Civil Engineering Reference
In-Depth Information
where
k
x
and
k
y
are the components of the wave number in the medium around the plate
(air). This expression has then to be a solution of the ordinary wave equation:
2
1
∂
p
2
∇ −
p
=
0.
(6.32)
2
2
ct
∂
0
p
(
x,y
)
y
x
λ
B
z
Figure 6.9
Sketch showing a plane bending wave on an infinitely large plate. The plate lies in the x-z plane and
the pressure is calculated in points (x,y).
for the sound field above the plate must be expressed by
== +
2
2
k
k
k
.
(6.33)
x
y
c
0
A further condition is that the component
v
y
of the particle velocity, i.e. the component
normal to the plate, must be equal to
u
B
at the surface of the plate (
y
= 0). Since
v
y
is
given by
ˆ
pk
1
∂
p
y
j(
ω
tkxky
−−
)
v
=−
⋅ =
⋅
e
x
y
,
(6.34)
y
j
ωρ
∂
y
ρω
0
0
we get when setting
y
= 0,
ˆ
pk
y
−
j
kx
−
j
kx
ˆ
ue
⋅
=
e
ρω
⋅
.
(6.35)
B
x
0
Hence,
ρω
0
ˆ
ˆ
p
=
⋅
u
and
k
=
k
.
x
B
k
y
The sound pressure may thereby be expressed as
ρ
cu
ˆ
22
B
j
kky
−
j(
ω
tkx
−
)
00
2
B
2
pxy
(, )
=
⋅
e
⋅
e
.
(6.36)
B
k
k
1
−
