Environmental Engineering Reference
In-Depth Information
−
a
that passes through the point
1
η
=
η
means
x
−
at
=
(
x
,
t
)
.
x
−
at
or
x
=
t
)
on the
Ox
t
-plane so it is a straight line that passes through the
x
−
a
(
t
−
with a slope of
1
a
.Onthe
Ox
t
-plane, the integral domain is thus an
isosceles triangle shown in Fig. 2.8.
Finally we obtain the solution of PDS (2.99)
point
(
x
,
t
)
t
d
t
x
+
a
(
t
−
t
)
x
1
2
a
x
,
t
)
d
x
,
u
(
x
,
t
)=
f
(
t
)
0
−
a
(
t
−
which is the same as Eq. (2.98).
Similarly we may also obtain the Kirchhoff formula of two or three dimen-
sional wave equations by using different methods.
2.8.4 Physical Meanings of the Poisson and Kirchhoff Formulas
Poisson Formula of Three-Dimensional Wave Equations
The Poisson formula (2.84) reveals that
u
(
M
,
t
)
depends only on initial disturbances
on
S
at
, and not on the initial disturbance outside
S
at
. For an initial
nonzero disturbance in a sphere
M
)
M
)
ϕ
(
and
ψ
(
Ω
of radius
R
,i.e.
=
M
∈
Ω
,
0
,
M
)
,
ψ
(
M
)=
ϕ
(
M
∈
Ω
,
=
0
,
0if
S
at
intersects with
generally
u
(
M
,
t
)
=
Ω
;Otherwise,
u
(
M
,
t
)=
0 (Fig. 2.9). Let
Fig. 2.8
Isosceles triangle
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