Design of a Fully Anechoic Chamber

Roman Rusz

Master’s Degree Project
TRITA-AVE 2015:36
ISSN 1651-7660

iii

iv

Preface
This Thesis is the final project within the Master of Science program MechanicalEngineering with specialization in Sound and Vibration at the Marcus Wallenberg
Laboratory for Sound and Vibration Research (MWL) at the department of
Aeronautical and Vehicle Engineering at the KTH Royal Institute of Technology,
Stockholm, Sweden.
This Thesis was conducted at Honeywell, in partnership with Technical University ofOstrava, Ostrava, Czech Republic under the supervision of Ing. Václav Prajzner.
Supervisor at KTH Royal Institute of Technology was Professor Leping Feng, Ph.D.

v

Acknowledgment
To begin with, I would like to thank Ing. Michal Weisz, Ph.D., the main acoustician of
the Acoustic department at Technical University of Ostrava. He was an excellent
support during the project. I would like to thank him for sharing his experiences andknowledge in acoustics.
I would also like to thank my supervisor Leping Feng, Ph.D. for his guidance in thisproject and for sharing his deep knowledge for the topic and all the teachers andprofessors at KTH Royal Institute of Technology for their support and assistance.
I would like to give special thanks to:
my beloved family. I thank you for your strong support during my studies at KTH RoyalInstitute of Technology.
my dear friend Jakub Cinkraut for being such an amazing friend. I thank you for thecountless hours we spent together in Sweden, Finland and Czech Republic. I would
not enjoy my studies without you.
my girlfriend Petra Gomolova for being such a wonderful partner. I thank you for all
your support and encouragement during my studies in Stockholm.

viTable of Contents
Preface ..................................................................................................................................... v
Acknowledgment .....................................................................................................................vi
Abstract ....................................................................................................................................ix
List of abbreviations ................................................................................................................. x
List of mathematical notations, indices and symbols ..............................................................xi
1.

Introduction ..................................................................................................................... 1
1.1 Fully Anechoic Chamber................................................................................................. 2
1.2 Objective ........................................................................................................................ 4

2.

Theory .............................................................................................................................. 6
2.1. Free Field (Direct field) .................................................................................................. 7
2.2 Chamber Shape and Dimensions ................................................................................... 9
2.3 Room Modes .................................................................................................................. 9
2.3.1 Calculation of the Room Modes............................................................................ 122.3.2 Preferred Room Dimensions According to Room Modes ..................................... 13
2.4 Wedge design ............................................................................................................... 14
2.4.1 Sound Absorption Material ................................................................................... 14
2.4.1.1 Main Absorber Categories ............................................................................. 14
2.4.1.2 Measurement Methods for Absorption and Impedance ............................... 16
2.4.1.3 The influence of air absorption ...................................................................... 20
2.4.2 Wedge Structure and Design ................................................................................ 21
2.4.3 Finite Element Method (FEM) Analyses of Wedge ............................................... 26
2.4.3.1 Properties of Bulk Reacting Material ............................................................. 28
2.4.3.2 Numerical Scheme ......................................................................................... 29
2.4.3.2 Design Curves for Reacting Wedges .............................................................. 30
2.5 Transmission loss ......................................................................................................... 37
2.5.1 Single wall.............................................................................................................. 38
2.5.1.1 Critical frequency (Coincidence frequency) ................................................... 38
2.5.1.2 Infinite panel .................................................................................................. 39
2.5.1.3 Finite panel..................................................................................................... 40
2.5.2 Double wall............................................................................................................ 41
2.5.3 Transfer Matrix Method (TMM)............................................................................ 41
2.5.3.1 Thin Elastic Panel............................................................................................ 42
2.5.3.2 Fluid layer ....................................................................................................... 43
2.5.3.3 Porous layer ................................................................................................... 44vii

2.5.3.4 Interface to/from porous layer ...................................................................... 44
2.5.3.5 Transmission factor from transfer matrix ...................................................... 45
2.5.4 Sound Propagation through Multiple Partitions ................................................... 46
2.6 Vibration Isolation ........................................................................................................ 46
2.6.1 Vibration Isolators ................................................................................................. 50
2.6.2 Isolator Selection................................................................................................... 53
2.7 Calibration of the Anechoic Chamber .......................................................................... 54
3. Design of the Anechoic Chamber ....................................................................................... 56
3.1 Shape and Dimensions of the Chamber Design ....................................................... 57
3.2 Room modes calculation .......................................................................................... 57
3.3 Sound Absorption Material Design .......................................................................... 59
3.4 Wedge Structure and Design ................................................................................... 60
3.4.1 Wedge structure ............................................................................................... 61
3.4.2 Wedge Dimensions Design................................................................................ 61
3.5 Transmission Loss (TL) .............................................................................................. 64
3.5.1 Wall Transmission Loss ..................................................................................... 64
3.5.2 Doors Transmission Loss ................................................................................... 66
3.5.3 Average Transmission Loss ............................................................................... 68
3.6 Vibration Isolation .................................................................................................... 68
3.7 Conclusions .............................................................................................................. 72
References.............................................................................................................................. 74
Appendix ................................................................................................................................ 76
1.

Vibration isolation ASHARE guide. [1] ................................................................ 76

2.

Villot TMM validation plot [19] .......................................................................... 78

3.

Measured absorption factors – different thickness ........................................... 79viii

Abstract
This thesis deals with fully anechoic chamber design. The main aim of this thesis is todesign fully anechoic chamber according to acoustics laws and customers(Honeywell’s) requirements. The fully anechoic chamber will be used for measuringsound and vibration quantities. This work is divided into two main parts. The first partdeals with the general anechoic chamber theory and all its related design aspects.
The second part, practical part, focus on specific design according to requirements.
The design of the chamber was performed using advanced design methods.

Key words:
Sound, Vibration, Design, Anechoic chamber, Acousticsix

List of abbreviations
1D – one-dimensional
2D – two-dimensional
3D – three-dimensional
ANSI – American National Standard Institute
ASTM – American Society for Testing and Materials
DUT – Device under Test
FEM – Finite Element Method
ISO – International Organization for Standardization
MPA – Micro-perforated Absorber
SAE – Society of Automotive Engineers
TL – Transmission Loss
TMM – Transfer Matrix MethodxList of mathematical notations, indices and symbols
A – Total equivalent absorption area [m2]B – Flexural rigidity [Pa· m3]; Surface boundary condition [-]D – Flexural rigidity [Pa · m3]
E – Young’s modulus [Pa]F – Force [N]; Fourier transform [-]H – Transfer function [-]; depth [m]I – Acoustic intensity [W/m2]
L – Length [m]; sound level [dB]
M – Modal overlap factor [-]N – Number of modes [-]Q – Sound volume flow rate [m3/s]
R – Pressure reflection factor [-], Sound reduction index [dB]
S – Surface area [m2]T – Transfer matrix [-], Reverberation time [s]V – Volume [m3]W – Acoustic power [W]
X – Displacement [m]Z – Acoustic impedance [Pa · s/m3]
a – Plate length [m]b – Plate width [m]c – Sound velocity [m/s]d – Diameter [m], Thickness [m]e – Euler number [-]f – Frequency [Hz]g – Gravitational acceleration [m/s2]
h – Thickness [m]k – Wavenumber [-]l – Length [l]m – Mass [kg], Vibration mode order [-]n – Modal density [-], Vibration mode order [-]p – Pressure [Pa]r – Radius [m]s – Base length [m]t – Time [s]; air gap behind the wedge [m]u – Particle velocity [m/s]w – Wedge width [m]Ω – Airway boundary condition [-]α – Absorption factor [-]γ – Complex wave propagation constant in the wedge material [-]xiδ – Phase angle [rad], Deflection [m]η – Damping [dB]
θ – Wave angle of incidence [rad]λ – Wavelength [m]ρ – Density of mass [kg/m3]
σ – Flow resistivity [N · s/m4]
τ – Transmission factor [-], Transmissibility [-]υ – Poisson ratio [-]ω – Angular frequency [rad/s]
𝜕 – Partial derivative [-]∇ - Laplace operator [-]xii

1. Introduction
In this section, the fully anechoic chamber is discussed and the objective of thismaster thesis is presented.

1

The requirements for measuring sound and vibration are nowadays higher than ever.
Customers demand more silent devices, whether it is a car, computer, vacuum
cleaner, washing machine or refrigerator. The customer’s needs are followed by the
ISO standards, which provides requirements, specifications, guidelines orcharacteristics that can be used consistently to ensure that products are fit for their
purpose.
Many acoustic ISO standards require the special room, fully anechoic or semi
anechoic chamber, where the acoustical measurement has to be performed. To
design and implement such a chamber, lot of funds and time have to be spent. Not
all companies can afford it, even though they need it.
In response to these needs, the request from Honeywell to design and construct afully anechoic chamber come into being. The Honeywell request to design a fully
anechoic chamber, which fulfill strict background sound pressure level requirements.
The background sound pressure level is closely associated with the transmission lossof the chamber’s walls, absorber design inside the chamber and vibration insulationof the whole chamber.

1.1 Fully Anechoic Chamber
Probably the best natural testing environment is outside with no boundaries to cause
reflections. However, temperature, pressure, humidity, wind and external noises may
significantly and unpredictably disturb the uniform radiation of sound waves. To
eliminate or to control aforementioned difficulties the special acoustic room, called
anechoic chamber, has to be designed.

Figure 1: Anechoic chamber illustrative picture (www.acoustics.salford.ac.uk)

2

The anechoic is defined as “free from echoes or reverberation”. An ideal chamber
would contain no reflections from its walls, ceiling, or floor and an acoustical freefield condition would exist. 99% of sound waves are absorbed by the speciallydesigned absorption lining.
In order to increase the measurement precision, the background sound pressure level
inside the anechoic chamber should be very low. To fulfill the background soundpressure limit, the chamber needs to be insulated from external noises which implies
an excellent walls, floor, ceiling and absorption lining design.
A practical anechoic chamber is expected to provide not only an acoustical free field,
but also an environment which meets other requirements, including the control oftemperature, pressure, humidity and ambient sound pressure level. Inside a welldesigned and executed anechoic chamber it is possible to make precise acousticalmeasurements of the sound output and frequency content of a source and its
directivity pattern.
For anechoic chambers to function as required to meet the various standards, anumber of acoustical, mechanical, electrical, and aerodynamic considerations apply.
This master thesis is intended to take in consideration some of the following
requirements:
 Test object size
 Anechoic chamber dimensions
 Anechoic treatment selection
 Absorption material design
 Wedge shape design
 Room modes consideration
 Cut-off frequency
 Transmission loss
 Vibration isolation
 Ventilation system requirements
 Visual requirements
The type of acoustical environment is generally specified in the appropriate SAE, ISO,ANSI or ASTM standards. In order to perform the measurement with greatest
accuracy, the aforementioned requirement limits must be fulfilled. [10]3

1.2 Objective
The objective of this Master Thesis is to design a fully anechoic chamber which will
met the requirements established by Honeywell. The table with all Honeywellrequirements is listed below.
MinimalItem
Requirements
requirements
Relevant standards
EN ISO 3745
IEC 60268-1
IEC 60268-5
EN 60534-8-2, Armature
EN 60534-2-3
ISO 13347
Anechoicity
Fully anechoic chamber
Min frequency [Hz]100 (89.1)
Max frequency [Hz]10000 (11220)
Max background10
10
acoustics pressure level[L Aeq dB]
DUT length [m]0,5DUT height [m]0,75DUT width [m]0,5Internal width [m]52Internal length [m]53Internal height above
3,5
2
sound passing floor [m]Floor height [m]31,5Internal height [m]52Internal net volume [m3] 125
37.5
Wedges length [m]11Wall thickness [m]0,50,5External width [m]85External length [m]86External height [m]852Floor area [m ]6430Internal floor height [m] 2
2Room connection- 4x DN 150 (two from opposite sides)- 10 to 15x ¼” fast connection forpressed air, vacuum, water- 4x refrigerant inlets- mains power supply- multiple signal wires
Room gate [cm]140x200
Room floor
Grill segments
DUT weight [kg]150

4

Chamber air
conditioning – heat gain
[kW]
Chamber airconditioning –[exchanges per hour]
Other equipment insidechamber

51Standard for labs

Pipes for testing according to EN
60534-8-2
Table 1: Honeywell’s requirements* DUT = Device under Test

Some characteristics listed in the table above (width of the wall, length of the wedges,
dimensions of the chamber, etc.) are merely Honeywell’s proposals. The mostimportant requirements (frequency limits, background noise, volume of thechamber, etc.) are highlighted (orange color) in the table. These parameters are
determinative and the chamber was designed in order to fulfill mainly theseparameters. In design of an anechoic chamber, not only Honeywell’s requirements
were followed but mainly standards, acoustical handbooks and scientific papers wereused as the main reference.

5

2. Theory
The following section intends to introduce the reader to the general principle ofanechoic chamber design. Primarily the transmission loss of the walls, wedge designand vibration isolation design is discussed.

62.1. Free Field (Direct field)
In the free field conditions the sound waves propagate directly from the source tothe receiver. Unimpeded sound is not subject to many influences such as reflection,
absorptions, deflections, diffraction, refraction, diffusion and it is not subjected toresonance effect. An approximate free field can exist in anechoic chamber. The freefield is also known as a direct field.

Figure 2: Free Field (Direct Field)

Due to spherical divergence the sound pressure decrease with the distance according
to following formula:
̅ 𝜌𝑐𝑄
𝑊
(1)
𝑝̃2 =
4𝜋𝑟 2
The sound pressure may be expressed in the logarithmic scale as a sound pressurelevel:
𝑄(2)
𝐿𝑝 = 𝐿𝑊 + 10log⁡= 𝐿𝑊 + 10log⁡(𝑄) − 20log⁡(𝑟) − 11
4𝜋𝑟 2
If the point source is considered, the sound waves are propagating in all directions inthe shape of sphere with the center in the point source. In spherical wave
propagation, sound power is divided over an ever-increasing area. The intensity
decreases to one fourth its original value for a doubling of the distance to the source,
and to one ninth when the distance is tripled. [3]Figure 3: Acoustic Power over an area (hyperphysics.phy-astr.gsu.edu)

7

It is complicated to measure the acoustic intensity (the acoustic power per unit
surface) because the special measurement device (intensity probe) is needed.
Therefore, the time average of the x-component of acoustic intensity can be
expressed using sound pressure. The acoustic intensity in the direction of the soundpropagation is proportional to acoustic pressure squared.
𝑝̃2
(3)
𝐼𝑥̅ =
𝜌𝑐
In practice, the sound pressure is often converted into sound pressure level. The
relationship between sound pressure level and the distance from the source isexpressed using following formula.
𝑟2
𝐿2 = 𝐿1 − 20𝑙𝑜𝑔(4)
𝑟1

Figure 4: Sound power level and the distance from the source relationshipThis implies that the sound pressure level and sound intensity level decrease by
6 dB with doubling the distance.

