A conical section in spherical coordinates.
A conical bore section in spherical coordinates
is depicted in Fig. 1. The wave equation in this geometric coordinate system is
Longitudinal wave motion in conical bores is possible along orthogonal trajectories to the principal axis.
While these transverse modes are only weakly excited in most musical instruments, they can become significant, for example, in the vicinity of a strongly flaring bell.
Assuming sinusoidal solutions, this equation is separable in spherical coordinates, and the resulting differential equations describe sinusoidal wave motion, or standing-wave distributions, along each of the spherical coordinate axes.
A complete general solution of the Helmholtz equation in spherical coordinates is then given by
is a Bessel function and
are associated Legendre functions.
The boundary condition at the cone wall, or at
which can be met by adjusting the values of m and n so that an extremum of
occurs at the wall.
Calculation of the Legendre functions for nonintegral n is nontrivial, and an accurate determination of cutoff frequencies for these modes is beyond the scope of this course. For axial symmetric waves (m=0), Hoersch (1925) has presented a method for determining values of n that satisfy the boundary condition in conical horns of various angles.
Typical values of for saxophones, oboes, and bassoons are and respectively (Nederveen, 1969). Using this procedure, the lowest values of n calculated for these angles are 109.27, 145.86, and 438.58, respectively.
Computer calculations of the associated Legendre functions for integer values of n support these values and further indicate that solutions with m=1 and n approximately equal to 53, 70, and 213, respectively are possible. These are nodal plane modes that correspond to the (1,0) mode in cylinders. Cutoff frequencies for these values of n are determined where the expression
The cutoff frequencies in a cone are dependent on x, so that waves of sinusoidal type having are only possible in the outer or wider portions of a cone (Benade and Jansson, 1974). Near the cone tip, any higher order modes that are excited will be evanescent.
For the m=1 mode solutions given above, the corresponding cutoff frequencies are approximately fc = 2.94/x kHz for saxophones, fc = 3.87/x kHz for oboes, and fc = 11.72/x kHz for bassoons, where x is given in meters.
Alto saxophones, oboes, and bassoons have approximate lengths of 1 meter, 0.64 meters, and 2.5 meters, and thus transverse modes can propagate well within the audio spectrum at certain locations in these instruments.
In comparison to cylindrical tubes, these higher modes propagate at much lower frequencies. However, excitation of the m=1 mode requires transverse circular motion, which will not occur with any regularity in musical instruments. As mentioned with regard to cylindrical bores, evanescent mode losses may occur in a woodwind instrument mouthpiece and near toneholes.