The noise directivity of railbuses

Applied Acoustics 72 (2011) 653–659

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The noise directivity of railbuses R. Gołe˛biewski ⇑ ´ , Umultowska 85, Poland Institute of Acoustics, Adam Mickiewicz University, 61-614 Poznan

a r t i c l e

i n f o

Article history: Received 24 August 2010 Accepted 1 March 2011 Available online 24 March 2011 Keywords: Railway noise Railbus

a b s t r a c t In this paper the noise generated by the railbuses is analysed. To find the directivity index of the railbuses a method of its estimation was proposed. This method is based on the assumption of symmetrical changes in the sound level during the passage of a railbus. The four directivity indices were analysed: monopole, cosine, squared cosine and cubed cosine. We find that the monopole directivity is the best to describe the noise generation by the moving railbuses. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The main parameters characterising the acoustic properties of noise sources are the noise source (or sources) position, the sound power level and directivity index of each of a partial source. For the traditional traction units the main noise sources are wheel/rail interactions (the vibration of the wheel – especially lateral surface and rail vibration), auxiliary equipment and the aerodynamic noise. The rolling noise (wheel/rail interaction) dominates at velocities below ca. 250 km/h, while the aerodynamic noise is dominant above this velocity. There are also other sources of noise generated by the railbuses (see Section 2). The sound power level, LWA, of the rail vehicles depends on: the type of track (type of the rail, sleepers and ballast), type of vehicles and velocity. In this paper we do not analyse the LWA parameter. Complete description of noise directivity of line sources includes the information on the vertical and horizontal directivity. The directivity index characterises the direction of noise emission by the source (acoustic power per unit angle in a specific direction). Most of the hitherto known models of railway noise generation, such Austrian, German, Swiss and Dutch model use both directivity indices (vertical and horizontal) to describe the sound radiation characteristics [1,2]. Additionally, most of noise prediction models have only a total directivity index (in dBA). Different types of horizontal and vertical directivity indices are presented in Fig. 3 (horizontal directivity) and Fig. 4 (vertical directivity) in Ref. [2]. All of the methods assume that for moving train, the majority of acoustic energy is emitted perpendicular to the moving direction. The choice of a specific directivity index (monopole, dipole, etc.) depends on the type of partial noise source, its position ⇑ Fax: +48 618295123. E-mail address: [email protected] 0003-682X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2011.03.001

and participation in the total noise. The participation of a partial noise sources in the total noise depends on many factors: the velocity of the source, the position above the railhead, etc. There are a few methods of the directivity index determination. The method which uses the microphone arrays is most often used. This method is very accurate but unfortunately it is expensive due to the number of microphones needed (at least a few microphones must be used). The main purpose of this paper is to propose a method of the horizontal directivity determination and to use it for estimation of the railbus noise directivity. There is no information about acoustic properties of railbuses because they are relatively new noise sources. In Polish regional and local rail transport, railbuses have become increasingly popular. This is an alternative type of rail transport – especially over relatively short distances. 2. Description of railbuses The railbus is a relatively new noise source in Poland. The oldest railbuses which are currently in use in Poland were manufactured after 2002 year. For this reason there is no information about the noise generated by these vehicles (directivity and sound power level). The railbus is the one- or two-spaced vehicle powered by one or two diesel engines. This type vehicles are mainly used on nonelectrified routes, where passenger flow is too low for traditional traction units. Because of their construction (one or two diesel engines located on the edges of the vehicle, the air conditioning systems located on the whole roof, the exhaust systems) the railbus cannot be rated among the other known passenger trains. The main noise sources of the railbuses are: – the rail/wheel contact, – the lateral surface of the wheel (wheel vibration),

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R. Gołe˛biewski / Applied Acoustics 72 (2011) 653–659

– the lateral surface of the vehicle (the vibration of the vehicle’s walls), – the diesel engines (the railbus has one or two engines – located on two cars), – the air conditioning systems (evenly placed on the roof), – the exhaust systems, – the lateral air-cooling of engines (symmetrical located in the vehicle). Usually in Poland, the railbuses are used on non-electrified routes (especially on single tracks) where passenger flow is too low for traditional traction units. In most cases, because of the condition of the tracks, the velocity does not exceed 70 km/h although the construction allowed velocity is 120 km/h. The results reported in this paper are based on the acoustic measurements performed at the line Poznan´ – Wa˛growiec. On this route, the railbuses manufactured by PESA Bydgoszcz type SA 13x, are used. The technical specifications of these vehicles are presented in Table 1. One of the railbuses is presented in Fig. 1.

