k Kowalczyk

Boundary and medium modelling using compact

finite difference schemes in simulations of room

acoustics for audio and architectural design

applications

Konrad KowalczykB.Eng., M.Sc.

Sonic Arts Research Centre

School of Electronics, Electrical Engineering and Computer Science

Queens University Belfast

Submitted for the Degree of Doctor of Philosophy

November 2008

To Kasia...

Abstract

Simulation of acoustic spaces with the aim of developing virtual immersive applications

and architectural design applications is one of the key areas in the field of audio signal

processing. In this thesis, a complete method for simulating room acoustics using compact

finite difference time domain (FDTD) schemes is presented.

A family of compact explicit and implicit schemes approximating the wave equation is

analysed in terms of stability, accuracy, and computational efficiency. The most accurate

and isotropic schemes based on a rectilinear nonstaggered grid are identified, and the

optimally efficient explicit schemes are indicated.

Novel FDTD formulations of frequency-independent and frequency-dependent bound-

aries of a locally reacting surface type are proposed, including a full treatment of corners

and boundary edges. In particular, it is proposed to model generally frequency-dependent

boundaries by local incorporation of a digital impedance filter (DIF), and the resulting

formulae for compact explicit schemes are provided. In addition, a numerical boundary

analysis (NBA) procedure is proposed as a technique for analytic evaluation of the numer-

ical reflectance of the presented boundary models. The digital impedance filter model is

also extended to model controllable surface diffusion based on the concept of phase grating

diffusers.

Results obtained from numerical experiments and numerical boundary analysis confirm

the high accuracy of the proposed boundary models, the reflectance of which is shown

to closely approximate locally reacting surface theory for different angles of incidence

and various impedances. Furthermore, the results indicate that boundary formulations

based on the identified accurate and isotropic schemes are also very accurate in terms of

numerical reflectance, and outperform directly related methods such as Yees scheme and

the standard digital waveguide mesh. In addition, one particular scheme - referred to as

the interpolated wideband scheme - is suggested as the best FDTD scheme for most audio

applications.

i

Acknowledgements

This research has been carried out at the Sonic Arts Research Centre, Queens University

Belfast between October 2005 and December 2008. I would like to express my gratitude to

my supervisor Dr. Maarten van Walstijn for letting me pursue a Ph.D. in this topic and

organising financial support for my research. I am deeply grateful for his dedicated and

consistent support, guidance, and patience in teaching me technical writing. Our weekly

discussions and his open-door policy greatly helped in the successful completion of this

thesis.

Many thanks go to my colleagues from SARC, both postgraduate students and staff,

for creating a very vibrant and inspiring environment, with a family-like atmosphere.

There are so many of you that it makes it impossible to mention you all here. I am

particularly grateful for your friendship in and outside SARC, and the very best social life

which successfully provided me with numerous pleasant distractions from this work; this

also includes late night jam sessions and Sunday football games.

I am thankful to Dr. Stefan Bilbao for his continuous interest in this work, many

helpful suggestions and personalised lectures introducing me to the concept of FDTD

methods. Very special thanks to Prof. Roger Woods from Queens who has provided

invaluable guidance and support throughout my graduate career.

I have had the great honour of being a visiting Ph.D. student at the Center for Com-

puter Research in Music and Acoustics (CCRMA), Stanford University, in 2007 and the

Audio Lab, Department of Electronics, University of York, in 2008. I am greatly indebted

to Prof. Julius O. Smith for a very inspirational stay at CCRMA, for many insightful dis-

cussions and for finding time to discuss my work despite the busy schedule. Many thanks

to Dr. Damian Murphy for warmly hosting me in York leading to a fruitful collaboration,

and for our highly interesting lunch conversations.

I would like to thank Prof. Peter Svensson, Patty Huang, Vasileios Chatziioannou, Dr.

Tapio Lokki, Prof. Rudolf Rabenstein, and Prof. Diemer de Vries for insightful discussions

related to my work on numerous occasions.

Sincere thanks go to my girlfriend and soon to become wife, Kasia, for her love, con-

tinuous encouragement and sharing ups and downs related to the Ph.D. experience. Last

iii

but definitely not least, I am indebted to my parents, sister Ela and all my friends, for

making me who I am and for being there for me.

I would also like to thank my football coach for giving me a place in the first team and

trusting my scoring skills, even when it seemed to me almost impossible to score a goal.

The financial support of the European Social Fund is acknowledged.

Contents

Abstract i

Acknowledgements iii

1 Introduction 1

1.1 Research Objectives and Applications . . . . . . . . . . . . . . . . . . . . . 3

1.2 Room Acoustics Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Related Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Fundamentals of Room Acoustics 11

2.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Sound Pressure Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Locally Reacting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Boundary Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Reflection at Normal Incidence . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Reflection at Oblique Incidence . . . . . . . . . . . . . . . . . . . . . 17

2.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Eigenmode Model (for Rigid Walls) . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Acoustical Porous Material . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.1 Maximum Length Sequence . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.2 Quadratic Residue Diffuser . . . . . . . . . . . . . . . . . . . . . . . 23

2.6.3 Modulated Quadratic Residue Diffuser . . . . . . . . . . . . . . . . . 25

2.6.4 Diffractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.5 Curved Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.6 Fractional Brownian Diffusers . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vi

2.7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7.2 Scattering Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Elements of Numerical Modelling 33

3.1 Room Acoustics Modelling Methods . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Geometrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 Wave-based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.3 Motivation for the Chosen Method . . . . . . . . . . . . . . . . . . . 37

3.2 The Finite Difference Time Domain Method . . . . . . . . . . . . . . . . . . 38

3.2.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Dispersion Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.3 Staggered FDTD Method . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.4 Digital Waveguide Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.5 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Frequency Warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Solving Tridiagonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Fractional Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Compact FDTD Schemes 68

4.1 2D Compact FDTD Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Special Cases of Explicit Schemes . . . . . . . . . . . . . . . . . . . . 70

4.1.2 Special cases of implicit schemes . . . . . . . . . . . . . . . . . . . . 72

4.1.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.4 Numerical dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.5 Dispersion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1.6 Accuracy and Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1.7 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2 3D Compact FDTD Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Special Cases of 3D Explicit Schemes . . . . . . . . . . . . . . . . . 88

4.2.2 3D Compact Implicit Schemes . . . . . . . . . . . . . . . . . . . . . 92

4.2.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2.4 Numerical Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2.5 Dispersion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.6 Accuracy and Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.7 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

vii

5 FDTD Formulation of Locally Reacting Surfaces 103

5.1 Locally Reacting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2 Frequency-independent Boundaries . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.1 2D Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.2 1D Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2.3 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3 Frequency-dependent Boundaries . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.1 Boundary Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3.2 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4 Boundaries in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113


5.4.2 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5 Numerical Boundary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.1 2D boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.5.3 3D Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.6.1 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.6.2 2D Frequency-independent Boundary . . . . . . . . . . . . . . . . . 123

5.6.3 2D Frequency-dependent Boundaries . . . . . . . . . . . . . . . . . . 126

5.6.4 3D Boundary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Modelling Frequency-Dependent Boundaries as Digital Impedance Fil-

ters 133

6.1 Digital Impedance Filter (DIF) . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2 2D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135


6.2.2 Other Rectilinear-grid Boundaries . . . . . . . . . . . . . . . . . . . 139

6.2.3 K-DWM Implementation . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.4 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3 3D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141


6.3.2 Corners and Boundary Edges . . . . . . . . . . . . . . . . . . . . . . 142


6.4.1 2D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.4.2 3D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

viii

6.5.1 1D Boundary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5.2 Impedance Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.5.3 Results of the 2D DIF Model . . . . . . . . . . . . . . . . . . . . . . 148

6.5.4 Results of the 3D DIF Model . . . . . . . . . . . . . . . . . . . . . . 153

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7 Compact Explicit Formulation of the DIF Model 158

