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Sound Radiation from a Loudspeaker Cabinet using the Boundary Element Method

Fernandez Grande, Efren

Publication date:
2008

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Citation (APA):
Fernandez Grande, E. (2008). Sound Radiation from a Loudspeaker Cabinet using the Boundary Element
Method. Department of Acoustic Technology, Technical University of Denmark.

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 Technical University of Denmark (DTU)
Acoustic Technology

Sound Radiation from a Loudspeaker Cabinet

using the Boundary Element Method

Master Thesis by
Efren Fernandez Grande

Supervisor: Finn Jacobsen

September 30, 2008

2
 Abstract

Ideally, the walls of a loudspeaker cabinet are rigid. However, in reality, the cabinet
is excited by the vibration of the loudspeaker units and by the acoustic pressure inside
the cabinet. The radiation of sound caused by such vibration can influence the overall
performance of the loudspeaker, in some cases becoming clearly audible. The aim of
this study is to provide a tool that can evaluate the contribution from the cabinet to
the overall sound radiated by a loudspeaker. The specific case of a B&O Beolab 9
early prototype has been investigated. An influence by the cabinet of this prototype
had been reported, based on subjective testing. This study aims to detect the reported
problem. The radiation from the cabinet is calculated using the Boundary Element
Method. The analysis examines both the frequency domain and the time domain
characteristics (in other words, the steady state response and the impulse response)
of the loudspeaker and the cabinet. A significant influence of the cabinet has been
detected, which becomes especially apparent in the time domain, during the sound
decay process.
2
Preface

This report presents the work carried out between February and September of 2008 as
a Master Thesis project at the Technical University of Denmark (DTU).

This Master Thesis was carried out at the Acoustic Technology department, under the
supervision of Finn Jacobsen. The project was done in collaboration with Bang &
Olufsen, with Sylvain Choisel as the contact person from the company.
4
Acknowledgments

Throughout this project, there have been several people that helped and kindly con-
tributed in different ways. Initially, I would like to thank Finn Jacobsen for his valuable
guidance and help during the whole project.
I would also like to thank Peter Juhl for his support and advice concerning the
Boudary Element Method, and along with Vicente Cutanda, for the development and
sharing of the OpenBEM software, that has been used in this project. Salvador Barrera
for his help with the Boundary Element Method calculations and explanations.
Thank the cooperation with Bang & Olufsen, especially with Sylvain Choisel, for
his interest and collaboration. Yu Luan (Spark) for sharing his work and helping
with the ANSYS modeling and Mogens Ohlrich for his advice and collaboration on
the vibration measurements. Finally, I would like to thank Ioanna Karagali, David
Fernandez and Kostas Komninos for the revision of the document and their priceless
support.
This work was granted with one of the Bang & Olufsen scholarships for 2008.
6
Contents

1 Introduction 6

2 Theoretical Background 9
2.1 The BEM in Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Sommerfeld’s radiation condition . . . . . . . . . . . . . . . . . . 10
2.1.3 The integral equation (BIE) . . . . . . . . . . . . . . . . . . . . . 11

3 Methodology 15
3.1 Vibroacoustic Measurements . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Measurement mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Procedure of the measurement . . . . . . . . . . . . . . . . . . . 18
3.2 Verification measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Sound pressure measurements . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Laser measurements . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Preliminary testing 23
4.1 Initial considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Preprocessing and Measurements 29

5.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2.1 Testing of the mesh . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 Boundary Conditions (Measurement) . . . . . . . . . . . . . . . . . . . . 33
5.3.1 Laser measurements . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.2 SPL measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.3 Measurement tests, validation tests . . . . . . . . . . . . . . . . . 35
5.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
CONTENTS CONTENTS

6 Results 39
6.1 Results from the vibration measurement . . . . . . . . . . . . . . . . . . 39
6.2 Sound pressure radiated by the loudspeakers cabinet . . . . . . . . . . . 42
6.3 Radiation by the cabinet compared to the total radiation . . . . . . . . 44
6.4 Contribution from the cabinet to the radiated sound . . . . . . . . . . . 47
6.4.1 Considerations regarding the cabinets radiation . . . . . . . . . . 53
6.5 Behavior of the cabinet in the time domain . . . . . . . . . . . . . . . . 54
6.5.1 Impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.5.2 Decay of the sound pressure . . . . . . . . . . . . . . . . . . . . . 58
6.6 Radiation at the back of the cabinet (time domain) . . . . . . . . . . . . 63
6.7 Radiation from the bottom of the cabinet . . . . . . . . . . . . . . . . . 67
6.7.1 Frequency domain (Bottom radiation) . . . . . . . . . . . . . . . 67
6.7.2 Impulse response (bottom radiation) . . . . . . . . . . . . . . . . 69
6.7.3 Decay of the sound pressure (Bottom radiation) . . . . . . . . . 69

7 Conclusions 73

Bibliography 77

List of Figures 79

A Non-linear vibration of the cabinet 83

B Deflection shapes of the cabinet 89

C Matlab codes 92
C.1 Calculation of the sound radiated by the cabinet . . . . . . . . . . . . . 92
C.2 Calculation of the sound radiated by the loudspeaker . . . . . . . . . . . 98
C.3 Time domain characteristics (Impulse response & decay) . . . . . . . . . 102
C.4 Testing of the method (Test Box) . . . . . . . . . . . . . . . . . . . . . . 109

D SHELL63 Elastic Shell 112

8
Chapter 1

Introduction

Since the first loudspeakers were developed, there has always been a great interest for
improvement, in pursuit of a higher sound quality. Generally, loudspeaker units have
been the main target of such development, because they are essentially the source of
sound radiation. To the contrary, less emphasis has been put on the cabinet’s design,
since its impact on the sound quality is not so immediate. However, a poor cabinet
design can clearly influence to a significant extent the overall quality of sound radiated
by the loudspeaker.

Ideally, the walls of a loudspeaker cabinet are rigid and unmoving. However, in
reality, such is not the case. The cabinet is excited mechanically by the reaction forces
caused by the motion of the loudspeaker units mounted on them, and by the acoustic
pressure developed within the cabinet [9]. These two sources of excitation, make the
cabinet vibrate, and thus radiate sound. It is of interest to know if this sound radiation
is of sufficient magnitude and time duration to contribute audibly to the loudspeaker’s
total acoustic output.

This problem has been studied by different authors, such as Tappan, Toole and
Olive, Capone, Lipshitz, Vanderkooy et al. and Karjalainen et al. However, sur-
prisingly little literature exists on the topic [2]. The early work on the subject was
generally based on steady-state measurements, where the contribution from poorly
designed cabinets would become apparent in the frequency response, as in [16]. Also
subjective testing was used as a tool for evaluation of the sound quality [17]. However,
in this early studies, the contribution from the loudspeaker enclosure would not be
estimated separately.
 1. Introduction

Later research has mainly been based on the the use of numerical methods, which
have proved to be very convenient for this problem. Such is the case in refs. [9], [2] or
[7] and other authors that have studied the problem, compared different methods and
procedures, and examined different cabinet designs. However, most of the work in this
area has primarily focused on the steady state response of the systems.

All formerly cited authors concluded that the acoustic radiation from a cabinet
can significantly affect the overall performance of a loudspeaker, mainly at the low-
est structural resonance frequencies of the cabinet. However, such influence depends
mainly on the design of the loudspeaker.

The particular case of this study focuses on the radiation of a prototype of the
Beolab 9 loudspeaker by Bang & Olufsen. During the development process of the
loudspeaker, a “coloring” effect from the loudspeaker cabinet was reported based on
subjective testing. The main purpose of this study is to provide a tool that can eval-
uate the radiation by a loudspeaker cabinet and verify if the reported problem can be
detected by such a tool.

It should be remarked that in the present case of study, the shape of the cabinet
is not a standard rectangular cabinet, but a curved, conical like shaped one. This is a
fundamental difference between the present case of study and the previously referred
work on the field.

In this report, first a brief fundamental explanation of the Boundary Element

Method is presented, followed by the description of the methodology used in this
project (Chapter 3). The results of some preliminary testing, done in order to verify
the procedure of calculations used, are presented in Chapter 4. In Chapter 5, the
model and preprocessing of the BEM is described. The results obtained are presented
and discussed in Chapter 6. Finally, Chapter 7 summarizes the main conclusions that
can be drawn from the study.

10
Chapter 2

Theoretical Background

The theoretical background of this project is mainly related to the Boundary Element
Method (BEM). The BEM has been the principal tool used in this project for the
calculations carried out. In this section the method is briefly presented,without going
deep into it. The aim of this section is to introduce some essential concepts and the
idea on which the method is based.

The BEM is a numerical method used to solve linear partial differential equations
(PDEs) formulated as integral equations. In general terms, a boundary value prob-
lem governed by a certain PDE is to be solved over a certain domain. The governing
PDE is formulated in an integral form by using a weighted integral equation and the
Green-Gauss theorem, until the so called boundary integral equation (BIE) is formu-
lated. The BIE consists only of boundary integrals. The problem is therefore reduced
from a three-dimensional problem to a two-dimensional one. By solving this boundary
integral equation, the solution to the problem over the entire domain is found.

2.1 The BEM in Acoustics

In the particular case of acoustics, which is the actual application here, the BEM makes
it possible to calculate the sound field over a certain domain (sound pressure, particle
velocity...), provided a certain set of boundary conditions.

The starting point is the wave equation for an homogeneous, inviscid compressible
fluid,

1 ∂ 2 pins
∇2 pins = (2.1)
c2 ∂t2
2.1. The BEM in Acoustics 2. Theoretical Background

where pins is the instant pressure variation, c is the speed of sound. Assuming time
harmonic waves, the wave equation can be expressed as the Helmholz equation, which
describes the propagation of the sound pressure in the medium.

∇2 p + k 2 p = 0 (2.2)
Due to the time-harmonic wave assumption, the time dependence has been removed
2
from the Helmholtz equation (2.2). Under this assumption, the term ∂ ∂tpins
2 from 2.1
2 jωt
can be expressed as −ω e pins . In the Helmholtz equation, the factor e jωt (which
would be the only time dependent term), is omitted. The wavenumber k is defined as
k = ω/c

2.1.1 Green’s function

If a point source (described by a delta function) is placed in the unbounded medium,
it can be expressed by the inhomogeneous equation,

(∇2 + k2 )Gk (~r, r~0 ) = −4πδ(~r − r~0 ) (2.3)

The solution to the above equation is the free space Green’s Function, which is
expressed as:

e−jk(~r−r~0 )
G(~r, r~0 ) = (2.4)
~r − r~0
This function represents the pressure in the medium at the observation point ~r,
generated by a point source placed at r~0 . This point source is a monopole, which is a
mathematical abstraction, but nevertheless an essential concept in acoustics, useful to
describe many important phenomena.

2.1.2 Sommerfeld’s radiation condition

The Sommerfeld’s radiation condition makes it possible to solve the Helmholtz equa-
tion uniquely. Therefore it is essential in acoustic radiation problems in unbounded
mediums. It is a boundary condition for infinite exterior domains. Basically, it ex-
presses the fact that the energy decreases towards infinity, decaying progressively to
zero. In other words, it expresses the fact that there are no reflections coming from
infinity. Quoting Sommerfeld’s own words [15] : “...the sources must be sources, not
sinks of energy. The energy which is radiated from the sources must scatter to infinity;
no energy may be radiated from infinity into the field”.

The Sommerfeld’s radiation condition for an acoustic problem may be stated as:

12
2. Theoretical Background 2.1. The BEM in Acoustics

limr→∞ [r(p − ρ0 cvr )] = 0 (2.5)

where r is the radial coordinate of the spherical system, ρ0 c is the characteristic

impedance of air, p is the sound pressure and vr is the radial component of the particle
velocity.

2.1.3 The integral equation (BIE)

As mentioned previously, given a boundary problem, the BEM makes it possible to
calculate the solution on the domain of the problem (for instance a general boundary
problem as below).

Figure 2.1: Simple Boundary Problem Sketch. In a sound radiation problem the domain is
governed by the wave equation. In this particular project the outer boundary condition
is the Sommerfield radiation condition -Infinity-, and the inner boundary condition is
the velocity of the body

To arrive to the integral equation, starting from the Helmholtz equation, the start-
ing point is the vector identity [5]:

G(∇2 + k2 )p − p(∇2 + k2 )G = ∇ · (G∇p − p∇G) (2.6)

Integrating equation (2.6) over the volume V (which is the volume of the sphere of
radius a that encloses all points outside S) yields,

13
2.1. The BEM in Acoustics 2. Theoretical Background

Z Z
2 2
− p(∇ + k )GdV = ∇ · (G∇p − p∇G)dV (2.7)
V V

Such a sphere of volume V is very large, its radius goes towards infinity, in order
to enclose all points of the domain.

The Gauss theorem 1 is applied to the right side of (2.7). Thus, it is possible to
transform the previous volume integral into a surface integral.
Z Z
2 2
− p(∇ + k )GdV = (G∇p − p∇G)ndS + IA (2.8)
V S

The term IA in (2.8) depends on the radius a of the sphere used to integrate the
volume in (2.7). The term IA can be shown to vanish towards infinity, because both
Green’s function and the pressure satisfy the Sommerfeld’s radiation condition (see
above: (2.1.2)) as in [5]. Thus, IA is zero. Equation (2.8) can be as well expressed as:
Z Z
∂G
4π p(r)δ(r − r0 )dV = (p(r) + jkz0 v(r)G)dS (2.9)
V S ∂n
The left side of (2.9) is the solid angle seen from the observation point, or field
point, which is:

0, if r0 is inside S
4πp , R if r0 is outside S
∂ 1
C(r0 ) = 4π + S ∂n ( R )dS , if r0 is in the boundary of S

Bearing all the previous in mind, the integral equation can be formulated as:

Z
∂G(P, Q)
C(P )p(P ) = (p(Q) + jkz0 v(Q)G(P, Q))dS (2.10)
S ∂n
The previous equation (2.10) relates the pressure on any point Q on the surface S
of the body with any point P in the domain, as well as the velocity of the body in its
surface (in the case that it is vibrating).

There is as well a more complete integral equation, in which the scattering case is
also taken into account. An additional term representing an incident wave is included.
In this case the previous equation (2.10) is then [6],

1
The Gauss theorem or divergence theorem is a conservation theorem that relates the flow through
a surface to the flow inside the surface. Intuitively, it states that the sum of all sources minus the sum
of all sinks inside a surface, equals the net flow through the surface (whether outwards or inwards).
The volume integral is equal to the surface integral over the boundary.

14
2. Theoretical Background 2.1. The BEM in Acoustics

Z
∂G(P, Q)
C(P )p(P ) = (p(Q) + jkz0 v(Q)G(P, Q))dS + 4πpI (P ) (2.11)
S ∂n

Therefore, this equation relates as well the pressure and velocity at any point Q be-
longing to the surface S of the body (Q is on S), with any point P in the domain. p(Q)
and v(Q) are respectively the sound pressure and vibration velocity at Q, G(P, Q) is
the free space Green’s function relating Q with a field point P . z0 is the characteristic
impedance of the medium (ρc). pI is the incident pressure in the case of a scattering
problem (if the problem is purely a radiation problem the term pI disappears).

This Boundary Integral Equation -BIE- (2.11), makes it possible to know the pres-
sure at any point P in the domain. However, this equation should be implemented
numerically in order to be solved. In general terms, equation (2.11) is discretized into
elements along the body surface, and the equation is expressed as a matrix equation.
By solving such matrix equation, the pressure on the nodes is calculated, that is the
pressure throughout the surface of the body. Once that the pressure at the surface is
known, the pressure at any field point can be as well obtained.

In this section, the numerical implementation of the BIE is not looked at, since the
aim of it is just to illustrate and briefly explain some of the mathematical foundations
in which the BEM is based (namely such an integral equation). For a more detailed
explanation refer to ref. [5] [1] [8].

15
2.1. The BEM in Acoustics 2. Theoretical Background

16
Chapter 3

Methodology

The aim of this project is to study the radiation by a loudspeaker cabinet, which is
vibrating and radiating sound simultaneously with the loudspeaker units. There is the
fundamental problem that it is difficult to measure the radiation from the loudspeaker
cabinet, due to the fact that the loudspeaker units are radiating simultaneously. Hence,
the BEM is a very appropriate solution to calculate the radiation from the cabinet
(without the contribution of the loudspeaker loudspeaker units).

The method presented here can be used for other radiation problems involving a vi-
brating body, not only a loudspeaker cabinet. It is applicable to many other cases, and
it is especially useful for problems in which the source of interest cannot be separated
from other sources radiating simultaneously (i.e: automotive problems, machinery,
etc).

In this section, the general procedure followed in order to calculate the sound pres-
sure radiated by a loudspeaker cabinet is presented. Initially, a general description is
provided, followed by a more detailed explanation of the measurements, testing and
calculations carried out.

In order to determine the sound radiated by the loudspeakers cabinet with the
BEM, it is necessary to know what is the velocity on its surface (it is the boundary
condition of the problem). The velocity of the cabinet is measured directly. When the
velocity is known, it is possible to calculate the sound pressure radiated by the cabinet
(without the direct contribution of the loudspeaker units).

Some verification measurements are performed as well, in order to check that the
calculations are correct and get a better understanding of the results. These verifi-
3.1. Vibroacoustic Measurements 3. Methodology

cation measurements consist, firstly of sound pressure measurements, and secondly of

laser measurements on the loudspeaker membrane (to also calculate the total SPL with
the BEM).

Therefore, the procedure used for the study can be divided into a measurement
stage (which comprises both vibroacoustic measurements and verification measure-
ments) and a numerical calculation stage, using the BEM.

3.1 Vibroacoustic Measurements

The vibroacoustic measurements are performed with accelerometers, placed directly
on the surface of the body. A measurement mesh along the surface of the body is
defined, consisting of a set of specific positions where the acceleration is measured.
These positions (the measurement mesh) are the same as the mesh used for the BEM
model. (See Sect. 5).

Generally, the effects of loudspeaker cabinets radiation manifest themselves primar-

ily at the cabinet’s resonance frequencies between 100 Hz and 300 Hz [2]. Concerning
the resolution, the measurement mesh was designed so that all the modes up to more
than 300 Hz can be measured and comprised in the study with sufficient resolution.
For this purpose a preliminary modal analysis of the body is very useful. It is helpful
to have an idea of how the vibration and the deflection shapes look like.

In the specific case of loudspeakers, the cabinet vibrates due to both structural
excitation from the drivers and acoustic excitation from the pressure within the cab-
inet [9]. The effect from both sources of vibration, is inherently accounted for by the
measurements of the normal velocity performed on the surface of the cabinet (BEM
boundary condition).

3.1.1 Measurement mesh

The loudspeaker cabinet was measured at more than 100 different measurement posi-
tions. The lower section of the loudspeaker, where the woofer of the loudspeaker lies,
was measured at four different heights. Each of the heights had between 16 and 18
positions around the circumference of the loudspeaker. The upper part (where the
midrange unit lies), was measured at three different heights with 8 positions at each of
the heights. Finally, the circumference just above the woofer, where the vibration lev-
els are the highest, was measured in a finer mesh, consisting of 32 positions along the
circumference of the cabinet. The measurement mesh is shown in the following figure
3.1 (note that the measurement mesh is the same as the BEM mesh shown in figure 5.3)

18
3. Methodology 3.1. Vibroacoustic Measurements

Figure 3.1: Measurement mesh of the Beolab 9. The velocity was measured at every node of
the mesh

Generally, it can be said that the circumference of the speaker was “sampled” with
at least 16 measurement positions (going up to 18 or 30 in some of the cases), and at
four different heights. The upper part of the speaker, despite it is not vibrating with
very high amplitudes was meshed with a similar resolution.

Based on the modal study carried out on the Beolab9 by Yu Luan [10], where the
deflection shapes of the cabinet were looked at, it can be assured that the mesh is
fine enough to go up to 300 Hz, with a resolution of approximately 6 measurement
positions/nodes per wavelength. This measurement mesh assures that all the modes
up to 300Hz are measured. However, the mesh can go up to higher frequencies (around
400 Hz and more), with less accuracy, but still providing acceptable results. In prin-
ciple, using this measurement mesh, deflection shapes with up to three wavelengths
in the radial direction of the structure (n=3) can be measured, with a resolution of 6
positions per wavelength (from [10] it can be seen that all the modes below 300 Hz
have a maximum of 3 wavelengths per circumference).

An example of a deflection shape (309 Hz) of a conical cabinet, from Yu Luan’s

work, is shown below [10].

19
3.1. Vibroacoustic Measurements 3. Methodology

Figure 3.2: Example of the deflection shape of a conical cabinet at 309 Hz. Image from [10]

It should as well be remarked that the most efficient modes in terms of sound ra-
diation from the cabinet are those in the lower frequency range, where big areas of the
cabinet are vibrating in phase. The acoustic radiation from a cabinet affects the over-
all performance of a loudspeaker only at the lowest structural resonance frequencies of
the cabinet, if at all [16] [2] . Therefore, the mesh designed is sufficient for the present
study, considering the frequency range of concern.

3.1.2 Procedure of the measurement

When studying the sound radiated by a specific body, the phase information is essen-
tial (the phase information tells us how the different parts of the body are vibrating,
and how the total sound radiation is built up with the contribution from each part of
the body). Therefore, in order to have a reliable phase information, it is necessary to
set a reference point in the body. Relative to this reference point, the phase of the rest
of the measurement positions can be determined (the phase is calculated relative to
the reference point). In other words, it could be said that the reference point has zero
phase, and the phase of all the remaining positions represent the “delay” between the
reference point and each position. It is essential that this reference is not placed in a
nodal line in any of the modes of interest.

The general procedure for the measurement is the following: The magnitude of
the acceleration of the reference point |R|, the magnitude of the acceleration of each
position |S|, and the complex transfer function between both positions Ĥ = Ŝ/R̂ are
measured. Based on those three measurements, the complex acceleration at each po-
sition Ŝ, which includes magnitude and phase information, can be obtained simply as:

20
3. Methodology 3.1. Vibroacoustic Measurements

Ŝ = Ĥ · |R|

In the previous expression, Ŝ is the complex acceleration of a measurement position

(where the phase information is included, referenced to the reference point). A more
thorough explanation of how to obtain it, is presented below:

Three quantities are measured:

-the magnitude of the velocity of the reference point: |R|

-the magnitude of the velocity of the measurement position: |S|
-the complex transfer function between the
reference and the measurement position: Ĥ = |H|ejφH

Considering that the magnitude of Ĥ is |H| = |S/R| and the phase is φH = φS − φR ,

the complex transfer function can be expressed as:

Ŝ |S| j(φS −φR )

Ĥ = |H|ejφ = = e (3.1)
R̂ |R|

Since R is the reference, it has phase zero, therefore

R̂ = |R|ejφR = |R|e0 = |R| (3.2)

The previous equation also shows that the phase of the transfer function and the
measurement position is the same, φH = φS .