Fig. 1. One of the railbuses studied.

3. The measurements Fig. 2. The measurement scenery.

100

D = 7.5m 1

90

D = 25m 2

80

Sound level [dB]

At the site where the acoustic measurements were performed, the terrain was flat, without trees and any other reflecting obstacles. The track was straight and level on an embankment of a height of about 0.5 m. The characterisation of the measurement site is given in Table 2. During the measurements, the temporal changes of sound level, LpA(t), of a single pass-by noise were measured at two distances (from the track’s centre): D1 = 7.5 m – at the height Hð1Þ o ¼ 1:4 m and D1 = 25.0 m – Hð2Þ o ¼ 4:0 m (Fig. 2). The time interval was Dt = 100 ms. During the measurements the speed V of each vehicle was measured. The atmospheric conditions during the measurements are described in Table 2.

70 60 50

4. Estimation of noise directivity 40

The railbus’s construction, e.g. the symmetrical positions of the main noise source, allows us to assume the symmetrical changes of the sound level. This assumption was used in this paper. An example of sound level as a time function during the passage of a railbus is presented in Fig. 3. Using the results of LpA(t) measurements made during the passage of the railbus, the following parameter, called as a shape factor, can be defined:

30 -10

-5

0

5

Fig. 3. An example of the sound level as a function of time during the passage of the railbus.

~2 ðt i Þ p ; pA ð0Þ

v~ i ¼ ~A2 Table 1 The technical specifications of the railbuses. Manufacturer

PESA Bydgoszcz

Axles system

B0 20 B0 (driving car with two axles, rolling car with two axles, driving car with two axles) 110

Number of seats Weight (tones) Length (mm) Width (mm) Height (mm) Wheel diameter (mm) Number of engines Power of engine (kW) Engine type Velocity (km/h)

77.5 41,700 2890 4135 840

2 350 MTU 6H1800R81 120

10

Time [s]

ð1Þ

where

~2A ðt i Þ ¼ p2o  100:1LpA ðti Þ p

~2A ð0Þ ¼ p2o  100:1LpA ðt¼0Þ ; po and p

¼ 2  105 Pa:

ð2Þ

The time t = 0 is the moment when the centre of the railbus passes ~ i is presented the receiver point (Fig. 4). An example of parameter v in Fig. 5. ~ i (Eq. (1)) was defined on the basis of sound The shape factor, v level measurements. In this chapter it will be shown how a shape factor can be calculated. The A-weighted squared sound pressure of noise, due to the unit length line source (lo = 1 m), may be written as:

p2A ¼

W A Q ðUÞqc 2

4p d

 GA ðd; U; c; HÞ;

ð3Þ

where WA (W) denotes the A-weighted sound power of the unit length line source, Q(U) characterises its directivity; qc is the char-

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R. Gołe˛biewski / Applied Acoustics 72 (2011) 653–659 Table 2 The characterisations of the measurement site. Measurement site

The track’s characteristics

Railway line Poznan´ – Wa˛growiec

The receiver points

Mean atmospheric conditions

Rail

Sleepers

Distance (m)

Height (m)

Wind,

Joint

Wooden

D1 = 7.5

H1o ¼ 1:4

2.1

D2 = 25.0

H2o ¼ 4:0

v (m/s)

Temperature, T (°C)

Humidity, h (%)

20.4

47

Fig. 4. The centre of the railbus passing the receiver point.

Fig. 6. Line source – receiver geometry in the horizontal plane.

1.5

D = 7.5m 1

In Eq. (3) we neglected other propagation effects such the air attenuation, scattering by the turbulence and acoustic refraction because the measurements were performed relatively close to the noise source. The A-weighted squared sound pressure (Eq. (3)) can be rewritten in the following form:

D = 25m 2

χi

1

p2A W A D2o ¼  HðUÞ  GA ðd; U; c; HÞ; p2o W o 4pd2

0.5

Do ¼ 1 m;

ð6Þ

and

Wo ¼ 0 -10

-5

0

5

10

Time [s] Fig. 5. The changes in vi parameter during the passage of the railbus.

