4.3 Bore Profiles of Brass Instruments
151
where B, x 0 and α are parameters which can be adjusted to provide an approximate
fit to a measured bell profile. The exponent α determines how rapidly the bell flares:
values of α between 0.5 and 0.65 have been found to give a similar rate of flaring
to that found in the expanding sections of modern trumpets and trombones (Benade
1976; Chaigne and Kergomard 2016). Once α has been fixed, the coefficients B and
x 0 determine overall shape and scale of the bore. With the input end of the horn
at x = 0, there is a singularity at x = x 0 where r → ∞; this is typically a few
centimetres beyond the bell exit plane.
It should be emphasised that the Bessel horn is a mathematical abstraction rather
than a practicable musical instrument. It would certainly not be possible to manufacture a playable Bessel trombone, since a slide section could not be incorporated
into the continuously expanding bore. However its theoretical simplicity makes it
a convenient vehicle for deriving and displaying some of the general principles of
sound propagation in flaring tubes. In Sect. 4.3.9 these principles will be applied to
a real brass instrument: the classic Conn 8H trombone.
The acoustical behaviour of the Bessel horn was explored in detail in a seminal
paper by Arthur Benade and Erik Jansson (1974). Following their example, we
consider a Bessel horn resembling the flaring section of a trombone. Unscrewing
the slide section of the Conn 8H instrument leaves a 1.13 m tube with an input
radius of 8 × 10 −3 m, flaring out to the bell end with radius 0.108 m. We choose an
exponent α = 0.65, similar to that adopted by Benade and Jansson. With α fixed,
substituting the values of r(x) at the entrance and the bell plane yields the remaining
two parameters in Eq. 4.66: B = 8.77 × 10 −3 m 1.65 and x 0 = 1.15 m.
The resulting Bessel horn profile is shown as the solid curve in Fig. 4.43a.
Comparison with the measured profile of the Conn 8H bell section, plotted as a
dashed line in Fig. 4.43a, shows that the Bessel horn profile reproduces the overall
shape of the bell well, but is not an exact fit. Bessel horn profiles have proved useful
in modelling brass instrument bells, but it is usually necessary to divide the bell into
several different sections, with a different set of parameters for each section. It is
nevertheless enlightening to consider the nature of the standing wave patterns and
mode frequencies in the idealised single Bessel horn shown in Fig. 4.43a.
Figure 4.43b shows the horn function U for the Bessel horn approximation to the
trombone bell, calculated using Eq. 4.60 with the assumption that the wavefronts in
the horn are spherical. The horn function is small in the first half of the horn, in
which the flare rate is very low. In the more strongly flaring part near the bell exit,
U increases rapidly with x, reaching a peak just under 5 cm inside the bell. It then
drops steeply, approaching zero around 2 cm beyond the bell exit.
The calculated input impedance for the Bessel horn is shown by the solid line
in Fig. 4.44. Each peak corresponds to a resonant mode of the contained air column
with a pressure antinode at the entrance. The values of k 2
0 for the first four modes are
marked by horizontal lines in Fig. 4.43b, and the pressure standing wave patterns for
these modes are are illustrated in Fig. 4.43c. The first mode has f (1) = 125.2 Hz;
the corresponding value of k 2
0 (1) = 2.28 m −2 is marked by the thin solid line in
Fig. 4.43b. This line intersects the horn function curve at x = 0.698; at that point the
151
where B, x 0 and α are parameters which can be adjusted to provide an approximate
fit to a measured bell profile. The exponent α determines how rapidly the bell flares:
values of α between 0.5 and 0.65 have been found to give a similar rate of flaring
to that found in the expanding sections of modern trumpets and trombones (Benade
1976; Chaigne and Kergomard 2016). Once α has been fixed, the coefficients B and
x 0 determine overall shape and scale of the bore. With the input end of the horn
at x = 0, there is a singularity at x = x 0 where r → ∞; this is typically a few
centimetres beyond the bell exit plane.
It should be emphasised that the Bessel horn is a mathematical abstraction rather
than a practicable musical instrument. It would certainly not be possible to manufacture a playable Bessel trombone, since a slide section could not be incorporated
into the continuously expanding bore. However its theoretical simplicity makes it
a convenient vehicle for deriving and displaying some of the general principles of
sound propagation in flaring tubes. In Sect. 4.3.9 these principles will be applied to
a real brass instrument: the classic Conn 8H trombone.
The acoustical behaviour of the Bessel horn was explored in detail in a seminal
paper by Arthur Benade and Erik Jansson (1974). Following their example, we
consider a Bessel horn resembling the flaring section of a trombone. Unscrewing
the slide section of the Conn 8H instrument leaves a 1.13 m tube with an input
radius of 8 × 10 −3 m, flaring out to the bell end with radius 0.108 m. We choose an
exponent α = 0.65, similar to that adopted by Benade and Jansson. With α fixed,
substituting the values of r(x) at the entrance and the bell plane yields the remaining
two parameters in Eq. 4.66: B = 8.77 × 10 −3 m 1.65 and x 0 = 1.15 m.
The resulting Bessel horn profile is shown as the solid curve in Fig. 4.43a.
Comparison with the measured profile of the Conn 8H bell section, plotted as a
dashed line in Fig. 4.43a, shows that the Bessel horn profile reproduces the overall
shape of the bell well, but is not an exact fit. Bessel horn profiles have proved useful
in modelling brass instrument bells, but it is usually necessary to divide the bell into
several different sections, with a different set of parameters for each section. It is
nevertheless enlightening to consider the nature of the standing wave patterns and
mode frequencies in the idealised single Bessel horn shown in Fig. 4.43a.
Figure 4.43b shows the horn function U for the Bessel horn approximation to the
trombone bell, calculated using Eq. 4.60 with the assumption that the wavefronts in
the horn are spherical. The horn function is small in the first half of the horn, in
which the flare rate is very low. In the more strongly flaring part near the bell exit,
U increases rapidly with x, reaching a peak just under 5 cm inside the bell. It then
drops steeply, approaching zero around 2 cm beyond the bell exit.
The calculated input impedance for the Bessel horn is shown by the solid line
in Fig. 4.44. Each peak corresponds to a resonant mode of the contained air column
with a pressure antinode at the entrance. The values of k 2
0 for the first four modes are
marked by horizontal lines in Fig. 4.43b, and the pressure standing wave patterns for
these modes are are illustrated in Fig. 4.43c. The first mode has f (1) = 125.2 Hz;
the corresponding value of k 2
0 (1) = 2.28 m −2 is marked by the thin solid line in
Fig. 4.43b. This line intersects the horn function curve at x = 0.698; at that point the