From 3.1 and 3.2, it is straightforward to express the complex quantity Ŝ as:

Ŝ = Ĥ · R̂ = Ĥ · (|R|ejφR ) = Ĥ · |R| (3.3)

where Ŝ is expressed only in terms of the measured quantities Ĥ and |R|.

From the derivation above, it is clear that Ŝ is the vibration at a certain position,
with a certain magnitude and a certain phase relative to the reference. The phase has
been referenced to a position of the cabinet (the reference point) instead of to the elec-
trical input. The reason for this is the stability of the measurement. By not choosing
the electrical reference, the influence of the phase distortion from the loudspeaker is
prevented, and the possible error is smaller.

These measurements provide the vibration velocity information necessary for the
numerical calculations to be carried out. They provide both the magnitude and the
phase at every position. These measured velocities at every position will be input into
the BEM, in order to calculate what is the radiated sound pressure from the body.

21
3.2. Verification measurements 3. Methodology

3.2 Verification measurements

There are some simple measurements that can be performed to verify the results ob-
tained. These measurements are not essential for the calculations, but they are very
useful for validation and for achieving a better understanding of the results.

3.2.1 Sound pressure measurements

According to what has been explained before, the SPL radiated by the body is also
measured. This is done in order to compare the calculated sound pressure radiated by
the body with the measured one. In the present study, the measurement corresponds
to the total sound pressure radiated by the speaker, with both the contribution from
the cabinet and the loudspeaker units.

Due to the fact that the BEM numerical study is done in an infinite medium, that
is in the free-field, the measurements should be performed preferably under anechoic
conditions (to avoid reflections from surfaces which are not accounted for in the numer-
ical model). If the measurements were not performed under such anechoic conditions,
the reflections from the surrounding bodies would influence the measurement.

3.2.2 Laser measurements

Another set measurements that could be performed for verification, involves measur-
ing the velocity of the loudspeaker’s membrane. Thus, the radiated pressure from the
loudspeaker can be calculated with the BEM and be compared with the pressure mea-
surements. These vibration measurements should be performed with a laser, since the
accelerometers are too heavy compared to the membrane, and their load would affect
the vibration severely.

3.3 Numerical calculations

The second part of the method is based in calculations by means of the Boundary
Element Method. The numerical study was carried out using the OpenBEM1 software
[6]. ANSYS and Solid Works were as well used for the preprocessing of the model. In
this section, the procedure to perform such calculations is explained.

Initially, a mesh is designed according to the geometry of the body. In this mesh,
the vibration velocity at each node should be known, thus, it is measured as explained
in 3.1. Each node of the mesh corresponds to a measurement position. Finally, the
1
OpenBEM is an open software based on Matlab, developed by Peter Juhl and Vicente Cutanda
from the University of Southern Denmark, Odense

22
3. Methodology 3.3. Numerical calculations

velocities are input into the BEM as the Boundary Conditions and the sound field is
obtained.

In this specific project, the meshes were defined manually in ANSYS and afterwards
exported to OpenBEM software. This manual procedure makes it possible to get a
better control over the mesh geometry, and guarantees a fairly good correspondence
between the actual body geometry and the mesh used for the numerical calculations
(hence, it guarantees a good correspondence between the nodes and the measurement
positions, which should be the same).

It is important to bear in mind the frequency range of interest, since a certain min-
imum resolution of the mesh is needed, in order for the results to be accurate enough.
A minimum of 6 nodes per wavelength should be used. Such resolution generally pro-
vides enough accurate results and a sufficient small error [5]. Below this limit, the
results are biased and can be severely affected. Above this resolution the results are
more precise (generally, the higher the mesh resolution, the more accurate the results
will be).

In this project, focused on the Beolab9 prototype, the frequency range of concern
goes up to around 250 or 300 Hz, where the cabinet is radiating most significantly. The
maximum element length was of approximately 0.1 m. A minimum density of 6 nodes
per wavelength (wavelength in the structure, mainly in the circumferential dimension)
was guaranteed for the frequency range of concern, which in principle is more than
enough. However, the resolution of the mesh at low frequencies, where the radiation
from the cabinet is higher, is much more fine (the deflection shapes are rather simple
-no more than one or two wavelengths around the circumference of the loudspeaker-
[10] so that the mesh resolution is very fine at the lower frequencies -much more than
6 elements/wavelength-). Thus, based on the wavelength of the vibration in the struc-
ture, the resolution of the mesh has been designed to provide enough accurate results.

23
3.3. Numerical calculations 3. Methodology

24
Chapter 4

Preliminary testing

It is important to test the calculation method and the procedure, to make sure that
the calculated sound pressure radiated by the loudspeaker’s cabinet is correct. It is
also important to examine the method in order to have a better understanding of the
results. Therefore, testing is an essential stage of the project, prior to the final inves-
tigation. It provides the wanted verification and evaluation of the method.

4.1 Initial considerations

In this project, the sound pressure radiated by a loudspeaker cabinet is calculated.
Since it is difficult to measure the cabinet’s radiation separately, the calculations can
not be verified by simple comparison with measurements1 . Therefore, a test case is
very convenient, prior to the study of the loudspeaker’s cabinet radiation itself.

The test case used for verification is a wooden box in which one of the walls is
vibrating. This structure is in principle similar to a vibrating loudspeaker cabinet. In
this case, all the sound pressure radiated is due to the vibration of the body itself,
without any other source (such as the loudspeaker units). Thus, it is possible to com-
pare the calculations with simple pressure measurements.

Another possible verification test is to measure the velocity of the loudspeaker

units, via laser measurements. This would make it possible to calculate the total radi-
ated pressure by the loudspeaker (cabinet + units) with the BEM, and compare it with
sound pressure measurements. However, it is preferable to do the testing based on the
1
The sound pressure radiated by the cabinet cannot be measured directly by simple measurements
because the loudspeaker units are radiating simultaneously.
4.2. Description 4. Preliminary testing

vibration of the box structure rather than on laser measurements of the loudspeaker
units. The reason is that we are interested in the radiation from the loudspeaker’s
cabinet, which is much lower than the radiation directly from the units. Thus, if the
radiation from the units is included in the calculations, the weight of the cabinet radi-
ation is minimized (and the weight of the units radiation is magnified). Furthermore,
considering that the unit’s radiation is calculated by a different procedure2 than the
cabinet’s radiation, the test box case is a much more reliable verification. Neverthe-
less, the calculation including both the radiation of the cabinet and the units (laser
measurements) will also be performed, since it is a useful verification of the results,
but it is not the best test to verify the calculation method.

4.2 Description
The purpose of the test is to examine the method explained in Chapter 3 - Method-
ology. For that purpose, the method is used to calculate the pressure radiated by a
vibrating box and compare the calculations with measurements.

The box used for the testing is a cubic box of 46x46x46cm, made of five rigid
wooden plates, and a sixth plate on top. This latter plate is a 1mm thick aluminum
plate with an inertial exciter attached in the center which excites the plate causing the
vibration. The test-box structure is shown in Figure 4.1.

Figure 4.1: Test Box used for the verification of the procedure
2
The sound pressure radiated by the unit is measured via laser measurements, instead of accelerom-
eters.

26
4. Preliminary testing 4.2. Description

The box is a convenient test structure for the method used in the project. If the
modeling body was a simple plate (instead of a box), near singular integration would be
involved in the calculations. If the box is present, such near singular integration is not
necessary, and the problem is simpler and more similar to the loudspeaker cabinet case.

Modeling the box is rather straightforward due to its simple geometry. The mesh-
ing has been done using triangular elements (SHELL63 in ANSYS) with a maximum
distance between nodes of 14cm. As it will be explained in section 4.3 the top frequency
limit of the calculation, due to resolution considerations, is approximately 150Hz. The
maximum “usable” frequency range of the box in the testing reaches to approximately
that same frequency 150-200 Hz (because due to the input levels used in the inertial
transducer in order to produce the wanted SPL, the transducer is overloaded, giving
rise to much distortion, especially above 200 Hz).

The BEM mesh of the test-box is shown in Figure 4.2

Figure 4.2: Mesh of the test box used for the BEM calculations. The maximum element length
is of 16cm

The measurement has been performed as explained in Chapter 3-Methodology. A

block diagram of the measurement setup is shown in figure 4.3

27
4.2. Description 4. Preliminary testing

Figure 4.3: Block diagram of the measurement

The normal velocity of the box has been measured at every node, via vibroacoustic
measurements. Both amplitude and phase have been measured. The reference point
was set near the exciter, because at that position all the modes are present.

Figure 4.4: Image of the measurement

The velocity of every single node is measured (one measurement per node of the
mesh). The measured data is input in the calculations as the normal velocity bound-
ary condition. Using the BEM, based on such measurements, the sound field in the
domain is calculated and the sound pressure at any point around the box is known.

In the BEM calculations, anechoic conditions are assumed (free-space); in other

words, no reflections are taken into account. The sound pressure measurements, in
order to be consistent with such anechoic conditions, have been performed inside an
anechoic chamber. Thus, for the sake of comparison, calculations and measurements

28
4. Preliminary testing 4.3. Results

are carried out under similar conditions.

4.3 Results
In Figure 4.5 the measured sound pressure and the sound pressure calculated with the
BEM are shown.

Test Box − Comparison between BEM and measurement (at 0.5m height)
60
Measurement
BEM

50

40
SPL [dB]

30

20

10

0
2
10
freq [Hz]

Figure 4.5: Comparison between the BEM calculation and the measured SPL

The agreement between calculations and measurement is generally very good, es-
pecially at low frequencies.

At higher frequencies the deviations are larger. This is related to the resolution of
the mesh. Based on some preliminary study of the wavelengths and natural frequencies
in the structure, and according to the rule of thumb of 6 nodes/wavelength, it can be
assumed that the top frequency limit of the calculation is below 150 Hz. This frequency
(actually 140 Hz) corresponds to the (1,3) mode of the plate, which has a resolution of
approximately 6 nodes/wavelength. Above this frequency, the resolution of the mesh
is not sufficient compared to the wavelengths in the structure, and greater deviations
occur as the frequency increases, especially above 180 Hz (where the resolution of the
mesh is becoming very coarse compared to the wavelength dimensions).

Apart from the coarser resolution of the BEM mesh, another explanation for this

29
4.4. Conclusion 4. Preliminary testing

greater deviation at higher frequencies (above 150 Hz) is the distortion of the box-
transducer. This is apparent from the rapid fluctuations in the frequency response
from 180 Hz on (the sound is very distorted in this frequency range). In the low fre-
quency range, where these fluctuations are less rapid, the calculations are very precise.
However, the overall agreement is good. If the box was behaving more linearly, the
agreement between measurement and calculations would be expected to be even better.

The load of the accelerometers was also considered and examined. The normal
velocity of the plate was measured with and without the load of an accelerometer, that
was attached at different positions. In principle it did not affect the vibration of the
plate of the box importantly.

4.4 Conclusion
A preliminary test was performed to verify that the calculation method used in the
project is valid. The results of this test are satisfactory. The agreement between mea-
surements and calculation is fairly good, especially at low frequencies, in the frequency
range where the mesh satisfies the 6 nodes/wavelength resolution. The method has
been proved to be appropriate for the purpose of this project, to the sound radiation
of a vibrating loudspeaker cabinet.

30
Chapter 5

Preprocessing and Measurements

In this section, the Boundary Element Method preprocessing required for calculating
the radiation from a loudspeaker’s cabinet is presented. The specific case of the B&O
Beolab 9 cabinet is investigated.

In first place, the geometry and the meshing of the cabinet’s model is presented.
Some testing of the mesh is also shown in this section. The Boundary Conditions of
the problem are as well examined. The required set of Boundary Conditions consists
basically of the velocity of the cabinet surface, which is obtained from empirical vi-
broacoustic measurements. Such measurements are briefly explained in this section.
After this preprocessing stage, the problem can be solved with the BEM.

In short, this section focuses on the specific case of study of the Beolab9. It basi-
cally presents the BEM preprocessing prior to the solution of the problem.

5.1 Geometry
The loudspeaker modeled in this study is an early prototype of the Beolab9 loudspeaker
manufactured by Bang & Olufsen. The loudspeaker is shown in Figure 5.1.

The cabinet has a characteristic curved shape, with an elliptical cross-section all
along its height, which is narrowing from the bottom to the top resembling somehow
a conical shape. There are three drivers in the speaker, a woofer and a midrange unit
placed directly on the surface of the cabinet, and the third driver is a tweeter placed
“on top” of the speaker, in the Acoustic Lens c device.
5.1. Geometry 5. Preprocessing and Measurements

Figure 5.1: Beolab9. Side view (left). Top view (centre). View without the screen cover
(right)

The actual cabinet of the loudspeaker is shown on the right side of Figure 5.1 (in
the other two figures, the cabinet is covered with a screen, which is the exposed visible
part). The cabinet is the structure to which the vibration of the driver is transmitted,
and presumably it is radiating sound.

The cabinet surface is basically smooth, except for the cavities to which the drivers
are attached. However, for simplicity, these irregularities in the geometry are disre-
garded in the modeling and the cabinet shape is simplified. Since the frequency range
of concern is basically low frequency (up to about 300 Hz), the shortest wavelength
considered is of more than 1m, and these irregularities can be neglected without any
significant effect. Thus, the geometry can be simplified to a smooth cabinet, as shown
in Figure 5.2.

Figure 5.2: Beolab9 simplified geometry in which the mesh was based

32
5. Preprocessing and Measurements 5.2. Meshing

5.2 Meshing
The mesh is designed using triangular elements (of three nodes per element). The
type of element used is SHELL63 (in ANSYS), triangular element of six degrees of
freedom at each node (see Apendix D). The element sizes vary along the surface. The
resolution of the mesh is finer around the woofer. The mesh is shown in Figure 5.3 1 .

Figure 5.3: Meshing of the Beolab 9 used for the BEM calculations

The maximum dimension of the elements (the maximum separation between nodes)
is of 13cm. According to the rule of thumb of 6 elements per wavelength, with this
mesh, considering the deflection shapes of the cabinet (see [10]) the resolution of the
mesh is sufficient below 300 Hz. The frequency range of concern goes up to no more
than this 300 Hz limiting frequency.

Depending on the number of wavelengths in the structure around the circumferen-

tial dimension, the resolution of the mesh is different. The mesh provides a minimum
resolution of about 18 nodes/wavelength if there is one wavelength in the structure
(n=1) in the circumferential dimension. If there are two wavelengths (n=2) around
the circumference, it provides a resolution of 8 to 10 nodes/wavelength. If there are
three wavelengths (n=3) in the structure, it provides a resolution of 5 to 6 nodes/wave-
length. Above this limit, the resolution is coarser (but the frequency range of concern
is nevertheless below it).

1
Note that the BEM mesh is the same as the measurement mesh

33
5.2. Meshing 5. Preprocessing and Measurements

Therefore, the meshing used is more than sufficient for the frequency range of in-
terest up to 300 Hz. Below this frequency limit, the resolution of the mesh is more
than sufficient.

The mesh was created manually in order to have a better control of the node
coordinates and achieve a better correspondence between the BEM mesh and the mea-
surement positions (since the velocity at each node is measured in order to determine
the Boundary Conditions). Therefore, it is important to make sure that the coordi-
nates of each measurement position match with the coordinates of each of the nodes.

The mesh was created in ANSYS, and afterwards exported to OpenBEM (in Mat-
lab), where the calculations were carried out.

5.2.1 Testing of the mesh

Once the mesh has been created, it is important to verify that it is working properly.
For this purpose a simple test was carried out. The test consists on placing a point
source in the origin of coordinates, and evaluate what the velocity is in the normal
direction at each node (in other words, calculate what is the normal velocity produced
by the point source at each node). Using those velocities as the set of Boundary condi-
tions, the sound pressure calculated by the BEM should equal that of the point source
calculated analytically.

Expressed in a more intuitive way, it could be said that the test consists on calcu-
lating what is the pressure produced by a point source with the BEM, using the mesh
as a ”midpoint” to verify that it is indeed “working” properly. The BEM results are
then compared with the analytical solution.

Figure 5.4 presents the results of the test. It is apparent from it that the mesh is
working properly, and that its resolution is clearly sufficient. The BEM results agree
with the analytical solution, and the error is very small. In the frequency range of
main interest, the error is sufficiently low. At 200 Hz the error is below 2.5% and at
300 Hz it is below 5%. At 400 Hz (frequency which is already out of the main range
of interest) the error is still below 10%. On the other hand, above 500 Hz, the mesh
resolution is not sufficient, and the results from the calculations break down (the error
at 700 Hz is of about 25%).

34
5. Preprocessing and Measurements 5.3. Boundary Conditions (Measurement)

−Testing of the Mesh− SPL 1m away of a point source placed iside the cabinet
150
149 BEM
148 Analytical
147
700Hz
146
145
144
143
142
400Hz
141
140
139 300Hz
SPL [dB]

138
137
136
200Hz
135
134
133
132
131
130
100Hz
129
128
127
126
125
0 50 100 150 200 250 300 350 400
Angle [deg]

Figure 5.4: Testing of the mesh. Sound pressure produced by a monopole placed at the origin.
The figure compares the sound pressure calculated analytically and with the BEM using
the loudspeaker mesh, for different frequencies.

The mesh has been shown to be working properly, and according to the previous
discussion, its resolution is appropriate for the case of study in this project.

5.3 Boundary Conditions (Measurement)

As it has been explained in Chapter 3 (Methodology), the Boundary Conditions of
the problem have to be determined via vibroacoustic measurements. In a radiation
problem such as the present, the boundary conditions consist of the velocity on the
surface of the radiating body. In the present case we are interested in the velocity of
the loudspeaker cabinet. This vibration velocity should be measured.

The detailed procedure of the measurement, and the measurement method have
been explained in section 3 (method). Here, only the setup used for measuring the
loudspeaker cabinet’s velocity is briefly explained.

As mentioned previously, the measurement grid (the measurement positions) along

the loudspeaker cabinet’s surface, should be the same as the nodes of the BEM mesh.

35
5.3. Boundary Conditions (Measurement) 5. Preprocessing and Measurements

In principle the velocity at each node is measured. Thus, there are 100 measurement
positions and a reference position (used to determine the phase of the vibration. See
Chapter 3 (Methodology). It is important that the coordinates of the measurement
positions and the coordinates of the nodes are the identical.

Figure 5.5: Some pictures of the vibration measurement

The reference position is selected to be just above the woofer, because the vibration
levels are dominant there and in principle every mode is present in that position.

A block diagram of the measurement is shown in Figure 5.6

Figure 5.6: Block diagram of the vibration measurement

The type of excitation used for the measurement was pseudorandom noise, with a
frequency span of 800 Hz and 800 lines. In this case, it is very convenient to use such
an excitation signal, due to the great amount of measurement positions involved (more

36
5. Preprocessing and Measurements 5.3. Boundary Conditions (Measurement)

than 100 positions). Pseudorandom noise measurements optimize very much the time
required for performing the measurements.

5.3.1 Laser measurements

In order to measure the velocity of the loudspeaker membrane, laser measurements
are required. Accelerometers cannot be used because the membrane is very light, and
their load would affect severely the vibration.

Based on the laser measurements, the total sound pressure radiated by the speaker
(cabinet + drivers) can be calculated with the BEM. These calculations can be com-
pared to the sound pressure measurements, for verification. It is interesting to inspect
the accuracy of the calculation of the total sound radiated by the speaker.

5.3.2 SPL measurements

Besides the vibroacoustic measurements, sound pressure measurements were also car-
ried out, in order to compare the calculations with the measurements, and have an
idea of what is the contribution of the cabinet, relative to the total radiated pressure.

The idea is to measure the total sound pressure radiated by the loudspeaker (both
drivers and cabinet), and compare it with the calculations. In order to do so, it is
necessary to measure under anechoic conditions, since for the BEM calculations free-
space is assumed (no reflections from other bodies). The pressure measurements were
therefore done in an anechoic chamber, to provide such free-space condition.

5.3.3 Measurement tests, validation tests

The results from the measurements were carefully processed and examined, position
by position. Apart from the calibration of the transducers, all the measured positions
were compared with the reference to make sure that the order of magnitude was correct
and that there were no errors in the measurement that could bias the results. There
were no apparent errors present in the data.

Estimation of the maximum expected pressure

Based on the vibration measurements, the maximum expected pressure radiated by
the cabinet can be estimated. Since the velocity on the surface of the body is known,
by evaluating the volume velocity of the body, the radiated sound pressure can be
estimated.

37
5.3. Boundary Conditions (Measurement) 5. Preprocessing and Measurements

The estimation was done to have an idea of the order of magnitude of the radiated
sound pressure. By doing so, the BEM calculations can be compared with the estima-
tion, to make sure that they are in the same range.

The volume velocity is expressed as the particle velocity u times the surface S:

Q = Su (5.1)
The volume velocity of the body can be approximated by multiplying each node
velocity by the surface around it. It is a discrete space approximation in which the
area surrounding each node is assumed to be vibrating with similar velocity. The loud-
speaker surface is therefore approximated by 101 surface “cells”. Since the frequency
range of concern is very low, this approximation is sufficiently accurate.

Once the volume velocity Q has been determined, the sound pressure radiated can
be also determined by

j2πf ρQcab −jkRf p

p= e (5.2)
4πRf p

where Q is the volume velocity, Rf p is the distance to the field point (the point where
the sound pressure is calculated), f and k are the frequency and the wavenumber
respectively. Therefore, it is straightforward to estimate the sound pressure radiated
from the body (the loudspeaker cabinet in this case).

The maximum expected radiation of the Beolab 9 cabinet (for an input of 100mV
amplitude broadband noise), is of approximately 46.5dB at 85 Hz and at 150 Hz. In
principle, considering the vibration velocity levels, it is a reasonable result.

This volume velocity approach is accurate if the velocity of all the positions is in
phase and the structure is smaller than a quarter of a wavelength in air. In the present
test, the low frequency condition is fulfilled. However, the velocities of all points are
not vibrating in phase. At the low frequency modes, which are the ones radiating the
most, the whole cabinet is vibrating almost in phase, so the estimation of the SPL is
reasonably good, but it is still overestimated by 3 or 4 dB (as will be apparent in the
following Chapter 6).