" GA ¼ b 1 þ c 

 2 #1 d ; H

ð4Þ

p2A ðt i Þ : p2A ð0Þ

ð8Þ

For a specific source – receiver geometry with the known ground coefficient c, the vi parameter depends on the directivity coefficient, Q(U), only. For different directivity indices, the envelope of the sound level (or squared sound pressure) as a function of time can be properly modelled to minimize the error defined as follows:

S¼

X ~ i Þ2 : ðvi  v

ð9Þ

i

where b characterises reflection from the ground beneath the source and c is the ground parameter. For d ? 0, the function GA ? b. The values of c for various ground surfaces and the method of its determination can be found in Refs. [5,6]. The quantity

Hs þ Ho H¼ ; 2

ð7Þ

~ i (Eq. (1)) can be obtained from the measureThe shape factor v ments of sound level during the passage of a railbus. A similar parameter based on the calculated values of A-weighted squared sound pressure can be defined in the following form:

vi ¼ acteristic impedance of air, and d expresses the instantaneous source-receiver distance (Fig. 6). Function GA (Eq. (3)) describes the ground effect and depends on the distance d and the type of the ground. In this study the simplified form of the ground effect was used [3,4]:

p2o D2o ¼ 1012 W: qc

ð5Þ

is the mean height of propagation (Hs – height of the source, Ho – height of the receiver).

In this paper the following directivity indices were tested:

Q 1 ðUÞ ¼ A — monople directivity;

ð10Þ

Q 2 ðUÞ ¼ A cos U — cosine directivity;

ð11Þ

Q 3 ðUÞ ¼ A cos2 U — squared cosine directivity;

ð12Þ

Q 4 ðUÞ ¼ A cos3 U — cubed cosine directivity:

ð13Þ

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R. Gołe˛biewski / Applied Acoustics 72 (2011) 653–659

e ¼

D  IðtÞ: Vto

ð20Þ

where

IðtÞ ¼

Z

U2 ðtÞ

U1 ðtÞ

p2A ðUÞ dU : p2o cos2 U

ð21Þ

The angles U1(t) and U2(t) are defined below (Fig. 7):

Vt  l=2 ; D

tg U1 ¼

tg U2 ¼

Vt þ l=2 : D

ð22Þ

Eq. (21) can be rewritten in the following form: Fig. 7. The definition of angles U1(t) and U2(t).

IðtÞ ¼

W A D2o  KðtÞ; W o 4D2

ð23Þ

The noise event (passage of the railbus) can be characterised by the sound exposure level:

where the propagation function K(t),

LAE ¼ 10 logðeÞ;

KðtÞ ¼

ð14Þ

1

Z

p

Z

þ1

p2A ðtÞ dt ; p2o t o

1

t o ¼ 1s;

ð15Þ

is the relative noise exposure (of the whole line noise source). Using Eq. (6) we get,

e¼

Z

þ1

Z l

1

 p2A ðxÞ dt dx ; p2o to

ð16Þ

or in other form

e¼le ;

ð17Þ

where l is the length of the railbus and

e ¼

Z

þ1

1

ð24Þ

;

describes the noise attenuation due to the reflection from the ground (track’s segment specified by the angles U1 and U2). 4.1. Monopole directivity Let’s assume that the railbus generates the noise in all directions, in the horizontal plane, with the same intensity (the monopole directivity):

Q ! Q 1 ðUÞ ¼ A and KðtÞ ! K 1 ðtÞ:





Q ðUÞ cos2 U

cðD=Do Þ2 þ cos2 U

U1

where

e¼

U2

ð25Þ

Substituting

a ¼ cðD=Do Þ2 ;

p2A ðtÞ dt ; p2o to

ð18Þ

denotes the noise exposure due to the unit length of the line source. Using the source – receiver geometry (Fig. 6), the quantity e equals

Z þp=2 2 D pA ðUÞ dU e ¼ : Vto p=2 p2o cos2 U

ð19Þ

The above equation was derived under the assumption that the source is moving on the infinite track’s segment. In the other cases, Eq. (19) must be rewritten:

ð26Þ

we get the function K1(t) in the following form

K 1 ðtÞ ¼

A

Z

p

U2

U1

cos2 U : a þ cos2 U

ð27Þ

Solving the last integral (Eq. (27)) we get

K 1 ðtÞ ¼

2 A4

p

8 0qffiffiffiffiffiffi1 0qffiffiffiffiffiffi193 rffiffiffiffiffiffiffiffiffiffiffiffi< aþ1 aþ1 = a a A  a tan @ a A 5: a tan @ ðU2  U1 Þ þ tg U2 tg U1 ; a þ 1: ð28Þ

1

1

V =30km/h V =40km/h V =50km/h V =60km/h

0.8

V =30km/h V =40km/h V =50km/h V =60km/h

0.8

0.6

χi

χi

0.6

0.4

0.4

0.2

0.2

0 -8

-6

-4

-2

0

2

4

6

8

Time [s] Fig. 8. The changes in vi parameter during passage of the railbus (monopole directivity, the distance D = 7.5 m).

