Chapter 6
Shocks and Surprises: Refining the
Elementary Model
In Chap. 5, an elementary model of a brass instrument coupled with a player was
developed and analysed. This elementary model does not attempt to include all the
complexities of human brass performance, and the three equations on which the
model is based rely on several simplifying approximations. It is assumed that the
propagation of sound waves in the air column of the instrument can be described
using linear acoustics, that the resonances of the player’s windway can be neglected,
that the player’s lips can be modelled as a one degree of freedom oscillator, and that
the walls of the instrument are completely rigid. These rather drastic approximations
are in accord with the philosophy of modelling explained in Sect. 2.2.1, which
seeks as a first step to find the simplest possible physical model which captures
at least some of the behaviour of the system under study. We saw in Chap. 5
that the elementary model is indeed capable of reproducing and explaining many
of the important features of brass instrument performance. The next stage in the
modelling approach is to reconsider the assumptions on which the elementary model
is based, with the aim of developing a refined model which incorporates some of the
previously neglected phenomena. In this chapter each of the four assumptions listed
above is revisited and discussed.
6.1 Why Brass Instruments Sound Brassy
In Sect. 6.1 it is explained that the linear approximation to the theory of sound
propagation is no longer adequate when describing a brass instrument played at
a level above mezzoforte. The distortion of the pressure waveform due to nonlinear
propagation is shown to be the main cause of the ‘brassy’ sound characteristic of
loudly played trumpets and trombones. An important result is the possibility of
deducing a parameter (the brassiness potential parameter B) which can be used
© Springer Nature Switzerland AG 2021
M. Campbell et al., The Science of Brass Instruments, Modern Acoustics and Signal
Processing, https://doi.org/10.1007/978-3-030-55686-0_6
271
Shocks and Surprises: Refining the
Elementary Model
In Chap. 5, an elementary model of a brass instrument coupled with a player was
developed and analysed. This elementary model does not attempt to include all the
complexities of human brass performance, and the three equations on which the
model is based rely on several simplifying approximations. It is assumed that the
propagation of sound waves in the air column of the instrument can be described
using linear acoustics, that the resonances of the player’s windway can be neglected,
that the player’s lips can be modelled as a one degree of freedom oscillator, and that
the walls of the instrument are completely rigid. These rather drastic approximations
are in accord with the philosophy of modelling explained in Sect. 2.2.1, which
seeks as a first step to find the simplest possible physical model which captures
at least some of the behaviour of the system under study. We saw in Chap. 5
that the elementary model is indeed capable of reproducing and explaining many
of the important features of brass instrument performance. The next stage in the
modelling approach is to reconsider the assumptions on which the elementary model
is based, with the aim of developing a refined model which incorporates some of the
previously neglected phenomena. In this chapter each of the four assumptions listed
above is revisited and discussed.
6.1 Why Brass Instruments Sound Brassy
In Sect. 6.1 it is explained that the linear approximation to the theory of sound
propagation is no longer adequate when describing a brass instrument played at
a level above mezzoforte. The distortion of the pressure waveform due to nonlinear
propagation is shown to be the main cause of the ‘brassy’ sound characteristic of
loudly played trumpets and trombones. An important result is the possibility of
deducing a parameter (the brassiness potential parameter B) which can be used
© Springer Nature Switzerland AG 2021
M. Campbell et al., The Science of Brass Instruments, Modern Acoustics and Signal
Processing, https://doi.org/10.1007/978-3-030-55686-0_6
271