38
5. Preprocessing and Measurements 5.4. Solution

5.4 Solution
Once the geometry and the meshing have been made and verified, and the necessary
set of boundary conditions for the problem have been measured, the problem can be
solved using the BEM. With the information collected, the sound pressure radiated by
the cabinet can be predicted at any point of the domain. The BEM software used is
OpenBEM, a software based on Matlab codes, developed by P.Juhl and V.Cutanda,
freely distributed by them [6].

The details on the implementation of the BEM and the computation process are
out of the scope of this report. Only the results of the BEM study are presented. The
codes produced in order to solve the problem using the BEM are included in Appendix
C (codes of Matlab). Some explanation is found in the comments of the codes. The
steps followed, the general procedure and the ideas behind the Matlab codes are re-
flected in the comments.

39
5.4. Solution 5. Preprocessing and Measurements

40
Chapter 6

Results

The aim of this project is to evaluate the contribution of the cabinet to the overall
sound radiated by the loudspeaker. In the present chapter, the results of the study are
presented and discussed.

In first place, the results from the measurements performed on the cabinet’s sur-
face are presented. Subsequently, the results obtained from the BEM calculations are
examined. These results calculated with the BEM show what is the actual radiation
from the loudspeaker’s cabinet. Therefore, they are essential in understanding the
contribution from the cabinet to the total radiated sound. Based on these results,
the problem is analyzed in both frequency and time domain. The sound radiated by
the cabinet is compared to the sound by the loudspeaker unit and to the total sound
radiated by the loudspeaker. Moreover, some additional measurements concerning the
non-linear behavior of the cabinet were done. However, these linearity measurements
are presented in Appendix A, since they are not the main focus of the study, but they
are nevertheless useful to understand better the problem studied in the present project.

For the sake of simplicity, from this point on we will refer to:
• the loudspeaker’s cabinet as “cabinet”
• the loudspeaker’s unit as “unit”
• the whole loudspeaker, consisting of both unit and cabinet as “loudspeaker”

6.1 Results from the vibration measurement

In this section, the results from the vibration measurements are presented. As ex-
plained in Chapter 5, the vibration on the loudspeaker’s cabinet was measured at
6.1. Results from the vibration measurement 6. Results

more than 100 positions (according to the measurement mesh designed). By analyzing
the vibration data obtained from the measurements, it is possible to get an idea of
which frequencies might be contributing more to the sound radiated by the cabinet,
which positions are vibrating with greater amplitudes, etc. In general terms, these
vibration data provide a first idea about how the vibration and the actual radiation
from the cabinet is.

Acceleration of the cabinet body (magnitude)

110

100

90

80
Acceleration [dB re.1 µ m/s ])
2

70

60

50

40

30

20

10

100 200 300 400 500 600 700 800

log f [Hz]

Figure 6.1: Acceleration on the cabinets surface at all the measurement positions

From Figure 6.1, it is possible to localize the natural frequencies of the cabinet
and to know at which ones it is vibrating with greater level. Such an investigation
provides an idea of the frequencies that could be more relevant to the sound radiation.
In principle, the main concern is in the low frequency modes (since their radiation
efficiency is expected to be greater).

It is straightforward to localize the main frequencies of vibration. Based on the

data of figure 6.1, the main frequencies concern are 85 Hz, 145-150 Hz, 185 Hz and
220 Hz.

42
6. Results 6.1. Results from the vibration measurement

An investigation on the deflection shapes of some natural frequencies was done.

By looking at the normal velocities at different positions, it was apparent that at low
frequency the cabinet is vibrating almost in phase, as a whole. Contrarily, at higher
frequencies the phase starts to change along the structure, and the deflection shapes
become more complicated.

In Appendix B, the deflection shapes of some modes of the cabinet are presented.
At 85 Hz and below, the cabinet is vibrating in a “pulsating mode” like shape. At
150Hz, the deflection shape resembles the one of n=2 circumferential modes (the front
and the back of the cabinet are vibrating in phase, while the sides are vibrating in
anti-phase with the front and back). Even though the deflection shapes are not clearly
defined (they do not correspond to the analytical standard shapes of simple cylindrical
bodies), it is interesting to see how the cabinet is vibrating at different natural fre-
quencies.

These results just provide an initial idea of the vibration levels, the deflection
shapes, what are the main frequencies that could be problematic, etc. This is useful
information, but it is no more than just a first insight in to the vibration, the inves-
tigation of the cabinet’s sound radiation -based on BEM calculations-, is presented in
the following sections.

43
6.2. Sound pressure radiated by the loudspeakers cabinet 6. Results

6.2 Sound pressure radiated by the loudspeakers cabinet

Based on the vibration measurements on the surface of the loudspeaker’s cabinet, the
sound pressure level radiated by the cabinet has been calculated with the BEM, as
explained in Chapter 3. The details about the preprocessing of the BEM model are
included in Chapter 5. In the present section, the results of the calculation are pre-
sented and discussed.

Figure 6.2: Field points at which the sound pressure was calculated

Figure 6.2 shows the field points defined in the domain at which the sound pres-
sure has been calculated. These field points are set around the azimuth angle (with a
resolution of 1 degree) at 1 meter distance from the radial direction and 35 cm height.
The calculation has been done for frequencies from 50 to 400 Hz (with 1 Hz resolution).

The sound pressure radiated by the cabinet is presented in a surface plot, showing
the SPL against the frequency and the polar angle (azimuth angle). The color corre-
sponds to the SPL. The phase can be plotted in a contour plot, where each line is an
isobar -all points of each line have the same phase-. Each transition between isobars
corresponds to a π/4 change in the phase.

Figures 6.3 and 6.4 show both the magnitude (in dB) and the phase of the sound
pressure.

44
6. Results 6.2. Sound pressure radiated by the loudspeakers cabinet

SPL radiated by the cabinet at 1m distance [freq vs azimuth angle]

150 35

100 30

50 25
azimuth angle [deg]

0 20

−50 15

−100 10

−150 5

50 100 150 200 250 300 350 400

f [Hz]

Figure 6.3: SPL radiated by the loudspeakers cabinet, at 1m distance and 0.35 m height,
around the azimuth angle (polar angle around the loudspeaker)

Figure 6.4: Phase of the pressure radiated by the loudspeakers cabinet, at 1m distance and
0.35 m height around the azimuth angle. The isobars represent a π/4 phase change

From Figure 6.3 it is evident that the highest sound pressure levels are in the fre-
quency range between 80 Hz and 180 Hz. The sound pressure is particularly high at
150 Hz, 85 Hz and 180 Hz. However, the most important sound radiation is at 150

45
6.3. Radiation by the cabinet compared to the total radiation 6. Results

Hz. The highest sound pressure levels are found towards the back of the cabinet. The
results presented here agree with conclusions drawn from the data shown in figure 6.1.

Based on the previous results, the frequencies at which the cabinet is radiating the
most have been identified. They can be plotted in a more conventional polar plot,
which shows the SPL along the polar angle (“around” the cabinet).

SPL of the main natural frequencies of the cabinet at 1 m

90º 40dB
120º 60º
30dB

150º 20 30º

10dB

180º 0º

210º 330º

240º 300º
270º 85Hz
150Hz
182Hz

Figure 6.5: SPL radiated by the loudspeakers cabinet, along the polar angle at 1m distance.
85 Hz (solid), 150Hz (dashed) and 182 Hz (dotted)

6.3 Radiation by the cabinet compared to the total radi-

ation
In order to estimate the contribution of the cabinet to the radiated sound, it is impor-
tant to know the total radiation from the loudspeaker. In this section, the total sound
pressure radiated by the loudspeaker is compared to the one radiated by the cabinet.

While the cabinet radiation has been calculated with the BEM (it is the same as
in the previous section 6.2), the total sound pressure from the speaker has been mea-
sured in an anechoic chamber - achieving the “free-space” condition also assumed in
the calculations -.

For the following comparison, three positions (three field points) are used as ob-
servation points. Those potitions are placed in the front, the side and the back of the
loudspeaker, at 1 m distance in the radial direction and 0.35 m height from the ground.

46
6. Results 6.3. Radiation by the cabinet compared to the total radiation

Figure 6.6: SPL radiated by the loudspeakers cabinet, at 1m distance, compared to the total
radiated pressure by the loudspeaker. The top figure shows the radiation in the front
direction -0o -, the medium figure shows the lateral radiation -90o -, and the bottom figure
the back radiation -180o-

Figure 6.6 shows the radiation from the cabinet compared to the total radiated
pressure. It is obvious that the radiation by the cabinet is in all cases much lower than
the total sound pressure. Thus, the influence of the cabinet does not seem too signif-
icant in terms of magnitude. The most important contribution from the cabinet can
be found in the back of the loudspeaker, in the frequency range between 80 and 150 Hz.

For a more clear comparison of the radiation from the speaker and the cabinet,

47
6.3. Radiation by the cabinet compared to the total radiation 6. Results

the results can be summarized in one same plot. In such plot it is apparent how the
radiation from the cabinet and the loudspeaker is changing differently depending on
the position of the observation point.

SPL radiated from the cabinet compared to the total

Cabinet
Total
60
0º

50 90º
180º

40
SPL [dB]

180º
30
0º

20

90º

10

2
10
log(f) [Hz]

Figure 6.7: SPL radiated by the loudspeakers cabinet, at 1m distance, compared to the total
radiated pressure by the loudspeaker. The front (blue), lateral (green) and back (black)
radiation are plotted together

It is obvious from figure 6.7 that the main radiation from the cabinet is taking
place in the front and in the back of the cabinet.

Results presented in section 6.2 and in figures6.6 and 6.7 indicated that the radi-
ation from the cabinet is greater in the back. Contrarily, the sound radiation of the
loudspeaker is weaker in the back1 . The combination of this two facts, result in an
increased influence from the cabinet towards the back of the loudspeaker.

Summarizing the results of this section, it should be highlighted that the SPL radi-
ated by the cabinet is much lower than the total radiated sound. However, the greater
radiation from the cabinet and the weaker radiation from the speaker towards the
1
These two facts are reasonable if we consider that most of the front area of the loudspeaker is
“occupied” by the loudspeaker units, while most of the cabinets net surface is in the back of the
speaker.

48
6. Results 6.4. Contribution from the cabinet to the radiated sound

back, result in a higher contribution from the cabinet at the back than at the front.
This contribution at the back of the cabinet seems to be the most important one, and
it could affect to some extent the total radiation from the speaker.

These results agree with those in refs. [9], [2] and [16]. However, there is a funda-
mental difference with ref.[9], since in their study, the contribution of the cabinet was
greater on axis (in front of the loudspeaker). This difference may be related with the
geometry of the cabinet, since their work focused only on rectangular shapes, where
the vibration transmitted to the back would be different than in the case of a curved
“elliptical” cabinet. Also the different radiation efficiency may be of importance. How-
ever, this hypothesis should be further investigated.

6.4 Contribution from the cabinet to the radiated sound

In the previous section, the radiation from the cabinet was compared to the total ra-
diated sound by the loudspeaker. It is clear that such a comparison is important in
order to get an idea of how the cabinet could influence the “final” sound radiated by
the loudspeaker. The results presented so far are useful, but it is difficult to withdraw
certain conclusions from them. In this section, a more thorough approach to the prob-
lem is formulated.

The total radiated sound by the loudspeaker can be determined by simple sound
pressure measurements (as was done in the previous section). However, another inter-
esting approach is to calculate the total sound radiation with the BEM. In this case the
calculations are based on the velocity of the loudspeaker unit, which is measured via
laser measurements (to avoid the influence of loading the membrane with accelerome-
ters).

Using the BEM, both the total pressure by the loudspeaker (including the radiation
from the cabinet and the loudspeaker unit) and the pressure radiated only by the unit
(as if the cabinet was completely rigid) can be calculated separately.

In the present study, both pressure measurements and BEM calculations have been
carried out and compared. Using the BEM, the pressure radiated by the cabinet, by
the loudspeaker units, and by the whole loudspeaker (cabinet and units) has been cal-
culated. These calculations are compared to the measurements of the total pressure
as well. The observation points used for the study were in the front, the side and the
back of the loudspeaker, at 1 m distance and 0.35 m height. In this section mainly the
results obtained for the front and the back positions are shown.

It should be noted that the BEM calculations require some more time than the
pressure measurements -since they involve laser measurements and the calculations

49
6.4. Contribution from the cabinet to the radiated sound 6. Results

themselves-, but they render it possible to obtain results which would not be easily
acquired otherwise. It is also very interesting to compare the BEM calculations with
the measurements, and it is a very useful verification tool.

SPL radiated from the loudspeaker −BEM compared to measurement−

64
BEM
Measurement
62

60

58
SPL [dB]

56

54

52

50
2
10
log(f) [Hz]

Figure 6.8: Comparison of measurements and BEM calculations of the total radiation from
the loudspeaker. The observation point is in front of the loudspeaker at 1 m distance
and 0.35 m height

In Figure 6.8, the total pressure radiated by the loudspeaker in the front direction
is presented,where both calculated and measured pressure are plotted. It seems that
there is a very good agreement between measurements and calculations, with a maxi-
mum difference of 1 dB SPL.

In the case where the observation point is placed in the back of the loudspeaker
(Figure 6.9), measured and calculated results are rather similar, but the agreement is
not as good as it was at the front of the loudspeaker (Figure 6.8).

50
6. Results 6.4. Contribution from the cabinet to the radiated sound

SPL radiated from the back of the loudspeaker −BEM compared to measurement−
60
BEM
59 Measurement

58

57

56
SPL [dB]

55

54

53

52

51

50
2
10
log(f) [Hz]

Figure 6.9: Comparison of measurements and BEM calculations of the total radiation from
the loudspeaker. The observation point is in the back of the loudspeaker at 1 m distance

At the back of the cabinet (Figure 6.8), the agreement is generally good, particu-
larly at low frequencies. However, there is a significant deviation around the 200 Hz
frequency range, where the measured pressure is 3 dB lower than the calculations. This
can be explained by the fact that the measurements were not performed in a strictly
free-space condition (free of any sort of reflections). The loudspeaker was standing
on a platform surface (instead of being suspended in the air), thus being sensitive to
reflection paths from the platform. The lower sound pressure around 200Hz seems
to be caused by a “ground reflection” from the platform2 . However the agreement
is fairly good, and despite the amplitude difference, both curves follow quite similar
fluctuations.

2
It is apparent from figure 6.8 that the reflection from the platform is not so present in the mea-
surements in the front of the speaker, because in this case it was placed in the edge of the platform,
not giving rise to the possibility of important reflections from it.

51
6.4. Contribution from the cabinet to the radiated sound 6. Results

Total SPL from the loudspeaker compared to the loudspeaker unit −BEM−
62
Total
Loudsp. unit
61

60

59
SPL [dB]

58

57

56

55
2
10
log(f) [Hz]

Figure 6.10: SPL of the loudspeaker unit compared to the total SPL (cabinet + unit) in front
of the loudspeaker calculated with the BEM. The observation point is at 1 m distance
(at the front)

The comparison between the pressure radiated by the unit and the total is shown
in figure 6.10. The observation point is placed in front of the loudspeaker, 1 m away.

There is a very slight difference between the radiation from the loudspeaker unit
and the total pressure in the front. However, such a difference is insignificant, showing
no apparent influence from the cabinet. If we consider that the SPL difference between
cabinet and loudspeaker is of more than 25 dB (see Figure 6.7), no significant influence
from the cabinet can be expected. For cabinet resonances to be visible in the frequency
response, a steady-state output of at least -20 dB relative to the output from the units
themselves is required [16]. Thus, it is clear than in this case, the total sound pressure
is mainly produced by the loudspeaker unit itself.

Similar calculations and measurements were performed as well in the back of the
loudspeaker, for an observation point placed at 1 meter distance (at 180o azimuth).
The results are presented in different figures, initially showing the calculations of the
loudspeaker unit and the total (figure 6.11), then compared with the measurements
(figure 6.12), and finally plotted along with the cabinet’s SPL (figure 6.13).

52
6. Results 6.4. Contribution from the cabinet to the radiated sound

Total SPL radiated from the back compared to the Loudspeaker unit −BEM calculations−
60
Total
59 Loudsp. unit

58

57

56
SPL [dB]

55

54

53

52

51

50
2
10
log(f) [Hz]

Figure 6.11: SPL of the loudspeaker unit compared to the total SPL (cabinet + unit) calcu-
lated with the BEM. The observation point is at 1 m distance (in the back).

SPL radiated at the back of the loudspeaker (180º) −BEM & measurements−
60
Total −BEM−
59 Loudsp. unit −BEM−
Total −Measured−
58

57

56
SPL [dB]

55

54

53

52

51

50
2
10
log(f) [Hz]

Figure 6.12: SPL of the loudspeaker unit compared to the total SPL (from the BEM calcula-
tions and from the measurements). The observation point is at 1 m distance (in the
back).

53
6.4. Contribution from the cabinet to the radiated sound 6. Results

SPL radiated at the back of the loudspeaker (180º) −BEM & measurements−
60

55

50
SPL [dB]

45

40

Total−BEM
Loudsp.unit−BEM
Total−Measured
35
Cabinet−BEM

30
2
10
log(f) [Hz]

Figure 6.13: SPL radaited by the loudspeaker unit compared to the total and to the cabinet.
The observation point is at 1 m distance (in the back).

On the contrary to figure 6.10, it is apparent from figures 6.11, 6.12 and 6.13 -
which show the pressure at the back of the cabinet-, that there is a visible influence
by the cabinet on the total radiated pressure. This influence is noticeable from the
peaks which appear around 150 Hz and 184 Hz in the total radiated pressure by the
loudspeaker. From figure 6.11, it is apparent that such peaks are not present in the
driver’s radiation, while they are present in the total sound pressure curve.

In figure 6.12, it can be seen how these peaks also appear in the measurements of
the total pressure, confirming that they are actually present in the total radiation (the
measurements are plotted along with the BEM calculations).

54
6. Results 6.4. Contribution from the cabinet to the radiated sound

Finally, in Figure 6.13, the calculated and measured data of the total sound radia-
tion is compared to the radiation of the cabinet. It is evident that the peaks appearing
in the curve of the total sound pressure correspond to the peaks of the cabinet’s fre-
quency response. Thus, there is little doubt that such peaks are due to the contribution
of the cabinet. This is a very interesting result, since it shows how the cabinet has
an influence on the final sound radiated by the loudspeaker, and confirms the greater
influence of the cabinet in the back side of it. However, the magnitude of the cabi-
net’s contribution is rather small. The magnitude of the peaks in the total pressure is
hardly 1 dB SPL. Therefore, it seems that the cabinet is indeed affecting the frequency
response of the loudspeaker, but not too drastically though.

6.4.1 Considerations regarding the cabinets radiation

It can be concluded from the results presented in section 6.3 and section 6.4 that the
cabinet radiation has not a drastical impact in the frequency response of the loud-
speaker, at least in terms of sound pressure level. However, there is still an influence
from the cabinet that should be examined more thoroughly.

All in all, it is important to bear in mind the fact that the influence of the cabinet
has only been inspected from the frequency domain so far. The contribution from the
cabinet to the total radiation of sound could be greater in the time domain. Particu-
larly, the narrow frequency peaks and dips present in the cabinet’s frequency response
(figures 6.6 and 6.7), point towards a low damping characteristic that might have an
influence in the behavior of the loudspeaker, which might be especially apparent in the
time domain.

It should be remarked that even for cases in which the disturbance from the cab-
inet to the steady-state response may be too small to be readily discerned, it is still
potentially audible [9].

55
6.5. Behavior of the cabinet in the time domain 6. Results

6.5 Behavior of the cabinet in the time domain

While in the previous section the results were based on the frequency domain, in the
present section they are based on the time domain.

So far, the frequency response of the loudspeaker and the cabinet have been exam-
ined. The cabinet has not a severe influence in the loudspeaker’s frequency response
(in terms of SPL). Nonetheless, the frequency response of the cabinet has some reso-
nances with a rather low damping which are of concern. These frequency peaks appear
at 80 Hz, 150 Hz, 184 Hz and 220 Hz.

The low damping, displayed as narrow peaks in the frequency domain, results in
a long decay of the sound in the time domain. In the present case of study, this low
damping can lead to the case where the impulse response of the cabinet is longer than
the impulse response of the unit. In other words, it may happen that while the con-
tribution of the cabinet is small relative to the total when reproducing a stationary
signal (such as the broadband noise used for the measurements), it becomes greater
in the decay of sound, after the loudspeaker unit stops vibrating. This can result in
a blurred, smeared sound, which would produce an unclear perception, and which is
especially disturbing when listening to sounds with some kind of transient or impulsive
components (which is generally the case in most signals).

Based on the frequency response of the loudspeaker unit and the cabinet calculated
with the BEM, the time domain behavior can be analyzed. If the frequency response
is transformed back to the time domain, by means of Inverse Fast Fourier Transforms
(IFFT), it is possible to obtain the impulse response, and hence know how the decay of
the sound from the loudspeaker is. The time domain characteristics are thus obtained
for free-space conditions as well.

6.5.1 Impulse response

The impulse response (IR) of a system is the output when a short transient signal,
mathematically described as a Dirac delta function (or Kronecker, in the case of dis-
crete systems), is presented to the system. A Dirac delta function consists of all
frequencies with equal amplitude. Since all frequencies are presented simultaneously
to the system, it is possible to calculate its frequency response by Fourier transforming
it. Similarly, the impulse response can be calculated from the frequency response by
inverse transforming it.

The impulse response provides useful information about the loudspeaker charac-
teristics. It is a good measure for evaluating the “transparency” of the loudspeaker
to sound. This property is related to how clear the sound of the loudspeaker is per-
ceived, or in other words, to which extent the loudspeaker is modifying the reproduced

56
6. Results 6.5. Behavior of the cabinet in the time domain

signal. Long impulse responses indicate unclear sound reproduction, while on the con-
trary, short impulse responses indicate clear reproduction, with little influence from
the speaker to the sound (flat frequency response, low phase inaccuracy, etc.).

In the frequency response of the Beolab9 cabinet, there are some peaks which in-
dicate a long impulse response. On the other hand, the frequency response of the
loudspeaker unit is rather flat, indicating a relatively short impulse response. If this
is the case, the impulse response of the cabinet could be longer than the loudspeaker
unit, and the sound from the cabinet may become audible. This situation would clearly
compromise the quality of the sound reproduction, and therefore it is of great concern.

