### More General One-Parameter Waves

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 :

ln

As we did for vibrating strings (§C.3.4), suppose the
pressure is sinusoidally driven so that we have
ln

Substituting into the momentum equation gives
ln ln

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
ln ln

Solving for the wave impedance gives
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 .

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Generalized Wave Impedance

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Wave Impedance in a Cone