116
4 After the Lips: Acoustic Resonances and Radiation
The input impulse response of a brass instrument can last for a significant
fraction of a second because of the successive reflections at the closed input.
In effect it consists of multiple acoustic scans of the instrument bore. A much
shorter signal which contains the same information about the instrument’s acoustical
response is obtained if the rigid plate sealing the input is replaced by a termination
which completely absorbs all the sound falling on it. There are then no secondary
reflections from the input plane, although multiple reflections may still exist within
the bore of the instrument. The mouthpiece pressure signal r(t) obtained in this way
is called the reflection function. Time domain models of wind instruments can be
made much more computationally efficient by using the reflection function rather
than the input impulse response to represent the linear acoustic behaviour of the
instrument (Schumacher 1981).
It is not experimentally feasible to provide a completely sound absorbing termination at the entrance to a brass instrument mouthpiece, although a long cylindrical
tube with the same diameter as the mouthpiece rim is an approximate solution
(see Sect. 4.2.4). The reflection function can be calculated from a knowledge of
the bore profile (Sharp 1996). It can also be derived from the frequency domain
measurements and calculations discussed in Sects. 4.2 and 4.7.
4.1.6 Input Impedance
While the reflection function is a useful representation of the acoustical behaviour of
an instrument for time domain calculations, the frequency domain input impedance
Z(ω) displays essentially the same information in a way which is much more
intuitively related to the musical properties of the instrument. The input impedance
can be defined directly by considering an experiment in which the mouthpiece
excitation signal is not a pressure impulse but a continuous sinusoidally varying
pressure
p(ω) = p a (ω)e
jωt
(4.22)
generated by a sinusoidal volume air flow
u(ω) = u a (ω)e
jωt−θ .
(4.23)
The amplitude of the mouthpiece pressure signal is p a (ω) Pa, and the amplitude
of the volume air flow is u a (ω) m 3 s −1 . The phase difference between the volume
flow and the resulting pressure is −θ radians. The input impedance Z(ω) is then
simply defined as the ratio of pressure to volume velocity, both quantities being
evaluated at the entrance plane of the tube:
Z(ω) =
p(ω)
u(ω)
=
p a (ω)
u a (ω)
e
jθ .
(4.24)
4 After the Lips: Acoustic Resonances and Radiation
The input impulse response of a brass instrument can last for a significant
fraction of a second because of the successive reflections at the closed input.
In effect it consists of multiple acoustic scans of the instrument bore. A much
shorter signal which contains the same information about the instrument’s acoustical
response is obtained if the rigid plate sealing the input is replaced by a termination
which completely absorbs all the sound falling on it. There are then no secondary
reflections from the input plane, although multiple reflections may still exist within
the bore of the instrument. The mouthpiece pressure signal r(t) obtained in this way
is called the reflection function. Time domain models of wind instruments can be
made much more computationally efficient by using the reflection function rather
than the input impulse response to represent the linear acoustic behaviour of the
instrument (Schumacher 1981).
It is not experimentally feasible to provide a completely sound absorbing termination at the entrance to a brass instrument mouthpiece, although a long cylindrical
tube with the same diameter as the mouthpiece rim is an approximate solution
(see Sect. 4.2.4). The reflection function can be calculated from a knowledge of
the bore profile (Sharp 1996). It can also be derived from the frequency domain
measurements and calculations discussed in Sects. 4.2 and 4.7.
4.1.6 Input Impedance
While the reflection function is a useful representation of the acoustical behaviour of
an instrument for time domain calculations, the frequency domain input impedance
Z(ω) displays essentially the same information in a way which is much more
intuitively related to the musical properties of the instrument. The input impedance
can be defined directly by considering an experiment in which the mouthpiece
excitation signal is not a pressure impulse but a continuous sinusoidally varying
pressure
p(ω) = p a (ω)e
jωt
(4.22)
generated by a sinusoidal volume air flow
u(ω) = u a (ω)e
jωt−θ .
(4.23)
The amplitude of the mouthpiece pressure signal is p a (ω) Pa, and the amplitude
of the volume air flow is u a (ω) m 3 s −1 . The phase difference between the volume
flow and the resulting pressure is −θ radians. The input impedance Z(ω) is then
simply defined as the ratio of pressure to volume velocity, both quantities being
evaluated at the entrance plane of the tube:
Z(ω) =
p(ω)
u(ω)
=
p a (ω)
u a (ω)
e
jθ .
(4.24)