l. godinho, j. antonio & a. tadeu fourier transforms.l. godinho, j. antonio & a. tadeu...

The scattering of 3D sound sources by rigid

barriers in the vicinity of tall buildings

L. Godinho, J. Antonio & A. TadeuDepartment of Civil EngineeringUniversity of Coimbra, Portugal

Abstract

The Boundary Element Method (BEM) is used to evaluate the acoustic scatteringof a three-dimensional sound source by an infinitely long rigid barrier in thevicinity of a tall building. The barrier is assumed to be non-absorbing and thebuildings are modeled as an infinite barrier. The calculations are performed inthe frequency domain and time signatures are obtained by means of InverseFourier Transforms.

The 3D solution is obtained by means of Fourier Transform in the directionin which the geometry does not vary. This requires solving a sequence of 2Dproblems with different spatial wavenumbers, k^. The wavenumber transform in

discrete form is obtained by considering an infinite number of virtual pointsources equally spaced along the z axis. Complex frequencies are used tominimize the influence of these neighboring fictitious sources.

Different geometric models, with barriers of varying sizes, are used. Thereduction of sound pressure in the vicinity of the buildings is evaluated and thecreation of shadow zones by the barriers analyzed.

1 Introduction

Diffraction-based methods have been used by many authors as a tool to analyzesound propagation in the presence of obstacles. Lam[l] introduced one suchmethod for the calculation of the acoustic energy loss produced by the insertionof simple, finite length three-dimensional acoustic barriers. This work was laterextended by Muradali & Fyfe[2] to include the modeling of two-dimensionalgeometries

Boundary Elements XXII, C.A. Brebbia & H. Power (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-824-4

524

Precision can be improved by using numerical methods like the BoundaryElement Method or the Finite Element Method to solve the wave-equation foreach frequency to be considered. Based on the theory of slender bodies, Filippi& Dumery[3] and Terai[4] developed a boundary integral equation technique toanalyze the scattering of sound waves by thin rigid screens in unboundedregions. This method was subsequently extended by Kawai and Terai[5] to allowthe prediction of sound attenuation by rigid barriers over a totally reflectiveground surface. Lacerda et al.[6] proposed a dual boundary element formulationfor analyzing the two-dimensional sound propagation around acoustic barriers,over an infinite plane, in which both the ground and the barrier were consideredto be absorptive. The three-dimensional propagation of sound around anabsorptive barrier was studied by Lacerda et al.[7], introducing a dual boundaryelement formulation that allowed the barrier to be modeled as a simple surface.

In the present work, besides the influence of the acoustic barrier, the presenceof very large buildings next to it is taken into account, and the pressure fieldgenerated by wave scattering at both objects is calculated using a standardBoundary Element Formulation. Both the acoustic barrier and the building areconsidered to be totally reflective.

In our model, the acoustic barrier is considered to be of infinite length, whilethe acoustic source takes the form of a point load. This situation is usuallyreferred to as a two-and-a-half-dimensional problem, for which solutions can beobtained by means of a spatial Fourier transform in the direction in which thegeometry does not vary (Tadeu & Godinho[8]).

The BEM model is used to compute the three-dimensional pressure fieldgenerated by a point pressure source in the vicinity of a rigid barrier placedbetween the source and a tall building, which is treated as an infinite rigidvertical plane surface. Simulation analyses are performed to investigate wavepropagation in the vicinity of such buildings in the presence of neighboringvarying-sized acoustic barriers. Both frequency and time domain responses areobtained to permit a quantitative study of the 3D eects of the scattering.

2 Problem definition

The pressure field generated by a harmonic point load in a spatially uniform fluidmedium is governed by

in which co is the oscillating frequency, (XQ , 0, 0) is the position of the load, the

subscript inc denotes the incident field, A is the wave amplitude, a is the

pressure wave velocity of the medium, and i = V - 1 .


Boundary Elements XXII 525

Fourier-transforming eqn 1 in the z direction, and using the effective

wavenumbers, k^ =J — --6^ with Imk^ <0, where k^ is the axialVa

wavenumber, one obtains

(2)

in which the //„'(...) are second Hankel functions of order n.

If one assumes the existence of an infinite set of evenly-spaced sources alongthe z direction, the former incident field may be written as

where L is the spatial source interval, and k^ = — m . Thus, the three-LI

dimensional pressure field may be obtained as the pressure irradiated by a sum ofharmonic (steady-state) line loads whose amplitude varies sinusoidally in thethird dimension. This sum converges and can be approximated by a finitenumber of terms.

