3.2 An Equation of Motion for the Lips
85
θ eq =
S io p m
2mlω 2
l
(3.10)
at which the restoring torque is equal to the torque due to the pressure difference
across the lips p = p m . In contrast to the case of the sliding door model, the
equilibrium position of the swinging door model increases with increasing mouth
pressure.
When the mouth pressure is raised just above the playing threshold, the lip will
start to oscillate about the threshold equilibrium angle. The relation between the
angle θ and the lip opening height is
h = h 0 + l(1 − cos θ).
(3.11)
Since this equation is nonlinear, simple harmonic motion of the swinging door does
not result in sinusoidal modulation of the lip opening. The behaviour of the lips
near the oscillation threshold can however be represented by assuming that around
the equilibrium position the movement of the mass in Fig. 3.22 is along the tangent
to its actual circular path. The small amplitude equation of motion for the swinging
door model can then be written in terms of the lip opening height as
d 2 h(t)
dt 2 +
ω l
Q l
dh(t)
dt
+ ω
2
l (h(t) − h eq ) = −
S io sin θ eq p(t)
2m
= −
p(t)
μ
,
(3.12)
where the effective mass per unit area of the vibrating lips is in this case μ =
2m/(S io sin θ eq ).
3.2.4 Inward-Striking and Outward-Striking Reeds
For a valve effect source to be capable of generating a continuous sound in a musical
wind instrument, it must be capable of supplying energy to sustain the standing
waves in the instrument’s air column. Before continuing with the development of
the elementary brass model, it is worth making a brief digression to examine the
different ways in which this energy supply requirement is met in reed woodwind
and lip-excited brass instruments.
The modelling of reed wind instruments was first put on a firm footing by the
great nineteenth century German physicist Helmholtz (1877). He understood that
the reed behaved as a pressure-controlled valve, modulating the flow of air from
the player’s mouth into the instrument. Helmholtz considered that there were two
distinct types of reed mechanism, depending on the geometry of the reed structure
relative to the direction of flow. He defined an ‘inward-striking’ (einschlagende)
reed as one in which the motion of the reed blade in the direction of flow resulted
in a decrease in the reed opening. An ‘outward-striking’ (aussschlagende) reed was
defined as one in which the motion of the reed blade in the direction of flow resulted
85
θ eq =
S io p m
2mlω 2
l
(3.10)
at which the restoring torque is equal to the torque due to the pressure difference
across the lips p = p m . In contrast to the case of the sliding door model, the
equilibrium position of the swinging door model increases with increasing mouth
pressure.
When the mouth pressure is raised just above the playing threshold, the lip will
start to oscillate about the threshold equilibrium angle. The relation between the
angle θ and the lip opening height is
h = h 0 + l(1 − cos θ).
(3.11)
Since this equation is nonlinear, simple harmonic motion of the swinging door does
not result in sinusoidal modulation of the lip opening. The behaviour of the lips
near the oscillation threshold can however be represented by assuming that around
the equilibrium position the movement of the mass in Fig. 3.22 is along the tangent
to its actual circular path. The small amplitude equation of motion for the swinging
door model can then be written in terms of the lip opening height as
d 2 h(t)
dt 2 +
ω l
Q l
dh(t)
dt
+ ω
2
l (h(t) − h eq ) = −
S io sin θ eq p(t)
2m
= −
p(t)
μ
,
(3.12)
where the effective mass per unit area of the vibrating lips is in this case μ =
2m/(S io sin θ eq ).
3.2.4 Inward-Striking and Outward-Striking Reeds
For a valve effect source to be capable of generating a continuous sound in a musical
wind instrument, it must be capable of supplying energy to sustain the standing
waves in the instrument’s air column. Before continuing with the development of
the elementary brass model, it is worth making a brief digression to examine the
different ways in which this energy supply requirement is met in reed woodwind
and lip-excited brass instruments.
The modelling of reed wind instruments was first put on a firm footing by the
great nineteenth century German physicist Helmholtz (1877). He understood that
the reed behaved as a pressure-controlled valve, modulating the flow of air from
the player’s mouth into the instrument. Helmholtz considered that there were two
distinct types of reed mechanism, depending on the geometry of the reed structure
relative to the direction of flow. He defined an ‘inward-striking’ (einschlagende)
reed as one in which the motion of the reed blade in the direction of flow resulted
in a decrease in the reed opening. An ‘outward-striking’ (aussschlagende) reed was
defined as one in which the motion of the reed blade in the direction of flow resulted