Civil Engineering Reference
In-Depth Information
ˆ
j
ω
pk
(
)
uk
ˆ()
=
x
.
(6.86)
(
)
x
4
4
B
Bk
−
k
x
We observe again, as when discussing the radiation factor of plates, the important
relationship between the acoustic wave number and the bending wave number. This
becomes more evident when we calculate the velocity, having a situation as sketched in
Figure 6.21.
For the sound pressure in the incident, reflected and transmitted wave,
respectively, we shall write:
ˆ
e
−
j s
ky
ϕ
−
j n
kx
ϕ
pp
=
e
y
>
0,
i
i
j s
ky
ϕ
−
j n
kx
ϕ
pp
=
ˆ e
e
y
>
0,
(6.87)
r
r
ˆ
e
−
j s
ky
ϕ
−
j n
kx
ϕ
pp
=
e
y
<
0.
t
t
Hence, the total pressure on the plate is (
y
= 0):
(
)
−
j n
kx
ϕ
ˆ
ˆ
ˆ
pxz
(,)
=+−
p
p
p
e
.
(6.88)
i
r
t
equation giving the relationship between the driving sound pressure and the resulting
velocity:
(
)
j
ω
ppp
ˆ
+−
ˆ
ˆ
i
r
t
ˆ
u
=
.
(6.89)
(
)
4
4
4
B
Bk
sin
ϕ
−
k
The ratio of the driving pressure to the velocity is generally known as
wall impedance
.
This quantity, for which we shall use the symbol
Z
w
, will be given by
ppp
B
ˆ
+−
ˆ
ˆ
(
)
4
4
4
i
r
t
Z
=
=
k
sin
ϕ
−
k
.
(6.90)
w
B
ˆ
u
j
ω
Under the condition
k > k
B
we shall always find an incident angle ϕ where
Z
w
is equal to
zero, making the velocity “infinitely” large. The plate will not present any obstacle for
the sound wave! The conditions determining this
trace matching
were discussed in
section
6.3.2
concerning sound radiation from a plate. The important point in this
connection is that the angle giving a maximum radiation is also the one giving maximum
excitation. This is an example of a general principle in acoustics, the so-called
reciprocity principle
, which we shall address in section
6.6.1
.
A further discussion on Equation
(6.90)
will be easier when introducing the critical
frequency
f
c
and also some energy losses by way of a complex bending stiffness
B
(1+j⋅η). We may then write
⎡
2
⎤
⎛⎞
f
(
)
⎢
4
⎥
Z
=
j
ω
m
1
−
⋅ +
1
j
ηϕ
sin
.
(6.91)
⎜⎟
w
⎢
f
⎥
⎝⎠
c
⎣
⎦
