PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Modeling and prediction of vehicular traffic noise through direct,
specular and diffuse paths at bus stops
Dayane Cristina Lima ESTERCIO1; Paulo Fernando SOARES2;
1

State University of Maringa, Brazil

2State University of Maringa, Brazil

ABSTRACTBus stops for public transport systems are structures where users are exposed to intense levels of noise andare poorly evaluated in surveys. Therefore, the goal of this research is to develop a mathematical model ofnoise level prediction at bus stops, in order to contribute to the characterization of the traffic noise level. For
this, a prediction model was developed based on direct waves and specular and diffuse reflections. A standard
bus stop model was adopted on a three-lane roadway, simulating the user standing and sitting. The soundsource was positioned on the axis of each strip, every five meters. In total, 126 source positions were
evaluated. The descriptors, sound pressure level (SPL) and sound equivalent level (LAeq) were calculated.
For the validation, the Anderson and Kurze model was applied, and later the Student t-test. The resultspresented better correspondence between the developed model and the reference model in the severe bands,
whereas in the 8 kHz frequency there was a greater variance between the models. As a preliminary study, theresearch presented satisfactory results, which may contribute to future research.
Keywords: prediction model, road traffic noise, radiosidy

1. INTRODUCTION
Traffic noise is the main source of urban noise. With the cities growth this problem has been
aggravated by the increase in demand for individual and collective transportation. Studies show that
exposure to high levels can cause harmful effects to people's health (1-3). For this reason, studies
related to urban noise have been extensively done (4-5).
In addition to studying and evaluating noise, studies are important tools in the planning anddevelopment of urban projects (6). Traffic noise prediction models are created from fieldmeasurements and analysis of different noise descriptors or traffic noise parameters. The developmentof noise prediction models allows the identification of critical points, the evaluation of population
exposure and the simulation of mitigation measures in noise maps (7).
The simplest models are based on the relationships between the number of vehicles and the levelof noise they emit (8-9). On the other hand, the more complex models calculate other parameters, suchas the speed, percentage of heavy vehicles, road configurations such as width and inclination, building
feedback, barriers, among others (9-11). Other studies have recently addressed the modeling of soundpropagation through specular or diffuse reflection analysis (12-15). The researches which use thisapproach allow the prediction of the behavior of sound propagation in different urban configurations,as well as the path of reflected sound waves (specular and diffuse).
Based on this, the main goal of this research is to develop a mathematical model of prediction of
the level of noise of road traffic, calculating the direct, specular and diffuse ways in bus stops, sincethese are spaces where the users are exposed to intense levels of and are poorly addressed in studies.
This research seeks to contribute to the studies of traffic noise prediction models and to predict thesound levels in which users of the public transport system are exposed to at bus stops.

1
2

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arqdayanestercio@gmail.com
pfsoares@uem.br2. METHODOLOGY
For the development of the research, a standard urban geometry model was elaborated. In the model,
a standard bus stop was dimensioned, inserted in the middle of the block. The modeled street has threelanes with three-story buildings (9 meters high) on each side of the track. In each band, a sound sourceof 100 dB sound power level, measured every 5 meters of distance, was calculated and positioned 0.5
m above the street. In total, in each range, 21 source positions were calculated, totaling 63 positionsin the three bands, as shown in Figure 1.

Figure 1 – Schematic model of the standard bus stop model and its insertion into the created geometry
Two scenarios were simulated separately, the first one simulating the user standing and the second
one the user seated. At each point, the sound pressure level (SPL) and the sound equivalent level(LAeq ) were calculated, in octaves of frequency. For the validation, the Anderson and Kurze model,
and later the Student's t-test, were applied to verify the variance between the models.
2.1 Theoretical Model
The theoretical model calculates the direct paths and specular and diffuse reflections, these paths
correspond to the pulses emitted from a sound source (Figure 2). The model simulates an urban street,
limited by buildings up to three decks. Each plane, presents absorption and scattering coefficients,
adopted later in the calculations of the sound reflections.

