80
3 Buzzing Lips: Sound Generation in Brass Instruments
y axis. The forces generated by the elasticity of the player’s embouchure are
represented by a single spring with stiffness k, which reacts to any displacement
of the mass from its equilibrium position y eq by applying a proportional restoring
force F R = −k(y − y eq ).
If the mass in Fig. 3.18 is pulled away from its equilibrium position and then
released, it will oscillate at its natural resonance frequency, which is
f r =
ω r
2π
=
1
2π
k
m
.
(3.2)
The amplitude of the oscillations will gradually diminish as the energy of motion
is reduced by viscothermal losses and internal friction in the spring. This process is
described as damping. In a similar way, the tensed lip of a brass player’s embouchure
would vibrate at its natural resonance frequency if plucked. The player is capable
of changing this resonance frequency over several octaves by adjustments of the
embouchure (see Sect.3.1.1). The oscillations of the plucked lip would die out more
quickly than those of a mass on a spring because the internal friction in the lip is
much greater than in a metal spring.
In the operation of the valve effect source, the most important feature of the lip
vibration is not the motion of one lip, but rather the modulation of the open area
S between the two lips. In Sect. 3.1.3 the equivalent rectangle was defined: at any
time t, this rectangle has width w(t) and height h(t), such that S(t) = w(t)h(t).
Making the additional simplifying assumptions that the width is constant and that
the motion of the lower lip is a mirror image of the motion of the upper lip, it is
possible to model the simultaneous motion of the two lips as one oscillator with a
single degree of freedom (along the y axis) (Cullen et al. 2000). The equation of
motion for this 1DOF oscillator can be written as
d 2 h
dt 2 +
ω l
q l
dh
dt
+ ω
2
l (h − h eq ) =
F
m
,
(3.3)
where ω l is the mechanical lip resonance frequency (assumed to be the same for
each lip), q l is the quality factor of the lip resonance (inversely proportional to the
strength of the damping) and h eq is the separation between the lips when they are
at rest. F is the component of the total external force acting in the +y direction on
the upper lip; a force of equal magnitude is assumed to act on the lower lip in the
−y direction. The effective mass m is equal to half the effective mass of each lip,
reflecting the fact that the lip displacement h in Eq. 3.3 is twice the displacement of
each lip.
The external force F acting on the brass player’s lips arises from the pressure
exerted on the lips by the surrounding air. When the player forms an embouchure
by pressing the lips against the mouthpiece rim, mouth and mouthpiece both initially
contain air at atmospheric pressure. Since the internal pressure of the lip tissue
is also equal to atmospheric pressure, there is no net pressure across any part
of the lip surface exposed to the air. To initiate the note, the player increases
3 Buzzing Lips: Sound Generation in Brass Instruments
y axis. The forces generated by the elasticity of the player’s embouchure are
represented by a single spring with stiffness k, which reacts to any displacement
of the mass from its equilibrium position y eq by applying a proportional restoring
force F R = −k(y − y eq ).
If the mass in Fig. 3.18 is pulled away from its equilibrium position and then
released, it will oscillate at its natural resonance frequency, which is
f r =
ω r
2π
=
1
2π
k
m
.
(3.2)
The amplitude of the oscillations will gradually diminish as the energy of motion
is reduced by viscothermal losses and internal friction in the spring. This process is
described as damping. In a similar way, the tensed lip of a brass player’s embouchure
would vibrate at its natural resonance frequency if plucked. The player is capable
of changing this resonance frequency over several octaves by adjustments of the
embouchure (see Sect.3.1.1). The oscillations of the plucked lip would die out more
quickly than those of a mass on a spring because the internal friction in the lip is
much greater than in a metal spring.
In the operation of the valve effect source, the most important feature of the lip
vibration is not the motion of one lip, but rather the modulation of the open area
S between the two lips. In Sect. 3.1.3 the equivalent rectangle was defined: at any
time t, this rectangle has width w(t) and height h(t), such that S(t) = w(t)h(t).
Making the additional simplifying assumptions that the width is constant and that
the motion of the lower lip is a mirror image of the motion of the upper lip, it is
possible to model the simultaneous motion of the two lips as one oscillator with a
single degree of freedom (along the y axis) (Cullen et al. 2000). The equation of
motion for this 1DOF oscillator can be written as
d 2 h
dt 2 +
ω l
q l
dh
dt
+ ω
2
l (h − h eq ) =
F
m
,
(3.3)
where ω l is the mechanical lip resonance frequency (assumed to be the same for
each lip), q l is the quality factor of the lip resonance (inversely proportional to the
strength of the damping) and h eq is the separation between the lips when they are
at rest. F is the component of the total external force acting in the +y direction on
the upper lip; a force of equal magnitude is assumed to act on the lower lip in the
−y direction. The effective mass m is equal to half the effective mass of each lip,
reflecting the fact that the lip displacement h in Eq. 3.3 is twice the displacement of
each lip.
The external force F acting on the brass player’s lips arises from the pressure
exerted on the lips by the surrounding air. When the player forms an embouchure
by pressing the lips against the mouthpiece rim, mouth and mouthpiece both initially
contain air at atmospheric pressure. Since the internal pressure of the lip tissue
is also equal to atmospheric pressure, there is no net pressure across any part
of the lip surface exposed to the air. To initiate the note, the player increases