Simpliﬁed prediction of the vibration reduction indices of double wall
junctions
Arne DIJCKMANS1 , Lieven DE GEETERE1 , Charlotte CRISPIN1
1 Belgian Building Research Institute, Belgium

Abstract
Heavyweight cavity walls are commonly used as party wall between row houses and between dwellings in apartment buildings in Belgium. The double wall junctions complicate the prediction of ﬂanking transmission. In thispaper, empirical rules for the vibration reduction indices of rigid double wall T- and X-junctions are proposed.
They are determined from Monte Carlo simulations carried out with a statistical model that was previously validated by laboratory measurements. In general, a double wall junction can be replaced by an equivalent single
wall junction in ﬂanking transmission models. The EN ISO 12354 approximation of a double wall T-junction byan equivalent single wall X-junction proves justiﬁed. For double wall X-junctions, the thickness of the equivalent single walls should be either the thickness of a single leaf or the total thickness of both double wall leafs,
depending on the transmission path of interest.
Keywords: Sound, Insulation, Flanking

1

INTRODUCTION

The vibration reduction index Ki j is an important input quantity in building acoustical models for ﬂanking
transmission. The standard ISO 12354-1 [1] gives empirical models based on measurements and simulations for
single plate junctions. According to the standard, a double wall T-junction (called Π-junction in this paper) canbe approximated by a single wall X-junction. No model is however given for double wall X-junctions (further
called H-junction), whereas this type of junction is commonly encountered in Belgium.
The vibration transmission across double wall junctions has been analyzed by means of measurements [2, 3]
and simulations [4, 5, 6, 7]. Both measurement [2] and simulation [5] results have proven that an equivalent
single wall X-junction can give good estimates for the vibration reduction indices of double wall junctions.
Because the experimental and numerical validation was only performed for a few particular case studies, little
is known about the general validity and accuracy of these single wall approximations. Crispin et al. [4] havedeveloped new expressions for the Ki j of H-junctions based on a large number of FEM calculations. The Ki j expressions are function of the PC-ratio [1] which incorporates both the surface mass ratio and the bending
stiffness ratio of the plates. The difﬁculty of determining the bending stiffness currently limits the applicabilityof these expressions.
Because simpliﬁed, empirical formulae as a function of mass ratio are available for the vibration reduction
indices of different types of single wall junctions [1], this paper investigates whether double wall Π- and Hjunctions can be modeled as equivalent single wall junctions to estimate their vibration reduction indices.
The ﬁgures in table 2 show diagrams of the Π- and H-junctions analyzed. Only symmetric Π- and H-junctions
are considered in this paper. This means that the properties of plate 1 and plate 2 are the same, as well as theproperties of plates 3 to 6.
For the double wall Π-junction, the following equivalent single wall junction has been investigated:• X’-junction where the single walls have the same thickness and density as one of the cavity leafs of the
double wall.
For the double wall H-junction, three equivalent single wall X-junctions have been investigated:234

• X’-junction where the single walls have the same thickness and density as one of the cavity leafs of the
double wall (blue results in ﬁgure 2)• X”-junction where the single walls have double the thickness (i.e. the total thickness of the cavity leafs)
and the same density as one of the cavity leafs of the double wall (red results in ﬁgure 2)• X”’-junction where the single walls have the same thickness and double the density as one of the cavity
leafs of the double wall (green results in ﬁgure 2).
To check the validity limits and general accuracy of the prediction rules, an extensive set of simulations has
been performed.