7.1 Compact Explicit DIF Formulation . . . . . . . . . . . . . . . . . . . . . . . 159

7.1.1 2D Compact Explicit DIF Boundary Model . . . . . . . . . . . . . . 159

7.1.2 2D Compact Explicit DIF Corners . . . . . . . . . . . . . . . . . . . 163

7.1.3 3D Compact Explicit DIF Boundary Model . . . . . . . . . . . . . . 165

7.1.4 3D Compact Explicit DIF Corners . . . . . . . . . . . . . . . . . . . 168


7.3 2D Boundary Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.3.1 Frequency-independent Results . . . . . . . . . . . . . . . . . . . . . 174

7.3.2 Frequency-dependent Results . . . . . . . . . . . . . . . . . . . . . . 176

7.4 3D Boundary Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.4.1 Frequency-independent Results . . . . . . . . . . . . . . . . . . . . . 178

7.4.2 Frequency-dependent Results . . . . . . . . . . . . . . . . . . . . . . 179

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8 A Phase Grating Approach to Modelling Surface Diffusion 186

8.1 A Method for Simulating Diffusive Surfaces . . . . . . . . . . . . . . . . . . 187

8.1.1 A Phase Grating Approach . . . . . . . . . . . . . . . . . . . . . . . 187

8.1.2 Relationship between the Well Depth and Delay Length . . . . . . . 188

8.1.3 Fractional Delay Implementation . . . . . . . . . . . . . . . . . . . . 189

8.1.4 Diffusion Parameter Control . . . . . . . . . . . . . . . . . . . . . . . 189

8.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2.1 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2.2 Modelled Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.2.3 Frequency-domain Results . . . . . . . . . . . . . . . . . . . . . . . . 194

8.2.4 Time-domain Results . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9 Conclusions and Recommendations 205

9.1 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.2 Future Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

ix

Bibliography 210

1Chapter 1

Introduction

Over the past two decades, various computer modelling techniques have been developed

for auralisation purposes. With the rise of the role of the audio part in many multimedia

applications, computational modelling of acoustic spaces has recently gained wider inter-

est. The level of accuracy to which the sound environment is modelled depends strongly

on a particular application and availability of computational resources for audio signal

processing. In the simplest case of real-time simulations in interactive multimedia appli-

cations and computer games, usually only the sound sources are rendered, leaving out the

acoustic effects of the surrounding environment. On the other hand, due to an increased

need for realism, many applications of computer-based modelling of room acoustics require

more details to be simulated. Previously, such more accurate and computationally expen-

sive modelling techniques were utilised in the creation of naturally sounding reverberation

units and room acoustics prediction for architectural design applications. However, due

to the increase in the computational power of commonly available processors, more de-

tailed modelling of sound propagation in acoustic spaces can now be integrated in generic

entertainment and multimedia applications.

The rapid development of virtual reality applications and multimedia technology has

stimulated the development and inclusion of acoustic modelling in numerous applications.

Therefore, there is a need for algorithms enabling the creation of virtual acoustic environ-

ments with multiple moving sound sources, which can be freely explored by the listener

or a group of listeners. The key feature of such systems is their perceptual immersiveness,

which can be defined as a feeling of realism experienced in a virtual acoustic space. Thus

the realistic quality of sound should be ensured, which can be obtained with perceptual

or physical approaches. With a perceptual approach, a plausible sound field is generated

using perceptual parameters. However, since the acoustics of the virtual space is not ex-

plicitly modelled, it is not suitable for room acoustics prediction. The main applications

include computer games, the creation of spatial effects for composers and plausible rever-

Chapter 1. Introduction 2

Sound source

modelling

Room acoustics

modelling

Receiver modelling according

to reproduction system

Virtual acoustic

application

Figure 1.1: Auralisation stages.

beration units in music production. The perceptual approach is also applied in the Spat

software [48] and for sound environment modelling in MPEG-4 scene description language

[117]. Conversely, the physical approach is based on modelling the acoustics of the virtual

enclosed space defined by physical parameters such as room shape and boundary material.

Consequently, it can be used to predict the acoustics of auditoria in architectural design

and are generally applicable to multimedia applications. Some example applications of

a physical approach include ODEON [78] software for room acoustics and virtual reality

applications such as DIVA [88] and [70].

The most popular approach to auralisation consists in computing one or more room

impulse responses of the modelled space and convolving them with a dry source signal.

The impulse responses are captured at a receiver position in a format defined by the

reproduction technique, and next the soundfield of the simulated acoustic space is repro-

duced in a listening environment [110]. Consequently, we can distinguish three modelling

components in an auralisation system, namely the source, room acoustics and listener, as

illustrated in Figure 1.1.

For a source of sound, the spatial localisation and directivity should be modelled.

Dry and monophonic input signal can then be fed into the system as a pre-recorded or

synthesised sound. Receiver modelling refers to the position and directivity of a listener

using available reproduction systems. For that purpose, various sound reproduction tech-

niques are available, including binaural techniques [74] for sound field reproduction for a

single listener using head-related transfer functions (HRTF) and multichannel loudspeak-

ers techniques. The latter have the advantage that a listener can freely move the head

without compromising the reproduced quality and include wavefield synthesis (WFS) [9],

Ambisonics [41] and vector-base amplitude-panning (VBAP) [83] techniques. Simulation

of the room acoustics is the main component of the modelling structure.


1.1 Research Objectives and Applications

The main goal of this research is to develop improved methods for the simulation of sound

propagation in acoustic spaces for architectural design and audio applications. Perceptual

realism and a high level of accuracy are increasingly required in these applications. This

research aims to develop numerical algorithms that are applicable to creating an immersive

acoustic environment, which allows the simulation of acoustic spaces of a complex shape

with multiple moving sound sources and listeners.

Computational modelling of acoustic spaces is fundamental for various auralisation

and room acoustics applications. Possible applications of room acoustic modelling are

architectural design software and the analysis and evaluation of existing acoustic spaces.

Non-real-time high accuracy simulations can be used to predict the soundfield in music

performance spaces, recording studio design, and for architectural design purposes. Such

a numerical tool is beneficial in the process of designing spaces with desirable acoustics, as

it enables predicting the performance before constructing the building. Similarly, a high-

accuracy simulation would be very beneficial in early stages of diffuser design, where sound

scattering from the boundary surface in time and space domain could be investigated. An

accurate acoustic model could also be applied to modelling of complex loudspeaker systems

with the emphasis on source directivity.

The methods developed for room acoustics could also be applied to virtual sound

environments and to the creation of spatial sound effects for multimedia applications.

Plausible sound field modelling could be applicable in naturally sounding reverberation

units. A good example of a multimedia application is the creation of a realistic reverberant

soundtrack for an animation or film.

In order to meet the aforementioned objectives, it is necessary to address the issues of

sound source modelling, acoustic space simulation and receiver modelling according to the

reproduction technique. However, in this thesis we focus entirely on room acoustics mod-

elling which constitutes the main modelling component. Sound propagation in an acoustic

space, room geometry, reflections at boundaries, occlusion, wave interference and diffusion

are key features that need to be reproduced in order to reach a high level of accuracy.

These objectives can be achieved with the use of numerical simulation techniques, which

is the main field of the undertaken research.

1.2 Room Acoustics Modelling

Research into numerical simulation of acoustic spaces is dominated by two distinct ap-

proaches, namely the geometrical and the wave-based approach. The former is based on

soundfield decomposition and is computationally relatively efficient. ODEON is a good


example of calculating impulse responses based on the geometrical approach [78]; it uses

the image source method [6] for early reflection modelling and the ray tracing technique for

modelling diffusive reverberation. Similar hybrid approaches are also described in the con-

text of virtual acoustics and auralisation in [85, 70]. The aim of DIVA (Digital Interactive

Virtual Acoustics) project is to create a real-time environment for full audiovisual experi-

ence [88]. In this case, the image source method is employed for modelling up to six early

reflections, while the late reverberation is generated using recursive filter structures to al-

low real-time processing. Although soundfield decomposition methods are efficient, their

formulation is not entirely physical, and consequently their predictive capacity is rather

limited. This limitation is generally apparent for low and middle frequency ranges, and

particularly so when applied to modelling small enclosures or rooms with highly nonrigid

walls [123, 110].

