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The concatenated tube model is useful because the acoustic behavior of a single tube of constant crosssectional area is quite simple to describe, in terms of a volume velocity , and a pressure deviation from the mean tube pressure. Provided wavelengths are long in comparison with the tube radius, and that pressures do not become too large (both these requirements are easily satisfied in the speech context), the timeevolution of the acoustic state of any single tube, such as that shown in Figure 1.2(a), will be described completely by

(1.1a) 
subject, of course, to initial conditions, and the effect of the boundary terminations on adjacent tubes. Given that the crosssectional tube area , the air density and the soundspeed are constant, the general solution to (1.1) can be written as

(1.2a) 
Here the physical pressure has been decomposed into a sum of a leftwardtraveling wave and a rightwardtraveling wave ; both are arbitrary functions of one variable. The volume velocity , which is dual to in the system (1.1), can be similarly expressed as a sum of leftward and rightwardtraveling velocity waves and . But these velocity waves are simply the pressure waves, scaled by the tube admittance, defined by
In addition, the rightwardtraveling wave component of the velocity is signinverted with respect to the corresponding pressure wave.
System (1.1) can be simplified to a single secondorder PDE in pressure alone,

(1.3) 
from which the traveling pressure wave solution is more easily extracted. The volume velocity satisfies an identical equation.
Consider one of the tube segments of length from Figure 1.1(b). It should be clear that we can represent the pressure travelingwave solution to (1.1) by using two delay lines, each of duration
; see Figure 1.2. We can obtain the physical pressure at either ends of the tube by summing the leftward and rightwardtraveling components, as per (1.2a). (The physical volume velocity can be obtained, from (1.2b), by taking the difference of and , and scaling the result by .) The discretetime implementation of this single isolated acoustic tube is immediate. Taking

(1.4) 
as the unit delay, or sampling period for our discretetime system,
we can see that there is no loss in generality in treating the paired shifts as digital delay lines, accepting and shifting discretetime pressure wave signals, at intervals of seconds. The discretetime model of the acoustic tube will still calculate an exact solution to system (1.1), at times which are integer multiples of . (This solution can be considered to be exact at all time instants as long as all signals in the network are assumed to be bandlimited to half of the sampling rate,
.)
Figure 1.2: (a) An acoustic tube and (b) a representation of the traveling wave solution; traveling pressure waves can be added together at either end of the tube to give the physical pressure, as per (1.2a).

Also note that because the traveling pressure and volume velocity waves are simply related to one another by a scaling, then in a computer implementation, it is only necessary to propagate one of the two types of wave in a given discrete tube sectionwe will assume, then, that pressure waves are our signal variables.
Next: Junctions Between Two Uniform
Up: Case Study: The KellyLochbaum
Previous: Concatenated Acoustic Tube Model
Stefan Bilbao
20020122