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Effects of Tilted Noise Barriers on Road Traffic Noise
Erik Salomons, Arno Eisses, Frits Eerden
To cite this version:
EFFECTS OF TILTED NOISE BARRIERS ON ROAD TRAFFIC NOISE
Erik Salomons
Arno Eisses
Frits van der Eerden
TNO Acoustics & Sonar, The Hague, The Netherlands erik.salomons@tno.nl
ABSTRACT
Noise barriers along highways in the Netherlands are often tilted, at an angle of typically 10 degrees from the vertical. In this way, it is prevented that sound reflections arrive at distant receivers along curved sound paths through the refracting atmosphere. In this paper we compare results of two types of calculation methods for sound reflections in such situations: i) numerical methods, and ii) an engineering method based on the concept of Fresnel zones. Based on the comparisons, the engineering method is optimized. Possible implementation in traffic noise prediction models, such as the Dutch standard model and the EU model Cnossos, is discussed. The variation of the reflected sound level with frequency and tilt angle is analyzed. For situations with tilted noise barriers on either side of the highway, the interaction between screening and reflection by the barriers is analyzed.
1. INTRODUCTION
With the Dutch standard model for road traffic noise (SRM) [1], reflections from vertical barriers are modeled by means of image sources. This is illustrated in Figure 1. The source (open circle) is replaced with image source S, and the sound transmission is calculated for the path from source S to receiver R. If the line S-R does not intersect the barrier, the reflection should be ignored. This intersection check is performed only in the horizontal projection plane, not in three dimensions. One might add the three-dimensional check to SRM, but a better option is to add the calculation method described in this paper. The method yields a continuous decrease of reflection level when the intersection point moves upward beyond the top of the barrier.
The calculation method can also be applied to situations with a tilted barrier. This is illustrated in Figure 2. In the situation shown in the figure, the image source S is located below ground level, and the sound path from S to R passes under the barrier. With increasing distance from the barrier, the calculated reflection level decreases.
Figure 1. Illustration of reflection from a vertical barrier, with image source S and receiver R.
Figure 2. Illustration of reflection from a tilted surface, with image source S and receiver R.
2. FRESNEL ZONES AND REFLECTION Figure 3 illustrates the concept of Fresnel zones for the modeling of sound reflection by a plane surface. The Fresnel zones are elliptic rings, such that contributions from adjacent Fresnel zones are 180 degrees out of phase. The idea here is that the reflection can be calculated as an integral over the reflecting surface, through the Kirchhoff-Helmholtz equation. The path length difference between the two sound paths shown in the figure is half a wavelength: r= /2. This applies also to sound paths along the edges of Fresnel zone n and Fresnel zone n+1. Thus, contributions from adjacent Fresnel zones cancel each other. What remains is a contribution from an area around the center of the Fresnel zones, i.e. the specular reflection point. We define this area also by the path length difference between the path along the edge and the path through the center:r= //nF. For parameter nF one may
use the value nF = 4, so that the area is about half the area
of the first Fresnel zone. In the calculation model Nord2000 [2] the value nF = 8 is used, and we will also use
this value here. For simplicity we refer to the reflection area with r= //nF as ‘the Fresnel zone’, although it is in
fact a part of the first Fresnel zone.
If the reflecting surface is smaller than the Fresnel zone, then the strength of the reflection is limited by the size of the reflecting surface. Then the surface integral is limited to the area of the reflecting surface, and consequently the reflection is weaker than in the case that the reflecting surface is larger than the Fresnel zone. The attenuation can be approximated by including a factor Sr/SF in the reflected sound pressure, where Sr is the area
of the reflecting surface and SF is the area of the Fresnel
zone. If Sr/SF is greater than 1, then we have the
unattenuated reflection. If Sr/SF is smaller than 1, then the
reflection is attenuated, with attenuation 20log10(Sr/SF) in
decibels.
