5.2 Crossing the Threshold: Small Amplitude Oscillating Solutions
223
5.2.1 Phase Relationships in the Lip Valve
The ultimate goal of a brass instrument model is to predict the nature of the sound
radiated by the instrument from a knowledge of the control parameters such as the
mouth pressure p m and the lip resonance frequency f r . The first step in this process
is to be able to predict the pressure p(t) generated in the instrument mouthpiece
under the influence of these control parameters. The simplest possible solution,
p(t) = 0, corresponds to the situation in which the mouth pressure is insufficient
to induce lip vibrations. From the musical point of view, solutions which do not
result in sound generation are not very interesting, but it is important to understand
the factors which determine the threshold pressure for self-sustained lip vibration.
Some insight into this question can be gained by examining the phase relationships
between air flow and pressure in the two different modes of lip valve operation
represented by the two different signs on the right-hand side of Eq. 3.15, in the
limiting case in which the blowing pressure is just above the threshold.
Although the single and double reeds of woodwind instruments fall unambiguously into the inward-striking class, the brass player’s lips have proved much harder
to categorise. Helmholtz considered that the lips were outward-striking, and this
view was also adopted by Fletcher (1979). In a seminal paper on regeneration
in brass instruments, Elliott and Bowsher also argued that the lip valve was
predominantly outward-striking, but noted that the Bernoulli force in the lip channel
introduced an alternative flow control mechanism with an inward-striking character.
This aspect of the model was further developed by Pelorson et al. (1994) and
Hirschberg et al. (1995). More recent studies (Yoshikawa 1995; Adachi and Sato
1996; Chen and Weinreich 1996; Ayers 2001; Cullen et al. 2000; Campbell 2004;
Boutin et al. 2015b) have confirmed that it is not possible to assign a single phase
characteristic to the lip valve: both inward-striking and outward-striking behaviours
have been observed in experiments and simulations.
For very low-amplitude oscillations, it can be assumed that the acoustic variables
are to a good approximation sinusoidal functions of time. The phase relationship
between mouthpiece pressure and valve opening height for the different valve
classifications can be found by inserting the expressions h(t) = ˆ
h cos(ωt +
φ h ), p(t) = ˆ
p cos(ωt) in Eq. 3.15 for each of the two signs of the right-hand
forcing term. The resulting variations of the phase φ h with frequency are shown
schematically in Fig. 5.2.
Figure 5.1 shows that the player and instrument are linked by a feedback loop
(see also Sect. 2.2.1). The arrow from the pressure in the mouthpiece back to the
control of the lip valve represents the feedback, which is similar in character to the
feedback from the loudspeaker of a public address system to the microphone whose
signal drives it. If the feedback is negative, an increase in the output results in a
decrease in the input, so the signal dies away; if the feedback is positive, an increase
in the output increases the input, and the buildup can result in an unpleasant howl of
self-sustained oscillation from the PA system. In the case of the musical instrument,
223
5.2.1 Phase Relationships in the Lip Valve
The ultimate goal of a brass instrument model is to predict the nature of the sound
radiated by the instrument from a knowledge of the control parameters such as the
mouth pressure p m and the lip resonance frequency f r . The first step in this process
is to be able to predict the pressure p(t) generated in the instrument mouthpiece
under the influence of these control parameters. The simplest possible solution,
p(t) = 0, corresponds to the situation in which the mouth pressure is insufficient
to induce lip vibrations. From the musical point of view, solutions which do not
result in sound generation are not very interesting, but it is important to understand
the factors which determine the threshold pressure for self-sustained lip vibration.
Some insight into this question can be gained by examining the phase relationships
between air flow and pressure in the two different modes of lip valve operation
represented by the two different signs on the right-hand side of Eq. 3.15, in the
limiting case in which the blowing pressure is just above the threshold.
Although the single and double reeds of woodwind instruments fall unambiguously into the inward-striking class, the brass player’s lips have proved much harder
to categorise. Helmholtz considered that the lips were outward-striking, and this
view was also adopted by Fletcher (1979). In a seminal paper on regeneration
in brass instruments, Elliott and Bowsher also argued that the lip valve was
predominantly outward-striking, but noted that the Bernoulli force in the lip channel
introduced an alternative flow control mechanism with an inward-striking character.
This aspect of the model was further developed by Pelorson et al. (1994) and
Hirschberg et al. (1995). More recent studies (Yoshikawa 1995; Adachi and Sato
1996; Chen and Weinreich 1996; Ayers 2001; Cullen et al. 2000; Campbell 2004;
Boutin et al. 2015b) have confirmed that it is not possible to assign a single phase
characteristic to the lip valve: both inward-striking and outward-striking behaviours
have been observed in experiments and simulations.
For very low-amplitude oscillations, it can be assumed that the acoustic variables
are to a good approximation sinusoidal functions of time. The phase relationship
between mouthpiece pressure and valve opening height for the different valve
classifications can be found by inserting the expressions h(t) = ˆ
h cos(ωt +
φ h ), p(t) = ˆ
p cos(ωt) in Eq. 3.15 for each of the two signs of the right-hand
forcing term. The resulting variations of the phase φ h with frequency are shown
schematically in Fig. 5.2.
Figure 5.1 shows that the player and instrument are linked by a feedback loop
(see also Sect. 2.2.1). The arrow from the pressure in the mouthpiece back to the
control of the lip valve represents the feedback, which is similar in character to the
feedback from the loudspeaker of a public address system to the microphone whose
signal drives it. If the feedback is negative, an increase in the output results in a
decrease in the input, so the signal dies away; if the feedback is positive, an increase
in the output increases the input, and the buildup can result in an unpleasant howl of
self-sustained oscillation from the PA system. In the case of the musical instrument,