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A numerical model for re-radiated noise in buildings from

underground railways

P. Fiala1,2, S. Gupta2, G. Degrande2, F. Augusztinovicz1

1Vibroacoustics and Audio Laboratory, Budapest University of Technology and Economics, Magyar

tudósok körútja 2. H-1117, Budapest, Hungary,

2K.U.Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001, Leuven, Belgium

Tel: +36 1 463 2543, Email: fiala@hit.bme.hu

Abstract

A numerical prediction model is developed to quantify vibrations and re-radiated noise due to

underground railways.

A coupled FE-BE model is used to compute the incident ground vibrations due to the passage of a

train in the tunnel. This source model accounts for three dimensional dynamic interaction between the

track, tunnel and soil. The incident wave field is used to solve the dynamic soil-structure interaction

problem on the receiver side, to determine the vibration levels along the essential structural elements

of the building. The soil-structure interaction problem is solved by means of a 3D boundary element

method for the soil coupled to a 3D finite element method for the structural part. An acoustic 3D

spectral finite element method is used to predict the acoustic response.

The Bakerloo line tunnel of London Underground has been modelled using the coupled periodic

FE-BE approach. The free-field response and the re-radiated noise in a portal frame office building is

predicted.Thenumericalsourcemodelaccountsformovingloadsandvariousexcitationmechanisms,

including quasi-static loads, random loads due to the rail and wheel unevenness and impact excitation

due to the rail joints. The effect of wall absorption on the internal pressure levels is investigated, and

the problem of modelling wall openings by absorbing impedance boundary conditions is dealt with.

1 Introduction

Ground-borne vibrations induced by underground railways are a major environmental concern in

urban areas. These vibrations propagate through the tunnel and the surrounding soil into nearby build-

ings, causing annoyance to people. Humans in buildings are affected both by vibrations of the struc-

ture (1-80 Hz) and through the re-radiated noise (> 80 Hz) from the walls and ceilings of the rooms.

For the prediction of ground-borne vibrations and re-radiated noise in buildings, a modular architec-

ture is adopted, which consists of the following subproblems: the dynamic vehicle-track-tunnel-soil

interaction problem, the dynamic soil-structure interaction problem, and finally the prediction of vi-

brations and primary re-radiated noise in the structures. The first two subproblems can be assumed

weakly coupled, if the distance between the source and the receiver is much larger than the dominant

wavelengths in the soil. The second and third subproblems are weakly coupled too, as the mechan-

ical impedance of the air inside the rooms is much smaller than the mechanical impedance of the

building’s walls.

In the first subproblem, the free field vibrations are predicted, by computing the contact force

generated by the wheel/track interaction and then solving the dynamic track-tunnel-soil interaction.

Fiala, Gupta, Degrande, Augusztinovicz

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The dynamic track-tunnel-soil interaction problem is tackled using the coupled periodic FE-BE model

developed within the framework of CONVURT [1]. The Floquet transform is used to exploit the

periodicity of the problem, which allows to model wave propagation over multiples of wavelengths

both in the longitudinal and transverse directions of the tunnel, which would have been impossible

with standard three-dimensional finite element and boundary element formulations. A finite element

method is used to model a periodic unit of the tunnel, while a boundary element method is used to

model the soil as a horizontally layered elastic half space [2].

Once the incident wave field in the soil has been determined, the dynamic response of a three-

dimensional building due to this incident wave field is computed. Similarly to the tunnel-soil interac-

tion, a subdomain formulation is employed where a finite element method is used for the structure and

boundary element formulation is used for the soil. This approach allows to investigate the influence of

dynamic soil-structure interaction on the structural response [3]. The kinematics of the structure are

described by its rigid and flexible modes. The Craig-Bampton substructuring is used to differentiate

between the foundation modes and their quasi-static transmission into the building, and the modes of

the superstructure on rigid base.

In the third subproblem, the computed structural displacements are used as a vibration input for the

computation of ground-borne noise in the building’s enclosures. For typical room dimensions in office

buildings or family dwellings, the targeted frequency range is relative low. Therefore, deterministic

3D methods can be used to solve the acoustic wave equation inside the enclosure. A spectral finite

element method is applied to the acoustic problem, which, for the case of low wall absorption, can

lead to a direct integral representation of the internal sound pressure.

