3.5 Volume Flow in Buzzing Lips
99
in the effective height of the lip channel known as the vena contracta (Hirschberg et
al. 1996a).
In the simplest model, the separation point is fixed at the downstream edge
for any values of the parameters. Such a jet flow is very unstable and becomes
turbulent, as illustrated schematically in Sect. 3.32. The turbulence implies a rapid
mixing of the jet flow with the surrounding stagnant air, leading to a deceleration of
the jet and a transfer of momentum. Some distance downstream of the lip channel
exit, the air particle velocity can be considered approximately constant across the
mouthpiece cross-section. Since the conservation of volume flow is still applicable
in this case, the volume flow rate in the mouthpiece is equal to that in the lip
channel. However the dissipative forces acting during the turbulent mixing render
the Bernoulli equation invalid in the mouthpiece cup, and the reduction in particle
velocity is not accompanied by a recovery of the pressure. The kinetic energy given
up by the decelerating air particles is not stored as potential energy, but converted
through turbulence into heat.
The absence of pressure recovery in the mouthpiece means that the pressure
difference p between the mouth pressure p m and the mouthpiece pressure p is
given by
p = p m − p = p m − p lc .
(3.25)
The acoustic volume flow rate u(t) into the mouthpiece can then be derived using
Eqs. 3.18 and 3.24:
u(t) = S lc (t) v lc (t)
= S lc (t)
2p(t )
ρ
(3.26)
= S lc (t)
2(p m − p(t))
ρ
.
(3.27)
Based on the simplified model of the brass player’s mouth and lips described in
Sect. 3.5.1, Eq. 3.27 predicts that for a fixed mouth pressure p m , the rate of flow of
air into the mouthpiece depends on two variables: the cross-sectional area of the lip
channel S lc and the mouthpiece pressure p. When a player buzzes the lips without
a mouthpiece, the air just in front of the lip channel exit is at atmospheric pressure,
corresponding to p = 0. The flow rate equation then simplifies to
u(t) freebuzz = S lc (t)
2p m
ρ
.
(3.28)
The flow rate is directly proportional to the cross-sectional area of the lip channel.
When the lips vibrate in such a way that the area varies sinusoidally, the flow leaving
99
in the effective height of the lip channel known as the vena contracta (Hirschberg et
al. 1996a).
In the simplest model, the separation point is fixed at the downstream edge
for any values of the parameters. Such a jet flow is very unstable and becomes
turbulent, as illustrated schematically in Sect. 3.32. The turbulence implies a rapid
mixing of the jet flow with the surrounding stagnant air, leading to a deceleration of
the jet and a transfer of momentum. Some distance downstream of the lip channel
exit, the air particle velocity can be considered approximately constant across the
mouthpiece cross-section. Since the conservation of volume flow is still applicable
in this case, the volume flow rate in the mouthpiece is equal to that in the lip
channel. However the dissipative forces acting during the turbulent mixing render
the Bernoulli equation invalid in the mouthpiece cup, and the reduction in particle
velocity is not accompanied by a recovery of the pressure. The kinetic energy given
up by the decelerating air particles is not stored as potential energy, but converted
through turbulence into heat.
The absence of pressure recovery in the mouthpiece means that the pressure
difference p between the mouth pressure p m and the mouthpiece pressure p is
given by
p = p m − p = p m − p lc .
(3.25)
The acoustic volume flow rate u(t) into the mouthpiece can then be derived using
Eqs. 3.18 and 3.24:
u(t) = S lc (t) v lc (t)
= S lc (t)
2p(t )
ρ
(3.26)
= S lc (t)
2(p m − p(t))
ρ
.
(3.27)
Based on the simplified model of the brass player’s mouth and lips described in
Sect. 3.5.1, Eq. 3.27 predicts that for a fixed mouth pressure p m , the rate of flow of
air into the mouthpiece depends on two variables: the cross-sectional area of the lip
channel S lc and the mouthpiece pressure p. When a player buzzes the lips without
a mouthpiece, the air just in front of the lip channel exit is at atmospheric pressure,
corresponding to p = 0. The flow rate equation then simplifies to
u(t) freebuzz = S lc (t)
2p m
ρ
.
(3.28)
The flow rate is directly proportional to the cross-sectional area of the lip channel.
When the lips vibrate in such a way that the area varies sinusoidally, the flow leaving