Figure 5: Inverse square law

This phenomenon is known as “inverse square law” in fact that the pressure isinversely proportional to the distance.
1(5)𝑝̃2 ~ 2𝑟
8

If more than one uncorrelated sources are in operation in free field conditions, theresulting sound pressure is a sum of the individual source’s sound pressures.
𝑁
2
𝑝̃𝑡𝑜𝑡
= ∑ 𝑝̃𝑛2(6)

𝑛=1

Then the total sound pressure level is given by following formula.
𝑁

𝐿𝑝𝑡𝑜𝑡 = 10𝑙𝑜𝑔 ∑ 10𝐿𝑝𝑛 /10

(7)

𝑛=1

In reality, it is very difficult to find a natural free field environment. There are lot of
aspects, illustrated in the picture below, that may disrupt the free field conditions.
[11] [3]







External noises
Reflections
Absorption by air
Absorption by material
Modes

Figure 6: Disturbing factors of a free field conditions

2.2 Chamber Shape and Dimensions
The dimension and shape of the chamber does not subject to any special
requirements or limits. Only requirement is determined by ISO 3745 standard whichprescribes the maximum volume of the object that can be measured in the chamber,which is 5% of the inside net volume of the chamber.
𝑉𝑜𝑏𝑗𝑒𝑐𝑡 ≤ 0.05 ∙ 𝑉𝑐ℎ𝑎𝑚𝑏𝑒𝑟(8)
The chamber dimensions are then dependent on the objects which are intended tobe measured in the chamber. [12]In order to avoid possible room modes which may disrupt the free field conditions, it
is better to design irregular shape of the chamber. The effect of the room modes isdiscussed in next section.

2.3 Room Modes
Room modes are the collection of resonances that exist in a room when the room is
excited by an acoustic source such as a loudspeaker. Most rooms have theirfundamental resonances in the 20 Hz to 200 Hz region, each frequency is related to9one or more of the room's dimensions or a divisor thereof. These resonances affectthe low-frequency and low-mid-frequency response of a sound system in the room
and are one of the greatest obstacle to accurate sound reproduction. If the anechoic
chamber is wrongly designed, the room modes affect the free field conditions(reflections occurred) and the chamber could not be validated (calibrated) according
to ISO 3745 standard. [1] [3]Room eigenmodes are divided into three categories:
1. Axial (1D) – generated between two facing surfaces

Acoustic pressure (interpolated)

Sound pressure level

Figure 7: Axial (1D) room mode visualisation

2. Tangential (2D) – generated between four surfaces10

Acoustic pressure (isosurfaces)

Sound pressure level

Figure 8: Tangential (2D) room mode visualisation

3. Oblique (3D) – include six surfaces crosswise

Acoustic pressure (isosurfaces)

Sound pressure level

Figure 9: Oblique (3D) room mode visualisation

The first eigenmode of the room is always the axial (1D) mode. The other modes(tangential, oblique) are not as strong as the axial mode. The energy in axial mode isabout 3 dB more than in the tangential mode and approximately 6 dB more than in
the oblique mode. [1] The visualization of acoustic pressure and sound pressure levelhas been done in COMSOL Metaphysics software.

112.3.1 Calculation of the Room Modes
If the three-dimensional equation in rectangular coordinates is considered as
𝜕𝑝2
𝜕𝑝2 𝜕𝑝2 𝜕𝑝2
2
(9)
=
𝑐
[
+
+
] = 𝑐 2 ∇2 𝑝
𝜕𝑡 2
𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2
the general solution takes the form of
𝒑 = 𝑨𝑒 𝑗(𝜔𝑡−𝑘𝑥 𝑥−𝑘𝑦 𝑦−𝑘𝑧 𝑧)(10)

If this expression is substituted into three-dimensional equation in rectangularcoordinates, the value for the wave numbers kx, ky and kz must satisfy the relationship𝜔(11)
𝑘 = = √𝑘𝑥2 + 𝑘𝑦2 + 𝑘𝑧2
𝑐0
The negative sign in Eq. 10 with one or more positive signs to obtain seven additional
equations, which represent the group of waves moving about the room and reflecting
off the boundaries. The rigid boundary conditions requires that the change inpressure with distance be zero at the boundary. After applying the boundary
conditions, the allowed values of the wave number are [1]
𝑛𝑖 𝜋
𝑘𝑖 =
(12)
𝑙𝑖
where i refers to the x, y, and z directions. The equation for the sound pressure
standing wave in the room is separable into three components
𝑛𝑦 𝜋𝑦
𝑛𝑥 𝜋𝑥
𝑛𝑧 𝜋𝑧 𝑖𝜔𝑡
(13)
𝒑 = 𝑨𝑐𝑜𝑠 () 𝑐𝑜𝑠 () 𝑐𝑜𝑠 ()𝑒𝑙𝑥𝑙𝑦𝑙𝑧
The natural frequencies are2𝑛𝑦
𝑐0 𝑛𝑥 2𝑛𝑧 2√𝑓(𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 ) =( ) +( ) +( )
2
𝑙𝑥
𝑙𝑦
𝑙𝑧

(14)

Thus, the room has an eigenfrequency and a corresponding eigenmode for every
combination of the indices nx, ny and nz. [3]The number of modes in a given frequency range can be determined by following
formula𝑑𝑁𝑓
𝑓2 𝜋 𝑓
𝐿(15)= 4𝜋𝑉 3 + 𝑆 2 +𝑑𝑓
𝑐0 2 𝑐0 8𝑐0
In high frequencies the density of modes is extremely large. The density of modes
may be also determined by modal overlap factor (M) which increase with the modal
12

density increasing. The dimensionless modal overlap factor, M, is the parameter that
describes dissipation of energy. It is also a very important characteristic of a dynamic
system. Modal overlap factor can be calculated using following formula [7](16)𝑀 = 𝜂𝜔𝑛𝑁(𝜔𝑢 ) − 𝑁(𝜔𝑙 ) ∆𝑁
(17)
𝑛=
=
𝜔𝑢 − 𝜔𝑙∆𝜔

2.3.2 Preferred Room Dimensions According to Room Modes
The recommended room dimensions considering the room modes are given in termsof the ratios of the lengths of the sides of a rectangular room. The one published in[8] is shown in Fig. 10.

Figure 10: Preferred room dimensions according to room modes [1]Recommendations such as those shown in Fig. 10 can be useful in designing of an
anechoic chamber, when a rectangular room is desired. Normal-mode calculation for
nonrectangular rooms is more complicated and can be performed using finite
element method. [1]132.4 Wedge design
In order to provide free field conditions inside the anechoic chamber an ideal
absorption lining have to be used. The acoustic energy impinging on the absorption
lining is dissipated and there are no reflected waves, which could affect the free field
conditions. In fact that the most important feature of the anechoic chamber is freefield condition, the absorption lining has to be designed to absorb 99% of the acoustic
energy in required frequency range and thus ensure the free field condition inside
the anechoic chamber.

2.4.1 Sound Absorption Material
When a sound wave (assuming plane wave propagation) hits the surface withdifferent properties (density, porosity, young’s modulus, etc.) a part of the energy isdeflected in different direction from that of the incident wave, a part of the energy istransmitted through the absorption material and the major part of the energy is
expected to be absorbed. Only the controlled part of the sound energy is absorbed,
which means that the sound energy is transformed into heat. The ratio of theabsorbed to the incident energy is than defined by the absorption factor (coefficient).
The absorption coefficient is dimensionless quantity, which can be measured using
different methods. [11] [1]2.4.1.1 Main Absorber Categories
Commonly used acoustic absorbers are divided into two main groups:
1. Porous absorbers - mineral wool, plastic foams, fabric etc.
2. Resonator absorbers - membrane or absorbers based on the Helmholtz
resonator principle
Porous material
Porous materials are often placed directly on to a hard surface. In order to increase
the absorption in low frequencies the porous material may be placed in front of thewall with a cavity behind.

Figure 11: Porous material application [11]14Porous materials are commonly products of plastic foams or mineral fibers. Widelyused are glass or stone wool blankets, which are manufactured in lot of different
thicknesses and sizes. The fibers are anisotropically distributed with a diameter in therange 2–20μ, commonly 4–10μ. The diameter for the plastic (polyester) fibers is
generally larger than for the mineral wool products, being of the order 20–50μ. The
fibers of plastic fibers products are anisotropically distributed as well. Other types ofporous materials, such as products containing glass or metals are also available butthese types of porous materials are less used. [11] [1]Membrane absorbers
The membrane absorber should not have stiffness but thin metal sheets are usuallyused as membrane absorbers. To achieve reasonably high absorption factor the low
surface weight and internal losses in the membrane material is required. If themembrane absorber is mounted at a certain distance from a hard wall or ceiling theabsorption factor 𝛼 in a limited frequency range is usually less than 0.5 – 0.6. Usually
the aluminum or steel plates are used as a membrane absorber. To achieve better
performance the plastic material is used. [11]Perforated plates
Perforated plates or Helmholtz resonator absorbers are based on simple spring-masssystem. The air in the holes of the plate represents a mass and the air volume of thecavity behind the plate represents the spring stiffness in an equivalent oscillator. If
the cavity behind the plate is filled with a porous material (resistive component), theacoustic energy is dissipated and the higher absorption is achieved.

Figure 12: Perforated plates application (no fabric) [11]Gluing a thin fabric to the plate an adjustment of the resistance can be reached.
Aluminum, steel, plasters or wood is commonly used as a fabric. In order to achieve
higher absorption, aforementioned method using porous material to fill the cavity
may be used.

Figure 13: Perforated plates application (fabric) [11]15Micro perforated absorber (MPA) is the special type of resonator absorber. The
perforation in plates or foils has a diameter of less than 1 mm. It is not necessary to
use any additional fabric in MPA since the viscous losses in the holes give thenecessary resistance. As an alternative to perforated plates, the thin slits may be
used. [11]Figure 14: Microperforated plates application [11]2.4.1.2 Measurement Methods for Absorption and Impedance
In order to define the absorption factor, the ratio of the absorbed and the incident
acoustic energy, the type of incident wave field and size of the specimen needs to beconsidered. According to the incident wave field and size of the specimen threedifferent methods are standardized:
Standing wave measurement (ISO 10534-1)
Standing wave measurement is applicable for a small specimens (in comparison withthe wavelength) exhibited to normal incidence acoustic waves (plane wave
propagation assumption). The absorption factor is determined by measuring the
maximum and minimum pressure amplitude in standing wave tube (Kundt’s tube) by
a loudspeaker.
From the ratio of the maximum and the minimum sound pressure amplitude thepressure reflection factor 𝑅𝑝 can be determined. These amplitudes are given by𝑝̂ 𝑖(𝑝̃)𝑚𝑎𝑥 =[1 + |𝑅𝑝 |]
(18a)
√2
𝑝̂𝑖
(𝑝̃)𝑚𝑖𝑛 =[1 − |𝑅𝑝 |](18b)√2
Pressure reflection factor 𝑅𝑝 is the complex quantity in which the phase angle δ isdetermined by the position of the first pressure minimum close to the specimen.
From these data both the input impedance 𝑍𝑔 and the absorption factor α is obtained
from the following equation.
1 + 𝑅𝑝
1 + 𝑅𝑝𝑍𝑔 = 𝜌0 𝑐0
= 𝑍0
(19)
1 − 𝑅𝑝1 − 𝑅𝑝

16

And
𝛼=

Where

𝑍𝑔
4𝑅𝑒 {𝑍 }
0

2

𝑍𝑔
𝑍𝑔
|𝑍 | + 2𝑅𝑒 {𝑍 } + 100𝑍0 = 𝜌0 ∙ 𝑐

(20)

(21)

It should be noted that the equations above does not take into account any possibleenergy losses in the medium in front of the specimen. [11] [3] [5]Figure 15: Standing wave measurement scheme (www.globalspec.com)

Transfer function measurement (ISO 10534-2)
Also the Transfer function method is based on a standing wave principle having justone frequency component. The basic idea is to express the relationship between the
wave components at two (or more) positions along the standing wave tube. The same
results may be obtained if the sound pressure and particle velocity are the simple
functions of time.
𝐹{𝑝𝑟 (𝑥, 𝑡)}(22)
𝑅(𝑥, 𝑓) =𝐹{𝑝𝑖 (𝑥, 𝑡)}
The arbitrary function is represented by pressure reflection factor R at any arbitrary
position x in the tube as being the transfer function. The pressure 𝑝𝑖𝑛 in the incident
plane wave represents the input variable and the pressure 𝑝𝑖𝑛 in the reflected wave
represents the output variable. The Fourier transform (symbol F in Eq. 22) is thanused to revert from the time domain to frequency domain.