The problem to be solved considers a spatially uniform acoustic mediumbounded by two perpendicular flat surfaces: one simulates the horizontal flatfloor, while the other models the tall building. A vertical rectangular rigidobstacle (acoustic barrier) is placed inside this medium. The pressure field(Green's function G(X, XQ,CD)) can then be computed by the following

expression, when the vertical plane and the horizontal plane are defined byx = 0 and y = 0 , respectively:

^ AG(x,Xo,4=E

;=1 8Pin which NS = 4,

3 Boundary Element formulation

Given the extensive literature currently available, the details of the BEMformulation required for the type of the scattering problem presented here areomitted. It is nevertheless important to state that each two-dimensional problemrequires the evaluation of the integral


526 Boundary Elements XXII

(5)i

in which H *' is the pressure velocity component at x^ due to pressure load at

x_i and HI is the unit outward normal for the /"* boundary segment C/ . The

required pressure velocity function is obtained by differentiating eqn (4) inrelation to the unit outward normal.

The required integrations in eqn (5) are performed by means of Gauss-Legendre quadrature, using six integration points. Since the acoustic barrier ismodelled as a thick object, numerical integrations on elements close to ordirectly facing each other are performed using a higher order Gauss-Legendrequadrature.

The scattered pressure field at any point inside the acoustic medium can thenbe calculated in relation to the previously calculated nodal pressure values.

4 Pressure in time-space

Responses in the time domain are calculated by means of an inverse fast Fouriertransformation in co, using an acoustic source, with temporal variation given by aRicker pulse:

wfT)=An-2T^g""^ (6)

in which A is the amplitude, r=(t-t,, )/t$ and t represents the time, with t,

being the time when the maximum occurs, while m$ is the dominant wavelet

period. By applying a Fourier transformation to this function, one obtains:

[/( w ) = A[2,J-'"*' ] # V (7)

where Q, = cotQ / 2 .

The Fourier transformations required are achieved by means of a summationof a finite number of terms, either in frequency or wavenumbers. This isequivalent to adding periodic sources at spatial intervals of L = 2n / Ak^ and

temporal intervals of T = In/Aco . In these expressions, Ak^ and Act) represent

the wavenumber and frequency increment, respectively. To prevent thecontamination of the response by the periodic sources, the spacing between themmust be large enough to ensure that their contribution arrives at times later thanT . This is achieved by shifting the frequency axis slightly downwards in thecomplex plane, using complex frequencies with an imaginary part of the form#c =O}-IT] (with rj=0.1Aco).

5 Validation of the BEM algorithm

The BEM algorithm was validated by taking a cylindrical circular rigid pipeplaced inside an unbounded homogeneous acoustic medium (a = 340 ra/s),


Boundary Elements XXII 527

illuminated by a harmonic point pressure load, for which the solution is knownin closed form (not illustrated). The required Green's function is obtained bysetting the NS parameter in eqn (4) to one.

Analysis of the results showed that the BEM approach for low frequencies isextremely accurate, and reveals only slight differences at high frequencies.

6 Numerical examples

The method described above has been used to study the influence of an acousticbarrier placed between a point sound pressure load and a very tall building. Thismodel simulates a possible point source in a highway in the vicinity of abuilding. In our examples, both the ground surface and the building wereconsidered to be non-absorbing, while the host acoustic medium had a pressurewave velocity of 340mA. A point acoustic source is placed 0.6m above theground and 25.0m away from a tall building. An acoustic barrier of height h isplaced 20.0m away from the building, to reduce the acoustic sound registered onits fagade. This geometry is represented in Figure 1.

20.0 m 5.0 mFigure 1: Geometry of the problem.

The acoustic barrier is modeled as a body 0.2m thick, and discretized usingan appropriate number of boundary elements, defined by the relation between thewavelength and the length of the boundary elements, which was set at & Thenumber of boundary elements used is never less than 32. To ensure accuracy,numerical integrations on elements close to each other are performed using ahigher order Gauss-Legendre quadrature.

