120
4 After the Lips: Acoustic Resonances and Radiation
approaches −π/2 at high frequency because of the lumped acoustic behaviour of
the mouthpiece volume (see Sect. 4.3.5).
4.2 Measuring Input Impedance
The input impedance Z(ω) of a brass instrument is defined in Eq. 4.24 as the ratio
between the acoustic pressure p(ω) in the mouthpiece and the acoustic volume flow
u(ω) into the mouthpiece. Both p(ω) and u(ω) are assumed to be sinusoidal signals
with frequency f = ω/2π . The most direct way to measure the input impedance
would therefore appear to involve simultaneous measurements of p(ω) and u(ω)
for each frequency of interest. However, although the input pressure can be easily
recorded using an adapted mouthpiece with a small microphone in the cup wall
(see Sect. 2.1.2), the direct measurement of the acoustic flow velocity inside the
mouthpiece is much more difficult. Studies of trombone input impedance have been
carried out using a hot wire probe in the mouthpiece throat to measure the flow
speed (Pratt et al. 1977; Elliott and Bowsher 1982), but this method is technically
difficult and insensitive to the direction of flow.
The Euler equation (Eq. 4.11) shows a basic fluid dynamic relationship between
the time derivative of the velocity and the spatial gradient of the pressure. It is
therefore possible in principle to replace the measurement of flow velocity by
pressure measurements from two or more spatially separated microphones. Most
of the techniques currently employed to measure the input impedance of wind
instruments employ at least two microphones. Depending on the method, calibration
is necessary using a number of acoustical systems with known impedance. The
calibrating systems used in practice range from cylindrical cavities a few millimetres
deep to an impressive 95 m long plastic tube used as an anechoic termination by Joe
Wolfe’s research group at UNSW, Sydney. It is even possible to carry out input
impedance measurements using a single microphone by an appropriate choice of
calibrating systems (Sharp et al. 2011).
Valuable and comprehensive reviews of input impedance measurement techniques and calibration methods have been published by Benade and Ibisi (1987),
Dalmont (2001a,b) and Dickens et al. (2007). Here we describe briefly three
different approaches which have proved useful in the study of brass instrument
acoustics.
4.2.1 Capillary-Based Methods
Many of the devices in current use for measuring input impedance rely on coupling a
cavity in front of a loudspeaker to the mouthpiece of the instrument through a narrow
diameter tube known as a capillary. The earliest mention of this idea in the scientific
literature is an article written by J. C. Webster (1947), then working in the research
4 After the Lips: Acoustic Resonances and Radiation
approaches −π/2 at high frequency because of the lumped acoustic behaviour of
the mouthpiece volume (see Sect. 4.3.5).
4.2 Measuring Input Impedance
The input impedance Z(ω) of a brass instrument is defined in Eq. 4.24 as the ratio
between the acoustic pressure p(ω) in the mouthpiece and the acoustic volume flow
u(ω) into the mouthpiece. Both p(ω) and u(ω) are assumed to be sinusoidal signals
with frequency f = ω/2π . The most direct way to measure the input impedance
would therefore appear to involve simultaneous measurements of p(ω) and u(ω)
for each frequency of interest. However, although the input pressure can be easily
recorded using an adapted mouthpiece with a small microphone in the cup wall
(see Sect. 2.1.2), the direct measurement of the acoustic flow velocity inside the
mouthpiece is much more difficult. Studies of trombone input impedance have been
carried out using a hot wire probe in the mouthpiece throat to measure the flow
speed (Pratt et al. 1977; Elliott and Bowsher 1982), but this method is technically
difficult and insensitive to the direction of flow.
The Euler equation (Eq. 4.11) shows a basic fluid dynamic relationship between
the time derivative of the velocity and the spatial gradient of the pressure. It is
therefore possible in principle to replace the measurement of flow velocity by
pressure measurements from two or more spatially separated microphones. Most
of the techniques currently employed to measure the input impedance of wind
instruments employ at least two microphones. Depending on the method, calibration
is necessary using a number of acoustical systems with known impedance. The
calibrating systems used in practice range from cylindrical cavities a few millimetres
deep to an impressive 95 m long plastic tube used as an anechoic termination by Joe
Wolfe’s research group at UNSW, Sydney. It is even possible to carry out input
impedance measurements using a single microphone by an appropriate choice of
calibrating systems (Sharp et al. 2011).
Valuable and comprehensive reviews of input impedance measurement techniques and calibration methods have been published by Benade and Ibisi (1987),
Dalmont (2001a,b) and Dickens et al. (2007). Here we describe briefly three
different approaches which have proved useful in the study of brass instrument
acoustics.
4.2.1 Capillary-Based Methods
Many of the devices in current use for measuring input impedance rely on coupling a
cavity in front of a loudspeaker to the mouthpiece of the instrument through a narrow
diameter tube known as a capillary. The earliest mention of this idea in the scientific
literature is an article written by J. C. Webster (1947), then working in the research