2

SIMULATIONS

2.1 SEA model
The vibration transmission across the single and double wall junctions is modeled using an SEA model. Whilethe inﬂuence of the modal behaviour of the plates may be important, like shown by ﬁnite element models[3, 4], spectral ﬁnite element models [7] and wave based models [3, 6], SEA modeling is suitable to investigate
general trends. The SEA model for the double wall junctions is described in detail in [5]. The model assumes
homogeneous, acoustically thin plates and unpinned junction lines. Each plate is represented by three subsystems to incorporate bending, quasi-longitudinal, and in-plane transverse waves. The coupling loss factors are
determined from wave theory for semi-inﬁnite plates under the assumption of diffuse vibration ﬁelds. An offset
has been taken into account for the double walls, equal to half the thickness of the continuous ﬂoor or wall, as
this proved important when validating the model with measurement results [3].
2.2 Input for Monte Carlo simulations
The direction average velocity level difference has been calculated for 16 different ﬂoors/walls (table 1), resulting in a set of 240 Π- and H-junctions. Originally, the model was used to investigate horizontal Π- andH-junctions composed of concrete ﬂoors and different types of double walls. Afterwards, the analysis was extended to Π- and H-junctions representing a vertical junction of single and double walls. In this way, also mass
ratio’s m3 /m1  1 are included in the investigation. As the cavity thickness does not signiﬁcantly inﬂuence thevibration reduction index [5], it was kept constant at 5 cm.
Table 1. Plate properties
Thicknesst[cm]
Concrete
12; 20; 28∗
Concrete bricks
9; 14
Aerated concrete
10; 15
Sandlimestone
15; 24
Sandlimestone brickwork
15
Gypsum blocks (lightweight)
10
Gypsum blocks (heavyweight)
10
Masonry bricks (lightweight)
9; 14
Masonry bricks (heavyweight)
9; 14∗ not used for the double walls (plates 3-6)

Densityρ[kg/m3 ]
2500
1700
650
1800
1300
900
1100
800
1200

Young’smodulus E[×109 Pa]
33
20
1.8
16
11
5.4
6.6
2.8
3.7

Poisson’s
ratio ν
[-]
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2

Loss factor
η
[-]
0.015
0.015
0.015
0.015
0.015
0.015
0.015
0.015
0.015

235

5
3

ϯ

5

ϰ

ϭ

4

Ϯ

estimation error [dB]estimation error [dB]

4
2
1
0
−1
−2
−3

−1

10

5

ϯ

0
−1

ϭ

−1

10

4

Ϯ

1
0
−1

0

10
mass ratio m /m [−]
3

5

ϰ

−2

(c)

1

(b)

1

2

−3

Ϯ

2

−3

1

10

estimation error [dB]estimation error [dB]010mass ratio m /m [−]
3

3

ϰ

−2

(a)
4

3

ϯ
ϭ

3

ϯ
ϭ

1

10

1

ϰ
Ϯ

2
1
0
−1
−2

−1

10

0

10
mass ratio m /m [−]
3

1

−3

1

10

(d)

−1

10

0

10
mass ratio m /m [−]
3

1

10

1

Figure 1. Estimation error as a function of the mass ratio m3 /m1 for the average vibration reduction indicesover 200 − 1250 Hz of the double wall Π-junction, when modeled as an equivalent single wall X-junction.
2.3 Simulation results
2.3.1 Double wall Π-junction
The double wall Π-junction can be relatively well approximated by an equivalent single wall X-junction ofwhich the walls have the same mass as the cavity leafs of the double wall. Figure 1 shows the estimation
error (i.e. Ki j,X  − Ki j,Π ) for the average vibration reduction index over the frequency bands 200 − 1250 Hz as afunction of the mass ratio m3 /m1 for the different ﬂanking paths.
For the transmission paths 1-2 (ﬁgure 1a), 1-3 (ﬁgure 1b) and 1-4 (ﬁgure 1c), the estimation error is negligible
when the double wall leafs are much lighter than the single walls (m3 /m1  1). The discrepancy between theΠ- and X’-junction becomes larger for increasing mass ratio m3 /m1 . For the path 1-3 (ﬁgure 1b) the equivalent
single wall X-junction gives an overestimation of the average vibration reduction index up to 3 dB. The averagevibration reduction index of path 1-4 (ﬁgure 1c) is estimated by the equivalent X’-junction with an accuracyof 2 dB. It can be noted that the difference Ki j,X  − Ki j,Π is always larger for the path 1-3 than for the path
1-4, because the vibration reduction index K14,Π is always larger than K13,Π and the same estimation K13,X  isused. For the transmission path 3-4 between the double wall leafs (ﬁgure 1d), the estimation error decreases
from +1.5 dB (i.e. an overestimation) for m3 /m1  1 to approximately −1 dB (i.e. an underestimation) form3 /m1  1.
2.3.2 Double wall H-junction
Figure 2 shows the estimation errors for the average vibration reduction indices for the three equivalent single
wall X-junctions used as approximation for the double wall H-junction in function of the mass ratio m3 /m1 .
Generally, the X-junction that uses an equivalent single wall with double the thickness of the double wall leafs(red) does not give a good estimation of the vibration reduction index when the transmission path includes at