On the other hand, wave-based methods simulate the acoustical equations directly

and therefore have the advantage of inherently modelling wave-related phenomena such

as diffraction, be it that the computational costs for wideband applications are high,

especially for modelling and auralisation of 3D spaces. The past few years have seen a

rise of interest in wave-based methods, partly driven by the steady increase of commonly

available processing power. These methods include finite difference time-domain (FDTD)

methods, digital waveguide mesh (DWM) modelling, the finite element method (FEM), the

boundary element method (BEM), and the functional transform method (FTM). Wave-

based techniques have the advantage of modelling acoustic spaces with great detail, which

results in highly accurate simulations. However, this is achieved at the expense of the

computational cost, which rises exponentially with increasing sampling frequency. It is

therefore important to formulate these models as efficient as possible.

This research focuses on FDTD modelling, which is a good choice for virtual acoustic

applications for the following reasons. Firstly, a wide body of knowledge and methods has

been developed since the 1960s in the field of electro-dynamics, the underlying equations of

which are identical to those of acoustic systems. Secondly, unlike finite element methods,

FDTD methods tend to use uniform grids, which are more suited to auralisation of virtual

spaces with moving sources and receivers. Note that in general irregular grid spacing

causes undesirable filtering effects. Finally, the formulation and implementation of FDTD

models is relatively straight-forward in comparison to some of the other approaches.

Since a substantially higher accuracy of wave-based methods is in general offset by a

much higher computational cost than for geometrical methods, hybrid approaches are con-

venient to address the problem of fine-tuning the balance between accuracy and efficiency

[110]. Such hybrids generally rely on a combination of a rigorous numerical technique

such as the FDTD method and a computationally efficient geometrical method for high

frequencies, examples of which can be found in [43, 62, 76]. In the long term, one may


expect that the burden of computational costs will be lessened by the growth in commonly

available processing power, where the development of multicore processors could be the

key step forward in bringing rigorous simulations of large 3D spaces within reach.

The main focus of this thesis is on modelling the acoustics of enclosed spaces, and

hence the main issues that should be addressed include modelling sound wave propaga-

tion, models of generally frequency-dependent boundaries, and modelling surface diffusion.

Note that other wave-related phenomena are inherently incorporated in the FDTD tech-

nique. Since the aim is to model multiple moving sound sources and listeners, the use of

off-line post-processing techniques such as frequency warping is excluded. Consequently,

throughout this thesis, we concentrate primarily on FDTD schemes with significantly re-

duced dispersion error without introducing much increase in computational cost, which

could be applied to on-line simulations. Furthermore, the considerations are constrained

to compact schemes on a rectilinear topology since it allows a straightforward fit of the

grid to rooms with parallel walls, which are dominant in real architecture.

The problem of developing accurate formulations of boundaries is an essential ingre-

dient in creating realistic and predictive FDTD simulations, especially given that realistic

boundaries are generally frequency-dependent. Strictly speaking, complete physical mod-

els of boundaries should include the transmission of waves in the wall. However, simulation

results in previous studies [15, 14] have suggested that in many practical cases there is no

significant difference if wave propagation in the wall is neglected. Therefore, in this thesis

it is assumed that any room surfaces are locally reacting, i.e. the reflective properties of

any point on the wall are completely characterised by a local impedance. Such generally

frequency-dependent boundary models with ensured scheme consistency between the room

interior and the boundary have a clear advantage that analytic prediction techniques can

be applied and the stability of the whole simulation is always guaranteed.

For auralisation purposes, strong simulation predictivity is in some cases of lesser

importance, the main objective shifting to enabling good control over the properties of

the simulated space, such as the overall room diffusivity. In the latter context, methods

for modelling controllable surface diffusion are required.

The issue of modelling directional sound sources and converting the output of the finite

difference grid according to various reproduction formats are not dealt with in this study.

Some interesting solutions to exciting the mesh include implementing transparent sources

in the FDTD grids [97] and modelling frequency-dependent directivity of sources in the

closely related digital waveguide mesh [45]. As far as receiver modelling is concerned, for

the low frequency range only, capturing pressure waves in points near the positions of a

listeners ears should be sufficient [110]. A more general approach is based on plane-wave

decomposition which can be post-processed for the most of reproduction systems, but it

is computationally heavy [108]. Therefore, simple solutions for a specified reproduction


technique might be provide a useful balance, such as capturing B-format channels [107].

1.3 Thesis Overview

This thesis is divided in two major parts, namely the background information that can

be found in the literature and the authors original contributions. An overview of the

basics of acoustics in enclosed spaces and the review of numerical techniques that are

applied in chapters to follow are presented in the first two chapters of this thesis. These

constitute a theoretical foundation for the work presented thereafter. The subsequent

chapters constitute the contributions to the field of FDTD modelling of room acoustics.

In Chapter 2, the fundamentals of acoustics related to sound propagation in enclosed

spaces are briefly reviewed, with the main focus on the properties of medium and bound-

aries. Basic acoustic laws are reviewed and the concept of locally reacting surfaces is dis-

cussed. In addition, this chapter provides an extensive overview of commercially available

diffusers and the standardised technique to measure the diffusivity of boundary surfaces.

Chapter 3 provides an overview of a number of numerical modelling techniques and

highlights the equivalence of various approaches. Methods based on the geometrical and

wave-based approaches are briefly discussed, and a more detailed motivation for the choice

of the FDTD method is provided. A number of techniques that are considered a subclass of

FDTD methods are reviewed, namely the digital waveguide mesh and the family compact

schemes based on staggered and nonstaggered grids. In particular, the analysis of stability

and dispersion of such methods is used to define the equivalence of these approaches.

For each technique, a short review of the boundary models available in the literature is

also provided. In addition, a compact implicit technique is discussed and the issue of a

computationally efficient implementation using the alternating direction implicit technique

is addressed.

Chapter 4 deals with modelling sound wave propagation in air, also referred to as

medium modelling. The family of compact finite difference time domain schemes based

on a nonstaggered rectilinear grid for approximating the 2D and 3D wave equation is

discussed. The issues of stability, accuracy and computational efficiency are investigated

for numerous special cases of a wide family of compact explicit and implicit schemes. The

presented analysis covers a wide range of techniques commonly used in the context of audio

such as the rectilinear digital waveguide mesh, the interpolated digital waveguide mesh,

and FDTD schemes such as the standard leapfrog, the octahedral, and the tetrahedral

schemes. As an alternative to these explicit techniques, the use of a fourth-order accurate

compact implicit finite difference technique is proposed for simulations in which very

high accuracy is required. The compact implicit formulation is presented for 2D and 3D

cases, including the most efficient splitting formulae for the alternating direction implicit


implementation. Compact explicit and implicit FDTD schemes are compared in terms of

numerical dispersion error, valid frequency ranges for accuracy and isotropy, computational

cost and overall efficiency.

The remaining chapters are about modelling the boundaries. Chapter 5 presents new

methods for constructing and analysing formulations of locally reacting surfaces that can

be used in FDTD simulations of acoustic spaces. Novel FDTD formulations of frequency-

independent and simple frequency-dependent impedance boundaries are proposed for 2D

and 3D acoustic systems, including a full treatment of corners and boundary edges. The

proposed boundary formulations are designed for virtual acoustics applications using a

rectilinear, nonstaggered grid, and apply to FDTD as well as Kirchhoff variable digital

waveguide mesh methods. These models include simple frequency-dependent boundaries

in which the wall is characterised by a complex impedance expression that incorporates lin-

ear resistance, inertia, and restoring forces. In addition, a new analytic evaluation method

that accurately predicts the reflectance of numerical boundary formulations is proposed.

The results obtained from numerical experiments and numerical boundary analysis (NBA)

are analysed in time and frequency domains in terms of the pressure wave reflectance for

different angles of incidence and various impedances. The proposed boundary models are

compared with the frequency-independent 1D boundary model commonly applied to ter-

minate the digital waveguide mesh [90] and Botteldoorens boundary model for staggered

Yees grid [15].

