METHOD OF ACOUSTIC CALCULATION OF TRAFFIC NOISE BARRIERS

Installation of noise barriers is an efficient and relatively cheap way of protecting residential areas from traffic noise. This paper proposes a comprehensive method of calculation of noise barriers. The method allows determining the optimal sizes of the barrier meeting acoustic and cost requirements. The calculation method takes into account the noise coming from the both parts of the traffic flow shielded and not shielded by the barrier, as well as sound reflected from the road surface, the opposite barrier, if present, and both barriers. Introduction High noise pollution of territories close to traffic flows is a serious environmental problem. Despite the reduction of car noise by creating low-noise engines, more efficient mufflers [15], etc., the desired effect is not achieved, and the problem of noise reduction remains relevant. There are many ways of protecting building close to traffic flow from noise. For example, there is personal protective equipment (ear plugs, headphones) and collective protective equipment, including soundproof windows and the use of noise-absorbing materials. However, noise barriers (NB) is the most effective and cost-efficient way of protecting buildings from noise. This paper proposes a comprehensive method of calculating the optimal parameters of NBs. This method ensures the required NB efficiency at the lowest cost. 1 Analysis of works on the research topic and setting a research task Literature review of methods of design and simulation of NBs revealed that many researchers neglect a number of factors influencing the NB efficiency or focus solely on individual design aspects without taking into account, for example, the noise coming from the traffic flow not covered by the NB. Researches have to make assumptions to simplify the design problem. For example, some studies consider infinitely long screens [6, 7], others consider only point sources [8, 9], while some studies offer calculation methods that ensure sufficient accuracy in a limited frequency range [10]. There are works that take into account the limited barrier length and the loss of acoustic energy during the propagation of sound from traffic * Corresponding author: vvtupov@mail.ru © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 320, 00034 (2020) https://doi.org/10.1051/matecconf/202032000034 ASO-2020


Introduction
High noise pollution of territories close to traffic flows is a serious environmental problem. Despite the reduction of car noise by creating low-noise engines, more efficient mufflers [1][2][3][4][5], etc., the desired effect is not achieved, and the problem of noise reduction remains relevant. There are many ways of protecting building close to traffic flow from noise. For example, there is personal protective equipment (ear plugs, headphones) and collective protective equipment, including soundproof windows and the use of noise-absorbing materials. However, noise barriers (NB) is the most effective and cost-efficient way of protecting buildings from noise. This paper proposes a comprehensive method of calculating the optimal parameters of NBs. This method ensures the required NB efficiency at the lowest cost.

Analysis of works on the research topic and setting a research task
Literature review of methods of design and simulation of NBs revealed that many researchers neglect a number of factors influencing the NB efficiency or focus solely on individual design aspects without taking into account, for example, the noise coming from the traffic flow not covered by the NB. Researches have to make assumptions to simplify the design problem. For example, some studies consider infinitely long screens [6,7], others consider only point sources [8,9], while some studies offer calculation methods that ensure sufficient accuracy in a limited frequency range [10]. There are works that take into account the limited barrier length and the loss of acoustic energy during the propagation of sound from traffic flow to the estimated point (EP) [11,12]. Paper [13] proposed a method of determining the acoustic efficiency of a noise barrier and its dimensions and proposed recommendations on improving the NB construction.
Thus, there is a need to develop a comprehensive methodology that will meet all the requirements for the construction of noise screens and the calculation of its effectiveness. The proposed method takes into account the influence of the wind and atmosphere turbulence on sound propagation, interaction of acoustic waves with the surfaces of the territory and wood lines. Also, the simulation takes into account the noise coming to the protected territory from part of the traffic flow closed by the barrier and the parts not closed by it, as well as sound reflected from the road surface, the opposite barrier (if present) and both barriers.

