We will first consider the elementary case of a *conical*
acoustic tube. All smooth horns reduce to the conical case over
sufficiently short distances, and the use of many conical sections,
connected via scattering junctions, is often used to model an
arbitrary bore shape [71]. The conical case is also
important because it is essentially the most general shape in which
there are exact traveling-wave solutions (spherical waves)
[360].

Beyond conical bore shapes, however, one can use a
*Sturm-Liouville formulation* to solve for one-parameter-wave
scattering parameters [50]. In this formulation, the
*curvature* of the bore's cross-section (more precisely, the
curvature of the one-parameter wave's constant-phase surface area) is
treated as a *potential* function that varies along the horn
axis, and the solution is an *eigenfunction* of this potential.
Sturm-Liouville analysis is well known in *quantum mechanics* for
solving *elastic scattering* problems and for finding the wave
functions (at various energy levels) for an electron in a nonuniform
potential well.

When the one-parameter-wave assumption breaks down, and multiple
acoustic modes are excited, the *boundary element method* (BEM)
is an effective tool [191]. The BEM computes the
acoustic field from velocity data along any enclosing surface. There
also exist numerical methods for simulating wave propagation in
varying cross-sections that include ``mode conversion''
[339,13,117].

This section will be henceforth concerned with non-cylindrical tubes in which backscatter and mode-conversion can be neglected, as treated in [320]. The only exact case is the cone, but smoothly varying horn shapes can be modeled approximately in this way.

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