Figure 2 – Profile of the model with direct paths and specular and diffuse reflection

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The direct path corresponds to the wave emitted by the source that arrives at the receiver directly
without reflections. It is obtained by calculating the intensity (Id) given by the relation between thesound power of the source and the distance between the source and the receiver. In equation 01 thedirectivity factor (Ω) and the losses due to the absorption of the air (ψ) are adopted.
ȳͶ‫ܫ‬ௗ ൌ ܹ௘ ൅ ͳͲ݈‫ ݃݋‬ቆ
ቇ൅݁൬
൰
െ߰݀ ்
Ͷȫ݀ ଶ்

(1)

The attenuation due to air absorption is calculated using the methodology of ISO 9613 -1: 1993.
The losses (ψ) are calculated by the sum of the classical losses due to thermal effects, rotational
relaxation, and vibrational relaxation of the oxygen and nitrogen molecules (16), as shown in Equation
2.
(2)

߰ ൌ ߰௖௟ ൅  ߰௥௢௧ ൅ ߰ைమ௩௜௕ ൅ ߰ேమ௩௜௕

The intensity of the specular path is obtained by means of the relation between the sound power ofthe source and the distance between the source and the receiver. In addition to the Ω directivity factor
and the losses due to the absorption of air ψ, the absorption and scattering coefficients of each planein which the sound wave is incident and reflected are calculated (Equation 3). For the calculation of
the specular path, the image source method is used, where the sound energy reflects at the same angle
as it is on the plane. In this method the mirror image of the sound source will have the same distanceon the opposite side of a reflecting plane (17).
ଵ଴ȳͶ‫ܫ‬௘ ൌ ܹ௘ ൅ ͳͲ݈‫ ݃݋‬ቆቇ൅݁൬൰ ෍ൣ൫ͳ െ ሺߙఈ ൅ ߞఈ ሻ൯൧െ߰݀ ்Ͷȫ݀ ଶ்

(3)ఈୀଵ

For the calculation of the specular and diffuse paths, absorption coefficients (α) and scattering (ζ)
were adopted for each material adopted in the model, as shown in Table 1. The absorption coefficient(α) is the ratio between the sound energy is not reflected and the incident sound energy, is the sum ofenergy absorbed and transmitted (18). The scattering coefficient (ζ) is the ratio of the reflected soundenergy in non-specific directions to the total reflected sound energy, the dispersion of the energy
distributed unevenly according to the roughness of the material (18).
Table 1 – Absorption coefficients and scattering
ABSORPTION COEFFICIENT (Hz)SCATTERING
COEFFICIENT

MATERIALS

Asphalt (street)
Concrete(sidewalk)

63

125

250

500

1000

2000

4000

8000

AVERAGE (Hz)

0.02

0.02

0.03

0.03

0.03

0.03

0.02

-

0.25

0.02

0.02

0.03

0,03

0.03

0.04

0.07

0.07

0.5

0.01

0.01

0.02

0.02

0.02

0.02

0.05

0.05

0.1

-

-

-

-

-

-

-

-

0.1

SmoothConcrete(external walls)
Bus stop(metal tile)

SOURCE: Adapted (19-21).

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For the calculation of the diffuse path, the first and second order radiosity model was adopted. The
radiosity model divides the space (model) into stretches and calculates path of noise from one stretch
to the other (18). In the first-order radiosity, the path between the source and the receiver is only adiffuse reflection in one of the planes of the model. In the section where the reflection is calculated
the sound power of the section of the plane is calculated as shown in Equation 4. In the calculation of
the sound power reflected by the section, the sound power emitted by the source, the scattering
coefficient and the area of the stretch. Subsequently, the diffuse path of the reflected segment iscalculated to the receiver.
ȳͶܹ௣ ൌ ܹ௘ ൅ ͳͲ݈‫ ݃݋‬ቆቇ൅݁ቆቇߞ ‫ܣ‬
ଶͶȫ݀௘௣െ߰݀௘௣ ‫ ݌‬௉

(4)

The intensity of the first-order diffuse path is obtained by means of the relation between the soundpower of the source of the "p" section and the distance between the first reflection section and thereceiver. In equation 5, the directivity factor (Ω) and the losses due to the absorption of the air (ψ) are
adopted for the calculation.
ȳͶ‫ܫ‬ଵ ൌ ܹ௣ ൅ ͳͲ݈‫ ݃݋‬ቆ
ቇ൅݁ቆ
ቇ
ଶ
Ͷȫ݀௣௥
െ߰݀௣௥

(5)

On the second order, radiosity path a double diffuse reflection is calculated, that is , from the soundsource, the stretch of the first reflection, the stretch of the second reflection, and then the receiver (r).
In the second order of radiosity the path between the first reflection and the second reflection is
calculated the diffuse energy radiated from the first stretch to the second, by means of the form factor(FF), as shown in Equation 6.
ȳ‫ܨܨ‬௝௜ ݁ ିటௗ೔ೕ
Ͷ‫ܫ‬ଶ ൌ ܹ௣ ൅ ͳͲ݈‫ ݃݋‬ቆ
ቇ൅݁ቆ
ቇ
ଶ
െ߰݀௝௥
Ͷȫ݀௝௥

