Cylindrical Sections: Frequency-Domain Approach

No wind instrument is constructed of a perfectly uniform cylindrical pipe. However, it is possible to approximate non-uniform bore shapes with cylindrical sections, as illustrated in Fig. 13. This approach can be pursued in either the frequency- or the time-domain.

Figure 13: A non-uniform bore (a) and its approximation in terms of cylindrical sections (b).

• A sinusoidal pressure disturbance at position $x$ in a cylindrical pipe of finite length is given by
    $$P(x,t) = \left[C^{+}e^{-jkx} + C^{-}e^{jkx}\right] e^{j\omega t},$$ (37)
  where $C^{+}$ and $C^{-}$ are complex traveling-wave amplitudes, $k = \omega/c$ is the wave number, $\omega$ is radian frequency, and $c$ is the speed of sound in air.

• The corresponding volume velocity is:
    $$U(x,t) = \frac{1}{Z_{c}}\left[C^{+}e^{-jkx} - C^{-}e^{jkx}\right] e^{j\omega t},$$ (38)
  where $Z_{c} = \rho c/A$ is the real-valued wave impedance of the pipe and $A$ is its cross-sectional area.

• For a pipe which extends from $x=0$ to $x=L$ and is terminated at $x=L$ by the load impedance $Z_{L}$, it is possible to derive (from the equations above) an expression for the impedance at $x=0,$ or the input impedance of the cylindrical pipe, given by
  

  $$Z_{in} = \frac{P(0,t)}{U(0,t)} = Z_{c}\left[\frac{C^{+} + C^{-}}{C^{+} - C^{-}}\right]$$ (39)

  $$= Z_{c}\left[\frac{Z_{L} \cos (kL) + j Z_{c} \sin(kL)}{jZ_{L} \sin(kL) + Z_{c} \cos(kL)}\right].$$ (40)

• This expression can also be formulated in terms of a transfer matrix (Keefe, 1981) which relates pressure and volume velocity at the input and output of a cylindrical section of length $L$ as
    $$\left[\begin{array}{c} P_{0} \\ U_{0} \end{array}\right] = \left[... ...d \end{array}\right] \left[\begin{array}{c} P_{L} \\ U_{L} \end{array}\right],$$ (41)
  where the lossless transfer-matrix coefficients are given by
  $$\begin{eqnarray*} a &=& \cos(kL) \\ b &=& j Z_{c} \sin(kL) \\ c &=& \frac{j}{Z_{c}}\sin(kL) \\ d &=& \cos(kL). \end{eqnarray*}$$

• In this way, a cylindrical pipe is represented by a transfer matrix, defined by its particular length and radius.

• By comparison with Eq. (40), the input impedance of the section can be calculated from the transfer-matrix coefficients as
    $$Z_{in} = \frac{b + aZ_{L}}{d + cZ_{L}}.$$ (42)

• For a sequence of $n$ cylindrical sections, the input variables for each section become the output variables for the previous section. The transfer matrices can then be cascaded as
  

  $$\left[\begin{array}{c} P_{0} \\ U_{0} \end{array}\right] = \left[\begin{array}{cc} a_{1} & b_{1} \\ c_{1} & d_{1} \end{arra... ...n} \end{array}\right] \left[\begin{array}{c} P_{L} \\ U_{L} \end{array}\right]$$

  $$= \left[\begin{array}{cc} A & B \\ C & D \end{array}\right] \left[\begin{array}{c} P_{L} \\ U_{L} \end{array}\right]$$ (43)

  where
    $$\left[\begin{array}{cc} A & B \\ C & D \end{array}\right] = \prod... ...{n} \left[\begin{array}{cc} a_{i} & b_{i} \\ c_{i} & d_{i} \end{array}\right],$$ (44)
  and the input impedance found for the entire acoustic structure as
    $$Z_{in} = \frac{B + AZ_{L}}{D + CZ_{L}}.$$ (45)

• Losses can be accurately accounted for in the transfer matrices by using a complex value of the wavenumber $k$.