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3.3 Acoustic network model
where A denotes the cross-sectional area <strong>of</strong> the element. Considering a timeharmonic
dependence <strong>of</strong> the acoustic variables (Eqn. (3.25)) Eqn. (3.43) yields:
i ω l eff u ′ i + [ p
′
ρ 0
+ u 0 u ′ ] j
i
+ u 0j ζu ′ j
= 0. (3.45)
The second necessary relation for a compact element can be derived using the
integral mass conservation equation:
d
d t
∫ ∫
ρ dV + ρ u d A= 0. (3.46)
Considering one-dimensional flows <strong>and</strong> linearizing this equation results in:
d
d t
∫ x j
x i
ρ ′ A d x+ [ (ρ ′ u 0 + ρ 0 u ′ )A ] j
i
= 0. (3.47)
Again a time-harmonic dependence (Eqn. (3.25)) <strong>of</strong> the acoustic variables
is assumed. If the speed <strong>of</strong> sound is constant one obtains introducing
∫ x j
x i
A(x)/A j ≡ l red <strong>and</strong> using Eqn. (3.18)
i ω
c i
p ′ [(
i
p
′
) ] j
l red A j +
c i c M+ ρ 0 u ′ A
i
= 0, (3.48)
where M denotes the Mach number (M = u 0 /c). The values <strong>of</strong> the first term
<strong>of</strong> the left h<strong>and</strong> side <strong>of</strong> Eqn. (3.48) <strong>and</strong> Eqn. (3.45) are in case <strong>of</strong> compact elements
small <strong>and</strong> are omitted in the following. A further simplification can
be made if the speed <strong>of</strong> sound <strong>and</strong> the density are assumed to be constant
over the element (c i = c j = c, ρ 0i = ρ 0j = ρ 0 ). Dividing Eqn. (3.45) by c <strong>and</strong>
Eqn. (3.48) by ρ 0 leads to the following relations for compact elements:
[ ( p
′
)] j
A
ρ 0 c M+ u′ = 0,
i
[ p
′
] j
ρ 0 c + M u′ + ζ j M j u ′ j
= 0. (3.49)
i
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