3. CALCULATION METHOD
The modeling of reflections with Fresnel zones has been applied previously in the calculation model Nord2000 [2]. The approach in Nord2000 was inspired by a theoretical analysis [3] of sound propagation over a ground surface consisting of areas with different acoustic impedances (for example, asphalt and grass). Later the calculation method with Fresnel zones has been applied in the European model Harmonoise [4].
Here we describe how the Dutch standard model SRM is extended with a Fresnel-zone model for reflections. The contribution of a reflection to the sound level at a receiver is written as follows:
L = LW + LSRM +LF + Labs
with
- LW the sound power level of the source,
-
LSRM the ‘normal’ contributions, such asgeometrical attenuations and ground attenuation, -
LF the geometrical reflection attenuation,-
Labs = 10log10(1-
) the absorption attenuation,with
the absorption coefficient of the barrier. The geometrical reflection attenuation is calculated with the following formula:LF = 20 log10 (Sr / SF).
In the previous section, Sr was the area of the reflecting
surface and SF the area of the Fresnel zone. With SRM,
however, we model the sound propagation in a vertical plane through the source and the receiver, and therefore we use heights in this plane rather than areas for Sr and SF.
So SR is the height of the barrier and SF is the height of the
Fresnel zone.
The calculation of Sr and SF consists of the following
five steps, which are illustrated in Figure 4. With steps 4 and 5, the effect of refractive ray curvature is taken into account.
Step 1. Determine the positions of the source (S), receiver (R), and the barrier (which may be tilted).
1The Fresnel ellipse is defined here as the set of points p that satisfy the equation |Sp|+|pR| - |SR| = /n
Step 2. Replace the source by the image source.
Step 3. Determine points A and B on the Fresnel ellipse1
around the straight source-receiver line. This yields the height SF of the Fresnel zone at the location of the foot of
the barrier, ignoring ray curvature.
Step 4. Calculate the vertical displacement z due to ray curvature at the location of the foot of the barrier:
z = RS RR / (26R)
with RS and RR the horizontal distances from S and R to
the foot of the barrier, and R=RS+RR.
Step 5. Raise the points A and B with 𝛿𝑧, determine SF = zB - zA, and determine height Sr of the part of the
barrier that is limited by the line segment AB.
Figure 4. Illustration of the five steps of the calculation of Sr (red) and SF (blue).
As noted before we use heights rather than areas for Sr
and SF. This approximation can be justified by
approximating the Fresnel ellipse in the vertical plane through the foot of the barrier by a rectangle. This implies that the ratio Sr / SF of two area scan be approximated by
the ratio of heights.
4. NUMERICAL EXAMPLES
The source is located at 10 m distance from a 5 m high barrier, which is tilted at an angle of 5 degrees. The receiver is located at 40 m distance from the barrier. The source height is 0.75 m and the receiver height is 5 m. The values of LF are -5.9 and -1.9 dB at the two frequencies.
The heights SF and Sr are indicated.
Figure 6 shows the result at 300 Hz for an angle of 15 degrees. In this case the Fresnel ellipse passes under the barrier (so Sr = 0) and we have LF = -.
Figure 7 shows the spectrum of LF for the situation
shown in Figure 5, with a barrier at an angle of 5 degrees. The results are given for a calculation with ray curvature and for a calculation without ray curvature.
Figure 8 shows the spectrum of LF for the same
situation but now for a barrier at an angle of 10 degrees. The value of LF decreases with increasing frequency and
is equal to - at high frequency because the Fresnel ellipse passes under the barrier.
Figure 5. Illustration of the calculation of Sr, SF,
and LF at two frequencies, for a situation with a
barrier at an angle of 5 degrees.
Figure 6. Illustration of the calculation of Sr, SF,
and LF at 300 Hz, for a situation with a barrier at
an angle of 15 degrees. In this case we have Sr = 0
and LF = -.
Figure 7. Spectrum of the relative sound level LF for
the situation shown in Figure 5, with a barrier at an angle of 5 degrees.
Figure 8. As Figure 7, but now for an angle of 10 degrees.
