O. as a function of CDwith y as a parameter and is based upon equation (22b). Figure 11 shows plots of the angle C$ as a function of @ with y as a parameter for a facade exposed to an infinite length line source. Inspection of Figures 9 and 10 yields a physical insight as to the nature of the equivalency as expressed by equation (22). For a facade orientation approaching grazing incidence (CD+ -9O”), the equivalent point source simulation is achieved by letting O. + 90” (as expected) and r, + co (a result that may not be evident). Of course, for Qi = - 90”, the facade is on the opposite side of the building from the line source and the incident intensity is zero. This is achieved, based on the above simulation, by requiring that the equivalent point source be located at an infinite distance from the facade in addition to using a grazing angle of incidence.

7. TIME-VARYING

NOISE

SOURCES

The above results have been derived by assuming that the line source is characterized by a constant sound power output per unit length. This section describes the corresponding results for time-varying noise. The line source, in this case, represents a straight line path. The model simulates the time-varying incident sound field by defining point noise sources that move along the straight line path with constant speed. Each moving source represents an incoherent omnidirectional point source with a steady or time-invariant source strength. In the case of time-varying noise, the facade sound transmission loss is then defined as TL=(AL)+101og[cos0]+101og(S,/A)+6

dB.

(23)

All other terms are as defined in equation (8). Equation (23) is based upon the following two assumptions: a diffuse interior sound field and an instantaneous response of the interior sound field to the incident exterior sound field. As described below, the time averages are integrations of the time-varying intensities. Hence, if the response time of the room is less than the time rate-of-change of the exterior sound field and if the integration time is long compared to the response time, the assumption of an instantaneous interior sound field is justified. By comparing equations (8) and (23), it is seen that the only difference is the interpretation of the term cos 0 for time-varying noise in relation to the appropriate time average (AL).

A DEFINITION

I

-90

OF BUILDING

415

7X.

I

I

I

I

1

I

-60

-30

0

30

60

90

Facade

Fieure 9. Plot of r,/~& “’ equ ayion (22a)).

FACADE

as a function

ormtotlon

of facade

1

orientation

I

I

angle,

@

angle

I

I

(degrees)

@ for an infinite length

1

I

y=O"

Figure 10. Plot of equivalent point source line source (see equation (22b)).

Figure Il. Functional length line source.

relationships

angle, @ (degrees)

angle of incidence,

Facade

orlentotlon

among the equivalent

=?Y

y=60°

,/

Focode orlentotlon

(see

1

I y

line source

angle,

@a, as a function

@ (degrees

of @ for an infinite

length

1

point source co-ordinates

x @, and 6 for an infinite

416

F. F. RUDDER,

7.1.

SINGLE

MOVING

POINT

JR

SOURCE

An omnidirectional point source of constant sound power output, W,,, moves along a straight line path at a constant speed, V. The speed. V. is assumed to be small relative to the speed of sound, c. The path-facade geometry is illustrated in Figure 1. By introducing the change of variable [= Vt or C$= tan-‘( Vt/R), where t denotes time, the incident mean square sound pressure is obtained by using equation (Al) and the normal component of the intensity is obtained by using equation (A3). These time-varying quantities are to the “spatial” integration integrated over the time interval t, 4 r G tz which corresponds over the interval 6, c C#J d 6 *, where (b, = tan’ (Vt,/R). In the context of the previous analysis, the time integration is limited to the time interval over which the facade is exposed to the moving point source. The integration is thus normalized by dividing by the integration time, T = t7- f,. By carrying out the above steps, the time average of the normal component of the intensity at a point on the facade is obtained as (Z,,(t)/Zref)“COS

0 10LJ’(‘,

(24)

where

1

‘2 L,,=lOlog [I

fl

b?iWde,l W(fz- tJ >

and cos is given by equation (A14). The equivalent exterior sound pressure level, Leq, is the incident sound pressure at the point on the facade averaged as indicated above. The time average of the interior sound pressure level, L_, is measured simultaneously with the exterior level. The overbar denotes that the interior sound pressure level is also averaged over the volume of the room. In this case, the appropriate time-average level difference, is given by the expression (AL)=L,,-Ee,

dB.

(25)

Since the averaging time, T, is identical for both L,, and Les the result Fukunishi’s time integration method for measuring the sound transmission facades [lo]. 7.2.

