Functional Phonology: chapter 3

# Chapter 3: Acoustical simulation

This chapter derives the aerodynamic equations needed in our articulation model, translates the myoelastic and aerodynamic equations into difference equations for numerical simulation, and presents the actual computer algorithm.

### Contents

3.1   The equation of continuity of mass flow
3.1.1   The integral equation of continuity
Figure 3.1 (moving boundaries and walls)
3.1.2   Pumping and sucking
3.1.3   Others' choices for the continuity equation
3.2   The equation of motion
Figure 3.3 (moving particle)
3.2.2   Bernoulli effect
3.2.3   Friction
Figure 3.4 (transverse velocity gradient)
Figure 3.5 (velocity profile)
Figure 3.6 (resistance)
3.2.4   Complete equation of motion
3.2.5   Others' choices for the equation of motion
3.3   The equation of state
3.4   Turbulence
3.4.1   Energy loss
Figure 3.7 (outlet)
3.4.2   Turbulence noise
3.5   Boundary conditions
3.5.1   At a closed boundary
Figure 3.8 (closed boundary)
3.5.2   At a boundary open to the atmosphere
Figure 3.9 (open boundary)
3.5.3   At a boundary between two tube sections
Figure 3.10 (two-way boundary)
3.5.4   At a three-way boundary
Figure 3.11 (three-way boundary)
3.6   Simplifying the aerodynamic equations
3.6.1   The aerodynamic equations in terms of continuous quantities
3.6.2   Eliminating the equation of state
3.6.3   A paradoxical factor of one half
3.7   Acoustic output
3.8   Digital simulation
3.9   The dissipative part of the equations
3.9.1   The exponential method
3.9.2   The first-order explicit method
3.9.3   The first-order implicit method
3.9.4   The second-order method
3.9.5   Which method should we use?
Figure 3.13 (four dampings)
3.10   The harmonic part of the myo-elastic equations
Figure 3.14 (four frequency and amplitude warpings)
3.10.1   The "explicit" method
3.10.2   The "exact" method
3.10.3   The "implicit" method
3.10.4   The "second-order" method
3.10.5   The amplitude of the periodic motion
3.10.6   Which method should we use?
3.11   The hyperbolic part of the aerodynamic equations
3.11.1   The Lax-Wendroff method
3.11.2   Stability, numerical damping, and frequency warping
Figure 3.15 (Lax-Wendroff warping)
Figure 3.16 (formant and bandwidth warping)
3.11.3   Four extensions to the Lax-Wendroff method
3.11.4   Stability, frequency warping, and numerical damping
3.11.5   Accuracy
3.12   The algorithm
3.13   Conclusion

Forward to chapter 4.
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