(6)

The form factor (Figure 3) calculates the fraction of energy that is radiated from one surface andreaches another surface (22), taking into account the distance, shape and relative orientation of thetwo planes. For the calculation of the FF used in Equation 6, the form factor was adopted from theHemi-cube.
This way of calculating the FF, is similar to the geometric analogy of Nusselt in that it replaces thehemisphere with a hemicube and dividing the faces into small flaps (pixels) (23). For the calculationof the FF, the radius between the planes and the cosine of the angles i and j is calculated. (Eqs. 7, 8,
9 and 10).

Figure 3 – Derivation of delta Form-Factor

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(7)

‫ ݎ‬ൌ ඥ‫ ݕ‬ଶ ൅ ‫ ݖ‬ଶ ൅ ͳܿ‫׎ݏ݋‬௜ ൌܿ‫׎ݏ݋‬௝ ൌ‫ ܨܨ‬ൌܼඥ‫ ݕ‬ଶ ൅ ‫ ݖ‬ଶ ൅ ͳͳඥ‫ ݕ‬ଶ ൅ ‫ ݖ‬ଶ ൅ ͳܿ‫׎ݏ݋‬௜ ൈ ܿ‫׎ݏ݋‬௝ȟ‫ܣ‬ߨ‫ ݎ‬ଶ

(8)

(9)

(10)

After calculating the paths, it is calculated the total intensity (It), which is the sum of all the pulses(paths) emitted by the source that arrive at the receiver, as shown in equation 11. The model was
calculated in octave bands, in the bands from 63Hz to 8kHz.
‫ܫ‬௧ ൌ ‫ܫ‬ௗ ൅ ෍ ‫ܫ‬௘ ൅ ෍ ‫ܫ‬ଵ ൅ ෍ ‫ܫ‬ଶ(11)Subsequently, the comparison between the methods was calculated the descriptor Level equivalentsound (LAeq ), as shown in Equation 12. For the validation the Anderson and Kurze model was applied.
The Anderson and Kurze model describe a basic equation (Eq. 13) of outdoor sound propagation (24).
In this model the solid angle (Ω) made available to the free propagation source is calculated. The
directivity (DIɵ) and A combined , corresponds to the attenuation between the source and the receiver.
௡ͳ‫ܮ‬஺௘௤ ൌ ͳͲ  ൜ ൤෍ ൫ͳͲ௅௜Τଵ଴ ൯൨ൠ݊௜ୀଵ‫ܮ‬௣ ሺ‫ݎ‬ǡ ߠሻ ൌ ‫ܮ‬ௐ െ ʹͲ  ‫ ݎ‬൅ ‫ܫܦ‬ఏ െ ͳͲ ȳെ ‫ܣ‬௖௢௠௕௜௡௘ௗ െ ͳͳ݀‫ܤ‬
Ͷߨ

(12)

(13)

To verify the similarity between the models, the t-Student test was adopted. The test consists of aprobabilistic model that calculates the variance between two samples. The sample size (n) is called
degrees of freedom, which corresponds to the value of n - 1 (25). The t-critical is obtained according
to the level of significance and degrees of freedom. If the t-critical is greater than or equal to thecalculated, we accept Hypothesis H 0 , that is, no significant variance between the samples (26). The
value of t is obtained from equation 14 (26).

‫ݐ‬ൌሺܺതଵ െ ܺതଶ ሻܵଶ ܵଶඨ ଵ൅ ଶ݊ଵ ݊ଶ(14)Where ܺതଵ ܺതଶ are the means of each group. S1 and S2. corresponds to the standard deviations of
the respective groups and n1 and n2 are the point numbers.

3. RESULTS AND DISCUSSION
The calculated results of the 126 source positions, 63 of the model developed and 63 of the modelof Anderson and Kurze, considered the users standing with height of 1.5 m and the users seated withheight of 1.1 m. The overall value of the sound spectrum as shown in Figure 4 shows the L Aeq valuesof the 126 source positions.
The results showed similarity between the models, in general the difference between the models
was higher in the 03 range presenting values ranging from 0.7 dB to 0.9 dB between the developedmodel and the reference model. On the other hand, bands 1 and 2 presented values between 0.2 dB to
0.4 dB, for band 1 and 0.1 dB to 0.7 dB in band 02.

