Civil Engineering Reference
In-Depth Information
In these equations
k
1
,k
2
and
k
3
are the components of the wave number vector
k
k
1
=
n
1
k
1
k
2
=
n
2
k
2
k
3
=
n
3
k
3
(3.3)
Using the wave equation (1.56), the components of the wave vector satisfy the equation
ρ
K
ω
2
k
1
+
k
2
+
k
2
=
k
2
=
(3.4)
Equation 3.2 can be rewritten as
A
Z
c
k
1
k
exp[
j(
υ
1
(x
1
,x
2
,x
3
,t)
=
−
k
1
x
1
−
k
2
x
2
−
k
3
x
3
+
ωt)
]
A
Z
c
k
2
k
exp[
j(
υ
2
(x
1
,x
2
,x
3
,t)
=
−
k
1
x
1
−
k
2
x
2
−
k
3
x
3
+
ωt)
]
(3.5)
a
Z
c
k
3
k
exp[
j(
υ
3
(x
1
,x
2
,x
3
,t)
=
−
k
1
x
1
−
k
2
x
2
−
k
3
x
3
+
ωt)
]
For the plane wave described by Equations (3.1) - (3.3),
k
is perpendicular to the
equiphase and equiamplitude planes. A more significant generalization can be achieved
by discarding Equation (3.3) and using only Equation (3.4) to define
k
1
,
k
2
and
k
3
.A
first simple example is considered where
k
2
is real in Equation (3.4).
The components of the wave number vector
k
are
k
1
=
l,
k
2
=−
jm, k
3
=
0
(3.6)
where the quantities
m
and
l
are real, positive and are linked by Equation (3.4), which
can be rewritten as
ρ
l
2
m
2
K
ω
2
−
=
(3.7)
From Equations (3.1) and (3.5), it follows that
p(x
1
,x
2
,x
3
,t)
=
A
exp[
j(
−
lx
1
+
ωt)
−
mx
2
]
(3.8)
Al
Z
c
k
exp[
j(
υ
1
=
−
lx
1
+
ωt)
−
mx
2
]
(3.9)
j
Am
υ
2
=−
Z
c
k
exp[
j(
−
lx
1
+
ωt)
−
mx
2
]
(3.10)
υ
3
=
0
(3.11)
From the above equations it is seen that the equiamplitude planes are parallel to the
x
2
=
j
in
υ
2
indicates a phase difference of
π
/2 between
υ
1
and
υ
2
. These two quantities cannot
be the geometrical projections of a vector on the
x
1
and
x
2
axes unless complex angles
are used to take into account the phase difference between the components.
Plane waves with distinct equiphase and equiamplitude planes are called inhomo-
geneous plane waves. Equations (3.3) can always be used to define the three complex
0 plane, and the equiphase planes are parallel to the
x
1
=
0 plane. The factor
−