0 -8

-6

-4

-2

0

2

4

6

8

Time [s] Fig. 9. The changes in vi parameter during passage of the railbus moving with different velocities (cosine directivity, the distance D = 7.5 m).

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R. Gołe˛biewski / Applied Acoustics 72 (2011) 653–659

1

V =30km/h V =40km/h V =50km/h V =60km/h

0.8

1

Q =A Q =Acos(Phi)

0.9

Q =Acos2(Phi)

0.8

Q =Acos3(Phi)

0.7 0.6

χi

χi

0.6

0.4

0.5 0.4 0.3

0.2

0.2 0.1

0 -8

-6

-4

-2

0

2

4

6

0 -8

8

-6

-4

-2

Time [s] Fig. 10. The changes in vi parameter during passage of the railbus moving with different velocities (squared cosine directivity, the distance D = 7.5 m).

For a source moving from 1 to +1 (U1 = p/2 and U2 = +p/2) we obtain,

rffiffiffiffiffiffiffiffiffiffiffiffi  a : K 1 ðtÞ ¼ A 1  aþ1

ð29Þ

2

4

6

8

Fig. 12. The changes in vi parameter during passage of the railbus with different directivity index (velocity V = 30 km/h, the distance D = 7.5 m).

qffiffiffiffiffiffiffiffi a1 D1 0:1ðLAe ðD1 ÞLAe ðD2 ÞÞ 1  a1 þ1 qffiffiffiffiffiffiffiffi ; 10 ¼ D2 2 1  a2aþ1

rffiffiffiffiffiffiffiffiffiffiffiffi  W A V o Do a : e ¼ l  e ¼ l A 1 aþ1 W o V 4D

ð30Þ

Performing the measurements at two different distances from the track, D1 and D2, the noise exposure can be obtained:

oraz eðD2 Þ ¼ 100:1LAE2 ;

ð31Þ

ð33Þ

where

a1 ¼ cðD1 =Do Þ2

Using Eqs. (17), (20), (23), (24), and (29)we get

eðD1 Þ ¼ 100:1LAE1

0

Time [s]

and a2 ¼ cðD2 =Do Þ2 :

ð34Þ

All parameters in Eq. (33) are known with the exception of c parameter. Unfortunately, this parameter cannot be determined directly (the equation does not have analytical solution) and must be determined numerically. We get the correct value of c when the left, L, and right R(c), sides of the above equation are equal. If there is no c which makes L = R(c) then we take c^ which meets the condition:

^Þj ¼ minimum: error ¼ jL  Rðc

where LAE1 and LAE2 is the sound exposure level at the distance D1 and D2, respectively. Dividing both quantities we get

qffiffiffiffiffiffiffiffi a1 eðD1 Þ D2 1  a1 þ1 qffiffiffiffiffiffiffiffi ; ¼ a2 eðD2 Þ D1 1  a2 þ1

ð32Þ

or

ð35Þ

As can be seen, using the sound exposure level measured at two different distances from the track during the passage of the railbus, the ground parameter c can be determined. Then the parameter vi can be calculated (at the distance Dk):

vi ¼

p2A ðt i Þ Kðti Þ Iðti Þ ¼ : ¼ p2A ð0Þ Kð0Þ Ið0Þ

ð36Þ

where

1

V =30km/h V =40km/h V =50km/h V =60km/h

0.8

1

Q =A Q =Acos(Phi)

0.9

Q =Acos2(Phi)

0.8

Q =Acos3(Phi)

0.7

0.6

χi

χi

0.6

0.4

0.5 0.4 0.3 0.2

0.2

0.1 0 -8

-6

-4

-2

0

2

4

6

8

Time [s] Fig. 11. The changes in vi parameter during passage of the railbus moving with different velocities (cubed cosine directivity, the distance D = 7.5 m).

0 -8

-6

-4

-2

0

2

4

6

8

Time [s] Fig. 13. The changes in vi parameter during passage of the railbus with different directivity index (velocity V = 30 km/h, the distance D = 25 m).

