Introduction to Acoustics

604 Part E Music, Speech, Electroacoustics
Part E 15.3
tion. Narrow-bore instruments can have a large number
of resonant modes below the cut-off, while instruments
with a large output bell, like many brass instruments,
have far fewer.
For a narrow-bore cylindrical bore wind instrument
with end radius a ≈ 1 cm, the cut-off frequency (ka ≈ 1)
is ≈ 5.5 kHz. Below this frequency the instrument will
support a number of relatively weakly damped resonant
modes, which will radiate isotropically from the ends
of the instrument or from open holes cut in its sides.
In contrast, for brass instruments the detailed shape and
size ot the flared end-bell determines the cut-off frequency.
The large size of the bell leads to an increase
in intensity of the higher partials and hence brilliance
of tone-color, especially when the bell is pointed directly
towards the listener. For French horns, much of
Input impedance
0 1000
2000
Excitation frequency
Input impedance
0 1000
2000
Excitation frequency
ρ
c0/Ain
Fig. 15.68 Input impedance of a length of 1 cm diameter
trumpet tubing with and without a bell attached to the output
end (after Benade [15.133])
the higher-frequency sound is therefore projected backwards
relative to the player, unless there is a strongly
reflecting surface behind.
For ka ≪ 1, the open end of a musical instrument acts
as an isotropic monopole source with radiated power P
given by
P = U 2 rms Rrad
2 ρ
= ω
8πc (Sωξ)2 . (15.95)
For a given vibrational displacement, the radiated power
therefore increases with the fourth power of both frequency
and radius. This very strong dependence on size
explains why brass instruments tend to have rather large
bells and why high-fidelity (HI-FI) woofer speakers and
the horns of public address loudspeakers tend to be rather
large. Conversely, it explains why the sound of small
loudspeakers, such as those used in PC notebooks, fall
off very rapidly below a few hundred Hz.
Acoustic radiation will lower the height and increase
the width of resonances in a cylindrical tube.
The resulting Q-values can be determined from
stored energy
Q = ω
radiated energy
1
4
= ω
ρSLω2ξ 2
ω4 (ρ/8πc) S2 = 2πcL/ωS . (15.96)
ξ2 Narrow-bore instruments will therefore have larger
Q-values and narrower resonances than wide-bore instruments
such as brass instruments, where the flared
end-sections enhance the radiated energy at the expense
of increasing the net losses.
The increased damping introduced by radiation from
the end of an instrument is illustrated in Fig. 15.68,
which compares the resonances of a length of 1 cmdiameter
trumpet tubing, first with a normal open
end and then with a bell attached (Benade [15.132],
Fig. 20.4). Attaching a bell to such a tube dramatically
increases the radiated sound from the higher partials and
perceived intensity, but at the expense of a cut-off frequency
at around ≈1.5 kHz and a significant broadening
of the resonances at lower frequencies. Audio
demonstrates the sound of a mouthpiece-blown length
of hose pipe with and without a conical chemical filter
funnel attached to its end.
Viscous and Thermal Losses
In addition to radiation losses, there can be significant
losses from viscous damping and heat transfer to the
walls, as discussed in detail in Fletcher and Rossing
([15.5], Sect. 8.2). Although simple models for waves