## Conical Bores: Modes of Propagation

• 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
 (2)

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 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.