118
4 After the Lips: Acoustic Resonances and Radiation
The reflection coefficient R(ω) of an instrument was introduced in Sect. 4.1.2
as the complex ratio of the pressure amplitudes of backward and forward travelling
waves at the input. The reflection coefficient is simply related to the input impedance
by the equation (Fletcher and Rossing 1998)
R(ω) =
Z(ω) − Z c
Z(ω) + Z c
.
(4.28)
The time domain reflection function is the inverse Fourier transform of the
reflection coefficient, while the input impulse response is the inverse Fourier
transform of the input impedance:
I F T [R(ω)] = r(t);
(4.29)
I F T [Z(ω)] = g(t).
(4.30)
The input impedance curve, which is a plot of input impedance magnitude as
a function of frequency, has become a standard and invaluable tool in displaying
and studying the linear acoustic properties of wind instruments. It is particularly
useful in the case of brass instruments, since it shows explicitly the frequencies
and amplitudes of the many acoustic resonances on which the played notes are
based (see Sect. 1.2). In Fig. 4.12 it can be seen immediately that peaks 2–12
have the approximately equal frequency spacing corresponding to a set of quasiharmonically related played notes. The first peak does not fit with this series,
suggesting that there is something acoustically different about the pedal note. The
peaks effectively disappear above 800 Hz, explaining the musical experience that
above the 15th natural note, continuous glissandi are possible.
Figure 4.12 displays only the magnitude of the input impedance Z. Information
about the phase of Z can also be enlightening, since it corresponds to the phase
difference between the acoustic pressure and the volume velocity. Figure 4.14 shows
the magnitude and phase of the input impedances of two similar brass instruments,
the tenor and bass trombones illustrated in Fig. 4.13.
Fig. 4.13 Two trombones by
Courtois. Above: bass
trombone. Below: tenor
trombone
4 After the Lips: Acoustic Resonances and Radiation
The reflection coefficient R(ω) of an instrument was introduced in Sect. 4.1.2
as the complex ratio of the pressure amplitudes of backward and forward travelling
waves at the input. The reflection coefficient is simply related to the input impedance
by the equation (Fletcher and Rossing 1998)
R(ω) =
Z(ω) − Z c
Z(ω) + Z c
.
(4.28)
The time domain reflection function is the inverse Fourier transform of the
reflection coefficient, while the input impulse response is the inverse Fourier
transform of the input impedance:
I F T [R(ω)] = r(t);
(4.29)
I F T [Z(ω)] = g(t).
(4.30)
The input impedance curve, which is a plot of input impedance magnitude as
a function of frequency, has become a standard and invaluable tool in displaying
and studying the linear acoustic properties of wind instruments. It is particularly
useful in the case of brass instruments, since it shows explicitly the frequencies
and amplitudes of the many acoustic resonances on which the played notes are
based (see Sect. 1.2). In Fig. 4.12 it can be seen immediately that peaks 2–12
have the approximately equal frequency spacing corresponding to a set of quasiharmonically related played notes. The first peak does not fit with this series,
suggesting that there is something acoustically different about the pedal note. The
peaks effectively disappear above 800 Hz, explaining the musical experience that
above the 15th natural note, continuous glissandi are possible.
Figure 4.12 displays only the magnitude of the input impedance Z. Information
about the phase of Z can also be enlightening, since it corresponds to the phase
difference between the acoustic pressure and the volume velocity. Figure 4.14 shows
the magnitude and phase of the input impedances of two similar brass instruments,
the tenor and bass trombones illustrated in Fig. 4.13.
Fig. 4.13 Two trombones by
Courtois. Above: bass
trombone. Below: tenor
trombone