17

In order to find unknown variables it is sufficient to measure only one single transfer
function between the total pressures at two positions. Having coordinates 𝑥1 and 𝑥2 ,the pressure reflection factor R is defined by
𝐹{𝑝𝑟 (𝑥1 , 𝑡)}
𝐹{𝑝𝑟 (𝑥2 , 𝑡)}(22a/b)𝑅(𝑥1 , 𝑓) =⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑅(𝑥2 , 𝑓) =𝐹{𝑝𝑖 (𝑥1 , 𝑡)}
𝐹{𝑝𝑖 (𝑥2 , 𝑡)}and a transfer function H12 for the total pressure in these two positions:
𝐹{𝑝(𝑥2 , 𝑡)} 𝐹{𝑝𝑖 (𝑥2 , 𝑡) + 𝑝𝑟 (𝑥2 , 𝑡)}
𝐻12 (𝑓) ==𝐹{𝑝(𝑥1 , 𝑡)} 𝐹{𝑝𝑖 (𝑥1 , 𝑡) + 𝑝𝑟 (𝑥1 , 𝑡)}(23)

Figure 16: Incident and reflected sound pressure in transfer function measurement [11]Transfer function may be defined correspondingly for the pressure in the incident
and reflected wave:
𝐹{𝑝𝑖 (𝑥2 , 𝑡)}
𝐹{𝑝𝑟 (𝑥2 , 𝑡)}[𝐻12 (𝑓)]𝑖 =(24a/b)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[𝐻12 (𝑓)]𝑟 =𝐹{𝑝𝑖 (𝑥1 , 𝑡)}
𝐹{𝑝𝑟 (𝑥1 , 𝑡)}
Eliminating 𝑅(𝑥2 , 𝑓) the 𝑅(𝑥1 , 𝑓) is derived:
𝐻12 − [𝐻12 ]𝑖𝑅(𝑥1 , 𝑓) =[𝐻12 ]𝑟 − 𝐻12(25)

In order to use transfer function method to determine absorption factor two
assumptions have to be defined:
1. Plane wave propagation(26a/b)[𝐻12 ]𝑖 = 𝑒 −𝑗𝑘12 ⁡∙⁡(𝑥2 −𝑥1) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[𝐻12 ]𝑟 = 𝑒 𝑗𝑘21 ⁡∙⁡(𝑥2 −𝑥1 )where 𝑘12 and 𝑘21 are wave numbers for the incident and reflected wave
2. No energy losses between the measurement positions
𝑘12 = 𝑘21 = 𝑘0

18

(27)

The equation 25 become
𝐻12 − 𝑒 −𝑗𝑘0 𝑑
𝑅(𝑥1 , 𝑓) = 𝑗𝑘 𝑑
𝑒 0 − 𝐻12
in which𝑑 = 𝑥2 − 𝑥1

(28)

(29)

When using this method the reflection factor, impedance and intensity may be
determined. Knowing the reflection factor, the Equation 19 from standing wave
method may be used and the impedance is calculated by following formula𝐻12 sin(𝑘0 𝑙) − 𝑠𝑖𝑛[𝑘0 (𝑙 − 𝑑)](30)
𝑍𝑔 (0, 𝑓) =𝑐𝑜𝑠[𝑘0 (𝑙 − 𝑑)] − 𝐻12 cos(𝑘0 𝑙)
where l is the distance from the first microphone to the specimen. [11]Reverberation time measurement (ISO 354)
This is the measurement of the absorption factor of larger specimens, which is
performed in a diffuse field – reverberation room. The reverberation time method isbased on the Sabine formula for the reverberation time in a room:55.3 𝑉 55.3
𝑉
𝑇=
∙ =
∙
(31)
𝑐0 𝐴
𝑐0 𝐴𝑆 + 4𝑚𝑉
where V is the volume of the room and A is the total equivalent absorption area. The
total absorption area has, as it is apparent from the expression, contributions 𝐴𝑠 fromthe surfaces and objects in the room together with the air absorption, the latter
specified by the power attenuation coefficient m. The determination of theabsorption factor is performed by measurements of the reverberation time before
and after the specimen is introduced into the room. Assuming the specimen to be aplane object having a total surface S, the absorption factor is expressed as
55.3𝑉 1 1
𝛼𝑆𝑎 =
( − )
(32)
𝑐0 𝑆 𝑇 𝑇0
where T and 𝑇0 are the reverberation times in the room with and without the
specimen, respectively. It is assumed, that the environment conditions are the same
in both measurements and the walls of the room are hard having negligible total
absorption. The specimen’s surface area required for the measurement is 10-12
square meters. The reason for these requirements is that the absorption factor
determined by this method always includes an additional amount due to the edge
effect, which is a diffraction phenomenon along the edges of the specimen. Thiseffect makes the specimen acoustically larger, which may result in obtaining
absorption factors larger than 1.0. Certainly, this does not imply that the energy
absorbed is larger than the incident energy!
19

It is possible to determine absorption factor using some other methods but none ofthese methods are standardized yet, and usually these methods are justcombinations of the three aforementioned standardized methods. [11] [5] [3]2.4.1.3 The influence of air absorption
In some cases, such as in large rooms or at high frequencies, acoustic energy is notdissipated only at the boundaries of the room but also the air itself may havesignificant contribution to the absorption. The most important effect caused thedissipation of the acoustic energy in the air is relaxation phenomena. Also viscousand thermal phenomena contribute to the total absorption but not that much. The
main principle of relaxation phenomena is the exchange of vibration energy between
the sound wave and the oxygen and nitrogen molecules. The energy of passing soundwave is extracted by molecules and released after some delay, which leads tohysteretic energy losses, an excess attenuation of the wave added to other energy
losses.
The relaxation process is critically dependent on the presence of water molecules,
which implies that the excess attenuation, also strongly dependent on frequency, isa function of relative humidity and temperature. In contrast with outdoors soundattenuation, in room acoustics, the power attenuation coefficient m is used. The
relation between the power attenuation coefficient m and absorption factor α isdescribed by equation below.
𝑑𝐵(33)
𝛼 = 𝐴𝑡𝑡𝑒𝑛𝑢𝑎𝑡𝑖𝑜𝑛⁡ ( ) = 10⁡ ∙ lg(𝑒) ∙ 𝑚 ≈ 4.343⁡ ∙ 𝑚
𝑚
The attenuation coefficient as well as absorption factor is treated more in detail in
ISO 9613 standard. The atmospheric attenuation coefficient for octave bands of noise
is listed below.
Atmospheric attenuation coefficient 𝜶, 𝑑𝐵/𝑘𝑚
Relative
Temperaturehumidity
Nominal midband frequency, Hz°C%63 125 250 500 1000 2000 4000 8000
1070
0.1 0.4
1.0
1.9
3.7
9.7
32.8 177
2070
0.1 0.3
1.1
2.8
5.0
9.0
22.9 76.630
70
0.1 0.3
1.0
3.1
7.4
12.7 23.1 59.3
15
20
0.3 0.6
1.2
2.7
8.2
28.2 88.8 202
1550
0.1 0.5
1.2
2.2
4.2
10.8 36.2 129
1580
0.1 0.3
1.1
2.4
4.1
8.3
23.7 82.8
Table 2: Atmospheric attenuation coefficient (ISO 9613)

20

The total absorption of the room is modified by an atmospheric (air) absorption by
an added term 4mV in Equation 31. The air absorption will be important in large
rooms, however, for fully anechoic chamber, in which the precise measurement isgoing to be performed, the air absorption should be considered. As it can be seenfrom the ISO 9613 table, in higher frequencies the air absorption may become anissue even in smaller rooms such as anechoic chamber. [1]2.4.2 Wedge Structure and Design
In order to reduce the generally reflected sound to a minimum, the surface must belarge and the absorption coefficient of the lining must approach the unity.
If the absorption material is well designed, the middle and high frequency sound isabsorbed. To ensure the absorption of sound energy also in low frequencies, theabsorption material has to be subjected to special geometrical treatment. For many
years lot of different types of acoustic wedges were tested. Beranek and Sleeper
suggested six types of wedge design to be tested. The results of the measurement
can be seen below. In expressing the performance of the acoustical structure under
test, it was decided not to adopt the conventional quantity, percentage sound energy
absorption, A, as an index of the absorbing efficiency. The percentage sound energy
absorption, A, is defined as 100 times the ration of the sound energy absorbed by the
structure to the sound energy incident upon it. Instead, it was decided to plot thepercentage sound pressure reflection, R, which is defined as 100 times the ratio ofreflected sound pressure to the incident sound pressure for sound normally incident
on the structure. This choice of ordinate is desirable because the region between 99
and 100 percent energy absorption corresponds to 10 to 0 percent pressurereflection, respectively. Thus, when using the latter scale, one obtains a moresensitive indication of difference among highly absorbent structures. [6]211. Harvard linear wedge structure

Harvard linear wedge structure

Pressure reflection [%]

30
25
20
15
10
5
0
50

500

Frequency [Hz]Figure 17: Pressure reflection in a Harvard linear wedge structure [6]2. Sheet layer structure

Sheet layer structure
50

Pressure reflection [%]

45
40
35
30
25
20
15
10
5
0
50

500

Frequency [Hz]Figure 18: Pressure reflection in a Sheet layer structure [6]223. Pyramidal structure

Pyramidal structure
40

Pressure reflection [%]

35
30
25
20
15
10
5
0
50

500

Frequency [Hz]Figure 19: Pressure reflection in a Pyramidal structure [6]4. Exponential pyramidal structure

Exponential pyramidal structure
30

Pressure reflection [%]

25
20
15
10
5
0
50

500

Frequency [Hz]Figure 20: Pressure reflection in an Exponential pyramidal structure [6]235. Exponential wedge structure

Exponential wedge structure

Pressure reflection [%]

20

15

10

5

0
50

500

Frequency [Hz]Figure 21: Pressure reflection in an Exponential wedge structure [6]6. Blanket layer structure

Blanket layer structure
50

Pressure reflection [%]

45
40
35
30
25
20
15
10
5
0
50

500

Frequency [Hz]Figure 22: Pressure reflection in a Blanket layer structure [6]24Summary:
60

Pressure reflection [%]

50
40
30
20
10
0
20

200

2000

Frequency [Hz]Sheet_layer_structure

Harvard_linear_wedge_structure

Pyramidal_structure

Exponential_pyramidal_structure

Exponential_wedge_structure

Blanket_layer_structure

Figure 23: Pressure reflection of all type of wedge structure [6]It can be seen from the plot above that the lining with wedge or pyramidal structure
perform well. In high frequencies almost all structures meet the free fieldrequirements. In low frequencies, the Harvard linear wedge structure of the lining
provides the highest absorption of the acoustical energy, therefore, this strucutre is
widely used in anechoic and semianechoic design.
The optimum dimensions for the structures are summarized as a function of desired
cut-off frequency in Figure 24. The cut-off frequency is defined as that frequency atwhich the pressure reflection rises to 10 percent of the pressure in a normally
incident sound wave. This corresponds to the frequency at which the absorption ofsound energy drops to 99 percent or at wchich there is a sound reduction of 20 dB
for a single reflection. [6]25Dimension [mm]

10000

1000

100

10
10

100

1000

Lower cut-off frequency [Hz]Air_space (L3)

Base_length (L2)Taper_length (L1)

Total_depth (D)Base_depth (d)Figure 24: Cut-off frequency – Wedge dimensions dependency [6]2.4.3 Finite Element Method (FEM) Analyses of Wedge
Six different design structures were tested experimentally. The independent
variables were taken as taper length, base length and air space. The dependent
variables are then the total depth and the base depth. The base depth, d, is specified
because experimentally this dimension is fairly critical, whereas a change in the ratioof base length to airspace is not quite as important.
In order to determine the importance of all parameters in wedge design, and to find
how they affect the cut-off frequency the Finite Element Method (FEM) isinvestigated. The finite element model predicts the reflection characteristics of awedge in an impedance tube. This model is based on the bulk reaction concept, thus
accounting for the wave propagation in the wedge material. The theoreticalpredictions are validated with experimental results.
26

A rectangle wedge placed in the duct (impedance tube) is shown in figure below. The
airway domain Ω1, is bounded by the boundary surface B1-B6. The wedge domain Ω2
is bounded by the surfaces B3, B4, B7, B8, and B9. The equation for the wave
propagation in the airway is
1 𝜕𝑝1
(34a)
∇2 𝑝1 = ( 2 ) 2
𝑐0 𝜕𝑡
And that in the wedge is
1 𝜕𝑝2∇2 𝑝2 = ( 2 ) 2
𝑐𝑚 𝜕𝑡

(34b)

p ≡ acoustic pressurec0 ≡ wave speed in the airway
cm ≡ wave speed in the material (complex value)
A time harmonic solution of the form 𝑒 𝑖𝜔𝑡 is then sought for an acoustical wave
propagating in the airway. The relations between the acoustic pressure p, acoustic
particle velocity u, and the particle displacement ξ in the airway and in the wedge are
1 𝜕𝑝1
𝑢1 = 𝑗𝜔𝜉1 = − ()(35a)
𝜌0 𝑗𝜔 𝜕𝑛𝑢2 = 𝑗𝜔𝜉2 = − (1𝜕𝑝2)
𝜌𝑚 𝑗𝜔 𝜕𝑛(35b)

Equation 34a and 34b are coupled by following boundary conditions
For the airway
𝜉1 = 0 normal to the surface B2, B5 and B6,
𝑝1 = 𝑝2 and 𝑛⃑1 ∙ ⁡ 𝜉1 = −𝑛⃑2 ∙ ⁡ 𝜉2 on B3, B4 and B9,Dirichlet boundary conditions (fixed boundary condition) on B1
For the wedge
𝜉1 = 0 normal to B7 and B8,
𝑝1 = 𝑝2 and 𝑛⃑1 ∙ ⁡ 𝜉1 = −𝑛⃑2 ∙ ⁡ 𝜉2 on B3, B4 and B9,Upon substiotution of the time dependence 𝑒 𝑖𝜔𝑡 in equation 34a and 34b following
is obtained2∇2 𝑝1 + 𝑘02 𝑝1 = 0 in Ω1,∇2 𝑝2 + 𝑘𝑚
𝑝2 = 0 in Ω2,(36a/b)27

Where k0 and km are the wavenumbers in the air and in the material, respectively,
given by𝜔

𝑘0 = 𝑐 ,𝑘𝑚 = −𝑗𝛾

0(37)and 𝛾 is the complex wave propagation constant in the wedge material. [9]2.4.3.1 Properties of Bulk Reacting Material
It is assumed that the wedge is made of foam or fibrous material, which ishomogeneous, isotropic and very porous. Delany and Bazley model of porous
material gave an empirical relationship for finding the complex wave propagation
constant as well as the characteristic impedance of fibrous materials in terms of theflow resistivity. The open celled foam has been lately extended by Astley andCummings. The relevant expressions are as follows: [9](38)𝑧𝑎 = 𝑟𝑎 + 𝑗𝑥𝑎
𝑧𝑎 ≡ the non-dimensionalized characteristic impedance of the material (nondimensionalized with respect to the characteristic impedance of air)𝑟𝑎 and 𝑥𝑎 are defined as
𝑟𝑎 = 1 + 𝑐1 𝜂𝑐2 , 𝑥𝑎 = −𝑐3 𝜂𝑐4 ,(39)
In which η is defined as
𝜂=

𝑓𝜌0
𝜎

(40)

f ≡ the frequency of excitationσ ≡ the flow resistivity of the material (defined as the pressure drop/ the flow velocity
per unit thickness of the material)
The wave propagation 𝛾 in the wedge is defined as𝛾= 𝛼 ′ + 𝑗𝛽 ′ ,
𝑘0

(41)

Where
𝛼 ′ = 𝑐5 𝜂𝑐6 ,𝛽 ′ = 1 + 𝑐7 𝜂𝑐8 .