To demonstrate the efficiency of the approach, the tunnel on the Bakerloo line of London Under-

ground is modelled using a coupled periodic FE-BE approach. The free field response is predicted in

the frequency range 1-150 Hz, and subsequently the re-radiated noise in a hypothetic nearby multi-

story portal frame office building is estimated (Fig 1).

uinc

pa

vs

x

z

Figure 1. Problem outline.

2 The incident wave field

The model of a moving vehicle on a track periodic or invariant in the longitudinal direction is used

to compute the incident wave field. The periodicity of the tunnel and the soil in the longitudinal y

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directionisexploitedusingtheFloquettransform,limitingthediscretizationefforttoasinglebounded

reference cell and formulate the problem in the frequency-wavenumber domain [4,2].

The vehicle is modelled as a set of concentrated masses representing the train’s unsprung masses,

connected to the track by Hertzian springs. The reference cell of the track and the tunnel is modelled

by means of a finite element method. The tunnel is embedded in a horizontally layered soil, charac-

terized by its shear wave velocity Cs, dilatational wave velocity Cp, material density ρ and hysteretic

material damping ratio β for each homogeneous layer.

First, the vehicle-track interaction problem is solved. The compliance matrices of the vehicle

ˆCv(ω) and the track-soil systemˆCt(ω) are used to compute the dynamic axle loads ˆ g(ω) arising

due to the rail unevenness ˆ uw/r(ω) as

[ˆCv(ω) +ˆCt(ω)]ˆ g(ω) = ˆ uw/r(ω)

(1)

The soil’s dynamic stiffness, which is incorporated in the track-soil compliance matrix, is computed

with a periodic boundary element formulation with Green-Floquet functions defined on the periodic

structure with period L along the tunnel [5,6,4].

In a second phase, the free-field soil vibrations ˆ ui(˜ x + nLey,ω) due to a set of axle loads ˆ gk(ω)

moving in a periodic domain are computed as [7]:

ˆ ui(˜ x + nLey,ω) =

1

2π

?∞

−∞ˆ gk(ω − kyv)e−iky(nL−yk)

?L/2

−L/2e−iky˜ y′˜ˆhzi(˜ y′,˜ x,κ0,ω)d˜ y′dky

(2)

where˜ˆhzi(˜ y′,˜ x,κ0,ω) is the Floquet transform of the transfer function in the frequency-wavenumber

domain relating the displacement at ˜ x to a unit load at the position {˜ x = 0, ˜ y′, ˜ z = 0}.

The transfer functions are computed by means of the coupled periodic FE-BE model using the

classical domain decomposition approach based on the finite element method for the tunnel and the

boundary element method for the soil [4,2,5,6].

(a)

rp

1.88m

ri

1.829m

re

1.953?m

ts

0.022m

hb

0.102m

(b)

Figure 2. (a) Cross section of the metro tunnel on the Bakerloo line at Regent’s Park. (b) Finite element model

of the reference cell.

The Bakerloo line tunnel of London Underground is a deep bored tunnel with a cast iron lining and

asingletrack,embeddedinLondonclayatadepthof28m.Thetunnelhasaninternalradiusof1.83m

and a wall thickness of 0.022 m (Fig. 2). There are six longitudinal stiffeners and one circumferential

stiffener at an interval of 0.508 m, resulting in a periodic structure. Dynamic soil characteristics have

been determined by in situ and laboratory testing [8]. For the numerical calculations, the tunnel is

assumed to be embedded in a layered soil consisting of a single shallow layer with a thickness of 5

Fiala, Gupta, Degrande, Augusztinovicz

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m on top of a homogeneous half space consisting of clay. The top layer has a shear wave velocity

Cs= 275 m/s, a longitudinal wave velocity Cp= 1964 m/s, a density ρs= 1980kg/m3and a material

damping ratio βs= 0.042. The underlying half space has a shear wave velocity Cs = 220 m/s, a

longitudinal wave velocity Cp= 1571 m/s, a density ρs= 1980kg/m3and a material damping ratio

βs= 0.039.

