A cylindrical pipe in cylindrical polar coordinates.
A short section of a cylindrical pipe in cylindrical polar coordinates is depicted in Fig. 5.
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
Assuming sinusoidal solutions, a complete general solution to the wave equation in circular polar coordinates is given by
where Jm is a Bessel function, a is the radius of the cylinder, and denotes the positive zeros of the derivative
The wavenumber for a sinusoidal disturbance propagating axially along the tube, kmn, varies with mode (m,n) as
One-dimensional plane-wave propagation corresponds to mode (0,0), for which
Higher modes will propagate only if kmn is positive, so that the frequency must exceed a cutoff value given by
For frequencies less than the mode is evanescent and decays exponentially with distance.
The plane-wave mode has a cut-off frequency of zero and no transverse wave motion.
The next two propagating modes are the (1,0) and (2,0) nodal plane modes, which have cutoff frequencies
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 26.85C, 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.