In the case of cylindrical tubes, the logarithmic derivative of the area variation, ln , is zero, and Eq.(C.145) reduces to the usual momentum conservation equation encountered when deriving the wave equation for plane waves [321,352,47]. The present case reduces to the cylindrical case when

If we look at sinusoidal spatial waves,
and
, then
and
, and the condition
for cylindrical-wave behavior becomes
, *i.e.*, the spatial
frequency of the wall variation must be much less than that of the
wave. Another way to say this is that the wall must be approximately
flat across a wavelength. This is true for smooth horns/bores at
sufficiently high wave frequencies.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University