Wave-based numerical investigation on di ff raction correction for a low-height barrier in energy-based sound propagation model for road tra ffi c noise

Yosuke Yasuda 1

Faculty of Architecture and Building Engineering, Kanagawa University 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686 Japan

Yu Kamiya 2

Graduate School of Engineering, Kanagawa University 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686 Japan Technical Research & Development Institute, Kumagai Gumi Co., Ltd. 1043 Onigakubo, Tsukuba-shi, Ibaraki, 300-2651 Japan

Makoto Morinaga 3

Faculty of Architecture and Building Engineering, Kanagawa University 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686 Japan

ABSTRACT In energy-based noise propagation prediction models such as CNOSSOS-EU in Europe and ASJ RTN-Model in Japan, sound propagation is calculated considering the noise attenuation by the geometrical divergence of the sound power from a sound source with various attenuation corrections such as di ff raction e ff ect and ground e ff ect. In ASJ RTN-Model, the insertion loss of a semi-infinite barrier with respect to a free field is generally used for the di ff raction correction for a single barrier, whereas the di ff raction correction for a low-height barrier is di ff erently defined; it is given as the insertion loss of the low-height barrier with respect to a semi-infinite barrier, the top height of which is at the ground surface level. The di ff raction correction is zero when the low-height barrier is 0 m high. However, its range of application is not clear. In this paper, the di ff raction correction for a low-height barrier is discussed by comparing the calculation results based on ASJ RTN-Model with the above-mentioned two di ff raction corrections and results based on a three-dimensional wave-based numerical analysis.

1. INTRODUCTION

Energy-based noise propagation prediction models, such as CNOSSOS-EU [1] in Europe and ASJ RTN-Model [2] in Japan, are generally based on the formula for distance attenuation due to

1 yyasuda@kanagawa-u.ac.jp

2 yu.kamiya@ku.kumagaigumi.co.jp

3 m-morinaga@kanagawa-u.ac.jp

point source S

plane of symmetry barrier (thickness: 0 m) 10

z [m]

h b prediction area R

0

ground

0 - 10 10 20 30 40 50 60 100 90 70 80 7.25

y [m]

Figure 1: Analysis model: cross-section ( yz plane).

plane of symmetry

100

30

prediction area R

20

y [m]

barrier (height: h b , thickness: 0 m)

10

7.25

point source S

0

0 10 20 30 40 50 60 100 90 70 80 x [m]

Figure 2: Analysis model: plan ( xy plane).

geometrical divergence from the sound source, to which various corrections such as di ff raction attenuation and ground surface attenuation are added. In ASJ RTN-Model, the di ff raction correction for semi-infinite barriers (Maekawa’s chart) [3,4] is used for di ff raction calculations for single and double barriers that can be considered knife wedges. On the other hand, for di ff raction calculations for low-height barriers, a di ff erent correction is given from that for general ones. This is given as the insertion loss when a low-height barrier is installed against a semi-infinite barrier whose top edge is at the ground surface level, and is designed so that the di ff raction correction is zero when the height of the low-height barrier is 0 m. However, its application range is not so clear. ASJ RTN-Model states that this correction is for “a barrier with a height of 1 m or less in flat terrain [2],” but does not explicitly state the basis or concept of this correction. In this paper, we clarify the characteristics of the di ff raction correction for a “low-height sound barrier,” which is calculated as the insertion loss with respect to the semi-infinite barrier, and discuss its range of application. We compare the results of road tra ffi c noise predictions for sound fields with a barrier calculated based on ASJ RTN-Model with di ff raction corrections for general barriers and for low-height barriers and those calculated based on a three-dimensional wave-based numerical analysis. The height of the barrier is varied to understand the characteristics of the di ff raction field for each height, as well as the characteristics of the above two types of di ff raction correction. Finally, we propose a specific application range of di ff raction correction for low-height sound barriers, taking into account practical aspects.

2. CALCULATION SETUP

2.1. Analysis Model Figures 1 and 2 show a cross-section and plan of the analysis model, respectively. A rigid barrier with zero thickness and infinite length (limited to 200 m in the three-dimensional wave-based numerical analysis described below) and a point source S (0 , 0 , 0) are placed on an infinite rigid plane that simulates the ground surface. The height of the barrier is varied from 0 to 4 m. The point source has the A-weighted sound power spectrum of a passenger car running at a steady speed of 60 m / h on a dense asphalt pavement road, the calculation formuli for which are shown in ASJ RTN-Model

2013 [5]. Prediction points are placed in the range of 0 ≤ x ≤ 100 m, 0 ≤ y ≤ 100 m, and 0 ≤ z ≤ 10 m with 2 m intervals. Near the ground surface 0 ≤ z ≤ 2 m, the interval is 0.1 m to examine the sound pressure level distributions in detail.