Calculation of direct and reflected sound levels from the traffic flow in the estimated point with NB
To calculate the levels of direct and reflected sound from the traffic flow in the estimated point with NB, we use the results of studies presented in [14]. First, we calculate the level of direct sound that is created by the traffic flow at the estimated point: where is the noise characteristic of the traffic flow; Δ is the noise reduction by NB; ΣΔ is the sum of all sound energy losses on the sound propagation path from is source to the estimated point.
During propagation from the source to the estimated point, the noise is reduced due to the following factors: losses from the surface of the protected territory, losses in the air, losses due to wind, atmosphere turbulence, influence of wood lines, losses due to the limited barrier length and divergence of acoustic waves. The influence on the noise level at the estimated point of reflected sound can be taken into account by introducing three imaginary sources each of which creates a sound level in the estimated point: where Δ , = 10lg( ) is the noise reduction of the i-th imaginary source when the traffic noise is reflected from a reflecting object; is the sound reflection coefficient (for reinforced concrete barriers ≈ 0,97 [15]; for asphalt and asphalt-concrete road surfaces ≈ 0.955 [16]); Δ is the sum of acoustic energy losses during propagation of sound from the i-th imaginary source to the estimated point; ∆ NB, is the noise barrier reduction of sound from the i-th imaginary sources (equations for calculating Δ NB, are shown below).
As a first approximation, the sum of reflected sound losses (ΣΔ i ) can be assumed equal, it is denoted as Δ . The total sound level in the estimated point created by three considered imaginary sources is given by: The sound level in the estimated point created by the direct and reflected sound: Σ = 10log(10 0,1 + 10 0,1 , ).
The increase of the sound level in the estimated point due to reflected sound is given by: 3 Calculation of reduction by the noise barrier of the sound reflected from the road surface To calculate reduction by the noise barrier of the sound reflected from the road surface, we use the results of study presented in [14]. Calculation diagram for determining the difference of the path lengths of the sound ray δ during reflection from the road surface is shown in Fig.  1. Reduction of the sound level reflected from the road surface by the noise barrier [17] is given by: , = 18,2 + 7,8 lg(δ + 0,02), dBA, (6) where δ is the difference of path lengths of the sound rays, m: -the distance between the acoustic center of the imaginary noise source and the upper edge of the barrier, m: -the distance from the highest edge of the barrier to the estimated point, m: -the shortest distance from the acoustic center of the imaginary source to the estimated point, m: = √( 1 + 2 ) 2 + (ℎ 1 + ℎ 2 ) 2 ; (10) ℎ -noise barrier height, m; ℎ 1 -height of the estimated point above the surface of the territory, m; ℎ 2 -height of the NS acoustic center above the road surface, m; 1 and 2 is the distance from the NB to axis of the far lane and the estimated point, accordingly, m. 4 Calculation of reduction by the noise barrier of the sound reflected from the screen located on the opposite side of the road To calculate of reduction by the noise barrier of the sound reflected from the screen located on the opposite side of the road, we use the result of [14]. The calculation diagram is shown in Fig. 2. Similarly, to (6), we determine of reduction by the noise barrier of the sound reflected from the parallel noise barrier on the opposite side of the road. The difference of path lengths of the sound rays δ is given by: δ = ′ + ′ -′ ; (11) ' -the distance between the acoustic center of the imaginary noise source and the upper edge of the barrier, m: ' -the distance from the upper edge of the barrier to the estimated point, m: ' -the shortest distance from the acoustic center of the imaginary noise source to the estimated point, m: ′ = √(2 − 1 + 2 ) 2 + (ℎ 1 − ℎ 2 ) 2 (14) where 1 and 2 -are the distances from the noise barrier to the axis of the closest traffic lane and the estimated point, accordingly, m; L the distance between parallel noise barriers installed on the opposite sides of the carriageway of the road, m.

Calculation of reduction by the noise barrier of the sound reflected from the NB and the barrier located on the other side of the road
For calculation of reduction by the noise barrier of the sound reflected from the noise barrier and the barrier located on the other side of the road, we use equations presented in [14]. The calculation diagram is shown in Fig. 3.  (6), we calculate the noise barrier reduction of noise reflected from both parallel noise barriers. The difference of the path lengths of the sound rays δ is given by: = ′′ + ′′ -′′ ; (15) ′′ -the distance between the acoustic center of the imaginary sound source and upper edge of the barrier, m: ′′ = √(2 + 1 ) 2 + (ℎ − ℎ 2 ) 2 ; (16) ′′ -the distance from the upper edge of the barrier to the estimated point, m: ′′ -is the closest distance from the acoustic center of the imaginary source to the estimated point, m: 1 and 2 -is the distance from the NB to axis of the far lane and the estimated point, accordingly, m.

