Conical Bores: 1D Spherical Wave Propagation

• For musical purposes, the principal mode of wave propagation in conical tubes is spherical and along the central axis of the tube.

• Spherical waves of sound can theoretically propagate without reflection or loss away from the apex and along the principal axis of an infinite conical bore, assuming the walls are rigid, perfectly smooth, and thermally insulating.

• One-dimensional spherical-wave propagation along the central axis of the cone is possible for $m=0$ and $\beta=0,$ in which case Eq. (2) reduces to a general solution of the form
    $$P(x) = \frac{C^{+}}{x}e^{-jkx},$$ (3)
  where $C^{+}$ is a constant and $k$ is the wave number in open air. Waves of this type will propagate at all frequencies.

• During a steady-state excitation, sinusoidal pressure at position $x$ in a finite length conic section is composed of superposed spherical traveling-wave components of the form
    $$P(x,t) = \left[\frac{C^{+}}{x}e^{-jkx} + \frac{C^{-}}{x}e^{jkx}\right] e^{j\omega t},$$ (4)
  where $C^{+}$ and $C^{-}$ are complex amplitudes and sinusoidal time dependence is assumed.

• The associated volume velocity in the cone is found by rewriting Newton's second law for one-dimensional spherical waves as
    $$\ensuremath{\frac{\partial {p}}{\partial {x}}} = - \frac{\rho}{A(x)} \ensuremath{\frac{\partial {U}}{\partial {t}}},$$ (5)
  where $A(x)$ is the surface area of a spherical cap that intersects the principal axis of the pipe at position $x$.

• For pressure waves of the form of Eq. (4), the corresponding volume flow is found using Eq. (5) as
  

  $$U(x,t) = \frac{A(x)}{x \rho c}\left[C^{+}\left(1 + \frac{1}{jkx}\right)e^{-jkx} -C^{-}\left(1 - \frac{1}{jkx}\right)e^{jkx}\right]e^{j\omega t},$$

  $$= \frac{1}{x} \left[\frac{C^{+}}{Z_{c}(x)}e^{-jkx} - \frac{C^{-}}{Z_{c}^{*}(x)}e^{jkx}\right] e^{j\omega t}.$$ (6)

• The wave impedance for spherical traveling-wave components propagating away from the cone apex is
    $$Z_{c}(x) = \frac{P(x)}{U(x)} = \frac{\rho c}{A(x)}\left(\frac{jkx}{1 + jkx}\right) = \frac{\rho c}{A(x)}\left(\frac{1}{1 + \frac{1}{jkx}}\right),$$ (7)
  which depends both on position $x$ and frequency $\omega.$

• The characteristic impedance for spherical traveling-wave components propagating in a cone toward its apex is given by $Z_{c}^{\ast}(x),$ or the complex conjugate of $Z_{c}(x).$

• For $kx \gg 1$, the spherical wave fronts become more planar in shape and $Z_{c}(x)$ approaches $\rho c/A(x),$ the wave impedance for plane waves in a duct of cross section $A.$

• Near the apex of the cone, however, the imaginary part of $Z_{c}(x)$ becomes increasingly dominant and in the limit as $x \rightarrow 0,$ the pressure and velocity traveling-wave components become $90^{\circ}$ out of phase at the cone tip.

• It is intuitively helpful to rewrite the characteristic impedance in the form
    $$Z_{c}(x) = \frac{1}{\frac{A(x)}{\rho c} + \frac{A(x)}{j \omega \rho x}},$$ (8)
  which is equivalent to the resistive wave impedance of a cylindrical bore in parallel with a lumped inertance of acoustic mass $\rho x / A(x)$.

• At low frequencies and near the conical apex, the inertance effectively shunts out the resistive element.