(42)

𝑐1 − 𝑐8 in all these empirical relations are constants
The complex density of the material is expressed as−𝑗𝜌0 𝑐0 𝑧𝑎 𝛾
𝜌𝑚 =
𝜔

28

(43)

2.4.3.2 Numerical Scheme
In order to find the solution the Galerkin method is used. The Galerkin method is the
special case of the method of weighted residuals (MWR) which are obtained byapproximating the solution to minimum. The approximation is done using set of̅𝑙 (𝑥, 𝑦, 𝑧)
shape functions so called trial functions 𝑁𝑙 (𝑥, 𝑦, 𝑧) in the domain and with 𝑁on the boundary. Then the domain Ω1 will be⁡⁡̅𝑙 (𝜕𝑝1 + 𝜌0 𝑗𝜔𝑢) ⁡𝑑𝐵 +∫ 𝑁𝑙 (∇2 𝑝1 + 𝑘02 𝑝1 )⁡𝑑𝛺 + ⁡ ∫ 𝑁
𝛺1
𝐵1
𝜕𝑛
⁡
⁡
𝜕𝑝1
𝜕𝑝1
̅𝑙 ( ) ⁡𝑑𝐵 +⁡ ∫ 𝑁
̅𝑙 ( + 𝜌0 𝑗𝜔𝑛̅ ∙ 𝑢̅) ⁡𝑑𝐵 +⁡ ∫⁡ 𝑁
̅𝑙 (𝜕𝑝1 +
⁡∫𝐵 𝑁
𝐵3
𝐵4
𝜕𝑛
𝜕𝑛
𝜕𝑛
2
⁡
⁡
𝜕𝑝1
𝜕𝑝1
̅𝑙 ( ) ⁡𝑑𝐵 +⁡̅𝑙 ( ) ⁡𝑑𝐵 +𝜌0 𝑗𝜔𝑛̅ ∙ 𝑢̅) ⁡𝑑𝐵 +⁡ ∫𝐵 𝑁
∫𝐵 𝑁
𝜕𝑛
𝜕𝑛
5
6
⁡
𝜕𝑝1
̅𝑙 ( + 𝜌0 𝑗𝜔𝑛̅ ∙ 𝑢̅) ⁡𝑑𝐵⁡ = 0⁡ ∫𝐵 𝑁𝜕𝑛(44)

9

If the first form of Green’s function is used a weak formulation of above equation is
obtained.
⁡⁡𝜕𝑝12− ∫ (∇𝑁𝑙 ∙ ∇𝑝1 − 𝑘0 𝑁𝑙 𝑝1 )⁡𝑑𝛺 + ⁡ ∫ 𝑁𝑙 (
) ⁡𝑑𝐵
𝜕𝑛
(45)
𝛺1
𝐵
+ 𝑡ℎ𝑒⁡𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦⁡𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙⁡𝑡𝑒𝑟𝑚𝑠 = 0
𝐵 = ∑𝑖 𝑏𝑖

Where̅𝑙 = −𝑁𝑙 , which yields
The shape functions on the boundary are chosen 𝑁⁡⁡∫ (∇𝑁𝑙 ∙ ∇𝑝1 − 𝑘02 𝑁𝑙 𝑝1 )⁡𝑑𝛺 + ⁡ ∫ 𝜌0 𝑗𝜔𝑁𝑙 (𝑛̅ ∙ 𝑢̅)⁡𝑑𝐵
𝛺1

𝐵3

⁡

⁡

+ ⁡ ∫ 𝜌0 𝑗𝜔𝑁𝑙 (𝑛̅ ∙ 𝑢̅)⁡𝑑𝐵 + ⁡ ∫ 𝜌0 𝑗𝜔𝑁𝑙 (𝑛̅ ∙ 𝑢̅)⁡𝑑𝐵
𝐵4
⁡

𝐵9

(46)

+ ⁡ ∫ (𝜌0 𝑗𝜔𝑁𝑙 )𝑢⁡𝑑𝐵 = 0
𝐵1

The corresponding equation may be written for the wedge⁡⁡∫ (∇𝑁𝑙 ∙ ∇𝑝2 − 𝑘02 𝑁𝑙 𝑝1 )⁡𝑑𝛺 + ⁡ ∫ 𝜌𝑚 𝑗𝜔𝑁𝑙 (𝑛̅ ∙ 𝑢̅)⁡𝑑𝐵
𝛺1

⁡

𝐵3

⁡

+ ⁡ ∫ 𝜌𝑚 𝑗𝜔𝑁𝑙 (𝑛̅ ∙ 𝑢̅)⁡𝑑𝐵 + ⁡ ∫ 𝜌𝑚 𝑗𝜔𝑁𝑙 (𝑛̅ ∙ 𝑢̅)⁡𝑑𝐵
𝐵4

(47)

𝐵9

=0
Combining the Eq. 46 and Eq. 47 and take into consideration fact that wedge airway
interface(48)𝑛̅1 ∙ 𝑢̅1 = −𝑛̅2 ∙ 𝑢̅2
the following relation is obtained
29

⁡

∫
𝛺1

(∇𝑁𝑙 ∙ ∇𝑝1 − 𝑘02 𝑁𝑙 𝑝1 )⁡𝑑𝛺 +
⁡

𝜌0 ⁡
2
∫ (∇𝑁𝑙 ∙ ∇𝑝2 − 𝑘𝑚
𝑁𝑙 𝑝2 )⁡𝑑𝛺2
𝜌𝑚 𝛺2

(49)

= −𝑗𝜔𝜌0 ∫ 𝑁𝑙 𝑢⁡𝑑𝐵
𝐵1

The pressure is expressed in terms of the nodal values 𝑝̃ as𝑛

𝑝 = ∑ 𝑝̃𝑙 𝑁𝑙

(50)

𝑙=1

where 𝑁𝑙 ⁡𝑎𝑛𝑑⁡𝑝̃ are the global trial function. The pressure continuity 𝑝1 = 𝑝2 on thewedge-airway boundary is ensured by selection of the trial function. Trial functions
N are continuous at all points. Substituting Eq. 50 in Eq. 49 yields [9]𝜌0𝜌0{[𝐾1 ] + ( ) [𝐾2 ] − 𝑘02 [𝑀1 ] − ( ) [𝑀2 ]} {𝑝̃} = [𝐹𝑙 ]
(51)
𝜌𝑚
𝜌𝑚
⁡

⁡

where [𝐾𝑖 ]𝑙𝑚 = ∫𝛺 (∇𝑁𝑙 ∙ ∇𝑁𝑚 )⁡𝑑𝛺 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[𝑀𝑖 ]𝑙𝑚 = ∫𝛺 (𝑁𝑙 𝑁𝑚 )⁡𝑑𝛺11⁡⁡[𝐹𝑖 ]𝑡 = ∫ (−𝜌0 𝑗𝜔𝑢)𝑁𝑙 ⁡𝑑𝐵 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑖 = 1, 2.
𝐵𝑡

2.4.3.2 Design Curves for Reacting Wedges
On the basis of experimental results the design curves for limited number of design
parameters are presented. The geometric parameters chosen for generating thedesign curves are:
1.
2.
3.
4.

The length l
The base length s
The air gap t
The wedge width w

30

Figure 25: FEM model used in the analysis (dimensions and boundary conditions notation) [9]Flow resistivity is also considered as a variable in fact that in practice manufacturers
offer absorption material with different flow resistivity. Besides, production
tolerances introduce some variation in acoustical properties such as the flow
resistivity. It is appropriate to study the effect of the type of material that constitutes
the wedge, because a wide range of materials are now available for selection as
wedge materials (foam, glass wool, mineral wool, etc.). [9]The range of values used for generating the curves is as follows (these are typical of
the wedges used in practice for limiting anechoic chambers):
Parameter
1st case
2nd case
3rd case
4th case
The length (l)0,75 m
1m
1,3 m
1,5 m
The base length (s) 0,1 m
0,2 m
0,3 m
0,4 m
The air gap (t)0m
0,05 m
0,1 m
0,2 m
The wedge width
0,2 m
0,3 m
0,4 m
(w)
Flow resistivity (σ) 4 x 103 Ns/m4 8 x 103 Ns/m4 2,4 x 104 1 x 105
Ns/m4
Ns/m4
Table 3: Values used in FEM analysis [9]* The highlighted quantities (green cells) are default values
311. Effect of the wedge length (l)It can be seen from the figure that as the length of the wedge increases, the cut-off
frequency is reduced. Furthermore, the performance of the wedges improves quite
notably over the entire frequency range as the wedge length increases. When thewedge length increases, there is a shift in the dip towards the origin. [9] [6]Figure 26: Effect of the wedge length (l)l=0,75 m (solid line), l=1 m (dashed line), l=1,3 m (dotted line), l=1,5 m (dot-and-dashcode)Remaining parameters are set to default values.
2. Effect of the wedge base (s)The figure shows that the change in the cut-off frequency with a change in the lengthof the wedge base is only marginal. For a wedge of a given length and air gap, thecut-off frequency of the wedge increases with an increase in the length of the wedge
base. The performance of a wedge with a wider base seems to be better after the
cut-off frequency limit. This implies that a compromise is to be effected in choosing
the dimension of the wedge base. [9] [6]32

Figure 27: Effect of the wedge base (s) [9]s=0,1 m (solid line), s=0,2 m (dashed line), s=0,3 m (dotted line), s=0,4 m (dot-anddash code)
Remaining parameters are set to default values.
3. Effect of the air gap (t)The effect of air gap on the cut-off frequency is marginal, as it can be clearly see fromthe figure below. The air gap slightly dampens the peaks and the troughs. The
reflection coefficients of the wedge with different air gaps are quite similar at higher
frequencies. [9] [6]Figure 28: Effect of the air gap (t) [9]t=0 m (solid line), t=0,05 m (dashed line), t=0,1 m (dotted line), l=0,2 m (dot-and-dashcode)Remaining parameters are set to default values.
33

4. Effect of the wedge width (w)When the wedge width increase the cut-off frequency increase too. There seems to
be only a shift in the curves with an increase in the width of the wedge. It can beobserved from the plot, that to get lower cut-off frequency in the anechoic chamber,
it is better to use wedges with smaller width. However, at higher frequencies thevalues of the reflection coefficients obtained for wedges of different widths are
identical. [9]Figure 29: Effect of the wedge width (w) [9]w=0,2 m (solid line), w=0,3 m (dashed line), w=0,4 m (dotted line)
Remaining parameters are set to default values.
5. Effect of the flow resistivity (σ)
For flow resistivity effect four different materials are studied. It is obvious from thefigure that, as the flow resistivity increase, the values of the corresponding cut-off
frequency increase as well. A factor of 10 can raise the cut-off frequency by as muchas 100%. The same effect can be observed also for fibrous materials. [9]34Figure 30: Effect of the flow resistivity () [9]σ=4x103 Ns/m4 (solid line), σ =8x103 Ns/m4 (dashed line), σ=2,4x104 Ns/m4 (dottedline), σ =1x105 Ns/m4 (dot-and-dash code)
Remaining parameters are set to default values.
6. Effect of the type of the wedge material
The FEM study was performed for four commonly used wedge materials, of which
the wave impedance and propagation constant is known. Namely they are fiberglass,polyurethane foam, pink foam and yellow foam. In the figure below it can be seenthat there is no qualitative variation in the behavior of wedges made out of different
materials. The effect of the type of material on the cut-off frequency is marginal. The
reflection coefficient at low frequencies for fibrous material and polyurethane foam
is almost similar. The performance of fibrous and polyurethane foam in low
frequencies seems to be better than for pink and the yellow foam. At higher
frequencies the type of wedge material does not affect the absorption coefficient somuch. [9]35Figure 31: Effect of the type of the wedge material [9]Fibrous material (solid line), Polyurethane foam (dashed line), Pink foam (dottedline), Yellow foam (dot-and-dash code)
Remaining parameters are set to default values.
In order to lower the cut-off frequency and to improve the performance of wedges
some other special types of wedges such as hybrid wedge or wedge made out of morethan one material may be used. It is very difficult and expensive to manufacture thistype of wedge. The second possibility is to use double wedge.
More than 20 years ago, the professor L. Cremer in Berlin, suggested, instead ofwedges, to use a series of absorbing cubes in different sizes and density. The
advantage of using separate cubes is the lack of large plane surfaces. The reflection
will decrease resulting in an increased absorption by turning the cubes in random
directions. [13]Figure 32: Special shape of the wedge [13]36

2.5 Transmission loss
To describe the acoustical properties of the structures, energy quantities are often
used. Concerning noise insulation, transmission loss quantity is preferred. The ratioof the transmitted power 𝑊𝑡 and the incident power 𝑊𝑖 on the surface is defined bytransmission factor τ.
𝑊𝑡
𝜏=
(52)
𝑊𝑖𝑛
The transmission loss is expressed in decibels by the sound reduction index R:
1
𝑊𝑖
𝑅 = 10𝑙𝑜𝑔 ( ) = 10𝑙𝑜𝑔 ( )
(53)
𝜏
𝑊𝑡
The main advantage of this measure is that it is only the property of the wall itself.
Therefore, transmission loss could be subsequently used to calculate expected soundpressure level in the room by its simple subtraction from the outside sound pressurelevel.
If the diffuse field in the sending and receiving room is assumed, the sound intensity
at the panel (wall) in the sending room is given by
𝑝̃𝑆2
(54)
𝐼𝑖 =
4𝜌0 𝑐0where 𝑝𝑆 denotes the sound pressure in the sending room. The acoustic sound powertransmitted through the structure element may be expressed using the soundintensity and the surface area (S) of the structure element.
𝑝̃𝑅2
(55)
𝑊𝑡 = 𝐼𝑡 ∙ 𝑆 =∙𝐴
4𝜌0 𝑐0 𝑅
where 𝑝𝑅 is the sound pressure in the receiving room. In the equation, the total
absorption area in the receiving room is designated as 𝐴𝑅 . Therefore, thetransmission factor is given by𝐼𝑡 ∙ 𝑆 𝑝̃𝑅2 𝐴𝑅
(56)
𝜏=
=
∙
𝐼𝑖 ∙ 𝑆 𝑝̃𝑆2 𝑆
The sound reduction index then becomes
1
𝑝̃𝑆
𝑆
𝑆
𝑅 = 10𝑙𝑜𝑔 ( ) = 20𝑙𝑜𝑔 ( ) + 10𝑙𝑜𝑔= 𝐿𝑆 − 𝐿𝑅 + 10𝑙𝑜𝑔
𝜏
𝑝̃𝑅
𝐴𝑅
𝐴𝑅

37

(57)

An alternative method for determination of the sound reduction index based on
measuring of the sound intensity may be used if the abovementioned classicalmethod does not work due to substantial flanking transmission effect. [1] [11] [3]2.5.1 Single wall
The general principle of sound transmission loss is described using the simplest
model, which is the infinite and finite panel (wall). Before the infinite and finite panels
is described the coincidence effect needs to explain.
2.5.1.1 Critical frequency (Coincidence frequency)
In simple terms, it is a frequency at which the wavelength of incidence wave in theair (trace velocity) match the wavelength of the wave in the structure (bending wave).
In other words: the phase speed 𝑐𝐵 in the solid medium is equal to the phase speed
𝑐0 in the surrounding medium (air). The coincidence effect affects the soundreduction index, which is at this frequency very poor.