236

6

4

4

4

2
0
−2
−4
−6

ϯ

−8

ϭ

−10
−12
−14

10

2
0
−2
−4
−6

ϯ

−8

ϭ

−10
−12

ϲ

0

−14

1

10
mass ratio m3/m1 [−]

10

ϰ
Ϯ

ϱ
−1

10

(b)

2
0
−2
−4
−6

−14

1

10
mass ratio m3/m1 [−]

10

8

8

6

6

4

4

4

0
−2
−4
−6

ϯ

−8

ϭ

−10
−12
−14

ϰ
Ϯ

ϱ
−1

10

0

10
mass ratio m3/m1 [−]

2
0
−2
−4
−6

ϯ

−8

ϭ

−10
−12

ϲ

−14

1

10

(e)

ϰ
Ϯ

ϱ
−1

10

0

10
mass ratio m3/m1 [−]

0
−4
−6

ϯ

−8

−14

10

(f)

1

10

−2

ϰ

ϭ

−12
1

ϲ

0

10
mass ratio m3/m1 [−]

2

−10

ϲ

Ϯ

ϱ
−1

10

(c)

6

ϰ

ϭ

−12

ϲ

0

8

2

ϯ

−8
−10

estimation error [dB]estimation error [dB]

Ϯ

ϱ
−1

(a)

(d)

ϰ

estimation error [dB]86estimation error [dB]86estimation error [dB]estimation error [dB]

8

Ϯ

ϱ
−1

10

0

10
mass ratio m3/m1 [−]

ϲ
1

10

Figure 2. Estimation error as a function of the mass ratio m3 /m1 for the average vibration reduction indicesover 200 − 1250 Hz of the double wall H-junction, when modeled as an equivalent X-junction with single walls
having the same thickness/density (blue), double thickness (red) or double density (green) compared to a single
cavity wall leaf of the H-junction.
least one double wall leaf (ﬁgures 2b-2f). The vibration reduction index is largely underestimated for small
mass ratio’s m3 /m1 . When the mass ratio increases, the underestimation becomes smaller for the transmission
paths including two double wall leafs (ﬁgures 2d-2f). For the transmission paths including only one double leaf
wall (ﬁgures 2b-2c), the equivalent X-junction with double thickness gives an overestimation of the vibration
reduction index when m3 /m1 > 1 .
The transmission paths that include at least one double leaf wall can be best approximated by the equivalent X-junctions with same thickness/density (blue) or double density (green) (ﬁgures 2b-2f). On average, thetransmission path 1-3 (ﬁgure 2b) is underestimated by 1 dB by the equivalent X-junction with same thickness/density. For the transmission paths 3-4 (ﬁgure 2d) and 3-5 (ﬁgure 2e), the vibration reduction indices are generally underestimated by 1 to 2 dB. For the transmission paths 1-4 (ﬁgure 2c) and 3-6 (ﬁgure 2f), the models
with same thickness/density (blue) or double density (green) give a slight underestimation for small mass ratio’s
m3 /m1 , but the average Ki j is underestimated by up to 5 dB for mass ratio’s close to 1.
2.4 Prediction rules
For Belgian construction practices, the cases for which the double wall leafs are lighter than the continuous
plate of ﬂoor (m3 /m1 < 1) are the most relevant. Table 2 gives empirical prediction rules that are derived from
the Monte Carlo simulation results in the range m3 /m1 < 2. The aim is to provide prediction rules as simple as237

Table 2. Prediction rules for double wall Π− and H-junctions with m3 /m1 < 2ɅͲũƵŶĐƚŝŽŶy͛ͲũƵŶĐƚŝŽŶϯ ϰ‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙ϭK12,Π = K12,X ϯ‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪

K34,Π = K34,X ‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙ϮK13,Π = K14,Π = K13,X ϭϮ‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪ϰ,ͲũƵŶĐƚŝŽŶy͛ͲũƵŶĐƚŝŽŶϯ ϰ‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙

ϭK12,H = K12,X ϯ‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙ϮK13,H = K13,X  + 1 dB
K13,X  + 1 dB
K14,H = max
K13,X ϭ

ϮK34,H = K34,X 
K35,H = K34,X  + 1 dB
‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪ϱ ϲ

