Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 81
Figure 7. Pressure amplitude close to the wall of the unmodified resonator section with
different amplitudes of the incident sound wave. U/c = 0.2, f=1033 Hz. From Ref. [29].
of the resonator section. The relatively involved relation between the far upstream
measured signal of the broad-band noise that was to be cancelled and the necessary
amplitude of the controlling sound signal was determined and adjusted by means of
an adaptive filter.
2.1.2 Pressure amplitude and static pressure in the resonator section
Considering the various types of instability in flow ducts with compliant walls, as
mentioned in the introduction, we have come to the conclusion that the sound amplification
is caused by such an instability. We assume that a convective instability
mode is excited by the incident sound at the leading edge of the resonator section.
The amplitude grows while the instability wave propagates through the lined duct
section, and from the large pressure oscillation at the end of the lined duct section
a large-amplitude sound wave is radiated into the rigid duct which is connected to
the rear of the resonator section. The spatial distribution of the pressure oscillation
in the resonator section supports this hypothesis. First of all it turns out that the
pressure field is composed of various modes which differ by the wavenumber and by
the radial dependence of the pressure. So rather involved interference patterns may
occur, e. g. the axial gradient of the phase may be positive along the axis of the duct
and be negative along the wall. The axial distributions depicted in the Figures 6 – 8
can be considered to be comparatively clear. Summarizing we arrive at the following
conclusions [29,33]:
(i) Besides evanescent modes which are excited at the ends of the resonator section
the pressure field contains a wave which exhibits all properties of an instability mode.
(ii) The phase velocity of the instability wave is O(U/2).
(iii) The parameters that determine the propagation of the instability mode strongly
vary along the axial coordinate; in particular the rate of exponential growth of the
wave decreases with the travelling distance and may even become negative (Fig. 8