Environmental Engineering Reference
In-Depth Information
Note that
B mnl β nl =
D mnl β nl =
A mnl ,
C mnl ,
α nl B mnl =
B mnl , α nl D mnl =
D mnl .
Thus
)= m , n , l { [( B mnl sin β nl t ) cos m ϕ
t W Φ (
r
, θ , ϕ ,
t
+(
D mnl sin
β nl t
)
sin m
ϕ ]+[(
A mnl cos
β nl t
)
cos m
ϕ
e α nl t P n (
+(
C mnl cos
β nl t
)
sin m
ϕ ] }
cos
θ )
j n (
k nl r
) .
(6.151)
Substituting Eqs. (6.149) and (6.151) into Eq. (6.148) and using the structure of
W ψ (
r
, θ , ϕ ,
t
)
in Eq. (6.146) thus yields the solution of PDS (6.147)
1
W Φ (
τ 0 +
B 2 W k nl Φ (
(
, θ , ϕ ,
)=
, θ , ϕ ,
)+
, θ , ϕ ,
) .
u
r
t
r
t
r
t
t
6.7.3 Solution from f
(
r
, θ , ϕ ,
t
)
Theorem 2. Let u
=
W ψ (
r
, θ , ϕ ,
t
)
be the solution of PDS (6.141). The solution of
u t
τ 0 +
B 2
A 2
u tt =
Δ
u
(
r
, θ , ϕ ,
t
)+
t Δ
u
(
r
, θ , ϕ ,
t
)
+
f
(
r
, θ , ϕ ,
t
) ,
0
<
r
<
a
,
0
< θ < π ,
0
< ϕ <
2
π ,
0
<
t
,
(6.152)
L
(
u
,
u r ) | r = a =
0
,
u
(
r
, θ , ϕ ,
0
)=
0
,
u t (
r
, θ , ϕ ,
0
)=
0
is
t
u
(
r
, θ , ϕ ,
t
)=
W f τ (
r
, θ , ϕ ,
t
τ )
d
τ ,
(6.153)
0
where f τ =
f
(
r
, θ , ϕ , τ )
. Thus Theorem 2 in Section 6.1 is also valid for spherical
coordinate systems.
 
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