# MSc thesis - Triple layer membrane structures - Appendices

## Triple-layer membrane structures

## Sound insulation performance and practical solutions

**APPENDICES**

## for

## MSc-thesis

## 26-08-2011

Made by: J.J.E. de Vries

In collaboration with:

Delft University of Technology

Faculty of Civil Engineering and Geo Sciences

Peutz bv, Zoetermeer

Tentech bv, Utrecht

## Table of contents

2A Basic acoustic theory

2A.1 Basics

2A.1.1 Basic relations

2A.1.2 Sound waves

2A.1.3 Vibration

2A.1.4 Energy density and intensity

2A.1.4 Ratiation

References

2B Sound reflection

References

2C Reverberation and steady-state energy density

References

2D Basic sound insulation quantities

References

3A Derivation and m-file (Matlab) - An impermeable triple-layer system

3B Derivation and m-file (Matlab) - Triple-leaf system with a permeable leaf

on sound incidence side

4A History of tent construction

4A.1 The early days

4A.2 Modern development

References

6A Peutz’ Laboratory for Acoustics

6B Standards and guidelines

6C Wooden frames

6D Cross-sectional connection frames to opening

6E Measurement information

6F Measurement data

6G Flow resistance measurements

7A Input file for the Multiple Layer Model

45

47

49

37

39

41

43

29

29

32

35

51

15

17

19

21

23

24

10

13

7

9

5

6

5

5

25

27

*3*

## Basic acoustic theory

2A

**2A.1 Basics**

**2A.1.1 Basic relations**

In the present paper we are less interested in sound propagation through the membrane material and more through gases, since we are concerned with air as the medium. Every complex sound field can be considered as numerous simple sound waves, called plane waves. For the description of the basic relations some assumptions should be made. First, we assume a free field (free of losses and obstacles). Furthermore, our medium is homogeneous in all directions and at rest. With all these assumptions made the (phase) velocity of sound in air is constant with reference to space and time and is given by:

*c*

0

=

(2A.1)

,where *T* is the temperature in degrees centigrade (between 15

0

and 30

0

centigrade). The density of air at equilibrium, , is also temperature dependent and can be calculated from:

ρ

0

### ρ

0

### =

### 273 +

*T kg m*

3

(2A.2)

In building acoustics the temperature is often assumed to be 20 o

*c*

0

centigrade, in which the velocity will become 343 m/s and the air’s density 1.21 kg/m

3

.

In any sound wave, the particles of the medium undergo vibrations around their “normal”

(mean) positions. Therefore, a sound wave can be described entirely by indicating the instantaneous displacements of these particles. Points in a sound field are at some locations pushed together and sometimes pulled apart. This causes variation of gas density and pressure. The difference between instantaneous pressure and the static pressure is called the sound pressure. Because of the changes of gas pressure caused by a sound wave, sound waves cause variations of the temperature. Density, pressure and temperature are therefore quantities characterizing a sound wave. The differential equation which governs the propagation of sound in any lossless fluid is called the ‘wave equation’:

*c*

0

2

### ∇

2

*p*

### =

### ∂

### ∂

*t*

2

2

*p*

### which means:

∂ 2

∂

*x*

2

*p*

+

∂ 2

∂

*y p*

2

+

∂

∂

2

*z*

2

*p*

−

*c*

1

0

2

∂ 2

∂

*t*

2

*p*

= 0 (2A.3) density and pressure respectively and the adiabatic (or isentropic) exponent ( = 1.4 for air).

*c*

2

=

γ

*p*

0

ρ

0

γ

ρ

0

γ

*5*

*BASIC ACOUSTIC THEORY*

**2A.1.2 Sound waves**

[1, 2]

**Plane waves**

Two types of waves can be distinguished, which both have a different solution for this differential equation. The plane longitudinal (harmonic) wave and the spherical wave.

Both solutions can be drawn from the general solution (eq. 2A.4), in which each term represents a progressive, plane wave.

*p x t*

### = ) ( − )

(2A.4)

, where *F* and *G* are arbitrary functions.

From eq. 2A.4 the sound pressure in a plane wave propagating in x, y and z-direction is described by an equation of the form [2]

*p x y z t*

### =

### exp( − )exp( − )exp( − ω

(2A.5)

### *p x t* =

### cos( ω

*t kx*

### )

*p*

=

*kc*

) related to the temporal period as . The equation also describes the ‘wavelength’ of the harmonic wave (through to period of a spatial harmonic vibration): . This is

λ =

ω

=

*c f f*

=

ω

2 π

=

1

*T*

λ related to the angular frequency by , where is the frequency of the vibration (Hz). The wave number is useful for aspects relating to spatial variation in both sound and vibration fields in terms of *kd*, where *d* is the distance between two points.

For real applications the first expression in eq. 2A.5 is not useful. By taking its real part that describes the sound pressure and ignoring its time dependence the solution becomes eq. 2A.6. Plane waves usually are harmonic waves, where the acoustic quantities follow from a sine or cosine function.

( , , ) = cos(

*k x*

)cos(

*k y*

)cos( )

(2A.6)

But when harmonic time dependence is used, the wave equation can be written in terms of the wave number by using eq. 2A.3.

∂ 2

∂

*x p*

2

+

∂ 2

∂

*y p*

2

+

∂ 2

∂

*z p*

2

= 0

(2A.7)

, which gives

*k*

2

### =

*k x*

2

### +

*k y*

2

### +

*k z*

2

(2A.8)

All formulae above only describe sound fields outside a room, because reflections are not taken into account. In other words, the equations don’t take the boundary conditions into account.

Because particle motion gives rise to sound pressure, the sound particle velocities can be

*6*

*SOUND WAVES*

related to the sound pressure by:

*u x*

=

### ωρ

0

### ωρ

0

### ωρ

0

(2A.9)

The ratio of sound pressure and velocity in a plane wave (in positive x-direction) is fre-

*p*

= ρ

0

*c*

For air at 20

0

centigrade its value is

*Z*

0

= ρ = 416 Pa m − 1

*s*

− 2 416 kg m − 2

*s*

− 1

(2A.10)

**Spherical waves**

The spherical wave has surfaces of constant pressure, i.e. ‘wave fronts’, in concentric spheres. In the centre a sound source (very small) should be imagined, a ‘point source’. At a bigger distance (for instance for traffic noise) these wave fronts will be almost (assumed) as plane waves (the further the source, the bigger the spherical surface and thus the smaller the surface curvature). Within rooms the sound field is mostly assumed diffuse, due to all reflections and here too, plane waves are used for calculations on boundary walls as a simplification. Since spherical waves only have a role in distances very small to the source, this will not be discussed in the present thesis.

**2A.1.3 Vibration**

When we look on a physical scale when sound impinges a surface of a solid, this solid starts vibrating and this vibration radiates sound energy through the wall or back in the room. Some theoretical background on vibration is thus needed. Three vector quantities are important with vibration: displacement w, velocity v and acceleration a. They related to each other by [2]:

*w t*

=

∫

( ) ( ) =

*dt*

=

∫

*a t dt*

( ) =

*dv t dt*

(2A.11)

For sound in air and other gases there is only one wave type that needs to be considered, namely longitudinal waves. In contrast, three wave types need to be considered for vibration on plates. They are quasi-longitudinal, transverse (shear) and bending waves (figure

2A.1). These three will be discussed here.

**Quasi-longitudinal waves**

According to e.g. Fahy [3] pure longitudinal waves (the direction of the particle displacement is purely in the direction of wave propagation) can only exist in solids that extend in all directions to distances large compared with a longitudinal wavelength. For bars or plates a different form of longitudinal waves exists: quasi-longitudinal waves [3]. The wave equation for plates is:

∂ 2

∂

*x w*

2

=

ρ( 1 −

*E v*

2 ) ∂

∂

*t*

2

*w*

2

(2A.12)

, and the corresponding, frequency-dependent phase speed is:

*7*

*BASIC ACOUSTIC THEORY c*

*L*

=

ρ( 1

*E*

−

*v*

2 )

1 2

(2A.13)

*Fig 2A.1 Transformation patterns for different variations of wave [3]*

These equations are only valid for frequency ranges in which the quasi-longitudinal wavelength greatly exceeds the cross-sectional dimensions of the structure.