5. COMPARISON WITH BEM
With the numerical boundary element method (BEM) [5] for calculating sound fields we can investigate the accuracy of the Fresnel-zone model described in Section 3.
We apply BEM in two dimensions, in the vertical plane through the source and the receiver. We ignore the ground surface and include only a thin tilted barrier in the system. With BEM we calculate the total (complex) sound pressure field p, i.e. the sum of the direct field pdir and the reflected
field prefl. The direct field can also be calculated
analytically (Hankel function), so we obtain the reflected field as follows:
prefl = p – pdir.
For the comparison with the Fresnel-zone model we convert this to a relative sound level:
Lrel = 20 log10 ( |prefl|/|pdir| )
Lrel = Dgeo + LF
where Dgeo is the difference in geometrical attenuation
between the reflected path and the direct path, i.e. the path image source → receiver and the path source → receiver: Dgeo =
Dgeo(image source → receiver) - Dgeo(source → receiver).
The formula Lrel = Dgeo + LF follows from the following
arguments. The reflected sound ray via the barrier is weaker than the direct from the real source by two effects:
i) the length of the reflected ray is larger, ii) the barrier has a finite height.
These two effects correspond with the two terms Dgeo and
LF.
For the comparison with the Fresnel-zone model we convert the BEM results for Lrel to values of LF, with the
relation LF = Lrel - Dgeo. The value of Dgeo for the
geometry in Figure 5 is 10lg(30/50)=-2.2 dB.
Figure 9 shows results of BEM calculations expressed as relative sound level Lrel. Figure 10 shows the same
results expressed as sound level Lrefl = 20 log10|prefl|. The
sound level Lrefl depends not only on the reflection
properties of the barrier but also on the source emission field (Hankel function), which has an amplitude that decreases with increasing frequency. The dependence on the source emission field is eliminated by expressing the results as Lrel (Figure 9).
Figure 11 shows a comparison of values of LF
calculated with BEM and with the Fresnel-zone model, for barrier angles of, 0, 5, 10, and 15 degrees. Figure 12 shows the same comparison, but now the BEM results have been converted to octave-band levels.
Figure 13 shows a comparison for a receiver at 100 m distance from the source, instead of 30 m. Figure 14 shows the same comparison, but now for a barrier at 30 m from the source instead of 10 m.
The BEM results in Figure 12 - Figure 14 confirm the trend that LF increases with frequency for a vertical
barrier (angle zero) and decreases with frequency for angles of 5-15 degrees (except for 5 degrees in Figure 12). For the larger angles, the values of LF calculated with
BEM are less negative than the values calculated with the Fresnel-zone model.
We have tried to adjust the Fresnel-zone model to improve the agreement with the BEM results.
First, we have varied the Fresnel-zone parameter nF.
With values of 4 or 16 instead of 8 the agreement did not improve. Therefore we decided to keep the value of nF at
8.
Next, we have eliminated the abrupt decrease to Lrel = - (see Figure 12 - Figure 14) by limiting the
decrease with frequency to a maximum of 3 dB per octave. The comparisons between BEM and the modified Fresnel-zone model are shown in Figure 15 - Figure 17. The agreement is certainly not perfect, but at least the trend with increasing range agrees. The modified Fresnel-zone model is considered as a practical engineering approach, and is used in the following sections.
Figure 9. BEM results expressed as relative sound level Lrel, for 250 Hz (angle 10 degrees).
Figure 10. BEM results from Figure 9, now expressed as sound level Lrefl = 20 log10|prefl|.
Figure 11. Comparison between BEM and Fresnel-zone model, for angles of 0, 5, 10, and 15 degrees (from top to bottom). For angle 15 degrees the Fresnel-zone model gives LF = - for frequencies
Figure 12. As Figure 11, but now the BEM results are converted to octave-band levels.
Figure 13. As Figure 12, but now for a receiver at 100 m from the source.
Figure 14. As Figure 13, but now for a barrier at 30 m from the source. The fourth red line for angle 15 degrees is absent, as we have LF = - for all
frequencies.
