Finite Length Cones

• At any particular position $x$ and frequency $\omega$, sinusoidal pressure and volume velocity traveling-wave components are related by
    $$P^{+}(x,\omega) = Z_{c}(x,\omega)\,U^{+}(x,\omega) \hspace{0.5in} P^{-}(x,\omega) = -Z_{c}^{*}(x,\omega)\,U^{-}(x,\omega)$$ (9)
  with
    $$P(x,\omega) = P^{+}(x,\omega) + P^{-}(x,\omega) \hspace{0.5in} U(x,\omega) = U^{+}(x,\omega) + U^{-}(x,\omega)$$ (10)

• The plus $(+)$ superscripts indicate wave components traveling in the positive $x$-direction or away from the cone apex, while negative $(-)$ superscripts indicate travel in the negative $x$-direction or toward the apex.

• These relationships are similar to those for the cylindrical bore, with the important difference that the characteristic impedance for waves traveling toward the cone tip is the complex conjugate of that for waves traveling away from the apex.

• In other words, wave propagation toward the apex is different from propagation away from the apex because a conical waveguide is nonsymmetric about its midpoint (Keefe, 1981, p. 70).

• Because the wave impedance for spherical-wave components is frequency dependent, these relationships are valid only for frequency-domain analyses.

  Figure 2: A divergent conical section and its associated dimensional parameters.

  

• Figure 2 illustrates a divergent conical frusta for which the apical section is truncated at $x=x_{0}.$ The frequency-domain pressure wave reflectance and transmittance for such a section, assuming a load impedance $Z_{L}$ at $x=L$, are given by
    $$\frac{C^{-}}{C^{+}} = e^{-2jkL}\left[\frac{Z_{L} Z_{c}^{*}(L) - Z_{c}(L) Z_{c}^{*}(L)}{Z_{L} Z_{c}(L) + Z_{c}(L) Z_{c}^{*}(L)}\right],$$ (11)
  and
    $$\frac{p(L,t)}{C^{+}} = \frac{e^{-jkL}}{L}\left[\frac{Z_{L} Z_{c}(L) + Z_{L} Z_{c}^{*}(L)}{Z_{L} Z_{c}(L) + Z_{c}(L) Z_{c}^{*}(L)}\right],$$ (12)
  respectively.

• The phase shift term $e^{-2jkL}$ in Eq. (11) represents wave propagation to $x=L$ and back and has unity magnitude. The length parameter in this term is $2L$ because the cone apex is defined at $x=0.$

• The $1/L$ factor in the transmittance is characteristic of conical waveguides and results from the spreading of pressure across an increasing surface area as the traveling-wave component of pressure propagates away from the cone apex (Benade, 1988).

• Equation 11 applies equally well to a convergent conical section and can be used to deduce the pressure reflectance at a conical apex by setting $Z_{L}=\infty$ and taking the limit as $x \rightarrow 0.$ In this case, the term in brackets reduces to negative one.

• Thus, while the apex of a cone is a pressure antinode, pressure traveling-wave components reflect from the apex with an inversion, a behavior that at first appears paradoxical.

• It should be remembered that the boundary condition at the tip must be met by the sum of the two traveling-wave components and their corresponding $1/x$ factors. The inversion of reflected pressure is necessary to maintain a finite pressure at $x = 0,$ which can then be determined by l'Hôpital's rule (Ayers et al., 1985).

• This behavior can also be explained in terms of pressure wave reflection at a rigid termination. Pressure wave components reflect from a rigid conical bore boundary with unity magnitude and a phase shift equal to $2 \, \angle Z_{c}.$ Since the angle of $Z_{c}(0)$ is $90^{\circ},$ pressure is reflected from the tip with a $180^{\circ}$ phase shift or with an inversion.