Input Impedance Calculations

• The waveguide “input impedance” $Z_{in}(x_{0}, \omega)$ for this system can be determined by calculating its response to a unit volume velocity impulse and transforming it to the frequency domain using the DFT.

• The continuous-time pressure response to a volume velocity impulse is given by the inverse Fourier transform of the characteristic impedance $Z_{c}(x, \Omega).$

• This function is given by
    $$h^{+}(x,t) = \frac{\rho c}{A(x)}\left[ \delta(t) - \frac{c}{x} \,e^{-(c/x)t}\epsilon(t)\right],$$ (26)
  where $\epsilon(t)$ is the Heaviside unit step function.

• It is necessary to determine a digital filter that appropriately models the response of $h^{+}(x,t)$ in the discrete-time domain.

• The spherical-wave characteristic impedance is given in terms of its Laplace transform as
    $$H_{c}^{+}(x,s) = Z_{c}(x,s) = \frac{\rho c}{A(x)} \left( \frac{x s}{c + x s} \right),$$ (27)
  where $A(x)$ is the spherical-wave surface area at $x.$

• Using the bilinear transform, an equivalent discrete-time filter is given by
    $$H_{d}^{+}(x,z) = \left( \frac{\rho c}{A(x)} \right) \left( \frac{\alpha x}{c + \alpha x} \right) \frac{1 - z^{-1}}{1 + a_{1} z^{-1}},$$ (28)
  where $\alpha$ is the bilinear transform constant and $a_{1}$ is equal to the allpass truncation filter coefficient given in Eq. (25).

• The impulse response $h(x_{0},n)$ of a closed-open truncated cone is found from the digital waveguide structure of Fig. 7 as the pressure response at $x=x_{0}$ to the impulse response $h^{+}(x_{0},n).$

• Because the cone is excited at $x=x_{0},$ output scale factors are appropriately weighted by $x_{0}.$

• Figure 8 is a plot of $h(x_{0},n),$ calculated using a Levine and Schwinger model of the open-end radiation and neglecting viscothermal losses.

  Figure 8: Impulse response of conical bore closed at $x=x_{0}$ and open at $x=L.$ Open-end radiation is approximated by a 2nd-order discrete-time filter and viscothermal losses are ignored.

  

• Salmon (1946) and Benade (1988) proposed a transmission-line conical waveguide model directly analogous to that described above.

• Shown in Fig. 9, Benade's model represents a conic section by a cylindrical waveguide, a pair of “inertance” terms, and a transformer whose “turns ratio” is equal to the ratio $P_{x_{0}}/P_{L}.$

  Figure 9: Equivalent circuit of a conical waveguide [after Benade (1988)].

  

• The digital waveguide truncated cone model implements the cylindrical waveguide section using delay lines, as previously described, and the transformer operation wherever a physical pressure measurement is taken.