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It should not seem strange that the acoustic impedance is unaltered by the change in cross-section because the characteristic impedance has changed and the acoustic impedance has therefore altered relative to the characteristic impedance. We may use equation (2.13) to work out the impedance at plane 0 and the effect of the new cross-section will then have an effect on the resulting value.

To further illustrate the point, we must deal with the forward and
backward going waves in each cylinder.
and are the complex pressure amplitudes for the waves
in the cylinder to the left of discontinuity, meaning the pressure
there from equation (2.2) is

(2.14) |

(2.15) |

From equation (2.7) the we volume velocity in plane 1 is

(2.17) |

(2.18) |

While the sum of the forward and backward waves is the same on either side of the discontinuity, the difference depends on the ratios of the cross-sections, hence the waves experience reflection and transmission coefficients.

In particular, consider if our pipes in figure 2.2 are infinite in
length. A pressure wave is
incident on the discontinuity from with amplitude . It is
partially reflected with amplitude back down the tube to and
partially transmitted to with amplitude .
since there is no backward going wave in . We can solve
(2.16) and (2.19) by eliminating
to get the reflection coefficient:

To summarise, impedance is not effected by changes in cross-section and equation (2.13) describes how impedance changes when projected along a pipe of given cross-section. Now we have the equations necessary to work out the impedance at one end of a instrument of known internal profile provided the impedance at the other end is known. The impedance at the open end will depend on the geometry of the opening and is treated in detail in chapter 3. Projecting the impedance down to the input (mouthpiece) end gives us the input impedance, the amount of pressure produced in the mouthpiece by a unit volume velocity source. Now we have derived a method of input impedance calculation in the plane wave approximation, we will go on to do the same for multimodal propagation.

This thesis has moved to Jonathan Kemp Thesis at http://www.kempacoustics.com/thesis

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