### VTAF 01 – Sound in Buildings and Environment

10. Building Acoustics 2 - Airborne sound insulation models

**NIKOLAS VARDAXIS**

DIVISION OF ENGINEERING ACOUSTICS, LTH, LUND UNIVERSITY

2021.05.11

### … recap from last lecture (I)

• Airborne sound insulation measurements (ISO standards)

⎟⎟⎠

⎞

⎜⎜⎝

+ ⎛

−

= ( ) ( ) 10log ( )

)

( *A* *f*

*f* *S*
*L*
*f*

*L*
*f*

*R* _{S}_{R}

Statement of results:

*• R’*_{w}*(C*_{50-3150}*; C** _{tr}*)

*• R*_{w}*(C*_{50-3150}*; C** _{tr}*)

63 125 250 500 1000 2000 4000 15

20 25 30 35 40 45 50 55 60

Frequency [Hz]

R [dB]

### … recap from last lecture (II)

• Impact sound pressure level measurements (ISO standards)

⎟⎠

⎜ ⎞

⎝

+ ⎛

= 10

) log (

10 ) ( )

( *A* *f*

*f*
*L*
*f*

*L*_{n}_{R}

0 10 20 30 40 50 60 70

100 160 250 400 630 1000 1600 2500

Ln[dB]

Frequency [Hz]

Statement of results:

*• L’*_{n,w}*(C*_{l,50-2500 }*)*

*• L*_{n,w}*(C*_{l,50-2500 }*)*

### Spectrum adaptation terms

Rw=36 dB C -1

Ctr -4

**Rw=36 (-1,-4)**

Rw=36 dB C -3

Ctr -11

**Rw=36 (-3,-11)**

• Example: Airborne sound insulation measurement

### Physics of sound transmission

• A sound field in a room (or a source on a floor) excites waves in the adjacent wall or on the floor.

• The structure vibrates and produced waves are transmitted through the structure. Eventually this event excites either other structures or the air in another room.

• Thinking about our SDOF, as a homogeneous solution, described in terms of natural frequencies (Eigen-).

• Eigenfrequencies and eigenmodes depend on boundary and coupling conditions. How a wall is mounted and the surrounding elements are key to determining final properties of a wall

• Field measurements (in-situ) and laboratory data differ (Rw vs. R’w).

### Physics of sound transmission

• The relationship between incident and transmitted waves depends on speed of sound in the two media (stiffness and density) and angle of incidence (we did not see that really, but it is so).

• Sound insulation of a material is an interplay between mass, stiffness and damping.

• Diffuse sound fields are typically assumed – but it is not the case at low frequencies, remember (Schröder frequency)!

### Outline

### Introduction

### Examples

Impact:

SS-EN 12354-2

### Flanking transmission

### Summary

Airborne:

SS-EN 12354-1

*Analytical models for calculation of R(f) in *

### single and double-leaf walls

### DEF: Coincidence – Critical frequency (I)

• The wavelength of a bending wave λ_{B} is dependent on frequency, bending
stiffness and mass density.

• When the wavelength of sound in air coincides with the structural wavelength à Coincidence phenomena

‒ Radiation efficiency becomes very high

‒ Poor insulation properties

Impinging wave

### DEF: Coincidence – Critical frequency (II)

### • Bending wave velocity in a plate

*• If f = f*

_{c}*thus c*

_{B}*= c*

_{o}*= 340 m/s (f*

_{c}### = critical frequency)

### • Or expressed as a function of the coincidence number

2 4

*m*
*f* *B*
*c*_{B}

= π ʹ′ʹ′

*B*
*m*
*f*_{c}*c* ʹ′ʹ′

= ^{2}π

2 0

*h*
*f** _{c}* =

*K*

NOTE: The condition for coincidence is that λ_{B}=λsin(φ). Therefore, if the incidence angle φ decreases,
the coincidence frequency f_{c} increases according to f_{c}(φ)=f_{c}/sin^{2}(φ). The lowest frequency at which
coincidence occurs (critical frequency) is at the incidence angle φ=90º.