**Transverse (shear) waves**

In figure (above) this type of waves is depicted. Unlike gases, solids can resist shear deformation, hence transverse shear waves. The shear modulus *G* of a solid is defined as the

2 1 +

*v*

)

τ

γ

. . The wave equation can be derived from the equation of transverse motion of an element having unit thickness and the stress-strain relationship of that element. The transverse wave equation and the frequency-independent phase speed are [3]:

∂

∂

2

*x w*

2

ρ

*G*

1 2

∂

2

*w*

∂

*t*

2

(2A.14)

*c s*

=

*G*

ρ

1 2

*E*

( +

*v*

)

1 2

(2A.15)

**Bending waves in thin plates**

Pure bending waves occur where the bending wavelength is large compared to the plate thickness; hence these waves only occur on ‘thin plates’ (‘thin’ when the thickness is less than a quarter of the bending wave’s length). A propagating bending wave causes both rotation and lateral displacement of the plate elements. The bending wave equation for thin plates is derived from the bending wave equation for bars by Fahy [3], and becomes:

*8*

*ENERGY DENSITY AND INTENSITY*

( 1

*EI*

−

*v*

2

)

∂ 4

*w*

∂

*x*

4

= −

∂ 2

∂

2

(2A.16)

, where *m *is the mass per unit area of the plate and *I *the second moment of area per

/ 12

*Eh*

which may be termed the bending stiffness of the plate. The phase speed is:

−

*v*

2 )

*c b*

=

ω

1

2

*D m*

1 4

or according to Hop kkins (2007):

(2A.17)

*c b*

=

4

ω 2

ρ

*s*

*B*

=

4

4 π 2

*f h E*

( −

*v*

2 )

=

2 π

*fhc*

*L*

12

The above formulae are for plates in only one direction (x). This can be supplemented to a thin plane lying in the xz-plane [4] to:

*D*

∂ 4

*w*

∂

*x*

4

+ 2

∂ 4

*w*

2 2

+

∂ 4

*w*

∂

*z*

4

∂ 2

∂ 2

(2A.18)

For bending waves traveling in every direction no simple solution can be given. An approximated equation is given by Fahy [3]:

*Dk b*

4

−

*m*

ω 2

= 0 (2A.19)

**k k**

=

**k**

+

**k k**

. ,i.e., in a direction at angle to the x-axis. Hence and

*k b*

=

## (

ω 2

*m D*

## )

1 4

**Bending waves in orthotropic plates**

According to Cremer et al. [4] the wave equation for bending waves on a thin homogeneous orthotropic plate was showed earlier. The bending stiffness in the xz-plane has elastic properties which are not always known. However, this can be approximated by the effective bending stiffness (for orthotropic plates) and is shown below.

*D xz*

≈

*D eff*

=

*D D z*

(2A.20)

**2A.1.4 Energy density and intensity**

The (sound) energy generated by a sound source is carried away in a sound wave. The energy density is the amount of energy in one unit volume of that wave. We can, like normal, distinguish potential and kinetic energy density [2, 4]:

*9*

*BASIC ACOUSTIC THEORY w pot*

=

2

*p eff*

ρ

0

*c*

2

2

and

*w kin*

=

ρ

0

2

**v**

2

The total energy density is then:

=

*pot*

+

*w kin*

(2A.21)

(2A.22)

The measure of the energy transported in a sound wave is called the sound intensity. It can be described as the amount of energy per second passing a virtual m

2

. Generally the intensity is a vector:

(2A.23)

When the particle velocity is expressed in terms of sound pressure, the energy density and intensity can be expressed by:

*w*

=

*p eff*

ρ

0

*c*

2

2

and

*I*

=

*p*

ρ

*eff*

0

*c*

2

(2A.24)

They are related by . Furthermore we can say that the total amount of energy stored in a volume *V* is given by . In a room the energy density is not uniformly distributed throughout space. Near room boundaries interference patterns occur and an increase in density can be seen close to the boundary. To take this into account the formula for *E* should be multiplied by a correction term [5]:

1 + *S*

8

*T*

*V*

λ

(2A.25)

*S*

*T*

*C*

logarithm of eq. 2A.25.

**2A.1.5 Radiation**

The sound insulation offered by a building element depends on two factors: 1) the dynamic response to the actual excitation (acoustic field or direct mechanical force) and 2) the efficiency as a sound radiator given the actual response pattern. Here, the latter is discussed.

The quantity that characterizes the efficiency of a given vibrating surface as a sound radi-

σ definition the ratio of the radiated power to the power radiated by a large baffled piston

(*ka >> 1*) [1, 2]:

σ =

ρ

*W rad c S u*

2

(2A.26)

*W*

ρ

*u*

2

*10*

*RADIATION*

(the RMS-value over all points on the surface).

To get a general insight into sound radiation, bending waves on an infinite plate is discussed briefly. This sets the scene for sound radiation from individual bending modes on finite vibrating plates. Before getting into this, the critical (coincidence) frequency in discussed.

**Critical frequency**

2A.17 [1, 2, 4].

*c*

=

*c*

called the critical or coincidence frequency and can be calculated according to equation

*f c*

=

*c*

0

2 π

2 ρ

*B s p*

=

*c*

0

π

2 ρ

(

*Et*

−

3

ν 2

)

=

*c*

0

π

2

*tc*

*L*

3

(2A.17)

*c*

*L*

λ

For (thin) homogeneous plates we may write:

λ

*f c*

≈

*c*

0

2

(2A.18)

*f f f f*

=

*f f*

(2A.19)

**Infinite plate theory**

** **[4]

*Fig 2A.2 Simple plane bending wave [4] u*

=

*u*

e

( )

*k*

*B*

number of the bending wave (section 2A.1.3) propagating in x-direction. Now the sound pressure above the plate can be expressed by:

*p x y*

=

ω ) (2A.20)

*11*

*BASIC ACOUSTIC THEORY*

This immediately shows that the wave number *k* above the plate must be expressed

*k*

=

ω

*c*

=

*k x*

2

+

*k*

2

*y*

pressure therefore is expressed by:

*u*

*B v*

=

ρ

1 −

*k k*

2

*B*

2

*e*

( ω − )

*e*

2

−

2

*B*

(2A.21)

Eq. 2A.21 shows that an important factor for the sound radiation is the ratio of the wave

λ

>

λ matching. In this case we may calculate the radiation factor from the radiated power, expressed as:

*k*

<

*k*

*W*

=

1

2

Re

{ }

=

ρ

2 1 −

*k k*

*B*

2

2

(2A.22)

The radiation factor is then given by:

σ =

1

1 −

*k k*

2

*B*

2

(2A.23)

**Finite plate theory**

In the above infinite plate model no radiation could occur for frequencies lower than the critical frequency. For real plate structures this is not the case. The sound radiation from individual bending modes can be calculated for a homogeneous, isotropic, rectangular plate with simply supported boundaries. It goes beyond the scope of this thesis to discuss this in detail, but to give an idea of the radiation factor the end result is given [4]:

σ

=

π

64

6

*n n z*

2

π

0

∫

2

π

0

∫

2

*n*

cos sin

α

*x*

π

2

α

2

1

⋅

⋅

cos sin

*n z*

β

π

2

β

2

1

2

(2A.24)

α =

*ka*

ϕ θ β =

*kb*

ϕ θ

*n n*

cos

, where , *a* and *b* the plate dimensions and

*12*

*REFERENCES*

**References**

4.

5.

1.

2.

3.

6.

Hopkins, K. (2007), Sound Insulation, First Edition, published by Elsevier Science

Publischers Ltd.

