Horns as Waveguides

Waves in a horn are typically analyzed as ``one-parameter waves,'' meaning that it is assumed that a coherent wavefront spreads out uniformly along the horn such that a ``surface of constant phase'' may be defined whose tangent plane is normal to the horn axis. For cylindrical tubes, the surfaces of constant phase are planar, while for conical tubes, they are spherical [322,142]. The key property of the horn is that a wave propagates from one end to the other with negligible ``back-scattering'' 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 back-scattering 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.

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 [74]. Tube shapes beyond conical, taking into account the continuous backscatter associated with a smoothly varying shape, may be analyzed using Sturm-Liouville methods [50]. This section will be henceforth concerned with non-cylindrical geometries in which backscatter can be neglected, as in [322].

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 [362]. You may have learned about the Huygens-Fresnel principle in a physics class when it covered waves [296]. 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 [120, 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).