Figures 6.14 and 6.15 show the impulse responses of the cabinet and the loud-
speaker unit at 1 m distance in front of the loudspeaker, based on the sound pressure
calculated with the BEM (presented in section 6.3 and 6.4).Figure 6.16 shows the im-
pulse response of the loudspeaker unit measured in the near field of the loudspeaker3 .
The fact that it was measured in the near field helps to minimize the influence of the
cabinet.

Impulse response of the cabinet −Pressure at 1m−

0.05

0.04

0.03

0.02
Sound Pressure (Pa)

0.01

−0.01

−0.02

−0.03

−0.04

−0.05
0 0.05 0.1 0.15 0.2 0.25 0.3
time (s)

Figure 6.14: Impulse response of the Beolab 9 cabinet. The impulse response is based on the
BEM calculations of the sound pressure at 1 m distance. The observation point is
placed in front of the loudspeaker (0o azimuth)

3
The impulse response of the loudspeaker included here was measured at Bang & Olufsen and
kindly provided by them to be used in the present project.

57
6.5. Behavior of the cabinet in the time domain 6. Results

Impulse response of the driver −Pressure at 1m−

2.5

1.5
Sound Pressure (Pa)

0.5

−0.5

−1

−1.5
0 0.05 0.1 0.15 0.2 0.25 0.3
time (s)

Figure 6.15: Impulse response of the Beolab 9 woofer (loudspeaker unit) 4 . The impulse
response is based on the BEM calculations of the sound pressure at 1 m distance. The
observation point is placed in front of the loudspeaker (0o azimuth)

Impulse response of the loudspeaker unit −Measured−

2.5

1.5
Sound Pressure (Pa)

0.5

−0.5

−1

−1.5
0 0.05 0.1 0.15 0.2 0.25 0.3
time (s)

Figure 6.16: Near field measurement of the impulse response of the Beolab 9 woofer.

4
The impulse response has been filtered at 300 Hz due to some noise from the laser measurements
present at 350Hz. The filtering does not influence the results, and it is necessary for a clear visualization
of the data.

58
6. Results 6.5. Behavior of the cabinet in the time domain

Figures 6.14 and 6.15 indicate that the sound from the cabinet is decaying at a
much slower rate than the sound by the unit. However, the amplitude scale of the
figures is not the same. For a better comparison, the impulse responses are plotted in
Figure 6.17 using the same amplitude scale and identical time span (until 300ms).

Impulse response of the driver and the cabinet −Pressure at 1m−

0.1

0.05
(Pa)

−0.05

−0.1
0 0.05 0.1 0.15 0.2 0.25 0.3
time (s)

0.1

0.05
(Pa)

−0.05

−0.1
0 0.05 0.1 0.15 0.2 0.25 0.3
time (s)

Figure 6.17: Comparison of the cabinet (up) and woofer (below) impulse responses. Note
than the amplitude axis is different than from figures 6.13, 6.14 and 6.15

It is apparent from figure 6.17 that the order of magnitude of the loudspeaker unit
and the cabinet is similar. This fact indicates that there is indeed an influence by the
cabinet. Moreover, the main frequency components of the decays are different -it is
apparent that the fluctuations in the case of the cabinet are much more rapid than in
the woofer-. If this would be the case, the problem could be magnified. This is studied
in the following section 6.5.2, where the spectrum of the signals throughout the decay
is analyzed.

59
6.5. Behavior of the cabinet in the time domain 6. Results

6.5.2 Decay of the sound pressure

From the impulse responses studied in section 6.5.1, it seems rather obvious that the
frequency components of the cabinet’s decay are rather different than the unit’s decay.
In other words, the distribution of energy during the decay is different in the case of
the loudspeaker unit and the cabinet.

In order to study the spectral characteristics of the sound throughout the decay,
the impulse responses can be segmented using short time windows, analyzing at differ-
ent instants the spectrum of the decay by means of Fast Fourier Transforms (FFTs).
Intuitively, this can be seen as “splitting” the impulse response in different consecu-
tive sections, to analyze them separately. It is thus a time-frequency representation of
the signal that shows which frequencies are dominating the decay of the sound pressure.

The time window used for the FFT analysis is shown in Ffigure 6.18. It is a co-
sine tapered window also known as Tukey window. This window provides a smooth
transition to zero at the beginning and the end of the window, without modifying too
much the amplitude of the original signal. The length of the time window is of 58ms,
which corresponds to 60samples. The consecutive windows were overlapped half the
size of the window. The signal has been analyzed in intervals of approximately 20ms,
starting from 0and30ms and reaching up to 150ms. For the FFT analysis, each of the
windowed signals is zero-padded to N=256 points (in order to achieve a better spectral
resolution).

cosine tapered window (tukey) −time− cosine tapered window (tukey) −frequency−
40
0

−5 20

−10
0
Amplitude

Mag [dB]

−15
−20

−20

−40
−25

−60
−30

−35 −80
0 0.01 0.02 0.03 0.04 0.05 0.06 0 50 100 150 200 250 300
time [s] f [Hz]

Figure 6.18: Time and frequency response of the cosine tapered window (or Tukey window)
used for the frequency analysis of the decay

In Figure 6.19 and 6.20 a Waterfall plot of the decay of the cabinet and the loud-
speaker unit is presented. The spectrum of the signals at 0, 30, 50, 70, 100 and 150
ms is shown.

60
6. Results 6.5. Behavior of the cabinet in the time domain

Figure 6.19: Waterfall plot of the cabinet’s decay. The spectrum sections are at 0, 30, 50,
70, 100 and 150 ms

Figure 6.20: Waterfall plot of the loudspeaker unit’s decay. The spectrum sections are at 0,
30, 50, 70, 100 and 150 ms

In Figure 6.21 the spectrum of the cabinet and the unit through the decay are
plotted at different instants. In this way, it is easier to compare the SPL at different
frequencies at different moments in time, and therefore get a better understanding of
how the cabinet could be influencing the decay.

61
6.5. Behavior of the cabinet in the time domain 6. Results

Spectrum of the decay of the cabinet and the driver at t=0ms Spectrum of the decay of the cabinet and the louspeaker unit at t=30ms
60
60 Loudsp. unit Loudsp. unit
Cabinet 55 Cabinet

50
50
45

40
40
SPL [dB]

SPL [dB]
35

30
30
25

20 20

15

10 10

5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Spectrum of the decay of the cabinet and the louspeaker unit at t=50ms Spectrum of the decay of the cabinet and the louspeaker unit at t=70ms
60 60
Loudsp. unit Loudsp. unit
55 Cabinet 55 Cabinet

50 50

45 45

40 40
SPL [dB]

SPL [dB]

35 35

30 30

25 25

20 20

15 15

10 10

5 5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Spectrum of the decay of the cabinet and the louspeaker unit at t=100ms Spectrum of the decay of the cabinet and the louspeaker unit at t=130ms
60 60
Loudsp. unit Loudsp. unit
55 Cabinet 55 Cabinet

50 50

45 45

40 40
SPL [dB]

SPL [dB]

35 35

30 30

25 25

20 20

15 15

10 10

5 5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Figure 6.21: Comparison of the cabinet’s -dashed line-, and the loudspeaker unit’s decay-solid
line- at different moments in time. 0 and 30 ms (top), 50 and 70 ms (middle), 100
and 130 ms (bottom)

62
6. Results 6.5. Behavior of the cabinet in the time domain

It is obvious that at 0ms (in the moment of the impulse), the cabinet is radiating
much less than the unit (it is equivalent to the stationary state situation). Nonetheless,
the decay rate of the cabinet is much slower than the one of the unit, and soon it is
taking over the decay process.

If the sequence of the decay is analyzed, it is clear that already at 50ms and until
the end of the decay, the cabinet is dominating in the frequency range from 100 to
250 Hz. At 50ms the SPL of the cabinet is 18 dB higher than the loudspeaker unit.
The energy of the unit’s decay is localized mainly in the low frequency end, at around
50 Hz (at the lower resonance frequency of the woofer). It is clear that the cabinet is
affecting to a great extent the total impulse response of the loudspeaker.

From the results presented here, there is no doubt that the cabinet is playing a
significant role in the radiated sound by the loudspeaker. Not only its contribution is
of the same order of magnitude as the loudspeaker’s unit (contributing to the total),
but in some cases the sound radiated by it is taking over the loudspeaker unit, being
much greater.

Moreover, the different frequency components of both decays may produce a differ-
ent outcome regarding the subjective perception. In terms of loudness sensation, given
the same SPL, the human hearing is more sensitive to the 100-250Hz frequency range
than to the 50 Hz range. This can be illustrated in figure 6.22 by the equal loudness
level contours for pure tones5 (the lines show at which SPL, pure tones of different
frequencies are perceived equally loud. For example: a 50 Hz pure tone of 50 dB is as
perceived as loud as a 30 dB 150 Hz tone) -ISO 226 (2003) [14] [11]-.

Figure 6.22: Equal loudness level contours. Sounds presented binaurally from the frontal
direction. The absolute threshold curve is also shown - from ISO 226 (2003) [14]

5
The curves in figure 6.22 cannot be used directly to predict the subjective loudness of complex
signals, because they do not reflect the effect of masking, temporal perception, etc.

63
6.5. Behavior of the cabinet in the time domain 6. Results

The results presented in this section so far, indicate that the cabinet of the loud-
speaker is clearly influencing the total sound radiated by the loudspeaker. Figure 6.23
shows the decay of the unit and the cabinet in a time-frequency representation, where
the SPL radiated by each is plotted as a surface. The figure illustrates how the cabinet
is dominant during most of the decay process for the frequency range above 100 Hz
(in such a plot it is easy to visualize which contribution of the two is greater).

The results obtained, suggest that even if the disturbance from the cabinet is not
clearly apparent in the frequency steady-state response, such disturbance can become
apparent in the time domain, provided that the damping of the cabinet’s natural fre-
quencies is low enough (long impulse response).

Figure 6.23: Surface plot illustrating the cabinet’s (red) and the loudspeaker unit’s decay
(green)

Throughout the section, the time domain characteristics of the loudspeaker have
been examined, and the contribution of the cabinet has been evaluated. Nevertheless,
the results studied so far in section 6.5, only deal with the case in which the observa-
tion point is placed in the front of the loudspeaker (0o azimuth). This is not a very
“critical” case, because in this position the influence of the loudspeaker unit is maxi-
mized, while the cabinet’s influence is minimized. It could be said it is the “best” of
the possible cases. In the following sections, some more critical cases are studied.

64
6. Results 6.6. Radiation at the back of the cabinet (time domain)

6.6 Radiation at the back of the cabinet (time domain)

It has been seen throughout the results of the project that the sound radiation from
the cabinet is greater in the back than in the front of the loudspeaker. When look-
ing at the time domain characteristics in section 6.5, only the position in front of the
loudspeaker has been investigated. In this section, the time domain characteristics of
the radiation of the cabinet and the unit are considered, when the observation point
is placed in the back of the cabinet.

Impulse response of the cabinet −Pressure at 1m−

0.05
0.04 at the back (180º)
0.03
Sound Pressure (Pa)

0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0 0.05 0.1 0.15 0.2 0.25 0.3
0.05
0.04
at the front (0º)
0.03
Sound Pressure (Pa)

0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0 0.05 0.1 0.15 0.2 0.25 0.3
time (s)

Figure 6.24: Comparison of the impulse responses of the cabinet in the back (top) and in the
front (bottom) of the loudspeaker

The impulse response in the back of the cabinet is shown in Figure 6.24. It is com-

65
6.6. Radiation at the back of the cabinet (time domain) 6. Results

pared to the impulse response in the front of the cabinet (the same as in figure 6.14).
It is obvious that the amplitude of the impulse response is greater at the back, and
the decay is somewhat longer, as would be expected from the spectral data previously
analyzed.

In a similar way as in section 6.5.2, the decay of the cabinet can be looked at in a
frequency-time representation, to see how the spectrum is changing over time. In this
way, the decay of the sound radiated by cabinet at the back and at the front of the
loudspeaker can be compared. Figure 6.25 shows such a comparison.

Figure 6.25: Comparison of the cabinet’s SPL decay in the back and at the front of the loud-
speaker

Figure 6.25 confirms the result that the magnitude of the sound radiated by the
cabinet is greater in the back of the loudspeaker than in the front during the decay of
the sound. Hence, considering that the unit is radiating less at the back, the cabinet
is expected to play an even more important role in the decay of the overall sound.

Figure 6.26 shows a comparison between the SPL from the cabinet and from the
unit at the back of the loudspeaker, at different instants of the sound pressure decay.

66
6. Results 6.6. Radiation at the back of the cabinet (time domain)

Spectrum of the decay of the cabinet and the driver at t=0ms Spectrum of the decay of the cabinet and the louspeaker unit at t=30ms
60 60
Loudsp. unit Loudsp. unit
55 Cabinet 55 Cabinet

50 50

45 45

40 40
SPL [dB]

SPL [dB]
35 35

30 30

25 25

20 20

15 15

10 10

5 5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Spectrum of the decay of the cabinet and the louspeaker unit at t=50ms Spectrum of the decay of the cabinet and the louspeaker unit at t=70ms
60 60
Loudsp. unit Loudsp. unit
55 Cabinet 55 Cabinet

50 50

45 45

40 40
SPL [dB]

SPL [dB]

35 35

30 30

25 25

20 20

15 15

10 10

5 5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Spectrum of the decay of the cabinet and the louspeaker unit at t=100ms Spectrum of the decay of the cabinet and the louspeaker unit at t=120ms
60 60
Loudsp. unit Loudsp. unit
55 Cabinet 55 Cabinet

50 50

45 45

40 40
SPL [dB]

SPL [dB]

35 35

30 30

25 25

20 20

15 15

10 10

5 5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Figure 6.26: Comparison of the cabinet’s -dashed line-, and the unit’s decay -solid line- at
different instants in time. 0 and 30 ms (top), 50 and 70 ms (middle), 100 and 120
ms (bottom). The observation point is in the back of the loudspeaker (180o azimuth)

67
6.6. Radiation at the back of the cabinet (time domain) 6. Results

It can be seen that from a very early stage of the decay (35ms) the amplitude of
the cabinet is greater than the amplitude of the unit, and very soon the cabinet is
radiating much more than the unit. Already after 50ms, the SPL of the cabinet at 150
Hz is 20 dB greater than the unit’s SPL. It is clear that the cabinet is very dominant in
the decay of the sound (it is a very short period of time, but there is a very significant
amplitude difference). This dominance is apparent in figure 6.27, which shows in a
surface plot the SPL by the cabinet (in red) and the unit (in green) during the decay.
The results shown in the figure make it easy to visualize the great contribution from
the cabinet to the total sound radiated by the loudspeaker.

Figure 6.27: Surface plot illustrating the decay of the of the cabinet (red) and the loudspeaker
unit (green) at the back of the cabinet

From the results shown in this section, it can be concluded that the greater contri-
bution of the cabinet towards the back of the loudspeaker is also apparent in the time
domain. This confirms that the problem of the cabinet being audible is found to a
greater extent at the back part of the loudspeaker, where the efficiency of the cabinet’s
radiation is higher. It is thus clear that the problem is even more critical in the back
region of the loudspeaker.

Generally, the rest of the conclusions that can be drawn from this section are similar
to those drawn in sections 6.5.2 and 6.5.3, when the observation point was placed in
the front of the speaker. The main difference is that in this case, the weight of the
radiation from the cabinet compared to the loudspeaker unit is even greater.

68
6. Results 6.7. Radiation from the bottom of the cabinet

6.7 Radiation from the bottom of the cabinet

The bottom of the loudspeaker’s cabinet is also vibrating, with a particularly high
amplitude at the lower frequency range (below 100 Hz). Such vibration levels could
in principle play a significant role in the radiation of sound by the cabinet. However,
the bottom vibration is not affecting too significantly the general characteristics of the
sound radiated by the rest of the cabinet.

The Beolab 9 loudspeaker is designed to be placed on the floor. In such a situation,

the bottom of the cabinet is not directly exposed to the medium, and not radiating
directly into it. Thus, it is interesting to consider separately the case where the bottom
is radiating directly into the medium, to understand better how it affects the sound
radiated by the cabinet.

Besides, it should be noted that in all the pressure measurements carried out
throughout the project, the loudspeaker was standing on a platform (and not sus-
pended in air, which would be a free-space condition). Thus, it is not clear that
including the vibration from the bottom in the calculations would model the measure-
ment conditions more accurately. Moreover, it was verified in section 6.4, that the
model considering the bottom of the cabinet not radiating to the medium, is a good
approximation to the measurement conditions. Therefore, after such verification, and
to avoid misleading results, the radiation from the bottom of the cabinet is considered
separately, to evaluate the possible contribution from it to the total cabinet’s radiation.

In this section, the loudspeaker is modeled as if it was suspended in air, and as

if the bottom of the cabinet was radiating normally into the medium. Based on an
initial investigation, the bottom is expected to play an influencing role only in the low
frequency end. Therefore, the resolution of the measurement mesh was designed to be
usable until 200 Hz (in order to economize the measurement and calculation time).

6.7.1 Frequency domain (Bottom radiation)

The BEM calculations were carried out both including the vibration from the bottom
and without including it. The sound pressure radiated in both cases is very similar.
The order of magnitude does not change generally, nor the spatial profile of how the
sound is radiated into the medium. The most important changes can be seen at low
frequencies, where the cabinet is vibrating as a whole, following a pulsating mode
pattern. However, the general sound radiation by the cabinet with and without the
contribution from the bottom is very similar.

In the following figure, the spectrum of the SPL radiated by the cabinet is shown.
The two cases of the cabinet with the bottom vibrating and considered as rigid are
shown for comparison.

69
6.7. Radiation from the bottom of the cabinet 6. Results

SPL radiated from the cabinet with and without radiation from the bottom
40

35

30
SPL [dB]

25

20

15
Vibrating bottom
Rigid bottom

10
2
10
log(f) [Hz]

Figure 6.28: Comparison of the sound pressure radiated by the cabinet when the bottom is
rigid (blue) and when it is vibrating (black)

It can be seen from Figure 6.28 that the radiation from the bottom is mainly af-
fecting the results below 90 Hz. Above 90 Hz, both curves follow a very similar pattern
. The vibration of the bottom is not changing the fundamental sound radiation char-
acteristics from the cabinet. However, there are some relevant differences which worth
to be mentioned.

Initially, the SPL at low frequencies is considerably greater (the bottom vibration
contributes to a greater radiation by the zeroth order mode -at 85 Hz-, and below 85
Hz). Also the level is greater at some other natural frequencies of the cabinet -such as
the 150 Hz one-). Furthermore, in most of the frequencies the overall SPL is slightly
higher (about 1 dB more) than before.

It is as well very relevant the fact that some of the main peaks of the frequency
response of the cabinet (such as 84 Hz and 150 Hz) are narrower. Narrower frequency
peaks would reveal a longer decay time of the modes, which could emphasize the sound
from the cabinet in the total sound radiated by the speaker. The best way to find out
to investigate it is by looking at the time domain response of the cabinet, as has been
done in previous sections.

70
6. Results 6.7. Radiation from the bottom of the cabinet

6.7.2 Impulse response (bottom radiation)

It has been seen that the influence of the bottom radiation in the frequency domain is
not too severe. However, the narrowing of some peaks in the frequency response and
the increased sound pressure of the low frequency end, are expected to play a role in
the radiation of the cabinet. In an analogous way as done in previous sections, the
impulse response was calculated from the spectrum of the SPL 1 meter away. The
Impulse response (with the bottom of the cabinet radiating in free-space) is shown in
Figure 6.29

Impulse response of the cabinet −Pressure at 1m−

0.08

0.06

0.04
Sound Pressure (Pa)

0.02

−0.02

−0.04

−0.06

−0.08

−0.1
0 0.1 0.2 0.3 0.4 0.5
time (s)

Figure 6.29: Impulse response of the loudspeakers cabinet (where the bottom has been assumed
to be radiating into the free-space domain)

If the impulse responses of the cabinet with and without the contribution from
the bottom (the bottom radiating or not radiating to the medium) are compared to
each other (see Figure 6.14), it is apparent that the impulse responses are similar,
but the response is slightly longer and of noticeably higher amplitude if the bottom is
contributing. This result could be expected, based on the spectrum shown in figure
6.28 -especially due to the low frequency boost-.

6.7.3 Decay of the sound pressure (Bottom radiation)

The decay of the sound pressure of the cabinet with the bottom radiating to the
medium, can be calculated, and plotted at different instants of time, compared to the
decay of the loudspeaker unit. The results are shown in Figure 6.30 (next page)

71
6.7. Radiation from the bottom of the cabinet 6. Results

Spectrum of the decay of the cabinet and the driver at t=0ms Spectrum of the decay of the cabinet and the louspeaker unit at t=30ms
60
60 Loudsp. Unit Loudsp. Unit
Cabinet 55 Cabinet

50
50
45

40
40
SPL [dB]

SPL [dB]
35

30
30
25

20 20

15

10 10

5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Spectrum of the decay of the cabinet and the louspeaker unit at t=50ms Spectrum of the decay of the cabinet and the louspeaker unit at t=70ms
60 60
Loudsp. Unit Loudsp. Unit
55 Cabinet 55 Cabinet

50 50

45 45

40 40
SPL [dB]

SPL [dB]

35 35

30 30

25 25

20 20

15 15

10 10

5 5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Spectrum of the decay of the cabinet and the louspeaker unit at t=100ms Spectrum of the decay of the cabinet and the louspeaker unit at t=120ms
60 60
Loudsp. Unit Loudsp. Unit
55 Cabinet 55 Cabinet

50 50

45 45

40 40
SPL [dB]

SPL [dB]

35 35

30 30

25 25

20 20

15 15

10 10

5 5
0 50 100 150 200 250 300 0 50 100 150 200 250 300
f [Hz] f [Hz]

Figure 6.30: Comparison of the cabinet’s -dashed line-, and the loudspeaker unit’s decay -
solid line- at different instants in time. 0 and 30 ms (top), 50 and 70 ms (middle),
100 and 120 ms (bottom). The cabinet is considered to be radiating in full free-space,
and the radiation from the bottom of the cabinet is included.

72
6. Results 6.7. Radiation from the bottom of the cabinet

It is clear from Figure 6.30 the fact that the cabinet is taking over an important
part of the decay of the sound. The results in this case are very similar to the case
where the radiation from the bottom is not considered (see figure 6.21), with the fun-
damental difference than in this case, the cabinet is much more present in the decay
at the lower frequency range (around 50-80 Hz ). Nevertheless, the results are pretty
much the same in both cases.

