200
4 After the Lips: Acoustic Resonances and Radiation
brass instruments such as those shown in Fig. 4.21b. One reason for this is that the
shapes of these instruments cannot be described by simple mathematical formulae.
Another problem is that the transition from the tube interior to the region outside
the bell cannot be treated analytically. Even for the case of a cylinder, Eq. 4.88
was obtained by neglecting sound radiation and assuming that the open end was
a pressure node.
An even more fundamental problem is that Eq. 4.1 was derived on the assumption
that there are no energy losses as the sound wave travels down the tube. In reality
there are various fluid dynamical loss mechanisms which drain energy from the
sound wave during its passage (Pierce 1989; Keefe 1984). The most important are
the viscothermal losses caused by frictional processes in the thin boundary layer
near the internal wall of the tube. These processes are governed by the Navier-Stokes
Equation, for which no analytical solutions are available.
Many numerical methods exist for finding approximate solutions to equations
which cannot be solved analytically. Techniques which have been applied successfully to the study of wind instruments include finite element (Lefebre 2010),
finite difference (Noreland 2002) and lattice Boltzmann (Kühnelt 2007) methods.
Increasingly powerful computers have made it possible to carry out simulations in
which the Navier-Stokes equation is solved directly using a three-dimensional finite
difference method (Giordano 2017). Here we restrict our discussion to a simple
approach, known as the transfer matrix method, which allows the input impedance
of a brass instrument bore to be calculated by representing it as a concatenation of
segments of simple geometry.
4.7.2 Lossless Plane Wave TMM Calculations
In the simplest version of the transfer matrix method (TMM), the flaring tube
of a brass instrument is modelled as a sequence of short cylinders with different
diameters, as shown in Fig. 4.90. It is assumed that only plane waves propagate in
the cylinders. If, in addition, viscothermal losses are ignored, Eq. 4.5 can be used to
derive an analytical espression for the input impedance Z in of a cylindrical section
in terms of the output impedance Z out .
The total pressure at a distance x along the cylinder axis is the sum of the
pressures in the forward and backward travelling waves, as shown in Eq. 4.15:
p(x, t) = p + (x, t) + p − (x, t) = Ae
j (ωt−kx)
+ Be
j (ωt+kx) .
The acoustic particle velocity in the forward travelling wave was given in
Eq. 4.13 as
v + (x, t) =
A
ρc
e
j (ωt−kx)
=
1
ρc
p + (x, t).
4 After the Lips: Acoustic Resonances and Radiation
brass instruments such as those shown in Fig. 4.21b. One reason for this is that the
shapes of these instruments cannot be described by simple mathematical formulae.
Another problem is that the transition from the tube interior to the region outside
the bell cannot be treated analytically. Even for the case of a cylinder, Eq. 4.88
was obtained by neglecting sound radiation and assuming that the open end was
a pressure node.
An even more fundamental problem is that Eq. 4.1 was derived on the assumption
that there are no energy losses as the sound wave travels down the tube. In reality
there are various fluid dynamical loss mechanisms which drain energy from the
sound wave during its passage (Pierce 1989; Keefe 1984). The most important are
the viscothermal losses caused by frictional processes in the thin boundary layer
near the internal wall of the tube. These processes are governed by the Navier-Stokes
Equation, for which no analytical solutions are available.
Many numerical methods exist for finding approximate solutions to equations
which cannot be solved analytically. Techniques which have been applied successfully to the study of wind instruments include finite element (Lefebre 2010),
finite difference (Noreland 2002) and lattice Boltzmann (Kühnelt 2007) methods.
Increasingly powerful computers have made it possible to carry out simulations in
which the Navier-Stokes equation is solved directly using a three-dimensional finite
difference method (Giordano 2017). Here we restrict our discussion to a simple
approach, known as the transfer matrix method, which allows the input impedance
of a brass instrument bore to be calculated by representing it as a concatenation of
segments of simple geometry.
4.7.2 Lossless Plane Wave TMM Calculations
In the simplest version of the transfer matrix method (TMM), the flaring tube
of a brass instrument is modelled as a sequence of short cylinders with different
diameters, as shown in Fig. 4.90. It is assumed that only plane waves propagate in
the cylinders. If, in addition, viscothermal losses are ignored, Eq. 4.5 can be used to
derive an analytical espression for the input impedance Z in of a cylindrical section
in terms of the output impedance Z out .
The total pressure at a distance x along the cylinder axis is the sum of the
pressures in the forward and backward travelling waves, as shown in Eq. 4.15:
p(x, t) = p + (x, t) + p − (x, t) = Ae
j (ωt−kx)
+ Be
j (ωt+kx) .
The acoustic particle velocity in the forward travelling wave was given in
Eq. 4.13 as
v + (x, t) =
A
ρc
e
j (ωt−kx)
=
1
ρc
p + (x, t).