Civil Engineering Reference
In-Depth Information
θ′
Fluid 1
Fluid 2
θ
M
3
M
2
M
1
X
3
d
X
1
Figure 3.4
A layer of fluid of thickness d backed by an impervious rigid wall.
Both determinations of the square root can be used if
d
is finite. At the impervious
rigid wall
Z(M
1
)
is infinite and Equation (3.36) becomes
Z
c
k
Z(M
2
)
=−
k
3
j
cotg
k
3
d
(3.39)
where
Z
c
is the characteristic impedance in fluid 1. The impedance
Z(M
2
)
at the boundary
in fluid 1 is equal to the impedance
Z(M
3
)
at the boundary in fluid 2, as in the case
for normal incidence. If the velocity in fluid 1 is much smaller than that in fluid 2, it
follows from Equation (3.22) that sin
θ
is close to zero and
k
3
is close to
k
for large range
of angles
θ
around normal incidence. Therefore,
Z(M
2
)
as given by Equation (3.39) is
close to its value at
θ
=
0, and weakly depends on the angle of incidence. Medium
1 is called a locally reacting medium (Cremer and Muller 1982). The locally reacting
property means that the material response at a given point on the surface is independent
on the behaviour at other points of the surface. It is a good approximation for several
engineered materials such as honeycombs and acceptable for a number of thin porous
materials. This approximation has been and is still widely used in finite element and
boundary element vibro-acoustic simulations. In these applications, the sound package
attenuation is accounted for in terms of a surface admittance, the latter is measured or
simulated in a plane wave incidence context. Chapter 13 replaces this approximation with
detailed modelling using poroelastic finite elements.
3.4.3 Impedance at oblique incidence in a multilayered fluid
A multilayered fluid is represented in Figure 3.5. Let
k
,
k
and
k
be the wave numbers
in fluids 1, 2 and 3, respectively. The angle of incidence is
θ
.The0
x
1
component of
the wave number vector is the same in the different media and is equal to
k
1
=
k
1
=
k
sin
θ
k
1
=
(3.40)
In fluid 1, the component
k
3
of
k
is
(k
2
k
1
)
1
/
2
k
3
=
−
(3.41)