CONTINUOUS

STREAM

OF

MOVING

POINT

corresponds to loss of building

SOURCES

A continous stream of moving point sources corresponds to a uniform stream of roadway traffic moving along the straight line path at an average travel speed, V. The term “uniform stream” denotes a constant spacing between vehicles. In this case, the cos 0 term appearing in equation (23) is defined on the basis of the site geometry and is independent of time. The time-varying noise may be characterized by an equivalent incident sound pressure level, Leq, defined by the relationship

3

dB.

The appropriate time-average level difference, (AL), is formally expressed as indicated in equation (25). However, it is emphasized that the averaging time, T, appropriate for the continuous stream of moving point noise sources is independent of the site geometry relating the path (source location) and the facade location. The spatially averaged interior with the exterior level. equivalent sound pressure level, Le,, is measured simultaneously

A DEFINITION

OF BUILDING

FACADE

TL.

417

FOR FIELD MEASUREMENTS 7.3. IMPLICATIONS Emphasis has been placed in the preceding discussion on the relationship between the appropriate L,, measurement and cos 0 for two different time-varying noise sources: a single moving point source and a continuous stream of moving point sources. The relationship appropriate for each of these types of time-varying noise sources implies certain conditions upon the field measurement of the facade sound transmission loss when the environmental noise at the site is used as the source. In this section these implications for such field measurements are discussed. For measurements with a single omnidirectional moving point source, the integration time, T, determines both L,, and cos 0. If the path is straight along its entire length, the relationship among the integration time, Leq, and cos is determined by the site geometry. If the path is curved, however, then one may approximate the path by a sequence of straight line segments. In this case, the integration time must be subdivided with each time interval, Ti, corresponding to the L,, measurement when the point source is moving along the path characterized by cos Oi (see equation (9)). If the source is highly directional, then the instantaneous intensity must be integrated as described by Fukunishi [lo]. In either event, the instrumentation and/or data reduction must provide an average over a time interval established by the instantaneous source location and the site geometry. For mesasurements with a continuous stream of moving point noise sources, further complications arise. As discussed above, the averaging time, T, is independent of the site geometry. However, the cos 0 term in equation (23) is established on the basis of only the line paths to which the facade is exposed. As a result, direct field measurement of the incident exterior level, Leq, can be realized only for the site geometry of a facade parallel to a single infinite-length line source. For all other site geometries comprising a non-parallel facade orientation, the building (once constructed) will shield the facade to some extent. As a result, the incident L,, cannot be measured directly prior to construction and can only be inferred from field measurements after construction since surface reflections and local scattering alter the sound field at the facade. From consideration of the expressions for the sound transmission loss given by equation (9), it appears that postconstruction field measurements are the more appropriate technique even though one must adjust the data to account for surface reflections and scattering [7,9]. Finally, it may be possible to utilize the equivalent point source locations described in section 6 as the basis for the development of in situ sound transmission loss measurements. This approach would apply only to a “continuous stream” source model such as highway traffic noise. Equation (21b) suggests one way to select the angular orientation so that the field measurement relates to the site geometry. This angle then corresponds to the angle of incidence used in standardized measurement methods [ll, 121. However, as suggested by equation (21a), it is also required to place the point source at an “equivalent” distance from the facade. This angle of incidence and distance are based upon establishing an equivalent exterior sound field at the facade. Of course, this approach must be validated by using field measurements. To the author’s knowledge, such data are not currently available to evaluate the suggested relationship.

8. CONCLUSIONS The analytical model described in this paper quantifies the effect of some building site parameters on defining the sound transmission loss of facades. This idealized model is based upon assumptions that must be examined when applying the results to a specific site. The most significant assumption, in this regard, is the perfect shielding provided by

418

F’.F. RUDDER.

