Flute acoustics: measurement, modelling and design - School of ...

CHAPTER 2. THEORY AND LITERATURE REVIEW 17
with
A = − sin(kL − θ 2)
sinθ 2
B = i ρc sin(kL)
S 2
C = i S 1
ρc
sin(kL + θ 1 − θ 2 )
sinθ 1 sinθ 2
(2.16b)
(2.16c)
(2.16d)
D = S 1
S 2
sin(kL + θ 1 )
sinθ 1
, (2.16e)
where θ 1 = tan −1 (kx 1 ) and θ 2 = tan −1 (kx 2 ). Fletcher & Rossing (1998) give expressions for the
input impedance of conical horns with various load impedances. Fletcher & Rossing also treat
other horn geometries, such as the exponential horn—these results are not reproduced here.
2.2.7 Wall losses
So far in our discussion we have assumed that the duct walls are perfectly smooth, rigid and
thermally insulating, and that acoustic waves propagate without loss along the duct. For most
woodwinds the walls are sufficiently rigid that mechanical vibration of the walls can be safely
neglected ∗ . However, waves within real woodwind instruments are subject to thermal dissipation
and viscous drag, leading to attenuation of the travelling wave.
As discussed by Nederveen (1998), the acoustic flow slows to zero near the tube walls, and
thermal exchange between the walls and the air removes some energy from the (adiabatic)
acoustic waves. For woodwind instruments, both of these effects occur only over a thin boundary
layer (approx. 0.05 mm at 1000 Hz in the case of the viscous effects), so the plane-wave
approximation remains valid (Nederveen 1998).
The effects of viscous drag and thermal exchange are discussed by Benade (1968). In general,
the effect of loss terms is to make Z 0 and k complex, which leads to attenuation of the
wave as it travels along the conduit. The small imaginary part of Z 0 can usually be neglected
for the bores of musical instruments. The complex wavenumber is given by
k = ω v − iα,
(2.17a)
where
and
[
v = c 1 − 1
]
]
(γ − 1)
1.65 × 10−3
− = c
[1 −
r v 2 r t 2 a f 1/2
α = ω c
(2.17b)
[ ]
1 (γ − 1)
+ = 3 × 10−5 f 1/2
. (2.17c)
r v 2 r t 2 a
Here r v and r t are the ratios of pipe radius to boundary layer thickness for viscous and thermal
losses respectively. Approximations for these ratios are given in Fletcher & Rossing (1998).
Equations (2.17b) and (2.17c) are valid for r v > 10. As shown by Keefe (1984) the viscothermal
effects on the characteristic impedance Z 0 become significant for r v < 10, as do terms of higher
order in r v and r t in the expressions for the phase velocity (2.17b) and attenuation coefficient
∗ According to measurements by Backus (1964) and Nederveen (1998), the walls of woodwind instruments vibrate
with an amplitude of about 1 µm and radiation due to this vibration is at least 40 dB smaller than the air signal.