2.2. Calculation Using ASJ RTN-Model Direct calculation for overall values

Basic equation The A-weighted sound pressure level L A [dB] for noise propagation from the source (vehicle) position to the prediction point is calculated as

L A = L W A − 8 − 20 log 10 r + ∆ L dif , (1)

where L W A is the A-weighted sound power level [dB] of a single running vehicle at the source position, and r is the direct distance [m] from the source position to the prediction point. ∆ L dif is the correction term for di ff raction [dB], which is replaced as ∆ L dif = ∆ L d , k or ∆ L dif = ∆ L dif , low , depending on the acoustical obstacle.

Correction term for di ff raction around a barrier ∆ L d , k The correction term for di ff raction around a simple plane barrier is equal to that around a knife wedge, ∆ L d , k , which is calculated as



− 20 − 10 log 10 ( c spec δ ) , c spec δ ≥ 1

− 5 − 17 . 0 sinh − 1 ( c spec δ ) 0 . 415 , 0 ≤ c spec δ < 1

(2)

∆ L d , k =

min[0 , − 5 + 17 . 0 sinh − 1 ( c spec | δ | ) 0 . 415 ] , c spec δ < 0

Here δ = SO + OP − SP is the path di ff erence for di ff raction considering source point S, di ff raction point O, and prediction point P, and defined as a negative value when S is visible from P. The function min[ a , b ] gives the smaller value of a and b . In this paper, the coe ffi cient c spec is set to 1 . 00 assuming a dense asphalt pavement.

Correction term for di ff raction around a low-height barrier ∆ L dif , low The correction term ∆ L dif , low for di ff raction around a barrier with a height of 1 m or less in flat terrain is given as the insertion loss of the barrier: ∆ L dif , low = ∆ L d , k , 1 − ∆ L d , k , 0 , (3)

where ∆ L d , k , 1 and ∆ L d , k , 0 are ∆ L d , k for the top edge of the barrier O 1 and that for the intersection of the barrier and the ground surface O 0 , respectively.

On reflection from ground surface When ∆ L d , k is used as the correction term for di ff raction, a mirror image of the prediction point relative to the ground surface is considered, and L A for the propagation path to this mirror image is also calculated by Equation (1), and then the energy is added in the same way as in Equation (5). When ∆ L dif , low is used as the correction term for di ff raction, it is assumed that the e ff ect of the reflection from the ground surface is taken into account only by setting ∆ L dif = ∆ L dif , low .

Calculation from frequency components

Basic equations The A-weighted sound pressure level L A [dB] at the prediction point is calculated using the following equations. L A in Equation (4) corresponds to that in Equation (1).

i 10 L A ( f c , i ) / 10 , (4)

L A = 10 log 10

M

m = 1 10 L A , m ( f c , i ) / 10 , (5)

L A ( f c , i ) = 10 log 10

L A , m ( f c , i ) = L W A ( f c , i ) − 11 − 20 log 10 r m + ∆ L dif , m ( f c , i ) , (6)

where f c , i is the center frequency [Hz] of the i th band, L A ( f c , i ) is the A-weighted sound pressure level [dB] for the frequency band of f c , i , L A , m ( f c , i ) is the A-weighted sound pressure level [dB] for propagation path m for the frequency band of f c , i , L W A ( f c , i ) is the sound power level [dB] for the frequency band of f c , i , r m is the direct distance [m] for propagation path m (from the sound source or its mirror image to the prediction point or its mirror image), and M is the number of the propagation paths. ∆ L dif , m ( f c , i ) is the correction term for di ff raction [dB] for propagation path m , which is replaced as ∆ L dif , m ( f c , i ) = ∆ L d , k ( f c , i ) or ∆ L dif , m ( f c , i ) = ∆ L dif , low ( f c , i ), depending on the acoustical obstacle.