Kuttruff, H. (2009), Room Acoustics, Fifth edition, published by Spon Press

Fahy, F. (1994), Sound and structural vibration - radiation, transmission and response,

First edition (1985) by Academic Press Ltd. Fourth printing in 1994

Vigran, T.E. (2008), Building Acoustics, First edition, published by Talyer & Francis

Waterhouse, R.V. (1955), Interference patterns in reverberant sound fields, Journal of the Acoustical Society of America, 27 (2), 247-258

Wallace, C.E. (1972), Radiation resistance of a rectangular panel, Journal of the

Acoustical Society of America, 51 (3), 946-952

*13*

## Sound reflection

2B

Sound reflection is best described for different kind of directions of incidence. Starting off with normal incidence, where the (plane) wall is normal to the incident wave and followed by oblique incidence where the angle of incidence may be any value between 0 and 90 o

θ

. Thereafter random incidence and reflection from a finite-sized surface will be discussed briefly.

o

**Normal incidence**

*Fig 2B.1 Reflection of a normally incident sound wave [1] p x t*

=

0

[

)

] to the wall, which is at *x=0 *(fig 2B.1).

=

ρ

0

*c*

[

ω )

] direction of the sound wave after reflection the sign of *k* changes and the sound pressure and particle velocity become (*r* is the reflection coefficient):

=

= −

0

ρ

*r p*

0

0

*c*

[

[

ω +

ω

)

]

+ )

]

(2B.1)

*r*

particle velocity in the plane of the wall are obtained by adding above expression and setting *x to 0*. The wall impedance is now given by dividing* p(0,t)* by *v(0,t)*:

*Z*

= ρ

0

*c*

1 +

1 −

*R*

*R*

And from this the reflection coefficient and factor are given:

= =

−

+

ρ

ρ

0

0

*c c*

=

ζ

ζ

− 1

+ 1

(2B.2)

(2B.3)

*15*

*SOUND REFLECTION*

Combining the above equation with the expression for the absorption coefficient we get

α

α = −

*R*

2 1

*r*

1

(

(

ζ

ζ

−

+

1

1

)

)

2

2

=

ζ

2

+

4

2 + 1

(2B.4)

ζ reflected wave is called a standing wave. The pressure amplitude is found by adding the equation for an incident wave’s sound pressure and the first expression in eq. 2B.1.

**Oblique incidence**

After inserting the transformation from *x to x’* and making it dependent on *y* and the angle of incidence , the wall impedance, reflection factor and absorption coefficient become:

θ

*Z*

=

ρ

0 cos

*c*

1 +

*R*

θ

1 −

*R*

(2B.5) cos cos

0

0

*c c*

=

ζ cos θ

ζ cos

θ

− 1

+ 1

(2B.6)

α =

ζ

2 cos

2

4

θ + 2

θ

θ + 1

(2B.7)

**Random incidence**

A typical sound field is composed by many plane waves, each with different amplitude, phase and direction. To find the effect of a wall on such a complicated sound field the reflection of each wave should be considered and all sound pressures should be added.

Here the end result is presented, please refer to [1] for the full derivation. The absorption coefficient for random or uniformly distributed incidence is:

α

*uni*

=

*E a*

=

*E i*

2

π

∫

0

2

( )cos sin

π

2

α θ θ θ θ = α θ

0

∫

(2B.8)

In some literature this is referred to as the ‘Paris’ formula’. For locally reacting surfaces the angular dependence of the absorption coefficient can be expressed according to equation 2.34.

α

*uni*

=

ζ

8

2 cos

µ ζ

+ sin

µ arctan

1 +

ζ sin

µ

ζ cos

µ

(

ζ cos

2

)

(2B.9)

µ =

ζ

ζ

**A finite-sized plane surface**

When a sound wave hits a free boundary it will become the origin of an additional sound wave. This additional sound wave is brought about by diffraction and is called a ‘diffrac-

*16*

*REFERENCES*

tion wave’. Examples for a rigid half-plane and small round disc are found in [1]. For the latter, or for a rigid strip with width *w*, a minimum frequency can be defined above which the disc can be considered as an efficient reflector:

*f*

min

*f*

= min

4

*cR a*

2

≈ 85

*a*

*R*

2

≈ 185

*w*

*R*

2

[Hz]

[Hz]

(2B.10)

(2B.11)

**References**

1. Kuttruff, H. (2009), Room Acoustics, Fifth edition, published by Spon Press

*17*

## Reverberation and steady-state energy density

2C

A time characteristic of a diffuse sound field is described by dividing it into three intervals, ‘onset’, ‘steady-state’ and ‘reverberation’ (see figure 2C.1). [1]

*dE*

*dt *= 0 is switched of the sound level falls again and the acoustic energy flows out. This is the last interval, the reverberation (sound decay).

The phenomenon reverberation can be described by sound produced in a room that will not disappear immediately after the sound source is shut off but remains audible for a certain period of time afterwards, although with steadily decreasing loudness. For the simple laws describing this process a diffuse sound field is assumed, which is quite a good approximation for the actual sound field.

*Fig 2C.1 Schematic time characteristics of a diffuse sound field [1]*

### θ φ

*dS *cosθ

4π

*w*

= 4π

*I c*

π *IdS* depend on the angle of incidence. This can be explained by the projection of *dS* in direction is (figure 2C.2).

*d*Ω

*I*

cosθ

*dSd*

Ω arriving per second on dS from the solid angle around the considered direction. By

*19*

*REVERBERATION AND STEADY-STATE ENERGY DENSITY*

*Fig 2C.2 Spherical polar coordinates [2]*

*E IdS d*

0

∫ ∫

0

φ cos sin

*d*

=

θ

*IdS*

*B*

= π

*I*

leads to the following important relation:

(2C.1) entire energy balance is however:

*dE*

*dt *= 0

*dE dt*

*P*

*V*

γ

*E*

(2C.2)

γ the energy needs to fall to a millionth of its initial value after switching off the sender.

This is equivalent to the time that the level needs to drop 60 dB. When taking this into account the room loss factor can be derived [1] and is:

(2C.3)

The most commonly used formula for the reverberation time *T* is Sabine’s formula (which, after setting* p = 0 *follows directly from above equations):

*T*

60

= ln( )

*V c A*

≈

*c*

0

*V*

*A*

= .

*V*

*A*

[s]

(2C.4)

, where *A *is the total absorption area in m

2

. Other formulae are developed which take into account the fact that reverberation is not a continuous process, but involves a stepwise reduction of the wave energy every time it hits a boundary. One of those formulae

*20*

*REFERENCES*

is from Eyring [3]:

*T*

*Ey*

=

*c*

0

−

*S*

*V*

ln( 1 −

α

)

α

=

Another one is from Millington-Sette [4, 5]:

1

*S*

∑

*i*

α

*i*

*S i*

*T*

*MS c*

0

−

∑

*i*

*S i*

*V*

ln( 1 − α

*i*

)

(2C.5)

(2C.6)

(2C.7)

**References**

4.

5.

1.

2.

3.

Möser, M. (2004). Engineering acoustics, an introduction to noise control. Published by

Springer-Verlag Berlin, Heidelberg. Translated by S. Zimmermann.

Kuttruff, H. (2009), Room Acoustics, Fifth edition, published by Spon Press

Eyring, C.F. (1930), Reverberation time in dead rooms, J. Acoust. Soc. Amer., Vol. 1,

217-241

Millington, G. (1930), A modified formula for reverberation, J. Acoust. Soc. Amer., Vol.

4, 69-82

Sette, W.H. (1933), A new reverberation time formula, J. Acoust. Soc. Amer., Vol. 4,

193-210

*21*

## Basic sound insulation quantities

2D

Some quantities which are relevant to sound insulation are listed here. These characterize sound insulation and are used in building codes and regulations and requirements for sound insulating properties of building elements.

