Back in section 8.5 we discussed a severely simplified model for a clarinet. Our model involved coupling the linear acoustics of the tube to a non-linear model of the mouthpiece and reed. We will look at wind instruments properly in Chapter 11, but we can anticipate the fact that almost all wind instruments involve a somewhat similar division between something non-linear happening in and around the mouthpiece, coupled to linear acoustics of the tube of the instrument.
The natural frequency response function to characterise this linear acoustics is the input impedance, defined as the ratio of pressure to volume flow rate at a plane defining the junction of tube and mouthpiece. In terms of measurement, you can supply the necessary input volume flow rate with some kind of loudspeaker, and you can measure the resulting pressure with a microphone. But the definition of input impedance presents a challenge: there is no easy and reliable sensor that can measure volume flow rate at acoustic frequencies. Over the years, considerable ingenuity has been devoted to the design of measurement methods that get around this difficulty.
In the world of wind instruments, measurements using continuous excitation are still the most usual, although we will come to an impulse-based method later. The earliest approach is illustrated in schematic form in Fig. 1. A loudspeaker is driven by a sine wave, and is connected to the instrument tube via a very thin capillary tube. The loudspeaker is enclosed in a robust container so that no sound escapes except into the instrument tube through the capillary. Two microphones, mounted flush with the surface, measure the pressure at the two ends of the capillary.
The significance of the capillary is that it presents a very high resistance to the acoustic flow, and this has the result that the volume flow rate is governed almost entirely by the pressure at the left-hand end, inside the loudspeaker cavity. The pressure at the other end, in the instrument tube, makes virtually no difference. This means that once the device has been calibrated, the pressure measured by the internal microphone can be converted directly into a volume flow rate entering the instrument, decoupled from the acoustic behaviour of the particular instrument being measured. The second microphone then measures the resulting pressure, and the combination of these two gives the input impedance of the instrument.
Unfortunately, the high resistance of the capillary means that such devices only inject rather small volume flow rates into the instrument. In theory this doesn’t matter, since the input impedance is a linear quantity, independent of the excitation amplitude. However, in practice low amplitude gives a reduction in signal-to-noise ratio. So it would be better to dispense with the capillary, and this leads to the next type of impedance-measuring device, shown schematically in Fig. 2. This time, the loudspeaker cavity is coupled directly to the instrument tube, with a microphone near the junction.
A second microphone is in the cavity behind the loudspeaker piston. Provided the walls of the device are sufficiently rigid, the volume flow rate into the instrument must be exactly equal and opposite to the volume flow rate into the rear cavity. This rate can be inferred from the pressure measured by the second microphone — provided a sufficiently elaborate and careful set of calibration tests has been carried out because the rear cavity will have resonances of its own which must be compensated for. It is a device of this type which is seen in use in Fig. 7 of section 10.4.
Another type of measurement strategy relies on making the connecting tube between the loudspeaker and the instrument very long. This long, parallel tube can only support plane acoustic waves, travelling in one direction or the other. By placing two (or more) microphones in the wall of this tube and then doing some computer processing, it is possible to separate the contributions from left-travelling and right-travelling waves. By comparing their amplitudes and phases, it is then possible to deduce the reflection coefficient of the instrument tube. There is a standard formula linking this reflection coefficient to the input impedance, and so the impedance can be determined.
Finally, there is a variant of this approach which we have already seen, back in section 10.2. If we drive our long connecting tube with a pulse, we can deduce the shape of the reflected pulse from the instrument. This pulse reflectometry information can be used for bore reconstruction, as we saw earlier. But it can also be processed to yield the input impedance, via a Fourier transform.
For more detail on all these methods, see Section 4.2 of the book by Campbell, Gilbert and Myers , and the extensive references contained therein.
 Murray Campbell, Joël Gilbert and Arnold Myers, “The science of brass instruments”, ASA Press/Springer (2021)