Input Admittance of a Conical Section

• The input admittance $Y(x)$ of a conical bore is more simply stated than its input impedance. For a conic section truncated at $x=x_{0},$ the input admittance is
    $$Y_{in}(x_{0}) = \frac{1}{Z(x_{0})} = \frac{U(x_{0},t)}{P(x_{0},t)... ...{0}} + \frac{C^{-}}{C^{+}}e^{jkx_{0}}} \right] + \frac{1}{j k x_{0}} \right\},$$ (13)
  where the pressure wave reflectance $C^{-}/C^{+}$ is determined by the length of the bore and the boundary conditions at the opposite end, as discussed above.

• Equation (13) applies equally well to bores of increasing and decreasing diameter by using either positive or negative values of $x$.

• Equation (13) may be interpreted as a parallel combination of an acoustic inertance and a term reminiscent of the impedance of a cylindrical waveguide (Benade, 1988).

• The impedance of the acoustic inertance, which has an equivalent acoustic mass $\rho x_{0} / A(x_{0}),$ approaches infinity as $x_{0} \rightarrow 0.$

• The input admittance seen from the open end (at $x=-L$) of a complete cone reduces to
  

  $$Y_{in}(-L) = \frac{A(L)}{\rho c} \left\{ \left[ \frac{e^{jkL} - \frac{C^{-}}{C... ...jkL}}{ e^{jkL} + \frac{C^{-}}{C^{+}}e^{-jkL}} \right] -\frac{1}{j k L} \right\}$$

  $$= \frac{A(L)}{\rho c} \left\{ \left[ \frac{e^{jkL} + e^{-jkL}}{ e^{jkL} - e^{-jkL}}\right] - \frac{1}{j k L} \right\}$$

  $$= \frac{A(L)}{j \rho c} \left[ \cot(kL) -\frac{1}{k L} \right],$$ (14)

  where the pressure reflectance at the cone apex ($x=0$) is negative one.

• An open end at $x=-L$ can be approximated by the low-frequency estimate $Y_{in} = \infty.$

• The resonance frequencies of a complete cone ideally open at its large end are thus found at the infinities of Eq. (14), which are given for $n = 1, 2, \ldots$ by
    $$f = \frac{n c}{2 L}.$$ (15)

• The complete cone with open mouth has a fundamental wavelength equal to two times its length and higher resonances that occur at all integer multiples of the fundamental frequency, as was observed for open-open cylindrical pipes.

• The anti-resonances of the complete cone, however, do not fall exactly midway between its resonances, but are influenced by the inertance term in Eq. (14).