Correction term for di ff raction around a barrier ∆ L d , k ( f ) The correction term for di ff raction around a simple barrier is calculated as [3,4]



− 13 − 10 log 10 N , 1 ≤ N

− 5 ∓ 9 . 08 sinh − 1 ( | N | 0 . 485 ) , − 0 . 324 ≤ N < 1

(7)

∆ L d , k ( f ) =

0 , N < − 0 . 324

where N = 2 δ/λ is the Fresnel number, λ = c / f is the wavelength [m], and c is the sound speed [m / s]. The sign ∓ is negative for N ≥ 0 and positive for N < 0.

Correction term for di ff raction around a low-height barrier ∆ L dif , low ( f ) The correction term ∆ L dif , low ( f ) for di ff raction around a low-height barrier is not described in ASJ RTN-Model. In this paper, similar to Equation (3), ∆ L dif , low ( f ) is calculated using the following equation:

∆ L dif , low ( f ) = ∆ L d , k , 1 ( f ) − ∆ L d , k , 0 ( f ) , (8)

where ∆ L d , k , 1 ( f ) and ∆ L d , k , 0 ( f ) are ∆ L d , k ( f ) for the top edge of the barrier O 1 and that for the intersection of the barrier and the ground surface O 0 , respectively.

2.3. 3-D Wave-Based Numerical Calculation Numerical method

The fast multipole boundary element method (FMBEM) [6] is used. To analyze objects of thickness 0, the hypersingular formulation (normal derivative form) is adopted. Since the infinite rigid ground surface is an acoustic specular surface and the cross-section orthogonal to the direction of the road through the point source is a plane of symmetry for the sound field, FMBEM for plane-symmetric acoustic problem [7] is applied.

Numerical setup

Boundary elements The boundary elements are constant and rectangular, and their size is ≤ 1 / 5 of the calculation wavelength.

Calculation frequencies The calculation frequencies are 1 / 9-octave band center frequencies in the five octave bands, the center frequencies of which are from 125 Hz to 2 kHz.

55 60 65 70 50 h b = 0.5 m

60 65 70 75 45 50 h b = 1 m

10

x = 0 m 0

55

z [m]

10

x = 40 m

55 60 50

50 55 60 45

0

10

45

x = 80 m

50

50

0 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

y [m]

h b = 2 m h b = 4 m

10

x = 0 m 0

55 60 65 70 75 35

40 45 50

45 50 55 6 0 6 5 70 75 40

z [m]

10

10 x = 40 m

50 55 60

40 45

55 60 40

45 50

35

0

40 45 50 55 35

x = 80 m

45 50

40

0

y [m] 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

Figure 3: Distributions of L FMBEM ( yz plane).

h b = 0.5 m h b = 1 m

10

x = 0 m 0

0.5

0.5 1

0

0

z [m]

10

0.5

x = 40 m

0

0

0

10

x = 80 m

0 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

0 0

y [m]

h b = 2 m h b = 4 m

10

x = 0 m 0

0.5 1 1.5 2 3 4 5

0.5 1 2

0

0

z [m]

10

0.5 1 1.5 2 2.5

0.5 1

10 x = 40 m

0

0

0

0.5 1

0.5

x = 80 m

0

0

0

y [m] 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

Figure 4: Distributions of d paths ( yz plane).

Point source At each calculation frequency f c , i , the volume velocity of the point source is given as its sound power level is equal to the A-weighted band power level L W A ( f c , i ) in Equation (6) minus 3 dB. The reason for subtracting 3 dB is to match the values of the sound pressure level obtained by ASJ RTN-Model, an energy-based calculation method, and a wave-based numerical method in a hemi-free field where only a point source is located on the infinite rigid plane.

Calculation of A-weighted sound pressure level

The A-weighted sound pressure level L A is calculated using Equation (4) with the sound pressure level L A ( f c , i ) obtained by FMBEM.