.IK

the vertical edge of the facade. The effect of local reflections and scattering at a point on the facade are not incorporated into the model. The importance of local reflections and scattering is recognized; however, the consideration of these effects is a topic described in references [7-9. 14, 151. The model however serves to demonstrate the significance of the cos 0 term in characterizing the incident sound field generated by a line source and defining the facade sound transmission loss. The major conclusion based upon the present analysis is that the effects of both line source-facade geometry and excess distance attenuation on facade sound transmission loss may be incorporated into a single term: cos 0. Based upon the numerical results presented in the paper, the magnitude of the term 10 log [cos 01 is of the order of, or less than, the effects attributable to local reflections and scattering at the facade [7]. This conclusion is expected since it has been confirmed by the measurements reported by Gilbert [14] and corresponds to the observations made by Schultz [lS]. However, the model defines a prediction method that can be used to develop a consistent field measurement method for the sound transmission loss of building facades exposed to a line noise source. Another result of the model is the definition of a point source location equivalent to a line source where the equivalence is based solely upon the exterior sound field. This result may be used to select the loudspeaker location for facade sound isolation measurements as defined by current [l l] and proposed [12] standards. Since the validity of this equivalence can be established only by future field measurements, no conclusions relative to the significance of this analogy can be drawn at the present time. Finally, the model provides a physical interpretation of the relationship between source and building site parameters for time-varying noise. This interpretation defines a relationship between the appropriate measurement of the time-varying sound pressure levels and the cos 0 term for defining the facade sound transmission loss.

REFERENCES 1. IS0 STANDARD 14O/III, 1978, Acoustic-Measurement of Sound Insulation in Buildings and of Building Elements-Part III, Laboratory Measurements of Airborne Sound Insulation of Building Elements. 2. ASTM STANDARD E90-75, Standard Method for Laboratory Measurement of Airborne Sound Transmission Loss of Building Partitions. 3. T. M. BARRY and J. A. REAGAN 1978 FHWA Highway Traffic Noise Prediction Model FHWA-RD-77-108. Washington, D.C.: U.S. Department of Transportation, Federal Highway Administration. 4. F. F. RUDDER, JR, D. F. LAM and P. CHUENG 1979 User’s Manual, FHWA Level 2 Highway Traffic Noise Prediction Model STAMINA 1.0 FHWA-RD-78-138. Washington, D.C.: U.S. Department of Transportation, Federal Highway Administration. 5. ANON. 1967 A Study of the Magnitude of Transportation Noise Generation and Potential for D.C.: U.S. Department Abatement, Volume 4, Train System Noise OST-71-1. Washington, of Transportation, Office of the Secretary. 6. D. E. BISHOP, A. P. HAYES, N. H. REDDINGIUS and H. SEIDMAN 1977 Calculation of Day-Night Levels (Ldn) Resulting from Civil Aircraft Operations EPA-555/9-77-450. Washington, D.C.: U.S. Environmental Protection Agency, Office of Noise Abatement and Control. 7. P. R. DONAVAN, D. R. FLYNN and S. L. YANIV 1980 Highway Noise Criteria Study: Outdoor/Indoor Noise Isolation NBS Tech Note 1113-2. Washington, D.C.: U.S. Department of Commerce, National Bureau of Standards. 8. S. LJUNGGREN 1972 Sound Insulation of Windows with Respect to Traffic Noise H-3065-A. Gothenburg, Sweden: Ingemassons IngenjGrbyrb AB. 9. P. T. LEWIS 1974 Journal of Sound and Vibration 33,127-141. A method for field measurement of the transmission loss of building facades.

A DEFINITION OF BUILDING FACADE I-L

419

10. T. FUKUNISHI and T. YAMAMOTO 1975 Infer-noise, Sendui 351-354. Field measurement of sound insulation of houses by the integral of sound energy. 11. IS0 STANDARD 140/V 1978 Acoustic-Measurement of Sound Insulation in Buildings and of Building Elements-Part V: Field Measurements of Airborne Sound Insulation of Facade Elements and Facades. 12. ASTM DRAFT PROCEDURE 1979 Field Measurement of Airborne Sound Insulation of

