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.
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.
a) Physical picture. b) Waveguide implementation.
Since a piecewise-cylindrical approximation to a general acoustic tube
can be regarded as a ``zeroth-order hold'' approximation. A piecewise
approximation then uses first-order
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
as traveling waves
. In first-order sections (conical
sections), spherical waves propagate as traveling waves
there are no traveling wave types for higher-order waveguide flare
, quadratic or higher) [357
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
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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