The wave impedance derivation above made use of known properties of waves in cones to arrive at the wave impedances in the two directions of travel in cones. We now consider how this solution might be generalized to arbitrary bore shapes. The momentum conservation equation is already applicable to any wavefront area variation :
Defining the spatially instantaneous phase velocity as
This reduces to the simple case of the uniform waveguide when the logarithmic derivative of cross-sectional area is small compared with the logarithmic derivative of the amplitude which is proportional to the instantaneous spatial frequency. A traveling wave solution interpretation makes sense when the instantaneous wavenumber is approximately real, and the phase velocity is approximately constant over a number of wavelengths .
Generalized Wave Impedance
Wave Impedance in a Cone