330
6 Shocks and Surprises: Refining the Elementary Model
that it describes the input impedance of a rigid tube. The modulus of this input
impedance shows peaks corresponding to the acoustic resonances of the closedopen air column.
The previous description does not take into account possible wall vibrations. In
order to describe those vibrations, it is necessary to consider that the tube behaves
as an elastic shell. Following the theoretical approach described in Sect. 6.7.1, the
instrument is modelled as a quasi-cylindrical shell, in this case clamped at one
end and free at the other. Structural vibrations and inner acoustic pressure are
coupled: the shell is excited by the internal acoustic pressure field, and the resulting
vibrations induce a disturbance of the initial pressure field. As a consequence of this
modification of the air column oscillation, the input impedance of the vibrating tube
can differ significantly from that of a perfectly rigid cylinder (Pico Vila and Gautier
2007).
The central point for modelling the acoustic input impedance of the vibrating
tube is to take into account a slight ellipticity of the tube, which models the
asymmetry which is in practice unavoidable. In a first approximation, its radius r
can be written using the polar equation, which is plotted in Fig. 6.38,
r(θ) = a[1 + cos(2θ)],
(6.45)
where a denotes the mean radius and an ellipticity parameter, which is small
compared to unity. With the notation of Fig. 6.38, is equal to (r max − r min )/2a.
The consequence of this asymmetry is the existence of a vibroacoustic coupling
between the acoustic plane wave and the ovalling modes of the structure. Ovalling
modes, characterised by a circumferential modal shape in a sin(2θ) form, are often
the modes of lowest frequency for geometries similar to those of wind instruments.
Physically, a section of the tube wall is subjected to a uniform pressure distribution
due to the plane wave. If the tube is perfectly circular, this tends to dilate and contract
it only in a cylindrically symmetric manner, and the section of the tube is only
subjected to tension forces. If the tube is oval shaped, then the isotropic pressure
distribution implies also bending forces that tend to round the tube by enlarging
the small diameter and shortening the bigger one. This movement is directly linked
to the ovalling deformation of the pipe. A detailed description of the vibroacoustic
coupling is given in Pico Vila and Gautier (2007). The conclusion of this is an
analytical expression for the acoustic input impedance of the vibrating tube, which
can be written as
Z(ω) = Z r (ω)[1 + C(ω)],
(6.46)
where C is a correction factor describing the wall vibration effect. Considering only
the interaction between the internal acoustic pressure and a single ovalling mode, it
is shown that
C(ω) ∝
2 /{(1 − e
−2jkL ) cos(kL)m μ [ω
2
μ (1 − jη μ ) − ω
2
]},
(6.47)
6 Shocks and Surprises: Refining the Elementary Model
that it describes the input impedance of a rigid tube. The modulus of this input
impedance shows peaks corresponding to the acoustic resonances of the closedopen air column.
The previous description does not take into account possible wall vibrations. In
order to describe those vibrations, it is necessary to consider that the tube behaves
as an elastic shell. Following the theoretical approach described in Sect. 6.7.1, the
instrument is modelled as a quasi-cylindrical shell, in this case clamped at one
end and free at the other. Structural vibrations and inner acoustic pressure are
coupled: the shell is excited by the internal acoustic pressure field, and the resulting
vibrations induce a disturbance of the initial pressure field. As a consequence of this
modification of the air column oscillation, the input impedance of the vibrating tube
can differ significantly from that of a perfectly rigid cylinder (Pico Vila and Gautier
2007).
The central point for modelling the acoustic input impedance of the vibrating
tube is to take into account a slight ellipticity of the tube, which models the
asymmetry which is in practice unavoidable. In a first approximation, its radius r
can be written using the polar equation, which is plotted in Fig. 6.38,
r(θ) = a[1 + cos(2θ)],
(6.45)
where a denotes the mean radius and an ellipticity parameter, which is small
compared to unity. With the notation of Fig. 6.38, is equal to (r max − r min )/2a.
The consequence of this asymmetry is the existence of a vibroacoustic coupling
between the acoustic plane wave and the ovalling modes of the structure. Ovalling
modes, characterised by a circumferential modal shape in a sin(2θ) form, are often
the modes of lowest frequency for geometries similar to those of wind instruments.
Physically, a section of the tube wall is subjected to a uniform pressure distribution
due to the plane wave. If the tube is perfectly circular, this tends to dilate and contract
it only in a cylindrically symmetric manner, and the section of the tube is only
subjected to tension forces. If the tube is oval shaped, then the isotropic pressure
distribution implies also bending forces that tend to round the tube by enlarging
the small diameter and shortening the bigger one. This movement is directly linked
to the ovalling deformation of the pipe. A detailed description of the vibroacoustic
coupling is given in Pico Vila and Gautier (2007). The conclusion of this is an
analytical expression for the acoustic input impedance of the vibrating tube, which
can be written as
Z(ω) = Z r (ω)[1 + C(ω)],
(6.46)
where C is a correction factor describing the wall vibration effect. Considering only
the interaction between the internal acoustic pressure and a single ovalling mode, it
is shown that
C(ω) ∝
2 /{(1 − e
−2jkL ) cos(kL)m μ [ω
2
μ (1 − jη μ ) − ω
2
]},
(6.47)