The conical acoustic tube is a one-dimensional waveguide which propagates circular sections of spherical pressure waves in place of the plane wave which traverses a cylindrical acoustic tube [22,352]. The wave equation in the spherically symmetric case is given by
and is the pressure at time and radial position along the cone axis (or wall). In terms of (rather than ), Eq. (C.142) expands to Webster's horn equation [360]:
where is the area of the spherical wavefront, so that (as discussed further in §C.18.4 below).
Spherical coordinates are appropriate because simple closed-form solutions to the wave equation are only possible when the forced boundary conditions lie along coordinate planes. In the case of a cone, the boundary conditions lie along a conical section of a sphere. It can be seen that the wave equation in a cone is identical to the wave equation in a cylinder, except that is replaced by . Thus, the solution is a superposition of left- and right-going traveling wave components, scaled by :
In cylindrical tubes, the velocity wave is in phase with the pressure wave. This is not the case with conical or more general tubes. The traveling velocity may be computed from the corresponding traveling pressure by dividing by the wave impedance. However, this impedance is frequency-dependent in cones, as we'll derive below in §C.18.4.