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 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
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:
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
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
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
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
and the input impedance found for the entire acoustic structure as
Losses can be accurately accounted for in the transfer matrices by using a complex value of the wavenumber k.