The decay of the cabinet and the loudspeaker unit are plotted in the following
figure, showing the SPL of both, and illustrating which one of the two is greater.

Figure 6.31: Surface plot illustrating the decay of the of the cabinet (red) and the loudspeaker
unit (green) when the radiation from the cabinet into the domain is considered.

The conclusions that can be drawn from the results presented in this section, are
that the bottom of the cabinet is not influencing too significantly the radiation from
the cabinet. It has only an important contribution in the very low frequency range
(50-100 Hz), but otherwise its influence is not too severe. However, it is preferable to
study it separately, since the measurement setup is not emulating a complete free-space
condition (the bottom of the cabinet is not radiating to the medium directly).

In any case, it seems clear that the contribution from the bottom of the cabinet
boosts the low frequency end of the sound pressure radiated by it.

73
6.7. Radiation from the bottom of the cabinet 6. Results

74
Chapter 7

Conclusions

The Boundary Element Method (BEM) has been proved to be an adequate tool for
studying the sound radiation by a loudspeaker cabinet. The BEM is particularly useful
in this case study due to the fact that it is difficult to measure the sound pressure by
the cabinet separately, because of the simultaneous radiation by the unit.

The normal velocity of the cabinet’s vibration has been examined, and some de-
flection shapes investigated. The maximum velocity levels were measured at 85, 150
and 182 Hz, in the front side of the speaker, close to the woofer. At low frequencies
the cabinet is vibrating as a whole, while at higher frequencies the deflection shapes
become more complicated.

The most significant radiation from the cabinet is in the frequency range between
80 Hz and 200 Hz. The SPL is particularly high at 150 Hz, 85 Hz and 185 Hz. The
low frequencies seem to be radiating more due to the higher vibration levels and the
higher radiation efficiency.

While the sound radiation from the cabinet is greater in the back of the loud-
speaker, the unit’s radiation is weaker there. The result of this, is in an increased
influence from the cabinet towards the back of the loudspeaker.

Studying the frequency steady-state response of the loudspeaker, no important in-

fluence from the cabinet was found in the front direction. To the contrary, in the back
side of the loudspeaker, there is a clear influence from the cabinet. The frequency
response at the back reveals some noticeable peaks at 140, 150, 185 and 220 Hz, which
correspond to the cabinet’s natural frequencies1 . However, such influence does not
1
Such peaks are not present in the units radiation, but they are indeed present in the total frequency
 7. Conclusions

seem too significant in terms of the SPL.

The results obtained, suggest that even if the disturbance from the cabinet is not
clearly apparent in the steady-state frequency response, such disturbance can become
apparent in the time domain, provided that the damping of the cabinet’s natural fre-
quencies is low enough (long impulse response).

In the present study , although the contribution from the cabinet to the total ra-
diated pressure is not too apparent in the frequency response, when investigating the
time domain, a very significant influence from the cabinet to the total radiation by the
speaker has been found. The cabinet’s impulse response is noticeably longer than the
unit’s.

The study revealed that just after a short period of time, the sound radiated by
the cabinet is dominating in the decay of the loudspeaker’s sound. Despite that the
SPL generated by the unit is much higher than that of the cabinet in a steady-state
situation (i.e. when reproducing a stationary signal, such as broadband noise), as soon
as the signal ceases, the sound pressure of the unit drops, while the cabinet keeps
“ringing” for a longer time. At the natural frequencies of the cabinet, its SPL was
found to be much greater than the unit’s during most of the decay process. Also in
the time domain, the influence of the cabinet is more severe in the back than in the
front.

An influence by the cabinet has been detected, especially apparent in the time
domain, in the frequency range between 80 and 180 Hz. Such an influence would in
principle compromise the quality of the sound reproduction. These results agree with
the subjective testing that initially indicated the problem in the developing process of
the loudspeaker.

The radiation from the bottom of the cabinet plays an important role at low fre-
quencies, especially below 100 Hz. However, it does not change the fundamental sound
radiation characteristics of the rest of the cabinet at higher frequencies.

The cabinet’s vibration is changing non-linearly depending on the input level to

the system. This non-linear behavior indicates that for increased reproduction levels,
the cabinet’s relative contribution might be greater, thus becoming even more audible.

The results obtained from this study, agrees fundamentally with those in refs. [9],
[2] and [16]. In this project, the decay of the sound has also been investigated using
a time-frequency representation, that reveals a different spectral distribution of the
energy in the case of the cabinet and the unit throughout the decay process.

response of the loudspeaker, indicating an apparent influence of the cabinet.

76
7. Conclusions

Recommendations for future work

In the present study, the Boundary Element Method calculations have been based on
vibration measurements performed on the cabinet. It would be interesting to study
the possibility of basing the BEM calculations on a FEM (Finite Element Method)
study instead of measurements. By doing so, the cabinet’s radiation could not only be
calculated, but also predicted before any prototype would have to be manufactured.
This approach would have the drawback of requiring a very thoroughly detailed and
“expensive” modeling.

This project focused on the sound radiation by the loudspeaker system, and the
influence of the cabinet. The vibration of the system was just briefly examined. A
greater insight into it could be very revealing. Basically, it would be useful to study
in greater detail how the structural properties and shape of the cabinet and the loud-
speaker system influence the sound radiation.

It should be noted that throughout the study, free-field conditions were assumed.
However, it could be of interest to study how would a different environment affect
the results. For instance, study the effect of placing the loudspeaker inside a listening
room (in a usual listening environment, with reflections from walls and ceiling, for
instance). Based on the measurements on the cabinet and the unit, it would be pos-
sible to model such a situation by introducing some reflecting surfaces. Such a study
would reveal if the effect of reflections would increase or not the influence of the cabinet.

The study carried out evaluated the contribution of the cabinet only based on
measurements and calculations. Such evaluation does not take into account subjective
matters of human perception. It should be noted that this type of study is more of
a quantitative approach, rather than a qualitative one. Therefore, it would be very
interesting to relate it with a subjective analysis of the influence of sound radiated by
loudspeaker cabinets.

77
 7. Conclusions

78
Bibliography

[1] FEM-BEM Notes. Bioengineering Institute of the University of Auckland, New

Zealand, 2005.

[2] Kevin J. Bastyr and Dean E. Capone. On the acoustics radiation from a loud-
speaker’s cabinet. J. Audio Eng.Soc., 51:234–243, April 2003.

[3] Finn Jacobsen and D. Bao. Acoustic decay measurements with a dual channel
analyzer. Journal of Sound and Vibration, 115(3):521–537, 1987.

[4] Finn Jacobsen and Peter M Juhl. Radiation of sound. Acoustic Technology
(Technical University of Denmark), Physics Department (Uiversity of Southern
Denmark), August 2006.

[5] Peter M. Juhl. The Boundary Elemet Method for Sound Field Calculations. PhD
thesis, Technical University of Denmark (DTU), August 1993.

[6] Peter M. Juhl and Vicente Cutanda Henriquez. OpenBEM -Open source Matlab
codes for the Boundary Element Method-. http://www.openbem.dk/, -.

[7] Matti Karjalainen, Veijo Ikonen, Poju Antsalo, Panu Maijala, Lauri Savioja, Antti
Suutala, and Seppo Pohjolainen. Comparison of numerical simulation models and
measured low-frequency behaviour of loudspeaker enclosures. Journal of the Audio
Engineering Society (JAES), 49:1148–1165, 2001.

[8] Stephen Kirkup. The Boundary Element Method in Acoustics. www.boundary-

element-method.com, 1998/2007.

[9] Stanley Lipshitz, Michael Heal, and John Vanderkooy. An investigation of sound
radiation by loudspeaker cabinets. J. Audio engineering Society (JAES), 19-22:F–
4, February 1991. Audio rsearch Group, Univ. of Waterloo, Ontario, Canada.

[10] Yu Luan. Modeling structural acoustic properties of the beolab 9 loudspeaker.

Master’s thesis, Acoustic Technology (AT), Technical University of Denmark
(DTU), January 2008.

[11] Brian C. J. Moore. An Introduction to the Psychology of Hearing. Academic

Press, 2003.
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[12] Philip M. Morse and K. Uno Ingard. Theoretical Acoustics. MIT, 1968.

[13] Mogens Ohlrich. Structure Borne Sound and Vibration. Acoustic Technology,
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[14] Torben Poulsen. Acoustic communication. hearing and speech (v2.0). Acoustic
Technology, Techical University of Denmark (DTU), Denmark, August 2005.

[15] Arnold Sommerfeld. Partial Differential Equations in Physics. Cambridge Uni-

versity Press, New York, 1949.

[16] Peter W. Tappan. Loudspeaker enclosure walls. Journal of the Audio Engineering
Society (JAES), 10:224–231, July 1962.

[17] F.E. Toole and S.E. Olive. The modification of timbre by resonances: Perception
and measurement. J. Audio Engineering Society., 36:122–142, March 1988.

80
List of Figures

2.1 Simple Boundary Problem Sketch. In a sound radiation problem the

domain is governed by the wave equation. In this particular project
the outer boundary condition is the Sommerfield radiation condition
-Infinity-, and the inner boundary condition is the velocity of the body . 11

3.1 Measurement mesh of the Beolab 9. The velocity was measured at every
node of the mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Example of the deflection shape of a conical cabinet at 309 Hz. Image
from [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 Test Box used for the verification of the procedure . . . . . . . . . . . . 24

4.2 Mesh of the test box used for the BEM calculations. The maximum
element length is of 16cm . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Block diagram of the measurement . . . . . . . . . . . . . . . . . . . . . 26
4.4 Image of the measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.5 Comparison between the BEM calculation and the measured SPL . . . . 27

5.1 Beolab9. Side view (left). Top view (centre). View without the screen
cover (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Beolab9 simplified geometry in which the mesh was based . . . . . . . . 30
5.3 Meshing of the Beolab 9 used for the BEM calculations . . . . . . . . . 31
5.4 Testing of the mesh. Sound pressure produced by a monopole placed
at the origin. The figure compares the sound pressure calculated ana-
lytically and with the BEM using the loudspeaker mesh, for different
frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.5 Some pictures of the vibration measurement . . . . . . . . . . . . . . . . 34
5.6 Block diagram of the vibration measurement . . . . . . . . . . . . . . . 34

6.1 Acceleration on the cabinets surface at all the measurement positions . 40

6.2 Field points at which the sound pressure was calculated . . . . . . . . . 42
6.3 SPL radiated by the loudspeakers cabinet, at 1m distance and 0.35 m
height, around the azimuth angle (polar angle around the loudspeaker) . 43
LIST OF FIGURES LIST OF FIGURES

6.4 Phase of the pressure radiated by the loudspeakers cabinet, at 1m dis-

tance and 0.35 m height around the azimuth angle. The isobars repre-
sent a π/4 phase change . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.5 SPL radiated by the loudspeakers cabinet, along the polar angle at 1m
distance. 85 Hz (solid), 150Hz (dashed) and 182 Hz (dotted) . . . . . . 44
6.6 SPL radiated by the loudspeakers cabinet, at 1m distance, compared to
the total radiated pressure by the loudspeaker. The top figure shows
the radiation in the front direction -0o -, the medium figure shows the
lateral radiation -90o -, and the bottom figure the back radiation -180o - . 45
6.7 SPL radiated by the loudspeakers cabinet, at 1m distance, compared to
the total radiated pressure by the loudspeaker. The front (blue), lateral
(green) and back (black) radiation are plotted together . . . . . . . . . . 46
6.8 Comparison of measurements and BEM calculations of the total radi-
ation from the loudspeaker. The observation point is in front of the
loudspeaker at 1 m distance and 0.35 m height . . . . . . . . . . . . . . 48
6.9 Comparison of measurements and BEM calculations of the total radia-
tion from the loudspeaker. The observation point is in the back of the
loudspeaker at 1 m distance . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.10 SPL of the loudspeaker unit compared to the total SPL (cabinet + unit)
in front of the loudspeaker calculated with the BEM. The observation
point is at 1 m distance (at the front) . . . . . . . . . . . . . . . . . . . 50
6.11 SPL of the loudspeaker unit compared to the total SPL (cabinet + unit)
calculated with the BEM. The observation point is at 1 m distance (in
the back). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.12 SPL of the loudspeaker unit compared to the total SPL (from the BEM
calculations and from the measurements). The observation point is at
1 m distance (in the back). . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.13 SPL radaited by the loudspeaker unit compared to the total and to the
cabinet. The observation point is at 1 m distance (in the back). . . . . . 52
6.14 Impulse response of the Beolab 9 cabinet. The impulse response is based
on the BEM calculations of the sound pressure at 1 m distance. The
observation point is placed in front of the loudspeaker (0o azimuth) . . . 55
6.15 Impulse response of the Beolab 9 woofer (loudspeaker unit) 2 . The im-
pulse response is based on the BEM calculations of the sound pressure at
1 m distance. The observation point is placed in front of the loudspeaker
(0o azimuth) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.16 Near field measurement of the impulse response of the Beolab 9 woofer. 56
6.17 Comparison of the cabinet (up) and woofer (below) impulse responses.
Note than the amplitude axis is different than from figures 6.13, 6.14
and 6.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.18 Time and frequency response of the cosine tapered window (or Tukey
window) used for the frequency analysis of the decay . . . . . . . . . . . 58
6.19 Waterfall plot of the cabinet’s decay. The spectrum sections are at 0,
30, 50, 70, 100 and 150 ms . . . . . . . . . . . . . . . . . . . . . . . . . . 59

82
LIST OF FIGURES LIST OF FIGURES

6.20 Waterfall plot of the loudspeaker unit’s decay. The spectrum sections
are at 0, 30, 50, 70, 100 and 150 ms . . . . . . . . . . . . . . . . . . . . 59
6.21 Comparison of the cabinet’s -dashed line-, and the loudspeaker unit’s
decay-solid line- at different moments in time. 0 and 30 ms (top), 50
and 70 ms (middle), 100 and 130 ms (bottom) . . . . . . . . . . . . . . 60
6.22 Equal loudness level contours. Sounds presented binaurally from the
frontal direction. The absolute threshold curve is also shown - from ISO
226 (2003) [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.23 Surface plot illustrating the cabinet’s (red) and the loudspeaker unit’s
decay (green) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.24 Comparison of the impulse responses of the cabinet in the back (top)
and in the front (bottom) of the loudspeaker . . . . . . . . . . . . . . . 63
6.25 Comparison of the cabinet’s SPL decay in the back and at the front of
the loudspeaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.26 Comparison of the cabinet’s -dashed line-, and the unit’s decay -solid
line- at different instants in time. 0 and 30 ms (top), 50 and 70 ms
(middle), 100 and 120 ms (bottom). The observation point is in the
back of the loudspeaker (180o azimuth) . . . . . . . . . . . . . . . . . . 65
6.27 Surface plot illustrating the decay of the of the cabinet (red) and the
loudspeaker unit (green) at the back of the cabinet . . . . . . . . . . . . 66
6.28 Comparison of the sound pressure radiated by the cabinet when the
bottom is rigid (blue) and when it is vibrating (black) . . . . . . . . . . 68
6.29 Impulse response of the loudspeakers cabinet (where the bottom has
been assumed to be radiating into the free-space domain) . . . . . . . . 69
6.30 Comparison of the cabinet’s -dashed line-, and the loudspeaker unit’s
decay -solid line- at different instants in time. 0 and 30 ms (top), 50 and
70 ms (middle), 100 and 120 ms (bottom). The cabinet is considered
to be radiating in full free-space, and the radiation from the bottom of
the cabinet is included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.31 Surface plot illustrating the decay of the of the cabinet (red) and the
loudspeaker unit (green) when the radiation from the cabinet into the
domain is considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.1 Velocity of the cabinet, measured in the reference point, for different
input levels (broadband noise was used). . . . . . . . . . . . . . . . . . . 84
A.2 Amplitude transfer function of the velocity of the loudspeaker cabinet
at 84 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.3 Amplitude transfer function of the velocity of the loudspeaker cabinet
at 150 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.4 Amplitude transfer function of the velocity of the loudspeaker cabinet
at 185 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A.5 Amplitude transfer function of the velocity of the loudspeaker cabinet
at 220 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

83
LIST OF FIGURES LIST OF FIGURES

A.6 Normalized velocity of the cabinet and the unit when reproducing a
tone of 80 Hz. The overtones are a result of the non-linear distortion
processes in the unit and cabinet. . . . . . . . . . . . . . . . . . . . . . . 87

B.1 Magnitude of the velocity around the circumference of the cabinet at 83

Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
B.2 Magnitude of the velocity around the circumference of the cabinet at 99
Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.3 Magnitude of the velocity around the circumference of the cabinet at
140 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.4 Magnitude of the velocity around the circumference of the cabinet at
182 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.5 Magnitude of the velocity around the circumference of the cabinet at
210 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

D.1 SHELL63 Elastic Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

84
Appendix A

Non-linear vibration of the

cabinet

An electrical input level of 100 mV amplitude has been used in all the measurements of
the Beolab 9. Such input level (using broadband noise) was chosen because it produces
a SPL of around 80 dB at 1m distance, which a reasonable listening level. However, it
is important to examine the influence of using different input levels on the loudspeaker
behavior.

In principle, the higher the input level, the more the cabinet vibrates (the vibration
of the loudspeaker unit increases and more vibration is transmitted to the cabinet).
Thus, the sound radiated by it is also expected to be greater1 . Just to get an idea of
this phenomenon, the vibration of the cabinet was measured at different input levels.
The vibration was measured in the reference point (which is situated just above the
woofer). This point was chosen because it is a representative point of the vibration of
the cabinet, since all the natural frequencies are present and the vibration levels are
rather high.

1
The SPL radiated by the cabinet at different levels was not calculated, since it would involve to
repeat all the measurements and calculations at different input levels.
 A. Non-linear vibration of the cabinet

Velocity of the reference point for different input levels

−2
10
25 mV
50 mV
100 mV
200 mV
−3
10 300 mV
400 mV
velocity[m/s]

−4
10

−5
10

−6
10
50 100 150 200
freq [Hz]

Figure A.1: Velocity of the cabinet, measured in the reference point, for different input levels
(broadband noise was used).

Figure A.1 shows the spectrum of the vibration velocity measured at the reference
point of the cabinet for different input levels ranging from 25 mV (which produces a
very low SPL) to 400 mV (which produces a very high SPL).

It is obvious from the results that the vibration of the cabinet is not linear. De-
pending on the input level of the vibration, the peaks of the frequency response are
shifting considerably. This is also apparent on the amplitude transfer function of the
vibration velocity, at different frequencies, as shown in figures A.2 to A.5.

86
A. Non-linear vibration of the cabinet

−3 velocity vs Input level transfer function at f=83Hz

x 10

0.8
velocity[m/s]

0.6

0.4

0.2
83Hz
linear reference
0
0 50 100 150 200 250 300 350 400
input level [mV]

Figure A.2: Amplitude transfer function of the velocity of the loudspeaker cabinet at 84 Hz.

−4 velocity vs Input level transfer function at f=150Hz

x 10

1.8

1.6

1.4
velocity[m/s]

1.2

0.8

0.6

0.4
150Hz
0.2 linear reference

0 50 100 150 200 250 300 350

input level [mV]

Figure A.3: Amplitude transfer function of the velocity of the loudspeaker cabinet at 150 Hz.

87
 A. Non-linear vibration of the cabinet

−5 velocity vs Input level transfer function at f=185Hz

x 10

14

12

10
velocity[m/s]

2 185Hz
linear reference

0 50 100 150 200 250 300 350

input level [mV]

Figure A.4: Amplitude transfer function of the velocity of the loudspeaker cabinet at 185 Hz.

−4 velocity vs Input level transfer function at f=220Hz

x 10

0.9

0.8

0.7
velocity[m/s]

0.6

0.5

0.4

0.3

0.2

0.1 220Hz
linear reference
0
0 50 100 150 200 250 300 350 400
input level [mV]

Figure A.5: Amplitude transfer function of the velocity of the loudspeaker cabinet at 220 Hz.

88
A. Non-linear vibration of the cabinet

It is apparent from figures A.2 to A.5 that the amplitude transfer function is clearly
non-linear, especially for the lower natural frequencies. At higher frequencies, there
is a certain non-linear characteristic, but less severe than at lower frequencies. It can
be seen from the figures that the vibration levels is increasing non-linearly with the
input level. Thus, the higher the input level, the relatively higher the vibration of the
cabinet will be.

However, it is reasonable to find a nonlinear vibration in the cabinet, especially con-

sidering that the loudspeaker unit is already vibrating nonlinearly (distortion, etc.).
To evaluate this phenomenon, a simple measurement was carried out. A tone of 80
Hz was presented, and both the vibration of the loudspeaker unit and the cabinet was
measurement (the loudspeaker unit was measured with a laser, and the cabinet using
an accelerometer). The distortion from both systems can therefore be compared, and
verify if the cabinet distortion is caused by the previous distortion of the unit, or if
the cabinet is also distorting by itself.

normalized velocity of a tone of 80 Hz

0
Lousp. unit
Cabinet

−20

−40
20log(V/Vmax)

−60

−80

−100

100 150 200 250 300 350 400 450

freq (Hz)

Figure A.6: Normalized velocity of the cabinet and the unit when reproducing a tone of 80
Hz. The overtones are a result of the non-linear distortion processes in the unit and
cabinet.

The background noise present in the “floor” of the cabinets vibration is due to the
fact that it was measured with accelerometers (while the unit was measured with the
laser). The level of the figures is normalized to the amplitude of each of the main tones
(at 80 Hz). In this way, the level of the overtones is relative to the main vibration of

89
 A. Non-linear vibration of the cabinet

the input tone.

Form Figure A.6 it is apparent that the cabinet is also contributing to the dis-
tortion. The level of the harmonics of the cabinet is greater than the level of the
harmonics of the unit. At low frequencies (160 Hz and 240 Hz), the distortion in the
harmonics of the cabinet is about 15 to 20 dB higher than the those of the unit.

It can be concluded from this section that the relative contribution from the cab-
inet to the radiated sound would increase with greater input levels. In other words,
the louder the signal reproduced by the loudspeaker, the greater the contribution by
the cabinet. Therefore, at high sound pressure levels, the problem of the cabinet being
audible would be magnified.

90
Appendix B

Deflection shapes of the cabinet

The appendix shows the magnitude of the velocity in the surface of the cabinet at a
height of 36 cm from the ground. The plots show the magnitude of the velocity around
the circunference of the cabinet. In the the nodes and antinodes can be identified, to
get an idea of how the deflection shapes look like.

