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# Boundary Conditions for Standing Waves in the Vocal Tract

Let us review a standard idealized model of the vocal tract. In this model, the vocal tract is represented by a lossless, uniform, 1-dimensional, acoustic tube. That is, the walls absorb no energy, the tube has a constant cross-sectional area along its length, and the air within the tube vibrates only in the direction of the length of the tube. The resonances of such a tube are identified with the standing waves that meet the boundary conditions of the endpoints of the tube.

Given a tube which is closed at one end and open at the other, what are the boundary conditions imposed on standing waves within the tube? The boundary condition imposed by the closed end of the tube is that air particles2.4 at the closed end cannot move back and forth. Because they are adjacent to a wall -- the closed end -- they may be compressed against the end of the tube, but they are not free to move. Thus pressure can fluctuate maximally at the closed end of an acoustic tube, but particle velocity is zero.

The boundary condition imposed at an open end of an acoustic tube is that air pressure must be equal to ambient air pressure. Air particles at the open end will respond to a pressure wave coming out of the tube not by compressing, since there is nothing to compress against; instead, they simply move back and forth. Thus at the open end of an acoustic tube, a standing wave has maximum fluctuation in volume velocity, but zero variation in air pressure.

In slightly different terminology, the velocity waveform2.5 has a node (a location of zero fluctuation) at the closed end, and an antinode (a location of maximum fluctuation) at the open end. A mnemonic for ``node'' is ``no delta'', where delta is the mathematical symbol for ``change''. In any period of a standing wave, there are two nodes and two antinodes alternating with each other.

The half-open (i.e., open at one end, closed at the other) tube, then, has a velocity node at the closed end, and a velocity antinode at the open end. The tube length represents, at the least, one quarter of the period of a standing wave; thus, the lowest resonance in a uniform acoustic tube is the quarter-wavelength standing wave. Other standing waves are also compatible with these boundary conditions. For example, the next higher-frequency standing wave has a node at the closed end, an antinode at the open end, and a node-antinode pair in between. Each successively higher-frequency standing wave has an additional node-antinode pair between the fixed endpoints. We will consider the first three of these, which in a uniform tube are resonances that may be called F1, F2, and F3.

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Thomas Veatch 2005-01-25