*Fig 2D.1 Sound intensity balance [1]*

Sound waves hitting a surface like a wall for instance are reflected, absorbed or transmitted (figure 2D.1) with the intensity balance of a building element). For a building element the sound reduction index *R* can be defined as:

*R*

= 10 lg

*I*

*I t i*

10 lg

*W*

*W i t*

10 lg

1

τ

[dB]

τ

From above expression the transmission factor can be written as:

(2D.1)

*W t i*

(2D.2)

In practice (or in a laboratory) the quantities for above equations cannot be measured easily, because intensity is hard to measure. Therefore another definition of the sound reduction index can be presented when a diffuse sound field in the source and sending room is assumed with intensity :

4πρ

0 0

*R*

(or

*R*

')

= −

*R*

+

10 lg

*S*

*A r*

[dB]

(2D.3)

, where *L*

*s*

* *and *L*

*r*

are the sound power level in the source and receiving room respectively,

*S* is the surface of the building element and is the total absorption in the receiving room.

*23*

*BASIC SOUND INSULATION QUANTITIES*

*L s*

* - L r *

is called the difference* D* in mean sound pressure level in the sending and receiving

2

can be calculated when rewriting Sabine’s formula for reverberation. The same formula can be used to express the apparent sound reduction index *R*, but now the focus lies on blaming the partition when other transmission paths contribute significantly to the level of the receiving room. For airborne sound insulation against outside noise the formula for R’ is a little different. First of all it should be noted that the NEN-EN 12342-3 (Dutch building code) requires a noise source with an angle of incidence of 45 o

. Furthermore *L* above equations should be reduced with 1.5 dB. Again it is different for traffic noise: the angle of incidence is not of importance anymore and the equation should be reduced with

3 dB.

*s*

* *will be including reflecting effects from the façade and

Alternatively the following equation can be given for the sound reduction index based on determination of the transmitted power to the receiving room by way of measuring the intensity. This is applied when the traditional method breaks down due to substation flanking transmission. Here* K*

*c*

is the Waterhouse correction seen earlier:

*R*

*I*

≈

*L pS*

−

*L*

*IR*

+ 10 lg

*S*

*S*

*R*

6 dB

(2D.4)

*R*

= +

*c*

=

*R*

*I*

+ 10 lg

1 +

8

*Vf*

0

(2D.5)

, where* L*

*Ps*

is the mean sound pressure level in the source room and *L*

*IR*

the mean intensity level over the surface *S*

*R*

Pa s/m.

ρ

*c*

**References**

1. Martin, H.J. (2007), Geluidisolatie, Collegedictaat TU Eindhoven behorend bij het college ‘Akoestiek 7S510’

*24*

## Derivation and m-file

## (Matlab)

3A

## An impermeable triple-layer system

**Derivation**

Equations 3.27-35 in section 3.2.4 are rewritten so that all terms including unknown parameters are brought to the left hand side of the equations and all terms including known parameters only are brought to the right hand side. The result is presented in equations

3A.1-9.

*Z*

*A*

ω

1

=

*Z*

*A p in*

(3A.1)

1 +

*Z*

*A p*

1

+ + ω

1

+

*Z*

*A p*

1

− = 0

(3A.2)

*p*

1

− +

*p*

1

+ − −

1

ω

1

=

*p in*

(3A.3)

−

1

+ ⋅

1 +

*Z*

*A*

1 +

*ikd*

1

*Z*

*A*

−

1

+

+

2

⋅

⋅

⋅

*ikd*

1

+

*ikd*

1

+

*Z*

*Z*

*A*

*A*

1

−

−

2

−

*ikd*

1

−

*ikd*

1

−

*ikd*

1

+ +

2

⋅

*ikd*

1

+ ⋅

+

+

−

*ikd*

1

ω

2

= 0

ω

2

= 0

−

*m*

2

ω 2

*w*

2

= 0

(3A.4)

(3A.5)

(3A.6)

−

1 −

*Z*

*A*

+

2

⋅

(

1

+

2

1

+

2

)

+

1 +

*Z*

*A*

2

)

− −

2

⋅

*t*

− (

1

*Z*

*A*

+

2

)

−

2

2

)

+

−

(

1

+

2

)

ω

3

= 0

+ ω

3

= 0

2

)

−

*m*

3

ω 2

*w*

3

= 0

(3A.7)

(3A.8)

(3A.9)

From above result a Matrix system **Ax=b** is made, where **x** are all unknown parameters in a 1x9 Matrix. These unknown parameters can have a random sequence, but chosen is below written sequence.

*x*

1

= ;

2

=

*p*

1

+

;

*x*

3

=

*p*

1

−

;

*x*

4

=

*p*

2

+

;

*x*

5

=

*p*

−

2

;

*x*

6

= ;

7

=

*w*

1

;

*x*

8

= ;

9

=

*w*

3

;

The 1x9 Matrix **b** is composed by all parameters on the right hand sight of eq. 3A.1-9 and 9x9 Matrix **A** are all left hand sight values (known) corresponding to the value in the **x**-Matrix. Equation 3A.10 shows this Matrix system.

*25*

*DERIVATION AND M-FILE (MATLAB) - IMPERMEABLE TRIPLE-LAYER SYSTEM*

*Z*

*A*

0

− 1

0

0

0

0

0

0

0

1 +

*Z*

1

*A*

1 +

*Z*

*A ep*

1

0

−

*ep*

1

00

0

0

0

*Z*

*A*

*Z*

1

*A eem*

1

0

−

*em*

1

0

0

0

0

0

0

0

1 +

*A*

*Z ep*

1

*ep*

1

1 −

*Z*

*A ep*

2

0

−

*ep*

2

0 0

*i*

ω

0

0

0

*A*

*Z em*

1

*em*

1

*Z*

*A em*

2

0

−

*em*

2

0

0

0

0

0

0

1 +

*A*

*Z ep*

2

*ep*

2

*i*

ω

−

*m*

1

ω 2

0

0

0

0

0

0

0

0

0

*i*

ω

*i*

ω

−

*m*

2

ω 2

0

0

0 −

*i*

ω

*i*

ω

*m*

0

0

0

0

0

0

3

2

*w*

2

*w*

3

*p w t*

1

*p p p*

1

−

2

++

2

−

*p p r*

1

+

⋅

=

00

0

0

0

0

*A*

⋅

*Z*

0

*p in*

0

*p in*

(3A.10)

*ep*

1 =

*e*

,

*em e*

,

*ep*

2 =

*e*

and

*em*

2 =

*e*

system gives solutions for all unknown parameters when varying the frequency. This is done by using Matlab, see below.

+

2

− ( +

2

**Corresponding m-file:**

clear all;

%vector with 9 unknowns:x=[pr;p1plus;p1min;p2plus;p2min;pt;w1;w2;w3] rho=1.21; c=343; teta=60; pin=100; A=0.0015; Z=rho*(c/cos(teta/180*pi)); m1=0.9; m2=0.8; m3=0.8; d1=0.05; d2=0.2; f=5000; om=2*pi*f; k= om/340; ep1=exp(i*k*d1); em1=exp(-i*k*d1); ep2=exp(i*k*(d1+d2)); em2=exp(-i*k*(d1+d2));

%A matrix for equation Ax=b

Amatrix=[(-1-A)/Z, 0,0,0,0,0, i*om,0,0;

0,(1+A)/Z, (-1+A)/Z, 0, 0,0, i*om, 0,0;

-1,1,1,0,0,0,-m1*om^2,0,0;

0,(1+A)*ep1/Z,(-1+A)*em1/Z, 0,0,0,0,i*om,0;

0,0,0,(1+A)*ep1/Z,(-1+A)*em1/Z,0,0,i*om,0;

0, -ep1,-em1,ep1,em1,0,0,-m2*om^2,0;

0,0,0,(1-A)*ep2/Z,(-1-A)*em2/Z,0,0,0,i*om;

0,0,0,0,0,(1+A)*ep2/Z,0,0,i*om;

0,0,0,-ep2,-em2,ep2,0,0,-m3*om^2]; b=[(-1+A)*pin/Z;0;pin;0;0;0;0;0;0]; x=Amatrix\b; absorb=1-(abs(x(1)/pin))^2; tau=(abs(x(6)/pin))^2;

R=-10*log(tau);

*26*

## Derivation and m-file

## (Matlab)

3B

## Triple-leaf system with a permeable leaf on sound incidence side

**Derivation**

For a triple-leaf system with a permeable leaf on sound incidence side the flow resistance for this first leaf should be modelled. Equations 3.36 and 3.37 (section 3.2.4) are rewritten as in Appendix 3A and shown in eq. 3B.1 and 3B.2. The rest of the nine equations are equal to those in Appendix 3A (eq. 3A.3-9).