3. CALCULATION RESULTS

3.1. Wave-Based Numerical Results Figure 3 shows distributions of L FMBEM , which is the A-weighted sound pressure level calculated using FMBEM. For low barriers such as h b = 0 . 5 m, the e ff ect of interference with reflected waves from the ground surface is significant, especially in the distant areas of the sound source. Interference

point source S

direct path: R dir d paths = R refl − R dir reflection path: R refl

z [m]

prediction point ground surface

y [m]

Figure 5: Propagation path di ff erence d paths = R refl − R dir .

makes a trough in the sound pressure level distribution, and the position of the trough is closer to the ground surface as the barrier height h b increases. In other words, the area where the interference e ff ect is significant is wider as the barrier height decreases. For the same prediction areas as in Figure 3, Figure 4 shows d paths = R refl − R dir , which is the di ff erence between the path R dir that reaches the prediction point directly from the top of the barrier and the path R refl that reaches the prediction point after being reflected by the ground surface on the prediction side (refer to Figure 5). Comparing to Figure 3, it can be seen that the sound pressure level trough occurs around the area where 0 . 1 ≤ d paths ≤ 0 . 2 m. Since the trough occurs around at d paths = λ/ 2, where λ is the wavelength, this indicates that the di ff racted sound has a large frequency component in the 1 kHz band where λ/ 2 is about 0.1 to 0.2 m. The higher the barrier, the closer the position where 0 . 1 ≤ d paths ≤ 0 . 2 m is to the ground surface, and also the closer the sound pressure level trough is.

3.2. Results Using ∆ L d , k Figure 6 shows distributions of L ASJ , OA , 2paths , which is the A-weighted sound pressure level calculated using ASJ RTN-Model with ∆ L d , k for the overall values. The distribution characteristics are significantly di ff erent from L FMBEM in the case where the barrier height h b is small, because interference does not occur in L ASJ , OA , 2paths , which is obtained using an energy-based calculation method. On the other hand, in the cases with large barrier height h b , L FMBEM and L ASJ , OA , 2paths are similar. Figure 7 shows the level di ff erence obtained by subtracting L FMBEM from L ASJ , OA , 2paths . At the position where the sound pressure level trough occurs in L FMBEM , L ASJ , OA , 2paths is approximately 2 dB larger, whereas near the ground surface L FMBEM is approximately 3 dB larger. This indicates that ASJ RTN-Model estimates the road tra ffi c noise near the ground surface lower. This is because the e ff ect of reflected waves near the ground surface is + 3 dB by the energy sum for L ASJ , OA , 2paths , while + 6 dB by the in-phase wave sum for L FMBEM . This di ff erence is problematic especially when the barrier height is small, because the a ff ected area is wider.

3.3. Results Using ∆ L dif , low Figure 8 shows distributions of L ASJ , OA , low , which is the A-weighted sound pressure level using ASJ RTN-Model with ∆ L dif , low for the overall values. Figure 9 shows the level di ff erence obtained by subtracting L FMBEM from L ASJ , OA , low . Similar to L FMBEM , interference-like patterns are seen in the distributions of L ASJ , OA , low . This is due to the fact that the value of − ∆ L d , k , 0 in Equation (3) has a maximum value of + 5 dB on the ground surface and decreases as the prediction point is away from the ground surface. On the ground surface, an increase from the sound pressure level of the incicent wave is + 6 dB in L FMBEM due to the wave sum with the in-phase reflected wave, + 3 dB in L ASJ , OA , 2paths due to the energy sum with the reflected wave energy, and + 5 dB in L ASJ , OA , low due to the value of − ∆ L d , k , 0 in Equation (3) as mentioned above. Therefore, L ASJ , OA , low is the closer to L FMBEM near the ground surface. In the case where the barrier height h b is small, the region where the incident and reflected waves are approximately in phase extends above the ground surface, and the region where L ASJ , OA , low is

h b = 0.5 m h b = 1 m

10

x = 0 m 0

45 50 55 60 65

50 55 60 65

z [m]

10

x = 40 m

50 55 45 50 55

0

10

x = 80 m 50 45 50

0 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

y [m]

h b = 2 m h b = 4 m

10

45 50 5 5 60 65 x = 0 m 0

40 45 50 55

z [m]

10

10 x = 40 m

40 45 50

45 50 55

0

x = 80 m

35 40 45

40 45 50

0

y [m] 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

Figure 6: Distributions of L ASJ , OA , 2paths ( yz plane).

