The waveguide “input impedance”
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
This function is given by
(26)
where is the Heaviside unit step function.
It is necessary to determine a digital filter that appropriately models the response of in the discrete-time domain.
The spherical-wave characteristic impedance is given in terms of its Laplace transform as
(27)
where is the spherical-wave surface area at
Using the bilinear transform, an equivalent discrete-time filter is given by
(28)
where is the bilinear transform constant and is equal to the allpass truncation filter coefficient given in Eq. (25).
The impulse response of a closed-open truncated cone is found from the digital waveguide structure of Fig. 7 as the pressure response at to the impulse response
Because the cone is excited at output scale factors are appropriately weighted by
Figure 8 is a plot of 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 and open at 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
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.