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Universidade Técnica De Lisboa Instituto Superior UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO ACOUSTIC MODELLING FOR VIRTUAL SPACES Diogo Gonçalo Franco Falcão Osório de Alarcão (Licenciado) Dissertação para obtenção do Grau de Doutor em Engenharia Electrotécnica e de Computadores Orientador: Doutor José Luís Bento Coelho Júri Presidente: Reitor da Universidade Técnica de Lisboa Vogais: Doutor Luís Manuel Braga da Costa Campos Doutor Jorge Viçoso Patrício Doutor José Luís Bento Coelho Doutor João Manuel Domingues Perdigão Doutor Afonso Manuel dos Santos Barbosa Novembro 2005 UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO ACOUSTIC MODELLING FOR VIRTUAL SPACES Diogo Gonçalo Franco Falcão Osório de Alarcão (Licenciado) Dissertação para obtenção do Grau de Doutor em Engenharia Electrotécnica e de Computadores Orientador: Doutor José Luís Bento Coelho Júri Presidente: Reitor da Universidade Técnica de Lisboa Vogais: Doutor Luís Manuel Braga da Costa Campos Doutor Jorge Viçoso Patrício Doutor José Luís Bento Coelho Doutor João Manuel Domingues Perdigão Doutor Afonso Manuel dos Santos Barbosa Novembro 2005 ii To my son Rodrigo and to my daughter Inês iii Abstract This thesis presents a new model for the sound energy propagation inside enclosed spaces. The new model is developed starting from the physical first principles and from rigorous mathematical definitions. The theoretical foundation allows the rigorous definition of the energy-based quantities used for describing sound energy propagation inside enclosures and the definition of a general energy balance equation for sound particles, applicable to any room. In addition, new theoretical tools for the study of sound energy fields in rooms are introduced, which are based on the language of functional analysis. The governing equations are thus expressed in terms of linear operators with very convenient properties, which are mathematically detailed. In addition, a new combined method that solves the derived equations is presented. This new combined method uses an extended mirror image source method solving for the propagation of the specularly reflected energy components inside enclosures, and a time-dependent hierarchical radiosity method solving for the propagation of the diffusely reflected energy components. New algorithmic refinements are introduced in the computer implementation of this combined method. Finally, it is shown that the new proposed method is fast, flexible and accurate enough to be applied as an alternative method for room acoustics simulation. Key words: room acoustics, sound particles, linear integral operators, diffuse and specular reflections, extended image source method, time-dependent hierarchical radiosity. iv Resumo A presente Tese descreve um novo modelo da propagação da energia sonora em recintos fechados. O novo modelo é desenvolvido a partir de primeiros princípios físicos e de definições matemáticas rigorosas. As fundações teóricas obtidas permitem definir de forma rigorosa as quantidades energéticas usadas na descrição da propagação de energia sonora dentro de salas bem como a definição de uma equação geral de balanço de energia para partículas sonoras, aplicável a uma sala qualquer. São introduzidas novas ferramentas teóricas para o estudo de campos de energia sonora em salas, baseadas na linguagem da análise funcional. As equação que governam a propagação de energia sonora são pois expressas em termos de operadores lineares com propriedades convenientes. Adicionalmente, é apresentado um novo método combinado para a resolução das equações derivadas. Este método combinado utiliza o método das imagens estendido para a solução da propagação das componentes reflectidas especularmente e o método de radiosidade hiérarquica, dependente do tempo, para obter a solução da propagação das componentes reflectidas difusamente. São ainda apresentados vários aperfeiçoamentos utilizados nos algoritmos implementados do método combinado. Finalmente, mostra-se que o novo método combinado é de computação rápida, fléxivel e suficientemente preciso estabelecendo-se num método alternativo para a simulação de acústica de salas. Palavras chave: acústica de salas, partículas sonoras, operadores integrais lineares, reflexões difusas e especulares, método das fontes imagem estendido, radiosidade hierárquica dependente do tempo. v ACKNOWLEDGMENTS I would like to sincerely thank my advisor Professor J. L. Bento Coelho for giving me the opportunity of conducting this work under his supervison. I thank him for the continuous encouragement and unhesitating support during this endeavour and for the the help in the preparation of this thesis. I must also thank my family, specially my wife Marta, for her love, understanding and patience at every step along the way. A very special thanks to my parents, especially to my mother, who was the first person that pressed me to start such a difficult task, giving me confidence and encouragement. Finally, to my young son and daughter, I thank them for their great patience when, many times, their father was wholly absorbed in his research and could not give them all the attention they deserve. This work was partially financially supported by FCT – Portuguese Foundation for Science and Technology under the III QCA of the EU. v i TABLE OF CONTENTS Chapter 1 – Introduction................................................................................................................................... 1 Chapter 2 – Basic Theory Concepts................................................................................................................ 7 2.1 Introduction............................................................................................................................. 7 2.2 Physical Descriptors............................................................................................................... 8 2.2.1 Characterisation of Acoustic Disturbances.............................................................. 9 2.2.2 The Wave Equation.................................................................................................... 11 2.2.3 Intensity and Energy Density ................................................................................... 16 2.2.4 Harmonic and Non-harmonic Sound Waves........................................................ 20 2.3 Sound Sources and Receivers............................................................................................. 22 2.4 Sound Radiation.................................................................................................................... 24 2.5 Sound Receivers.................................................................................................................... 25 2.6 Propagation of Sound Waves............................................................................................. 26 2.6.1 The Sound Field in Front of a Wall......................................................................... 27 2.6.2 Diffraction of Sound Waves ..................................................................................... 39 2.6.3 The Attenuation of Sound in Air............................................................................. 41 2.6.4 Sound Absorbers......................................................................................................... 43 2.6.4.1 Membrane Absorbers..........................................................................................43 2.6.4.2 Porous Absorbers ................................................................................................45 2.6.4.3 Resonant Absorbers ............................................................................................46 2.6.4.4 People and Furniture...........................................................................................47 2.7 Sound Perception ................................................................................................................. 47 2.7.1 Fundamental Properties of the Human Ear .......................................................... 48 2.7.1.1 Intensity Perception of the Auditory System. Thresholds. ..........................49 2.7.1.2 Equal Loudness Contours..................................................................................50 2.7.1.3 Frequency Perception of the Auditory System...............................................51 2.7.1.4 Critical Bandwidths and Masking......................................................................52 2.7.1.5 Time Perception...................................................................................................53 2.7.1.6 Spatial Sound Perception....................................................................................53 Chapter 3 – Sound Fields in Enclosures....................................................................................................... 57 3.1 Introduction........................................................................................................................... 57 3.2 The Wave Equation and Boundary Conditions.............................................................. 58 3.3 Natural Modes for the Rectangular Room ...................................................................... 59 3.4 Steady-state and Transient Sound Fields inside Enclosures......................................... 65 3.5 Reverberation Time..............................................................................................................69 3.5.1 Time Distribution of Reflections............................................................................
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