Civil Engineering Reference
In-Depth Information
The acoustic density
ξ
defined by Equation (4.40) is
1
v
p
κ
ρ
o
P
o
p
ρ
o
T
o
(
1
−
j)δ
B
ξ
=
−
−
(5.B.7)
and by the use of Equation (4.39), the bulk modulus of air in the material can be written
ρ
o
p
ξ
γP
o
K
=
=
−
1
)
1
−
j)
δ
B
(5.B.8)
γ
−
(γ
(
1
−
Appendix 5.C: Calculation of the characteristic length
for
a cylinder perpendicular to the direction of propagation
The cylinder in the acoustic field is represented in Figure 5.C.1. The fluid is nonviscous,
and the velocity field can be calculated with the aid of the conformal representation
(Joos 1950).
The displacement potential is equal to
jω
x
3
1
R
2
x
1
+
v
0
ϕ
=
+
(5.C.1)
x
3
υ
o
being the modulus of the velocity far from the cylinder. The two components of the
velocity are
υ
o
1
+
R
2
x
1
+
2
R
2
x
3
(x
1
+
∂ϕ
∂x
3
=
υ
3
=
x
3
−
(5.C.2)
x
3
)
2
R
2
(x
1
+
∂ϕ
∂x
1
=−
υ
1
=
2
υ
o
x
1
x
3
(5.C.3)
x
3
)
2
The squared velocity
υ
2
at the surface of the cylinder is
4
−
4
x
3
R
2
υ
2
·
υ
o
=
(5.C.4)
u
0
θ
x
1
C
x
3
Figure 5.C.1
A cylinder having a circular cross-section is placed in a velocity field
such that the velocity far from the cylinder is perpendicular to the cylinder.