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Figure 4 – LAeq of each point
The values obtained (Figure 4) were closer between the models at the points closest to the receiver,
but were the ones that presented the most variation between the bands. This greater variation in points
near the receiver occurred due to the distance between the source and the receiver being smaller,
presenting a lower attenuation due to the atmospheric absorption and the absorption of the scatteringof the materials. The comparison between the two scenarios (standing and sitting users) shows thatstanding users are more exposed to the noise emitted by the source.
The total value of the sound spectrum in the developed model presented Leq values of 78.3 dB,
75.5 dB and 73.9 dB, in lanes 1, 2 and 3 respectively, for the standing user, and 77.9 dB, 75.3 dB and
73.8 dB for the seated user. In the reference model of Anderson and Kurze, the values were 78.4 dB,
75 dB and 73.2 dB, in lanes 1, 2 and 3 for the standing user, for the seated user were 77.7 dB, 74 .7
dB and 73 dB, respectively. The variance between methods was shown to be lower in the bass
frequencies and in the bands closer to the receiver.
The values obtained show the level at which users using public transport systems are exposed atbus stops. The values (Figure 4) become more intense as they approach the bus stop. Exposure tolevels above 70 dB can lead to cardiovascular problems and changes in the auditory system (27).
Student's t-test was used to determine the significance level of 5% (α = 0.05). For the calculation,
21 samples corresponding to the 21 source positions in each range were adopted. In the similarity
analysis, the developed model and the reference model were compared. Thus, the samples of eachmodel were added, totaling 42 samples. It is observed that for each group analyzed we calculated (n 1). In this way the value of 40 degrees of freedom was obtained.
The first analysis of similarity of the two models, the global value of the sound spectrum of eachband was calculated for the user sitting and standing. The test as shown in Table 2, presented values
below the t-critical value. For the two-tailed test with 40 degrees of freedom the t-critical value
corresponds to 2.021.

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Table 2 – Bi-caudal t test of similarity of methods
SCENARIOS

MODELAVAREGEVARIENCE

Standing

Developed

72.62

43.68

Lane

personReference

72.40

44.98

01

Sitting

Developed

72.62

41.57

personReference

72.29

42.41

Standing

Developed

72.07

29.52

Lane

personReference

71.57

29.88

02

Sitting

Developed

72.08

28.02

personReference

71.44

28.28

Standing

Developed

71.55

28.38

Lane

personReference

70.85

21.65

03

Sitting

Developed

71.56

20.32

personReference

70.73

20.55

T

P(T<=T)

0.105

0.917

0.164

0.871

0.300

0.766

T CRITICAL

2.021
0.386

0.701

0.496

0.623

0.596

0.555

The "t" values (Table 2) show that there is no significant difference between the models. Thus
presenting similarity between the developed model and the reference model.
The second analysis was calculated considering the users standing and seated in each b ands in eachfrequency range. The calculated results (Figure 5) showed greater similarity between the models at
low frequencies and in the band closest to the receiver.

Figure 5 – Teste t-student by frequency lane
Analysis by frequency range, presented values of "t" inferior to the value t -critical, for 40 degrees
of freedom. This means that the difference between the models is less than 5%, that is, the values arein the range of -2.021> t <2.021. In this way the hypothesis of Ho is accepted, in which it is assumed
that there are no significant differences between the developed model and the Anderson and Kurze.

4. CONCLUSIONS
It is concluded that the prediction model based on the calculations of the paths (direct, spec ular
and diffuse) allowed to predict the sound levels at the bus stop. The sound levels calculated in themodel presented more intense values in the bands near the receiver, thus showing intensity that theusers are exposed, and that can be harmful to health. The application of the t-Student test to evaluate
the similarity, allowed to validate the developed model, comparing with the reference model ofAnderson and Kurze. The difference presented between the models was less than 5%, that is, themodel shows similarity higher than 95% with the reference model.
As a preliminary study, the research presented satisfactory results in the prediction of road noiselevels reaching the receiver (standing or sitting user), emitted from a sound source. It is hoped thatthe study may contribute to evaluations of vehicle traffic noise characterization at bus stops and also
to the application in other contexts.

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ACKNOWLEDGEMENTS
The present work, developed at the State University of Maringá, was carried out with the supportof the Coordination of Improvement of Higher Education Personnel - Brazil (CAPES) and, forparticipation in the Congress, obtained the support of the "Youth Conference Grants Program
Scientists from ICA-ASA-DEGA ".

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