### 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 . The conical case is also important because it is essentially the most general shape in which there are exact traveling-wave solutions (spherical waves) .

Beyond conical bore shapes, however, one can use a Sturm-Liouville formulation to solve for one-parameter-wave scattering parameters . 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 . 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 . 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 . You may have learned about the Huygens-Fresnel principle in a physics class when it covered waves . 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).

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Conical Acoustic Tubes
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Faust Implementation