- A conical bore section in spherical coordinates
is depicted in Fig. 1. The wave equation in this geometric coordinate system is

(1) - 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

where is a Bessel function and are associated Legendre functions. - The boundary condition at the cone wall, or at
is
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 becomes positive. - 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*f*_{c}= 2.94/*x*kHz for saxophones,*f*_{c}= 3.87/*x*kHz for oboes, and*f*_{c}= 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.

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