We have seen how a tract of cylindrical tube is governed by a partial differential equation such as (7) and, therefore, it admits exact simulation by means of a waveguide section. When the tube has a conical profile, the wave equation is no longer (1), but we can use the equation for propagation of spherical waves [27]:
In the equation (73) we can evidentiate a term in the first derivative, thus obtaining
Let us put a complex exponential eigensolution in (73), with an amplitude correction that accounts for energy conservation in spherical wavefronts. Since the area of such wavefront is proportional to , such amplitude correction has to be inversely proportional to , in such a way that the product intensity (that is the square of amplitude) by area is constant.
The eigensolution is
(80) |
We can define the two wave admittances
Wave propagation in conical ducts is not lossless, since . However, the medium is passive in the sense of section II, since the sum is positive semidefinite along the imaginary axis.
As compared to the lossy cylindrical tube, the expression for wave admittance is structurally unchanged, with the only exception of the sign inversion in the shunt inductance. This difference is justified by thinking of the shunt inductance as a representation of the signal that does not propagate along the waveguide. In the case of the lossy tube, such signal is dissipated into heat; in the case of the cone, it fills the shell that is formed by interfacing a planar wavefront with a spherical wavefront.
The discrete-time simulation of a length- cone tract having the (left) narrow end at distance from the apex is depicted in figure 8.