Lower Bound on Frequency Validity of Energy-Stress Tensor Based Diuse Sound Field Model

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Energy Density and Sound Intensity

More recent approaches considered the mathematical and statistical behavior of sound elds themselves. The energy density of a eld that satis es the acoustic wave equation is de ned in terms of the kinetic and potential energy at a given location and point in time resulting from the compression and rarefaction of gas. Energy density is not equivalent to sound pressure, but acts in a similar fashion, re ecting the tendency of energy throughout a space to return to equilibrium. Derivation of these quantities in the acoustic context is given in Morse and Feshbach (1953) and Morse and Ingard (1968). Though energy density and ux, which in acoustics we call the sound intensity, cannot be used to directly model pressure, application of these concepts in general acoustical elds such as that of Schi rer and Stanzial (1994) encouraged interest in energetic approaches to acoustics outside of noise control studies, and as we show later on, can still be used for the synthesis of perceptually equivalent representations of the stochastic reverberation. One particularly popular use of the sound intensity is as a predictor of average direction of arrival for individual pressure wavefronts in a spatial impulse response, as described by Merimaa and Pulkki (2005).
Concurrently with the development of energy-based acoustics, statistical approaches based on the trajectories of particles of sound were being developed. The concept is similar to ray tracing, but the goal is not to collect discrete rays at a listener position, but rather, to understand the statistical properties of such a system throughout time. As demonstrated in Polack (1992), under this formalism, the idea of the transition time could be tied to develop-ment of the di use eld, leading to the concept of di usion as the mechanism for modeling the stochastic reverberation. From these ideas, a model of di use sound elds based on par-ticle di usion was developed by Picaut et al. (1997), which would come to be known as the di usion equation method, or DEM. Later improvements by Jing and Xiang (2007) were a result of computing the energy balance at the boundaries assuming an isotropic distribution of incidence, making it applicable for a wider variety of absorption coe cients. The use of the DEM continues in recent research, especially for large structures as in Su Gul et al. (2019).
The thread that ties these two somewhat distinct concepts together is that of the di use sound eld itself, which comes with assumptions about the nature of the mean energy ow within a space. Thus, predicting the stochastic reverberation with the DEM is inherently tied to the behavior of the energy density and sound intensity within the space. The main issue with the method is that the resulting conservation equation for sound intensity does not explicitly involve time due to the assumption that the energy density in a space is nearly isotropic, resulting from the observation that sound intensity tends to zero much more quickly than energy density. Working from Morse and Feshbach (1953), it turns out that explicitly introducing time variance to the sound intensity requires the consideration of the wave-stress tensor, which generalizes the energy density and sound intensity and allows for conservation of sound intensity to be de ned, but also greatly complicates computation of the resulting eld. In the next section, the derivation of these relationships is performed, and the collection of the resulting terms into a single conserved quantity, the energy-stress tensor, is demonstrated.

ENERGY-STRESS TENSOR METHOD 

wave-stress tensor, such that Exy = Eyx = 0. While these assumptions are not necessary for the 1-dimensional cases we will study over the course of this thesis, we mention them here to re ect back upon in the nal chapter, when a computational study of a sound eld may allow us to con rm or deny whether these assumptions are valid.
Now, we proceed with a derivation of the reduction of the 3-dimensional EST to a 1-dimensional solution, following from Dujourdy et al. (2017). A hallway is a good candidate for a 1-dimensional reduction, as its primary axis is much longer than either its width or height. We further presume that the width and height are on the order of the characteristic length of the EST, meaning that our discretization need not exceed one sample, and allowing us to treat the entire space according to linear samples along its length.
To begin, we consider an arbitrary hallway as a rectangular solid of dimensions lx ly lz, where lx is the length, ly the width and lz the height of the corridor, on average, implying a cross-sectional area S = lylz. Note that despite the fact that we are interested in reducing the problem to modeling only the propagation of sound along the length of the corridor, we still must consider the 3-dimensional nature of the hallway, and thus, begin from the 3-dimensional version of the energy-stress tensor. Energy balance We return rst to the continuity equation for energy density, Equation 2.6: @tEtt + @xExt + @yEyt + @zEzt = 0.