Calculation of the levels of noise on the protected territory generated by unshielded parts of the traffic flow
To calculate the levels of noise on the protected territory generated by the parts of the unshielded parts of the traffic flow, we use the results presented in [18]. The calculation diagram is shown in Fig. 4. As a first approximation, the noise characteristic of the traffic flow [19] is given by an empirical equation: (19) where N -traffic intensity (number of traffic units passing in both directions per one hour), 1/h; V -weighted average car traffic speed, km/h; r -the share of freight vehicles and public transport vehicles in the traffic flow, %; s -number of traffic lanes; q -longitudinal slope of the road, %; P -road surface type (P = 0 for asphalt concrete, P = 3 dBA for cement concrete). We determine the angles of visibility of parts of the unshielded traffic flow from the estimated point: -to the left of the barrier: -to the right of the barrier: where γ = π 2 ⁄ − γ 0 ; (23) -right: Let us determine the initial number of elements on these sections: -left: -right: Let us round up the values n 1 and n 2 to the integer values of m 1 and m 2 and recalculate the element lengths ∆ 1 , ∆ 2 for sections S 1 , S 2 : -left: -right: The finite elements are numbered as follows: -left: i 1 = 1, 2…m 1 ; -right: i 2 = 1, 2…m 2 .
Let us calculate the distance from the estimated point to the i-th element of the traffic flow model: -left: -right: where = √(ℎ 1 − ℎ 2 ) 2 + 1 2 -the distance from the estimated point to the acoustic center of a vehicles measured in direction normal to the traffic flow, (35) h 1 and h 2 -the height of the estimated point and the height of the acoustic center of the noise source above the surface of the territory, m.
Let us determine the reduction of the sound level due to its divergence from the i-th elements of the traffic flow model: where K -the index for writing equations in the general form, while K=1 is for the parts of the traffic flow to the left of the NB and K=2 is for the traffic flow parts to the right of the NB.
We determine the noise reduction in air during propagation of acoustic waves from i-th elements to the estimated point: We determine the reduction of sound due to wind and atmosphere turbulence: We determine the losses from the interaction of the acoustic waves and the territory surface using equations derived in [19]: -for acoustically rigid surface: ∆ , ( ) = 0; -for acoustically soft surface: where ( ) = 5 [ ( ) ℎ 1 ⁄ ] − 3,45ℎ 2 . The distance travelled by the sound waves from the i 1 -th element and the i 2 -the element to the estimated point above the ground surface measured from the shoulder of the road is given by: where ∆ 1 = ∆ 1 2 1 ⁄ and ∆ 2 = ∆ 2 2 1 ⁄ -lengths of the traffic flow elements relative to the closest shoulder of the road; 2 = + -the distance from the estimated point to the shoulder of the road, Fig. 4, m; 2 = 2 ⁄ and 2 = 2 ⁄ -length of the noise barrier and the length of its part relative to the shoulder of the road closest to the estimated point.

