### Horns as Waveguides

Waves in a horn can be analyzed as *one-parameter waves*, meaning
that it is assumed that a constant-phase wavefront progresses
uniformly along the horn. Each ``surface of constant phase''
composing the traveling wave has tangent planes normal to the horn
axis and to the horn boundary. For cylindrical tubes, the surfaces of
constant phase are planar, while for conical tubes, they are spherical
[357,317,144]. The key property of a
``horn'' is that a traveling wave can propagate from one end to the
other with negligible ``backscattering'' of the wave. Rather, it is
smoothly ``guided'' from one end to the other. This is the meaning of
saying that a horn is a ``waveguide''. The absence of backscattering
means that the entire propagation path may be simulated using a pure
delay line, which is very efficient computationally. Any losses,
dispersion, or amplitude change due to horn radius variation
(``spreading loss'') can be implemented where the wave exits the delay
line to interact with other components.

#### Overview of Methods

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)
[357].

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 [190]. 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''
[336,13,117].

This section will be henceforth concerned with non-cylindrical tubes in which backscatter and mode-conversion can be neglected, as treated in [317]. The only exact case is the cone, but smoothly varying horn shapes can be modeled approximately in this way.

#### Back to the Cone

Note that the cylindrical tube is a limiting case of a cone with its apex at infinity. Correspondingly, a plane wave is a limiting case of a spherical wave having infinite radius.

On a fundamental level, all pressure waves in 3D space are composed of
spherical waves [357]. You may have learned about the
*Huygens-Fresnel principle* in a physics class when
it covered waves [295]. The Huygens-Fresnel principle states that the
propagation of any wavefront can be modeled as the superposition of
spherical waves emanating from all points along the wavefront
[122, page 344]. This principle is especially
valuable for intuitively understanding *diffraction* and related
phenomena such as *mode conversion* (which happens, for example,
when a plane wave in a horn hits a sharp bend or obstruction and
breaks up into other kinds of waves in the horn).

**Next Section:**

Conical Acoustic Tubes

**Previous Section:**

Faust Implementation