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## Sound pressure, intensity and radiance

The propagation of sound wave causes the gas pressure changes rapidly. In room acoustics, the sound pressure p at the measurement point is dened as the dierence between the instantaneous pressure and the static pressure p0: p = pins − p0. (1.2)

The uctuation of sound pressure from its static level is much smaller than the static pressure. The static pressure p0 is 1.013 · 105 Pa in normal laboratory conditions, whereas the sound pressure p rarely exceeds 10 Pa.

The normally rapidly oscillating sound pressure p is usually more useful for theoretical analysis than for practical purpose. In practice the Root Mean Square (RMS) of the sound pressure is often used, which is the square root of the average of the square of the pressure of the sound signal over a given duration. This is also what we can read on our measurement tools (or what our ears perceive).

### Representation of Sound Propagation

The propagation of a sound within an enclosed space undergoes several reections and diraction

at walls and obstacles. This physical property creates the so-called reverberation. The reverberation eects depend on the size and shape of the room, and the construction material of its surfaces. It sometimes also causes coloration eects by the repetition of sound caused by reections. The coloration depends on the position of the sound source and listener. Thus it is important to discuss the representation of the sound propagation for the purpose of reverberation eect analysis and sound rendering.

**Wave-based Numerical Acoustics**

Wave-based numerical methods are the most accurate modeling methods for the simulation of sound wave propagation in bounded environments. The direct numerical solution of the wave equation, achieved through the use of discrete element methods, provides a means of emulating the behavior of sound elds that inherently incorporateswave properties. At the same time, they are the most computationally-demanding methods too.

The currently used wave-based acoustic modeling methods can be classied into three categories: Finite Element Methods (FEM), Boundary Element Methods (BEM) and Finite-Dierence Time-Domain methods (FDTD) [Botteldooren, 1995; Savioja et al., 1995]. In FEM, the complete space considered need to be discretized into volumetric elements, while in BEM, only the boundary of the space are discretized. Typically, the requirements of computation resource andmemory capacity for the FEM are higher than for the BEM. While the FEM and the BEM typically operate in the frequency domain, the FDTD simulates the room acoustic in the time domain. The main principle of the FDTD is that the derivatives in the wave equations are replaced by the corresponding nite dierences [Strikwerda, 2004].

However, the wave-based methods are also the most computationally-demanding methods. Thus, in practice, they are usually only used formodeling the roomacoustics in the lowfrequency range [Kleiner et al., 1993; Pietrzyk, 1998]. For real-time auralization, they are only used tomodel the direct sound and the early reections. With consideration of simplifying the computation, the FDTD can be suitable in real-time auralization [Savioja, 2010].

#### Ray-based Geometric Acoustics

In geometrical room acoustics where the room size is large compared to the wavelengths of sound, the concept of a wave is of minor importance; it is replaced instead by the concept of a sound ray [Kuttru, 2009]. This condition is often met in room acoustics. For example in a room with dimension of 4×4×3, ifwe consider a sinusoid sound of frequency 1 kHz, its corresponding wavelength is 34 cm, which is much smaller than the dimension of the walls of the room. The sound ray, like the light ray, is a concept which simplies the sound wave to the limiting case of vanishingly small wavelengths. The sound ray has well-dened direction of propagation, and is subject to specular and diuse reections when it hits the reective surfaces. The limited velocity of sound requires that the propagation time cannot be neglected when simulating the propagation process using sound rays. Many acoustical eects are related to the propagation time, such as echoes, reverberations and so on.

