2.5-DIMENSIONAL FINITE-DIFFERENCE TIME- DOMAIN ANALYSIS AND ITS APPLICATION TO ROAD TRAFFIC NOISE PROBLEMS. Shinichi Sakamoto 1, Rino Otani 2

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1 nvited Paper.5-MNONL NT-RN TM- OMN NLY N T PPLTON TO RO TR NO PROLM hinichi akamoto, Rino Otani ffiliation: nstitute of ndustrial cience, The University of Tokyo OYH orporation sakamo@iis.u-tokyo.ac.jp rino@iis.u-tokyo.ac.jp bstract method of calculating transient responses in a three-dimensional (3-) sound field with a constant sectional geometry from a - solution obtained by the finite-difference time-domain analysis, based on uhamel s theor is presented. The method was applied to a road traffic noise problem and a building acoustic problem. s a road traffic noise problem, noise radiation characteristics from semi-underground road structures were calculated and they were compared with scale model experiments. Regarding a building acoustics, noise attenuation due to eaves/louvers attached on a building façade was investigated. rom the above examinations, an applicability of the presented calculation method to noise control problems was confirmed. Keywords:.5-dimensional finite-difference time-domain analysis, semi-underground road, building façade ntroduction s an efficient method of calculating sound pressure in a field with uniform geometry and characteristics along its length, uhamel showed a method in which a three-dimensional (3- ) sound field for a point source is obtained from two-dimensional (-) solutions using a ourier-type integration []. This technique is useful for predicting the effectiveness of various types of noise barriers under a point source, and incoherent line source conditions and several applications have been examined [-4]. s a method of obtaining - solutions, the frequency-domain boundary element method (M) is usually used. When using this method, - solutions at a large number of discrete real and imaginary frequencies should be prepared for the ourier-type integration in order to obtain a 3- solution at a target frequency. Therefore, the computational cost is relatively high. On the other hand, the - sound field can also be solved in the time domain. The finite-difference time-domain (T)

2 NTRNO JUN 3-6 LON PORTUL method has recently been applied to problems regarding outdoor noise propagation[5-7]. rom the solutions obtained by such a time-domain analysis, frequency components of the solutions can be easily obtained by the fast ourier transform (T). Therefore, once a transient response at a receiving point is obtained by time-domain analysis, its frequency components can be obtained with considerably low computational cost, and the resultant frequency components can be efficiently applied to uhamel s transformation technique from - to 3-. n this paper, such a calculation method using a time-domain solution is described at first and its applications to road traffic problems are introduced. Theory We suppose a 3- sound field with a point source and obstacles with rigid surfaces, of which the cross-sectional shapes are constant along their length, as show in ig.. n ig., Ω 3 and Γ 3 denote a 3- domain and 3- boundary with a rigid surface, Ω and Γ denote a - domain and a - boundary with a rigid surface, Ω is the boundary of a - fluid domain and denotes a point source. Under such a geometrical condition, uhamel showed the following transformation from a - to 3- solution []: Φ ( x z, k) = Φ ( x y k α ), + 3, j α z, e dα, () π where Φ 3 (x,z,k) is a 3- solution at position (x,z) in artesian coordinates with k as the wave number (=ω/c, ω is the angular frequency and c is the speed of sound), and Φ (x,k ) is a - solution with k as the wave number. The - solution, Φ, satisfies the following exterior problem equation and outgoing wave condition at infinity: Φ Φ, ( + k ) Φ = in Ω, + inc = on Ω j = H 4 ( k r) Φ, inc n n (-), (-) where Φ,inc is the incident sound and H (x) is the Hankel function of the first kind. Here, Φ,inc is the fundamental solution of the - Helmholtz equation, so the required solution in the - field, which is applied to q. (), is the solution for the sound source condition described as the fundamental solution. Ω 3 Γ 3 Ω y z Ω Γ x 3-dimensional geometry -dimensional geometry igure 3- sound field under consideration, Ω 3 being the 3- domain, Γ 3 the boundary of the 3- domain, and Ω and Γ the - domain and its boundary in a cross-sectional plane.

