Generalized Wave Impedance

Figure C.49 depicts a section of a conical bore which widens to the right connected to a section which narrows to the right. In addition, the cross-sectional areas are not matched at the junction.

The horizontal axis (taken along the boundary of the cone) is chosen so that corresponds to the apex of the cone. Let denote the cross-sectional area of the bore.

Since a piecewise-cylindrical approximation to a general acoustic tube
can be regarded as a ``zeroth-order hold'' approximation. A piecewise
*conical *approximation then uses *first-order* (linear)
segments. One might expect that quadratic, cubic, etc., would give
better and better approximations. However, such a power series
expansion has a problem: In zero-order sections (cylinders), plane
waves propagate as traveling waves. In first-order sections (conical
sections), spherical waves propagate as traveling waves. However,
there are no traveling wave types for higher-order waveguide flare
(*e.g.*, quadratic or higher) [360].

Since the digital waveguide model for a conic section is no more expensive
to implement than that for a cylindrical section, (both are simply
bidirectional delay lines), it would seem that modeling accuracy can be
greatly improved for non-cylindrical bores (or parts of bores such as the
bell) essentially for free. However, while the conic section itself costs
nothing extra to implement, the *scattering junctions* between
adjoining cone segments are more expensive computationally than those
connecting cylindrical segments. However, the extra expense can be small.
Instead of a single, real, reflection coefficient occurring at the
interface between two cylinders of differing diameter, we obtain a *first-order reflection filter* at the interface between two cone
sections of differing taper angle, as seen in the next section.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University