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Boundary conditions on ends of the hallway

The nal element that is needed in order to fully represent the EST in the hallway are the boundary conditions at the ends. The process of dimensional reduction by integration meant that the boundary conditions along the oor, ceiling, and two walls were all integrated into the volume equations written above as the telegraph equation, however, the physical behavior of the EST at the ends of the hallways remains unde ned. Thus, we must introduce conditions for the ends of the hallway in terms of the energy balance de ned earlier.
As before, Dujourdy et al. (2017) contend that the absorption at the wall is given by the energy density in front of it and the normal sound intensity incident on it: Jx = AE = ArE; (3.6).
where A is again the modi ed absorption coe cient, and Ar denoting this special instance of the coe cient to distinguish it in the nal system of equations. Introducing the sign of the normal n = 1 at each boundary according to the x coordi-nate, we may rewrite the second member of Equation 3.3 and replace the sound intensity with the expression above. 1 D n@xE = ( c @t + )nJ.

Finite di erence time domain discretization

Dujourdy et al. (2017) discretizes the telegraph equation (Equation 3.5) with regularly spaced spatial and temporal samples. Choosing a given spatial sample step x and time sample step t means that the time evolution of the energy density in a particular space may be modeled with a grid of sample values, where a space of length l implies l= x spatial samples, and where the number of temporal samples Ns is chosen such that the simulation runs from t = 0 to a termination time at t = Ns t. We refer to a speci c spatio-temporal sample of the energy density, then, as Ein, where i indexes the spatial samples and n indexes the temporal samples.
The second order continuous spatial and temporal derivatives are approximated with second order central nite di erences in time and space. Ignoring truncation error results in the following expressions that may be substituted into the telegraph equation: t En+1 2En + En 1 i i i @ttEjx = ; t2 En 2En + En (3.8) t i i i @xxEjx = : x2.
First order derivatives are also approximated with a central nite di erence. Again, ignoring the truncation error, the approximations are: t Ein+1 Ei 1 @xEjx = 0.

Table of contents :

Abstract
Resume
Resume Long
Acknowledgments
Notation
Acronyms
1 Introduction 
1.1 Reverberation
1.2 Energy-based methods
1.3 Motivation
1.4 Outlook
I Energy-Stress Tensor Method 
2 Background 
2.1 Introduction
2.2 Statistical acoustics
2.2.1 Energy Density and Sound Intensity
2.3 Energy-Stress Tensor Method
2.3.1 Theory
2.3.2 Scattering and diusion
2.3.3 Dimensional reduction
2.3.4 Limitations
2.3.5 Advantages
3 Frequency Dependence and Validity of a 1D Model 
3.1 Introduction
3.2 Parameter tting
3.2.1 Telegraph equation
3.2.2 Boundary conditions on ends of the hallway
3.2.3 Finite dierence time domain discretization
3.2.4 Discussion
3.3 Measurements
3.3.1 Geometry
3.3.2 Details
3.4 Results
3.4.1 Physical hallways
3.4.2 Numerical model
3.4.3 Alcove hallway
3.4.4 Plain hallway
3.5 Discussion
3.6 Future work
3.7 Conclusion
II Finite Volume Approaches 
4 Sources and Finite Volume Formulation 
4.1 Introduction
4.2 Sources
4.2.1 1-dimensional EST
4.3 Finite volume model
4.3.1 Spatial discretization
4.3.2 Time domain discretization
4.4 Evaluation and commentary
4.5 Conclusion
5 Auralization 
5.1 Introduction
5.2 Hybrid Model
5.2.1 EST method
5.2.2 Low-frequency reverberation
5.2.3 Direct path and early reections
5.3 Calibration
5.3.1 Between simulation methodologies
5.3.2 Between simulated results and measurements
5.4 Evaluation
5.5 Results
5.6 Future work
5.7 Conclusion
6 Energy-Stress Tensor Quantities 
6.1 Introduction
6.2 Toward 3-dimensional prediction
6.3 Pressure domain simulation
6.3.1 Ambisonic microphone approach
6.4 Derivation of EST terms in FVTD formalism
6.5 Measurements
6.6 Preliminary results
6.6.1 Passage sections
6.6.2 Junction and alcove sections
6.6.3 Discussion
6.7 Riemannian tessellation
6.8 Discussion
6.9 Future work
6.10 Conclusion
III Conclusions 
7 Conclusions 
7.1 Review
7.2 Future work
7.3 Perspectives
IV Appendices 
A Publications 
A.1 Lower Bound on Frequency Validity of Energy-Stress Tensor Based Diuse Sound Field Model
A.2 Implementation of Sources in an Energy-Stress Tensor Based Diuse Sound Field Model
A.3 Auralization of a Hybrid Sound Field using a Wave-Stress Tensor Based Model
A.4 Riemannian Space Tessellation with Polyhedral Room Images
B Code listing 
B.1 Introduction
B.2 FVTD
B.3 Remote code execution
Bibliography

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