The extension to modelling generally frequency-dependent boundary models is pro-

posed in chapter 6. The proposed approach allows direct incorporation of a digital

impedance filter (DIF) in the multidimensional (i.e. 2D or 3D) FDTD boundary model

of a locally reacting surface. An explicit boundary update equation is obtained by care-

fully constructing a suitable recursive formulation. The method is analysed in terms of

pressure wave reflectance for different wall impedance filters and angles of incidence. Its

performance is compared with the performance of the 1D model in which a reflectance

filter is combined with the FDTD room interior implementation using KW-pipes [53, 75].

In Chapter 7, the formulation of a novel digital impedance filter model for any member

of the family of 2D and 3D compact explicit FDTD schemes is proposed. Since the interpo-

lated scheme equation represents the most general form of compact explicit schemes, such

a boundary formulation is in this thesis referred to as the interpolated digital impedance

filter model. Such a formulation naturally encompasses the boundary models for all other

compact explicit schemes, which are obtained by setting the values of the respective free

parameters.

In Chapter 8, a method for modelling diffusive boundaries in finite difference time

domain (FDTD) room acoustics simulations with the use of digital impedance filters is

proposed. The presented technique is based on the concept of phase grating diffusers,


and is suitable for modelling scattering from small irregularities in the boundary surface

and diffusers consisting of narrow wells. A range of diffuser types is investigated through

numerical experiments, generally giving good agreement with theory. It is proposed that

irregular surfaces are modelled by shaping them with Brownian noise, giving good control

over the sound scattering properties of the simulated boundary through two parameters,

namely the spectral density exponent and the maximum well depth.

1.4 Contributions

The main contributions of this thesis are as follows:

A new compact explicit FDTD scheme is identified - named the interpolated wide-band scheme - which provides the full bandwidth in 2D and 3D simulations, exhibits

no dispersion error in axial directions, and is shown to be an excellent choice regard-

ing accuracy and efficiency.

The formulation of the boundary condition in terms of pressure only, which appliesto schemes based on unstaggered grids, is proposed.

A new frequency-independent boundary model of a locally reacting surface for afamily of compact explicit schemes is proposed.

Novel formulation of simple frequency-dependent walls incorporating linear resis-tance, inertia and restoring forces is proposed.

The digital impedance filter (DIF) boundary model is introduced - a new methodfor modelling generally frequency-dependent boundaries of a locally reacting surface

type. A structurally stable and efficient explicit boundary formulation is constructed

by carefully combining the boundary condition in the direction normal to the bound-

ary surface with the compact explicit update equation.

All the boundary models proposed in this thesis include physically-correct formula-tions of corner and boundary edge nodes, which appears to have never been addressed

in the literature on FDTD/DWM room acoustics simulations.

Numerical Boundary Analysis (NBA) is proposed - a new analytic method for theexact prediction of the numerical boundary reflectance of multidimensional boundary

models, such as those proposed in this thesis, thus removing the need for carrying

out elaborate numerical experiments to evaluate the boundary performance.

A useful method for modelling phase grating diffusive boundaries by designingboundary impedance filters from normal-incidence reflection filters with added delay


is proposed. These added delays, that correspond to the diffuser well depths, are

varied across the boundary surface, and implemented using Thiran allpass filters.

This technique is suitable for modelling high frequency diffusion caused by small

variations in the surface roughness and, more generally, diffusers characterised by

narrow wells with infinitely thin separators.

In addition, this work also includes the following minor contributions:

The application of the fourth-order accurate nonstaggered compact implicit schemeimplemented using alternating direction implicit technique is proposed for the first

time in the field of audio and acoustics. This method constitutes an efficient al-

ternative to explicit methods when an extremely high accuracy of the 2D or 3D

simulations is required.

The most accurate and isotropic compact schemes are identified, and most efficientones indicated.

The most accurate and isotropic in numerical reflectance digital impedance filterboundary models are identified.

A method to control sound scattering properties in numerical simulations to matchdiffusion coefficient data by shaping surface roughness with a Brownian noise is

proposed.

1.5 Related Publications

Some parts of the work presented in this thesis have been published in the form of con-

ference proceedings and journal articles.

Conference proceedings

1. K. Kowalczyk and M. van Walstijn, On-line simulation of 2D resonators with re-duced dispersion error using compact implicit finite difference schemes, Proc. IEEEInt. Conf. on Acoustics, Speech and Signal Process. (ICASSP), pp.285-288, April2007, Honolulu, Hawaii.

2. K. Kowalczyk and M. van Walstijn, Formulation of a locally reacting wall in finitedifference modelling of acoustic spaces, Int. Symp. on Room Acoustics (ISRA),pp.1-6, September 2007, Seville, Spain.

3. K. Kowalczyk and M. van Walstijn, Virtual room acoustics using finite differencemethods. How to model and analyse frequency-dependent boundaries?, Proc. IEEEInt. Symp. on Communications, Control and Signal Process. (ISCCSP), pp.1504-1509, March 2008, St. Julians, Malta.


4. K. Kowalczyk and M. van Walstijn, Modeling frequency-dependent boundaries asdigital impedance filters in FDTD and K-DWM room acoustics simulations, 124thConvention of the Audio Eng. Soc., prepring no. 7430, May 2008, Amsterdam, TheNetherlands. An extended manuscript appeared in the Journal of the AES.

5. M. van Walstijn and K. Kowalczyk, On the numerical solution of the 2D wave equa-tion with compact FDTD schemes, Int. Conf. on Digital Audio Effects (DAFx),pp. 205-212, September 2008, Espoo, Finland.

Journal articles

1. K. Kowalczyk and M. van Walstijn, Modeling frequency-dependent boundaries asdigital impedance filters in FDTD and K-DWM room acoustics simulations, J.Audio Eng. Soc., vol. 56, No. 7/8, pp. 569-583, July/August 2008.

2. K. Kowalczyk and M. van Walstijn, Formulation of a locally reacting wall inFDTD/K-DWM modeling of acoustic spaces, Acta Acustica united with Acustica,accepted for publication in the special issue on Virtual Acoustics, vol. 94, No. 6,pp. 891-906, November/December 2008.

3. K. Kowalczyk and M. van Walstijn, Wideband and isotropic room acoustics simu-lation using 2D interpolated FDTD schemes, IEEE Trans. on Audio, Speech andLanguage Processing, accepted for publication.

4. K. Kowalczyk, M. van Walstijn, and D.T. Murphy, A phase grating approach tomodelling surface diffusion in FDTD room acoustics simulations, IEEE Trans. onAudio, Speech and Language Processing, submitted for publication.

11

Chapter 2

Fundamentals of Room Acoustics

The aim of this chapter is to briefly review the basics of acoustics related to sound prop-

agation in enclosed spaces. The main focus is on the properties of the medium, leading

to the wave equation, and next on the boundary condition and analytic formulae for

the boundary impedance of a locally reacting surface. The final part reviews currently

available diffusers and the measurement setup for capturing diffusion coefficient.

This chapter is structured as follows. Firstly, the most important acoustic laws appli-

cable to room acoustics are presented in Section 2.1, followed by sound pressure level defi-

nition in Section 2.2. The theoretical formulation of a locally reacting surface is presented

in Section 2.3, including the definition of the boundary impedance and the derivation of

a reflection coefficient. In Section 2.4, an analytic method to calculate modal frequencies

of rectangular room with completely rigid walls is discussed. Section 2.5 provides analytic

formulae for the impedance of acoustic porous materials. An overview of available diffusers

is provided in Section 2.6, focusing on their structure and diffusive properties. Finally, the

setup for diffusion coefficient measurements is discussed in Section 2.7.

2.1 Wave Equation

When a sound wave propagates, the particles of the medium undergo vibrations about

their mean positions. In some regions they may be pushed together, whereas in others

they are pulled apart. Once the wave has passed, the particles return to their original

state. Consequently, the variations of both pressure and velocity occur as functions of time

and space. Sound pressure is defined as a difference between the instantaneous pressure

and the static pressure. The velocity of particle displacement is yet another important

quantity characterising a travelling sound wave.

Even though in large concert halls some variations of temperature cannot be avoided

and air conditioning systems may cause air not to be completely at rest, such inhomo-

Chapter 2. Fundamentals of Room Acoustics 12

geneities are relatively small and can be neglected. Therefore, it seems justified to assume

that the air in the interior of the room can in ideal conditions be regarded as homogeneous

and at rest.

