Sound Propagation
A detailed acoustic theory must consider the effects of the following:
·	Time variation of the vocal tract shape
·	Losses due to heat conduction and viscous friction at the vocal tract walls
·	Softness of the vocal tract walls
·	Radiation of sound at the lips
·	Nasal coupling
·	Excitation of sound in the vocal tract
Let us begin by considering a simple case of a lossless tube:
For frequencies that are long compared to the dimensions of the vocal tract (less than about 4000 Hz, which implies a wavelength of 8.5 cm), sound waves satisfy the following pair of equations:
     or     
where
	is the variation of the sound pressure in the tube
	is the variation in the volume velocity
	is the density of air in the tube (1.2 mg/cc)
	is the velocity of sound (340 m/s)
	is the area function (about 17 cm long)


For frequencies that are long compared to the dimensions of the vocal tract (less than about 4000 Hz, which implies a wavelength of 8.5 cm), sound waves satisfy the following pair of equations:
     or     
where
	is the variation of the sound pressure in the tube
	is the variation in the volume velocity
	is the density of air in the tube (1.2 mg/cc)
	is the velocity of sound (35000 cm/s)
	is the area function (about 17.5 cm long)

Uniform Lossless Tube
If , then the above equations reduce to:

The solution is a traveling wave:


which is analogous to a transmission line:



What are the salient features of the lossless transmission line model?
where
Acoustic Quantity	Analogous Electric Quantity
p - pressure	v - voltage
u - volume velocity	i - current
r/A - acoustic inductance	L - inductance
A/(rc2) - acoustic capacitance	C - capacitance


The sinusoisdal steady state solutions are:

where  is the characteristic impedance.
The transfer function is given by:

This function has poles located at every . Note that these correspond to the frequencies at which the tube becomes a quarter wavelength: .
Is this model realistic?
Effects of Losses

What do we predict the effects of yielding walls to be?
Use perturbation analysis:

We can develop a model that relates dA(x,t) to pressure:

and solve for the new transfer function. But we can easily predict the effect of this:
What would you expect to be the effect of friction and thermal losses?
Lip Radiation
How is the sound pressure wave within the vocal tract coupled into the air?
Net effect is to place a complex load on the system:
     and     
where  and , and  is the radius of the opening.
This impedance acts as a short circuit at low frequencies, and an imaginary impedance at high frequencies. The next effect on the volume velocity is to act as a highpass filter and to attenuate low frequencies. Lip radiation introduces a zero in the spectrum at DC and broadens the bandwidths at higher frequencies.
Radiation from a spherical baffle
Radiation from an infinite plane baffle
Nasal Coupling
How is the sound pressure wave within the vocal tract coupled into the air?
We also must worry about the nasal cavity, especially for labial sounds for which the mouth is closed during sound production.
This is the equivalent of placing a transmission line in parallel with the vocal tract (oral cavity). What will the effect be?
The net effect is to produce a zero in the spectrum at about 1 kHz. As a result, nasal sounds (such as "m" and "n" in American English) have very little high frequency energy.
Piecewise Linear Approximations For The Vocal Tract
Consider the following approximation to the vocal tract area function:
Recall,

For the kth section, if we apply the boundary conditions:

We can combine these two equations to show:

where .
We can define a reflection coefficient for the kth junction:

It is easy to show that the reflection coefficients are bounded: .
The velocity can be expressed in terms of the reflection coefficients:

Ultimately, we will relate  to a discrete model of the velocity profile.






Resonant Frequencies
Consider a two tube approximation to the vocal tract:
The frequency response of this system is:

What does this tell us about the frequency response?
If we consider the case :
For this system, the poles are located at values that satisfy the equation:

How does this compare to a single lossless tube?

Poles must be found through numerical analysis - nonlinear equation.










	Resonator Geometry	Formant Patterns






Three-Tube Models
	Resonator Geometry	Formant Patterns
invdicates the fundamental resonance of the front cavity
Transfer Function of the Lossless Tube Model
Recall, . In the discrete domain, we can write: .
Following our derivation of the wave equation, we can express the transfer function for a lossless tube as follows:

where
     and     
The combined transfer function is a product of these matrices.
The net result is a transfer function that can be expressed as:

where

We can write  in a simpler form:


Why is this important?