−4 Velocity magnitude on the surface of the cabinet [83 Hz]

x 10

2
magnitude(m/s)

0
−4 −3 −2 −1 0 1 2 3 4
polar angle (aprox.)

Figure B.1: Magnitude of the velocity around the circumference of the cabinet at 83 Hz
 B. Deflection shapes of the cabinet

−4 Velocity magnitude on the surface of the cabinet [99 Hz]

x 10
1.2

0.8
magnitude(m/s)

0.6

0.4

0.2

0
−4 −3 −2 −1 0 1 2 3 4
polar angle (aprox.)

Figure B.2: Magnitude of the velocity around the circumference of the cabinet at 99 Hz

−5 Velocity magnitude on the surface of the cabinet [140 Hz]

x 10
10

7
magnitude(m/s)

1
−4 −3 −2 −1 0 1 2 3 4
polar angle (aprox.)

Figure B.3: Magnitude of the velocity around the circumference of the cabinet at 140 Hz

92
B. Deflection shapes of the cabinet

−5 Velocity magnitude on the surface of the cabinet [182 Hz]

x 10
6

4
magnitude(m/s)

0
−4 −3 −2 −1 0 1 2 3 4
polar angle (aprox.)

Figure B.4: Magnitude of the velocity around the circumference of the cabinet at 182 Hz

−5 Velocity magnitude on the surface of the cabinet [210 Hz]

x 10
1.2

0.8
magnitude (m/s)

0.6

0.4

0.2

0
−4 −3 −2 −1 0 1 2 3 4
polar angle (aprox.)

Figure B.5: Magnitude of the velocity around the circumference of the cabinet at 210 Hz

93
Appendix C

Matlab codes

C.1 Calculation of the sound radiated by the cabinet

[cabinetradiation.m]

%% V i b r a t i n g l o u d s p e a k e r , b a s e d i n measurements
%% INITIALIZATION
%c l e a r ;
clc ;
open ( ’ w o r k s p a c e v i b r a t i n g c a b i n e t (24 −6 −08) . mat ’ ) ; %Opens t h e s c r i p t
which
%c o n t a i n s t h e d a t a o b t a i e d from t h e measurements
v c a b i n e t=ans . v c a b i n e t ;

%p r e s s u r e measured 1m i n f r o n t o f t h e s p e a k e r

p 3 5=ans . p 3 5 ; p 3 5=p 35 ’ ;
p b a c k 3 5=ans . p b a c k 3 5 ; p b a c k 3 5=p ba ck 3 5 ’ ;
p s i d e 3 5=ans . p s i d e 3 5 ; p s i d e 3 5=p s i d e 3 5 ’ ;

p F P a l l f=zeros ( 3 6 0 , 6 0 0 ) ;

%%
for f =1:600

Rfp =1; % r a d i u s o f t h e arc o f f i e l d p o i n t s ( h a l f a c i r c l e i n t h e z−

y plane )

k=2∗pi ∗ f / 3 4 3 ; % Wavenumber , m−1

nsingON =1; % Deal w i t h near−s i n g u l a r i n t e g r a l s
C. Matlab codes C.1. Calculation of the sound radiated by the cabinet

% CONDICIONES AMBIENTALES
pa = 1 0 1 3 2 5 ; % P r e s i o n A t m o s f e r i c a (Pa )
t = 20; % Temperatura ( C )
Hr = 5 0 ; % Humedad r e l a t i v a (%)
[ rho , c , c f , CpCv , nu , a l f a ]=amb2prop ( pa , t , Hr , 1 0 0 0 ) ;

%% Read Geometry ( mesh )

% Read n odes and t o p o l o g y .
no des=r e a d n o d e s ( ’ N o d e l i s t o k . l i s ’ ) ;
e l e m e n t s=r e a d e l e m e n t s ( ’ E l e m e n t l i s t o k . l i s ’ ) ;

M=s i z e ( nodes , 1 ) ; N=s i z e ( elements , 1 ) ;

% c h e c k geomet ry and add body numbers
[ nodesb , to po lo g yb , t o p o s h r i n k b , tim , segmopen ]= b o d y f i nd ( nodes , e l e m e n t s )
; axi s e q u a l ;
t o p o l o g y b=t o p o s h r i n k b ;

%% BC’ s : V e l o c i t y ( v i b r a t i n g c a b i n e t )

% C a l c u l a t e t h e BEM m a t r i c e s and s o l v e t h e p r e s s u r e s on t h e s u r f a c e
%[A, B, CConst ]= TriQuadEquat ( nodesb , t o p o l o g y b , t o p o l o g y b , z e r o s (N, 1 ) , k ,
nsingON ) ; % q u a d r i l a t e r a l
[ A, B, CConst]= TriQuadEquat ( nodesb , to po lo g yb , to po lo g yb , o nes (N, 1 ) , k ,
nsingON ) ; % t r i a n g u l a r
B=i ∗k∗ rho ∗ c ∗B ;
p f=A\(−B∗ v c a b i n e t ( : , f +1) ) ;

% Field points

%This i s f o r f i e l d p o i n t s around t h e l o u d p s e a k e r , p o l a r diagram

%0 d e g r e e s i s t h e f r o n t o f t h e l o u d s p e a k e r
Mfp=360; % Number o f f i e l d p o i n t s
h e i g h t = 0 . 3 5 ; %h e i g h t o f t h e f i e l d p o i n t s . A p p a r e n t l y a t 35cm i s t h e
h i g h e s t sound r a d i a t i o n
t h e t a=l i ns pace ( 0 , 2 ∗ pi , Mfp ) ’ ;
%xyzFP=Rfp ∗ [ c o s ( t h e t a ) h e i g h t ∗ on es ( Mfp , 1 ) s i n ( t h e t a ) ];%my t r y
xyzFP=Rfp ∗ [ s i n ( t h e t a ) h e i g h t ∗ o nes ( Mfp , 1 ) cos ( t h e t a ) ] ; %my t r y
%p l o t f i e l d p o i n t s , t o make s u r e t h a t t h e y are w e l l d e f i n e d
% f i g u r e ; p l o t 3 ( xyzFP ( : , 1 ) , xyzFP ( : , 2 ) , xyzFP ( : , 3 ) ) ; x l a b e l ( ’ x ’ ) ; y l a b e l
( ’y ’) ; zlabel ( ’z ’) ;

% Calculate f i e l d points
%[ Afp , Bfp , Cfp ]= p o i n t ( nodesb , t o p o l o g y b , t o p o l o g y b , z e r o s (N, 1 ) , k , xyzFP ,
nsingON ) ; % q u a d r i l a t e r a l
[ Afp , Bfp , Cfp ]= p o i n t ( nodesb , to po lo g yb , to po lo g yb , o nes (N, 1 ) , k , xyzFP ,
nsingON ) ; % t r i a n g u l a r
pfFP=(Afp∗ p f+j ∗k∗ rho ∗ c ∗ Bfp ∗ v c a b i n e t ( : , f +1) ) . / Cfp ;
p F P a l l f ( : , f )=pfFP ;%a l l f r e q u e n c i e s i n c l u d e d h e r e . t h e rows are t h e
a n g l e s from 0 t o 2 p i and t h e columns t h e f r e q u e n c y from 1 t o 500Hz

95
C.1. Calculation of the sound radiated by the cabinet C. Matlab codes

save ( ’ v i b r a t i n g c a b i n e t l o o p t e m p . mat ’ ) ;
cl os e a l l ;

end

%
% PLOTTING,POSTPROCESSING AND VERIFICATION
%

%p l o t f i e l d p o i n t s , t o make s u r e t h a t t h e y are w e l l d e f i n e d
% f i g u r e ; p l o t 3 ( xyzFP ( : , 1 ) , xyzFP ( : , 2 ) , xyzFP ( : , 3 ) ) ; x l a b e l ( ’ x ’ ) ; y l a b e l
( ’y ’) ; zlabel ( ’z ’) ;
%open ( ’ w o r k s p a c e v i b r a t i n g c a b i n e t (24−6−08) . mat ’ )
PALL=ans . p F P a l l f ;

save ( ’ w o r k s p a c e A l l c a b i n e t . mat ’ ) ;

%PLOT i s a dummy v a r i a b l e c r e a t e d j u s t t o p l o t t h e p o l a r p l o t s w i t h
the 0
%a n g l e r e p r e s e n t e d i n t h e c e n t e r o f t h e p l o t
PLOT= c i r c s h i f t (PALL, −180) ;

fi gure ;
imagesc ( 5 0 : 6 0 0 , ( 1 8 0 / pi ) ∗ t h e t a , 2 0 ∗ log10 ( abs (PALL ( : , 5 0 : 6 0 0 ) ) /20 e −6) , [ 0
40]) ;
t i t l e ( ’SPL r a d i a t e d by t h e c a b i n e t a t 1m d i s t a n c e [ f r e q vs azimuth
angle ] ’ )
xlabel ( ’ f ’ ) ; ylabel ( ’ azimuth a n g l e [ deg ] ’ ) ;
colormap ( [ l i ns pace ( 1 , 0 . 2 , 1 0 ) ’ l i ns pace ( 1 , 0 . 2 , 1 0 ) ’ l i ns pace ( 1 , 0 . 2 , 1 0 )
’]) ;
xlim ( [ 5 0 3 5 0 ] ) ; colorbar ; shading i n t e r p ;

fi gure ;
imagesc ( 5 0 : 6 0 0 , ( 1 8 0 / pi ) ∗ ( t h e t a −pi ) , 2 0 ∗ log10 . . .
( abs (PLOT( 1 : 3 6 0 , 5 0 : 6 0 0 ) ) /20 e −6) , [ 0 4 0 ] ) ;
t i t l e ( ’SPL r a d i a t e d by t h e c a b i n e t a t 1m d i s t a n c e [ f r e q vs azimuth
angle ] ’ )
xlabel ( ’ f ’ ) ; ylabel ( ’ azimuth a n g l e [ deg ] ’ ) ;
colormap ( [ l i ns pace ( 1 , 0 . 2 , 1 0 ) ’ l i ns pace ( 1 , 0 . 2 , 1 0 ) ’ l i ns pace ( 1 , 0 . 2 , 1 0 )
’]) ;
xlim ( [ 5 0 3 5 0 ] ) ; colorbar ; shading i n t e r p ;

fi gure ;

96
C. Matlab codes C.1. Calculation of the sound radiated by the cabinet

contour ( 5 0 : 6 0 0 , ( 1 8 0 / pi ) ∗ ( t h e t a −pi ) , 2 0 ∗ log10 ( abs (PLOT . . .

( : , 5 0 : 6 0 0 ) ) /20 e −6) , 1 0 : 1 : 4 0 ) ;
t i t l e ( ’SPL r a d i a t e d by t h e c a b i n e t a t 1m d i s t a n c e −I s o b a r=1dB− ’ ) ;
xlabel ( ’ f ’ ) ; ylabel ( ’ azimuth a n g l e [ deg ] ’ ) ;
xlim ( [ 5 0 3 5 0 ] ) ; colorbar ;
colormap ( [ l i ns pace ( 1 , 0 . 2 , 1 0 ) ’ l i ns pace ( 1 , 0 . 2 , 1 0 ) ’ l i ns pace ( 1 , 0 . 2 , 1 0 )
’]) ;

fi gure ; contour ( 1 : 5 0 0 , l i ns pace (1 8 0 , − 1 8 0 ,3 6 0 ) , angle (PLOT( : , 1 : 5 0 0 ) ) , 8 ) ;

t i t l e ( ’ Phase a t 1m from t h e c a b i n e t [ f r e q vs p o l a r a n g l e ] −( I s o b a r=p i
/ 4 )− ’ )
; xlabel ( ’ f ’ ) ; ylabel ( ’ azimuth a n g l e [ rad ] ’ ) ;
xlim ( [ 5 0 3 5 0 ] ) ; colormap ( [ 0 0 0 ; 0 0 0 ] ) ; grid on ;

%% COMPARISON WITH THE TOTAL PRESSURE OF THE LOUDSPEAKER

Cabinet SPL =20∗log10 ( abs (PALL ( 1 , : ) ) / (2 0 e −6) ) ;

Total SPL=20∗log10 ( sqrt ( abs ( p 3 5 ) ) / (2 0 e −6) ) ;

fi gure ; semilogx ( 1 : 5 0 0 , Cabinet SPL ( 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 )

hold on ; plot ( 1 : 5 0 0 , Total SPL ( 1 : 5 0 0 ) , ’−− ’ , ’ l i n e w i d t h ’ , 2 ) ;
t i t l e ( ’SPL r a d i a t e d from t h e c a b i n e t compared t o t e t o t a l SPL ’ ) ;
legend ( ’ Ca binet ’ , ’ To ta l ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 6 0 0 ] ) ; grid on ;

% S i d e r a d i a t i o n o f t h e c a b i n e t ( a t −90deg )
fi gure ;
semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( abs (PALL( 2 7 1 , 1 : 5 0 0 ) ) /20 e −6) , ’ l i n e w i d t h ’ , 2 )
hold on ;
semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( sqrt ( abs ( p s i d e 3 5 ( 1 : 5 0 0 ) ) ) /20 e −6) , ’−− ’ , ’
linewidth ’ ,2)
t i t l e ( ’SPL r a d i a t e d from t h e s i d e o f t h e c a b i n e t compared t o t h e
total ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 6 0 0 ] ) ; grid on ;
Legend ( ’ Ca binet ’ , ’ To ta l ’ )

%Back r a d i a t i o n o f t h e c a b i n e t
fi gure ;
semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( abs ( PALLall ( 1 8 0 , 1 : 5 0 0 ) ) /20 e −6)+1 , ’ l i n e w i d t h ’
,2 , ’ color ’ , ’ r ’ )
hold on ;
semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( sqrt ( abs ( p b a c k 3 5 ( 1 : 5 0 0 ) ) ) /20 e −6) , ’−− ’ , ’
linewidth ’ ,2)
t i t l e ( ’SPL r a d i a t e d from t h e back o f t h e c a b i n e t compared t o t h e
total ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 6 0 0 ] ) ;
grid on ; Legend ( ’ Ca binet ’ , ’ To ta l ’ )

97
C.1. Calculation of the sound radiated by the cabinet C. Matlab codes

%% POLAR PLOT
fi gure ; polar ( t h e t a , 2 0 ∗ log10 ( abs (PALL ( : , 8 4 ) ) /20 e −6) ) ;
hold on ; polar ( t h e t a , 2 0 ∗ log10 ( abs (PALL ( : , 1 5 0 ) ) /20 e −6) , ’−− ’ ) ;
hold on ; polar ( t h e t a , 2 0 ∗ log10 ( abs (PALL ( : , 1 8 2 ) ) /20 e −6) , ’ : ’ ) ;
Legend ( ’ 85Hz ’ , ’ 150Hz ’ , ’ 182Hz ’ )
% t i t l e ( ’ SPL o f t h e main n a t u r a l f r e q u e n c i e s o f t h e c a b i n e t a t 1 m’ )
;
% Legend ( ’ 8 5 Hz ’ , ’ 1 5 0 Hz ’ , ’ 1 8 2 Hz ’ , ’ 2 2 0 Hz ’ )

%% PLOT THE PHASE & MAGNITUDE OF THE VELOCITY

% THIS IS DONE IN ORDER TO SEE THE DEFLECTION SHAPES MORE OR LESS
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

%phase=at an ( imag / r e a l )
%At 150Hz , f o r t h e n odes a t 18cm h e i g h t
%f i g u r e ; c o n t o u r ( 1 5 0 , 1 7 : 3 2 , a n g l e ( v c a b i n e t ( 1 7 : 3 2 , 1 5 0 ) ) , 8 ) ; t i t l e ( ’
Phase a t 1m from t h e c a b i n e t [ f r e q v s p o l a r a n g l e ] −( I s o b a r=p i /4)
− ’) ; x l a b e l ( ’ f ’ ) ; y l a b e l ( ’ azimu t h a n g l e [ rad ] ’ ) ;

f e r n a n d e z= c i r c s h i f t ( angle ( v c a b i n e t ( 4 9 : 6 4 , 8 3 ) ) , 1 9 ) ;%f e r n a n d e z=unwrap

( fernandez ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) ) ; grid on ;
t i t l e ( ’ V e l o c i t y pha se on t h e s u r f a c e o f t h e c a b i n e t [ 8 3 Hz ] ’ ) ; xlabel
( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ pha se ( rad ) ’ ) ;
f e r n a n d e z= c i r c s h i f t ( abs ( v c a b i n e t ( 4 9 : 6 4 , 8 3 ) ) , 1 9 ) ;%f e r n a n d e z=unwrap (
fernandez ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) ) ; grid on ;
t i t l e ( ’ V e l o c i t y magnitude on t h e s u r f a c e o f t h e c a b i n e t [ 8 3 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ magnitude (m/ s ) ’ ) ;

f e r n a n d e z= c i r c s h i f t ( angle ( v c a b i n e t ( 4 9 : 6 4 , 9 9 ) ) , 1 9 ) ;%f e r n a n d e z=unwrap

( fernandez , 2 ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) + 1 .5 ) ; grid on ;
t i t l e ( ’ V e l o c i t y pha se on t h e s u r f a c e o f t h e c a b i n e t [ 9 9 Hz ] ’ ) ; xlabel
( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ pha se ( rad ) ’ ) ;
f e r n a n d e z= c i r c s h i f t ( abs ( v c a b i n e t ( 4 9 : 6 4 , 9 9 ) ) , 1 9 ) ;%f e r n a n d e z=unwrap (
fernandez ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) ) ; grid on ;
t i t l e ( ’ V e l o c i t y magnitude on t h e s u r f a c e o f t h e c a b i n e t [ 9 9 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ magnitude (m/ s ) ’ ) ;

f e r n a n d e z= c i r c s h i f t ( angle ( v c a b i n e t ( 4 9 : 6 4 , 1 4 0 ) ) , 1 9 ) ; f e r n a n d e z=unwrap
( fernandez ) ;

98
C. Matlab codes C.1. Calculation of the sound radiated by the cabinet

fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) +8) ; grid on ;

t i t l e ( ’ V e l o c i t y pha se on t h e s u r f a c e o f t h e c a b i n e t [ 1 4 0 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ pha se ( rad ) ’ ) ;
f e r n a n d e z= c i r c s h i f t ( abs ( v c a b i n e t ( 4 9 : 6 4 , 1 4 0 ) ) , 1 9 ) ;%f e r n a n d e z=unwrap (
fernandez ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) ) ; grid on ;
t i t l e ( ’ V e l o c i t y magnitude on t h e s u r f a c e o f t h e c a b i n e t [ 1 4 0 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ magnitude (m/ s ) ’ ) ;

f e r n a n d e z= c i r c s h i f t ( angle ( v c a b i n e t ( 4 9 : 6 4 , 1 8 2 ) ) , 1 9 ) ;%f e r n a n d e z=
unwrap ( f e r n a n d e z ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z + 1 .5 ) ) ; grid on ;
t i t l e ( ’ V e l o c i t y pha se on t h e s u r f a c e o f t h e c a b i n e t [ 1 8 2 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ pha se ( rad ) ’ ) ;
f e r n a n d e z= c i r c s h i f t ( abs ( v c a b i n e t ( 4 9 : 6 4 , 1 8 4 ) ) , 1 9 ) ;%f e r n a n d e z=unwrap (
fernandez ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) ) ; grid on ;
t i t l e ( ’ V e l o c i t y magnitude on t h e s u r f a c e o f t h e c a b i n e t [ 1 8 2 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ magnitude (m/ s ) ’ ) ;

f e r n a n d e z= c i r c s h i f t ( angle ( v c a b i n e t ( 4 9 : 6 4 , 2 1 0 ) ) , 1 9 ) ; f e r n a n d e z=unwrap
( fernandez ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) −4.5) ; grid on ;
t i t l e ( ’ V e l o c i t y pha se on t h e s u r f a c e o f t h e c a b i n e t [ 2 1 0 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ pha se ( rad ) ’ ) ;
f e r n a n d e z= c i r c s h i f t ( abs ( v c a b i n e t ( 4 9 : 6 4 , 2 1 0 ) ) , 1 9 ) ;%f e r n a n d e z=unwrap (
fernandez ) ;
fi gure ; plot ( l i ns pace(−pi , pi , 1 6 ) , ( f e r n a n d e z ) ) ; grid on ;
t i t l e ( ’ V e l o c i t y magnitude on t h e s u r f a c e o f t h e c a b i n e t [ 2 1 0 Hz ] ’ ) ;
xlabel ( ’ p o l a r a n g l e ( aprox . ) ’ ) ; ylabel ( ’ magnitude (m/ s ) ’ ) ;

99
C.2. Calculation of the sound radiated by the loudspeaker C. Matlab codes

C.2 Calculation of the sound radiated by the loudspeaker

[loudspeaker-radiation.m]
%% V i b r a t i n g l o u d s p e a k e r , b a s e d i n l a s e r measurements
% s t i l l t o do : c h e c k t h e v i b r a t i o n l e v e l s , t h e v i b r a t i o n v e l o c i t i e s
s h o u l d be
%around 0 . 6 5 8 5 m/ s a t 80Hz
%a l s o measure t h e phase o f t h e s p e a k e r ,

%% INITIALIZATION
cl ear ;
clc ;

open ( ’ V e l o c i t y s p e a k r . mat ’ ) ;
v d r i v e r=ans . v d r i v e r ;
v c a b i n e t=ans . v c a b i n e t ;
open ( ’ V e l o c i t y c a b . mat ’ ) ;
%v c a b i n e t =ans . v c a b i n e t ;
v s p e a k r=v c a b i n e t+v d r i v e r ;
%v s p e a k e r c o n t a i n s t h e v e l o c i t y o f t h e c a b i n e t and o f t h e d r i v e r s

%% Get measurements o f t h e p r e s s u r e
%p r e s s u r e measured 1m i n f r o n t o f t h e s p e a k e r
p 3 5=ans . p 3 5 ; p 3 5=p 35 ’ ;
p b a c k 3 5=ans . p b a c k 3 5 ; p b a c k 3 5=p ba ck 3 5 ’ ;
p s i d e 3 5=ans . p s i d e 3 5 ; p s i d e 3 5=p s i d e 3 5 ’ ;

