Introduction to Acoustics

propagating from a central point. Such waves satisfy
the wave equation, which may be written in spherical
coordinates as
∇ 2 (rp) = 1
c2 ∂
0
2 (rp)
∂t2 . (15.101)
We therefore have standing-wave solutions for rp that
are very similar to those of a cylindrical tube, with
sin kr
p = C e
r
iωt . (15.102)
Note that the pressure remains finite at the apex of the
cone, r = 0, where sin(kr)/r → k. For a conical tube
with a pressure node p = 0 at the open end, we therefore
have standing wave modes with kn L = nπ and
fn = nc0/2L,wherenis any integer. The frequencies of
the excited modes are therefore identical to the modes
of a cylindrical tube of the same length that is open at
both ends. The lowest note at f1 = c0/2L for a conical
tube instrument with a reed at one end (e.g. the oboe and
saxophone) is therefore an octave above a reed instrument
of the same length with a cylindrical bore (e.g. the
clarinet) with a fundamental frequency of c0/4L.
The flow velocity U is determined by the acceleration
of the air resulting from the spatial variation of the
pressure, so that
ρ ∂U
∂t = ∂(r2 p)
∂r = C (sin kr + kr cos kr) eiωt .
(15.103)
Figure 15.71 illustrates the pressure and flow velocity
for the n = 5 mode of a conical tube. Unlike the modes
of cylindrical tube, the nodes of U no longer coincide
with the peaks in p, which is especially apparent for
the first few cycles along the tube. Furthermore, the
amplitude fluctuations increase with distance r from the
apex (∼ r), whilst the fluctuations in pressure decrease
≈ 1/r. A conical section therefore acts as an acoustic
transformer helping to match the high impedance at the
input mouthpiece end to the low impedance at the output
radiating end.
Attaching a mouthpiece or reed to the end of a conical
tube requires truncation of the cone, which will
clearly perturb the frequencies of the harmonic modes.
However, using a mouthpiece or reed unit having the
same internal volume as the volume of the truncated
section removed will leave the frequencies of the lowest
modes unchanged. Only when the acoustic wavelength
becomes comparable with the length of truncated section
will the perturbation be large.
20
15
10
5
0
–5
–10
–15
Pressure p
Musical Acoustics 15.3 Wind Instruments 607
Flow U
–20
0 0.2 0.4 0.6 0.8 1
r/L
Fig. 15.71 Pressure and flow velocity of the n = 5 mode
along the length of a conical tube
Fletcher and Rossing ([15.5], Sect. 8.7) consider the
physics of the truncated conical tube and give the input
impedance derived by Olson [15.139]
� �
Zin = ρc0
S1
� �
sin(kL−θ2) ρc0
iZL sin + sin kL
θ2 S2
ZL sin(kL+θ1−θ2)
� � ,
ρc0 sin(kl+θ1)
sin θ1 sin + j θ2 S2 sin θ1
(15.104)
where x1 and x2 are the distances of the two ends from
the apex of the truncated conical section. The length L =
x2 − x1, the end areas are S1 and S2, with θ1 = tan −1 kx1
and θ2 = tan −1 kx2.
a) b)
Resonant frequencies
4
3
2
1
0.01 0.1 1.0
Input/output diameters
0.01 0.1 1.0
Input/output diameters
Fig. 15.72a,b The first four resonant frequencies of truncated
cones with (a) both ends open, and (b) the input end
closed, as a function of the ratio of their input to output
diameters (after Ayers et al. [15.138])
Part E 15.3