For the first simulation, the response was evaluated at a grid of receivers,placed along a vertical plane 0.5 m away from the rigid wall, equally spaced at a

distance of 1.0m and 4.0m apart along the vertical and longitudinal directions,

respectively. The calculations were performed for frequencies ranging from 2Hzto 256%, with a frequency increment of 2//z, determining a total time windowresponse of 0.5j. The source time dependence is assumed to be a Ricker waveletwith a characteristic frequency of


528 Boundarv Elements XXII

0 20 40

h-Om

20.0m 5.0mFigure 2: Sound pressure level along a vertical plane 0.5m away from the

building.

Figure 2 presents the sound pressure level (10/0g/? / (2xlO~^) where p

refers to the maximum amplitude of the time responses Iculated for eachreceiver) obtained when there is no barrier. These plots use a gray scale wherelighter and darker shades are ascribed to higher and lower values respectively.The results show that the maximum sound pressure level field is not obtained atz =0.0m. This can be explained by the effect of the directly incident pulses

being added to those reflected on the building. This behavior disappears if thegrid of receivers is positioned at ;c = 0.0m (not illustrated). A decrease in the

general sound pressure level is noted as the distance of the receiver to the sourceincreases.

h=2m

h—4m

h=6m

Figure 3: Sound pressure along a vertical plane 0.5m away from the building:a) Pressure level, b) Pressure attenuation.

Figure 3a and 3b display the sound pressure level, and its attenuation when abarrier is inserted between the source and the building. For a barrier 2.0 m tall,

these results indicate that the barrier performs less well for receivers placed


Boundary Elements XX 11 529

closer to the ground, owing to the interaction of the direct field diffracted by thebarrier with that reflected by the ground floor. The performance of the barrierimproves as receivers are placed at a greater distance from the floor, reaching itsmaximum efficiency approximately 8 m above the ground for z - 0.0 m . Above

this value the barrier loses efficiency, even being outperformed by the casewhere there is no barrier, at greater heights above the ground. The performanceof the acoustic barrier is not constant along the z axis. A line following thepoints of maximum efficiency, for consecutive vertical z planes, is inclined,indicating better performances at receivers placed further above the floor as zincreases. It appears that the reflections on the ground close to the building,mentioned above, increase in importance as z increases.

As the height of the barrier increases from 2.0m, to 4.0m and then to

6.0m, the sound pressure level attenuation increases for the full domain of

receivers. The above-defined line of maximum efficiency is closer to the groundsurface and less inclined. It seems that the reflected waves on the ground close tothe building and the direct field diffracted by the barrier, lose importance forreceivers closer to the floor as the barrier increases in height.

Next, to better illustrate the propagation of the sound pressure from its sourceto the receivers, the time responses are presented for receivers placed atz = 0.0 m and z = 30.0m.

Figure 4a displays the time responses at receivers placed at z = 0.0 m (source

plane), for barriers of height 0.0 m , 2.0 m, 4.0 m and 6.0 m . These records in

the time domain exhibit a series of incident pulses and the result of theirreflections on the ground, wall and barrier. The results show that the first set ofpulses is recorded at later times at receivers closer to the ground, as taller barriersare used. Therefore, a pulse takes longer to travel from the source to the edge ofthe barrier and then to these receivers, as the height of the obstacle increases. Thearrival times of the different pulses agree with those calculated using the acousticray theory, allowing identification of the travel paths followed by the differentpulses shown in the response. It can further be observed that the amplitude of theresponse recorded at these receivers becomes smaller as a result of a diffractioneffect at the edge of the barrier. As described above, the efficiency of theacoustic barrier is seen to increase as its height increases.

A second pulse is clearly visible in the time responses just after the firstarrival when an acoustic barrier is present. This second pulse is clearly separatedfrom the first when the barrier is taller and the receivers are placed at greaterdistances from the ground. This second pulse is caused by a prior reflection fromthe rigid ground.

As time progresses, a third pulse appears when there is an acoustic barrierinserted between the source and the building. This pulse originates in the energytrapped between the barrier and the building, which allows secondaryreverberation effects to occur. Other pulses occur at later times (not visible in thetime window presented) as a result of this interaction between the barrier and therigid wall. The approximate time difference between these pulses, showing


530 BozWory

progressively lower amplitude as energy dissipates, is given by

40.0 m/340.0 ms^ = 1 17.6 ms .

h=0m

h-4m

h=6m

a) b)Figure 4: Time response at receivers 0.5m away from the building:

a) z = 0.0m. b) z = 30.0m.