Impinging wave

### Critical frequency for common materials

Material Coincidence number (K) Thickness [mm] *f** _{c }*[Hz]

Concrete 18 160 110

Light concrete 38 70 540

Gypsum 32 10 3200

Steel 12-13 1 12000

Glass 18 3 6000

*K* _{60000} *E*ρ

=

• For a homogeneous isotropic plate of uniform thickness, the coincidence number is:

### Outline

### Double-leaf wall

### Exact method

*Analytical models for calculation of R in * single and double-leaf walls

### Single-leaf wall

### Approximate method

### Wall types

### Sound reduction index of single-leaf partitions

### • Exact method

*‒ Region I: Stiffness-controlled region (f < f** _{11}*)

*‒ Region II: Mass-controlled region (f*_{11}*< f < f** _{c}*)

*‒ Region III: Damping-controlled region (f**c*< f)

### Sound reduction index of single-leaf partitions

*• Region I: Stiffness-controlled region (f < f** _{11}*)

‒ Panel vibrates as a whole (considered thin)

### ( )

### ( )

2 2 2

3 8

2

2 2

1 1

) 1

( 768

4 ) (

1 ln log 1 10

log 10 )

(

⎟⎠

⎜ ⎞

⎝

⎛ +

= −

=

+

⎟⎟−

⎠

⎞

⎜⎜⎝

= ⎛ ^{−}

*b*
*Eh* *a*

*C*

*C*
*c*
*f*
*f*

*K*

*K* *K*
*f*

*R*

*s*

*s*
*F*
*F*
*S*

*S*
*S*

π

υ ρ π

C_{s}: mechanical compliance for a rectangular plate
E: Young’s modulus of the material the wall is made of
h: wall thickness

a, b: plate dimensions : Poisson’s ratio of the wall

: density of the surrounding fluid (F), i.e. air
c_{F}: wave propagation speed in the fluid (F), i.e. air

c_{Lplate}wave propagation speed in the plate (longitudinal wave)

υ
ρ*F*

⎟⎠

⎜ ⎞

⎝

⎛ +

= _{2} _{2}

11

1 1

3

4 *c* *h* *a* *b*

*f* π _{Lplate}

For a simply supported
*plate of dimensions a x b*

### Sound reduction index of single-leaf partitions

*• Region II: Mass-controlled region (f*_{11}*< f < f** _{c}*)

‒ Sound reduction independent of stiffness (controlled by mass inertia)

‒ Some energy transmitted and part reflected at panel surface

*c* *dB*
*f* *fm*

*R*

*F*
*F*

'' 5 1

log 10 )

(

2

⎟−

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞

⎜⎜⎝

+⎛

= ρ

π

m’’= h is the surface mass of the panel^{ρ}

*NOTE: Although the above equation is valid for frequencies up to f** _{c}*, it yields only accurate results for f

*≤ 0.5f*_{c}*. The mathematical expresion around f** _{c}* is mathematically cumbersome and rarely used, so
approximate methods were developed.

Mass law >>1

Random incidence correction

### Sound reduction index of single-leaf partitions

*• Region III: Damping-controlled region (f*

_{c}### < f)

‒ Curve “dip” controlled by internal material damping

‒ Important for design (low insulation)

*‒ Contribution from plate reverberant field dominates (only above f** _{c}* )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞

⎜⎜⎝

+⎛

=

⎟⎟−

⎠

⎞

⎜⎜⎝

+ ⎛

+

=

'' 2

1 log 10 ) (

7 . 5 log

22 . 33 ) log(

10 ) ( ) (

*F*
*F*

*c*
*c*

*c*
*c*

*c*
*m*
*f* *f*

*R*

*f* *dB*
*f* *f*

*R*
*f*

*R*

ρ π

η

is the total loss factor or damping of the panel η

*Lplate*
*F*

*c* *hc*

*f* *c*
π

2 3

=

### Notes: Sound reduction index of single-leaf partitions

• As the frequency of the incident sound wave increases, the wavelength of the bending-waves in the plate approaches the wavelength of sound waves in air (remember that bending waves are dispersive).

• When the wavelengths match, the panel offer little resistance and vibrates with same velocity as incident sound wave

‒ Curve “dip” controlled by internal material damping

‒ Important for design (low insulation)

*Lplate*
*F*

*c* *hc*

*f* *c*
π

2 3

=

### Notes: Sound reduction index of single-leaf partitions

• Design in respect to coincidence frequency

‒ Make coincidence either very low (100-125 Hz) or very high (3150-4000 Hz). 100 Hz – 4000 Hz is important for human hearing.

