4.7 Going Further: Calculating Input Impedance
209
and on the model of Hélie and Rodet has shown that the pulsating sphere model,
together with a spherical wave approach to propagation in the bell section, gives
much closer agreement with experiment than a plane wave model (Eveno et al.
2012).
4.7.6 Multimodal Calculations
The acoustical disturbances which travel through a brass instrument tube and radiate
from the bell cannot be exactly represented as either purely plane or purely spherical
waves (Benade and Jansson 1974). To arrive at a more accurate representation of
the bulging wavefronts, it is necessary to find more general solutions of the threedimensional wave equation
p =
1
c 2
∂ 2 p
∂t 2 .
(4.1)
Since most brass instrument bores are to a good approximation cylindrically
symmetric, it is convenient to describe the pressure and velocity fields in the tube
using a cylindrical polar coordinate system in which x represents the distance along
the tube axis, r the displacement perpendicular to the axis and θ the angle of rotation
about the axis (Félix et al. 2012). In this system Eq. 4.1 can be written
⊥ p +
∂ 2 p
∂ 2 x
=
1
c 2
∂ 2 p
∂t 2 ,
(4.123)
where
⊥ =
1
r
∂
∂r
r
∂
∂r
+
1
r 2
∂ 2 p
∂θ 2
(4.124)
is the transverse Laplacian (Pagneux et al. 1996).
For a hard walled uniform cylindrical tube of radius a without losses, solutions
of Eq. 4.123 must satisfy the boundary condition that the acoustic particle velocity
(and therefore the radial pressure gradient) must be zero at r = a. An infinite set of
such solutions exists, each representing a mode with its own characteristic patterns
of pressure and flow velocity. These modes form a complete orthogonal set, and
any type of acoustic disturbance can in principle be reproduced by adding together
members of the set. This is the basis of the multimodal method of impedance
calculation, and the process of identifying the neccessary modes is described as
modal decomposition.
Multimodal treatments of brass instruments have been described by several
authors (Pagneux et al. 1996; Amir et al. 1997; Kemp 2002; Braden 2006; Félix
et al. 2012). The pressure in a forward travelling wave is derived as a sum of the
209
and on the model of Hélie and Rodet has shown that the pulsating sphere model,
together with a spherical wave approach to propagation in the bell section, gives
much closer agreement with experiment than a plane wave model (Eveno et al.
2012).
4.7.6 Multimodal Calculations
The acoustical disturbances which travel through a brass instrument tube and radiate
from the bell cannot be exactly represented as either purely plane or purely spherical
waves (Benade and Jansson 1974). To arrive at a more accurate representation of
the bulging wavefronts, it is necessary to find more general solutions of the threedimensional wave equation
p =
1
c 2
∂ 2 p
∂t 2 .
(4.1)
Since most brass instrument bores are to a good approximation cylindrically
symmetric, it is convenient to describe the pressure and velocity fields in the tube
using a cylindrical polar coordinate system in which x represents the distance along
the tube axis, r the displacement perpendicular to the axis and θ the angle of rotation
about the axis (Félix et al. 2012). In this system Eq. 4.1 can be written
⊥ p +
∂ 2 p
∂ 2 x
=
1
c 2
∂ 2 p
∂t 2 ,
(4.123)
where
⊥ =
1
r
∂
∂r
r
∂
∂r
+
1
r 2
∂ 2 p
∂θ 2
(4.124)
is the transverse Laplacian (Pagneux et al. 1996).
For a hard walled uniform cylindrical tube of radius a without losses, solutions
of Eq. 4.123 must satisfy the boundary condition that the acoustic particle velocity
(and therefore the radial pressure gradient) must be zero at r = a. An infinite set of
such solutions exists, each representing a mode with its own characteristic patterns
of pressure and flow velocity. These modes form a complete orthogonal set, and
any type of acoustic disturbance can in principle be reproduced by adding together
members of the set. This is the basis of the multimodal method of impedance
calculation, and the process of identifying the neccessary modes is described as
modal decomposition.
Multimodal treatments of brass instruments have been described by several
authors (Pagneux et al. 1996; Amir et al. 1997; Kemp 2002; Braden 2006; Félix
et al. 2012). The pressure in a forward travelling wave is derived as a sum of the