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R. Gołe˛biewski / Applied Acoustics 72 (2011) 653–659

1

Q =A Q =Acos(Phi)

1

Q =Acos2(Phi)

0.8

0.6

χi

χi 0.4

0.4

0.2

0.2

-6

-4

-2

0

2

4

6

0 -8

8

Time [s]

I1 ðt i Þ ¼ U2 ðDk ; t i Þ  U1 ðDk ; t i Þ ffi 1 ffi 19 8 0 qffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffi ak þ1 ak þ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi< = ak ak ak A  a tan @ A ; þ a tan @ ak þ 1: tg U2 ðDk ; ti Þ tg U1 ðDk ; ti Þ ; ð37Þ where k = 1 (at the distance D1) or 2 (at the distance D2). The angles U1(Dk, ti) and U2(Dk, ti) are defined by the following equations:

  Vt  l=2 ; U1 ðDk ; ti Þ ¼ arctan Dk

 Vt þ l=2 : U2 ðDk ; ti Þ ¼ arctan Dk



ð38Þ The quantity ak equals

ak ¼ cðDk =Do Þ2 :

ð39Þ

4.2. Cosine directivity

ð40Þ

we get

A

Z

p

-4

-2

0

2

4

6

8

Fig. 15. The changes in vi parameter during passage of the railbus with different directivity index (velocity V = 60 km/h, the distance D = 25 m).

D1 0:1ðLAe ðD1 ÞLAe ðD2 ÞÞ 10 ¼ D2

pffiffiffiffiffiffiffiffi 1þa þ1 1 1  paffiffiffiffiffiffiffiffi ln pffiffiffiffiffiffiffiffi1 2 1þa1 1þa1 1 pffiffiffiffiffiffiffiffi ; 1þa þ1 a2 1  pffiffiffiffiffiffiffiffi ln pffiffiffiffiffiffiffiffi2 2

1þa2

ð45Þ

1þa2 1

where a1 and a2 are described by Eqs. (34). For the known value of c, the vi can be determined from

v2i ¼

I2 ðt i Þ I2 ð0Þ

ð46Þ

:

where

ak I2 ðt i Þ ¼ sin U2 ðDk ; t i Þ  sin U1 ðDk ; t i Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ak þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a1 þ 1  sin U1 ðDk ; t i Þ ak þ 1 þ sin U2 ðDk ; t i Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ak þ 1  sin U2 ðDk ; t i Þ a1 þ 1 þ sin U1 ðDk ; t i Þ ð47Þ where k = 1 (at the D1) or 2 (at the D2). The angles U1(Dk, ti) and U2(Dk, ti) are defined by Eqs. (38) and ak – by Eq. (39).

Using the cosine directivity:

Q ! Q 2 ðUÞ ¼ A cos U;

-6

Time [s]

Fig. 14. The changes in vi parameter during passage of the railbus with different directivity index (velocity V = 60 km/h, the distance D = 7.5 m).

4.3. Squared cosine directivity For the squared cosine directivity:

U2

U1

cos3 U : a þ cos2 U

ð41Þ

where a is described by Eq. (26). For the U1 = p/2 and U2 = +p/2 (the source moving from 1 to +1) the function K2 can be written as:

K 2 ðtÞ ¼

Q =Acos3(Phi)

0.6

0 -8

K ðtÞ ¼

Q =Acos2(Phi)

0.8

3

Q =Acos (Phi)

2

Q =A Q =Acos(Phi)

2A

Z þp=2

p

p=2

cos3 U : a þ cos2 U

ð42Þ

Using the dependence: 2

cos2 U ¼ 1  sin U;

ð43Þ

Q ! Q 3 ðUÞ ¼ A cos2 U;

ð48Þ

we get

K 3 ðtÞ ¼

A

Z

p

U2

U1

cos4 U : a þ cos2 U

ð49Þ

By analogy to the previous sections, the ground parameter can be determined numerically from the equation:

qffiffiffiffiffiffiffiffi a1 D1 0:1ðLAe ðD1 ÞLAe ðD2 ÞÞ 1  2a1 þ a1 a1 þ1 qffiffiffiffiffiffiffiffi : 10 ¼ D2 2 1  2a2 þ a2 a2aþ1

