Уравнения математической физики. Мицик М.Ф. - 11 стр.

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
=
=
.sin
;cos
ϕ
ϕ
ry
rx

( )
truu ;;ϕ=
,  (34)
:
( )
0;;
0
=tru ϕ
. (35)
,

( )
2
2
11
ϕ
u
r
ur
r
uu
r
ryyxx
′′
+
′
′
=
′′
+
′′
.
, 
:
( )
0
11
2
2
2
=
′′
+
′
′
−
′′
ϕ
u
r
ur
r
au
r
rtt
. (36)
, . 
 (36), 
 (35) :
( ) ( ) ( )
tTrRu ϕΦ=
. (37)

u
 (10)  (36), :
( )
0
1
2
2
=
Φ
Φ
′′
+
′
′
−
′′
rrR
Rr
a
T
T
.
, :
( )
2
2
22
1
, νν −=
Φ
Φ
′′
+
′
′
−=
′′
r
rR
Rr
a
T
T
, (38)

−
2
ν
.  (38) 
:
atBatAtT νν sincos +=
,
.  (38)

( )
+
′
′
−=
Φ
Φ
′′
22
ν
rR
Rr
r
,
, 
( )
c
rR
Rr
r =
+
′
′
−=
Φ
Φ
′′
22
, ν
.
                             x = r cos ϕ ;
                            
                             y = r sin ϕ .
                                                        u = u (r ;ϕ ; t ) ,              (34)
         :
                                       u (r0 ; ϕ ; t ) = 0 .                               (35)
                                                                                                  ,


                            u ′xx′ +u ′yy′ =
                                               1
                                                 (rur′ )r ′ + 12 uϕ′′2 .
                                               r              r
                ,
                                   :
                                        1         ′ 1        
                            utt′′ − a 2  (ru ′r )r + 2 uϕ′′2  = 0 .                      (36)
                                        r           r        
                                                                              ,   .
                                                                 (36),
       (35)                                    :
                            u = R(r )Φ(ϕ )T (t ) .                                         (37)
         u                        (10)                         (36),                        :
                                          ′
                            T ′′   2 (rR ′)    1 Φ′′ 
                                 −a         + 2      = 0.
                            T        rR     r Φ
                                                     
                                                         ,                        :
             T ′′
                  = − a 2ν 2 ,
                                            (rR′)′ +     1 Φ′′
                                                               = −ν 2 ,                    (38)
             T                                 rR       r Φ
                                                          2

ν2 −                                                     .                        (38)
                        :
                            T (t ) = A cosν at + B sinν at ,
                                                                       .                 (38)

                                            ′     
                            Φ ′′     2 (rR ′)
                                 = −r         +ν  ,
                                                 2

                            Φ          rR        
                    ,
                               ′     
             Φ ′′       2 (rR ′)
                  = , −r         +ν  = c .
                                    2

             Φ            rR        

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