In such a homogeneous isotropic loss-free medium, sound velocity is constant with

reference to time and space. Under these conditions, the magnitude of sound velocity c in

m/s is given as [64]

c = (331.4 + 0.6), (2.1)

where is the temperature in centigrade. Sound wave propagation in air is governed by

two basic laws, namely the conservation of mass and the conservation of momentum [64].

The former is expressed by

p+ ut

= 0, (2.2)

and the latter is given byp

t+ u = 0, (2.3)

where p denotes the acoustic pressure, u is the vector particle velocity, is the air density,

c is the sound velocity, and the adiabatic exponent is given by

= c2. (2.4)

In these equations, we assume that time dependent changes in particle velocity are small

compared to the static values and that particle velocity is substantially smaller than the

sound velocity. These linear equations are typically used to describe practical conditions

for room acoustics. Note that this assumption is typically made in room acoustics, and it

does not hold for high-amplitude sound such as that produced by jet engines. The wave

equation can be derived by eliminating the particle velocity from Equation (2.2) using

Equation (2.3), which yields2p

t2= c22p, (2.5)

where 2p is given as2p =

2p

x2, (2.6)

2p = 2p

x2+2p

y2, (2.7)

2p = 2p

x2+2p

y2+2p

z2, (2.8)

in a 1D, 2D, and 3D acoustic system, respectively; x, y, and z are directions of an x-

y-z Cartesian coordinate system. This differential equation is fundamental in the field


of acoustics and applies to waves of any type of the wavefront. Furthermore, it holds

not only for sound pressure variations but also for density and temperature variations.

By applying the Fourier transform to the wave equation given by Equation 2.5, the time

invariant version of the wave equation, known as the Helmholtz equation, results

p+ k2 p = 0, (2.9)

where k denotes the wave number of the wave that is given by

k =

c, (2.10)

and is the angular frequency. The frequency of vibration is given as f = 2/ in Hz,

whereas the wavelength can be calculated from

=c

f. (2.11)

The wave equation in theory determines the sound pressure at all positions and at all

times. However, it can only be solved analytically for very special cases for predescribed

boundary conditions. Therefore, numerical techniques are necessary to approximate solu-

tions of the wave equation for more general acoustic spaces.

2.2 Sound Pressure Level

As has been mentioned in Section 2.1, sound pressure is a difference between the pressure

caused by the passing wave and the ambient pressure at a particular point in space. The

effective sound pressure is the root mean square (RMS) of such a pressure difference

measured over a period of time at the point in space, according to

prms =

p12 + p22 + ...+ pn2

N, (2.12)

where p1, p2, ..., pn is the instanteous pressure in Pa measured over N samples. Due to

the nature of the human hearing, sound pressure is often expressed on a logarithmic scale

in relation to a reference pressure value po by

L = 20logprmspo

, (2.13)where L denotes the level difference between two sound pressure values, measured in dB.

If the reference pressure value is taken as po = 2 105N/m2, which is the threshold of

hearing at a frequency of 1kHz, the resulting value L is the sound pressure level.


2.3 Locally Reacting Surfaces

In general, room acoustics concerns sound propagation in enclosures, where a medium is

bounded by side walls, floor and ceiling. Most boundaries in rooms reflect some portion of

the impinging energy, and some fraction of energy is absorbed. In this section, we consider

a reflection of a plane sound wave from a single surface.

Concerning the shape of the incident wave, we assume an incident wave to be plane,

that is the wave is propagating in one direction only. The name - plane wave - stems from

a planar surface of a constant phase which is perpendicular to the propagation direction.

Plane waves are actually hard to encounter in reality. In real room acoustics, we primarily

deal with spherical waves or sections of spherical waves [64]. The most similar to plane

waves is the wavefield caused by an infinite number of monopoles distributed along a line,

for which a cylindrical wave results [64]. However, when a reflecting wall is sufficiently

far away from the source position, the curvature of the wavefront can be neglected and

the resulting error caused by substituting a spherical wavefront with a plane wave can be

considered negligible.

The wall considered in this section is assumed to be unbounded and plane. However,

very small irregularities (i.e., roughness of the surface that is much smaller than the wave-

length of the incident sound wave) along a wall that is much larger then the wavelength

are not excluded in this consideration.

A wave reflected from a wall has both phase and amplitude that differ from those of

an incident wave. The incident and reflected waves interfere with each other creating (at

least partially) a standing wave. We can express such changes with a reflection coefficient

R defined as a function of frequency, herein also referred to as reflectance in order to

emphasize its frequency-dependency. Such a reflectance completely defines the acoustic

properties of a wall for any frequency and angle of incidence.

2.3.1 Boundary Impedance

Two prime quantities in room acoustics theory are sound pressure and particle velocity.

The first one is a scalar, whereas the second is a vector quantity. For convenience, the scalar

value defined as the normal component of the particle velocity has been introduced. The

term normal refers to both wavefront or the boundary surface when a sound wave encoun-

ters a wall. At a right boundary (depicted in Figure 2.1), the ratio between the pressure

and the normal component of the particle velocity defines the boundary impedance

Zw =p

ux, (2.14)


where p denotes pressure and ux is the velocity component that is normal to the surface

of the boundary. The boundary impedance is generally complex as the reflection alters

both amplitude and phase of an incident wave. The boundary impedance is often divided

by the characteristic impedance of air

w =Zwc

, (2.15)

in which case it is referred to as the specific acoustic impedance. The typical value for the

characteristic impedance of air at normal condition is [64]

oc = 414kgm2s1. (2.16)

The inverse of the specific acoustic impedance is the specific acoustic admittance defined

as

Yw =1

w. (2.17)

The intensity of the reflected wave is reduced by |R|2 in comparison to the incidentwave. Based on this property, an alternative quantity to the reflection coefficient can be

formulated. This quantity is referred to as the absorption coefficient and is given as

= 1 |R|2. (2.18)

Note that the absorption coefficient only defines the change in amplitude but does not

include any information about the phase change.

2.3.2 Reflection at Normal Incidence

In this section, we explore the relationship between the boundary impedance and reflection

coefficient at normal incidence, largely following the derivation in [64]. Let us first consider

a wall parallel to the x-axis of a rectangular coordinate system x-y. An incident plane wave

is travelling in the positive x-direction, as depicted in Figure 2.1, for which the pressure

and velocity component normal to the wall are given respectively as

pi(x, t) = Po ej(tkx) (2.19)

and

ui(x, t) =Poc

ej(tkx). (2.20)

When such an incident wave interacts with the boundary at normal incidence, the

propagation direction is reversed. In addition, the amplitude of the reflected wave de-

creases due to boundary absorption and phase undergoes a change. In practice, phase


x=0

pi

pr

Figure 2.1: Plane wave reflection for a right boundary located at x = 0 at normal angle of incidence.

alterations occur for nonrigid walls such as curtains, light nonstiff walls, wall and floor

coverings. Both these changes are fully defined by the reflection coefficient R. Further-

more, the flow is reversed, and hence a change in sign of the particle velocity is required.

Thus, the pressure and particle velocity of the reflected wave are given as

pr(x, t) = R Po ej(t+kx) (2.21)

ur(x, t) = R Poc

ej(t+kx), (2.22)

respectively. The total sound pressure and particle velocity in front of the wall are obtained

by adding the respective values of the incident and reflected waves. In addition, we assume

that the boundary is located at x = 0 for simplicity. Consequently, the total pressure and

particle velocity in the plane of the wall is

p(0, t) = (1 +R) Po ejt (2.23)

u(0, t) = (1 R) Poc

ejt. (2.24)

Dividing the pressure value by the normal component of the particle velocity u yields the

boundary impedance

Zw = c1 +R

1R, (2.25)

from which the reflection coefficient R can be found as

R =Zw cZw + c

=w 1w + 1

. (2.26)

Three extreme cases for the value of the boundary impedance are:


x=0

pi

pr

Figure 2.2: Plane wave reflection for a boundary located at x = 0 at oblique angle of incidence.

