Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 85
power spectral density
coherence function
10
1
.1
1
0.5
0
−0.5
−1
1
0.5
0
a
n=
m=
b
0
1
0
2
1
0
0
3
0 1
4 1
0 1 2 0
5 2 0 6
1
3
2 0
1 7
−0.5
c
−1
0 1 2 3 4 5
frequency [kHz]
Figure 11. Average of the data of Fig. 10 over 0.13 ≤ U/c ≤ 0.25. The cut-on frequencies
of the various modes in the resonator section are marked on the bottom of Fig. (a) where
the radial order of the modes are denoted by n, and the expected coherence functions (see
text) are shown as dashed curves in Figs. (b) and (c).
by a polynomial which was fitted to the logarithm of the power spectral density
(psd), and only the difference between the actual psd and this ‘background spectrum’
is presented as a function of the frequency and the flow velocity in figure 10a.
As in Fig. 2(b) the peaks A 1 , A 2 , and A 3 are the the most outstanding ones also in
Fig. 10(a) where the fourth harmonic A 4 is included.
In addition quite a few less prominent peaks appear in Fig. 10(a) which have no
correspondence in Fig. 2(b). These peaks are therefore supposed to originate from
higher-order modes which are evanescent in the rigid pipes. With one exception
(C), the mid-frequencies of the additional peaks do not depend on the flow velocity.
So the clearness of these peaks can be increased by averaging the spectra over a
range U/c = 0.13 · · · 0.25 of flow velocities, within which the less prominent peaks
more or less dominate over the A-peaks. The average is shown in Fig. 11(a), and
moreover the cut-on frequencies of the higher-order modes in the resonator section
are marked on the frequency axis; the cut-on frequencies are only weakly dependent
on the flow velocity and therefore have been computed for air at rest. A steep increase
of the power spectral density is found at the low-order cut-on frequencies. So it is
conjectured that the contribution to the pressure fluctuations by a definite mode is
particularly large at frequencies just above the respective cut-on frequency.