3 NTRNO JUN 3-6 LON PORTUL rom q. (), one can find that the solutions for the - domain at both real and imaginary frequencies are necessary. Here, q. () is expressed as z k) = Φ (3-) 3, + k α ) cosαzdα k = π Φ (3-) j α k ) cosαzd, Φ α k = (3-) π The integral term is obtained from solutions in real frequencies and is obtained from those in imaginary frequencies. Now, we assume a solution in the time domain, denoted by φ (x,t). Then, its real frequency component can be obtained by the ourier transform of the solution as follows., ( x k) = φ t) jωt Φ e dt (4) n practice, the solution in the time domain, φ (x,t), is numerically obtained as discrete data with a finite number of lengths defined at discrete points in time. ts real frequency components can be calculated by the T. On the other hand, let us consider the case of imaginary frequency. ubstituting jω into ω in q. (4) leads to Φ, ( x jk) = φ t) e ωt dt. (5) quation (5) means that the imaginary frequency components of Φ can be obtained by the Laplace transform of the transient solution. The integral operation of q. (5) must be performed numerically. Therefore, the integration transformation described by q. () can be calculated using the results of the ourier and Laplace transforms of a transient solution. n order to adapt the T results obtained from the transient solution, which is defined at discrete frequencies with constant intervals of Δf = f s /N, f s and N being the sampling frequency and the number of T points, respectivel to the integration procedure, q. (3-) is transformed through a change of variables, k α = β, as follows. k = Φ cos π β ) ( k β z) β dβ. (6) k β Here, we discretize the frequency axis of β in a constant interval of Δβ, and we assume that the frequency component included in a segment between β[i] and β[i+] is a constant value equal to the average of the frequency components at β[i] and at β[i+]. Then we obtain the following approximation of the integral term of. or z β[] i ) + Φ β[ i + ] ) M Φ == [] sin k β i z sin k β[ i + ] π i z or z = z = (7-) β[] i ) + Φ β[ i + ] ) { k β[] i k β[ i + ] } M Φ = π i= (7-) 3

4 NTRNO JUN 3-6 LON PORTUL n the same manner, the integral term of for imaginary frequency components can be approximated using the values calculated through the Laplace transform of the transient solution as follows. Here, note that the change of variables applied to q. (3-) is α k = β. or z jβ[] i ) + Φ jβ[ i + ] ) { sin( k + β[ i + ] z) ( k + β[] i z) } Φ = M sin π i= z or z = Φ β[] i ) + Φ β[ i + ] ) { k + β[ i + ] k + β[] i } = i= π (8-) (8-) n the formulation mentioned here, the frequency component for each segment, which ranges from β[i] to β[i+], is approximated as a constant, and therefore, a discretization error caused by this approximation can occur. The discretization error can be reduced by increasing the number of T points, N, in order to make the frequency resolution of Δβ sufficiently small. Using qs. (3-), (7-), (7-), (8-) and (8-), the 3- sound field solution in the frequency domain is calculated. fter the components at equally placed frequencies are obtained, the 3- sound field solution in the time domain can be obtained by the inverse ourier transform process. 3 pplication to Road Traffic Noise Problem or road traffic noise problems, a condition of noise source is usually modeled as cylindrical in energy base. uch an assumption is true when the sound propagation is treated in energy base, but when wave-based treatment is required, the assumption cannot be applied. The calculation method proposed in the previous section is efficiently utilized in such cases where wave phenomena should be taken into consideration. 3. Noise radiation characteristics from semi-underground roads s a means of mitigating the propagation of road traffic noise, depressed or semiunderground roads are often constructed in Japan. n such cases, it is difficult to predict the noise propagation from the road structures to the roadside areas, because noise propagation inside the structure is very complicated owing to multiple reflections and diffraction. or predicting noise radiation from a semi-underground road, in this stud the -dimensional finite-difference time-domain analysis is applied and the obtained solutions are transformed from - to 3- according to the proposed signal processing procedure. 3.. emi-underground Road under nvestigation straight semi-underground road with a symmetric cross section was considered as shown in ig.. eometrical parameters characterizing this type of road structure are the width of the road, R, width of the mouth, W, height of the underground part of the road structure, H, and the thickness of the overhangs, T. The value of H was made constant (= 5 m) in this study. 4

5 NTRNO JUN 3-6 LON PORTUL H: Height of underground part( =5 m) W: Width of mouth T: Thickness of overhang R: Width of road igure ross sectional shape of semi-underground road under investigation. 3.. T nalysis -dimensional T analysis was performed on the cross sectional shape of the semiunderground road shown in ig. 3. ound field inside a semi-underground road and scattering field with width of 5 m and height of 5 m were modeled for the analysis. round the scattering field, perfectly matched layers[8,9] were set in order to realize non-reflecting termination. s T calculation conditions, a discrete spatial grid size and a time step interval were made.5 m and. ms, respectively. n order to see directivity characteristics of sound radiation from a mouth part of a semi-underground road, receiving points in 3-dimenaional space were distributed on a quarter spherical surface of m radius as shown in ig. 4. R/4 z PML 5 m θ x 5 m r = m T igure 3 etting of domain for - T analysis. W R 5 m igure 4 eometrical setting of a semi-underground road, sound sources and receiving points. y W R ϕ T : : ource point x 5