‒ Very low: thick wall, low density, high Young’s modulus: e.g. 15 cm concrete.

‒ Very high: thin wall, high density, low Young’s modulus: e.g. 13 mm plasterboard.

‒ Both these construction elements are typical indeed!

*Lplate*
*F*

*c* *hc*

*f* *c*
π

2 3

=

### Notes: Mass law simplification

### • Simplification:

⎟⎟⎠

⎞

⎜⎜⎝

≈ ⎛

*F*
*F**c*
*f* *fm*

*R* ρ

π 2 log '' 20 )

(

*c* *dB*
*f* *fm*

*R*

*F*
*F*

'' 5 1

log 10 )

(

2

⎟−

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞

⎜⎜⎝

+⎛

= ρ

π

Mass law

Random incidence correction

0 10 20 30 40 50 60 70

Sound reduction(dB)

Frequency (Hz)

concrete leight-weight wall mineral wool

Mass law (6 dB/octave)

Doubled weight leads to 6 dB increased insulation

>>1

### Sound reduction index of a single leaf wall – Summary

nollmods område

fåmods område R [dB]

frekvens [Hz]

+6 dB/oktav
*masslagen R**0*

mindre dämpning mångmods

område

* kritisk frekvens, f**c*

### Outline

### Double-leaf wall

### Exact method

*Analytical models for calculation of R in * single and double-leaf walls

### Single-leaf wall

### Approximate method

### Sound reduction index of single-leaf partitions (I)

• Approximate method – Just a review (not used)

‒ *Region I: Mass-controlled region (f < f** _{1}*)

‒ *Region II: “Plateau” (f*_{1}*< f < f** _{2}*)

‒ *Region III: Stiffness-controlled region (f** _{2}*< f)

Hyphotesis: Infinite panel and diffuse field excitation

NOTE: f_{1}and f_{2}are not the resonance and coincident frequency
explained in the exact method!

### Sound reduction index of single-leaf partitions (II)

*• Region I: Mass-controlled region (f < f** _{1}*)

‒ Transmission independent of panel stiffness

### ( )

^{m}### ( )

^{f}

^{c}

^{dB}*f*

*R*( ) 20log ´´ 20log 20log ^{F}* ^{F}* ⎟−5

⎠

⎜ ⎞

⎝

− ⎛

+

= π

ρ

### Sound reduction index of single-leaf partitions (III)

*• Region II: “Plateau” (f*_{1 }*< f < f** _{2}*)

‒ Governed by internal damping

‒ Height of the plateau depends on material

*‒ f*_{1 }*and f** _{2}* are the lower and upper limits of the plateau

» Calculated with expresions of adjoining regions

### Sound reduction index of single-leaf partitions (IV)

*• Region III: Mass-controlled region (f*_{2 }*< f )*

‒ Governed by stiffness of the panel

### ( )

2log 22 . 33 )

( )

( *f* *R* *f*_{2} _{f}^{f}

*R* = +

NOTE: The slope of the expression (10 dB/octave) should just be used only for the 2
octaves above f_{2}. For the following octaves, one should use a slope equal to 6 dB/octave,

i.e. “20log(f/f_{2oct})” instead of “33.22log(f/f_{2})”, where f_{2oct }is the frequency where the 3^{rd}
octave above f_{2 }starts.

### Example – Approximate method

### 1. A glass window has a thickness of 8.3 mm. Using the “approximate”

### method, determine the sound reduction index as a function of

### frequency in octaves over the range from 63 Hz to 8000 Hz.

### Outline

### Double-leaf wall

### Exact method

*Analytical models for calculation of R in * single and double-leaf walls

### Single-leaf wall

### Approximate method

### Introduction

• How to improve single-wall? Add a second wall!

– Double-leaf wall literature is extensive

– Theoretical analysis, less developed due to complexity

– Analyses often carried out using advanced methods in software:

• FEM (Finite Elements Method), SEA (Statistical Energy Analysis)…

• Several theoretical derivations of sound transmission

– Double-leaf wall without mechanical coupling – Double walls with structural connections

### Sound reduction index of double-leaf walls

• Approximate empirical model for a double leaf wall without structural connections, with cavity filled with porous absorber (Sharp 1978)

