- 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,

(20) - The continuous-time traveling-wave solution to this equation is

(21) *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)

(22) - 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.
**Figure 4:**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.

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