A hard wall, in which case the boundary is infinitely rigid (|Zw| ). Thus thereflection coefficient amounts to R = 1 and total reflection occurs.

A soft wall is characterised by Zw = 0, in the case of which R = 1. This time,reflection is also total but out of phase.

When the boundary impedance equals the characteristic impedance of the mediumZw = c, R = 0 and a completely absorbent wall is obtained.

2.3.3 Reflection at Oblique Incidence

Let us consider an incident wave propagating in the direction x, for which the deviationfrom the direction normal to the wall is given by an angle , as illustrated in Figure 2.2.

Again, without the loss of generality this problem can be treated in two dimensions. Such

a propagation direction is related to the x-y coordinate system by

x = (x cos + y sin ). (2.27)

The pressure pi and the particle velocity component ui of an incident wave that is normal

to the boundary are given respectively by

pi(x, y, t) = Po ejt ejk(x cos +y sin ) (2.28)

ui(x, y, t) =Poc

cos ejt ejk(x cos +y sin ). (2.29)

Similarly to the case of normal incidence reflection, the sign of x is reversed in the

reflected wave because of the change of the propagation direction, and amplitudes are

amended respectively. Consequently, the pressure and normal velocity component of the


reflected wave are

pr(x, y, t) = R Po ejt ejk(x cos +y sin ), (2.30)

ur(x, y, t) = R Poc

cos ejt ejk(x cos +y sin ). (2.31)

By setting x = 0 at the wall, and adding the pressure and velocity components of both

incident and reflected waves, the total values along the boundary surface are obtained

p(0, y, t) = (1 +R) Po ejt ejky sin , (2.32)

u(0, y, t) = (1R) Poc

cos ejt ejky sin (2.33)

Finally, the boundary impedance is obtained by dividing the total pressure by the total

normal velocity component

Zw =c

cos

1 +R

1R, (2.34)

from which the reflection coefficient is obtained as

R =Zw cos cZw cos + c

, (2.35)

or, expressed in terms of the specific boundary impedance, it is given as

R =w cos 1w cos + 1

. (2.36)

2.3.4 Discussion

Most of the boundaries in enclosed spaces are solids, such as concrete brick walls, in

which case additional shear waves are excited for an oblique-incident sound wave [82].

Furthermore, there are several types of waves travelling on the boundary surface, Rayleigh

waves are good examples. Several types of transverse wave motions of the solid should be

taken into account when the width of the solid is much smaller than boundary dimensions

[8]. Consequently, the reflection coefficient would have to be replaced with a reflection

function that defines the reflected wave at one position when the boundary is excited at

another position. The theoretical treatment of a vibrating panel on a wall in a rectangular

room is provided in [82]. However, such nonlocally reacting walls are hard to treat properly

in practice due to the lack of detailed measurement data and modelling problems.

In the context of room acoustics, the boundary impedance is often assumed to be in-

dependent on the angle of the incident sound wave. This simplification is only true for

walls in which the particle velocity at the boundary surface depends solely on the sound

pressure in front of the wall element, and not on the pressure of neighbouring elements

[64]. Such walls are referred to as locally reacting surfaces. The locally reacting surface is


encountered when a wall itself and the space behind the wall does not allow wave prop-

agation in the direction parallel to the boundary surface; seat and floor coverings, heavy

curtains, and light nonstiff walls are good examples. In the context of computer simula-

tions of room acoustics, the assumption of local reaction reduces the diffusive properties of

simulated walls, decreasing the overall diffusiveness of the simulated space. Consequently,

techniques for modelling diffusion can be used to compensate for the lack of real nonlocally

reacting boundaries.

2.4 Eigenmode Model (for Rigid Walls)

This section deals with searching for the solutions of the wave equation using series of

eigenmodes and eigenfunctions. Even though eigenmodes occur for enclosures of arbitrary

shapes, analytic solutions can only be found for special cases of the room geometry and

simple boundary impedance values. For instance, complex eigenvalues result for a complex

boundary impedance, and the solution cannot generally be analytically found without

further approximations.

For a rigid boundary (w ), the normal velocity component is zero, and thus theboundary condition reduces to

p

n= 0. (2.37)

Consider an acoustic space of a rectangular geometry and dimensions (Lx, Ly, Lz), in

which all walls are parallel to the axis of the Cartesian coordinate system. The Helmholtz

equation can be solved by separation of variables and composing the solution of three

factors

p(x, y, z, ) = px(x, )py(y, )pz(z, ), (2.38)

each depending solely on one propagation direction. The Helmholtz equation is split into

three ordinary equations. For instance px satisfies equation

pxx

+ k2xpx = 0, (2.39)

where kx denotes the wavenumber in x-direction. Furthermore, the separation of variables

applies to the boundary condition, i.e.

pxx

= 0 (2.40)

for x = 0 and x = Lx. Analogous conditions apply to y- and z-directions, and the

respective directional wavenumbers are related to each other by

k2 = k2x + k2y + k

2z . (2.41)


The solution that satisfies the boundary condition given by Equation (2.40) is px =

cos(kxx), in which the allowed values for the wavenumber kx are

k2x =nx

Lx, (2.42)

where nx is a nonnegative integer. Consequently, the eigenvalues of the wave equation are

given by

knxnynz = [(nx

Lx

)2+(nyLy

)2+(nzLz

)2] 12

, (2.43)

and associated eigenfunctions are given as

pnxnynz(x, y, z) = cos(nxx

Lx

)+ cos(

nyy

Ly

)+ cos(

nz

Lz

). (2.44)

Equation (2.44) multiplied by ejt represents a 3D standing wave. Standing waves are

referred to as room modes and their respective frequencies as modal frequencies. The

eigenfrequencies corresponding to the eigenvalues are given as

fnxnynz =c

2knxnynz . (2.45)

2.5 Acoustical Porous Material

A common practice in room acoustics is taming unwanted reflections in an enclosed space.

For that purpose an absorptive material is often applied to walls and other reflective

surfaces. The most popular choices include dense porous materials such as polyurethane

foam and fiberglass. Carpet and drapes are examples of soft fibrous materials, which are

commonly applied as room fittings in order to absorb high frequencies. The sound is

absorbed in porous material by converting the acoustical energy into small amounts of

heat. In order to simulate such materials in computer simulators, information about the

boundary impedance of porous materials is necessary.

Delany and Bazley [29] have proposed empirical expressions for characteristic impedance

Zw() and the propagation constant () for porous materials. Their formulation is based

on a large number of measurements of fibrous materials with porosities. The porosity is

defined as the ratio of the fluid volume occupied by the continuous fluid phase to the total

volume of the porous material.

If we denote the static air flow resistivity as expressed in Nm4s, the expressionsproposed by Delany and Bazley for the impedance and the propagation constant at normal


room conditions are as follows

Zw() = c[1 + 0.0571

(f

)0.754 j0.087

(f

)0.732 ], (2.46)

() =j

c

[1 + 0.0978

(f

)0.70 j0.189

(f

)0.595 ]. (2.47)

According to [29], such boundary expressions are valid in the following range of the

flow resistivity values

0.01 2w. (2.59)

On the other hand, good scattering should not be expected for frequencies even half an

octave below the design frequency [99]. The depths of the quadratic residue diffuser are

given by

dn =o2

snN, (2.60)

where o is the design wavelength. An example quadratic residue diffuser for the sequence

of length 7 is depicted in Figure 2.4. The majority of commercially available QRDs are 1D,

that is their well depths are changing in one direction only (see Figure 2.5). Such diffusers

are mainly used for lateral scattering. Thin rigid separators between individual wells are

very important in the scattering process, especially for oblique incidences [99]. The lack

of these fins would greatly diminish scattering properties resulting in poor diffusion.

Alternatively, QRD could be constructed as a reflecting planar hard wall with vary-

ing local impedance in a periodic fashion, where the reflection coefficients are given by

Equation (2.57). However, such rapid changes in the boundary impedance along the wall

surface cannot be manufactured [99].

