Cylindrical Pipes: Modes of Propagation

Figure 6: A cylindrical pipe in cylindrical polar coordinates.

• A short section of a cylindrical pipe in cylindrical polar coordinates $(r, \phi, x)$ is depicted in Fig. 6.

• Longitudinal wave motion is possible along the principal axis, as well as in planes orthogonal to the principal axis. However, it will be seen that these transverse modes are only weakly excited in musical instrument bores of small diameter.

• The wave equation in this geometric coordinate system is
    $$\frac{1}{r}\frac{\partial}{\partial r}\left(r \ensuremath{\frac{\... ...{2}}} = \frac{1}{c^{2}}\ensuremath{\frac{\partial^{2} {p}}{\partial {t}^{2}}}.$$ (13)

• Assuming sinusoidal solutions, a complete general solution to the wave equation in circular polar coordinates is given by
    $$P_{mn}(r, \phi, x) = \Phi(\phi)R(r)X(x) = P_{0}\cos(m\phi)J_{m}\left(\frac{\alpha_{mn}r}{a}\right)e^{-j k_{mn}x},$$ (14)
  where $J_{m}$ is a Bessel function, $a$ is the radius of the cylinder, and $\alpha_{mn}$ denotes the positive zeros of the derivative $J_{m}^{\prime}(\alpha_{mn}).$

• The wavenumber for a sinusoidal disturbance propagating axially along the tube, $k_{mn},$ varies with mode $(m,n)$ as
    $$k^{2}_{mn} = \left( \frac{\omega}{c} \right)^2 - \left( \frac{\alpha_{mn}}{a} \right)^2,$$ (15)
  where mode paramter $m$ refers to nodal diameters and $n$ refers to nodal circles (see Fig. 7 below).

• One-dimensional plane-wave propagation corresponds to mode $(0,0),$ for which $k_{00} = k = \omega/c.$

• Higher modes will propagate only if $k^{2}_{mn}$ is positive (or $k_{mn}$ is real), so that the frequency must exceed a critical (often called “cutoff”) value given by
    $$\omega_{c} = \frac{\alpha_{mn}c}{a}.$$ (16)

• For frequencies less than $\omega_{c},$ the mode is evanescent and decays exponentially with distance (see animation).

• The plane-wave mode has a cut-off frequency of zero and no transverse wave motion.

  Figure 7: Pressure (upper) and flow (lower) patterns for the lowest three higher-order modes in a cylindrical pipe (from Fletcher and Rossing (1991)).

  

• The next two propagating modes are the $(1,0)$ and $(2,0)$ nodal plane modes, which have cutoff frequencies $\omega_{c} = 1.84 c/a$ and $\omega_{c} = 3.05 c/a,$ respectively.

• A typical clarinet has a radius of about 7.5 millimeters for a majority of its length, while that of a flute is about 8.5 millimeters. With the speed of sound approximated by $c = 347.23$ meters per second for a temperature of $26.85^{\circ}$C, these cutoff frequencies are 13.56 kHz and 22.5 kHz for the clarinet and 11.96 kHz and 19.8 kHz for the flute.

• The first propagating transverse mode is well within the range of human hearing. However, excitation of this mode requires transverse circular motion, which will not occur with any significance in musical instruments.

• Evanescent mode losses may be possible in turbulent regions, such as in the mouthpiece and near toneholes.

  Figure 8: Acoustic flow patterns and pressure maxima and minima for modes in a cylindrical duct. The lowest two systems represent evanescent, non-propagating mode patterns (from Fletcher and Rossing (1991)).