6 NTRNO JUN 3-6 LON PORTUL 3..3 nalysis of alculation Results n the transformation from - to 3-, the number of T points was 54,88 ( 9 ) and consequently the frequency resolution for the integration became.9 Hz. ased on the obtained 3- impulse responses, directivities of -weighted sound pressure level assuming road traffic noise as the sound source are examined. n the calculation, -weighted single event sound exposure level was calculated from the 3- impulse response, under the condition where -weighted sound energy level of a sound source was d. To simulate road traffic noise, the following spectral characteristics was applied. f ΔL RTN ( f ) = log +, (9) 5 where, ΔL RTN is relative correction term, f is a frequency. The function of frequency shown in q. (9) is specified as spectral characteristics of vehicle noise in J RTN-Model 8[] omparison with xperimental Result The calculated sound radiation characteristics were compared with experimental results obtained with a / scale model in order to validate the applicability of the calculation. igure 5 (a) and 5 (b) show comparison between calculation and experiment for two cases with different geometries. n these figures, single event sound exposure levels on a receiving surface when the sources had a d sound energy level are shown in a form of net graph. We can see that the radiation directivity varies with the difference of the sectional shape. n all cases, the directivity in x-y plane is sharp in upper and oblique directions, whereas that in the longitudinal section of y-z plane is rather gentle. greement between calculation results and experimental results are fairly good. 6 xperiment 6 6 xperiment PL alculation PL PL alculation PL PL PL R=, T=, W= PL igure 5 omparison between calculation and a scale model experiment PL R=, T=, W= 5 R=, T=, W=5 R=, T=, W= 3. Noise shielding effects of eaves/louvers attached on building façade s a second example of the applications of.5-dimensional T analysis to noise problems, noise shielding effects of eaves/louvers attached on a building façade is 6

7 NTRNO JUN 3-6 LON PORTUL introduced here. everal types of eaves/louvers are attached to building façade for the aim of mitigating thermal load by solar radiation as shown in ig. 6. rom the viewpoint of noise control, such eaves and louvers are expected to provide noise reduction effects. o such effects were analyzed using.5-dimensional T method. 3.. onditions of façade under investigation To apply the calculation method described in the former chapter, a three dimensional sound field having a homogeneous cross-section consisting of a hard flat ground, a perpendicular building wall and eaves/louvers attached on the building wall was assumed in this study as shown in ig. 7 (a). n numerical analysis, a cross-section including receiving points was modeled in -dimension as shown in ig. 7 (b) and the sound field was analyzed by the T method. n the ig. 7 (a) and 7 (b), x s is a distance between a building wall and a lane of sound source, z s is a depth of the source in z-direction, H s is height of a target room and h c is a story height. n this stud x s and h c were assumed to be m and 4 m, respectively. Three kinds of eaves /louvers were selected : flat eaves, incline eaves and louvers were assumed to be attached on the building wall as shown in ig. 8. s receiving points, 7 points ( to in ig. 8) were distributed on the surface of the wall. y R z h c =4m y Perfectly Matched Layer z H R x x ssumed model (3 ) alculation model ( ) igure 6 n example of (a) 3-dimensional model (b) -dimensional model used eaves attached on a building assumed in this study for numerical analysis igure 7 alculation model e e 3 cm 5 cm Receiving point Type : lat Type : nclined Type : Louvers igure 8 Variation of eaves/louvers θ 5 cm x x 7