ð50Þ

and solving the integral (42) we obtain 2

K ðtÞ ¼

2A

p



pffiffiffiffiffiffiffiffiffiffiffiffi  a 1þaþ1 1  pffiffiffiffiffiffiffiffiffiffiffiffi ln pffiffiffiffiffiffiffiffiffiffiffiffi : 2 1þa 1þa1

The shape factor equals

ð44Þ

To determine parameter c the following equality must be numerically solved:

v2i ¼

I3 ðt i Þ I3 ð0Þ

;

where the function I3(ti), at the distance Dk is

ð51Þ

R. Gołe˛biewski / Applied Acoustics 72 (2011) 653–659 Table 3 The number of railbus passages for analysed directivity index. Distance, D (m)

5. The results

Directivity index

7.5 25

2

3

A

A  cos U

A  cos U

A  cos U

38 22

2 16

2 2

0 2

I3 ðti Þ ¼ ð0:5  ak Þ½U2 ðDk ; t i Þ  U1 ðDk ; ti Þ þ 0:5½sin 2U2 ðDk ; t i Þ ffi 0 qffiffiffiffiffiffiffi 1 ak þ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðctg U2 ðDk ; ti Þ  U1 ðDk ; t i ÞÞ ak ak A:  ak a tan @ 1 þ ak 1 þ akaþ1  tg U1 ðDk ; t i Þ  tg U2 ðDk ; t i Þ k

ð52Þ

4.4. Cubed cosine directivity

Q ! Q 4 ðUÞ ¼ A cos3 U;

ð53Þ

we get

Z

p

U2

U1

cos5 U : a þ cos2 U

ð54Þ

The ground parameter can be determined numerically from the equation:

D1 0:1ðLAe ðD1 ÞLAe ðD2 ÞÞ 10 D2

pffiffiffiffiffiffiffiffi 0:5a21 a1 þ1þ1 2=3  a1 þ pffiffiffiffiffiffiffiffi ln pffiffiffiffiffiffiffiffi a1 þ1 a1 þ11 pffiffiffiffiffiffiffiffi : ¼ 0:5a22 a2 þ1þ1 2=3  a2 þ pffiffiffiffiffiffiffiffi ln pffiffiffiffiffiffiffiffi a2 þ1

ð55Þ

v2i ¼

I4 ð0Þ

ð58Þ

i

~ i and vi showed that in most cases the monopole Comparison of v directivity index gives the smallest error between these parameters. At the distance D = 7.5 m, in 38 cases (the total number of railbus passages was 42) the monopole directivity index was the best and at the distance D = 25 m – in 22 cases (Table 3).

;

The main purpose of this paper was to describe a method of the horizontal directivity estimation of the railbuses. In the paper four directivity indexes were analysed: monopole, cosine, squared cosine and cubed cosine. Using these directivities the shape factors were calculated and compared to a similar parameter determined on the basis of on the acoustic measurements of the single pass-by noise. Analysis of the obtained results of this comparison allowed us to point out the monopole directivity as the best directivity of moving railbuses. This directivity index gives the smallest error (defined by Eq. (58)). This choice can be explained by the partial noise sources (a lot of these sources), symmetrically located in the vehicles analysed. Acknowledgements

a2 þ11

The shape factor is

I4 ðti Þ

X ~ i Þ2 : ðvi  v

6. Conclusion

For the cubed cosine directivity:

A

Using the results of the measurements performed, for each pas~ i parameter was determined (at two dissage of the railbus the v tances D1 and D2). Then the vi parameters were calculated – using the method presented in this paper, for the four directivity indices analysed (monopole, cosine, squared cosine and cubed co~ i and vi) the following error sine). For each pair of shape factors (v was determined:

S¼

 sin 2U1 ðDk ; t i Þ

K 4 ðtÞ ¼

659

The author is grateful to Agata Kłos and Michał Kaczmarek for their help during the field measurements.

ð56Þ References

and the function I4(ti), at the distance Dk is

I3 ðti Þ ¼ ðak  1Þ½sin U1 ðDk ; ti Þ  sin U2 ðDk ; ti Þ a2k 3 3 ffi þ 1=3ðsin U1 ðDk ; ti Þ  sin U1 ðDk ; t i ÞÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ a2k pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 þ ak þ sin U2 ðDk ; ti Þ 1 þ ak  sin U1 ðDk ; ti Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ak  sin U2 ðDk ; ti Þ 1 þ ak þ sin U1 ðDk ; ti Þ ð57Þ Examples of the calculated shape factors vI, for different directivity indices and source velocities are presented in Figs. 8–15.

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