Figure 4b illustrates the responses obtained for the same set of receivers forz = 30.0 m . The different pulses arrive at later times as they travel along longer

paths. However, the results exhibit similar features to the ones observed forz = 0.0 m . Comparison of the responses for z = 0.0 m and z = 30.0 m reveals

the existence of a very slight drop in amplitude. This can be explained by the


Boundai"v Elements XXII 531

effect of the directly incident pulses being added to those reflected on thebuilding, already mentioned above.

Analyses in the frequency domain were performed for the same geometries.Sound pressure levels are shown in figure 5, for the frequencies of 125 Hz and

10007/z . In the absence of the acoustic barrier, the pressure field results fromthe direct incident field interacting with that reflected by the floor and thebuilding. Thus, the total field is given by the sum of waves with different phasesleading to a spatially variable sound pressure level, distinguishable as a patternof darker and lighter zones. This phenomenon grows more complex as thefrequency increases.

a) b)Figure 5: Sound pressure along a vertical plane 0.5m away from the building:

a) Frequency of 125%. b) Frequency of 1000%.

Again, the performance of the barrier 2.0 m tall is poorer at receivers placed

closer to the floor when the excitation frequency is low. The reflected field onthe ground and the trapped energy between the barrier and the building areresponsible for this behavior. As the frequency increases, the barrier creates a"shadow" zone behind it, leading to a pronounced attenuation of the soundpressure field at the lower receivers. However, the influence of the reflections atreceivers very close to the ground is maintained.

The "shadow" created behind the barrier becomes more intense as the heightof the barrier changes from 2.0 w to 6.0m , and as the frequency increases from

125.0 Hz to 1000.0 Hz. This behavior is expected since higher frequency waves

have smaller wavelengths, and are easily influenced by smaller obstacles.

7 Conclusions

The analysis of the sound pressure level obtained over a plane parallel to thebuilding indicates that there is no uniform performance on the part of the


532 Boundary Elements XXII

acoustic barrier. The maximum efficiency of the barrier was observed atreceivers placed at higher distances from the floor as z increases. As the barrierbecomes taller the sound pressure level attenuation increases, for the full domainof receivers, and a maximum efficiency is found for receivers nearer the ground.

It was shown that the time arrivals of the different pulses at the receiversagree with the travel path of the incident pulses and their reflections on theground, wall and barrier. The results confirm that the efficiency of the acousticbarrier is poorer as the receiver gets closer to the ground due to the effect of theinteraction of the different pulses: diffracted by the barrier and reflected on theground.

Results obtained in the frequency domain confirm that the performance of alow barrier is poorer at receivers placed closer to the floor when the frequency islow. For higher frequencies the barrier creates a "shadow" zone behind it,leading to a pronounced attenuation of the sound pressure field at lowerreceivers. The efficiency of the barrier improves further if its height increases,creating a more intense "shadow" zone behind the barrier.

References

[1] Lam, Y. W. Using Maekawa's chart to calculate finite length barrier insertionloss. Appl Acoust, 42, pp. 29-40, 1994.

[2] Muradali, A. & Fyfe, K.R. A study of 2D and 3D barrier insertion loss usingimproved diffraction-based methods. Appl Acoust, 53, pp. 49-75, 1998.

[3] Filippi, P. & Dumery, G. Etude theorique et numerique de la diffraction parun ecran mince. Acustica, 21, pp. 343-359, 1969.

[4] Terai, T. On calculation of sound fields around three-dimensional objects byintegral equation methods, /. Sound Vib., 69, pp. 71-100, 1980.

[5] Kawai, Y. & Terai,T. The application of integral equation methods to thecalculation of sound attenuation by barriers, Appl Acoust, 31, pp. 101-117,1990.

[6] Lacerda, L.A., Wrobel, L.C. & Mansur, W.J. A dual boundary elementformulation for sound propagation around barriers over an infinite plane, J.&)tW Mb, 202, pp. 235-347, 1997.

[7] Lacerda, L.A., Wrobel, L.C., Power, H. & Mansur, W.J. A novel boundaryintegral formulation for three-dimensional analysis of thin acoustic barriersover an impedance plane, J. Acoust. Soc. Am., 104(2), pp. 671-678, 1998.

[8] Tadeu, A. & Godinho, L. 3D wave scattering by a fixed cylindrical inclusionsubmerged in a fluid medium. Eng. An. Bound. El, 23, pp. 745-756, 1999.


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