Building Facades and Facade Elements 7th Draft (E-33: 03-E). 13. R. MAKAREWICZ 1980 Acustica 46, 325-329. Equivalent level (L,,) of noise inside an enclosure generated by sources moving outside. 14. P. GILBERT 1972 An Investigation of the Protection of Dwellings from External Noise through et Technique du Batiment. (Translated from the Facade Walls. Paris: Centre Scientifique French in National Bureau of Standards Technical Note 710-2, Washington, D.C.: U.S. Department of Commerce, National Bureau of Standards.) 15. T. J. SCHULTZ 1979 Applied Acoustics 12, 231-239. Variation of the outdoor noise level and the sound attenuation of windows with elevation above the ground. 16. Y. Y. MA and F. F. RUDDER, JR 1978 Statistical Analysis of FHWA Traffic Noise Data Base FHWA-RD-78-64. Washington, D.C.: U.S. Department of Transportation, Federal Highway Administration. Book 17. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill Company. See p. 140. 18. M. ABRAMOWITZ and A. STEGUN (Editors) 1965 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Math Series 55. Washington, D.C.: U.S. Department of Commerce, National Bureau of Standards, Third Printing. Book 19. L. L. BERANEK (Editor) 1971 Noise and Vibration Control. New York: McGraw-Hill Company. See pp. 281-287. 20. B. H. SHARP, P. K. KASPER and M. L. MONTROLL 1980 Sound Transmission Through WashBuilding Structures-Review and Recommendations for Research, NBS-GCR-80-250. ington, D.C.: U.S. Department of Commerce, National Bureau of Standards.

APPENDIX

A: EXCESS

DISTANCE

ATTENUATION

In this appendix the detailed steps used to develop the functional relationships discussed in the main text concerning excess distance attenuation are presented. The assumptions used in this development are presented in section 2 of the main text and will not be repeated here.

A. 1. ATTENUATION MODELS In the paper three idealized distance attenuation models are used in estimating the sound transmission loss of building facades exposed to line noise sources. Each of these models is quantified by defining a function, E(r), that represents attenuation of the acoustic intensity with distance in excess of ideal geometric spreading of the waves. At a large distance from a point source, the intensity may be approximated as that of a plane wave and is lo(r) =pi(r)/pc=

( W,,/47r)E(r)/r2

watt/m2,

(Al)

where W,, is the source sound power (watt), r is the distance (meters) and the subscript 0 denotes a point source. For a line source, the acoustic intensity is expressed as I(R)

where W is the source sound power (watt/meter) and r2 = R2 + &* (meter2). clature for equation (A2) is defined in Figure 1. The integration indicated

642) The nomenin equation

420

P. F.

RUDDER.

.IK

(AZ) is over the length of the line source. In each model, the source sound power radiates into free space with the resulting factor of 47r. The acoustic intensities given by the above expressions are relative to the direction i for a point source and l? for the line source. With the facade introduced into the analysis, one must then obtain expressions for the acoustic intensity relative to the normal to the facade plane at the point of interest. With O. denoting the angle between the normal to the facade plane and the incident plane wave direction of propagation, the acoustic intensity for a point noise source may be expressed as I,,,,(r) = ( wt,/477-) cos @,,E(r)lr-

’ watt/m’,

(A3)

where cos O,, =cos y sin @ cos &--cos CDsin c#J,,,and the subscript On denotes the normal component for the point source. The nomenclature for equation (A3) is defined by equation (Al) and Figure 8. For a line noise source, the acoustic intensity relative to the normal to the facade plane may be expressed as

L,(R) =G

w

I

cos @E(r)

d.$

r2

watt/m’,

where cos 0 = cos y sin @ cos C#J -cos @ sin 4, r7 = R’+ 5’ = R’ sec2@, and the subscript n denotes the normal component for the line source. The nomenclature for equation (A4) is defined by equation (A2) and Figure 1. The integration limits for equation (A4) are, in general, different from the limits for equation (A2). In the case of equation (A4), the integration is over the length of the line source exposed to the facade surface as indicated in Figure 1. Equations (A3) and (A4) are the intensity components of the incident sound field normal to the facade plane. functions discussed in the The function E(r), for the three excess distance attenuation main text, geometric, power law and exponential, is defined, respectively as E(r) = 1,

E(r) = (R/r)‘“,

(ASA7)

E(r)-exp[-v(r-R)].

The distance l? introduced in these equations is an arbitrary reference distance less than r. These attenuation models are now used to obtain explicit functional expressions for both the sound pressure level and the acoustic intensity level in terms of the source and site characteristics. A.2.

SOUND

PRESSURE

LEVEL

The sound pressure level variation with distance from the source and the other source/site parameters is obtained from equations (Al) and (A2) for the point source and the line source, respectively. For the point source, the sound pressure level may be expressed as -L,(r) = L++- 20log(r)+lOlog[E(r)]-lOlog(47r)

dBrep,,,,

where LQ= 10 log ( W,/ W,,,) dB re Wrer. In terms of the sound distance R from the point source, the point source pressure expressed as

L,,(r) = L,,(R)

+ AL,,(r)

(Ag)

pressure level at a level may also be

dB re prrf,

where AL,,(r) = 20 log (R/r) + 10 log [E(r)]. The appropriate attenuation incorporated in the above results by using equations (A5)-(A7).

