Cylindrical Pipes: Plane Wave Propagation

• 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
    $$P(x,t) = C^{+} e^{j(\omega t - kx)},$$ (17)
  which is a solution to the one-dimensional wave equation
    $$\ensuremath{\frac{\partial^{2} {p}}{\partial {x}^{2}}} = \frac{1}{c^{2}} \ensuremath{\frac{\partial^{2} {p}}{\partial {t}^{2}}}.$$ (18)

• 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
    $$\ensuremath{\frac{\partial {p}}{\partial {x}}} = - \frac{\rho}{A} \ensuremath{\frac{\partial {u}}{\partial {t}}},$$ (19)
  where $A$ is the cross-sectional area of the pipe and $\rho$ is the mass density of air.

• For pressure waves given by Eq. (17), the associated volume flow is found from Eq. (19) as
    $$U(x,t) = \left(\frac{A}{\rho c}\right)C^{+}e^{j(\omega t - kx)}$$ (20)
  and the characteristic or wave impedance is
    $$Z_{c} = \frac{P(x, t)}{U(x, t)} = \frac{\rho c}{A}.$$ (21)

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