Environmental Engineering Reference
In-Depth Information
exists. In three-dimensional space,
S
a
(
t
+
Δ
t
)
does not contain
S
at
.The
D
at
is however
apartof
D
a
(
t
+
Δ
t
)
and
⎧
⎨
d
a
,
=
0
,
for
t
<
Effect of initial disturbances
has not yet propagated to point
M
0
,
for
d
D
a
,
u
(
M
0
,
t
)=
=
0
,
a
<
t
<
Effect of initial disturbances has arrived
at point
M
0
,
⎩
D
a
,
=
0
,
for
t
>
Wave front has passed, but left a lasting effect.
Therefore, there is no wave rear. The phenomenon of having a lasting effect is called
wave dispersion
. The mechanismbehind this difference can be understood by noting
that a two-dimensional plane is a special case of three-dimensional space so
ϕ
=
0
and
0 in an infinitely long cylinder that is parallel to the
z
-axis and with
D
as the
cross-plane. Therefore, waves propagate as cylindrical waves in two-dimensional
cases. Similarly, they propagate as plane waves in one-dimensional cases.
ψ
=
Kirchhoff Formula of Three-Dimensional Wave Equations
in
V
at
.
The structure of Eq. (2.85) is the same as that of potential functions. The Newton
potential of a body
M
,
(
,
)
(
)
The Kirchhoff formula (2.85) shows that the
u
M
t
comes from
f
t
Ω
at point
M
is defined in mechanics as
M
)
r
ρ
(
r
=
MM
,
v
(
M
)=
d
v
,
Ω
M
)
where
in
Eq. (2.85) is also called a potential function. The value of
u
at time instant
t
de-
ρ
(
is the density of the body. Because of this similarity,
u
(
M
,
t
)
pends on the value of
f
at time instant
t
a
. The effect of
f
is thus deferred by a
r
−
r
a
in its propagation to point
M
. Therefore,
u
period of
(
M
,
t
)
in Eq. (2.85) is called
the
retarded potential
.
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