K36,H = K34,X  + 2 dBϰy͟ͲũƵŶĐƚŝŽŶ

ϯʹ‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙‫ݐ‬௙ ǡ ‫ܧ‬௙ ǡ ߩ௙ϭϮʹ‫ݐ‬௪ ǡ ‫ܧ‬௪ ǡ ߩ௪ϰpossible while limiting the estimation errors. Especially a large overestimation of the vibration reduction index
needs to be avoided for robust design practice.
For the double wall Π-junction, the rules of table 2 concur with the approximation given in annex J of ISO
12354-1 [1]. While these rules generally give accurate estimations for the vibration reduction indices Ki j,Π when
m3 /m1 < 2 (with a maximum overestimation of 1 dB), the overestimation is more signiﬁcant when m3  m1 .
For transmission path 1 − 3, the overestimation can already reach 2 dB when m3 /m1 > 1.
For the double wall H-junction, the newly developed prediction rules give a maximum overestimation of 1 dB
when m3 /m1 < 2. The underestimation can be larger, with values up to 3 dB in some speciﬁc cases. The
transmission paths 1 − 4 and 3 − 6 proves most difﬁcult to predict, as all three equivalent X-junction give large238estimation error [dB]

4
3

ϯ

ϰ

ϭ

Ϯ

2
ϱ

ϲ

−1

10
mass ratio m /m [−]

1
0
−1
−2
−3

10

0

3

1

10

1

Figure 3. Estimation error as a function of the mass ratio m3 /m1 for the average vibration reduction indices
K14,H over 200 − 1250 Hz of the double wall H-junction when using the estimation of table 2
underestimations when m3 < m1 (ﬁgures 2c and 2f). To minimize the underestimation of K14,H , it is suggested
to use the maximum of (i) the Ki j -value for the case with same thickness/density (blue) plus 1dB and (ii) theKi j -value for the case with double thickness (red). By use of this new proposal, the underestimation can belimited to 3 dB (ﬁgure 3). For the path 3 − 6, it is suggested to use the Ki j -value for the equivalent X’-junction
with same thickness/density augmented by 2 dB.

3EXPERIMENTAL VALIDATION

To validate the models and prediction rules of table 2, vibration reduction index measurements have been carriedout on a half scaled test bench (ﬁgure 4a). A ﬁrst series of measurements (ﬁgure 4b) was performed onjunctions composed of a continuous concrete ﬂoor (t f = 0.10 m, m1 = 230 kg/m2 ) and single and double gypsum
block walls (tw = 0.07 m, m3 = 65 kg/m2 ) [3]. A second series (ﬁgure 4c) was performed on junctions with acontinuous concrete ﬂoor (t f = 0.10 m, m1 = 230 kg/m2 ) and concrete block walls (tw = 0.09 m, m3 = 198 kg/m2 ).
The measured average vibration reduction indices over the frequency range 400 Hz − 2.5 kHz (corresponding
to 200 Hz-1250 Hz in real scale) for the double wall junctions are summarized in table 3, together with theestimated values from the measured vibration reduction indices for the equivalent single wall X’-junction andthe estimation errors. Because an equivalent X”-junction with double thickness walls has not been measured,
the prediction rules for paths 1 − 2 and 1 − 4 of the H-junction cannot be validated.
In general, the vibration reduction indices of the gypsum block double wall junctions are slightly overestimated

(a)

(b)

(c)

Figure 4. Measurement set-up of a gypsum block double wall Π-junction (b) and a concrete block double wall
H-junction (c) on a half scaled test bench for vibration transmission measurements [3].

239(on average 1.2 dB), while the simulations suggest a better agreement for this small mass ratio of m3 /m1 = 0.28.
The largest estimation error is found for the wall-wall path K34,Π . This could be expected as the simulations
also indicate that the prediction rule for K34,Π gives an overestimation for the smaller mass ratio’s.
The absolute values of the estimation error are larger for the concrete block double wall junctions, but generally on the safe side (i.e. an underestimation of the vibration reduction index). This larger uncertainty is inaccordance with the larger spread in the simulation results around this mass ratio m3 /m1 = 0.86.
Table 3. Measurement results (arithmetic average of Ki j between 400 Hz and 2.5 kHz)double wallmeasurement
K12,Π = 6.8 dB
K13,Π = 15.1 dB
K14,Π = 15.9 dB
K34,Π = 18.2 dB
K13,H = 16.2 dB
K15,H = 18.3 dB
K34,H = 20.5 dB
K35,H = 22.1 dB
K36,H = 23.6 dB

