Figure P 16.36 shows the two lowest resonances recorded in...

Figure $\mathrm{P} 16.36$ shows the two lowest resonances recorded in the vocal tract of the eastern towhee, a small songbird.
a. Is this bird's vocal tract better modeled as an open-open tube or an open-closed tube?
b. Estimate the length of the towhee's vocal tract.

Key Concepts

Acoustic Resonance in Tubes

This concept involves the natural frequencies at which standing waves are established in a tube. The pattern of nodes and antinodes depends on whether the tube ends are open or closed, which in turn determines the harmonic series and spacing of resonant frequencies.

Boundary Conditions – Open-Open vs. Open-Closed Tubes

The characteristics of the tube’s endpoints (open or closed) govern its resonant behavior. An open-open tube supports pressure nodes at both ends, resulting in a harmonic series where each mode is an integer multiple of the fundamental frequency, while an open-closed tube has a pressure antinode at the closed end and a node at the open end, leading to resonances at odd multiples of the fundamental frequency.

Vocal Tract as an Acoustic Resonator

The vocal tract can be modeled as an acoustic tube in order to study its sound filtering properties. Depending on the anatomical configuration, it may more closely resemble either an open-open or an open-closed tube, which affects the resonant frequencies produced during vocalization.

Estimating Tube Length from Resonant Frequencies

The relationship between the resonance frequencies and the physical length of the tube is given by formulas that depend on the tube's boundary conditions. For example, in an open-open tube the fundamental frequency is related to length by f = v/(2L), while for an open-closed tube it is f = v/(4L), where v is the speed of sound. This principle allows one to estimate the length of the vocal tract based on its resonance frequencies.

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