Civil Engineering Reference
In-Depth Information
p
r
Z
g
p
i
x
= 0
Figure 3.10
Incident and reflected plane wave at a surface.
velocity in the incident wave as
j(
ω
tkx
−
)
ˆ
pxt
(,)
=⋅
p
e
i
i
(3.64)
ˆ
p
j(
ω
tkx
−
)
i
and
vxt
( , )
=
⋅
e
,
i
ρ
c
00
where
ρ
0
c
0
is the characteristic impedance of the medium. For the reflected wave we get
j(
ω
tkx
+
)
j(
ω
tkx
+
)
ˆ
ˆ
pxt
(,)
=⋅
p
e
= ⋅
Rp
e
r
r
p
i
(3.65)
ˆ
p
j(
ω
tkx
+
)
i
and
vxt
( , )
=−
R
⋅
e
.
r
p
ρ
c
00
There will be a change in sign for the wave number due to the change of direction of the
wave. At the same time the particle velocity is changing sign as the gradient of the
pressure is changing sign along with the wave number. The total pressure at the
boundary surface (
x
= 0) will be
( )
(
)
j
( )
( )
e
ωt
ˆ
pt
0,
=
p t
0,
+
p t
0,
=
p
1
+
R
⋅
(3.66)
i
r
i
p
and the particle velocity:
p
(
)
j
i
ω
t
vt
(0, )
=
v t
(0, )
+
v t
(0, )
=
1
−
R
⋅
e
.
(3.67)
i
r
p
ρ
c
00
Inserting these expressions into Equation
(3.63)
we get