Figure 33: Critical frequency (Coincidence effect) [1]

The velocity of bending waves in a panel is frequency dependent.
4 𝐷𝜔 2𝑐𝐵 = √
𝜌ℎ

Where

(58)

𝐸ℎ3

𝐷 = 12(1−𝜐2 )Since the pressure maxima and minima are spatially matched, energy is easilytransmitted from the air into the panel and vice versa. The frequency at whichcoincidence occurs varies with the angle of incidence.

38𝑓𝑐𝑜 (𝜃) =

𝑐02
𝑚𝑠
√
2𝜋 sin 𝜃 𝐵(59)

Figure 34: Transmission loss and coincidence frequency - angle of incidence dependency [1]For normal incidence, the coincidence frequency is infinite, so there is no coincidence
effect. The minimum value of the matching frequency occurs at grazing incidence andis called the critical frequency. This fact implies that the coincidence effect always
occurs at or above the critical frequency.
𝑓𝑐 =𝑐02 𝑚𝑠𝑐02 12(1 − 𝜎 2 )𝜌𝑚
√
√
=
2𝜋 𝐵
2𝜋ℎ
𝐸

(60)

Where 𝜌𝑚 is a bulk density of the panel material (kg/m3)
The two frequencies are related by means of𝑓𝑐𝑜 (𝜃) =

𝑓𝑐
sin 𝜃

(61)

The coincidence frequency is dependent on angle of incidence. The critical frequency
is the lowest coincidence frequency when the angle of incidence Θ is [1]𝜋𝜃 = = 90°2
2.5.1.2 Infinite panel
For an infinite panel, at low frequencies, the wavelength in the plate is smaller than
the wavelength in air. There is no angle of incidence at which the coincidence effectmay occur. The incidence wave travel through the obstacle in form of mass (mass per

39

unit area m”) which is excited with a forced vibration. This type of transmission iscalled non-resonant transmission, or mass-law, and the transmission factor comesfrom the plate impedance [16]𝑍𝑝 = 𝑗𝜔𝑚"(62)
2.5.1.3 Finite panel
For a finite panel the edges give an increase in radiation efficiency just below thecritical frequency. Generally, three cases may occur:
1. The wavelength in air is larger than the wavelength in the panel in both
directions.
2. The wavelength in air is smaller than the wavelength of the plate just in
one direction.
3. Radiation above the critical frequency.

Figure 35: Plate radiation efficiency [14]Due to a finite size of the panel there are other effects, such as eigenmodes or
stiffness of the plate, which may deteriorate the transmission loss. Neglecting thepossible effects, which may lower the transmission loss, the sound reduction index isapproximated by following formulas2𝜋𝑓𝑓√𝑎𝑏)] + 20 log10 [1 − ( )] + 5⁡𝑑𝐵,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑓 < 𝑓𝑐
𝑅𝑑 − 10 log10 [𝑙𝑛 (
𝑐0
𝑓𝑐
𝑅
(63)
𝑓
𝑅𝑑 + 10 log10 (2𝜂 ) ⁡𝑑𝐵⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑓 > 𝑓𝑐
𝑓𝑐{
where 𝑅𝑑 denotes the diffuse field mass-law
𝑅𝑑 = 20 log10 (𝑚"𝑓) − 47⁡𝑑𝐵
The mass-law gives an increase in reduction index of +6 dB/octave. [14] [16]

40

(64)

2.5.2 Double wall
The transmission loss of a double panel is based on a theory that considers panels of
infinite extent, not structurally connected. It is assumed that the air cavity is filledwith batt insulation, which damps the wave motion parallel to the wall. The airspace(air cavity) separating the panels acts as a spring and at a given frequency, a massair-mass resonance occurs.
1 1
1𝑓𝑟𝑒𝑠 = 59.8⁡ (√ ( +))𝑑 𝑚1 𝑚2(65)

At this frequency the sound reduction index is lowered. If the frequency of the soundincident on a double panel is higher than the resonance frequency, the air chamberabsorbs part of the sound energy, resulting in greater acoustic insulation than isobserved in a single element with the same mass. Thus, it could be relevant to havethe resonance frequency located at relatively low frequencies. [1]For a double wall transmission loss theory the infinite panel is considered, therefore,
the bending stiffness is neglected and the panel does not radiate the energy, which
is not true in reality. In order to calculate sound reduction index more precisely andin order to take into account also bending stiffness of the panel, more advanced
method needs to be introduced.

2.5.3 Transfer Matrix Method (TMM)
A plane wave propagating in air hits the face of the material with thickness d, at anincidence angle Θ.

Figure 36: Plane wave impinging the surface of the material [15]The material is considered to be infinite in 𝑥1 and 𝑥2 direction. The incoming soundwaves give rise to a wave field in the finite medium. The 𝑥1 components of the wave

41

number in finite medium and in air are equal 𝑘𝑎 sin 𝜃 where 𝑘𝑎 is the wave numberin free air.
The sound propagation and the acoustic field is described using following equation(66)
𝑽(𝑀) = [𝑇]𝑽(𝑀′ )where the sound propagation is designated by transfer matrix [𝑇] and the vector
V(M) describe the acoustic field at point M and M’. If the pressure p and particle
velocity v is assumed as state variables of the vector V, the relationship between thestate variables at point M and M’ is written as𝑝(𝑀)𝑇𝑇12 𝑝(𝑀′ )(67)[] = [ 11][]
𝑇21 𝑇22 𝑣3 (𝑀′ )𝑣3 (𝑀)The 𝑥3 component of the particle velocity is considered to be the state variable, whichmeans that the velocity is normal to the surface layer. For isotropic and homogenous
material the following equations hold. [15]𝑇11 = 𝑇22
𝑇11 𝑇22 − 𝑇21 𝑇12 = 1(68)
The transfer matrix is invertible in fact that the latter of two abovementioned
equations may be stated as det(𝑇) = 1 ≠ 0. For a multi-layered structure, thetransfer matrix of each layer is multiplied and the relationship between the statevariables on the input and output side is obtained.
2.5.3.1 Thin Elastic Panel
If the thin elastic panel (wall or plate) with impedance 𝑍𝑝 =𝑝1 −𝑝2𝑣𝑝

with the normal

velocity (𝑣1 = 𝑣2 = 𝑣𝑝 ) on both sides of the panel layer is considered, the transfer
matrix is written as
𝑝1
1 𝑍𝑝 𝑝2
𝑇𝑝𝑎𝑛𝑒𝑙 = [𝑣 ] = [
][ ]
(69)
1
0 1 𝑣2The impedance of the panel is described by following formula
𝐵 4 4
𝑍𝑝 =
(𝑘 𝑠𝑖𝑛 𝜃 − 𝑘𝑝4 )
𝑗𝜔 𝑎

(70)

where B is the bending stiffness of the panel, 𝑘𝑎 the wave number in air and 𝑘𝑝 thewave number of bending waves in the panel. As it is apparent from Equation 70, there
exist an incidence angle 𝜃 for 𝑘𝑎 > 𝑘𝑝 where impedance of the panel is equal to zero
which makes the panel velocity infinitely large. In reality, there are some energy

42losses, which may be introduced in terms of a complex bending stiffness 𝐵(1 + 𝑗𝜂).
Considering the critical frequency 𝑓𝑐 the impedance of the panel is expressed as
𝑓 2
𝑍𝑝 = 𝑗𝜔𝑚" [1 − (1 + 𝑗𝜂)𝑠𝑖𝑛 𝜃 ( ) ]
𝑓𝑐
4

(71)

In the equation above the mass-law behaviour at low frequencies (below the criticalfrequency) can be observed, whereas, above the critical frequency the bending
stiffness of the panel plays a role. According to thin plate theory, the transmission
loss for single panel is calculated by following equation [16]1 + 𝑍𝑝 ∙ cos 𝜃(72)
𝑇𝐿 = 10 log10 || ⁡𝑑𝐵(2𝜌𝑐)2
2.5.3.2 Fluid layer
For a sound wave with incidence angle θ into a fluid layer of thickness d, with wave
number k and characteristic impedance 𝑍𝑐 , the pressure and velocity in x-direction is
written as(73)
𝑝(𝑥) = 𝐴𝑒 −𝑗𝑘∙𝑐𝑜𝑠𝜃𝑥 + 𝐵𝑒 𝑗𝑘∙𝑐𝑜𝑠𝜃𝑥𝑣𝑥 (𝑥) =𝑐𝑜𝑠𝜃
(𝐴𝑒 −𝑗𝑘∙𝑐𝑜𝑠𝜃𝑥 − 𝐵𝑒 𝑗𝑘∙𝑐𝑜𝑠𝜃𝑥 )𝑍𝑐

implementing the boundary conditions x=0 and x=d leads to following
Pressure 𝑝1 = 𝑝(0) = 𝐴 + 𝐵
𝒑𝟏
x=0 Velocity 𝑐𝑜𝑠𝜃
𝑣1 = 𝑣𝑥 (0) =(𝐴 − 𝐵)
𝑍𝑐
𝒗𝟏
Pressure 𝑍𝑐 sin(𝑘𝑑⁡𝑐𝑜𝑠𝜃)𝑝2 = 𝑐𝑜𝑠(𝑘𝑑⁡𝑐𝑜𝑠𝜃)𝑝1 − 𝑗
⁡𝑣1
𝒑𝟐
cos 𝜃
x=d Velocity cos 𝜃 sin(𝑘𝑑⁡𝑐𝑜𝑠𝜃)𝑣2 = 𝑐𝑜𝑠(𝑘𝑑⁡𝑐𝑜𝑠𝜃)𝑣1 − 𝑗
⁡𝑝1
𝑍𝑐
𝒗𝟐

(74)

(75)
(76)
(77)
(78)

Table 4: Fluid layer boundary conditions [16]Putting above equation into matrix form and inverting it gives the transfer matrix [16]𝑍𝑐 sin(𝑘𝑑⁡𝑐𝑜𝑠𝜃)
𝑐𝑜𝑠(𝑘𝑑⁡𝑐𝑜𝑠𝜃)
𝑗
𝑝1
𝑝2
cos 𝜃
(79)
𝑇𝑓𝑙𝑢𝑖𝑑 = [𝑣 ] =[𝑣 ]cos 𝜃 sin(𝑘𝑑⁡𝑐𝑜𝑠𝜃)12𝑗𝑐𝑜𝑠(𝑘𝑑⁡𝑐𝑜𝑠𝜃)
[
𝑍𝑐
]

43

2.5.3.3 Porous layer
Porous material is a frame permeated by a network of pores filled with a fluid. It isdifficult to describe or to modelled porous material, nevertheless, some theories are
known from which two theories are predominantly used: [17]1. Biot theory- The frame is elastic and can take the motion.
- The frame is coupled with the surrounding media.
- Six variables are needed – two velocity components, one velocity
component of the fluid, two components of the stress tensor of the frame,and one in the fluid
2. Equivalent fluid model- More simple model- Frame of the porous layer is considered motionless (heavy, rigid) – nodisplacement, no deformation- May be used when the frame is elastic but the solid-fluid coupling is
negligible- Transfer matrix equivalent of a fluid layer- The losses are taken into account by a flow resistivity r
2.5.3.4 Interface to/from porous layer
At the interface of a porous layer the pressure and volume flow is continuous. The
porous layer is defined with porosity σ, which is typically close to unity.
𝑉𝑎
𝜎=
(80)
𝑉𝑡𝑜𝑡
In formula above the 𝑉𝑎 is the volume of air and 𝑉𝑡𝑜𝑡 is the total volume of porous
material. The volume flow continuity is described with the pressure and the velocity
by following equations
𝑝1 = 𝑝2
𝑣1 𝑆 = 𝑣2 𝜎𝑆(81)
where index 1 denotes the pressure and velocity at the air side 𝑥 = 0− , and the index
2 denotes the porous side 𝑥 = 0+ . The S in equation above is the cross section is of
the interface. Putting above equations into matrix form yields the transfer matrix ofthe interface into a porous layer. Transfer matrix of the interface from a porous layer
is basically the inverse.
1 0
𝑇𝑡𝑜⁡𝑝𝑟𝑜𝑢𝑠 = [
]
(82)
0 𝜎
−1
𝑇𝑓𝑟𝑜𝑚⁡𝑝𝑜𝑟𝑜𝑢𝑠 = 𝑇𝑡𝑜⁡𝑝𝑜𝑟𝑜𝑢𝑠

44

(83)

The total transfer matrix of a porous layer with porosity σ is then𝑇𝑝𝑜𝑟𝑜𝑢𝑠 = 𝑇𝑡𝑜⁡𝑝𝑜𝑟𝑜𝑢𝑠 ∙ 𝑇𝑓𝑙𝑢𝑖𝑑 ∙ 𝑇𝑓𝑟𝑜𝑚⁡𝑝𝑜𝑟𝑜𝑢𝑠
𝑍𝑐 sin(𝑘𝑑 cos 𝜃)𝑐𝑜𝑠(𝑘𝑑⁡𝑐𝑜𝑠𝜃)𝜎 cos 𝜃=sin(𝑘𝑑 cos 𝜃)
𝜎 cos 𝜃𝑐𝑜𝑠(𝑘𝑑⁡𝑐𝑜𝑠𝜃)
[
𝑍𝑐
]

(84)

When the panel consisting of two panels with porous layer in between the totaltransfer matric is obtained by multiplying the transfer matrix of each element. [17]𝑇𝑡𝑜𝑡𝑎𝑙 = 𝑇𝑝𝑎𝑛𝑒𝑙⁡1 ∙ 𝑇𝑝𝑜𝑟𝑜𝑢𝑠 ∙ 𝑇𝑝𝑎𝑛𝑒𝑙⁡2(85)
2.5.3.5 Transmission factor from transfer matrix
The transmission factor of a structure is defined as the ratio of the transmitted power1

and the incident power. Sound power is written as 𝑊 = 2 𝑹𝒆(𝑝∗ 𝑣). With𝑝

𝑍𝑐 = 𝑣

the power is written as
|𝑝|2
1
𝑝∗ 𝑝
1
𝑊 = 𝑹𝒆 {}=𝑹𝒆 { }
2𝑍𝑐
2𝑍𝑐
Insertion the power in the definition of transmission factor yields
1|𝑝𝑡 |2 𝑅𝑒 {𝑍𝑐,2 }𝜏=1|𝑝𝑖 |2𝑅𝑒 {𝑍 }
𝑐,1

(86)

(87)

where Zc,1 is the characteristic impedance on the input side and Zc,2 the characteristic
impedance on the output side. The transfer matrix of the structure withcorresponding characteristic impedance is illustrated in the scheme below.

