- A sinusoidal pressure disturbance at position
*x*in a cylindrical pipe of finite length is given by

(32) *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) *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

- 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)

- 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)

(40) - Losses can be accurately accounted for in the transfer matrices by using a complex value of the wavenumber
*k*.

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