The track is a non-ballasted concrete slab track with Bull head rail supported on hard wooded

sleepers nominally spaced at 0.95 m with cast iron chairs. Both ends of a sleeper are concreted into

the invert and the space between the sleepers is filled with shingle. Resilience is mainly provided

by the timber sleepers, as the rails are not supported by rail pads. The rails are modelled as Euler-

Bernoulli beams and are attached to the invert via a continuous elastic support. The smeared stiffness

of the support is assumed as¯ktr= 80MN/m2, while a value of 12 × 104Ns/m2is assumed for the

distributed damping in the support ¯ ctr.

The train consists of seven cars: a driving motor car, a trailer car, two non-driving motor cars, two

trailer cars and a driving motor car. The length of a motor car is 16.09 m, while the length of the trailer

car is 15.98 m. The bogie and axle distances on all cars are 10.34 m and 1.91 m, respectively. The

distance between the first and the last axle of the train is 108.33 m. The wheels are of the monobloc

type and have a diameter of about 0.70 m. The tare mass of a motor car is 15330 kg, while the bogie

mass is 6690 kg and the mass of wheel set is 1210 kg. The tare mass of a trailer car is 10600 kg,

while the bogie mass is 4170 kg and the mass of a wheel set is 950 kg. For the computations, only the

unsprung mass of the cars is considered to act on the rails. The speed of the train is 50 km/h.

The rail roughness is expressed as a stochastic process characterized by a single-sided power spec-

tral density (PSD)˜Sw/r(ky), written as a function of the wavenumber ky= ω/v = 2π/λy[9]:

˜Sw/r(ky) =˜Sw/r(ky0)

?ky

ky0

?−w

(3)

The parameters˜Sw/r(ky0) and w depend on the quality of the rail. For rail unevenness, ky0= 1 rad/m

and w = 3.5 are commonly assumed. An artificial profile uw/r(y) is generated from the PSD-curve

based on the superposition of simple random processes with known statistical properties. The PSD

ˆSw/r(ω) of the unevenness in the frequency domain is found as 1/v˜Sw/r(−ω/v) and increases with

the vehicle speed proportional to v2.5, where w = 3.5 in equation (3). Thus, an increase in speed is

expected to give rise to higher dynamic axle loads and higher vibration levels.

(a)

−10 −8 −6 −4 −2 0 2 4 6 8 10

Time [s]

−5

0

5

10

x 10

−5

Velocity [m/s]

(b)

0 50

Frequency [Hz]

100 150

0

0.5

1

1.5

2

2.5

3

3.5x 10

−5

Velocity [m/s/Hz]

Figure 3. (a) Time history and (b) frequency content of the vertical (blue) and horizontal (red) component of

the incident wave field in the soil at coordinates {−5 m,−7.5 m,0 m}.

Fig. 3 shows the horizontal (x) and vertical component of the free-field incident velocity computed

Fiala, Gupta, Degrande, Augusztinovicz

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in the corner point of the building at coordinates {−5 m,−7.5 m,0 m}. The dominating part of the

frequency content is between 20 Hz and 80 Hz with a peak at 50 Hz, which corresponds to the

wheel-track resonance frequency. The bogie passages are not clearly visible in the time history as the

tunnel is situated at a considerable depth, however, due to the specific train composition, the observed

velocity spectrum is quasi-discrete. The computed incident wave field has been validated by means

of the experiments performed on the Bakerloo line [8,10].

3 Dynamic soil-structure interaction

A weak coupling between the incident wave field and the structure is assumed, meaning that the

presence of the building has no effect on the vibration generation mechanism and the free field dis-

placements are applied as an excitation on the coupled structure-soil model.

The decomposition method proposed by Aubry et al. [11] and Clouteau [12] is used to formu-

late the dynamic soil-structure interaction problem. The finite structure is modelled in the frequency

domain by a 3D structural finite element method. The equation of motion of the building is:

Kss

Ksb

KbsKbb+ˆKg

bb

− ω2

MssMsb

MbsMbb

ˆ us

ˆ ub

=

0

ˆfb

(4)

where the structural displacements are separated to the displacement DOF of the foundation ˆ uband

the remaining DOF of the superstructure ˆ us. M and K denote the finite element mass and stiffness

matrices, andˆKg

the impedance of the soil are computed using a 3D boundary element method [12] in the frequency

domain, using the Green’s functions of a layered half-space.

bbstands for the frequency dependent impedance matrix of the soil. The forceˆfband

Figure 4. Finite element mesh of the office building.