⎟⎠

⎜ ⎞

⎝

+ ⎛

+

=

⇒

⎟⎟⇒

⎠

⎞

⎜⎜⎝

+ ⎛

−

=

⇒

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞

⎜⎜⎝

+ ⎛

−

=

⎟⎟⎠

⎞

⎜⎜⎝

+ ⎛

−

=

*S*
*R* *A*

*R*
*R*

*A*
*L* *S*

*L*
*R*

*A*
*L* *S*

*L*
*R*

*A*
*L* *S*

*L*
*R*

*DoubleWall*
*DoubleWall*

2 2 1

3 3

1

3 3

2 2

2 2

1 1

log 10 log 10 log

10 log 10

⎪⎩

⎪⎨

⎧

>

+ +

<

<

−

⋅ +

+

<

=

*d*

*d*
*M*

*f*
*f*
*dB*

*R*
*R*

*f*
*f*
*f*

*dB*
*d*

*f*
*R*

*R*

*f*
*f*
*R*

*R*

; 6

; 29

) log(

20

;

2 1

0 2

1

0 ⎟⎟

⎠

⎞

⎜⎜⎝

⎛

+ ʹ′ʹ′

= ʹ′ʹ′

2 1

0

1 1

2 *d* *m* *m*

*f* *c* ρ* ^{F}*
π

*f** _{d}* 55

*d*

=

*R*_{M}*denotes the mass law with M=m*_{1}*+m*_{2}

*R*_{1 }*and R** _{2 }*denote the individual sound reduction index for each leaf

*d: distance between the two leaves i.e. (cavity thickness)*

NOTE: Diffuse field assumed in both rooms

• Improvement in the sound reduction index of a double-leaf wall respect to a single wall, and also when including insulation in the cavity.

### Examples (i)

### Examples (ii)

0 10 20 30 40 50 60 70 80

50 80 125 200 315 500 800 1250 2000 3150

frekvens [Hz]

Reduktionstal [dB]

R3 R2 R1

135 mm

Gipsplatta, 13 mm Mineralull

Stålreglar c/c 600 R1= tomt hålrum mm

R3 = 140 mm mineralull R2 = 30 mm mineralull Rw [dB]

R3: 55 R2: 49 R1: 43

• Variation in the sound reduction index of a double-leaf wall when varying parameters in the cavity (inclusion of insulation and its thickness).

### Examples (iii)

### Examples (iv)

### Examples (v)

### Examples (vi)

*“Rule of thumb”: decoupled structures perform much better à*

### acoustic bridges eliminated

### Outline

### Introduction

### Examples

Impact:

SS-EN 12354-2

### Flanking transmission

### Summary

Airborne:

SS-EN 12354-1

*Analytical models for calculation of R(f) in *

### single and double-leaf walls

### Review of flanking transmission treatments

### • Sound transmission

– Airborne

– Structure-borne

### • Transmission paths

*– Direct transmission (D)*
*– Flanking paths (F** _{i}*)

– In total: 13 paths (1 direct / 12 flanking)

### • Flanking: cause of problems related with acoustic comfort

– Difference between lab and in-situ measurements ~4 dB

»Estimation methods described in SS-EN 12354:2000

– Acoustic performance as sum of individual contributions

### Remember…

*… Laboratory vs. Field situation (flanking transmission comes into play)*

[REF] Vigran(2008)

ISO 717-1:2013 ISO 10140-2:2010

*R*_{w}

ISO 717-1:2013 ISO 16283-1:2014

*R’*_{w}

SS-EN12354-1:2000

*Prediction of R’**w**from the individual acoustic performances (R**w*) of the
elements involved in the junction, as sum of individual contributions

### Design example – Decoupling structural elements

### Design example: timber volume elements

### Design example: elastic interlayers

### Summary

### • Analytical calculation methods of reduction sound index – Single-leaf wall

» Exact method

» Approximate method (not used anymore)

### – Double-leaf wall

### – There is more complex and effective solutions

### References: SS-EN 12354:2000 series

• SS-EN12354-1:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 1: Airborne sound insulation between rooms (2000).

• SS-EN12354-2:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 2: Impact sound insulation between rooms (2000).

• SS-EN12354-3:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 3: Airborne sound insulation against outdoor sound (2000).

• SS-EN12354-4:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 4: Transmission of indoor sound to the outside (2000).

• SS-EN12354-5:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 5: Sound levels due to service equipment (2000).

• SS-EN12354-3:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 6: Sound absorption in enclosed spaces (2000).