Starting from the radiation impedance matrix, the equations above can be used to project the impedance matrix along cylinders and across discontinuities alternately until the input end is reached, giving the input impedance matrix. Remembering that the element gives the contribution to the pressure of the th mode due to a given volume velocity of the th mode, the corner value () gives the component of the plane pressure due to a given plane volume velocity.
If the air column is driven by a plane piston vibrating sinusoidally at the input end with a velocity amplitude of , we know the acoustic volume velocity will be planar with a volume velocity amplitude of . Physically, the corner value thus corresponds to the ratio of the plane component of the pressure and the (planar) volume velocity. The objects we are interested in (such as brass musical instruments) are much narrower at the input end than any wavelength present in the excitation meaning that higher modes will not propagate there. On the other hand, in most brass instruments there is a fairly rapid change of area in the mouthpiece. This causes some loss of energy to evanescent higher modes.
In chapter 5 we will verify our theoretical results using the experimental technique of acoustic pulse reflectometry. It is difficult to put significant energy into the main bore of the instrument using this technique when the mouthpiece is in place. In order to compare the experiment with theory effectively we will therefore perform all experiments and theoretical calculations in the absence of the mouthpiece.
The objects we will be looking at will therefore be both narrow and approximately cylindrical at the input. Higher modes will not be excited at the input and both the pressure and volume velocity are plane there. The corner value of the input impedance matrix can then be labelled as the input impedance without ambiguity. The theoretical results which follow plot the corner value of the input impedance matrix calculated at a range of frequencies of musical interest, showing maxima at resonances of the air column.
Because there are, in principle, an infinite number of modes in a cylindrical pipe, the vectors and matrices used in the calculation should ideally be infinite. In practice, however, at high enough mode numbers the pressures represented are insignificant to the propagation of sound. This is because they have very high cut-off frequencies so are strongly damped and have negligible coupling with the plane mode.
The matrices can then be truncated to exclude all modes with mode numbers above a certain value. Note that we cannot exclude all mode numbers which are in cut-off; energy lost from propagating modes to the many non-propagating modes with lower mode numbers must still be modelled for an accurate analysis. In practice, the number of propagating modes depends on the maximum radius in the object while the number of significant non-propagating modes also depends on the flare rate. The calculation must be repeated including more and more modes until the answer converges (ie. adding more modes only changes the answer by a small factor corresponding to the percentage error required). The length of time taken to perform the calculation increases exponentially as the number of modes increases since matrix multiplication is involved.
In addition to the number of modes under consideration, the number of cylinders used in approximating the bore is another numerical consideration. Obviously the more cylinders used in approximating the bore, the closer the approximate bore will resemble the actual bore of the instrument. The values in the impedance matrix converge on a final answer as the number of cylinders increase with the computational load increasing linearly. The number of cylinders required depends on the flare rate.
As mentioned previously, viscothermal losses may be included in the calculation by using a lossy wavenumber. The effect of losses may be determined by performing calculations with and without losses.