**Table of contents :**

Résumé

Abstract

Remerciements

**Introduction en Français **

0.1 Introduction de la Thèse

0.2 Techniques de la Modélisation Acoustique

0.3 Filtres Réverbérants Basé sur la Géométrie

0.4 Filtre Réverbérant avec des Retards Variables

0.5 Modélisation des Premières Réexions

0.6 Réseaux de Rendu Acoustiques

0.7 Conclusions et Travaux Futurs

Introduction

**1 Representation of Sound **

1.1 The Physics of Sound

1.1.1 Sound propagation

1.1.2 Sound pressure, intensity and radiance

1.1.3 Velocity of sound

1.1.4 Sound reections and diraction

1.2 Representation of Sound Propagation

1.2.1 Room Impulse Response

1.2.2 Plenacoustic Functions

1.2.3 Echogram

1.3 Perception of Room Acoustics

1.3.1 Perceived Sound Intensity

1.3.2 Frequency Perception of Sound

1.3.3 Spatial Perception

1.3.4 Objective Parameters of Room Acoustics

**2 Acoustic Modeling Techniques **

2.1 Wave-based Numerical Acoustics

2.2 Ray-based Geometric Acoustics

2.2.1 Image Source Method

2.2.2 Deterministic Ray Tracing Methods

2.2.3 Beam Tracing

2.2.4 Radiance Transfer Method

2.3 Perception-based Statistical Acoustics

2.3.1 Schroeder Filter

2.3.2 Feedback Delay Networks

2.4 Hybrid Methods

**3 Geometric-Based Late Reverberator **

3.1 Radiance Transfer Method

3.1.1 Analytical Formulation

3.1.2 Numerical Formulation

3.1.3 Form Factor

3.1.4 Simulation Procedure

3.2 From RTM to FDN

3.2.1 Relation between RTM and FDN

3.2.2 Grouping the patch-to-patch interactions

3.2.3 Delay Lengths of the FDN

3.3 RTM Model-Based Late Reverberator

3.3.1 System structure

3.3.2 Parameter estimation

3.4 Stability of the Reverberator

3.4.1 System transfer functions

3.4.2 Choice of the feedback matrix

3.4.3 Even-energy grouping scheme

3.5 FDN-RTM Algorithm

3.6 Experimental Results

3.6.1 Experimental geometries

3.6.2 Simulation examples

3.6.3 Evaluation of the decaying trend

3.6.4 Computation performance

3.6.5 Subjective evaluation using listening tests

3.6.6 Frequency-dependent reverberation for realistic materials

3.7 Conclusions

**4 Reverberator Using Variable Delays **

4.1 Radiance Transfer Model Using Variable Delays

4.1.1 Variable delays of the radiance transfer model

4.1.2 Modication to the computation procedure

4.2 Feedback Delay Networks Using Variable Delays

4.2.1 Delay distribution of a room

4.2.2 Delay-varying comb lter

4.2.3 Recursive networks using variable delay lengths

4.3 Experiments

4.3.1 Simulated RIRs

4.3.2 Computational comparison

4.3.3 Sound quality

4.4 Conclusion

**5 Modeling Early Reections **

5.1 Acoustic Rendering Equation

5.1.1 Kajiya’s Rendering Equation

5.1.2 Temporal Propagation Operator

5.1.3 Reection Kernel

5.1.4 Room Acoustic Rendering Equation

5.1.5 Discretized Room Acoustic Rendering Equation

5.1.6 Relation to the Radiance Transfer Method

5.2 Numerical Simulation Using Ray-Tracing

5.2.1 Pre-computation

5.2.2 Run-time computation

5.3 Final Gathering Schemes

5.4 Experimental Results

5.4.1 Experiment

5.4.2 Evaluation Criterion

5.4.3 Results

5.5 Conclusion

**6 Acoustic Rendering Networks**

6.1 Feedback Delay Network for Articial Reverberation

6.2 Modication of the Acoustic Rendering Equation

6.2.1 Discretization of directions

6.2.2 Simplication of the reection kernel

6.3 Acoustic Rendering Networks

6.3.1 Design overview

6.3.2 Feedback delay components

6.3.3 Initial and nal components

6.3.4 Relation to previous works

6.4 Numerical Evaluation

6.4.1 Early reections

6.4.2 Late reverberation

6.4.3 Sound quality evaluation

6.4.4 Computation performance

6.5 Conclusion

**7 Applications **

7.1 Virtual Auditory Environments

7.1.1 Interactive auditory scene generation

7.1.2 Binaural auralization

7.1.3 Applications

7.2 3D Audio Rendering in REVERIE

7.2.1 The REVERIE research project

7.2.2 3D audio rendering

**8 Conclusions **

8.1 Conclusions

8.2 Contributions

8.3 Future work

**Bibliography**