212
4 After the Lips: Acoustic Resonances and Radiation
the waves a few diameters on either side of the transition may be close to planar, the
representation of the acoustic field in the immediate vicinity of an abrupt change in
radius will require a large number of (probably evanescent) higher modes.
In the multimodal approach, the pressures and volume velocities p n and u n are
the elements of two column vectors P and U, respectively. The impedance matrix Z
is defined by
P = ZU.
(4.131)
The diagonal elements Z ii relate pressure and volume flow in the same mode and are
called direct impedances. The off-diagonal elements Z ij , i = j , relate the pressure
in one mode to volume flow in another mode and are called coupled impedances.
As in the TMM methods previously discussed, the calculation begins with
a known radiation impedance, in this case represented by a matrix Z rad . The
multimodal radiation impedance derived by Zorumski (1973) for a cylindrical
duct surrounded by an infinite baffle has been shown to provide an acceptable
approximation to Z rad for the case of a brass instrument bell (Amir et al. 1997;
Kemp 2002; Braden 2006), although it does not take into account the radiation field
in the hemisphere behind the bell. Propagating the impedance from the bell to the
input through a set of matrix multiplications yields the input impedance Z in . The
matrix equations describing propagation along a cylindrical element and across a
discontinuity between two elements are presented in Pagneux et al. (1996).
For numerical treatment it is necessary to truncate the set of modes at a finite
number N = n max + 1. It is not possible to exclude all evanescent modes, since
the reflection of a propagating mode at a discontinuity is influenced by coupling to
evanescent modes. The number of modes required depends on the shape and scale
of the duct and in practice is found by carrying out successive calculations with
increasing N until the results converge. Kemp (2002) showed that for a trumpet
bell, acceptable convergence was achieved with seven modes. Since the process
involves repeated multiplication of N × N matrices, the computational load is
strongly dependent on the number of modes, and the time taken for a multimodal
TMM input impedance calculation is typically several orders of magnitude greater
than for the equivalent plane or spherical wave calculation.
4.7.7 Bends in Brass Instruments
For practical reasons almost all brass instruments include sections in which the tube
axis is curved or bent rather than straight (see Sect. 1.2.11). Many of the instruments
illustrated in Chap. 7 have tuning slides with tight 180 ◦ bends of the type shown
in Fig. 4.93a, while sound waves are obliged to undergo several rapid changes of
direction when travelling through valves. Alpine pastures offer ample space for the
majestic length of the alphorn, but the more crowded environments of the orchestra
pit and bandstand have led to the evolution of the folded and curved forms of the
4 After the Lips: Acoustic Resonances and Radiation
the waves a few diameters on either side of the transition may be close to planar, the
representation of the acoustic field in the immediate vicinity of an abrupt change in
radius will require a large number of (probably evanescent) higher modes.
In the multimodal approach, the pressures and volume velocities p n and u n are
the elements of two column vectors P and U, respectively. The impedance matrix Z
is defined by
P = ZU.
(4.131)
The diagonal elements Z ii relate pressure and volume flow in the same mode and are
called direct impedances. The off-diagonal elements Z ij , i = j , relate the pressure
in one mode to volume flow in another mode and are called coupled impedances.
As in the TMM methods previously discussed, the calculation begins with
a known radiation impedance, in this case represented by a matrix Z rad . The
multimodal radiation impedance derived by Zorumski (1973) for a cylindrical
duct surrounded by an infinite baffle has been shown to provide an acceptable
approximation to Z rad for the case of a brass instrument bell (Amir et al. 1997;
Kemp 2002; Braden 2006), although it does not take into account the radiation field
in the hemisphere behind the bell. Propagating the impedance from the bell to the
input through a set of matrix multiplications yields the input impedance Z in . The
matrix equations describing propagation along a cylindrical element and across a
discontinuity between two elements are presented in Pagneux et al. (1996).
For numerical treatment it is necessary to truncate the set of modes at a finite
number N = n max + 1. It is not possible to exclude all evanescent modes, since
the reflection of a propagating mode at a discontinuity is influenced by coupling to
evanescent modes. The number of modes required depends on the shape and scale
of the duct and in practice is found by carrying out successive calculations with
increasing N until the results converge. Kemp (2002) showed that for a trumpet
bell, acceptable convergence was achieved with seven modes. Since the process
involves repeated multiplication of N × N matrices, the computational load is
strongly dependent on the number of modes, and the time taken for a multimodal
TMM input impedance calculation is typically several orders of magnitude greater
than for the equivalent plane or spherical wave calculation.
4.7.7 Bends in Brass Instruments
For practical reasons almost all brass instruments include sections in which the tube
axis is curved or bent rather than straight (see Sect. 1.2.11). Many of the instruments
illustrated in Chap. 7 have tuning slides with tight 180 ◦ bends of the type shown
in Fig. 4.93a, while sound waves are obliged to undergo several rapid changes of
direction when travelling through valves. Alpine pastures offer ample space for the
majestic length of the alphorn, but the more crowded environments of the orchestra
pit and bandstand have led to the evolution of the folded and curved forms of the