98
3 Buzzing Lips: Sound Generation in Brass Instruments
=
1
2
ρv
2
lc
1 −
v 2
m
v 2
lc
(3.22)
=
1
2
ρv
2
lc
1 −
S 2
lc
S 2
m
.
(3.23)
The open area S m of the mouth just behind the lips depends on the position of
the tongue. There is evidence that some players deliberately constrict this area to
assist high register playing (see Sect. 6.3). For low notes Farkas (1962) recommends
a low tongue position corresponding to an ‘oh’ vowel, corresponding to an area
S m 500 mm 2 (Story et al. 1996). The lip opening area S lc in trombone playing is
typically less than 50 mm 2 (see Fig. 3.7); in this case S 2
lc /S 2
m 1, and
p m − p lc
1
2
ρv
2
lc .
(3.24)
.
3.5.2 Acoustic Volume Flow Equation
The left-hand side of Fig. 3.32 illustrates the mouth air flow pattern described in
Sect. 3.5.1. As the streamlines converge into the narrow lip channel, the air particle
velocity increases, and the pressure drops. If the flow emerging from the lip channel
into the mouthpiece followed the same pattern, the streamlines would diverge
smoothly from the channel exit, resulting in an increase of pressure and a decrease in
the particle velocity. The final mouthpiece pressure p and particle velocity v would
depend on the cross-sectional area of the mouthpiece cup, but not on the dimensions
of the lip channel. If the lips started to oscillate, the changing cross-sectional area
of the lip channel would have no effect on the downstream pressure and flow unless
the amplitude of oscillation were enough to completely close the channel.
The flow control which makes the oscillating lips an effective valve effect source
relies on subtle fluid dynamic effects localised in the viscothermal boundary layers
close to the walls of the lip channel. Even when air is flowing through the lip
channel with a speed greater than 10 m s −1 , the air in contact with the lip surface
remains at rest; there is a rapid velocity gradient across the boundary layer, which is
typically much less than a millimetre thick. At a certain point along the lip channel,
whose location depends on the channel profile and the air speed, instabilities in the
boundary layer cause the flow to separate from the wall and become a free jet. The
flow separation point occurs earlier in a diverging channel. Since the profile of the
lip channel changes during the course of one cycle of lip vibration, the separation
point is also likely to vary during the vibration cycle. There may also be a secondary
separation point at the upstream entrance to the lip channel, giving rise to a reduction
3 Buzzing Lips: Sound Generation in Brass Instruments
=
1
2
ρv
2
lc
1 −
v 2
m
v 2
lc
(3.22)
=
1
2
ρv
2
lc
1 −
S 2
lc
S 2
m
.
(3.23)
The open area S m of the mouth just behind the lips depends on the position of
the tongue. There is evidence that some players deliberately constrict this area to
assist high register playing (see Sect. 6.3). For low notes Farkas (1962) recommends
a low tongue position corresponding to an ‘oh’ vowel, corresponding to an area
S m 500 mm 2 (Story et al. 1996). The lip opening area S lc in trombone playing is
typically less than 50 mm 2 (see Fig. 3.7); in this case S 2
lc /S 2
m 1, and
p m − p lc
1
2
ρv
2
lc .
(3.24)
.
3.5.2 Acoustic Volume Flow Equation
The left-hand side of Fig. 3.32 illustrates the mouth air flow pattern described in
Sect. 3.5.1. As the streamlines converge into the narrow lip channel, the air particle
velocity increases, and the pressure drops. If the flow emerging from the lip channel
into the mouthpiece followed the same pattern, the streamlines would diverge
smoothly from the channel exit, resulting in an increase of pressure and a decrease in
the particle velocity. The final mouthpiece pressure p and particle velocity v would
depend on the cross-sectional area of the mouthpiece cup, but not on the dimensions
of the lip channel. If the lips started to oscillate, the changing cross-sectional area
of the lip channel would have no effect on the downstream pressure and flow unless
the amplitude of oscillation were enough to completely close the channel.
The flow control which makes the oscillating lips an effective valve effect source
relies on subtle fluid dynamic effects localised in the viscothermal boundary layers
close to the walls of the lip channel. Even when air is flowing through the lip
channel with a speed greater than 10 m s −1 , the air in contact with the lip surface
remains at rest; there is a rapid velocity gradient across the boundary layer, which is
typically much less than a millimetre thick. At a certain point along the lip channel,
whose location depends on the channel profile and the air speed, instabilities in the
boundary layer cause the flow to separate from the wall and become a free jet. The
flow separation point occurs earlier in a diverging channel. Since the profile of the
lip channel changes during the course of one cycle of lip vibration, the separation
point is also likely to vary during the vibration cycle. There may also be a secondary
separation point at the upstream entrance to the lip channel, giving rise to a reduction