−6 [dB] −5 −4 −3 −2 −1 0 1 2 3 4 5 6

h b = 0.5 m h b = 1 m

10

50 55 60 65

x = 0 m 0

50 55 60 65

z [m]

10

50 55 50 55

x = 40 m

0

10

x = 80 m

0 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

y [m]

h b = 2 m h b = 4 m

10

x = 0 m 0

35 40 45 50 55

45 50 55 60 65

z [m]

10

35 40 45 50

10 x = 40 m

45 50 55

0

x = 80 m

0

y [m] 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

Figure 7: Distributions of L ASJ , OA , 2paths − L FMBEM ( yz plane).

accurate also becomes wide. As a result, for barriers of about h b ≤ 1 m, L ASJ , OA , low is clearly more accurate than L ASJ , OA , 2paths in the region from the ground surface to about 5 m above. On the other hand, L ASJ , OA , 2paths is more accurate for barriers of h b ≥ 2 m except for the region very close to the ground surface.

3.4. Results Calculated from Frequency Components We obtained L ASJ , Freq , 2paths and L ASJ , Freq , low , which are the results calculated from frequency components using Equations (4) to (6). The tendencies of L ASJ , Freq , 2paths and L ASJ , Freq , low were similar to those of L ASJ , OA , 2paths and L ASJ , OA , low . However, the values were about 0.5 dB higher in the whole areas. This may be due to the fact that the component of the 4 kHz octave band is not included in L ASJ , Freq , 2paths and L ASJ , Freq , low whereas it is included in ∆ L d , k .

h b = 0.5 m h b = 1 m

10

50 55 60 65

x = 0 m 0

50 55 60 65

z [m]

10

x = 40 m

50 55 50 55

0

10

x = 80 m

50 45 50

0 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

y [m]

h b = 2 m h b = 4 m

10

x = 0 m 0

35 40 45 50 55

45 50 55 60 65

z [m]

10

35 40 45 50

10 x = 40 m

45 50 55

0

35 40 45

45 50

x = 80 m

0

y [m] 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

Figure 8: Distributions of L ASJ , OA , low ( yz plane).

−6 [dB] −5 −4 −3 −2 −1 0 1 2 3 4 5 6

h b = 0.5 m h b = 1 m

10

x = 0 m 0

z [m]

10

x = 40 m

0

10

x = 80 m

0 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

y [m]

h b = 2 m h b = 4 m

10

x = 0 m 0

z [m]

10

10 x = 40 m

0

x = 80 m

0

y [m] 0 10 20 30 40 50 60 70 80 90 100 y [m] 0 10 20 30 40 50 60 70 80 90 100

Figure 9: Distributions of L ASJ , OA , low − L FMBEM ( yz plane).

4. DISCUSSION ON APPLICATION RANGE OF ∆ L DIF , LOW

Based on the above results, we propose a specific range of application of ∆ L dif , low , taking its practical aspects into consideration.

4.1. Relation to Barrier Height h b It is the most practical and convenient to determine the application range of ∆ L dif , low based only on the height of the barrier. From the results of the previous section, it is generally recommended to use ∆ L dif , low for barriers of h b ≤ 1 m and ∆ L d , k for those of h b ≥ 2 m. For barriers of 1 < h b < 2 m, there is no significant di ff erence between both correction terms for di ff raction.

4.2. Relation to Propagation Path Di ff erence d paths As mentioned above, di ff racted sound fields around a low-height barrier are considerably a ff ected by interference with the reflected waves from the ground surface in the prediction region. In order to see

L ASJ,OA,2paths 　 L FMBEM L ASJ,OA,low 　 L FMBEM 㸫 㸫

−6 −5 −4 −3 −2 −1 0 1 2 3 4 0.25 ≤ h b ≤ 1.25 m 0.25 ≤ h b ≤ 1.25 m

Difference [dB]

0.25 m

0.5 m

0.75 m

1 m

1.25 m

−5 −4 −3 −2 −1 0 1 2 3 4

1.5 ≤ h b ≤ 4 m 1.5 ≤ h b ≤ 4 m

Difference [dB]

1.5 m

1.75 m

2 m

3 m

4 m

−6 0.1 1 10 0.01 0.001 0.1 1 10 0.01 0.001

d paths [m] d paths [m]

Figure 10: Scatter plots of L ASJ , OA , 2paths − L FMBEM and L ASJ , OA , low − L FMBEM in relation to d path .

