4.7 Going Further: Calculating Input Impedance
203
simplicity, we will assume here that each has a length L. The entrance to the first
section is at x = 0, and the exit plane is at x N = NL, as shown in Fig. 4.90.
The radiation impedance Z(x N ) = Z rad is assumed known. For a unit volume flow
u(x N ) = 1 at the exit, p(x N ) = Z(x N ), so the pressure and volume flow at the
output are represented by the vector [Z rad 1] T .
At the input to the Nth section,
P (x N −1 )
u(x N −1 )
= T N
Z rad
1
.
(4.99)
This then becomes the output vector for the preceding section, whose input is
found from
P (x N −2 )
u(x N −2 )
= T N −1 T N
Z rad
1
.
(4.100)
This process continues until the pressure and volume velocity at the entrance of the
instrument are found:
p(0)
u(0))
= T 1 T 2 . . . T N −1 T N
Z rad
1
.
(4.101)
The input impedance of the instrument is then
Z(0) = p(0)/u(0).
(4.102)
4.7.3 Including Losses in TMM Calculations
The model of a wind instrument presented in Sect. 4.7.2 is described as lossless
because it is assumed that waves propagate in the tube without viscothermal losses.
To obtain agreement with experimentally measured input impedance curves, it is
necessary to include these losses in the model.
The two principal energy loss processes take place near the tube wall, in a
boundary layer which for musical instruments is typically around 0.1 mm thick
(Benade 1968). The air is assumed at rest next to the wall, but moving with the
speed of the bulk flow just outside the boundary layer. Frictional shearing forces
exist within the boundary layer because of the viscosity of the air. The rise and fall
of temperature due to the passage of a sound wave also results in flow of heat to
and from the wall. These two processes depend on the frequency of the wave and
also on the ratios of the radius of the tube to the viscous and thermal boundary layer
thicknesses, which are slightly different (Zwicker and Kosten 1949; Bruneau 2006;
Pierce 1989).
203
simplicity, we will assume here that each has a length L. The entrance to the first
section is at x = 0, and the exit plane is at x N = NL, as shown in Fig. 4.90.
The radiation impedance Z(x N ) = Z rad is assumed known. For a unit volume flow
u(x N ) = 1 at the exit, p(x N ) = Z(x N ), so the pressure and volume flow at the
output are represented by the vector [Z rad 1] T .
At the input to the Nth section,
P (x N −1 )
u(x N −1 )
= T N
Z rad
1
.
(4.99)
This then becomes the output vector for the preceding section, whose input is
found from
P (x N −2 )
u(x N −2 )
= T N −1 T N
Z rad
1
.
(4.100)
This process continues until the pressure and volume velocity at the entrance of the
instrument are found:
p(0)
u(0))
= T 1 T 2 . . . T N −1 T N
Z rad
1
.
(4.101)
The input impedance of the instrument is then
Z(0) = p(0)/u(0).
(4.102)
4.7.3 Including Losses in TMM Calculations
The model of a wind instrument presented in Sect. 4.7.2 is described as lossless
because it is assumed that waves propagate in the tube without viscothermal losses.
To obtain agreement with experimentally measured input impedance curves, it is
necessary to include these losses in the model.
The two principal energy loss processes take place near the tube wall, in a
boundary layer which for musical instruments is typically around 0.1 mm thick
(Benade 1968). The air is assumed at rest next to the wall, but moving with the
speed of the bulk flow just outside the boundary layer. Frictional shearing forces
exist within the boundary layer because of the viscosity of the air. The rise and fall
of temperature due to the passage of a sound wave also results in flow of heat to
and from the wall. These two processes depend on the frequency of the wave and
also on the ratios of the radius of the tube to the viscous and thermal boundary layer
thicknesses, which are slightly different (Zwicker and Kosten 1949; Bruneau 2006;
Pierce 1989).