(A9) model may be

A DEFINITION

OF BUILDING

FACADE

421

7-L

For the line source, the sound pressure level may be expressed as L,(R)=&-lOlog(RR)+lOlog

[I

E(R set 4) d4

I

- 10 log (4r)

dB rep,,,, (AIO)

where L,+ = 10 log ( Wl?/ Wret) dB reWref. Expressed in terms of the sound pressure level at a distance R from the line source, an alternative form of the sound pressure level is L,(R)=L,(R)+&(R)

dB rep,,,,

(All)

where AL,(R)

= 10 log (E/R)

+ 10 log

[I

E(Rsecb)d+]-lOlog[~E(Rsec~)d~].

In equations (AlO) and (All), the integrations are over the length of the line source as The appropriate attenuation model may be defined for the variable C#J = tan-’ (t/R). incorporated in the above results by using equations (A5)-(A7). In the case of geometric attenuation, the integrations can be expressed in terms of the arctangent function. For either the power law model or the exponential model, however, the integrations can only be expressed in terms of tabulated functions. These results are discussed below. A.3. INCIDENT INTENSITY LEVEL The component of the incident intensity in the direction normal to the plane of the facade is given by equations (A3) and (A4) for the point source and the line source, respectively. From equations (A3) and (A8), the incident intensity level for the point source is L,,(r) =L,(r)+lOlog(cos

0,)

dB re Iref,

(AI2)

where L,(r) is given by equation (A8), and cos O0 is given in equation (A3). From equations (A4) and (AlO), the incident intensity level for the line source is expressed as L,(R)=L,(R)+lO log[cos 01 dB reIref, (AI3) where L,(R) is given by equation (AlO). The mean or average value of the cosine of the angle of incidence, cos 0, is defined by the relationship 6 cos 0 =

I-6

42 cos OE(R

set 4) d4

=cos ysin@cos4-cos@Z$,

/I

E(R

set 4) dd

61 (AI4)

where cos 0 is defined in equation (A4) and cos C$and sin have obvious definitions. The integration limits appropriate for use in equations (A13) and (A14) are indicated in Figure 2. The appropriate attenuation model may be incorporated into the above expressions. However, other than in the case of geometric attenuation, the required integrations lead to tabulated functions as described below. A.4. EVALUATION OF INTEGRALS In this section the evaluation of the integrals appearing in equations (AlO) and (A14) for the cases in which explicit results are obtained in terms of tabulated functions is discussed.

F. F. RUDDER.

422

JK

Geometric attenuation

A.4.1.

For geometric

attenuation.