%p F P a l l f=z e r o s ( 3 6 0 , 6 0 0 ) ; %t h i s i s f o r a l l t h e p o l a r a n g l e , around
the speaker
p F P a l l f=zeros ( 3 , 6 0 0 ) ; %t h i s i s f o r f e w e r p o i n t s ( f r o n t s i d e back )

for f =1:400

Rfp =1; % r a d i u s o f t h e arc o f f i e l d p o i n t s ( h a l f a c i r c l e i n t h e z−

y plane )
k=2∗pi ∗ f / 3 4 3 ; % Wavenumber , m−1
nsingON =1; % Deal w i t h near−s i n g u l a r i n t e g r a l s
% CONDICIONES AMBIENTALES
pa = 1 0 1 3 2 5 ; % P r e s i o n A t m o s f e r i c a (Pa )
t = 20; % Temperatura ( C )
Hr = 5 0 ; % Humedad r e l a t i v a (%)
[ rho , c , c f , CpCv , nu , a l f a ]= amb2prop ( pa , t , Hr , 1 0 0 0 ) ;

%% Read Geometry ( mesh )

% Read n odes and t o p o l o g y .
no des=r e a d n o d e s ( ’ N o d e l i s t o k . l i s ’ ) ;
e l e m e n t s=r e a d e l e m e n t s ( ’ E l e m e n t l i s t o k . l i s ’ ) ;

M=s i z e ( nodes , 1 ) ; N=s i z e ( elements , 1 ) ;

100
C. Matlab codes C.2. Calculation of the sound radiated by the loudspeaker

% c h e c k geomet ry and add body numbers

[ nodesb , to po lo g yb , t o p o s h r i n k b , tim , segmopen ]= b o d y f i nd ( nodes , e l e m e n t s )
; axi s e q u a l ;
t o p o l o g y b=t o p o s h r i n k b ;

%% BC’ s : V e l o c i t y ( v i b r a t i n g c a b i n e t )

% C a l c u l a t e t h e BEM m a t r i c e s and s o l v e t h e p r e s s u r e s on t h e s u r f a c e
%[A, B, CConst ]= TriQuadEquat ( nodesb , t o p o l o g y b , t o p o l o g y b , z e r o s (N, 1 ) , k ,
nsingON ) ; % q u a d r i l a t e r a l
[ A, B, CConst]= TriQuadEquat ( nodesb , to po lo g yb , to po lo g yb , o nes (N, 1 ) , k ,
nsingON ) ; % t r i a n g u l a r
B=i ∗k∗ rho ∗ c ∗B ;
%p f=A\(−B∗ v s p e a k r ( : , f +1) ) ;
p f=A\(−B∗ v d r i v e r ( : , f +1) ) ;

% Field points
%This i s f o r f i e l d p o i n t s 1m away from t h e l o u d s p e a k e r , i n t h e
positions
%t h a t were measured . In t h e f r o n t i n t h e s i d e and i n t h e back o f i t .
Three p o i n t s :
%f r o n t −s i d e −back r e s p e c t i v e l y
xyzFP =[0 0 . 3 5 1 . 2 ; 1 . 2 0 . 3 5 0 ; 0 0 . 3 5 −1.2 ] ;
%p l o t f i e l d p o i n t s , t o make s u r e t h a t t h e y are w e l l d e f i n e d
%f i g u r e ; p l o t 3 ( xyzFP ( : , 1 ) , xyzFP ( : , 2 ) , xyzFP ( : , 3 ) ) ; x l a b e l ( ’ x ’ ) ; y l a b e l ( ’
y ’) ; zlabel ( ’z ’) ;

% Calculate f i e l d points
[ Afp , Bfp , Cfp ]= p o i n t ( nodesb , to po lo g yb , to po lo g yb , o nes (N, 1 ) , k , xyzFP ,
nsingON ) ; % t r i a n g u l a r
%pfFP=(Afp ∗ p f+j ∗ k ∗ rho ∗ c ∗ Bfp ∗ v s p e a k r ( : , f +1) ) . / Cfp ;
pfFP=(Afp∗ p f+j ∗k∗ rho ∗ c ∗ Bfp ∗ v d r i v e r ( : , f +1) ) . / Cfp ;
p F P a l l f ( : , f )=pfFP ;%a l l f r e q u e n c i e s i n c l u d e d h e r e . t h e rows are t h e
p o s i t i o n s fron , s i d e , back . The columns are t h e f r e q u e n c i e s
cl os e a l l ;
save ( ’ w o r k s p a c e v i b r a t i n g d r i v e r s 2 ’ , ’ p F P a l l f ’ , ’ pfFP ’ )
end

open ( ’ w o r k s p a c e v i b r a t i n g c a b i n e t (24 −6 −08) . mat ’ )

PALL=ans . p F P a l l f ( [ 1 270 1 8 0 ] , : ) ; %t h e d i v i s i o n i s done f o r
calibration
cl ear ans
open ( ’ w o r k s p a c e v i b r a t i n g d r i v e r s . mat ’ ) ;
PALLsp=ans . p F P a l l f ;

%%

%%

101
C.2. Calculation of the sound radiated by the loudspeaker C. Matlab codes

%% PLOTTING
%%

fi gure ; semilogx ( 1 : 8 0 1 , 2 0 ∗ log10 ( abs ( v d r i v e r ( 6 , : ) ) / ( 1 e −9) ) , ’ l i n e w i d t h

’ ,2) ;
hold on ; semilogx ( 1 : 8 0 1 , 2 0 ∗ log10 ( abs ( v d r i v e r ( 7 6 , : ) ) / ( 1 e −9) ) , ’
linewidth ’ ,2)
t i t l e ( ’ v i b r a t i o n v e l o c i t y o f t h e Woofer and t h e Midrange ’ ) ;
xlabel ( ’ l o g f [ Hz ] ’ ) ; ylabel ( ’dB ( r e f 1nm/ s ) ’ )
grid on ; xlim ( [ 5 0 5 0 0 ] ) ; ylim ( [ 5 0 1 5 0 ] )

Driver SPL =20∗log10 ( abs (PALLsp) / (2 0 e −6) ) ;

Total SPL=20∗log10 ( sqrt ( abs ( p 3 5 ) ) / (2 0 e −6) ) ;
Total BEM=20∗log10 ( abs (PALL+PALLsp) / (2 0 e −6) ) ;

%f r o n t r a d i a t i o n
fi gure ; semilogx ( 1 : 5 0 0 , Total BEM ( 1 , 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ )
hold on ; plot ( 1 : 5 0 0 , Total SPL ( 1 : 5 0 0 ) , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ ) ;
t i t l e ( ’SPL r a d i a t e d from t h e l o u d s p e a k e r −BEM compared t o
measurement− ’ ) ;
legend ( ’BEM’ , ’ Measurement ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 3 5 0 ] ) ; grid on ;
ylim ( [ 5 0 6 4 ] ) ;

fi gure ; semilogx ( 1 : 5 0 0 , Total BEM ( 1 , 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ )

hold on ; plot ( 1 : 5 0 0 , Driver SPL ( 1 , 1 : 5 0 0 ) , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k
’);
t i t l e ( ’ To ta l SPL from t h e l o u d s p e a k e r compared t o t h e l o u d s p e a k e r
unit ’ ) ;
legend ( ’ To ta l ’ , ’ Loudsp . u n i t ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 3 5 0 ] ) ; grid on ;
ylim ( [ 5 5 6 2 ] ) ;

% S i d e r a d i a t i o n o f t h e c a b i n e t ( a t −90deg )
% f i g u r e ; s e m i l o g x ( 1 : 5 0 0 , Total BEM ( 2 , 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 ) ;
% h o l d on ;
% s e m i l o g x ( 1 : 5 0 0 , 2 0 ∗ l o g 1 0 ( s q r t ( a b s ( p s i d e 3 5 ( 1 : 5 0 0 ) ) ) /20 e −6) , ’ − − ’)
% t i t l e ( ’ SPL r a d i a t e d from t h e s i d e o f t h e l o u d s p e a k e r (90 deg ) ’ ) ;
% x l a b e l ( ’ l o g ( f ) [ Hz ] ’ ) ; y l a b e l ( ’ SPL [ dB ] ’ ) ; x l i m ( [ 5 0 3 5 0 ] ) ;
% g r i d on ; Legend ( ’BEM’ , ’ Measurement ’ )
% ylim ([50 60]) ;

%Back r a d i a t i o n o f t h e c a b i n e t
fi gure ; semilogx ( 1 : 5 0 0 , Total BEM ( 3 , 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ )
hold on ;
semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( sqrt ( abs ( p b a c k 3 5 ( 1 : 5 0 0 ) ) ) /20 e −6) , ’−− ’ )
t i t l e ( ’SPL r a d i a t e d from t h e back o f t h e l o u d s p e a k e r ’ ) ;

102
C. Matlab codes C.2. Calculation of the sound radiated by the loudspeaker

xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 3 5 0 ] ) ; ylim ( [ 5 0

60]) ;
grid on ; Legend ( ’BEM’ , ’ Measurement ’ )

fi gure ; semilogx ( 1 : 5 0 0 , Total BEM ( 3 , 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ )

hold on ; semilogx ( 1 : 5 0 0 , Driver SPL ( 3 , 1 : 5 0 0 ) , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r
’ , ’k ’ )
t i t l e ( ’ To ta l SPL r a d i a t e d from t h e back compared t o t h e Lo udspea ker
unit ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 3 5 0 ] ) ; ylim ( [ 5 0
60]) ;
grid on ; Legend ( ’ To ta l ’ , ’ Loudsp . u n i t ’ ) ;

fi gure ; semilogx ( 1 : 5 0 0 , Total BEM ( 3 , 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ )

hold on ; semilogx ( 1 : 5 0 0 , Driver SPL ( 3 , 1 : 5 0 0 ) , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r
’ , ’k ’ )
hold on ; semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( sqrt ( abs ( p b a c k 3 5 ( 1 : 5 0 0 ) ) ) /20 e −6) , ’
: ’)
t i t l e ( ’SPL r a d i a t e d a t t h e back o f t h e l o u d s p e a k e r (1 8 0 ) ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 3 5 0 ] ) ; ylim ( [ 5 0
60]) ;
Legend ( ’ To ta l −BEM− ’ , ’ Loudsp . u n i t −BEM− ’ , ’ To ta l −Measured− ’ )

fi gure ; semilogx ( 1 : 5 0 0 , Total BEM ( 3 , 1 : 5 0 0 ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ )

hold on ; semilogx ( 1 : 5 0 0 , Driver SPL ( 3 , 1 : 5 0 0 ) , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r
’ , ’k ’ )
hold on ; semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( sqrt ( abs ( p b a c k 3 5 ( 1 : 5 0 0 ) ) ) /20 e −6) , ’
: ’)
hold on ; semilogx ( 1 : 5 0 0 , 2 0 ∗ log10 ( abs (PALL( 3 , 1 : 5 0 0 ) ) /20 e −6) , ’ −. ’ )
t i t l e ( ’SPL r a d i a t e d a t t h e back o f t h e l o u d s p e a k e r (1 8 0 ) ’ ) ;
xlabel ( ’ l o g ( f ) [ Hz ] ’ ) ; ylabel ( ’SPL [ dB ] ’ ) ; xlim ( [ 5 0 3 5 0 ] ) ; ylim ( [ 3 0
60]) ;
Legend ( ’ Total−BEM’ , ’ Loudsp . u n i t −BEM’ , ’ Total−Measured ’ , ’ Cabinet−BEM’
)

103
C.3. Time domain characteristics (Impulse response & decay) C. Matlab codes

C.3 Time domain characteristics (Impulse response & de-

cay)

[cabinetradiation-TimeDomain.m]
%%TIME DOMAIN RESPONSE
%This s c r i p t p r e f o r m s an IFFT on t h e r e s p o n s e o f t h e c a b i n e t , i n
order to
%know what i s t h e t ime domain r e s p o n s e . In p r i n c i p l e p l o t t i n g t h e
frequency
%a g a i n s t t h e t ime ( i . e . w a t e r f a l l p l o t )

%The phase s h o u l d be r e f e r e n c e d t o t h e e l e c t r i c a l i n p u t
%For t h a t reason , t n e x t r a t r a n s f e r f u n c t i o n phase s h o u l d be added
from t h e
%e l e c t r i c a l r e s p o n s e t o t h e v i b r a t i o n r e f e r e n c e p o i n t on t o p o f t h e
woofer .
%Such a f u n c t i o n i s c a l l e d H0 ( which i s Ref / Rel )

cl ear ; c l c ;

open ( ’ w o r k s p a c e v i b r a t i n g d r i v e r s . mat ’ )
p f d r i v e r=ans . p F P a l l f ( 1 , 1 : 3 5 0 ) ; cl ear ans ;
%i n t h e c a s e o f t h e back d i r e c t i o n ;
%p f d r i v e r=ans . p F P a l l f ( 3 , 1 : 3 5 0 ) ; c l e a r ans ;
%f i g u r e ; p l o t ( a b s ( p f d r i v e r ) ) ;
open ( ’ w o r k s p a c e v i b r a t i n g c a b i n e t (24 −6 −08) . mat ’ )
H 0 r e a l=ans . H0 r ea l ’ ;
H0imag=ans . H0imag ’ ;
H0mag=sqrt ( ( H 0 r e a l . ˆ 2 ) +(H0imag . ˆ 2 ) ) ;
H0ph=angle ( H 0 r e a l+i ∗H0imag ) ; H0phase=unwrap( H0ph) ; cl ear H0ph ;
PALL=ans . p F P a l l f ; cl ear ans ;

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%CASE OF ALL THE CABINET VIBRATING ( w i t h t h e Bottom )

%t h i s doesn ’ t work b e c a u s e t h e imp r e s p o n s e i s t o o l o n g and t h e r e i s
t ime
%a l i a s i n g . Thus , t h e r e i s a n o t h e r s c r i p t c a l l e d Cabbottom ( which i s
exactly
%t h e same as t h i s one , w i t h a l o n g e r t ime span ( o f two s e c o n d s ) )
%open ( ’ V e l o c i t y c a b A l l . mat ’ )
%PALL=ans . p F P a l l f ; c l e a r ans ;
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

104
C. Matlab codes C.3. Time domain characteristics (Impulse response & decay)

%PALL c o n t a i n s t h e f r e q domain i n f o r m a t i o n o f t h e complex f r e q u e n c y

over
%a l l a n g l e s i n 1 d e g r e e r e s o l i t i o n and 1Hz r e s o l u t i o n PALL ( 0 , : ) i s
the front
%a n g l e
%DEFINITION OF THE PARAMETERS INVOLVED IN THE ANALYSIS
di r =1;%d i r e c t i o n i n d e g r e e s f o r which t h e c a l c u l a t i o n i s done
%complex p r e s s u r e on t h e a n g l e d i r w i t h t h e e l e c t r i c a l phase
p f =(PALL( dir , 1 : 3 5 0 ) ’ ) . ∗ ( H0phase ( 1 : 3 5 0 ) . / abs ( H0phase ( 1 : 3 5 0 ) ) ) ;
p f ( 3 5 1 : 7 0 0 )=conj ( p f ( 3 5 0 : − 1 : 1 ) ) ;

%VERIFICATION OF THE LOADED DATA

%f i g u r e ; p l o t ( a b s ( p f ) ) ;
%f i g u r e ; p l o t ( a n g l e ( p f ) ) ; f i g u r e ; p l o t (20∗ l o g 1 0 ( a b s ( p f ) /20 e −6) ) ;
%f i g u r e ; p l o t ( a b s ( p f d r i v e r ) ) ;
%f i g u r e ; p l o t ( a n g l e ( p f d r i v e r ) ) ; f i g u r e ; p l o t (20∗ l o g 1 0 ( a b s ( p f d r i v e r ) /2 e
−5) ) ;

%%

%% IMPULSE RESPONSE OF THE CABINET − Using f r e q . up t o 300Hz

d f =1;%Hz
N=1024;
f s=N/ d f ;

ptime =(350) ∗ i f f t ( pf , N, ’ symmetric ’ ) ; %350= f a c t o r f o r t h e i f f t

amplitude
fi gure ; plot ( 0 : 1 / f s :(1 −1/ f s ) , r e a l ( ptime ) , ’ l i n e w i d t h ’ , 1 . 5 ) ;%p l o t
u n t i l 1 sec
xlabel ( ’ time ( s ) ’ ) ; ylabel ( ’ Sound P r e s s u r e ( Pa ) ’ ) ;
t i t l e ( ’ Impulse r e s p o n s e o f t h e c a b i n e t −P r e s s u r e a t 1m− ’ )
grid on ; xlim ( [ 0 300 e − 3 ]) ; ylim ( [ − 0 . 0 5 0 . 0 5 ] ) ;

% %i n t e r p o l a t i o n ( f o r b e a u t y r e a s o n s )−n ot n e c e s a r y and n ot u sed

% %i n any c a s e f o r r e s u l t s −

% t i n t =1/4096;% f i n t =1/ t i n t ;
% t i n t e r p =(0: t i n t :(1 − t i n t ) ) ’ ;
% p t i m e i n t=i n t e r p 1 ( ( 0 : 1 / f s :1 −1/ f s ) , ptime , t i n t e r p , ’ s p l i n e ’ ) ;% ’ l i n e a r
’) ;
% f i g u r e ; p l o t ( 0 : t i n t :(1 − t i n t ) , r e a l ( p t i m e i n t ) , ’ l i n e w i d t h ’ , 1 . 5 ) ;% p l o t
u n t i l 1 sec
% x l a b e l ( ’ t ime ( s ) ’ ) ; y l a b e l ( ’ Sound P r e s s u r e (Pa ) ’ ) ;
% t i t l e ( ’ I m p u l s e r e s p o n s e o f t h e c a b i n e t −P r e s s u r e a t 1m− ’)
% g r i d on ; x l i m ( [ 0 300 e −3]) ; y l i m ([ − 0.05 0 . 0 5 ] ) ;

105
C.3. Time domain characteristics (Impulse response & decay) C. Matlab codes

% V e r i f i c a t i o n of the impulse response obt ain ed

%c h e c k= f f t ( ( pt ime ) ,N) ;
%f i g u r e ; p l o t ( 0 : d f :N/2−1, a b s ( c h e c k ( 1 :N/2) ) ) ;

%f i g u r e ; p l o t ( 0 : 1 / f s : 1 , ( a b s ( pt ime ( 1 : ( f s +1) ) ) / a b s (max ( pt ime ) ) ) ) ;

%g r i d o n ;% n o r m a l i z e d decay
fi gure ; plot ( 0 : 1 / f s :(1 −1/ f s ) , r e a l ( ptime ( 1 : f s ) ) , ’ l i n e w i d t h ’ , 1 . 5 ) ;%
u n t i l 1 sec
xlabel ( ’ time ( s ) ’ ) ; ylabel ( ’ Sound P r e s s u r e ( Pa ) ’ ) ;
t i t l e ( ’ Impulse r e s p o n s e o f t h e c a b i n e t −P r e s s u r e a t 1m− ’ )
grid on ; xlim ( [ 0 300 e − 3 ]) ;%y l i m ([ − 1.5 e−3 1 . 5 e −3]) ;

%%

%% IMPULSE RESPONSE OF THE DRIVER − Using f r e q . up t o 300Hz

%%%%%%%%%%%%%%%%%%%%%%%%%%% GOOD %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Nd=1024;
f s d r i v e r=Nd/ d f ;
%f s d=Nd/ d f ;% which means f s d r i v e r=Nd
p d r i v e r t i m e =(175) ∗ i f f t ( p f d r i v e r , Nd , ’ nonsymmetric ’ ) ;
%175= f a c t o r f o r t h e i f f t a m p l i t u d e ( s i n c e 175 l i n e s are u sed )

fi gure ; plot ( 0 : 1 /Nd:(1 −1/Nd) , r e a l ( p d r i v e r t i m e ) , ’ l i n e w i d t h ’ , 1 . 5 ) ;%

u n t i l 1 sec
xlabel ( ’ time ( s ) ’ ) ; ylabel ( ’ Sound P r e s s u r e ( Pa ) ’ ) ;
t i t l e ( ’ Impulse r e s p o n s e o f t h e l o u d s p e a k e r u n i t −P r e s s u r e a t 1m− ’ )
grid on ; xlim ( [ 0 300 e − 3 ]) ;%y l i m ([ − 1.5 e−3 1 . 5 e −3]) ;

% V e r i f i c a t i o n of the impulse response obt ain ed

% c h e c k 2 =(1/175) ∗ f f t ( ( p d r i v e r t i m e ) ,Nd) ;
% f i g u r e ; p l o t ( 0 : f s d r i v e r /Nd : Nd−1, a b s ( c h e c k 2 ) ) ;%
% f i g u r e ; p l o t ( 0 : f s d r i v e r /Nd : Nd−1 ,20∗ l o g 1 0 ( a b s ( c h e c k 2 ) /2 e −5) ) ;
%
%i n t e r p o l a t i o n ( t o have t h e same d i m e n s i o n s as t h e c a b i n e t )
tdinter p =(0:(1/1024) :(1 −(1/1024) ) ) ’ ;
p d r i v e r t i m e i n t=interp1 ( ( 0 : 1 / Nd:1 −1/Nd) , p d r i v e r t i m e , t d i n t e r p , ’ l i n e a r ’
);
% f i g u r e ; p l o t ( t d i n t e r p , r e a l ( p d r i v e r t i m e i n t ) ) ;% p l o t u n t i l 1 s e c
% x l i m ( [ 0 300 e −3]) ; y l i m ([ −1 1 ] ) ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LOAD & COMPARE IMPULSE RESPONSE BY B&O
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

open ( ’ workspace impulseBO . mat ’ ) ;

106
C. Matlab codes C.3. Time domain characteristics (Impulse response & decay)

impulseBO=ans . impulseBO eq ;%we u se t h e e q u a l i z e d one , s i n c e i t i s

t h e one
%implemen t ed when t h e measurement was
done
%impulseBO=ans . impulseBO ;
cl ear ans ;
%remove t h e o f f s e t ;
o f f s e t =40;%s a m p l e s o f o f f s e t u n t i l t h e maximum
fi gure ;
plot(− o f f s e t / 5 0 0 0 : 1 / 5 0 0 0 : ( ( 2 0 0 0 − o f f s e t ) /5000 −1/5000) , ( impulseBO ) ∗1 e
−4) ;
xlabel ( ’ time ( s ) ’ ) ; ylabel ( ’ Sound P r e s s u r e ( Pa ) ’ ) ;
t i t l e ( ’ Impulse r e s p o n s e o f t h e l o u d s p e a k e r u n i t −Measured− ’ ) ;
grid on ; xlim ( [ 0 300 e − 3 ]) ;%y l i m ([ − 1.5 e−3 1 . 5 e −3]) ;