Figure 15. As Figure 12, but now with the modified Fresnel-zone model (see text).
Figure 16. As Figure 13, but now with the modified Fresnel-zone model.
3D BEM versus 2D BEM
As described above we have used 2D BEM results as a reference for improving the Fresnel-zone model. In general, 2D and 3D values of the relative sound level (i.e. the sound level minus the free-field level) are equal in good approximation [6]. We have checked this by a comparison of 2D and 3D BEM results, for reflection from a rectangular surface of height 4 m and width 20 m.
6. COMPARISON WITH PE
We have also compared results of the Fresnel-zone model with results of the numerical parabolic-equation model (PE model; see [7]), for situations with reflection from a vertical barrier. In this section we present one of these comparisons.
With PE we can calculate LF directly; the correction
Dgeo is not required. We consider a situation with a source
at height 100 m, and a vertical barrier with a 2 m wide gap also centered at height 100 m (see Figure 18). The distance between source and barrier is 10 m. We model the effect of the barrier on the PE sound field by the Kirchhoff approximation (zero sound pressure directly behind the barrier [7]). The sound field on the right-hand side of the barrier represents the reflected sound generated by a source located 10 m to the right of a 2 m high barrier (height 99-101 m).
The upper graph in Figure 18 shows the field of LF
calculated with PE at 125 Hz. The lower graph shows the value of LF at height 100 m as a function of range, for
31.5, 125, and 1000 Hz. Also included are the predictions of the Fresnel-zone model. Differences are about 2 dB at most.
Figure 18. Comparison between PE and Fresnel-zone model.
7. IMPLEMENTATION IN DUTCH MODEL We have implemented the (modified) Fresnel-zone model in a Matlab code for the Dutch standard model SRM for road traffic noise [1]. The geometrical reflection
attenuation LF was added to the sound contributions of
reflected sound rays.
We present here some results of calculations with vertical and tilted barriers. Figure 19 shows the calculated sound field for a road without barriers. We note that SRM assumes moderate downwind conditions. The road length is 200 m, and is located at x = 0 and y = -100 to +100 m in the graph (red line). We used the following parameters for the calculation.
- traffic (light/medium/heavy vehicles):
o flow 1000/50/50 vehicles/hour (7-19h), o flow 300/15/15 vehicles/hour (19-23h), o flow 100/5/5 vehicles/hour (23-7h), o speeds 100/80/80 km/h.
- road surface type: dense asphalt concrete, - road halfwidth 3 m,
- ground parameter B=1 (absorbing), - receiver height 4 m.
At 200 m distance from the road (x = 200 m, y = 0), we find a sound level Lden of 46.2 dB. With commercial
software tool Geomilieu we also find 46.2 dB.
Figure 19. Result of a calculation with the Dutch standard model (SRM), for a 200 m long road (red line at x=0). The color represents the day-evening-night sound level Lden.
Next we added a 5 m high vertical barrier to the situation, at 10 m from the road. We assumed a value of 0.8 for barrier reflection coefficient Figure 20 shows the sound field calculated for this situation. The sound levels are 15-20 dB lower than in the situation without the barrier (note the difference in color scales between and). The level at 200 m distance is 28.7 dB. With Geomilieu we also find 28.7 dB.
Figure 21 shows the result of a calculation with a second 5 m high (vertical) barrier on the other side of the road (also with =0.8). The sound levels are higher than in the situation with a single barrier, due to the reflection by the second barrier. At 200 m from the road, the sound level is 33.2 dB, which is 4.5 dB higher than in the situation with a single barrier. With Geomilieu we find a level of 33.4 dB.
Figure 21. As Figure 20, but now with a second barrier of 5 m height on the other side of the road, also at distance 10 m.
