## 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. 10. This approach can be pursued in either the frequency- or the time-domain.

• A sinusoidal pressure disturbance at position x in a cylindrical pipe of finite length is given by
 (32)

where C+ and C- are complex traveling-wave amplitudes, is the wave number, is radian frequency, and c is the speed of sound in air.

• The corresponding volume velocity is:
 (33)

where 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 ZL, 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
 = (34) = (35)

• 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
 (36)

where the lossless transfer-matrix coefficients are given by

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

• By comparison with Eq. (35), the input impedance of the section can be calculated from the transfer-matrix coefficients as
 (37)

• 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
 = = (38)

where
 (39)

and the input impedance found for the entire acoustic structure as
 (40)

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