Figure 37: Transfer matrix method system [18]𝑝1
𝑇[𝑣 ] = [ 11𝑇211

𝑇12 𝑝2][ ]𝑇22 𝑣2(88)

The pressure and velocity on the input and output side is written summarized in thetable below.
45

Pressure (p)Input side

𝑝1 = 𝑝𝑖 + 𝑝𝑟

Output side

𝑝2 = 𝑝𝑡

Velocity (v)(𝑝𝑖 − 𝑝𝑟 )
𝑣1 =
𝑍𝑐,1
𝑝𝑡
𝑣2 =
𝑍𝑐,2

(89)
(90)

Table 5: Pressure and velocity on an input and output side [18]After a few mathematical operations, the expression for transmission factor is [18]
−2
𝑇12
(91)
𝜏 = 4 |𝑇11 ++ 𝑍𝑐,1 𝑇21 + 𝑇22 |𝑍𝑐
The transfer matrix method is a strong tool to predict a multi-layered structuretransmission factor of infinite structures. In order to take into consideration the finite
size of the structure, the spatial windowing technique may be used. Villot haspresented and validated this technique in 2000 [19] (see the plot in Appendix).

2.5.4 Sound Propagation through Multiple Partitions
Often, instead of being completely homogeneous, the area of the partition may infact consist of distinct sub-areas (sub-elements), each of which differs markedly fromadjacent areas in its insulating characteristics; for example, a wall with windows issuch a composite partition. In order to determine the transmission loss of acomposite partition, it is assumed that the incident sound power is uniformly
distributed over the area of all sub-elements of the partition. A composite
transmission loss may be calculated using following formula. [3]𝑆
𝑇𝐿 = 10𝑙𝑜𝑔(92)∑𝑛 𝑆𝑛 10−𝑅𝑛/10
Using this expression, it soon becomes clear that the component having the lowest
transmission loss will control the process. [1] [3 ]2.6 Vibration Isolation
The anechoic chamber is going to be build up in the industrial environment,
therefore, not only air born sound but also structure born sound may affect therequired free field condition inside the chamber. The main cause of possible structureborn noises could be the machines producing vibrations, which are placed in ancillary
workshop. Those unwanted vibrations have to be minimized by specially designed
vibration isolation system. The main aspects in the vibration isolation system design
are the total mass of the chamber, the predicted frequency spectrum of the
disturbing vibrations and the possibilities of the market.

46

When a simple harmonic force is applied to a spring mass system, it induces a
response that reaches a maximum at the resonant frequency of the system. Figure
38 shows a plot of the response of the system. As the driving frequency moves toward
the resonant frequency the output increases—theoretically reaching infinity atresonance for zero damping. The damping not only limits the maximum excursion atresonance but also shifts the resonant peak downward in frequency.

Figure 38: Normalized Excursion vs Frequency for a forced simple harmonic system with
damping (Thomson, 1965) [1]

The forces are transmitted to the support structure through the spring and shock
absorber system. The formulas remain the same whether the mass is resting on
spring or hung from springs. The force on the support can be solved as follow
𝐹𝑡 = √(𝑘⁡𝑋)2 + (𝑐⁡𝜔⁡𝑋)2(93)

Using the expression for relation between the applied force and the displacement
amplitude
𝑋
1
=
(94)
𝑋0 √[1 − (𝜔/𝜔𝑛 )2 ]2 + [2𝜂⁡(𝜔/𝜔𝑛 )]2
where

𝐹

𝑋0 = 𝑘0

the ratio of impressed and transmitted forces can be solved as47

𝐹0 √1 + (

𝐹𝑡 =

𝑐⁡𝜔 2
)
𝑘

2

2
2
√[1 − ( 𝜔 ) ] + (2𝜂⁡ 𝜔 )
𝜔𝑛
𝜔𝑛

(95)

The Equation 95 can be expressed in terms of transmissibility, which is the ratio of
the transmitted to the imposed force.
𝜏=

𝐹𝑡
=
𝐹0

√1 + (2𝜂⁡

𝜔 2
𝜔𝑛 )

2

2
2
√[1 − ( 𝜔 ) ] + (2𝜂⁡ 𝜔 )
𝜔𝑛
𝜔𝑛

(96)

Figure 39: Transmissibility of a viscous damped system [1]The transmissibility decreases above a given frequency √2𝑓𝑛 as the frequency of thedriving force increases, therefore, the transmitted force is decreased. This is thefundamental principle behind vibration isolation.
The isolation is dependent on frequency ration, the lower the resonant frequency,
the greater the isolation for a given excitation frequency. The natural frequency ofthe spring mass system is mass and spring constant dependent. The natural

48

frequency is expressed in terms of the static deflection of the vibration isolator under
the weight of the supported object.
𝑓𝑛 =

𝜔𝑛
1 𝑘
1 𝑔
√ =
√
=
2𝜋 2𝜋 𝑚 2𝜋 𝛿

(97)

A fundamental principle for effective isolation is that the greater the deflection of theisolator, the lower the resonant frequency of the spring mass system, and the greater
the vibration isolation.
In vibration isolation system design the mechanical stability must be considered.
Generally, very soft mounts are less stable than stiff ones. To increase the deflection,
the load on each isolator must be increased; so a few point-mount isolators are
preferable to a continuous mat or sheet. Thick isolators are generally more effective
than thin isolators since thick isolators can deflect more than thin ones. Finally,
trapped air spaces under isolated objects should be avoided and, if unavoidable, thenwide spaces are better than narrow spaces, because the trapped air acts like anotherspring. Note that the greater the damping, the less the vibration isolation, but thelower the vibration amplitude near resonance. This leads to a second important
point, which is that damping is incorporated into vibration.
Assuming that the frequency ratio is greater than √2 and the damping is zero, thetransmissibility is given by𝜔 2𝜏 ≅ [( ) − 1]
𝜔𝑛
where

−1

(98)

𝑔

𝜔𝑛 = 𝛿

then transmissibility becomes(2𝜋𝑓)2 ⁡𝛿
𝜏≅[
− 1]
𝑔

−1

(99)

𝑔 ≡⁡the acceleration due to gravity𝛿⁡ ≡⁡the static deflection of the spring under the load of the supported mass
The formula above (Eq. 99) is sometimes expressed as an isolation efficiency orpercent reduction in vibration as it can be seen in the plot below.

49

Figure 40: Isolation efficiency for a flexible mounts [1]This simplification is occasionally encountered in vibration isolation specifications
that call for a given percentage of isolation at the operating point. It is better tospecify the degree of isolation indirectly by calling out the deflection of the isolator,
which is directly measurable by the installing contractor, rather than an efficiency
that is abstract and difficult to measure in the field. This simple formula is applicable
only for a single degree of freedom system.
When the excitation force is applied directly to the supported object or when it isself-excited through eccentric motion, vibration isolators do not decrease theamplitude of the driven object but only the forces transmitted to the support system.
When the motion of the support base excites the supported object, there is a similar
reduction in the forces transmitted to the object. For a given directly applied
excitation force, an inertial base consisting of a large mass, such as a concrete slab
placed between the vibrating equipment and the support system, can decrease theamplitude of the supported equipment, but not the amplitude of the transmitted
force. [11] [3]2.6.1 Vibration Isolators
Commercially available vibration isolators fall into several general categories:
resilient pads, neoprene mounts, and a combination of a steel spring and neoprene
pad.

50

Figure 41: Types of vibration isolators [11]Isolators are offered by the manufacturer with a range of rated loads and a static
deflection, which is the deflection under the maximum rated load. Most isolators will
tolerate some loading beyond their rated capacity, often as much as 50%; however,
it is good practice to check the published load versus deflection curve to be sure. An
isolator must be sufficiently loaded to achieve its rated deflection, but it must also
remain in the linear range of the load versus deflection curve and not bottom out.
Isolation Pads (Type W, WSW)
Isolation pads of felt, cork, neoprene impregnated fiberglass, or ribbed neoprene
sometimes sandwiched by steel plates usually have about a 1 mm deflection(𝑓𝑛 = 14⁡𝐻𝑧) and are used in noncritical or high frequency applications. Typically,
these products are supplied in small squares, which are placed under vibrating
equipment or piping. Depending on the stiffness of the product, they are designed to
be loaded to a particular weight per unit area of pad. For 40 durometer neoprenepads, for example, the usual load recommendation is about 35 000 kg/m2. Where
higher deflections are desired or where there is a need to spread the load, pads are
sandwiched with thin steel plates. Such pads are designated WSW or WSWSW
depending on the number of pads and plates. [11]Neoprene Mounts (Type N, ND)
Neoprene isolators are available in the form of individual mounts, which have abouta 6 mm rated deflection, or as double deflection mounts having a 10 mm deflection.
These products frequently have integral steel plates, sometimes with tapped holes,that allow them to be bolted to walls or floors. They are available in neoprene ofvarious durometers from 30 to 60, and are color-coded for ease of identification in
the field. The double deflection isolators can be used to support floating floors in
critical applications such as recording studios. [11]51Steel Springs (Type V, O, OR)A steel spring is the most commonly used vibration isolator for large equipment. Steelsprings alone can be effective for low-frequency isolation; however, for broadband
isolation they must be used in combination with neoprene pads to stop highfrequencies. Otherwise these vibrations will be transmitted down the spring. Springs
having up to 13 cm static deflection are available, but it is unusual to see deflections
greater than 8 cm due to their lateral instability. Unhoused open-spring mounts (TypeO) must have a large enough diameter (at least 0.8 times the compressed height) toprovide a lateral stiffness equal to the vertical stiffness. Housed springs have theadvantage of providing a stop for lateral (Type V) or vertical motion and an integral
support (Type OR) for installing the equipment at or near its eventual height, but are
more prone to ground out when improperly positioned. These stops are useful during
the installation process since the load of the equipment or piping may vary;
particularly if it can be filled with water or oil. Built in limit stops are not the same as
earthquake restraints, which must resist motion in any direction. Threaded rods,
allowing the height of the equipment to be adjusted and locked into place withdouble nuts, are also part of the isolator assembly.
Spring isolators must be loaded sufficiently to produce the design deflection, but not
so much that the springs bottom out coil to coil. A properly isolated piece ofequipment will move freely if one stands on the base, and should not be shorted out
by solid electrical or plumbing connections. [11]Air Mounts (AS)Air springs consisting of a neoprene bladder filled with compressed air are alsoavailable. These have the disadvantage of requiring an air source to maintainadequate pressure along with periodic maintenance to assure that there is noleakage. The advantage is that they allow easy level adjustment and can provide
larger static deflections than spring isolators for critical applications. [11]Support Frames (Type IS, CI, R)
Since the lower the natural frequency of vibration the greater the vibration isolation,
it is advantageous to maximize the deflection of the isolation system consistent withconstraints imposed by stability requirements. If the support system is a neoprene
mount—for example, under a vibrating object of a given mass—it is generally best to
use the fewest number of isolators possible consistent with other constraints. It isless effective to use a continuous sheet of neoprene, cork, flexible mesh, or othersimilar material to isolate a piece of equipment or floating floor since the load perunit area and thus the isolator deflection is relatively low. Rather, it is better to space
the mounts under the isolated equipment so that the load on each mount is52

maximized and the lowest possible natural frequency is obtained. A structural frame
may have to be used to support the load of the equipment if its internal frame is not
sufficient to take a point load. Integral steel (IS) or concrete inertial (CI) or rail frame(R) bases are used in these cases. A height-saving bracket that lowers the bottom of
the frame to 25 to 50 mm above the floor is typically part of an IS or CI frame. Brackets
allow the frame to be placed on the floor and the equipment mounted to it before
the springs are slid into place and adjusted.
When equipment is mounted on isolators the load is more concentrated than withequipment set directly on a floor. The structure beneath the isolators must be
capable of supporting the point load and may require a 100 to 150 mm housekeeping
pad to help spread the load. Equipment such as small packaged air handlers mounted
on a lightweight roof can be supported on built up platforms that incorporate a thin(3”) concrete pad. Lighter platforms may be used if they are located directly above
heavy structural elements such as steel beams or columns. In all cases the ratio ofstructural deflection to spring deflection must be less than 1:8 under the equipment
load. [11]Figure 42: Support frames [11]2.6.2 Isolator Selection
A number of manufacturers, as well as ASHRAE, publish recommendations on theselection of vibration isolators. By and large these recommendations assume that thebuilding structure consists of concrete slabs having a given span between columns.
One of the most useful is that published by Vibron Ltd. (Allen, 1989). This particular
guide is reproduced as Tables 11.2 (see Appendix) through 11.4. (see Appendix) To
use it, first determine the sensitivity of the receiving space, the floor thickness, andspan. The longer the span, the more the deflection of the floor, the lower its resonant
frequency, and the harder it is to isolate mechanical equipment that it supports. From
step one we obtain an isolation category, a number from 1 to 6, which is a measureof the difficulty of successfully isolating the equipment. The charts is then enteredinto Tables 11.3 or 11.4 (see Appendix) and pick out the base type and isolator
deflection appropriate to the type of equipment and the isolation category.