all necessary

integrations

are readily

conducted

to obtain

“2 E(R

~~~j7/2GCj1<(6~c-7T/2.

set 4) d4 = 24,

(AlSa)

I d*, d_? cos OE(R set 4) d4 = 2 sin $cos I- dD, wherecos@=cosysin

@cos$-cos

@sin1$,&=((6;~-&,)/2,and

0,

(A15b) 4=(&+$,)/2.

Power law attenuation

A.4.2.

For the power law attenuation model, the integration may be expressed in terms of the incomplete beta function [ 18, p. 9441. As discussed in the text, only specific conditions of a facade either perpendicular or parallel to an infinite length line source result in explicit functional relationships. The required integrations are 77/z

I

(cos 4)” d+ =&I

o

(F)/zr(Y),

(Al6a)

V/Z sin+(cos4)^dg=I(F)/21.(+)=&,

(Al6b)

where I’(x) is the gamma function. For other facade orientations or for a finite length line source, one must numerically evaluate the integrals for the appropriate value of cr.

Exponential attenuation

A.4.3.

For the exponential attenuation model, the integration required to evaluate equation (AlO) is the Sievert integral and is tabulated [18, pp. lOOO-lOOl]. The integrations required to evaluate the numerator of equation (A14) must be determined numerically. As discussed in the text, for facades perpendicular and parallel to an infinite length line source, all of the integrations may be expressed in terms of tabulated functions. For these cases, one uses the results Tr/7 cos C$exp [ - x set 41 d4 = Ki,(x)F(x), I0

r/z exp[-xsec$]d4=Ki,(x), I0 rrl2

sin C#Jexp [ - x set ~$1d4 = Ez(x),

(Al7a-c)

i0 whereF(x) = x[{K, (x)/Ki, (x)} - 11. In equation (Al 7a), Ki, (x) is the first iterated integral of the Bessel function KC1 and is tabulated [18, pp. 492-4931. The function F(x) is evaluated in terms of Ki,(x) and the Bessel function K,(x) [18, pp. 417-4221. The function E*(X) is an exponential integral [ 18, pp. 245-2481. For x + 0, the functions F(x) and E2(x)/Ki,(x) each approach the limiting value of 2/ 7r (see Figures 6 and 7).

APPENDIX

B: NOMENCLATURE

Definitions A

total sound absorption

C

speed of sound in air, m/s distance illustrated in Figure

D

of receiving 1, m

room,

m2

A DEFINITION

E(r) EArI F(x) G2 Zref

b”(f) K,(x) Ki,(x) L,(R)

G,,(r) L,(R)

LP

LJ r) L,(R) LW L WC, L_v L AT

AL,(R) AL,,(r) P(t)

P”(l) Pref R R’

R r

rn SW T TL V W WI WO Wref cos

0

cos

0

cos

0,

cos

0

a

Ux)

0

5 517 P 0

52

OF BUILDING

FACADE

7-L

423

excess distance attenuation functions: equations (A5), (A6), (A7) an exponential integral: equation (Al7c) function defined in equation (A17b) cos2 @ + cos2 y sin’ @ reference intensity, = lo-r2 watt/m’ normal component of incident intensity at the facade for a single moving point source, watt/m’ a modified Bessel function first iterated integral of the Bessel function K, acoustic intensity level at a point on the facade plane for a line source, dB acoustic intensity level at a point on the facade plane for a point source, dB incident sound pressure level at a point on the facade plane for a line source, dB spatial average interior sound pressure level, dB incident sound pressure level at a point on the facade plane for a point source, dB exterior sound power level at the facade wall for a line source, dB sound power level of a line source referenced to the distance I?, dB sound power level of a point source, dB the equivalent sound pressure level, dB: equation (24) the spatially averaged interior equivalent sound pressure level, dB the instantaneous difference between the time-varying incident exterior sound pressure level and the interior sound pressure level, dB change in sound pressure level relative to the reference distance I? for a line source, dB change in sound pressure level relative to the reference distance I? for a point source, dB time-varying sound pressure, N/m2 sound pressure of a moving point noise source (time-varying at the receiver), N/m2 reference sound pressure, = 2 X 1O-5 N/m2 distance illustrated in Figures 1, 2 and 8, m variable distance defining line source geometry to points within the facade wall area SW,m a constant reference distance, m distance from a point source or from a point on a line source to a point on the facade plane, m equivalent point source distance from the facade, m facade wall area of receiving room, m2 averaging time for Leq, s time, s sound transmission loss of facade wall, dB constant speed of a moving point noise source, m/s sound power per unit length of line source, watt/m total exterior sound power of line source at facade wall of area S,, watt sound power of point source, watt reference sound power, = lo-l2 watt defined by equation (2) defined by equation (4) cos y sin @ cos & - cos @ sin & cosysin@cos+cos@sin4 parameter defining the excess distance attenuation for the power law model, equation (‘46) the gamma function elevation angle illustrated in Figures 1, 2 and 8 parameter defining the excess distance attenuation for the exponential model, equation (A7) angle of incidence of plane wave measured from normal to the facade, Figures 1, 2 and 8 distance measured along line source, Figure 1, m limits defined by integration over length of the line source, Figures 1 and 2, m density of air, kg/m3 facade orientation angle relative to the line source, Figures 1, 2 and 8

424 $Y,d;, $ *

F. F. RUDDER.

JR

angular orientation of fixed exterior point source, Figure 8 angles defining the integration over the length of the line sources, (4*+&w (A--+I)/2

Subscripts I 0” On P

Symbols z S ( )

denotes denotes denotes denotes denotes

acoustic intensity a component normal a point source a component normal sound pressure

approximately defined by time average

equal to

to the facade plane for a line source to the facade plane for a point source

Figure

2