Sound propagation in conical air columns can be well modeled by one-dimensional spherical waves traveling along the length of the cone.
The one-dimensional wave equation for spherical pressure waves,
accurately represents lossless pressure wave propagation along the central axis of a conical tube, subject to the boundary conditions at both its ends.
The continuous-time traveling-wave solution to this equation is
where the functions and are completely general and continuous and can be interpreted as arbitrarily fixed waveshapes that travel in opposite directions along the x-axis with speed c.
This expression is similar to that for plane waves with the exception that spherical pressure traveling-wave components are inversely proportional to their distance from the cone apex.
A solution of this form can be discretized in time and space and given by (Smith, 1991)
The behavior of a finite length conical bore can be approximated for low-frequency sound waves by assuming that pressure is equal to zero at an open end. This boundary condition is met with an inversion of traveling-wave pressure components at the open end.
Figure 4 represents the digital waveguide implementation of ideal, lossless spherical-wave propagation in an ideally terminated conical tube.
Digital waveguide implementation of ideal, lossless spherical-wave propagation in a conical tube.
Aside from the 1/x scale factors, which are implemented at observation points, the cylindrical and conical waveguide implementations are exactly the same.
Further, if the system input and output are measured at the same location, the 1/x scale factor is unnecessary.