- 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

(15)

(16) - Thus, traveling-wave components of pressure and velocity are in-phase and related by a purely resistive wave impedance.

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