−

*Rh*

*Z*

− 1

*r p r*

+

*p*

1

+

+

*p*

1

−

+ ⋅

ω

1

= −

*Z*

*Rh*

− 1

*p in*

*Z*

*Rh*

+ 1

*p*

1

+

+

−

*Rh*

*Z*

+ 1

*p*

1

−

+ ⋅

ω

1

=

*p in*

(3B.1)

(3B.2)

Again (Appendix 3A) using the same sequence of unknown parameters and writing the system of 9 unknown parameters and 9 equations into the Matrix system **Ax=b** (eq.

3B.3).

−

*Rh*

*Z*

− 1

−

−

0

0

0

0

0

0

1

1

1

*Z*

*Rh*

+ 1

1

1 +

*Z*

*AA ep*

1

0

−

*ep*

1

0

0

0

1

*Z*

−

*Rh*

*Z*

1

+ 1

*A em*

1

0

−

*em*

1

0

0

0

0

0

0

0

1 +

*A*

*Z ep*

1

*ep*

1

1 −

*Z*

*A ep*

2

0

−

*ep*

2

0

0

0

0

*A*

*Z em*

11

*em*

1

*Z*

*A em*

2

0

−

*e*

2

0

0

0

0

0

0

0

1 +

*A*

*Z ep ep*

2

2

ω

−

*m*

1

ω 2

0

0

0

0

0

0

ω 0

0

0

*i*

ω

*i*

ω

−

*m*

2

ω 2

0

0

0

*i i*

0

0

0

0

0

0

ω

ω

−

*m*

3

ω 2

*p p*

−

2

+

2

*p w t*

1

*pp r p p*

1

+

1

−

*w*

2

*w*

3

⋅ =

−

*Z*

*Rh*

− 1

*p in p in*

0

0

⋅⋅

0

0

0

0

*p in*

(3B.3)

*ep*

1 =

*e*

,

*em e*

,

*ep*

2 =

*e*

and

*em*

2 =

*e*

system gives solutions for all unknown parameters when varying the frequency. This is done by using Matlab, see below.

+

2

+

2

*27*

*DERIVATION AND M-FILE (MATLAB) - PERMEABLE LEAF SOUND INCIDENCE*

**Corresponding m-file:**

clear all;

%vector met 9 onbekenden:x=[pr;p1plus;p1min;p2plus;p2min;pt;w1;w2;w3] rho=1.21; c=343; teta=60; pin=100; A=0.0015; Z=rho*(c/cos(teta/180*pi)); m1=0.7; m2=0.8; m3=0.8; d1=0.05; d2=0.2; f=200; om=2*pi*f; k= om/c; Rh=3000; ep1=exp(i*k*d1); em1=exp(-i*k*d1); ep2=exp(i*k*(d1+d2)); em2=exp(-i*k*(d1+d2));

%A matrix voor vergelijking Ax=b

Amatrix=[(-Rh/Z-1), 1,1,0,0,0, Rh*i*om,0,0;

-1,(Rh/Z+1), (-Rh/Z+1), 0, 0,0, Rh*i*om, 0,0;

-1,1,1,0,0,0,-m1*om^2,0,0;

0,(Rh/Z-1)*ep1,(-Rh/Z-1)*em1, ep1,em1,0,0,Rh*i*om,0;

0,-ep1,-em1,(Rh/Z+1)*ep1,(-Rh/Z+1)*em1,0,0,Rh*i*om,0;

0, -ep1,-em1,ep1,em1,0,0,-m2*om^2,0;

0,0,0,(1-A)*ep2/Z,(-1-A)*em2/Z,0,0,0,i*om;

0,0,0,0,0,(1+A)*ep2/Z,0,0,i*om;

0,0,0,-ep2,-em2,ep2,0,0,-m3*om^2]; b=[-Rh*pin/(Z-1);pin;pin;0;0;0;0;0;0]; x=Amatrix\b; absorb=1-(abs(x(1)/pin))^2; tau=(abs(x(6)/pin))^2;

R=-10*log(tau);

*28*

## History of tent construction

4A

The true breakthrough for tent structures started in 1954 when Frei Otto developed his first fragile tent structures and wrote his dissertation on “the suspended roof” (and after an essay entitled “Tent” in 1961). But long before that forms of tent structures have been used for different kind of purposes. It started with creating a simple shelter with a minimum of materials, mostly for military camps and later (in the civilized world) as emergency shelter during catastrophes.

*Fig. 4A.1 *

*Emergency shelter in Adaparzi, Turkey after an earthquake in 1999 [1]*

**4A.1 The early days**

Awnings have been used in the south for a long time to protect against solar radiation. The alleys of the bazaars in Arabia and

North Africa are (and were) spanned by stretched lengths of material. The umbrella is a mobile example of this type of material use for protecting against climate influences as solar radiation and rain.

Even as early as 500 BC the umbrella was carried in Asia as a symbol of rank. Common characteristics of tents, awnings and umbrellas lie in the way they offer protection against climatic influences and the similarities between the ways they are made (combination of stabilising linear or network-like structures with areas of textile).

*Fig. 4A.2*

*Jonas Hanway and his umbrella [1]*

**Tents: archetypes**

Architectural remains of 40.000-year-old encampments in Siberia suggest that archetypical forms of tents were constructed with straight and curved wood and a covering of

*Fig. 4A.3 *

*A 28.000 BC reconstruction of remains found in the Ukraine [2] *

*29*

*HISTORY OF TENT CONSTRUCTION*

*Fig 4A.4 *

*Transportable tent of the Nganasan people of Siberia [1]*

leaves, bark or skins (figures 4A.3 an 4A.4). Preserved (and more recent) tents have been found by the North American Indians, Inuit and Lapps. But also in the more hot regions of the earth tent structures have been found, showed by the example of the Bedouin tents in North Africa (figure 4A.5). Here, due to a lacking material for masts, the skin covering

*Fig 4A.5 *

*Bedouin tent from North Africa [1]*

forms part of the load-bearing structure.

The yurt (figure 4A.6) represents another basic tent-form. This kind of tent is found in areas extending from Mongolia to western Anatolia and from the Urals to Afghanistan.

*Fig 4A.6 *

*Mongolian yurt (18th century) [2]*

The yurt uses a frame made up of several wooden parts in a domed shaped form with a covering of sheep’s wool.

During these times tents were used mainly for public and courtly occasions and for military purposes. Examples are the Nineveh military camp of the Assyrian King Sennacherib (705-681 BC) (figure 4A.7) and the Egyptian royal tent of the Battle of Kadesh

(275 BC), which was said to have been supported by cedar poles and covered an area of roughly 8.200 square meters.

*30*

*THE EARLY DAYS*

*Fig 4A.7*

*Nineveh tent, ca 700 BC [2]*

Europe, where no previous independent development can be found, was influenced by many different cultures. From the Middle Ages down the 19th century tents were mainly for military purposes in central Europe. One famous example was at the Field of Cloth of

*Fig 4A.8*

*The Field of the Cloth of Gold [3]*

Gold in 1950, where Henry VIII of England met the French King Francis I near Calais.

More than 400 tents were assembled there to accommodate 5000 people and 3000 horses

(figure 4A.8).

Another purpose for tents was popular around 1770/80. Tapissiers, the upholsterer and outfitter of tents, designed a lot of garden buildings. An example is the drawings of the

Kassel court architect Jussow for designs of small garden pavilions.

