Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 91
is indeed justified because the pressure drop is determined by the static pressure far
ahead and far behind the resonator section where the velocity profile is fully adapted
to the smooth rigid wall of the pipe. The pressure drop which would arise along a
smooth rigid pipe between the measuring points was subtracted from the measured
pressure difference. So only the pressure drop due to the additional wall shear stress
in the resonator section is recorded. With Eq. (7) wherein τ dU/dr has been replaced
by means of the Eqs. (5) and (8), one obtains for the circular pipe
� L
∆p = −
=
� L�
R�
dpτ 1
′′ dU
dx = − τ
0 dx U 0 0 dr + τ �
dU 2rdr
µ
dx
dr R2 ��
U
�
ρ
S U 2 u′2 + v ′2
� �L dS
+
S 0
� L
� L�
� �
∂(S)
1
dU dS
+
dx + ɛµ − τµ dx . (9)
S
U
dr S
0
p ′ v ′ w
U
First we discuss the most simple case and omit all dissipative effects which are comprised
in the second and the third term on the right-hand side of Eq. (9). Then
∆p is given by the axial change of the flux of kinetic energy contained in the fluc-
tuation of the velocity. The acoustically excited part of the kinetic energy density
(ρ/2)u ′2
ac + v ′2
ac is proportional to the square of the amplitude of the instability wave,
and since the acoustic energy density is very small at the leading edge of the resonator
section, the acoustically induced pressure drop ∆pac is proportional to the square of
the pressure amplitude at the rear end the of resonator section which on its part
determines the pressure amplitudes of the transmitted sound and in the backmost
resonator cavity. So the experimental results shown in Fig. 9 and described by Eq. (1)
can be understood for medium sound amplitudes.
The dissipative terms in Eq. (9) exhibit the same quadratic dependency on the
amplitude of the instability wave as the first term on the righthand side of the
equation as long as the dispersion relation of the instability wave is not too much
affected by the nonlinearity of the wave propagation. However, when the growth
of the instability wave becomes saturated at high sound amplitudes (see Fig. 7),
the kinetic energy at the rear end of the resonator section is more affected by the
nonlinearity than the dissipative terms in Eq. (9). So while the pressure drop further
increases with the incident sound amplitude the radiated pressure and the pressure
in the backmost cavity begin to stagnate, i. e. the data points more and more lie
above the 45 o line in Fig. 9. In addition one has to take into account that the
ratio between the amplitudes of the pressure and the velocity is influenced by the
considered nonlinearity as well, however there is no simple way to predict how the
function DP{· · ·} in Eq. (1) is affected by this implication of the nonlinearity.
4 Theoretical approaches to the wave propagation in the lined duct
In order to substantiate the physical explanation of the considered phenomena we
need a deeper insight into the properties of the instability wave. Various aspects of
the wave propagation in the lined duct have been studied for this purpose [34,35].
0
S