The 1D quadratic residue diffuser can be extended to 2D by replacing a quadratic

residue sequence sn in Equation (2.60) with two quadratic residue sequences sk and sl,


Figure 2.5: 1D quadratic residue diffuser.

which yields

dn =o2

sk + slN

, (2.61)

where sl = l2 and sk = k

2, l = 0, 1, 2, ... and k = 0, 1, 2, ....

2.6.3 Modulated Quadratic Residue Diffuser

Quadratic residue sequences are of finite length, and so is the size of a standard manufac-

tured quadratic residue diffuser. Therefore, in order to cover a large wall area, individual

diffusers are typically concatenated. However, repetition of the sequence causes a more

concentrated reflection of energy into gratings, the sharpness of which depends on the

number of repeats. The more repeats, the sharper the lobes. A solution similar to mod-

ulating the carrier in communication systems has been proposed by Angus [7], which is

based on using a pseudorandom spreading sequence.

Modulated Phase Reflection Grating

The modulated phase reflection grating is a modulation technique which is based on using

a normal and inverted version of the basic sequence according to the pseudorandom binary

sequence, such as MLS or Barkley codes [7]. As a result, the diffusion comparable to the

scattering of a single diffuser is achieved. Such an inverted diffuser can be viewed as an


upside down version of the original diffuser. Furthermore, an inverted sequence is modulo

N equivalent to the normal sequence if zero depths in the inverted diffuser are replaced

with maximum depths [7]. For instance, the original quadratic residue sequence of length

5 is

(0, 4, 1, 1, 4), (2.62)

and its inverted version is

(5, 1, 4, 4, 1). (2.63)

Sequence inversion modulated gratings greatly reduce the effect of narrow energy lobes

caused by periodicity of the sequence. However, the diffusion characteristic is not enhanced

compared to having just one quadratic residue diffuser, and hence high scattering can only

be expected at integer multiplies of the design frequency.

Orthogonal Modulation

Orthogonal modulation is based on interconnecting two quadratic residue diffusers of dif-

ferent lengths but with the same maximum depth. This way a variation in the step

size is introduced [7]. Similarly to the modulated phase reflection grating technique, the

maximum length sequence is typically applied as a pseudorandom binary sequence.

The well depths of the orthogonally modulated quadratic residue diffuser are given by

dn = n2modN(

dmaxN

) n = 0, 1, 2, ..., N (2.64)

where N is a sequence length and dmax is the well depth that corresponds to a sequence

value equal to N . For instance, an orthogonal modulation of sequences of lengths 5 and 7

reads

dn =(0,1

5,4

5,4

5,1

5, 0,

1

7,4

7,2

7,2

7,4

7,1

7

)dmax. (2.65)

Such a composite structure, which results from orthogonal modulation, generally gives

better diffusion for higher frequencies since there are now two integer multiplications of

the design frequency [7]. In fact, it does not really behave like a flat panel up to the

frequency determined by the lowest common multiple of the sequence lengths. Such a flat

panel frequency is given by [7]

fflatplate =N1 N2 c

2dmax, (2.66)

where N1 and N2 are the two respective quadratic residue sequence lengths. A more

stringent limit at high frequencies may come from the widths of the diffuser wells.


Figure 2.6: Lateral cross section through a diffractal based on a quadratic residue sequence N = 7.

2.6.4 Diffractals

Addressing the issue of diffusion for bass and high frequencies simultaneously is possible

with the use of diffractals (i.e., diffusing fractals). Diffractals are constructed by embedding

small scaled versions of the quadratic residue diffuser at the bottom of each well of a larger

quadratic residue diffuser [26]. An example diffractal is illustrated in Figure 2.6. A small

scaled version of the diffuser scatters mid-high frequencies, whereas a larger diffuser deals

with the scattering of the bass frequencies. Both diffusers are orthogonal and independent

in performance as large wavelengths are unaware of the small surface irregularities of the

high frequency diffuser, and the polar distribution for short wavelengths remains unaffected

by the phase changes introduced by the low frequency diffuser. Diffractals provide an easy

means of introducing an additional low-frequency absorption along with low frequency-

diffusion by fitting damped diaphragmatic membranes or thick porous layers to the rear

of the diffractal [26].

2.6.5 Curved Diffusers

Schroeder diffusers scatter sound uniformly for oblique angles of incidence. However,

strong equalising flows develop between the elements of the quadratic residue diffuser

causing an additional absorption at low frequencies [40], [65]. Such extra bass absorption

is usually advantageous in studio spaces, but in large concert halls additional absorption

may actually be disadvantageous. Therefore, simple curved reflectors based on the arc

of a circle have been suggested as an alternative, and the optimisation techniques have

been proposed in order to find the best diffusers for covering large wall areas [20]. It

is important to reduce the number of curvatures in the design of such surfaces so that

additional absorption is not introduced.

Through optimisation, a rough semicircular shape has been found the best curved

diffuser if the width of a diffuser does not exceed 1m [20]. Such a simple semicircular

shape produces fairly uniform diffusion for on-axis sources but scattering is less uniform

for oblique incidences. Furthermore, concatenating diffusers that are semicircular in shape


again reduces their diffusive properties. Therefore, for wider diffusers (e.g. diffusers which

are 4m wide), the optimised curvature structures prove better scatterers than a simple

semicircular shape [20].

2.6.6 Fractional Brownian Diffusers

Good sound scatterers can be realised with the use of the Fourier synthesis technique, in

which a Gaussian white noise signal is spectrally shaped with the use of a linear roll off

filter [22]. The decrease in the spectral content of the input signal depends on the gain

of the filter at each frequency band. Since the shape generated by the Fourier synthesis

technique represents Brownian motion, such sound scatterers are referred to as fractional

Brownian diffusers. A simple roll off filter is given as

A(f) =1

f/2, (2.67)

where is the spectral density exponent. For = 1, pink noise results in which the

roll off is of 3dB/octave. For higher values of the spectral density component, Brownian

noise is obtained in which the sharpness of spikes decreases with . For instance, a roll

off of 6dB/octave is achieved for = 2. In order to obtain the best performance at low

frequencies, the spectral density component should be large, leading to a smoother shape.

Good high frequency scattering results for a more spiky shape which is obtained for low

values of .

The results presented in [22] indicate that the sound scattering properties of fractional

Brownian diffusers at high and low frequencies are determined to a great extent by the

roll off filter. However, the diffusive quality does not solely depend on the spectral density

component but also on the white noise sequence. Therefore, the optimisation technique has

been proposed in order to have a total control of the diffusive properties. Such optimisation

is also useful for manufacturing purposes, in which a spiky shape is hard to produce.

However, in practice a number of fractional Brownian diffusers have to be repeated to

cover large wall areas, inevitably leading to the decrease in the diffusive properties caused

by the pattern repetition, in a similar fashion to concatenated Schroeder diffusers. By

analogy to fins of Schroeders diffusers, the flow of sound energy around sharp spikes may

increase absorption of fractional Brownian diffusers [22].

2.7 Diffusion Coefficient

The diffusion coefficient is a quantity developed in order to evaluate the quality of the

diffuse reflections from a scattering surface. It is a measure of the uniformity of the polar

response of sound scattering across the range of reflection directions. All mechanisms of


Figure 2.7: Setup for the measurement of the diffusion coefficient according to AES standard [3].Squares indicate a changing position of a source and circles indicate receivers.

the diffuse reflection are simultaneously evaluated, including scattering effects caused by

the roughness of the boundary surface and the edge diffraction caused by finite-size of the

boundary element.

The definition and guidelines concerning the measurement of the diffusion coefficient

are provided in the Audio Engineering Society information document for room acoustics

and sound reinforcement systems entitled Characterisation and measurement of surface

scattering uniformity [3]. Polar response information, that is the distribution of reflected

energy across all possible reflection directions for a given angle of incidence, is necessary

to compute the diffusion coefficient value. According to [3], scattering properties are

generally evaluated on a single 2D plane of reflection or across the hemisphere depending

on the diffuser. For a single plane diffuser, the measurement should be undertaken in the

plane of maximum diffusion. The source positions and receivers positions are placed on

a semicircle in the front plane of the diffuser. In order to measure the random-incidence

diffusion coefficient, the maximum angular resolution of receivers should not exceed 5o,

covering a semicircle around the reference normal. On the other hand, source positions

should be measured with a maximum resolution of 10o on a semicircle with a larger

radius, as illustrated in Figure 2.7. The measurements should ideally be undertaken in

an anechoic chamber to exclude reflections from other surfaces such as walls, floor and

ceilings. Alternatively, a room sufficiently large compared with the object under test can

be chosen as long as an appropriate window is applied to the measured impulse response

so that unwanted side-wall reflections are removed from the measured signal.

