4.3 Bore Profiles of Brass Instruments
133
Fig. 4.27 Spherical wavefronts in a conical tube
Fig. 4.28 Input impedance Z calculated for a conical tube with length 2834 mm, entrance radius
6.9 mm and output radius 106.5 mm
Examples of spherical wavefronts in a conical tube are shown in Fig. 4.27 as thin
blue lines. In each case the wavefront is a section of the surface of a sphere centred
at the vertex of the cone. In a playable instrument, the cone must be truncated by the
removal of a section like that shown dotted in Fig. 4.27 to allow for the insertion of
a mouthpiece.
Figure 4.22 showed an example of a truncated cone. In this hypothetical
instrument the length, input diameter and final bell diameter were chosen to be the
same as those of the Conn 8H trombone, but the tube is a cone of constant taper.
This bore profile is in fact very similar to that of the B ophicleide illustrated in
Fig. 7.28b. The calculated input impedance is shown in Fig. 4.28. Comparison with
the input impedance curve for a cylinder of the same length in Fig. 4.26a shows that
while the first impedance peak is the largest in the cylindrical tube, the first peak in
the cone’s impedance curve has remarkably low amplitude, and the third peak is the
largest.
Another distinction between the properties of the impedance peaks in cones
and cylinders, which is even more important from the musician’s perspective, is
that the frequencies of the peaks in the cone impedance curve are very close to a
complete harmonic series with fundamental frequency 54.7 Hz. This is almost twice
133
Fig. 4.27 Spherical wavefronts in a conical tube
Fig. 4.28 Input impedance Z calculated for a conical tube with length 2834 mm, entrance radius
6.9 mm and output radius 106.5 mm
Examples of spherical wavefronts in a conical tube are shown in Fig. 4.27 as thin
blue lines. In each case the wavefront is a section of the surface of a sphere centred
at the vertex of the cone. In a playable instrument, the cone must be truncated by the
removal of a section like that shown dotted in Fig. 4.27 to allow for the insertion of
a mouthpiece.
Figure 4.22 showed an example of a truncated cone. In this hypothetical
instrument the length, input diameter and final bell diameter were chosen to be the
same as those of the Conn 8H trombone, but the tube is a cone of constant taper.
This bore profile is in fact very similar to that of the B ophicleide illustrated in
Fig. 7.28b. The calculated input impedance is shown in Fig. 4.28. Comparison with
the input impedance curve for a cylinder of the same length in Fig. 4.26a shows that
while the first impedance peak is the largest in the cylindrical tube, the first peak in
the cone’s impedance curve has remarkably low amplitude, and the third peak is the
largest.
Another distinction between the properties of the impedance peaks in cones
and cylinders, which is even more important from the musician’s perspective, is
that the frequencies of the peaks in the cone impedance curve are very close to a
complete harmonic series with fundamental frequency 54.7 Hz. This is almost twice