Civil Engineering Reference
In-Depth Information
In this equation
ξ
is the acoustical density,
P
o
and
T
o
are the ambient mean pressure
and mean temperature. The variations of
τ
in the
x
3
direction parallel to the tube are
smaller than in the radial directions
x
1
and
x
2
, and Equation (4.2) reduces to
∂
2
τ
∂x
1
+
∂
2
τ
∂x
2
−
jω
τ
j
ω
ν
=−
κ
p
(4.30)
where
ν
is given by
κ
ρ
0
c
p
ν
=
(4.31)
This equation has the same form as Equation (4.9) for
v
3
. Moreover, the boundary
conditions are the same,
τ
being equal to zero at the surface of the tube.
An angular frequency
ω
defined by
ω
=
ω
η
ρ
o
ν
(4.32)
will be used instead of
ω
in Equation (4.30) which then becomes
∂
2
τ
∂x
1
+
∂
2
τ
jω
υ
ρ
o
p
κη
∂x
2
−
jω
ρ
o
τ
η
=−
(4.33)
The quantity
η/(ρ
0
ν
)
is the Prandtl number, and will be denoted by
B
2
. We rewrite
Equation (4.9) as
∂
2
υ
3
∂x
1
+
∂
2
υ
3
∂x
2
−
jωρ
o
η
1
η
∂p
∂x
3
υ
3
=
(4.34)
By comparing Equations (4.33) and (4.34), it follows that
τ
and
v
3
are equal to
pν
κ
ψ(x
1
,x
2
,B
2
ω)
τ
=
(4.35)
∂p
∂x
3
ψ(x
1
,x
2
,ω)
1
jωρ
o
υ
3
=−
(4.36)
where
ψ(x
1
,x
2
,ω)
is a solution of the equation
∂
2
ψ
∂x
1
+
∂
2
ψ
∂x
2
−
jω
ρ
o
jωρ
o
η
η
ψ
=−
(4.37)
with the boundary condition that
ψ
vanishes at the surface of the tube.
The effective density
is calculated from the average value
υ
3
of
υ
3
over
the
cross-section of the tube (Equation 4.17)
∂p
∂x
3
1
jωυ
3
=
ρ
0
ψ(x
1
,x
2
,ω)
ρ
=−
(4.38)
Using Equation (1.57) and the linearized continuity equation, the bulk modulus is
given by
p/
¯
ρ
0
p/ξ
K
=−
∇
u
=
(4.39)