*Fig. 4A.9, 4A.10 *

*Designs for garden tents (from the album *

*‘Décoration intérieure et jardins de Versailles’ and a Polygonal Garden Tent from *

*Jussow ca 1810) [1]*

**Circus tents**

In a special volume of Handbuch der Architektur, the major encyclopaedia of architecture around 1900, the following definition can be found: “Easiest to move around are the tentlike circus systems as frequently used in recent times by travelling companies of stunt

*31*

*HISTORY OF TENT CONSTRUCTION*

riders. They can be erected in only a few hours and taken down in even less time. But such special systems hardly belong to the field of architecture.” [4]

Those first ‘chapiteau’s’ came around 1860 and was made of sailcloth raised with the masts and stretched externally over storm and roundel rods. A modern chapiteau with a diameter of 50 meters has four main masts (25 meters high) and are made of the light filigree lattice girders. They are tied by steel cables and are made up of several sections of membrane.

**4A.2 Modern development**

In those early days tent construction was neglected as a constructional form in the theory of architecture, but in the 19th century architects did turn their attention to this form of structure (mostly in a decorative way). Then in 1861 Gottfried Semper described fabric forms of construction as one of the four basic elements of building and in 1937 Le Corbusier designed a exhibition pavilion for the Paris World Fair with a square tent surrounded by a fence with material stretched between poles and stabilized by guy ropes.

Due to the developments (social changes and growth of the middle-class) in the 19th century leisure activities like sports, singing, etc. needed economical temporary structures.

Temporary halls where also required for exhibitions of industrial products (in cities that could not afford the crystal palaces of London and Paris).

**Frei Otto and his philosophy**

Down to the 1950s, despite their increasing dimensions and spans, tents were still constructed largely with timber frames and cotton fabric covering. All this changed after

1954, when Frei Otto designed elements subject to tension forces in their most effective structural form. This was aided by the fact that the tent-building industry was ready for new technologies [5]. Improvements in textile membranes, consisting of natural and artificial fibres (polyester, PVC, glass fibre, etc.) facilitated larger spans and guaranteed greater durability and safety. Nowadays, cable nets (wood, metal or plastics) and tents

(glass-fibre textiles) can last just as long as conventional forms of construction.

*32*

*Fig. 4A.11 German Pavilion at the EXPO’67 in Montreal (Otto) [6]*

*MODERN DEVELOPMENT*

Since 1954 most of the construction forms described by Frei Otto have been realized and developed in a variety of ways. Aided nowadays by new manufacturing methods, computerized means of calculation and form finding.

Frei Otto’s long career in lightweight structures includes the development of stressed tensile sails for the Lausanne EXPO64, the famous German Pavilion’s membrane and prestressed cable nets for EXPO67 in Montreal (figure 4A.11), the Munich Olympic Roofs in

1972 (figure 4A.12) and the Gridshell at Mannheim in 1975 (figure 4.1A3). The conceptual design studies for these and many other projects were carried out at the Institut fur

LeichteFlachentragwerke (IL). He is one of the leading innovators in the development of lightweight structures and therefore off course in membrane structures. Public and professional acceptance of tensile structures as permanent systems was enhanced by these large and historically important prestressed cable-net structures.

*Fig. 4A.12 *

*One of Münich Olypic roofs in 1972 (Otto) *

*[1]*

In both the developed and developing world, widespan enclosures are increasingly required to house and facilitate many of the collective activities of society. Aims here are not using excessive quantities of scarce construction material or using unnecessary quantities of energy in their fabrication and operation, but off course also a delight in their occupation. Fundamentals for securing this aim are the beauty of architecture, efficiency of structural form and appropriateness of material usage.

*Fig. 4A.13 *

*Gridshell at Mannheim in 1975 (Otto) [6]*

**Movable roofs, pneumatic structures and large climatic shells**

The retractable roof was, and still is, a particular aspect of tent and membrane building.

In the mid-twentieth century it was technically possible to provide large spaces with an opening and closing roof. Examples from Schlaich Bergermann und Partner are from 1987

*33*

*HISTORY OF TENT CONSTRUCTION*

and 1990 with opening membrane roofs.

Pneumatic (inflatable) structures as we know it now originate from the development of steerable airships and balloon travel. The cotton spanning over the rigid frame and held together by internal gas pressure was invented by Count Zeppelin in 1891. Richard

Buckminster Fuller, also known from his work on zeppelins, was closely involved in the development of large wide-spanning building shells. In the 1950s and ‘60s interest arose for creating a climate in an artificial environment. Examples are Fuller’s dome over Manhattan and Otto’s Antarctic vision.

*Fig. 4A.14*

*American Pavilion at Osaka World Fair ‘70 *

*[1]*

The realisation of large pneumatic structures was set by the American pavilion at the

Osaka World Fair in 1970 (figure 4A.14) with its air-supported membrane. Today pneumatic structures are realized by thin flexible membrane, which is pre-stressed by internal pressure so it can withstand external forces.

In the 1970s advances were made in low profile air-supported membrane structures by

Geiger Berger Associates, and, coupled with the development of PTFE coated glass fibre membrane as a new membrane material, they led to the emerge of enormous, permanent membrane structures in the US.

**EXPO’s and sports facilities**

After Le Corbusier’s Pavilion in 1937 and

Otto’s German pavilion at the EXPO’67 in Montreal, membrane constructions were presented at the EXPO’s of Osaka (1970),

Seville (1992) and Hanover (2000). A beautiful example at Hanover was Shigeru Ban’s

Japanese Pavilion (figure 4A.15).

A particular domain of membrane construction is that of the sports arenas. Large areas and no intermediate supports are desirable.

A steel structure covered with an efficient, generally translucent membrane has become a standard solution.

*Fig 4A.15 Japanese Pavilion at Hanover [1]*

*34*

*REFERENCES*

**References**

4.

5.

6.

1.

2.

3.

Koch K.-M. (2004), Membrane structures, Published by Prestel Verlag

Burkhardt B., Geschichte das Zeltbaus. Detail – Zeitschrift fur Architektur + Baudetail

(Serie 2000, nr. 6, 960-964).

House Greydragon, http://www.greydragon.org/library/tentpics/

Schmitt E., Handbuch der Architektur, Darmstadt, p. 76

Otto F. (edited by) (1962 – 1966), Tensile structures – Design, structure and calculation of buildings of cables, nets and membranes, MIT Press

Gabriel A. (interviewed by), “… noch vieles ist moglich…” – Frei Otto zur Zukunft des

Zeltbaus. Detail – Zeitschrift fur Architektur + Baudetail (Serie 2000, nr. 6, 965-970).

*35*

## Peutz’ Laboratory for Acoustics

6A

*Fig 1. Laboratory’s overview*

*37*

*PEUTZ’ LABORATORY FOR ACOUSTICS*

*38*

*Fig. 2 Insulation measurement rooms, window opening*

## Standards and guidelines

6B

*ISO 140-3:1995*

Acoustics – Measurements of sound insulation in buildings and of building elements –

Part 3: Laboratory measurements of airborne sound insulation of building elements

Other standards which are referred to in this report:

*ISO 140-1:1997*

Acoustics – Measurements of sound insulation of building elements – Part 1: Requirements for laboratory test facilities with suppressed flanking transmission

*ISO 140-2:1991*

Acoustics – Measurements of sound insulation of building elements – Part 2: Determination, verification and application of precision data

*ISO 717-1:1996*

Acoustics – Rating of sound insulation in buildings and of building elements – Part 1:

Airborne sound insulation

*ISO 717-1:1996/A1:2006*

Acoustics – Rating of sound insulation in buildings and building elements – Part 1: Airborne sound insulation – Amendment 1: Rounding rules related to single number ratings and single number quantities

And Dutch codes:

*NEN 5079:1990*

Geluidwering in woongebouwen

Het weergeven in één getal van de geluidisolatie van bouwelementen, gemeten in het laboratorium

*39*

## Wooden frames

6C

*Fig. 1 Small frame*

*Fig. 2 Big frame*

*41*

## Cross-sectional connection frames to opening

6D

*Fig. 1 (Lightweight) glass wool variant *

*2x Small frame & 1x Big frame*

*Fig. 2 Polyester wool variant*

*2x Small frame & 1x Big frame*

*43*

*CROSS-SECTIONAL CONNECTION FRAMES TO OPENING*

*Fig. 3 *

*Aerogel variant on Small Special frame*

*44*

## Measurement information

Membrane types and their number (section 5.2.1).