The purpose of using a tilted barrier is to reduce the sound level increase due to the reflection. Figure 22 shows that an angle of 5 degrees is already effective in this case. The sound level decreases from 33.4 dB to 29.2 dB, which is only 0.5 dB higher than the level of 28.7 dB for the situation with a single barrier. The graph shows results of four calculations:
- SRM for a single vertical barrier, - SRM for two vertical barriers, - SRM+ for two vertical barriers, - SRM+ for two tilted barriers.
Here SRM+ stands for the calculation method with the Fresnel-zone model implemented.
Figure 23 is similar to Figure 22, but now for an angle of 10 degrees instead of 5 degrees. We see that with 10 degrees the level increase due to the reflection is completely eliminated, and we are back at the level of 28.7 dB for the situation with a single barrier.
Figure 22. Spectra of sound level Lden, calculated
with SRM and SRM+ for four situations (see legend).
Figure 23. As Figure 22, but now for angle 10 degrees. The dashed and solid blue lines coincide.
8. CONCLUSIONS
In this paper we have analyzed and improved a calculation method for reflections by vertical and tilted noise barriers. The method can be used in combination with the Dutch standard calculation method for traffic noise (SRM).
The method yields a geometrical reflection attenuation LF, which is calculated from the ratio of the sizes of the
reflecting surface and the Fresnel zone. Based on comparisons with BEM calculations we have improved the method by limiting the decrease of LF with increasing
frequency to a maximum of 3 dB per octave. This approach resulted in a better agreement with the BEM results.
We have implemented the Fresnel-zone model in the Dutch standard calculation model (SRM) for road traffic noise. Calculations for situations with vertical and tilted barriers along a road showed that a tilt angle of 5 or 10 degrees was sufficient to eliminate the sound level increase due to barrier reflections. In other situations, a tilt angle of 5 or 10 degrees may not be sufficient.
It is of interest to consider implementation of the Fresnel-zone model in the European calculation model Cnossos [8]. The model description of Cnossos indicates that “reflections on vertical obstacles are dealt with by means of image sources”. And further the text mentions that “an obstacle is considered to be vertical if its slope in relation to the vertical is less than 15 degrees”. Reflections from obstacles with a slope larger than 15 degrees are ignored with Cnossos. This implies that the effect of tilting a noise barrier up to 15 degrees is currently neglected with Cnossos. For future versions of Cnossos one may consider implementation of the Fresnel-zone model for reflections. In this way, it will become possible to model with Cnossos reflections from tilted barriers.
Acknowledgement
This project has been funded by the Dutch institute RIVM.
9. REFERENCES
[1] Dutch standard traffic noise model (calculation method SRM2), Reken- en meetvoorschrift geluid 2012 (RMG2012), Staatscourant Nr. 11810, 27 juni 2012. An English translation of the model description is available through the Electronic Physics Auxiliary Publication Service (EPAPS) of the American Institute of Physics: see EPAPS Document No. E-JASMAN-126-051911.
[2] Birger Plovsing, Jorgen Kragh. “Nord2000. Comprehensive Outdoor Sound Propagation Model. Part 1: Propagation in an Atmosphere without Significant Refraction”, 31 December 2001. See page 99.
[3] D.C. Hothersall and J.N.B. Harriot: “Approximate models for sound propagation above multi-impedance plane boundaries,” Journal of the Acoustical Society of America, Vol. 97, pp. 918-926, 1995.
[4] R. Nota, R. Barelds, D. Van Maercke. “Harmonoise WP3. Engineering method for road traffic and railway noise after validation and fine-tuning”, HAR32TR-040922-DGMR10, 22 December 2004. [5] R. D. Ciskowski and C. A. Brebbia (eds.): Boundary
Element Methods in Acoustics, Elsevier, London, 1991.
[6] T. van Renterghem, E.M. Salomons, D. Botteldooren, “Efficient FDTD-PE model for sound propagation in situations with complex obstacles and wind profiles”, Acta Acustica, Vol. 91, pp. 671-679, 2005. See Annex A.
[7] E.M. Salomons: Computational Atmospheric Acoustics, Kluwer, Dordrecht, 2001.
