6.7 Going Further: Analytical Modelling of Vibroacoustic Coupling in Ducts
327
mechanical forces, so that wall vibration can arise only from coupling to the
acoustic field. The basic theory is explained in Sect. 6.7.1 and used in Sect. 6.7.2
to develop an analytical expression for the input impedance of a duct whose crosssection is slightly elliptical. Some experimental results obtained using clarinet-like
instruments are compared with predictions of the vibroacoustic model in Sect. 6.7.3.
The theoretical and experimental results presented in this section come principally
from Backus and Hundley (1966), Gautier et al. (2007), Pico Vila and Gautier
(2007) and Nief et al. (2008).
6.7.1 Basic Vibroacoustic Theory
The easiest way to take into account the influence of wall vibrations on the internal
acoustic field is to model the pumping effect due to changes in the cross-sectional
area of the duct (Backus and Hundley 1966). We consider a tube with yielding
walls such that under internal pressure p, the fractional change in area S from the
undistorted area S 0 is given by
(S − S 0 )/S 0 = βp.
(6.39)
The ‘yield’ parameter β, which is dimensionally an inverse pressure, is assumed
to have a value such that βp 1. As a consequence of Eq. 6.39, the conservation
of mass is modified, and after some mathematics the classical wave equation is
obtained with a speed of sound c slightly different from its value c 0 in a rigid-walled
tube:
c = c 0 /
1 + βρ 0 c 2
0 ≈ c 0 (1 − βρ 0 c
2
0 /2).
(6.40)
This correction implies that the resonance frequencies of a cylindrical tube are
shifted from those of a rigid tube of the same length by a fractional amount
f/f = 0 = −βρ 0 c
2
0 /2.
(6.41)
For a copper tube, Backus found that β = 1.210 −10 , which implies a frequency
change of 0.0084%: this corresponds to a pitch difference of 0.14 cents, which is too
small to be significant.
Are there situations where strong effects of coupling between vibrating walls
and the internal pressure field can be visible, for example, on the acoustic input
impedance? A thin stretched membrane in the form of a cylindrical tube can exhibit
strong effects on sound waves propagating through the tube. In some frequency
ranges, the waves can become evanescent, leading to the presence of stop bands
(see the experimental and theoretical work based on Korteweg’s equation in Gautier
327
mechanical forces, so that wall vibration can arise only from coupling to the
acoustic field. The basic theory is explained in Sect. 6.7.1 and used in Sect. 6.7.2
to develop an analytical expression for the input impedance of a duct whose crosssection is slightly elliptical. Some experimental results obtained using clarinet-like
instruments are compared with predictions of the vibroacoustic model in Sect. 6.7.3.
The theoretical and experimental results presented in this section come principally
from Backus and Hundley (1966), Gautier et al. (2007), Pico Vila and Gautier
(2007) and Nief et al. (2008).
6.7.1 Basic Vibroacoustic Theory
The easiest way to take into account the influence of wall vibrations on the internal
acoustic field is to model the pumping effect due to changes in the cross-sectional
area of the duct (Backus and Hundley 1966). We consider a tube with yielding
walls such that under internal pressure p, the fractional change in area S from the
undistorted area S 0 is given by
(S − S 0 )/S 0 = βp.
(6.39)
The ‘yield’ parameter β, which is dimensionally an inverse pressure, is assumed
to have a value such that βp 1. As a consequence of Eq. 6.39, the conservation
of mass is modified, and after some mathematics the classical wave equation is
obtained with a speed of sound c slightly different from its value c 0 in a rigid-walled
tube:
c = c 0 /
1 + βρ 0 c 2
0 ≈ c 0 (1 − βρ 0 c
2
0 /2).
(6.40)
This correction implies that the resonance frequencies of a cylindrical tube are
shifted from those of a rigid tube of the same length by a fractional amount
f/f = 0 = −βρ 0 c
2
0 /2.
(6.41)
For a copper tube, Backus found that β = 1.210 −10 , which implies a frequency
change of 0.0084%: this corresponds to a pitch difference of 0.14 cents, which is too
small to be significant.
Are there situations where strong effects of coupling between vibrating walls
and the internal pressure field can be visible, for example, on the acoustic input
impedance? A thin stretched membrane in the form of a cylindrical tube can exhibit
strong effects on sound waves propagating through the tube. In some frequency
ranges, the waves can become evanescent, leading to the presence of stop bands
(see the experimental and theoretical work based on Korteweg’s equation in Gautier