%%

%% COMPARISON − PLOTTING

fi gure ; subplot ( 2 , 1 , 1 ) ;
plot ( 0 : 1 / f s :(1 −1/ f s ) , r e a l ( p d r i v e r t i m e i n t ) , ’ l i n e w i d t h ’ , 2 ) ;%p l o t
u n t i l 1 sec
grid on ; xlim ( [ 0 300 e − 3 ]) ; ylim ([ −1 e−1 1 e − 1 ]) ;
xlabel ( ’ time ( s ) ’ ) ; ylabel ( ’ ( Pa ) ’ ) ;
t i t l e ( ’ Impulse r e s p o n s e o f t h e d r i v e r and t h e c a b i n e t −P r e s s u r e a t 1
m− ’ )
subplot ( 2 , 1 , 2 )
plot ( 0 : 1 / f s :(1 −1/ f s ) , r e a l ( ptime ) , ’ l i n e w i d t h ’ , 1 . 5 ) ;%p l o t u n t i l 1 s e c
grid on ; xlim ( [ 0 300 e − 3 ]) ; ylim ([ −1 e−1 1 e − 1 ]) ;
xlabel ( ’ time ( s ) ’ ) ; ylabel ( ’ ( Pa ) ’ ) ;

fi gure ;%dB−SPL decay

plot ( 0 : 1 / f s :(1 −1/ f s ) , 2 0 ∗ log10 ( abs ( p d r i v e r t i m e i n t ( 1 : f s ) ) / ( 2 e −5) ) ) ;
grid on ;
hold on ;
plot ( 0 : 1 / f s :(1 −1/ f s ) , 2 0 ∗ log10 ( abs ( ptime ( 1 : f s )+p d r i v e r t i m e i n t ( 1 : f s ) )
/ ( 2 e −5) ) ) ;
t i t l e ( ’ To tla SPL decay compared t o t h e l o u d s p e a k e r u n i t −SPL a t 1m− ’
)
grid on ; xlim ( [ 0 100 e − 3 ]) ; xlabel ( ’ time ( s ) ’ ) ; ylabel ( ’SPL(dB) ’ ) ;%y l i m
([0 50]) ;
legend ( ’ D r i v e r o n l y ’ , ’ To ta l ( d r i v e r+c a b i n e t ) ’ ) ; grid on ;%dB−SPL decay

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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C.3. Time domain characteristics (Impulse response & decay) C. Matlab codes

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%

%
%%SPECTROGRAM−WATERFALL PLOT C a l c u l a t i o n ( from t h e Time domain )
%

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Try t o compare t h e decay u s i n g f f t s

n=256;
t=1e −3;
twin=29e −3;%h a l f o f t h e s l i d i n g t ime window u sed

%Compare t h e t ime windows t h a t c o u l d be u sed

win1=Hann( 2 ∗ twin ∗ f s ) ;
win2=r e c t w i n ( 2 ∗ twin ∗ f s ) ;
win3=tukeywin ( 2 ∗ twin ∗ f s + 1 , 0 . 1 5 ) ;
win4=tukeywin ( 2 ∗ twin ∗ f s + 1 , 0 . 4 ) ;
wvto o l ( win1 , win2 , win3 , win4 ) ;
wvto o l ( win4 ) ;
%f i n d t h e b e s t compromise f o r t h e window . t h e one w i t h n arrower
main lobe
%and s m a l l e r s i d e −l o b e s .
win=win4 ;
fi gure ;
subplot ( 1 , 2 , 1 )
plot ( ( 0 : 1 / f s : length ( win ) / f s −1/ f s ) , 2 0 ∗ log10 ( win ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r
’ , ’k ’ ) ;
t i t l e ( ’ c o s i n e t a p e r e d window ( tukey ) −time− ’ ) ;
grid on ; xlabel ( ’ time [ s ] ’ ) ; ylabel ( ’ Amplitude ’ ) ; ylim ([ −35 1 ] ) ;
subplot ( 1 , 2 , 2 )
plot ( 0 : f s /n : f s −1/n , 2 0 ∗ log10 ( f f t ( win , n ) ) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ ) ;
xlim ( [ 0 3 0 0 ] ) ; grid on ; t i t l e ( ’ c o s i n e t a p e r e d window ( tukey ) −
frequency−’ ) ;
grid on ; xlabel ( ’ f [ Hz ] ’ ) ; ylabel ( ’Mag [ dB ] ’ ) ;
%t =0ms

t =[1 30 50 70 100 130 1 5 0 ] ∗ 1 e −3;

108
C. Matlab codes C.3. Time domain characteristics (Impulse response & decay)

for co unt = 1 :( length ( t ) )

i f co unt==1
e s t r i n g=win ( twin ∗ f s : ( 2 ∗ twin ) ∗ f s ) . ∗ ptime ( 1 : ( t ( co unt )+twin ) ∗ f s
) ;%j u s t a s t r i n g c o n t a i n i n g t h e d a t a t r a n s f o r m e d , t o a v o i d
problems with the in dex es
deca yca b ( : , co unt )=f f t ( e s t r i n g , n ) ; deca yca b ( : , co unt )=deca yca b
( : , co unt ) / ( n / 2 ) ;%( l e n g t h ( e s t r i n g ) ) ;
e s t r i n g d=win ( twin ∗ f s : ( 2 ∗ twin ) ∗ f s ) . ∗ p d r i v e r t i m e i n t ( ( 1 : ( t (
co unt )+twin ) ∗ f s ) ) ;
d e c a y d r i v ( : , co unt )=f f t ( e s t r i n g d ( : , co unt ) , n ) ; d e c a y d r i v ( : ,
co unt )=d e c a y d r i v ( : , co unt ) / ( n / 2 ) ;%( l e n g t h ( e s t r i n g d ) ) ;
fi gure ; plot ( 0 : f s /n : ( f s / 2 )−1/ f s , 20∗ log10 ( abs ( d e c a y d r i v ( 1 : n
/ 2 , co unt ) ) /2 e −5) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ ) ;
hold on ; plot ( 0 : f s /n : ( f s / 2 )−1/ f s , 20∗ log10 ( abs ( deca yca b ( 1 : n
/ 2 , co unt ) ) /2 e −5) , ’ l i n e s t y l e ’ , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k
’);
xlim ( [ 0 3 0 0 ] ) ; ylim ( [ 5 6 3 ] ) ;
Legend ( ’ Loudsp . u n i t ’ , ’ Ca binet ’ ) ; grid on ; xlabel ( ’ f [ Hz ] ’ ) ;
ylabel ( ’SPL [ dB ] ’ ) ; t i t l e ( ’ Spectrum o f t h e decay o f t h e
c a b i n e t and t h e d r i v e r a t t=0ms ’ ) ;
else
e s t r i n g=win . ∗ ptime ( round ( ( t ( co unt )−twin ) ∗ f s : ( t ( co unt )+twin ) ∗
f s ) ) ;%j u s t a s t r i n g c o n t a i n i n g t h e d a t a t r a n s f o r m e d , t o
avoid problems with the in dex es
deca yca b ( : , co unt )=f f t ( e s t r i n g , n ) ; deca yca b ( : , co unt )=deca yca b
( : , co unt ) / ( n / 2 ) ;%( l e n g t h ( e s t r i n g ) ) ;
e s t r i n g d=win . ∗ p d r i v e r t i m e i n t ( round ( ( t ( co unt )−twin ) ∗ f s : ( t (
co unt )+twin ) ∗ f s ) ) ;
d e c a y d r i v ( : , co unt )=f f t ( e s t r i n g d , n ) ; d e c a y d r i v ( : , co unt )=
d e c a y d r i v ( : , co unt ) / ( n / 2 ) ;%( l e n g t h ( e s t r i n g d ) ) ;
fi gure ; plot ( 0 : f s /n : ( f s / 2 )−1/ f s , 20∗ log10 ( abs ( d e c a y d r i v ( 1 : n
/ 2 , co unt ) ) /2 e −5) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ ) ;
hold on ; plot ( 0 : f s /n : ( f s / 2 )−1/ f s , 20∗ log10 ( abs ( deca yca b ( 1 : n
/ 2 , co unt ) ) /2 e −5) , ’ l i n e s t y l e ’ , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k
’);
xlim ( [ 0 3 0 0 ] ) ; ylim ( [ 5 6 0 ] ) ;
Legend ( ’ Loudsp . u n i t ’ , ’ Ca binet ’ ) ; grid on ; xlabel ( ’ f [ Hz ] ’ ) ;
ylabel ( ’SPL [ dB ] ’ ) ; t i t l e ( [ ’ Spectrum o f t h e decay o f t h e
c a b i n e t and t h e l o u s p e a k e r u n i t a t t= ’ , num2str ( t ( co unt ) ∗1
e3 ) , ’ms ’ ] ) ;
t ( co unt )
end
end

fi gure ;
for co unt = 1 :( length ( t ) −1)
subplot ( 3 , 2 , co unt ) ; plot ( 0 : f s /n : ( f s / 2 )−1/ f s , 20∗ log10 ( abs (
d e c a y d r i v ( 1 : n / 2 , co unt ) ) /2 e −5) , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ ) ;

109
C.3. Time domain characteristics (Impulse response & decay) C. Matlab codes

hold on ; plot ( 0 : f s /n : ( f s / 2 )−1/ f s , 20∗ log10 ( abs ( deca yca b ( 1 : n / 2 ,

co unt ) ) /2 e −5) , ’ l i n e s t y l e ’ , ’−− ’ , ’ l i n e w i d t h ’ , 2 , ’ c o l o r ’ , ’ k ’ ) ;
grid on ; xlim ( [ 0 3 0 0 ] ) ; t i t l e ( [ ’ t= ’ , num2str ( t ( co unt ) ∗1 e3 ) , ’ms ’ ] ) ;
end
Legend ( ’ Ca binet ’ , ’ D r i v e r ’ ) ;%x l a b e l ( ’ f [ Hz ] ’ ) ; y l a b e l ( ’ SPL [ dB ] ’ ) ;

f = (0 : f s /n : ( f s / 4 )−1/ f s ) ’ ;
[T F]=meshgrid ( t , f ) ;
fi gure ; mesh(T, F, 2 0 ∗ log10 ( abs ( d e c a y d r i v ( 1 : n / 4 , : ) ) /2 e −5) , ’ MeshStyle ’ , ’
column ’ , ’ l i n e w i d t h ’ , 2 ) ; a lpha ( . 1 )
t i t l e ( ’ Sequency o f t h e decay o f t h e l o u d s p e a k e r u n i t ’ ) ;
colormap(1−white ) ; xlabel ( ’ time [ s ] ’ ) ; ylabel ( ’ f [ Hz ] ’ ) ; z l a b e l ( ’SPL [
dB ] ’ ) ;
view ( [ 3 5 2 5 ] ) ;%z l i m ( [ 0 7 0 ] )

fi gure ; mesh(T, F, 2 0 ∗ log10 ( abs ( deca yca b ( 1 : n / 4 , : ) ) /2 e −5) , ’ MeshStyle ’ , ’

column ’ , ’ l i n e w i d t h ’ , 2 ) ; a lpha ( . 9 )
t i t l e ( ’ Sequency o f t h e decay o f t h e c a b i n e t ’ ) ;
colormap(1−white ) ; xlabel ( ’ time [ s ] ’ ) ; ylabel ( ’ f [ Hz ] ’ ) ; z l a b e l ( ’SPL [
dB ] ’ ) ;
view ( [ 3 5 2 5 ] ) ;%z l i m ( [ 0 7 0 ] ) ;

f = (0 : f s /n : ( f s / 4 )−1/ f s ) ’ ;
[T F]=meshgrid ( t ( 2 : 5 ) , f ) ;
fi gure ; mesh(T, F, 2 0 ∗ log10 ( abs ( d e c a y d r i v ( 1 : n / 4 , 2 : 5 ) ) /2 e −5) , ’ MeshStyle ’
, ’ column ’ , ’ l i n e w i d t h ’ , 2 ) ; a lpha ( 0 . 0 0 1 )
hold on ; mesh(T, F, 2 0 ∗ log10 ( abs ( deca yca b ( 1 : n / 4 , 2 : 5 ) ) /2 e −5) , ’ MeshStyle
’ , ’ column ’ , ’ l i n e w i d t h ’ , 2 , ’ l i n e s t y l e ’ , ’−− ’ ) ; a lpha ( . 0 0 1 )
colormap(1−white ) ; xlabel ( ’ time [ s ] ’ ) ; ylabel ( ’ f [ Hz ] ’ ) ; z l a b e l ( ’SPL [
dB ] ’ ) ; Legend ( ’ Loudsp . u n i t ’ , ’ c a b i n e t ’ ) ;
t i t l e ( ’ Comparison o f t h e decay o f t h e l o u d s p e a k e r u n i t and t h e
cabinet ’ )
view ( [ 2 7 2 5 ] ) ;

fi gure ; mesh(T, F, 2 0 ∗ log10 ( abs ( d e c a y d r i v ( 1 : n / 4 , 2 : 5 ) ) /2 e −5) , ’ f a c e c o l o r ’

, ’ g r e e n ’ , ’ e d g e c o l o r ’ , ’ none ’ ) ; a lpha ( 0 . 9 )
hold on ; mesh(T, F, 2 0 ∗ log10 ( abs ( deca yca b ( 1 : n / 4 , 2 : 5 ) ) /2 e −5) , ’ f a c e c o l o r
’ , ’ r ed ’ , ’ e d g e c o l o r ’ , ’ none ’ ) ; a lpha ( 0 . 9 )
colormap(1−white ) ; xlabel ( ’ time [ s ] ’ ) ; ylabel ( ’ f [ Hz ] ’ ) ; z l a b e l ( ’SPL [
dB ] ’ ) ; Legend ( ’ Loudsp . u n i t ’ , ’ c a b i n e t ’ ) ;
t i t l e ( ’ Comparison o f t h e decay o f t h e l o u d s p e a k e r u n i t and t h e
cabinet ’ )
view ( [ 4 2 2 5 ] ) ;

110
C. Matlab codes C.4. Testing of the method (Test Box)

C.4 Testing of the method (Test Box)

[testbox-radiation.m]

%% V i b r a t i n g t e s t box , b a s e d i n measurements
%%
cl ear ;
clc ;
open ( ’ V e l o c i t y b o x . mat ’ ) ; %Opens t h e s c r i p t V e l o c i t y c a b . mat . . .
% . . . which c o n t a i n s t h e d a t a o b t a i e d from t h e measurements
v box=ans . v box ;
p r e s 0 2 5=ans . p r e s 0 2 5 ;
p r e s 0 5=ans . p r e s 0 5 ;
p F P a l l f=zeros ( 2 , 6 0 0 ) ;
fmax =600;

for f =1:500
k=2∗pi ∗ f / 3 4 3 ; % Wavenumber , m−1
nsingON =1; % Deal w i t h near−s i n g u l a r i n t e g r a l s
% CONDICIONES AMBIENTALES
pa = 1 0 1 3 2 5 ; % P r e s i o n A t m o s f e r i c a (Pa )
t = 20; % Temperatura ( C )
Hr = 5 0 ; % Humedad r e l a t i v a (%)
[ rho , c , c f , CpCv , nu , a l f a ]=amb2prop ( pa , t , Hr , 1 0 0 0 ) ;

%% Read Geometry ( mesh )

% Read n odes and t o p o l o g y .
no des=r e a d n o d e s ( ’ NLISTbox . l i s ’ ) ;
e l e m e n t s=r e a d e l e m e n t s ( ’ ELISTbox . l i s ’ ) ;

M=s i z e ( nodes , 1 ) ; N=s i z e ( elements , 1 ) ;

% c h e c k geomet ry and add body numbers
[ nodesb , to po lo g yb , t o p o s h r i n k b , tim , segmopen ]= b o d y f i nd ( nodes , e l e m e n t s )
; axi s e q u a l ;
t o p o l o g y b=t o p o s h r i n k b ;

%% BC’ s : V e l o c i t y ( v i b r a t i n g c a b i n e t )

% C a l c u l a t e t h e BEM m a t r i c e s and s o l v e t h e p r e s s u r e s on t h e s u r f a c e
%[A, B, CConst ]= TriQuadEquat ( nodesb , t o p o l o g y b , t o p o l o g y b , z e r o s (N, 1 ) , k ,
nsingON ) ; % q u a d r i l a t e r a l
[ A, B, CConst]= TriQuadEquat ( nodesb , to po lo g yb , to po lo g yb , o nes (N, 1 ) , k ,
nsingON ) ; % t r i a n g u l a r
B=i ∗k∗ rho ∗ c ∗B ;
p f=A\(−B∗ v box ( : , f +1) ) ;
% Field points
%c o o r d i n t a e s o f t h e f i e l d p o i n t
%c e n t r e a t two h e i g h t s 50cm and 25cm (+44cm h e i g h t o f t h e box )=94 &
69 cm

111
C.4. Testing of the method (Test Box) C. Matlab codes

xyzFP = [ 0 . 2 2 1 0 . 9 4 0 . 2 2 1 ; 0 . 2 2 1 0 . 6 9 0 . 2 2 1 ] ;
% Calculate f i e l d points
%[ Afp , Bfp , Cfp ]= p o i n t ( nodesb , t o p o l o g y b , t o p o l o g y b , z e r o s (N, 1 ) , k , xyzFP ,
nsingON ) ; % q u a d r i l a t e r a l
[ Afp , Bfp , Cfp ]= p o i n t ( nodesb , to po lo g yb , to po lo g yb , o nes (N, 1 ) , k , xyzFP ,
nsingON ) ; % t r i a n g u l a r
pfFP=(Afp ∗ p f+j ∗k∗ rho ∗ c ∗ Bfp ∗ v box ( : , f +1) ) . / Cfp ;
p F P a l l f ( : , f )=pfFP ;%a l l f r e q u e n c i e s i n c l u d e d h e r e . t h e rows are each
f i e l d p o i n t and t h e columns t h e f r e q u e n c y from 1 t o 500Hz

%% P l o t s o l u t i o n
% figure ;
% %s u b p l o t ( 2 , 1 , 1 )
% p l o t ( t h e t a ∗180/ pi , 2 0 ∗ l o g 1 0 ( a b s ( pfFP ) /(20 e −6) ) ) ; g r i d
% x l a b e l ( ’ Angle [ deg ] ’ ) ; y l a b e l ( ’ | p r e s s u r e on f i e l d p o i n t s [ dB ] | ’ ) ;
% t i t l e ( [ ’ C o n t r i b u t i o n o f t h e l o u d s p e a k e r c a b i n e t , f = ’ , num2str ( f ) , ’
Hz a t r = ’ , num2str ( Rfp ) , ’m and h e i g h t = ’ , num2str ( h e i g h t ) , ’m’ ] ) ;%
su bplot (2 ,1 ,2)

%p l o t f i e l d p o i n t s , t o make s u r e t h a t t h e y are w e l l d e f i n e d
% f i g u r e ; p l o t 3 ( xyzFP ( : , 1 ) , xyzFP ( : , 2 ) , xyzFP ( : , 3 ) ) ; x l a b e l ( ’ x ’ ) ; y l a b e l
( ’y ’) ; zlabel ( ’z ’) ;

save ( ’ v i b r a t i n g b o x l o o p t e m p . mat ’ ) ;
cl os e a l l ;
end

save ( ’ w o r k s p a c e v i b r a t i n g b o x . mat ’ )

PALL=p F P a l l f ;
p F P a l l f= c i r c s h i f t ( p F P a l l f , [ 0 1 ] ) ;

fi gure ; plot ( 1 : 8 0 1 , 2 0 ∗ log10 ( sqrt ( abs ( p r e s 0 2 5 ) ) / (2 0 e −6) ) , ’ l i n e w i d t h ’

,2) ;
hold on ; plot (2 0 ∗ log10 ( abs ( p F P a l l f ( 2 , : ) ) / (2 0 e −6) ) , ’ −. ’ , ’ l i n e w i d t h ’
,2) ;
t i t l e ( ’ Test Box−Comparison between BEM and measurement ( a t 0 . 2 5m
height ) ’ )
ylabel ( ’SPL [ dB ] ’ ) ; xlabel ( ’ f r e q [ Hz ] ’ ) ;
grid on ; legend ( ’ Measurement ’ , ’BEM’ )
fi gure ; plot ( 1 : 8 0 1 , 2 0 ∗ log10 ( sqrt ( abs ( p r e s 0 5 ) ) / (2 0 e −6) ) , ’ l i n e w i d t h ’ , 2 )
;
hold on ; plot (2 0 ∗ log10 ( abs ( p F P a l l f ( 1 , : ) ) / (2 0 e −6) ) , ’ −. ’ , ’ l i n e w i d t h ’
,2) ;
t i t l e ( ’ Test Box − Comparison between BEM and measurement ( a t 0 . 5m
height ) ’ )
ylabel ( ’SPL [ dB ] ’ ) ; xlabel ( ’ f r e q [ Hz ] ’ ) ;
grid on ; legend ( ’ Measurement ’ , ’BEM’ )

112
C. Matlab codes C.4. Testing of the method (Test Box)

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MODES IN THE TEST BOX − v e r y approx imat e , j u s t t o have an idea −

%44 by44by44 cm box made o u t o f f i b e r board , on t h e t o p s u r f a c e t h e r e
is a
%1mm t h i c k aluminium p l a t e d r i v e n by an i n e r t i a l e x c i t e r
h=1e −3;
l =0.42%l e n g t h o f t h e s i d e s
n1 =1;%mode
n2 =3;
%aluminium
rho =2.7 e3 ;
E=7.1 e10 ; v = 0 . 3 3 ;
m=rho ∗h ;
B=(E∗h ˆ 3 ) /(12∗(1 − v ˆ 2 ) ) ;
wn=(( n1 ∗ pi / l ) ˆ2+(n2∗ pi / l ) ˆ 2 ) ∗ sqrt (B/m) ;
f=wn/ (2 ∗ pi ) ;

113
Appendix D

SHELL63 Elastic Shell

SHELL63 has both bending and membrane capabilities. Both in-plane and normal
loads are permitted. The element has six degrees of freedom at each node: trans-
lations in the nodal x, y, and z directions and rotations about the nodal x, y, and
z axes. Stress stiffening and large deflection capabilities are included. A consistent
tangent stiffness matrix option is available for use in large deflection (finite rotation)
analysis. See Section 14.63 of the ANSYS Theory Reference for more details about
this element. Similar elements are SHELL43 and SHELL181 (plastic capability), and
SHELL93 (mid-side node capability). The ETCHG command converts SHELL57 and
SHELL157 elements to SHELL63.

Figure D.1: SHELL63 Elastic Shell