The portal frame office building has the dimensions 15 m × 10 m × 9.6 m and is symmetrically

placed on the free surface above the tunnel. The finite element mesh of the modelled portal frame

structure is shown in Fig. 4. The three story superstructure is supported by a 0.3 m thick reinforced

concrete raft foundation. The basic structure consists of a reinforced concrete portal frame structure

Fiala, Gupta, Degrande, Augusztinovicz

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containing vertical columns of cross sectional dimensions 0.3 × 0.3 m and horizontal beams of di-

mensions 0.3 × 0.2 m. This frame structure supports 0.3 m thick horizontal slabs. The structure has

a reinforced concrete central core which surrounds the stair-case. The thickness of the core walls is

0.15 m. The structural model is extended with the in-fill walls of three rooms besides the core. Room

1 has dimensions 5 × 6 × 3 m, and is located on the first floor, behind the core wall; room 2, which

has the dimensions, is located on the second floor; a smaller room 3 with dimensions 5 × 4 × 3 m

is located on the first floor, besides the core. The masonry in-fill walls are 0.06 m thick. The finite

element size is chosen as 0.5 m, which is fine enough for computations up to 100 Hz. A constant

hysteretic structural damping of βs= 0.025 is assumed.

A Craig-Bampton modal decomposition method is used in order to reduce computational costs.

Thedisplacementvectorofthefoundationiswrittenasasuperpositionofrigidandflexiblefoundation

modes Φb, while the displacement vector of the superstructure is decomposed into the quasi-static

transmission Φs

superstructure with a clamped base:

bof the foundation modes to the superstructure and the flexible modes Φsof the

ˆ us

ˆ ub

=

ΦsΦs

b

0 Φb

ˆ αs

ˆ αb

(5)

This substructuring method has the advantage that the foundation and the superstructure are decou-

pled. If modifications are made to the building’s superstructure, the forces resulting from the incident

wave field do not have to be recomputed.

(a)

mode 7, 3.72 Hz

mode 8, 3.82 Hz mode 80, 162.30 Hz

(b)

mode 2, 6.65 Hzmode 4, 11.93 Hz

mode 200, 98.27 Hz

Figure 5. (a) Quasi-static transmissionof flexible foundation modes on the superstructureand (b) flexible modes

of the superstructure with clamped foundation.

According to the Rubin criterium [13], all the modes up to 1.5fmaxhave to be taken into account in

the modal superposition in order to have a kinematic base that is sufficient up to a frequency fmax. In

the present study, all the foundation and superstructure modes up to 200 Hz have been accounted for.

A few modes are displayed in Fig. 5. The lowest mode of the superstructure with a clamped base is at

2.60 Hz, and only 12 modes of the superstructure have been found under 20 Hz. These low frequency

modes are the global torsional and bending modes of the whole building. Above 50 Hz, however,

Fiala, Gupta, Degrande, Augusztinovicz

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(a)

050

Frequency [Hz]

100150

0

1

2

3

4

5

6x 10

−5

Velocity [m/s/Hz]

(b)

0 50

Frequency [Hz]

100150

0

1

2

3

4

5

6x 10

−5

Velocity [m/s/Hz]

Figure 6. Frequency content of the vertical structural velocity in points Q1 (a) and Q2 (b).

the modal density tends to be very high and the high frequency modal shapes show local bending

modes of the floor slabs and the core walls. The number of superstructure and foundation modes

increases linearly with frequency in the higher frequency range. The total number of superstructure

and foundation modes that is accounted for is equal to 482 and 96, respectively.

In the following, the structural response of the office building to the passage of the metro is pre-

sented. Fig. 6 displays the structural velocities in two points (Q1 and Q2) of the building. The point

Q1 is located on the ground level, Q2 is located on the floor of room 1, both at horizontal coordinates

x = −3 m, y = 0 m. The vibration levels in the point Q1 are very similar to the incident wave field,

presented in Fig. 3. This indicates that, in the present case, dynamic soil-structure interaction plays a

negligible role in the vibration transmission between the soil and the building. A significant vibration

amplification can be observed between the foundation and the first floor due to the first local bending

modes of the floor slab in the frequency range 20 − 30 Hz. The ground vibrations above 70 Hz are

not transmitted up to the first floor, which is an effect of structural damping.