1.

2.

3.

4.

5.

6.

Duraskin B18656 PTFE-coated fiberglass mesh

Duraskin B3704 142 PVC-coated PES (polyester)

ACOUSTIS-50 Perforated fiberglass

Duraskin B4951 286 PVC-coated PES

Duraskin B4617 286 PVC-coated PES

Duraskin B18039 PTFE-coated fiberglass

**Phase 1: Glass and polyester wool measurements**

Triple layer systems

**Measurement # Variant **

1

2

3

4

5

Glass-A

Glass-B

Glass-C

Pol-A

Pol-B

Pol-C

Glass-D

Glass-E

Glass-F

Pol-D

**Membrane configuration**

1-6-6*

3-6-6

5-6-6

1-6-6

3-6-6

5-6-6

2-4-4

6

7

8

9

10

11

12

Pol-E

Pol-F

2-4-5

4-4-5

2-4-4

2-4-5

4-4-5

**Numbers refer to above listed membranes and section 5.2.1.*

6E

700 g/m2

480 g/m2

410 g/m2

800 g/m2

900 g/m2

800 g/m2

Single membranes

**Measurement # Variant **

13

14

15

Single-4

Single-5

Single-6

**Membrane type**

PVC-coated PES (800 g/m2)

PVC-coated PES (900 g/m2)

PTFE-coated fibreglass (800 g/m2)

**Phase 2: Aerogel measurements**

**Measurement # Variant **

16

17

18

19

Single-3

Aero-CS

GLasswool-CS

Aero-CB

**Membrane type or configuration**

Perforated fibre glass (410 g/m2)

5-6-6 and small cavities

5-6-6 and small cavities

5-6-6 and big cavities

*45*

800

1000

1250

1600

2000

2500

3150

4000

5000

**Frequency [Hz] **

100

125

160

200

250

315

400

500

630

## Measurement data

6F

**Single layer and aerogel triple layer membranes**

Measurement number 13 (Appendix 6E): Perforated fibre glass (410 g/m

2

)

Measurement number 14:

PVC-coated PES (800 g/m

2

)

Measurement number 15:

PVC-coated PES (900 g/m

2

)

Measurement number 16: PTFE-coated glass fibre (800 g/m

2

)

Measurement number 17:

Aero-CS

Measurement number 18:

Glasswool-CS

Measurement number 19: Aero-CB

3.7

3.8

3.8

3.9

3.7

3.7

3.8

3.9

3.8

4.2

2.7

2.8

3

3.1

3

**#13 #14 #15 #16 #17 #18 #19**

3.7

4.4

5

3

4.6

6.2

4

4.3

5.5

3.6

4.7

6

14

13.3

15

9.4

11.1

11.5

6.8

10.2

10.3

5.1

6.2

7

8.6

9.9

5

5.4

6.5

8

10

5.1

6.1

6.8

8.3

9.3

13.7 11.4 13.8

12.5 11.2 17.1

14.4

17.9

18.4

12.8

15.6

17

21.4

24.9

29.2

11.3 11.5 10.6 18.9 20.4 34.6

12.8 12.7 12.3 22.7 30.4 37.7

15.5 14.6 13.9 26.2 41.1 39.2

15.3 15.6 14.7 31.8 50.2 44.7

16.2 17.1 15.6 37.7 55 51.2

17.9 18.8 17.1 43.7 56.8 58.2

20.1 20.8 19.2 52.3 59.4 66.1

21.8

23.1

22.6

24

20.9 59.1 60.6 68.6

22.2 59.7 59.5 64.3

25.4 26.4 24.8 58.5 57.5 60.1

**Triple layer membranes**

Measurement number 1: Glass-A Measurement number 7: Glass-D

Measurement number 2: Glass-B Measurement number 8: Glass-E

Measurement number 3: Glass-C Measurement number 9: Glass-F

Measurement number 4: Pol-A

Measurement number 5: Pol-B

Measurement number 6: Pol-C

Measurement number 10: Pol-D

Measurement number 11: Pol-E

Measurement number 12: Pol-F

*47*

*MEASUREMENT DATA*

800

1000

1250

1600

2000

2500

3150

4000

5000

**Frequency [Hz] **

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

**Frequency [Hz] **

100

125

160

200

250

315

400

500

630

**#1 #2 #3 #4 #5 #6**

8.5 7.4 7.7 6.1 7.2 7.3

12.6 12.6 13.8 10.3 11.6 12.9

14.6 14.6 15.9 12.1 13.3 14.3

17.6 17.5 18.6 14.3 15.2 16.6

20.1 20.9 21.7 17.3 18.2 19.3

22.8 24.6 24.8 20.9 21.7 22.5

27.4 29.1 30.3 25.5 26.3 26.9

32.6 33.5 37.4 29.2 30.1 33

37.8 39.2 45.4 32.7 33.9 39.7

44.3 45.7 55.1 37.5 38.8 47.6

51.1 52.8 63.4 42.9 44.9 55.4

58.3 59.9 68.6 48.5 50.6 63.1

64.7 66.4 74 53.7 56.1 69.1

69.8 71.3 75.6 58.8 61.2 73.3

71.3 71.9 73.2 64 66 72.8

71.1 70.3 70.3 66.7 67.7 69.9

64.9 64.9 65.2 64.1 64.1 64.5

60.6 61.8 61.8 61.7 61.8 62.1

**#7 #8 #9 #10 #11 #12**

6.8 7.2 7.6 7.4 6.8 6.6

11.1 11.8 13.2 10.4 11.1 12.1

14.4 14.7 16 12.6 13.3 14.8

17

20.4

17.5 18.7 14.9 15.1 17

21.6 22 18.3 18.7 20.1

24.7 25.3 25.8 22.1 22.9 24.2

29.4 30.1 31.6 26.9 27.5 29

34.1 35 38.7 30.7 31.3 34.7

39.5 40.3 47.2 34.4 35.3 41.8

45.9 46.7 56.5 38.8 40 49.5

52.9 53.7 64.6 44.5 45.5 57.4

59.7 60.4 69.4 50.1 51 64.6

66.2 66.4 74.1 55.2 56.3 70.6

71.4 71.4 75.4 60.3 61.2 74.1

72.8 72 72.9 65.2 65.9 73.3

69.8 69.6 69.1 67.5 67.5 70.7

64.1 64.6 63.7 64.5 63.9 65

61.7 62 61.8 61.3 60.9 61.6

*48*

## Flow resistance measurements

6G

The flow resistance measurements are performed in the Peutz’ Laboratory for Acoustics in Mook (Appendix 6A; figure 1). Samples of glass wool, polyester wool and aerogel

(fig. 6G.1) are measured. The measurements are performed according to ISO 9053:1991

“Acoustics – Materials for acoustical applications – Determination of airflow resistance”.

The samples are placed in a sample holder (fig. 6G.2) and by using a compressor air is pressured through the holder. The airflow (and derived from that the velocity) through the sample and the pressure difference over the sample are measured.

For glass and polyester wool a sample of 100 mm is taken and for aerogel 10 mm.

For each filling material the three samples are taken, put in the tube and the pressure difference is read from the pressure meter. The reading of the measurement tube (B) is done for five values with corresponding flow [*L/h*] and airflow speed [m/s].

From these parameters and the averaged pressure difference the flow resistance is calculated. The measurement results are given in table 6G.1.

According to the ISO 9053:1991 inter laboratorial comparing measurements are still to be carried out to determine the accuracy of the measurements.

