88
3 Buzzing Lips: Sound Generation in Brass Instruments
Fig. 3.25 A schematic
diagram of a one-mass lip
model exhibiting linear
simple harmonic motion
along transverse and axial
directions. From Cullen et al.
(2000)
these two diagrams explains why some designs of inward-striking reed behave like
outward-striking reeds when the direction of flow is reversed. The free metal reeds in
accordions and harmonicas can function as either (−, +) or (+, −) valves depending
on the configuration and the blowing pressure (Fletcher and Rossing 1998).
Figure 3.25 is an alternative schematic representation of the lip valve, in which
the lip is assumed to move along a straight line in the (x, z) plane at an angle θ to the
y axis (Cullen et al. 2000). Although this is a more abstract picture than the swinging
door model, it allows for the representation of linear simple harmonic motion in
both axial and transverse directions. A decrease in the mouthpiece pressure p,
corresponding to an increase in the pressure difference ¯
P m − p, drives the lip
upwards because of the geometrical constraint; the corresponding force F D tends
to increase the lip opening H , displaying an outward-striking character. The drop
in p also implies an additional downward force F B on the lip surface; this is the
Bernoulli force with its inward-striking character, tending to reduce the lip opening.
A generalised dynamical equation describing the small amplitude motion of both
outward- and inward-striking lip valve models can be written as
d 2 h(t)
dt 2 +
ω l
Q l
dh(t)
dt
+ ω
2
r (h(t) − h eq ) = ±
p(t)
μ
,
(3.15)
with the right hand forcing term positive for the inward-striking model and negative
for the outward-striking model.
The simple 1DOF lip model with an outward-striking character, whose small
amplitude behaviour is represented by Eq. 3.12, corresponds to the scientific obser-
3 Buzzing Lips: Sound Generation in Brass Instruments
Fig. 3.25 A schematic
diagram of a one-mass lip
model exhibiting linear
simple harmonic motion
along transverse and axial
directions. From Cullen et al.
(2000)
these two diagrams explains why some designs of inward-striking reed behave like
outward-striking reeds when the direction of flow is reversed. The free metal reeds in
accordions and harmonicas can function as either (−, +) or (+, −) valves depending
on the configuration and the blowing pressure (Fletcher and Rossing 1998).
Figure 3.25 is an alternative schematic representation of the lip valve, in which
the lip is assumed to move along a straight line in the (x, z) plane at an angle θ to the
y axis (Cullen et al. 2000). Although this is a more abstract picture than the swinging
door model, it allows for the representation of linear simple harmonic motion in
both axial and transverse directions. A decrease in the mouthpiece pressure p,
corresponding to an increase in the pressure difference ¯
P m − p, drives the lip
upwards because of the geometrical constraint; the corresponding force F D tends
to increase the lip opening H , displaying an outward-striking character. The drop
in p also implies an additional downward force F B on the lip surface; this is the
Bernoulli force with its inward-striking character, tending to reduce the lip opening.
A generalised dynamical equation describing the small amplitude motion of both
outward- and inward-striking lip valve models can be written as
d 2 h(t)
dt 2 +
ω l
Q l
dh(t)
dt
+ ω
2
r (h(t) − h eq ) = ±
p(t)
μ
,
(3.15)
with the right hand forcing term positive for the inward-striking model and negative
for the outward-striking model.
The simple 1DOF lip model with an outward-striking character, whose small
amplitude behaviour is represented by Eq. 3.12, corresponds to the scientific obser-