the relationship between the aforementioned propagation path di ff erence d paths (see Figure 5) and the values calculated using each correction term, scatter plots of L ASJ , OA , 2paths − L FMBEM and L ASJ , OA , low − L FMBEM in relation to d path are shown in Figure 10. Here the prediction points are limited in the range of 0 ≤ x ≤ 80 m and 8 ≤ y ≤ 100 m, with 10 m interval in the x -axis direction (the direction of the road). Generally, L ASJ , OA , low is less di ff erent from L FMBEM when d path is about 0.1 m or less, and L ASJ , OA , 2paths is less di ff erent when d path is about 0.1 m or more. This is true not only for low-height barriers of about 1 m height or less, but also for higher barriers. This indicates that ∆ L dif , low can be used for more accurate prediction, depending on the propagation path di ff erence d path . In other words, by using the propagation path di ff erence d path as a criterion, the application range of ∆ L dif , low can be set more appropriately. Figure 11 shows distributions of L ASJ , OA , 2paths − L FMBEM and L ASJ , OA , low − L FMBEM on the prediction plane z = 1 . 2 m, a typical measurement point height, and the propagation path di ff erence d path , indicated by solid lines. Red lines indicate d path = 0 . 1. When observed in detail, the boundary value of d path to determine whether L ASJ , OA , 2paths or L ASJ , OA , low should be calculated is slightly smaller than 0.1 for lower-height barriers and slightly larger than 0.1 for higher-height ones. However, there seems to be no significant problem if the boundary value of d path is considered to be 0.1.

5. SUMMARY

In order to clarify the characteristics and the application range of the di ff raction correction which is calculated as the insertion loss with respect to a semi-infinite barrier, road tra ffi c noise propagated above a barrier was predicted based on ASJ RTN-Model with two correction terms, i.e. , the correction term for di ff raction around a general simple barrier ∆ L d , k and that for a low-height barrier ∆ L dif , low . The results were compared with those based on a three-dimensional wave-based numerical analysis. The findings are summarized as follows.

1. In general, the interference of direct waves from the top of the barrier and reflected waves from the ground surface produces a sound pressure level trough in the prediction region. The lower the barrier is, the wider and more pronounced the interference trough is, and the higher the barrier is, the more limited it is to a very close vicinity of the ground surface.

2. Since ∆ L dif , low is calculated as the insertion loss of the barrier with respect to a semi-infinite

L ASJ,OA,2paths L FMBEM

−6 [dB] −5 −4 −3 −2 −1 0 1 2 3 4 5 6

h b = 0.5 m h b = 1 m h b = 2 m h b = 4 m

100

90 80 70 60 50 40 30 20 10

0.1

y [m]

0.1

0.1

0.1

0.5 1

0.5 1

0.5 0.5

0

0 10 203040 5060708090100 x [m] 010 203040 5060708090100 x [m] 0 10 203040 5060708090100 x [m] 0 10 203040 5060708090100

x [m]

L ASJ,OA,low L FMBEM

h b = 0.5 m h b = 1 m h b = 2 m h b = 4 m

100

90 80 70 60 50 40 30 20 10

0.1

y [m]

0.1

0.1

0.1

0.5 1

0.5 1

0.5 0.5

0

0 10 203040 5060708090100 x [m] 010 203040 5060708090100 x [m] 0 10 203040 5060708090100 x [m] 0 10 203040 5060708090100

x [m]

Figure 11: Distributions of L ASJ , OA , 2paths − L FMBEM and L ASJ , OA , low − L FMBEM ( z = 1 . 2 m). Solid lines indicate d paths .

barrier, it produces a distribution including a wave-like interference trough. Therefore, when the barrier is low and the interference trough appears over a wide area, ∆ L dif , low can be used to obtain highly accurate results.

3. When ∆ L d , k is used, no interference troughs appear in principle. Therefore, if the barrier is high and the interference trough is limited to a very close vicinity of the ground surface, ∆ L d , k can be used to obtain highly accurate results.

4. The most convenient way to determine the application range of ∆ L dif , low is by the height of the barrier h b . It is generally appropriate to use ∆ L dif , low for the barriers of h b ≤ 1 m and ∆ L d , k for the barriers of h b ≥ 2 m. For the barriers of 1 < h b < 2 m, there is no significant di ff erence regardless of which correction term for di ff raction is used.

5. If the application range of ∆ L dif , low is to be determined for each prediction point, the di ff erence between the path that reaches directly from the top of the barrier to the prediction point and the path that reaches after reflection by the ground surface (propagation path di ff erence d paths ) can be used as a criterion. It is generally appropriate to use ∆ L dif , low if d paths ≤ 0 . 1 m and ∆ L d , k if d paths > 0 . 1 m.

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