*Figure 6G.1 Material samples fttb: glass wool, polyester wool and aerogel*

*Figure 6G.2 Sample holder (Peutz’ Laboratory for Acoustics)*

*49*

*FLOW RESISTANCE MEASUREMENTS*

**Meas. Reading Cal. Flow Speed Polyester wool tube [mm] values [L/h] [m/s]**

B

30

60

90

120

150

1.32

2.82

4.32

5.77

7.15

79.2 3.92

169.2 8.38

M1

0.5

1.2

259.2 12.84 1.8

346.2 17.15 2.5

428.4 21.22 3.1

M2

0.5

1.2

1.8

2.5

3.2

M3

0.5

1.1

1.8

2.5

3.1

**Glass wool **

M1

2.6

5.6

8.8

12

14.9

*Table 6G.1 Pressure difference observations for three samples per filling material*

M2

2.2

4.4

6.9

9.3

11.9

M3

3

5

7.9

10.6

13.1

**Aerogel**

M1

6.3

15

24.6

34.5

42.2

The averaged results are presented in table’s 6G.2-4. The specific airflow resistance *Rs*,

=

⋅ 1000

3

(or Pa.s/m).

*v*

thickness of the sample gives the airflow resistivity per unit thickness in Ns/m

4

. In most literature and papers the ‘specific airflow resistance’ is referred to as the ‘airflow resistance’ and is compared with this parameter. In the MLM model however the airflow resistivity should be entered.

**Reading [mm] Speed [m/s] **

30

60

90

120

150

3.92

8.38

12.84

17.15

21.22

**Pressure [Pa] Rs [Ns/m**

2.6

5

7.9

10.6

13.3

663

597

613

620

627

**3**

**] r [kNs/m**

**4**

**]**

6.6

6

6.1

6.2

6.3

**correction**

**Averaged r**

**Averaged Rs at 0.5 mm/s**

0.328 -

638

6400

Ns/m

3

Ns/m

4

*Table 6G.2 Averaged specific flow resistance and resistivity for three samples of glass wool (100 mm)*

**Reading [mm] Speed [m/s] Pressure [Pa] Rs [Ns/m**

**3**

30

60

90

120

150

3.92

8.38

12.84

17.15

21.22

0.5

1.2

1.8

2.5

3.1

128

139

140

146

148

**] r [kNs/m**

**4**

**]**

1.3

1.4

1.4

1.5

1.5

**correction**

**Averaged Rs at 0.5 mm/s**

**Averaged r**

0.944 -

127 Ns/m

3

1300 Ns/m

4

*Table 6G.3 Averaged specific flow resistance and resistivity for three samples of polyester wool (100 mm)*

M2

5.2

M3

5.8

13.8 13.6

24.2 22.6

36 31.8

48.4 41.6

**Reading [mm] Speed [m/s] Pressure [Pa] Rs [Ns/m**

**3**

30

60

90

120

150

3.92

8.38

12.84

17.15

21.22

5.8

14.1

23.8

34.1

44.1

1471

1687

1854

1988

2077

**] r [kNs/m**

**4**

**]**

147.1

168.7

185.4

198.8

207.7

**correction**

**Averaged Rs at 0.5 mm/s**

**Averaged r**

0.990 -

1389 Ns/m

138900 Ns/m

3

4

*Table 6G.4 Averaged specific flow resistance and resistivity for three samples of aerogel (10 mm)*

*50*

## Input file for the MLM

7A

*// *************** INVOERFILE VOOR HET MEERLAGENMODEL ************************

*//*

*// G-Glass1: triple-leaf construction*

*// Achter naar voor: Glass fibre, glass wool, glass fibre, glass wool, x1*

*//*

*// *************** Eerst de startlaag aan de achterzijde **********************

*// ********** Direct gevolgd door het medium aan de zendzijde *****************

*//*

*// De achterzijde is altijd een liquid, slechts gegeven door rho en c*

*// De voorzijde is een liquid, gegeven door rho en c.*

*//*

*// Voor- en achterzijde lucht:*

*1.21 342.6*

*1.21 342.6*

*//*

*// 0.09 1259.0 // Waterstof*

*// 5.51 144.0 // Freon = zeer zwaar gas*

*// 1000.0 1600.0 // Water*

*//*

*// ******************* Hoeken start, stap en stop *****************************

*//*

*// Bij berekening ALZIJDIG*

*//*

*// 0.5 1.0 89.5*

* 1 2 89*

*// 2.5 5.0 87.5*

*//*

*// Bij berekening HOEK*

*//*

*// 5 10 85*

*// 0 0 0*

*// 85 0 85*

*//*

*//*

*// *************** Weegfunktie van de verliesfactor (verl/500)^macht **********

*// Macht = 0 betekent dat het verlies oploopt evenredig met de frekwentie.*

*// Macht = -1 geeft een constante waarde over alle frekwentie.*

*//*

*// Tot en met MLM_35 stond macht = -1 hard geprogrammeerd.*

*//*

*// Lees het derde artikel voor het NVBV-Bouwfysica_blad*

*//*

*// -2.0*

*-1.0*

*//*

*// *************** Weegfunktie met sin * cos^(1+weging) ***********************

*// Weging = 0 betekent dus geen weging.*

*//*

*// Staal doet het wat mooi met 0.7*

*// Voor algemeen gebruik komt 0.5 meer in aanmerking*

*//*

*// 0.7*

*0.5*

*51*

*INPUT FILE FOR THE MLM*

*//*

*// ****************************************************************************

*// Frekwenties start, stap en stop*

*//*

*//*

*// Bij berekening ALZIJDIG*

*//*

*//*

*// Voor 8 oktaven, 62.5 125 250 500 1k 2k 4k 8k*

*//*

*// Beginnend met 2 lijnen in band 62.5*

*// 55.2427 22.0971 11302.66*

*//*

*// Idem beginnend met 1 lijn in band 62.5, dus halve aantal*

*// 66.2913 44.1942 11291.618*

*//*

*// Idem beginnend met 2 lijnen in band 62.5, maar slechts paar oktaven*

*// 55.2427 22.0971 1403.176*

*//*

*// Beginnend met 1 dummie-lijn onder de 62.5-band.*

*// Daarmee wordt ook een minimum op de eerste lijn in 62.5*

*// gedetecteerd.*

*//*

*// AANBEVOLEN (bij alzijdig)*

*//*

* 33.1456 22.0971 11302.66*

*//*

*//*

*// Bij berekening HOEK*

*//*

*// 100 100 4000*

*//*

*// *************** Dan alle tussenlagen van achteraf ***************************

*//*

*// Zie ook de help-file*

*//*

*//*

*// Glass-C (3x impermeable)*

*// Ongedempt*

*// sol 0.001 800 0.9e9 0.00 0.01 0.30 // 6*

*// liq 0.2 1.21 1.42e05 0.00 0.00 6400.0 // glass wool*

*// sol 0.001 800 0.9e9 0.00 0.01 0.30 // 6*

*// liq 0.05 1.21 1.42e05 0.00 0.00 6400.0 // glass wool*

*// sol 0.001 900 0.9e9 0.00 0.01 0.30 // 5*

*// Glass-A*

*// Ongedempt*

* sol 0.001 800 0.9e9 0.00 0.01 0.30 // 6*

* liq 0.2 1.21 1.42e05 0.00 0.00 6400.0 // glass wool*

* sol 0.001 800 0.9e9 0.00 0.01 0.30 // 6*

* liq 0.05 1.21 1.42e05 0.00 0.00 6400.0 // glass wool*

* sol 0.001 700 0.9e9 0.00 0.01 0.25 // 1*

*// Pol-C (3x impermeable)*

*// Ongedempt*

*// sol 0.001 800 0.9e9 0.00 0.01 0.30 // 6*

*// liq 0.2 1.21 1.42e05 0.00 0.00 1300.0 // pol wool*

*// sol 0.001 800 0.9e9 0.00 0.01 0.30 // 6*

*// liq 0.05 1.21 1.42e05 0.00 0.00 1300.0 // pol wool*

*// sol 0.001 900 0.9e9 0.00 0.01 0.30 // 5*

*END*

*//**********************************************************************************

*52*

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