Introduction to Acoustics

For travelling waves,ei(ωt±kx) , this results in a ratio between
the pressure and flow rate, defined as the tube
impedance
Z = p
U =±ρc0 , (15.94)
S
where the plus and minus signs refer to waves travelling
in the positive and negative x-directions, respectively.
There is a very close analogy with an electrical transmission
line, with pressure and flow rate the analogue
of voltage and current, as discussed later. Because the
impedance is inversely proportional to area, it can be
appreciably higher at the input end of a brass or wind
instrument than at its flared output end. The flared bore
of a brass instrument or the horn on an old wind-up
gramophone can therefore be considered as an acoustic
transformer, which improves the match between the high
impedance of the vibrating source of sound to the much
lower impedance presented by air at the end of the instrument.
There is clearly an optimum matching, which
enhances the radialed sound without serious degredation
of the excited resonant modes.
Acoustic Radiation
In elementary textbook treatments, the pressure at the
endofanopenpipeisassumedtobezeroandtheflow
rate a maximum, so that Zclosed = p/U = 0. However,
in practice, the oscillatory motion of the air extends
1
0.1
0.01
0.001
ρc0
π a 2
X~ω
R~ω 2
0.0001
0.01 0.1 1
10
ka
Fig. 15.66 Real and imaginary components of f , the
impedance at the unbaffled open end of a cylindrical tube
of radius a, in units of ρc0/πa 2 ,asafunctionofka (after
Beranek [15.136])
–10dB
–20dB
–30dB
θ
Musical Acoustics 15.3 Wind Instruments 603
ka = 0.5 ka = 1.5 ka = 3.83
Fig. 15.67 Polar plots of the intensity radiation from the
end of a cylindrical pipe of radius a for representative ka
values, calculated by Levine and Schwinger [15.137]. The
radial gradations are in units of 10 dB. The intensities in the
forward direction (θ = 0) relative to those of an isotropic
source are 1.1, 4.8 and 11.8dB(Beranek [15.136])
somewhat beyond the open end, providing a pulsating
source that radiates sound, as described by Rayleigh
([15.3] Vol. 1, Sect. 313). Such effects can be described
by a complex terminating load impedance, ZL = R + jx.
Figure 15.66 shows the real (radiation resistance) R and
imaginary (inertial end-correction) x components of ZL
as a function of ka, wherea is the radius of the openended
pipe. The impedance is normalised to the tube
impedance pc0/πa 2 .
When ka ≪ 1, the reactive component is proportional
to ka and corresponds to an effective increase in
the tube length or end-correction of 0.61a. At low frequencies,
the real part of the impedance represents the
radiation resistance Rrad = ρc/4S(ka) 2 .Inthisregime,
the sound will be radiated isotropically as a monopole
source of strength U e iωt , illustrated in Fig. 15.67 by the
polar plots of sound intensity as a function of angle and
frequency.
When ka is of the order of and greater than unity,
the real part of the impedance approaches that of a plane
wave acting across the same area as that of the tube.
Almost all the energy incident on the end of the tube is
then radiated and little is reflected. For ka ≫ 1, sound
would be radiated from the end of the pipe as a beam
of sound waves. The transition from isotropic to highly
directional sound radiation is illustrated for a sequence
of ka values in Fig. 15.67. The ripples in the impedance
in Fig. 15.66 arise from diffraction effects, when the
wavelength becomes comparable with the tube diameter.
For all woodwind and brass instruments, there is
therefore a crossover frequency fc ∼ c0/2πa, below
which incident sound waves are reflected at the open
end to form standing waves. Above fc waves generated
by the reed or vibrating lips will be radiated from the
ends of the instrument strongly with very little reflec-
Part E 15.3