An analytic Green's function for a lined circular duct containing ...

SPL, dB
50
40
30
20
10
0
−10
−20
−30
−40
−50
0.8 0.85 0.9
Radius (r)
0.95 1
ARTICLE IN PRESS
S.W. Rienstra, B.J. Tester / Journal of Sound and Vibration 317 (2008) 994–1016 1011
28+
28−
24+
24−
20+
20−
16+
16−
12+
12−
8+
8−
4+
4−
0+
0−
Total+
Total−
Free−field
−50
0.8 0.85 0.9
Radius (r)
0.95 1
Fig. 12. Radial variation without flow of analytic and numerical Green’s function amplitudes per azimuthal mode, SPL ¼ 20 logðjGmjÞ,
and total field SPL v. free-field Green’s function þ reflected SPL (o ¼ 31:354, M ¼ 0, Z1 ¼ 2:5, r0 ¼ 1, analytic: cut-off ratio ¼ 0:07,
numerical: limited to 40 cut-off modes). Analytical (a), numerical (b). NB: The left and right running modal (analytic and numerical)
Green’s functions, labelled Totalþ, Total , are indistinguishable in these no-flow results and oscillate about the Free-field Green’s
function, which decays monotonically away from the source at r ¼ 1.
The issue of a possible instability is still an open question. Causality considerations suggest, at least in the
linear model, the presence of convective or even absolute instabilities. This has found support by numerical
results in time domain [13]. Also experimentally there is evidence [14,15], at least for impedances of small
resistance. In most cases, however, the exponentially large fields that would result have little resemblance with
physical reality. Therefore, we have not included any such instability in the presented numerical results and
adopted stability of the solution as modelling assumption.
Comparison of our analytical mode amplitudes (for the hollow duct version of our solution) with the
numerical solution of Alonso et al. showed a good agreement without mean flow, irrespective of how
many modes are included in the matrix inversion for the numerical mode amplitudes. A large number of
modes are required for convergence near the source. With flow, the mode amplitudes do not agree but
as the number of modes included in the matrix inversion is increased, enough to include all important surface
SPL, dB
50
40
30
20
10
0
−10
−20
−30
−40