The Equivalent Circuit

The acoustic properties of a conic frustum can be represented by an equivalent circuit consisting of a uniform transmission line, two acoustic inertances, and a transformer, as shown in Fig. 14 (Benade, 1988).

Figure 14: Equivalent circuit of a conical waveguide.
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This representation suggests that a conical air column model can be implemented using a cylindrical waveguide, a scalar “turns ratio” multiplier, and appropriately designed inertance components at each end of the waveguide.
An ideal open end, represented by a load impedance of zero, will “short-circuit” an input/output inertance.
The equivalent circuit for an open-open conic frustum then reduces to a uniform transmission line and a scalar transformer term.
This confirms the fact that a cylindrical pipe open at both ends and an open-open conic section, each of length , have equivalent longitudinal mode frequencies given by $f_{n} = n c / 2 L, \: n = 1, 2, 3, \ldots$ , where is the speed of wave propagation within the structures.
A pressure-controlled wind instrument excitation mechanism, such as a saxophone reed-mouthpiece or trumpet player lip-reed, functions as a nearly rigid, time-varying termination at an air column input.
When attached to the input of a conical waveguide, the parallel driver impedance and input inertance ( $M_{o}$ ) combination plays a significant role in determining the overall behavior of the air column.