Conical Bores: Modes of Propagation

Figure 1: A conical section in spherical coordinates.

• A conical bore section in spherical coordinates $(x, \phi, \theta)$ is depicted in Fig. 1. The wave equation in this geometric coordinate system is
    $$\frac{1}{x^{2}}\frac{\partial}{\partial x}\left(x^{2} \ensuremath... ...{2}}} = \frac{1}{c^{2}}\ensuremath{\frac{\partial^{2} {p}}{\partial {t}^{2}}}.$$ (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
    $$P_{mn}(x, \phi, \theta) = \Phi(\phi)\Theta(\theta)X(x) = \frac{P_... ...(kx)^{\frac{1}{2}}}\cos(m\phi)\Theta_{n}^{m}(\cos\theta)J_{n+\frac{1}{2}}(kx),$$ (2)
  where $J_{n+\frac{1}{2}}(kx)$ is a Bessel function and $\Theta_{n}^{m}(y)$ are associated Legendre functions.

• The boundary condition at the cone wall, or at $\theta = \theta_{0},$ is $\partial p/\partial \theta = 0,$ which can be met by adjusting the values of $m$ and $n$ so that an extremum of $\Theta_{n}^{m}(\cos\theta)$ 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 $\theta_{0}$ for saxophones, oboes, and bassoons are $2^{\circ},$ $1.5^{\circ},$ and $0.5^{\circ},$ 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 $k^{2} -\beta^{2}/x^{2}$ becomes positive.

• The cutoff frequencies in a cone are dependent on $x,$ so that waves of sinusoidal type having $\beta \neq 0$ 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.