Given the previous analysis, wave motion in cylindrical woodwind bores is primarily planar and along the principal axis of the air column. The equation of motion for a pressure wave propagating in this way along the x-axis with sinusoidal time dependence has the form
as found by substitution in Eq. (10) for the (0,0) mode.
Plane waves of sound can theoretically propagate without reflection or loss along the principal axis of an infinite cylindrical pipe, assuming the walls are rigid, perfectly smooth, and thermally insulating.
From Newton's law, pressure and volume velocity in a cylindrical pipe are related by
where A is the cross-sectional area of the pipe and is the mass density of air.
For pressure waves given by Eq. (13), the associated volume flow is found from Eq. (14) as
and the characteristic or wave impedance is
Thus, traveling-wave components of pressure and velocity are in-phase and related by a purely resistive wave impedance.