4 Re-radiated noise in the structure

After determining the structural response of the building, the acoustic radiation problem can be

now solved. As the impedance of the radiating walls is much larger than that of the internal acoustic

space, a weak coupling between structural and acoustic vibrations is assumed. The acoustic pressure

inside the room has no effect on the vibration of the walls and the computed structural vibration

velocity is applied as a boundary condition in an acoustic boundary value problem.

The internal acoustic space is characterized by the speed of sound Ca= 343 m/s and the material

density of the air ρa= 1.225 kg/m3. The absorbing surfaces of the rooms are characterized by the

acoustic impedance Za, relating the acoustic pressure ˆ pato the difference of normal structural and

acoustic velocities ˆ vsand ˆ vaof the acoustic boundary Γa:

ˆ p = Za(ˆ vs− ˆ va)

(6)

At relative low frequencies, the acoustic impedance can be computed from the walls’ acoustic ab-

sorption coefficient α, which gives the ratio of the absorbed and the incident acoustic energy when a

normal incident acoustic plane wave is reflected from the surface. The relation between the acoustic

Fiala, Gupta, Degrande, Augusztinovicz

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absorption coefficient α and the wall’s impedance Zacan be approximated as:

Za= ρaCa1 +√1 − α

1 −√1 − α

(7)

An acoustic spectral finite element method [14] is used to express the internal pressure ˆ pa(x,ω) in

terms of the acoustic room modes:

ˆ pa(x,ω) =

?

n

Ψn(x)ˆβn(ω)

(8)

whereΨn(x)denotesthen-thacousticmodeoftheshoe-boxshapedinteriordomainwithrigidbound-

ary conditions andˆβn(ω) is the corresponding modal coordinate. The application of the spectral finite

element method results in a system of linear equations for the acoustic modal coordinatesˆβn:

?

Λ + iωD − ω2I

?ˆβ = iωˆF

(9)

where Λ = diag{ω2

n} contains the eigenfrequencies of the acoustic domain, I is a unit matrix,

?

Za

is the modal damping matrix related to the wall absorption, and

Dnm= ρaC2

a

Γa

ΨnΨm

dΓ

(10)

ˆFn= ρaC2

a

?

ΓaΨnˆ vsdΓ

(11)

denotes the modal load vector. For the case of constant wall absorption on the boundary, the element

Dnmcan be expressed analytically. Moreover, the off-diagonal elements can be truncated with a

relative small error [15], resulting in a very fast algorithm for the acoustic computations. In this case

the spectral finite element method results in a direct boundary integral representation of the acoustic

modal coordinates.

The dimensions of room 1 and room 2 are 5 m × 6 m × 2.8 m, while the size of room 3 is

5 m × 4 m × 2.8 m. The absorption is assumed to be constant on the rooms’ surface and over the

whole frequency range. Two different absorption coefficients are considered for the three rooms:

α = 0.03 stands for a strongly reflecting room with uncovered concrete walls and an uncarpeted

floor, while α = 0.15 is typical for an unfurnished, carpeted room [16]. Using Sabine’s formula, the

reverberation time trevcan be approximated from the room’s volume V , the surface area A and the

average absorption coefficient α as:

trev= γV

where γ = 0.16 s/m. For the case of the larger rooms (room 1 and room 2), the absorption coefficients

α = 0.03 and α = 0.15 result in reverberation times of 3.3 s and 0.66 s, respectively, while for the

case of the smaller room 3 these reverberation times are 3.68 s and 0.73 s.

A modal base including all the acoustic modes up to 200 Hz has been used for the spectral finite

element method. Referring to the frequency range of structural vibrations on the first floor, it is clear

that the dominant frequencies of the acoustic response will be determined by the first few acoustic

modes.