53

When a concrete inertial (type CI) base is required, the thickness may be calculatedusing the normograph given by ASHARE. Using such a table is a practical way of
selecting an appropriate isolator for a given situation. Although these tabular design
methods are simple in practice, there is a great deal of calculating and experiencethat goes into their creation.
The ASHRAE isolator selection guide is listed in Appendix. [11]2.7 Calibration of the Anechoic Chamber
In order to calibrate the anechoic chamber and to verify a free field conditions at least
three different methods recommended to be used.
1. The decrease in sound pressure
If there are no reflection inside the anechoic chamber, the intensity of the soundwave ought to decrease according to the inverse square law out to distances as
great as the dimensions of the chamber will allow – the sound pressure decrease
inversely with distance.
2. Using highly directional microphone
The anechoic chamber may be calibrated measuring the ratio of the sound, which
travels directly to the microphone from the source, to the sound which arrives at
the microphone from all other directions. In case of reflections, the sound waves
will arrive from directions other than that of the source and can be measured
accordingly.
3. Impulse method
Measuring the echo of short impulse caused by reflection from the walls
occurring at a later time.
In the plot below the typical measured inverse square law curve may be seen. The
solid straight lines represents theoretical inverse square law model whereas the dots
represents the measured values. The anechoic chamber is validated if the measured
values deviate from theoretical curve in accordance with ISO 3745 ( ±1 dB for
800-5000 Hz or ±1.5 dB for frequencies above or below this range). [12] [20]54Figure 43: Inverse square law measured and theoretical curve for 180 to 260 Hz [20]55

3. Design of the Anechoic Chamber
In this section the fully anechoic chamber is designed on the basis of the theory part.
The design parameters of the chambers were chosen always regarding to Honeywellrequirements, market possibilities and ISO standards. The design of the chamber was
discussed with Honeywell specialists, acousticians from Technical University ofOstrava, supervisor of this thesis Mr. Leping Feng and other experienced acousticiansfrom KTH Royal Institute of Technology.

56

3.1 Shape and Dimensions of the Chamber Design
According to Honeywell’s chamber net volume requirement, 37.5 m3, the objects
with maximum volume 1.88 m3 (5% of the net volume inside the chamber) are
expected to be measured inside the chamber. This limit was calculated according toISO 3745 formula, which set the limit for the maximum volume of the measurable
object by following formula
𝑉𝑜𝑏𝑗𝑒𝑐𝑡 ≤ 0.05 ∙ 𝑉𝑐ℎ𝑎𝑚𝑏𝑒𝑟
𝑉𝑜𝑏𝑗𝑒𝑐𝑡 ≤ 0.05 ∙ 37.5
𝑉𝑜𝑏𝑗𝑒𝑐𝑡 ≤ 1.88⁡𝑚3
Based on the Honeywell data it is possible to build an anechoic chamber withmaximum net volume around 367.5 m3. The maximum chamber net volume estimate
was done due to the available space for the chamber, which is 8x8x8 m. The thicknessof the wall (0.5 m) was roughly estimated at this design stage. If the chamber was
designed with the maximum estimated net volume, then the limit for the maximum
volume of the measurable object is around 18.4 m3. This limit is enough high tomeasure the objects in size of a car in such a chamber.
Regardless the object volume limit, which is set by ISO 3745 standard the anechoic
chamber, was designed as large as possible.

3.2 Room modes calculation
The rectangular shape of the chamber is used so that the recommendation plot from
Section 2.3.2 to design inner dimensions of the chamber with respect to the roommodes may be used. According to recommendations the minimum dimensions ratioof the chamber should be following
𝑋: 𝑌: 𝑍 = 1: 1,2: 1,4

Figure 44: Design of the room dimensions of the chamber according to room modes
57

Despite abovementioned recommended dimensions of the chamber, the chamber isis designed as large as possible. If the wall thickness is estimated to be 0.5 m thickthen the anechoic chamber have following dimensions:
Length: 7 m
Width: 7 m
Height: 7.5 m
Inner net volume of the chamber: 367.5 m3
According to above calculated inner dimensions of the chamber the first ten roommodes were calculated using Eq. 14.
Mode order Natural frequencies (Hz) Mode type
1
22,9
Axial
2
24,5
Axial
3
33,5
Tangential
434,6Tangential
541,5Oblique
6
45,7
Axial
7
49
Axial
8
67
Tangential
9
68,6
Axial
10
73,5
Axial
Table 6: Calculated room modes of the chamber of dimensions 7x7x7.5 m

According to Honeywell requirements, the lowest third octave band for which theanechoic chamber should be validated is the third octave band with central frequency
100 Hz. The lower frequency limit of this band is 89.1 Hz. From the table of calculatedroom modes it can be seen that the strongest, therefore, most important roommodes are below this frequency. For higher order modes the modal density is higher,
therefore, those room modes are not so important. Moreover, the anechoic chamberis going to be lined with acoustic wedges so no room mode is going to disrupt freefield conditions.
Hypothetically, if the dimensions of the anechoic chamber will be slightly smaller, thenatural frequencies will be a little increased. The dimensions of the chamber may bechanged due to design of the wall in following sections.

58

Finally, it can be stated that it is not necessary to follow the recommendation plot
with suggested chamber dimensions. More important are the calculated room modes
which shows the possible threat.

3.3 Sound Absorption Material Design
Before the shape of the acoustic wedge is discussed, it is necessary to design a soundabsorption material from which the acoustic wedges will be made. The market
research was conducted and few materials were chosen for further analysis.
According to the material properties, availability of the material and price the porous
material-mineral wool was identified as the best material for acoustical wedges. The
absorption factors of four different material densities were measured using
reverberation time method. The results may be seen in the plot below.
1,05

Alpha [-]

1
0,95
0,9
0,85
0,8
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Frequency [Hz]Ortech45

Ortech65

Ortech90

Ortech110

Figure 45: Measured absorption factor of four different materials (100 mm thick sample)

Centralfrequency(Hz)α Ortech45 (-)α Ortech65 (-)α Ortech90 (-)α Ortech110 (-)

100
125
160

0.27
0.34
0.74

0.29
0.38
0.78

0.34
0.43
0.81

0.41
0.48
0.75

200
250
315

1
1
1

1
1
1

1
1
1

0.86
0.87
0.88

400
500

1
1

1
1

1
1

0.93
0.96

59

630

1

1

1

0.95

800
1000
1250

1
1
1

1
1
0.98

1
1
1

0.95
0.99
0.98

1600
2000
2500

1
1
1

0.99
0.99
0.98

1
1
0.99

0.97
0.98
0.99

3150
4000
5000

1
1
1

0.99
1
1

1
1
1

0.99
0.98
0.98

Table 7: Table of absorption factors of four measured materials (100 mm thick sample)

The properties of the material samples are listed in the table below.
Material
Density (kg/m3) Thickness (mm)Ortech4545100Ortech65
65
100
Ortech9090100Ortech110110100
Table 8: Table of the measured material properties (density and thickness)

Also the absorption factor of the above mentioned materials with different sample
thickness was measured. The resulted plots are listed in the Appendix as well as thetables with absorption factor values.
The measurement shows, that the best performance from four measured materials
has the Oretech90. For higher density materials the reflections occur and it is notpossible to reach a unity absorption factor. The absorption factor of the Oretech90
reached unity already in the 200 Hz third octave band. The absorption factor values
below this band are also satisfactory. The absorption factors in low frequencies will
be improved by wedge shape which help to fulfill the standard requirement thatprescribes the 99% absorption of the acoustic energy to ensure the free field
conditions inside the anechoic chamber.

3.4 Wedge Structure and Design
In the anechoic chamber design the wedges are the most important design element.
As it was mentioned in the previous section, in order to higher the absorption in low
frequencies the wedges have to be well designed.

60

3.4.1 Wedge structure
According to wedge structure comparison in Section 2.4.2, the pyramidal and Harvard
linear wedge structure is the most suitable structure for wedge design. The Harvard
linear wedge structure was chosen for wedge design in fact that it is the commonlyused wedge structure and the final price for this type of wedges is lower than for
pyramidal wedge structure (lower waste during water jet cutting).

Figure 46: Designed wedge structure

Some special structures were considered, nevertheless, despite the better
performance it is very expensive to produce inhomogeneous wedges or wedges withmore advanced shape.
3.4.2 Wedge Dimensions Design
The most important parameter in the wedge design is its length. The wedge length isdesigned relating the lowest frequency of interest, in our case it is 89.1 Hz. In orderto ensure effective dissipation of acoustic energy, in other words, to reach absorption
coefficient α=1 in all frequency bands of interest, the following condition have to be
fulfilled.
𝐷(100)≥ 0,25𝜆Where λ is the wavelength of the lowest frequency of interest and the D is the lengthof the wedge.
𝜆=𝑐0 343== 3.85⁡𝑚
𝑓 89,1𝐷 ≥ 𝜆 ∙ 0.25
𝐷 ≥ 0,96⁡𝑚

61

Above mentioned calculation is a simplification of deep Beranek research. In order todesign the remaining wedge dimensions the Beranek’s plot was used. The green
dashed line shows the lowest frequency of interest due to which all wedge
parameters were derived.

Figure 47: Design of the wedge dimensions according to Beranek’s plot – green dashedvertical line

From the Beranek’s plot it can be seen (green dashed vertical line) that the wedge
length (total depth) should be 1 m. According to Beranek’s plot, the total depth
includes also air space between the wedge and the wall. After the discussion with
Honeywell acousticians, the wedge was designed with no air space between thewedge and the wall. The base length (L2) was designed with length of base depth (D).
Moreover, the FEM analysis shows only marginal effects when air space changing.
Filling the air space with the porous material leads to achieving the higher absorption.
The base dimensions were designed after thorough study of FEM analysis. Thechange in base length does not affect the absorption so much, therefore, the baselength was designed due to experiences of the company which is going to cut thewedges. In the scheme drawing below all parameters are dimensioned.

62

Figure 48: Designed wedge dimensions

The total number of acoustic wedges was approximated to 5723 pcs (with 5% margin
in case of wedge damage). The wedges are going to be installed in following patternwhich has no acoustical substantiation.

Figure 49: Installation wedge pattern (anechoicchamberdesign.blogspot.com)

As it was discussed in previous section, the mineral wool Ortech90 will be used foracoustic wedge manufacturing. It is not possible to use higher density material
because of the fact that the higher density material reflect some acoustic energy andthe standardized 99% absorption condition would not be fulfilled. Moreover, the63deflection of the wedge’s free end is another aspect which should be considered. The
deflection measurement results of 1 m wedge long may be seen in the plot below.

Deflection [mm]

300
250
200
150
100
50
0
80

90

100

110

120

130

Density [kg/m3)]

Figure 50: Deflection of the free end of the wedge

The plot above shows, that the deflection is higher for lower density materials,
therefore, it is not possible to use very low density material for wedge design eventhough it has a good absorption properties.

3.5 Transmission Loss (TL)
The anechoic chamber is going to be placed in an industrial environment so that someexternal noises may influenced and disrupt the free field condition inside thechamber. In order to design the parameters of the wall and doors the software
Maine3A based on transfer matrix method has been used.

Light concrete

Acoustic foam

Hard concreteLight concrete

Air

Light concreteLight concrete

3.5.1 Wall Transmission Loss
The transmission loss of three different walls was calculated. The cross sections of
the walls may be seen in the schemes below.
Double wall with
Single wall
Light concrete double wallacoustical treatment

Figure 51: Three different walls cross section used in the transmission loss calculation

64

The dimension 250 mm for the concrete panels was chosen according to building
material catalog. The change in wall thickness changed the inner dimensions of thechamber so that the inner dimensions of the chamber are following
Length: 6.8 m
Width: 6.8 m
Height: 7.4 m
Inner net volume of the chamber: 342.2 m3Material parameter used in transmission loss calculation:
Young’s modulus
Material
Density [kg/m3]
Poisson ration [-][MPa]
Hard Concrete
2300
2500
0.2
Light concrete
850
2000
0.2
Material
Air flow resistivity [N.m.-4.s]
Porosity
Acoustic Foam
40000
0.97
Table 9: Material parameters of the wall used in transmission loss calculation (Maine3A)

With the change of wall thickness the natural frequencies of the room modes will beslightly lowered.
180
160
140

TL [dB]

120
100
80
60
40
20
0
50

80

125

200

315

500

800

1250

2000

3150

Frequency [Hz]Single_light_concrete_wall
Double_light_concrete_wall
Double_light_and_hard_concrete_wall_porous

Figure 52: Calculated transmission loss of three different walls types (see Figure 51)

The comparison analysis of transmission loss of three different wall types shows thatthe double wall with one hard concrete wall, one light concrete wall and acoustic
foam in between those panels has the best performance. Due to this fact, this wall65

type was chosen for anechoic chamber design. It can be seen from the plot abovethat the transmission loss has the highest values over all third octave bands incomparison with other wall types.
It is expected that except the imperfect sealing no factor will deteriorate thetransmission loss in fact, that the room modes should not occur inside the chamber(absorption assumption).

Steel

Acoustic foam

Steel

3.5.2 Doors Transmission Loss
The transmission loss is always driven with the weakest element such as window,
ventilation or doors. In our case the weakest elements will be doors, ventilation andtechnological intakes, which have to be designed so well so it will not affect the totaltransmission loss. In order to improve the transmission loss and to reach better
acoustical conditions inside the chamber, a special doors system will be used. The
doors of dimension 2115 mm x 1000 mm are going to be installed in the outer coat
(hard concrete wall). For this special purpose, the doors with following cross section
was found on the market.

Figure 53: Cross section of the doors used in transmission loss calculation

Material parameters used in transmission loss calculation:
Young’s
Poisson ration [Material
Density [kg/m3]
modulus [MPa] ]
Steel
7800
200 000
0.3
Material
Air flow resistivity [N.m.- Porosity
4.s]
Acoustic Foam
40000
0.97
Table 10: Material parameters of the doors used in transmission loss calculation (Maine3A)

66

120
100

TL [dB]

80
60
40
20
0
50

80

125

200

315

500
800
Frequency [Hz]

1250

2000

3150

Single_doors

Figure 54: Calculated transmission loss of the doors (see Figure 53)

It can be seen from TMM model, the transmission loss in high frequencies is quite
high. The average transmission loss will be slightly deteriorate by doors just in low
frequencies. The TL dip in 2500 Hz third octave band is caused by coincidence
frequency of the steel panels which was calculated according to Eq.61 to be
2563.2 Hz. The air cavity (in between the doors) resonance could not be clearly seenin the plot in fact that its value is around 20 Hz. The air cavity in between the steel
plates could not be seen as well in fact, that it is filled with acoustic foam, whichdamped it.
A few factors (imperfect sealing, solid connectors, frame, dynamic behavior, etc.)
may deteriorate the transmission loss of the doors from which one can be thedynamic behavior of the steel plate. It is expected that it would not be a problem infact, that the first few natural (modal) frequencies of the steel plate are under thefrequency for which the chamber should be valid. The modal frequencies of the steel
plate may be calculated according to following formula.
𝑓𝑛𝑚 (𝐻𝑧) =𝜋 𝑛3 𝑚 2𝐷( 2 + 2)√
2 𝑎
𝑏
𝜌ℎ

(101)

The resulting frequencies will be lower in reality in fact, that the equation is valid forsimply supported rectangular plate. The boundary conditions of steel plates
embedded in the doors are expected to be between simply supported and freely
supported boundary conditions.

