Oscillations, Waves, and Interactions - GWDG

84 D. Ronneberger et al.
(flow velocity) / (speed of sound)
0.3
0.2
0.1
0.3
0.2
0.1
0.3
0.2
a
b
c
C
A 1
0.1
0 1 2 3 4 5
frequency [kHz]
A 2
Figure 10. Power spectral density and coherence of the pressure in the backmost cavity
of the unmodified resonator section plotted as a function of the frequency and of the flow
velocity. (a) power spectral density (normalized to the background spectrum); (b) coherence
between two locations the azimuthal coordinates of which differ by ∆ϕ = 180 o ; (c) like (b)
with ∆ϕ = 120 o .
2.2 Higher-order modes
Regarding the possible technical application of the sound-induced static pressure
drop, the excitation and the effect of higher-order modes have been studied. It is anticipated
that some of these propagate in the resonator section, but are evanescent in
the rigid ducts so that the mostly unwelcome propagation of the amplified controlling
sound is suppressed. We are particularly interested in circumferential modes which
are easier to handle than radial modes. Then the amplitudes of the flow velocity and
the pressure depend on the azimuthal coordinate ϕ according to (û, ˆv, ˆp) ∼ exp(imϕ)
wherein m is the azimuthal wavenumber and |m| is the azimuthal order of the mode.
Because of the evanescence of the considered modes we have to abandon the radiated
pressure fluctuations and the acoustic transmission coefficient as sources of
information in this study; instead we analyze the pressure in the backmost cavity.
The spectum of the pressure is more complex, and it contains more peaks than the
radiated pressure spectra shown in Fig. 2; in addition the ‘background spectrum’ on
top of which the peaks are observed exhibits a stronger dependency on the frequency
than the radiated pressure. The background spectrum was therefore approximated
A 3
A 4
dB
110
00
−10 −1
1.0
0.5
0
−0.5
−1.0