Fig. 7 shows the pressure response in room 1 during the passage of the train for the two absorption

coefficients. The dominant one-third octave bands are those containing the room’s resonance frequen-

cies at 28.6 Hz, 34.3 Hz, 57.2 Hz, 61.25 Hz and 68.6 Hz. Due to the frequency dependent sensitivity

αA,

(12)

Fiala, Gupta, Degrande, Augusztinovicz

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(a)

−10 −8 −6 −4 −2 0 2 4 6 8 10

Time [s]

−0.2

−0.1

0

0.1

0.2

Pressure [Pa]

(b)

16 20 25 31 40 50 63 80 100 125

One−third octave band [Hz]

30

40

50

60

70

Sound pressure level [dB]

Figure 7. (a) Time history and (b) one-third octave band spectra of the sound pressure in room 1 during the

passage of the train, with α = 0.03 (blue) and α = 0.15 (red).

of the human ear, the observed noise is determined by the 63 Hz peak. The one-third octave band

spectra show a difference of 5 dB between the two wall absorptions above the first acoustic resonance

of the room. In the time histories, the difference in reverberation times is clearly visible.

The sound pressure levels due to the passage of the train in rooms 1 and 3, which have different

dimensions, are compared in Fig. 8. As the first eigenfrequencies of room 3 (at 34.3 Hz, 42.85 Hz and

61.3 Hz) are distributed uniformly in the frequency scale, the one-third octave band spectra are more

balanced between 31 Hz and 80 Hz.

(a)

16 20 25 31 40 50 63 80 100 125

One−third octave band [Hz]

30

40

50

60

70

Sound pressure level [dB]

(b)

16 20 25 31 40 50 63 80 100 125

One−third octave band [Hz]

30

40

50

60

70

Sound pressure level [dB]

Figure 8. One-third octave band spectra of the sound pressure response in (a) room 1, (b) room 3 during the

passage of the train for the case of α = 0.03 (blue) and α = 0.15 (red).

As the described spectral finite element method can only handle closed shoe box shaped acoustic

volumes, the influence of wall openings on the re-radiated noise can not be taken directly into account.

However, as it is usual in acoustics to model wall openings with absorbing boundary conditions, the

investigation of their influence can be incorporated in the current model.

Four wall openings are modelled in room 3, as shown in Fig. 9. The dark gray wall is the core wall,

three windows are placed on the exterior walls of the room, covering a total area of 4.5 m2, and one

door is placed in the remaining wall covering an area of 2 m2. The absorbing boundary condition for

these walls is an acoustic impedance Za= ρaCa. Fig. 10 displays the acoustic response in the room

to the passage of the metro, for the case of the lower absorption α = 0.03, with and without wall

openings. Comparing Figs. 8(b) and 10(b), it can be stated that the influence of this large open area to

the re-radiated noise in the room is similar to the influence of the absorption increase from α = 0.03

to α = 0.15.

Fiala, Gupta, Degrande, Augusztinovicz

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x

y

Figure 9. Wall openings defined on room 3

(a)

−10 −8 −6 −4 −2 0 2 4 6 8 10

Time [s]

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Pressure [Pa]

(b)

16 20 25 31 40 50 63 80 100 125

One−third octave band [Hz]

30

40

50

60

70

Sound pressure level [dB]

Figure 10. (a) Time history and (b) one-third octave band spectra of the sound pressure response in room 3

during the passage of the train without (blue) and with (red) wall openings (α = 0.03).

5Conclusions

A 3D numerical model has been presented that is capable of predicting subway induced vibrations

and re-radiated sound in buildings.

The Bakerloo line tunnel of London Underground has been modelled using the coupled periodic

FE-BE model and subsequently the structural and acoustic response in a hypothetic three-story portal

frame office building has been predicted in the frequency range 1 − 150 Hz.

It has been demonstrated that the frequency content of the free-field vibrations strongly depends

on the wheel/rail interaction, which is governed by the dynamic behaviour of the vehicle and the rail.

The dominant frequencies of the traffic induced acoustic response are basically determined by the

first acoustic resonances of the room. The effect of wall absorption on the sound pressure has been

investigated, and above the first acoustic resonance, a difference of 5 dB has been found between

typical wall absorptions for concrete and carpeted walls. The room dimensions are found to impor-

tantly effect the sound pressure levels. It has been showed how wall openings can be modelled with

absorption boundary conditions.

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