67

3.5.3 Average Transmission Loss
Considering the single doors system and double wall with one wall made from hardconcrete, one wall from light concrete and the acoustic foam in between, theresulting transmission loss is plotted below.
180
160
140

TL [dB]

120
100
80
60
40
20
0
50

80

125

200

315

500
800
Frequency [Hz]Average_transmission_loss

1250

Doors_TL

2000

3150

Wall_TL

Figure 55: Total transmission loss (wall +doors)

The acoustic lining (acoustic wedges) was not considered in transmission loss
calculation since it is recommended to measure the transmission loss for anechoic
chambers without the acoustic lining. The acoustic lining will improve the final
transmission loss approximately of about 20 Hz in low frequencies and much more inhigher frequencies.
One should be aware that above calculated transmission losses for wall and doors
does not take into account the solid connections between the panels that transmit
the vibrations. The imperfect sealing is another aspect, which above calculations does
not take into consideration and which deteriorate the transmission loss a lot. Abovecalculations assume ideal conditions which are very difficult to reach in real situation.

3.6 Vibration Isolation
Acoustic field may be disrupt not only by sound but also by vibrations, especially low
frequency vibrations. The higher frequency vibrations will be damped by concretebase plate, which does not transmit it. In vibration isolation design the total weightof the chamber and frequency spectrum of possible disturbing vibrations.
The total weight of the chamber was determined using 3D CAD software where allpreviously defined material parameters was used. The simple 3D visualization of thechamber may be seen in the picture below.
68Figure 56: 3D visualisation of the anechoic chamber

Item
Support plate
Coat (walls, ceiling)
Acoustic lining (wedges) with related structures
Floor grid
Equipment

Wight
47 000 kg
321 000 kg
18 000 kg
500 kg
500 kg

TOTAL WEIGHT

387 000 kg (387 t)
Table 11: Anechoic chamber weight

In order to isolate unwanted vibrations, 25 spring isolators will be used from which
4 spring isolators will be corneous. In the middle of this corneous isolators there is an
integrated viscous isolator. The main function of the viscous isolator is to reducetransmitted force in resonant frequencies. This isolators will be placed in the corners
under concrete base plate.

Figure 57: Corner spring isolator (Farrat catalog)
69

Parameters:
Load perNaturalIsolator height
unit
freq.
SpringLength Width
Dampingcharacteristics Min. Max. Min. Max.
Min. Max. Freeload load load load
mm
mm
kN/mm
kN
kN
Hz
Hz
mm mm mm
C/Cc
850
305
6,8
56
170
5,5
3,2
157 140 165
0,15
Table 12: Corner spring isolator parameters (Farrat catalog)

The rest 21 spring isolators will be equally distributed under the concrete base plate.
This isolators are made from welded plates with the coil spring system in between.

Figure 58: Spring isolator (Farrat catalog)

Parameters:
Load perNaturalIsolator height
unit
freq.
SpringLength Width
Dampingcharacteristics Min. Max. Min. Max.
Min. Max. Freeload load load load
mm
mm
kN/mm
kN
kN
Hz
Hz
mm mm mm
C/Cc
725
305
6,8
56
170
5,5
3,2
157 140 165
0,15
Table 13: Spring isolator parameters (Farrat catalog)

Generally, it is better to use spring isolators instead of some rubber mat isolators,
therefore, the above described spring isolators were chosen from not so wide market
portfolio. The design of the vibration isolation was made also on the basis of Vibron
recommendation (see Appendix).
Using the isolator’s parameters the natural frequency of the system was calculated
according the Eq. 97.
𝑓𝑛 =
𝑓𝑛 =

𝜔𝑛
1 𝑘
1 𝑔
√ =
√
=
2𝜋 2𝜋 𝑚 2𝜋 𝛿

1 𝑘
1 25 ∙ 6.8 ∙ 106√√ == 3.3𝐻𝑧2𝜋 𝑚 2𝜋38700070

The natural frequency of the system was calculated to be 3.3 Hz. One should be awarethat the weight of the anechoic chamber was approximated, therefore, the natural
frequency may be slightly changed in reality. In order to calculate transmissibility thestatic deflection may be derived from the equation above.
1 𝑔
√
𝑓𝑛 =
2𝜋 𝛿
𝑔
9.81
𝛿= 2
=
= 0.02⁡𝑚
𝑓𝑛 ∙ 4𝜋 2 3.32 ∙ 4 ∙ 𝜋 2
𝑓Then the transmissibility is defined with respect to driving frequency ratio 𝑓 by𝑛

Eq. 98.
130Transmissibility [-]

110
90
70
50
30
10
-10 0
-30

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

Driving Frequency Ratio [-]Figure 59: Transmissibility – driving frequency ratio dependency

To see how effective the isolator is, the effectiveness in percentage is plotted.
120

Isolator effectivness [%]

100
80
60
40
20
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Disturbing Frequency [Hz]Figure 60: Isolator effectiveness71

From the plots above it can be seen, that the disturbing frequencies above the natural
frequency of the system (3.3 Hz) will be very well isolated. After the discussion with
Honeywell’s specialists, it is expected that if some unwonted vibrations will occurred
the frequency spectrum will be above the 20 Hz. Moreover, the chamber should bevalid from 89.1 Hz, therefore, it is expected that all unwanted vibrations will be
isolated and the acoustic free field inside the chamber will be retained.

3.7 Conclusions
The measurement of sound and vibration is not an easy task and the special acoustic
conditions have to be fulfilled to perform the correct measurement. In order to met
the special requirements for sound and vibration correct measurement, the anechoic
chamber has to be designed.
The most important assumption for precise sound and vibration measurement,
consequently, the most important design parameter in anechoic chamber design isthe acoustic free field condition. The acoustic free field condition depends on many
aspects, which should be take into consideration in anechoic chamber’s design. In
this thesis not only acoustic laws had to be followed but also customers (Honeywell’s)
requirements had to be satisfied.
Acoustic wedge design is one of many design parameters to which the special
attention should be paid. For the wedge’s design the Finite Element Method wasused. The acoustic wedges with Harvard linear wedge structure of length 1 m were
designed. Finally, the mineral wool was chosen for wedges manufacturing.
The second design element of the chamber, which should not be neglected istransmission loss of the chamber’s shell. The double wall shell with air cavity in
between was designed using Transfer Matrix Method. The thickness of the walls was
designed to be 0.25 m. To improve the attenuation of air born sound, the air cavityof thickness 0.1 m will be filled with acoustic porous material.
Last but not least design parameter that should be well designed is vibration
insulation of the chamber. Based on the calculated weight of the chamber, 25 springs
isolator were designed to attenuate the unwanted structure born sound. These
springs are going to be equally distributed below the concrete base plate.
After the design of the other elements of the chamber, such as inlets, electricalwiring, lights, etc. the chamber is going to be built and validated.

72

Despite the fact, that the wedges has been designed using FEM analysis, it isrecommended to do additional deep research in FEM designed of porous material infact that it is very complicated to build an accurate FEM model of porous material.

73References[1] LONG, Marshall. DRIVE. Architectural Acoustic. UK: Elsevier Academic press,
2006. 1. ISBN 13:978-0-12-455551-8.
[2] TADEU, António J.B. a Diogo M.R. MATEUS. Sound transmission through single,
double and triple glazing. Experimental evaluation. Applied Acoustics [online]. 2001,
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Available on: http://linkinghub.elsevier.com/retrieve/pii/S0003682X00000323[3] BODÉN H., Carlsson U., Glav R., Wallin H., Abom M. , Sound and Vibration course
book, KTH, Stockholm , 2003
[4] ORRENIUS, Ulf, Hao LIU, Andrew WAREING, Svante FINNVEDEN a Vincent
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5th print (2011)[6] BERANEK, Leo L. The Design and Construction of Anechoic Sound Chambers. The
Journal of the Acoustical Society of America. 1946, vol. 18, issue 1, s. 140-. DOI:
10.1121/1.1916351. Available on:
http://scitation.aip.org/content/asa/journal/jasa/18/1/10.1121/1.1916351[7] FINNVEDEN Svante, A brief introduction to statistical Energy Analysis. Lecturenotes (2014), Sweden.
[8] R. H. BOLT, “Note on The Normal Frequency Statistics in Rectangular Rooms,” J.
Acoust. Soc. Am., vol. 18, pp. 130–133 (1946).
[9] EASWARAN, V. and M.L. MUNJAL. Finite Element Analysis of Wedges Used In
Anechoic Chambers. Journal of Sound and Vibration. 1993, vol. 160, issue 2, s. 333350. DOI: 10.1006/jsvi.1993.1027. Available on:
http://linkinghub.elsevier.com/retrieve/pii/S0022460X83710278[10] BUELOW, R., "The Design Considerations of an Anechoic Chamber," SAE
Technical Paper 1999-01-1832, 1999, doi:10.4271/1999-01-1832.

74

[11] VIGRAN, Tor Erik. Building acoustics. New York: Taylor & Francis, 2008, xix, 362
p. ISBN 0203931319.
[12] ISO 3745-2012, Acoustics -- Determination of sound power levels and soundenergy levels of noise sources using sound pressure -- Precision methods foranechoic rooms and hemi-anechoic rooms.
[13] SCHØYEN NIELSEN, M. B. Anechoic vs. Semianechoic chamber. Hørsholm
Denmark. Scientific paper. SESSIONS.
[14] M. J. Crocker and A. J. Price. Sound transmission using statistical energy
analyis. Journal of Sound and Vibration, 9(3):469{486, 1969.
[15] B. H. Song and J. S. Bolton. A transfer-matrix approach for estimating thecharacteristic impedance and wave numbers of limp and rigid porous materials.
J. Acoust. Soc. Am., 107(3):1131{1152, 2000.
[16] T. E. Vigran. Building Acoustics. Taylor & Francis, 2008.
[17] J.-F. Allard and N. Atalla. Propagation of sound in porous media: modellingsound absorbing materials. John Wiley & Sons, second edition, 2009.
[18] B. Brouard, D. Lafarge, and J.-F. Allard. A general method of modellingsound propagation in layered media. Journal of Sound and Vibration,
183(1):129{142, 1995.
[19] M. Villot, C. Guigou, and L. Gagliardini. Predicting the acoustical radiationof finite size multi-layered structures by applying spatial windowing on infinite
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[20] OLSON, N. Acoustic Properties of Anechoic Chamber: The Journal of Acoustic
Society of America. 1961, 6(33).

75Appendix
1. Vibration isolation ASHARE guide. [1]76Figure 61: ASHARE vibration isolation guide772. Villot TMM validation plot [19]Figure 62: Sound transmission index for a multi-layer system; measured values (blankcircles-full line), TMM infinite system (full line), TMM spatially windowed (dashed line)783. Measured absorption factors – different thickness

80 mm
1,2
1

Alpha [-]

0,8
0,6
0,4
0,2
0
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Frequency [Hz]Ortech45

Ortech65

Ortech90

Ortech110

Figure 63: Measured absorption factor of four different materials (80 mm thick sample)
Freq

Or45
Or65
Or90
Or110
100
0.16
0.17
0.19
0.25
125
0.25
0.28
0.29
0.27
160
0.56
0.53
0.62
0.62
200
0.95
0.96
0.96
0.89
250
1
1
1
0.92
315
1
1
1
0.93
400
1
1
1
0.97
500
1
1
1
0.98
630
1
1
1
0.98
800
1
1
1
0.97
1000
1
1
1
0.96
1250
1
1
1
0.96
1600
1
0.98
0.99
0.97
2000
1
0.98
0.99
0.96
2500
1
0.98
0.99
0.96
3150
1
1
1
0.97
4000
1
1
1
0.98
5000
0.98
0.98
0.99
0.97
Table 14: Table of absorption factors of four measured materials (80 mm thick sample)

79

60 mm1,2
1Alpha [-]

0,8
0,6
0,4
0,2
0
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Frequency [Hz]Ortech45

Ortech65

Ortech90

Ortech110

Figure 64: Measured absorption factor of four different materials (60 mm thick sample)

Freq

Or45
Or65
Or90
Or110
100
0.11
0.13
0.13
0.14
125
0.17
0.2
0.2
0.17
160
0.35
0.38
0.38
0.41
200
0.57
0.65
0.75
0.75
250
0.75
0.84
0.9
0.85
315
0.89
0.97
1
0.89
400
1
1
1
0.94
500
1
1
1
0.95
630
1
1
1
0.96
800
0.99
1
1
0.94
1000
1
1
0.99
0.96
1250
0.98
0.99
0.99
0.97
1600
0.99
1
0.98
0.97
2000
0.97
0.97
0.98
0.98
2500
0.98
0.98
0.99
0.97
3150
0.99
1
0.98
0.97
4000
1
0.99
0.99
0.97
5000
0.97
0.99
0.98
0.97
Table 15: Table of absorption factors of four measured materials (60 mm thick sample)

80

40 mm1,2
1Alpha [-]

0,8
0,6
0,4
0,2
0
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Frequency [Hz]Ortech45

Ortech65

Ortech90

Ortech110

Figure 65: Measured absorption factor of four different materials (40 mm thick sample)

Freq

Or45
Or65
Or90
Or110
100
0.08
0.07
0.07
0.07
125
0.11
0.11
0.11
0.1
160
0.19
0.18
0.17
0.22
200
0.29
0.31
0.35
0.4
250
0.4
0.44
0.51
0.57
315
0.58
0.66
0.73
0.74
400
0.74
0.84
0.92
0.85
500
0.82
0.95
0.97
0.87
630
0.92
0.96
0.99
0.91
800
0.94
0.98
1
0.92
1000
0.96
1
1
0.95
1250
0.94
0.99
0.99
0.94
1600
0.95
0.99
0.98
0.94
2000
0.95
0.99
0.97
0.95
2500
0.95
0.97
0.97
0.94
3150
0.94
0.97
0.97
0.95
4000
0.94
0.98
0.96
0.94
5000
0.96
0.97
0.96
0.94
Table 16: Table of absorption factors of four measured materials (40 mm thick sample)

81