4

Gypsum blocks
estimation from
Ki j,X  measurement
K12,X  = 8.0 dB
K13,X  = 16.4 dB
K13,X  = 16.4 dB
K34,X  = 21.7 dB
K13,X  + 1 = 17.4 dB
K14,X  + 1 = 20.0 dB
K34,X  = 21.7 dB
K34,X  + 1 = 22.7 dB
K34,X  + 2 = 23.7 dB

overestimation+1.1 dB+1.3 dB+0.5 dB+3.5 dB+1.1 dB+1.7 dB+1.2 dB+0.6 dB+0.1 dB

double wallmeasurement
K12,Π = 6.4 dB
K13,Π = 11.1 dB
K14,Π = 12.1 dB
K34,Π = 14.8 dB
K13,H = 14.6 dB
K15,H = 12.7 dB
K34,H = 18.5 dB
K35,H = 17.3 dB
K36,H = 18.4 dB

Concrete blocks
estimation from
Ki j,X  measurement
K12,X  = 8.4 dB
K13,X  = 11.0 dB
K13,X  = 11.0 dB
K34,X  = 14.4 dB
K13,X  + 1 = 12.0 dB
K14,X  + 1 = 13.5 dB
K34,X  = 14.4 dB
K34,X  + 1 = 15.4 dB
K34,X  + 2 = 16.4 dB

overestimation+2.0 dB−0.2 dB−1.2 dB−0.4 dB−2.6 dB+0.8 dB−4.1 dB−1.9 dB−1.9 dB

SUMMARY AND CONCLUSIONS

Simpliﬁed formulae are proposed for the vibration reduction index of rigid double wall Π- and H-junctions.
They allow to estimate the vibration reduction index of double wall junctions using existing semi-empiricalformulae for rigid, single wall junctions. The empirical rules are determined from Monte Carlo simulations
carried out with a statistical model. The simulation results indicate that the estimations are most accurate
when the mass ratio m3 /m1 is small, meaning that the double wall junction can be well represented by anequivalent single wall junction as long as the double wall leafs are relatively light compared to the continuous
single ﬂoor or wall. The simpliﬁed formulae have been validated by two sets of vibration reduction index
measurements. For the double gypsum block walls, the rules give an average overestimation of the double
wall vibration reduction index of 1.2 dB. For the double concrete block walls, the vibration reduction index isgenerally underestimated with an average absolute estimation error of 1.7 dB. Future research should investigate
whether these estimations can also be used for asymmetric junctions and junctions including elastic interlayers.

ACKNOWLEDGEMENTS
The results presented in this paper have been obtained within the frame of the project A-light II (Integrating
lightweight concepts in acoustical standardization). The authors are grateful for the ﬁnancial support from theFederal Public Service Economy of Belgium.

REFERENCES[1] ISO, International Standard ISO 12354-1: Building acoustics - Estimation of acoustic performance of buildings from the performance of elements - Part 1: Airborne sound insulation between rooms, 2017.
[2] Schoenwald S, Zeitler B, Sabourin I. Analysis on structure-borne sound transmission at junctions of solid

240

wood double walls with continuous ﬂoors. Proceedings of 7th Forum Acusticum, Krakow, Poland, September 7-12, 2014. In CD-ROM.
[3] Crispin C, Mertens C, Dijckmans A. Detailed analysis of measurement results of ﬂanking transmission
across junctions composed of double walls carried out on a half scaled test bench. Proceedings of the 24th
International Congress on Sound and Vibration (ICSV), London, UK, July 23-27, 2017. In CD-ROM.
[4] Crispin C, De Geetere L, Ingelaere B. Extensions of EN 12354 vibration reduction index expressions bymeans of FEM calculations. Proceedings of Inter-Noise 2014, Melbourne, Australia, November 16-19, 2014,
pp 5859-5868.
[5] Dijckmans A. Vibration transmission across junctions of double walls using the wave approach and statistical energy analysis. Acta Acustica United with Acustica, 102 (3), 2016, pp 488-502.
[6] Dijckmans A. Wave based modeling of vibration transmission across junctions composed of rectangularsingle and double walls. Acta Acustica United with Acustica, 102(6), 2016, pp 1011-1026.
[7] Poblet-Puig J, Guigou-Carter C. Catalogue of vibration reduction index formulas for heavy junctions basedon numerical simulations. Acta Acustica United with Acustica, 103(4), 2017, pp 624-638.

241