One obvious improvement to the measurement procedure presented in [3] would be to

undertake measurements with and without the test sample and the application of a longer

window that also allows first-order reflections from surrounding boundaries. Subtracting

these two impulse responses would remove first-order reflections from side walls. This way

longer input signals characterised by reach spectral content could be applied.

Concerning the test sample, the entire diffuser structure applied in a real acoustic

space should be tested in order to ensure that all the surface roughness diffraction and


edge effects are characterised. However, in the vast majority of cases the whole diffuser

is too large to be fitted in an anechoic chamber. Therefore, according to [3] the size of

the diffuser sample should be large enough so that at least 4 complete repete sequences

of a periodic test surface are included in a sample. For nonperiodic diffusers, the size of

the sample should be sufficient so that surface scattering effects are more prominent than

edge diffraction effects.

Ideally, diffusion coefficient measurements should be undertaken under true far field

conditions for on-axis scattering, which are met when [3]

r Dmax,r

Dmax Dmax

, (2.68)

where Dmax is the largest dimension of the diffuser, o is the wavelength, r1 is the distance

from the source to the reference point, r2 is the distance from the receiver to the reference

point, and

r =2r1r2r1 + r2

. (2.69)

True far field conditions require large measuring distances, often much larger than can

be realistically achieved. Fortunately, true far field conditions are not necessary. If true

far field conditions cannot be achieved, the general requirement is that at least 80% of

receivers are placed outside of the specular zone, which is defined as the region over which

a geometric reflection occurs [23].

Directional diffusion coefficients for a particular source position are calculated from

impulse responses measured at all receiver positions. The impulse response with the test

surface present h1(t) and without the test surface h2(t) are obtained in two successive

measurements. This way the unwanted background reflections can be removed by sub-

tracting h1(t) from h2(t). Furthermore, an impulse response h3(t) from loudspeaker (used

as a source) to microphone (used as a receiver) should be measured, and next deconvolved

from the subtracted impulse response as

h4(t) = IFT[FT [h1(t) h2(t)]

FT [h3(t)]

], (2.70)

where FT denotes a forward Fourier transform and IFT denotes an inverse Fourier trans-

form. A rectangular window is next applied, the size of which is determined such that the

full test response is obtained and side-wall reflections are excluded. A windowed impulse

response should be Fourier-transformed and RMS pressure amplitude levels are calculated

in each one-third octave band. Such frequency ranges should conform to the ISO 266

standard [1]. Sound pressure levels, defined as power in each frequency band, serve as a


basis for calculating the directional diffusion coefficient. For a fixed source position and

sound pressure levels Li from n receivers, the directional diffusion coefficient is given as

[3]

d =

(ni=1

10Li/10

)2

ni=1

(10Li/10

)2(n 1)

ni=1

(10Li/10

)2 . (2.71)Finally, the diffusion coefficient is calculated as an arithmetic average of all directional

diffusion coefficients. If the measurement criteria concerning angular resolutions of source

and receivers positions are met, then the random-incidence diffusion coefficient dri is given

as the arithmetic mean of directional diffusion coefficients [3]

dri =

180oi=0o,i=i+10o d

19. (2.72)

2.7.1 Discussion

One of the consequences of the diffusion coefficient measurement technique is that large

diffusion coefficient values are obtained at low frequencies even for a completely flat panel.

This is due to edge diffraction of the finite size of the panel. It is therefore a good practice,

when presenting the diffusion coefficient data, to provide the diffusion coefficient values

for a flat plane surface of the same dimensions measured under the same test conditions

as a reference [23].

The finite width of the test sample additionally imposes a constraint on the lowest

frequency that is effectively reflected from the sample. For incident waves that are sub-

stantially longer than the width of the diffuser sample, almost no reflection of energy is

observed and the edge diffraction starts to dominate. Therefore, for a given sample width

one can calculate the lowest cut-off frequency below which the diffusion coefficient data

should not be regarded valid [23].

As a measure of correlation between sound waves scattered in different directions,

the diffusion coefficient is a useful measure of spatial diffusion. However, it does not

monitor temporal dispersion properties. The amount of the sound dispersed in time in the

impulse response of the reflected sound is very important in the sound scattering process.

In particular, time-spreading property of the diffuser is important in the suppression of

flutter echo [21].


2.7.2 Scattering Coefficient

Apart from the diffusion coefficient, there exists a second quantity that deals with scat-

tering properties, namely the scattering coefficient [124, 2]. It has been introduced for

geometrical room acoustic models since the diffusion coefficient cannot be directly applied

to a simplified geometrical approach. The scattering coefficient differentiates between the

portion of energy that is reflected specularly and the energy that is reflected in a diffuse

way. It is defined as the ratio between the scattered energy and the total reflected sound

energy. The scattered energy is usually distributed according to the Lamberts law, in

which the diffuse sound energy is scattered in a random direction.

As the scattering coefficient contains only partial information about the spatial distri-

bution of the scattered sound, it does not differentiate between dispersion and redirection.

Consequently, high scattering coefficient values can be obtained for a slightly tilted flat

surface. When treating an echo problem, even surfaces with high scattering values may

simply redirect echoes from one place to another, instead of dispersing it. Since the scat-

tering coefficient is only concerned with how much of the reflected energy is moved from

specular directions, it is not considered a good measure of the quality of the diffusive prop-

erties. Consequently, it cannot be used as a quality measure in the design and evaluation

of real diffusers [21].

Furthermore, the wave-based room acoustic simulators are incompatible with the for-

mulation of the scattering coefficient. This is due to the wavelengths that are not physically

consistent with the incoherent energy approach, and consequently the diffusion coefficient

is the only measure of diffusion that can be applied in wave-based room acoustic models,

such as finite difference time domain simulations of sound propagation in acoustic spaces.

2.8 Summary

In this chapter, the basics of room acoustics were briefly reviewed, with the main focus

on fundamental acoustic laws describing sound wave propagation in air and reflections

from boundaries. It is well understood how to define the boundary impedance and the

reflection coefficient for locally reacting surfaces, and therefore only such boundaries are

considered for simulations of room acoustics in this thesis. In addition, analytic formulae

for the impedance of porous materials were discussed, which are used for the evaluation

of boundary models proposed in chapters to follow. Finally, an extensive overview of

commercially available diffusers was presented and the standardised technique to measure

their diffusive properties was described. The remaining question is if we can simulate those.

One of the questions addressed in subsequent chapters is if we can simulate diffusive and

nondiffusive walls.

33

Chapter 3

Elements of Numerical Modelling

There exist numerous numerical techniques that can be applied to modelling acoustic

spaces. Although these techniques have been commonly applied in the context of room

acoustics simulations, research efforts seem to be independent of each other. This is partic-

ularly true for the finite difference time domain technique (FDTD), which can be divided

into subclasses based on rectilinear staggered and unstaggered girds or implemented as

a digital waveguide mesh (DWM). These methods are usually treated separately in the

literature and may sometimes send a confusing message to an inexperienced reader that

these constitute totally distinct techniques. The main purpose of this chapter is to show

that these are not distinct approaches but rather they fall into one larger category of finite

difference time domain methods. This chapter provides an overview of various numerical

modelling techniques and highlights the equivalence of various approaches.

The chapter is structured as follows. An overview of computer-based methods appli-

cable to simulations of room acoustics is provided in Section 3.1, including the motivation

for the chosen technique. Section 3.2 presents the basics of the finite difference time do-

main method. In particular, a basic approximation to the wave equation by means of a

standard leapfrog scheme is presented, followed by stability and dispersion error analysis.

A second type of FDTD schemes based on a stag

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