Introduction to Acoustics

626 Part E Music, Speech, Electroacoustics
Part E 15.3
a) b)
0.3
0.2
0.1
1 1
0
0 0.2 0.4 0.6 0.8 1
p
pmouth
0
t
0.3
0.2
0.1
1 4
4
0
0 0.2 0.4 0.6 0.8 1
p
pmouth
0
p
pmouth
Fig. 15.97a,b Large-amplitude Helmholtz pressure fluctuation
of (a) a cylinder and (b) a truncated cone with length
to apex of 1/4 of its length, illustrating the dependence of
fluctuation amplitudes as a function of mouthpiece pressure.
For the cylinder, the mouthpiece pressure is single
valued for a given flow rate, but for the truncated cone there
are two possible solutions referred to as the standard and
inverted Helmholtz solutions
end of the instrument, where it is reflected with change
of sign. On return to the reed it reverses the pressure
difference across the reed, which switches to the reduced
pressure state. The subsequent reflection of the
reduced pressure wave then switches the reed back
to its original high-pressure state and the process re-
Reed
p(t)
l p(t)
p(t)
Reed
p(t)
Winds Strings
L
2L
v(t)
L
t
t
0
l
v(t)
v(t)
v(t)
Fig. 15.98 Analogy between large-amplitude pressure
waves in the bores of wind instruments and the transverse
velocity of Helmholtz waves on a bowed stretched string,
where the reed position is equivalent to the bowing position
L
t
1
t
peats indefinitely, with a periodic time of 4L/c0, as
expected.
The dependence of the square-wave pressure fluctuations
on the applied pressure can be obtained by the
simple graphical construction illustrated in Fig. 15.97.
The locus of the static pressure required to excite
square-wave pressure fluctuations above and below the
mouth pressure is shown by the solid line drawn from
pmouth = 1/3 to1/2pmax, which bisects the high and
low pressures for a given flow rate. If losses are taken
into account, the horizontal lines are replaced by load
lines with a downward slope given by the real part of
the instrument’s input admittance (Fletcher and Rossing
[15.5], Fig. 15.9). At large amplitudes, the solutions
can then involve periods during which the reed is completely
closed. The transition from small-amplitude to
large-amplitude solutions is clearly of musical importance,
as it changes the sound of an instrument, and
remains an active area of research [15.154].
Analogy with Bowed String
In recent years, an interesting analogy has been noted
between the large-amplitude pressure fluctuations of
a vibrating air column in a cylindrical or truncated conical
tube and simple Helmholtz waves excited on a bowed
string (Dalmont and Kergomard [15.162]). For example,
the square-wave pressure fluctuations at the output
of the reed attached to a cylindrical tube are analogous
to the velocity of the bowed Helmholtz transverse waves
of a string bowed at its centre, illustrated schematically
in Fig. 15.98. Helmholtz waves could equally well be
excited on a bowed string by a transducer with a squarewave
velocity output placed halfway along the length
of the string, in just the same way that the reed with
a square-wave pressure output excites Helmholtz sound
pressure waves into a cylinder, which acts like half the
string length.
The analogy is particularly useful in discussing the
large-amplitude pressure fluctuations in conical-bore
instruments such as the oboe or saxophone. As described
earlier, the conical tube has the same set of
resonances as a cylindrical tube that is open at both
ends. Therefore, in addition to having the same set of
standing-wave sinusoidal solutions for the transverse oscillations
of a stretched string, a conical tube can also
support Helmholtz wave solutions. For the bowed string,
the closer one gets to either end of the string the larger
becomes the mark-to-space ratio between the regions of
high to low transverse velocity. The same is also true for
the switched fluctuations in pressure in a lossless conical
tube, shown schematically in Fig. 15.97. Hence, if one