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 :
As we did for vibrating strings (§C.3.4), suppose the pressure is sinusoidally driven so that we have
where , , and is the driving frequency. The partial derivatives become
Substituting into the momentum equation gives
Because the medium is linear and time-invariant, the velocity must be of the form , and we can define the spatially instantaneous wave impedance as
The corresponding instantaneous wave admittance is then . Then , and the momentum equation becomes
Solving for the wave impedance gives
Expressing in exponential form as
where may be complex, we may define the instantaneous spatial frequency (wavenumber) as
and since ln , we have
Defining the spatially instantaneous phase velocity as
we have