Calculation of the levels of sound in the estimated point with diffraction of sound on the top and side edges of the noise barrier
To calculate the levels of sound in the estimated point with diffraction of sound on the top and side edges of the noise barriers, we use the results presented in [19]. First, we calculate the sound level and the sound pressure level (SPL) in each j-th octave band without the noise barrier: where ∆ -reduction of sound due to divergence of acoustic waves over distance; ∆ -losses during sound propagation in air from the sound source to the estimated point; ∆ -absorption of sound by the territory surface; ∆ -influence of the air turbulence and wind on the propagation of acoustic waves from the sound source to the estimated point; ∆ -sound attenuation in the green belt; ∆ -correction related to the limited angle of visibility of a part of the road covered by the barrier from the estimated point. ∆ , -correction of the traffic noise spectrum shape in each j-th octave band. Correction values are either determined experimentally or taken from Second, we calculate the noise characteristic of the traffic flow using empirical equation (19).
Using empirical formulas, we calculate noise reduction due to divergence of the sound waves over distance ∆ , the losses due to sound propagation in air ∆ and absorption of sound by the territory surface ∆ using data from [20]: The error of the calculation using equation (52) relative to the initial data in the range 20 ≤ 1 < 1000 m is less than 0,5 dBA.
The correction term ∆ / , that takes into account turbulence of air and the influence of wind on the propagation of sound [13], is given by: The attenuation of sound in the green belt, if present, is approximated [13] as follows: where 10 ≤ ≤ 100 m is the width of the green belt. For regular green areas, their noise attenuation is neglected, i.e. ∆ = 0. We calculate the correction term related to the limited angle of visibility of the part of the road shielded by the noise barrier from the estimated point: where α -the angle of visibility of the barrier, rad.
The study presented in [19] obtained equations for calculating acoustic efficiency of the noise barrier in each j-th octave band with diffraction of sound solely on the top edge of the noise barrier: where = • δ 1 , Hz • m; Relative to the initial data, the error of calculation using equation (58) is less than 0,5 dBA.
Based on the data from [21], paper [19] obtained relations for evaluating the acoustic energy losses due to interaction of the sound waves with the territory surface with diffraction of sound the top edge of the noise barrier: -for acoustically hard ground surface: Relative to initial data, the error of calculating ∆L A surf using formula (59) is less than 0,1 dBA in the range of 1,45 < δ1 ≤ 1,75 m.
By plugging in the results of calculating ∆L A surf using formulas (59) and (60) in the equation (50) instead of equation (54) and passing over to the sound pressure level in each jth octave band using equation (51), we calculate the spectrum of noise at the estimated point created by the sound radiation of the traffic flow diffracted over the noise barrier: Having summed the energy of octave band SPL levels corrected with the "A" characteristic, we calculate the corresponding sound level:   The difference of the lengths of the paths of the sound rays coming from the center of the i-th element to the estimated point directly and by going around the edges of the noise barrier is given by: The equivalent sound level in the estimated point generated by the i-th element of the traffic flow without taking into account noise attenuation during propagation: We determine individual the j-th octave sound pressure levels in the calculation level from the i-th element of the traffic flow with diffraction of sound on the side edges of the barrier to the left and right: where ∆ ,2,3, , ∆ ,2,3, ⁄ and ∆ are calculated using formulas (54), (55) and (56), accordingly; ∆ , -see Table 1; Based on the data presented in [19] derived equations for calculating the reduction of noise with diffraction on the side edges of the noise barrier:  Relative to the initial data, the error of calculating ∆ 2,3, , NB using equation (77) is less than 0,3 dB.
Having summed the energy SPL in the estimated point for all m elements of the traffic flow in each j-th octave band, we obtain spectra of the noise diffracted on the left and right edges of the noise barrier:

Calculation of total noise level in the estimated point
Using the formulas presented above, we determine the total noise level in the estimated point taking into account reduction of acoustic energy due to diffraction on the top and side edges of the noise barrier of the sound reflected from the road surface, from the noise barrier, the noise barrier on the opposite side of the road, and the parts of the traffic flow not shielded by the noise barrier., = 10 [10 0,1 ,Σ + 10 0,1 1 + 10 0, 1 2 ].
where ,Σ -the total level of sound reflected from the road surface, from the noise barrier, from the barrier on the opposite side of the road, if present, as well as from both barriers installed on both sides of the road; 1 -noise level on the protected territory created by the parts of the traffic flow not shielded by the noise barrier; 2 -noise level in the estimated point with diffraction of sound on the top and side edges of the noise barrier.

Conclusion
The proposed comprehensive acoustic calculation method enables designing noise barriers taking into account different factors. These factors include noise diffracted along the whole outer perimeter of the barrier, as well as the sound reflected from the opposite barrier, if present, and parts of the traffic flow unshielded by the noise barrier. During design process, the designer can identify where most of the noise is coming from, whether from the top of the screen or from its sides, including the noise from the parts of the traffic flow unshielded by the noise barrier to the left and to the right of the barrier, if the noise levels violate regulations. Thus, there is a possibility of rationally choosing geometric dimensions of the barrier by either increasing its height or increasing its length in either direction.
The design process determines the optimal geometric dimensions of the barrier. The proposed approach allows reducing design time of noise barriers and reduce noise barrier installation expenses.