8 NTRNO JUN 3-6 LON PORTUL 3.. nalysis The noise shielding effects were evaluated by an insertion loss of -weighted overall level with frequency range between 35 Hz and khz in /3 octave bands. lso in this calculation, the spectral characteristics of vehicle noise expressed by q. (9) were assumed Parametric study on length of eaves ifference of noise shielding effects due to the length of the eaves, e, was investigated for Type and. for Type, angle of the inclination, θ, was made to be 45 degrees. The lengths of m, 6 cm and cm were investigated under the condition of z s = m here. alculation results for H R =6 m (corresponding to 5 th floor) and H s =56 m (corresponding to 5 th floor) are shown in ig. 9 (a) and 9 (b). or both cases, noise shielding effect of Type is larger than that of Type and the effects become high as the lengths of the eaves become longer. The insertion loss at receiving points and is positive and large, which is caused by shielding effect by eaves attached under the stor whereas the insertion loss at higher receiving points of, and is negative due to the reflection from the eaves attached upper the story. or Type on 5 th floor, relatively large effects of 8 d for 6 cm and over d for m length can be seen. y a short eave with length of cm, however, the noise shielding effect is not so large although the effect does not become negative. lat eaves cm 6cm cm ncline eaves cm 6cm cm lat eaves cm 6cm cm ncline eaves cm 6cm cm nsertion loss nsertion loss nsertion loss nsertion loss (a) Hs=6 m (5 th floor) (b) Hs=56 m (5 th floor) igure 9 alculation result on difference of noise shielding effect due to eave length 3..4 Parametric study on story height Under the condition of e = m, difference of noise shielding effect due to story height was investigated. The story heights were varied in five cases as 4 m ( nd story), 6 m (5 th story), 36 m ( th story), 56 m (5 th story) and 76 m ( th story). alculation results for z s = m and z s =96 m are shown in ig. (a) and (b). On the whole, the effects of Type are much larger than those of Type and. or Type on the stories higher than th stor large effects over 8 d can be seen Parametric study on inclination angle of Type eave rom the investigations described above, it is indicated that an inclined eave such as Type can be effective for noise reduction against road traffic noise. However 45 degrees of inclination angle may not be favorable from the viewpoint of architectural design. o in the 8

9 NTRNO JUN 3-6 LON PORTUL next investigation, inclination angle was varied and noise reduction effects for respective angles were calculated. The calculation results at a receiving point of under the condition of z s = m are shown in ig.. n the figure, insertion loss is arranged in relationship with incidence angle of noise from the source. rom the result, for the conditions where the incident angle is larger than 65 degrees, noise attenuation effects over 5 d are seen for the cases of the inclination angle over 5 degrees. On the other hand, little effect can be obtained when the incident angle is small, even if the inclination angle of the eave is made large. 4 ummay n this paper, a method of calculating 3-dimensionnal sound field with a homogeneous geometry in a longitudinal direction based on a -dimensional solution for the cross section using the T method was firstly described and two applications of the proposed method on road traffic noise problems were introduced. The proposed method can be efficiently used for prediction of road traffic noise and examination of countermeasure against road traffic noise when the effect of wave phenomena such as multiple diffraction cannot be ignored on the wave propagation. 5 5 lat eaves ncline eaves Louver nsertion loss nsertion loss (a) z s = nsertion loss nsertion loss nsertion loss (b) z s =96 m igure alculation result on difference of noise shielding effect due to story height nsertion loss lat eaves ncline eaves Louver 9

10 NTRNO JUN 3-6 LON PORTUL ncline eaves ncident sound ncident angle nsertion Loss 5 5 ngle of eaves: ~ ncident angle [deg] igure alculation result on difference of noise shielding effect due to inclination angle of Type eave. References []. uhamel, fficient calculation of the three-dimensional sound pressure field around a noise barrier, J. ound Vib., 97, (996). [] P. Jean, variational approach for the study of outdoor sound propagation and application to railway noise, J. ound Vib.,, (998). [3] P. Jean, J. efrance and Y. abillet, The importance of source type on the assessment of noise barriers, J. ound Vib., 6, -6 (999). [4]. J. Martin and.. Hothersall, Numerical modeling of median road traffic noise barriers, J. ound Vib., 5, (). [5]. akamoto, T. eimiya and H. Tachibana, Visualization of sound reflection and diffraction using finite difference time domain method, coust. ci. & Tech., 3, (). [6] T. Yokota, Y. Hirao and K. Yamamoto, fficient calculation on outdoor sound propagation by T and P methods, coust. ci. & Tech., 7, (6). [7] T. Van Renterghen,. alomons and. ottledooren, Parameter study of sound propagation between city canyons with a coupled T-P model, pplied coustics, 67, (6). [8] J.-P. erenger, perfectly matched layer for the absorption of electromagnetic waves, J. omput. Phys., 4, 85- (994) [9] 5 Q. Qui and T. L. eers, valuation of the Perfectly Matched Layer for computational acoustics, J. omput. Phys., 39, (998) [] Kohei Yamamoto, Road traffic noise prediction model J RTN-Model 8 : Report of the Research ommittee on Road Traffic Noise, coust. ci. & Tech., 3, - 55 ()