Уравнения математической физики. Мицик М.Ф. - 12 стр.

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
c
, 
( )
ϕΦ

−
π
2
. , 
,....2,1,0,
2
=−= kk
. 
( )
ϕϕϕ kCk sincos
21
+=Φ
,

−
21
,C
.

( )
rR

( )
222
k
rR
Rr
r −=
+
′
′
−ν
,

0
1
2
2
2
=
−+
′
+
′′
R
r
k
R
r
R ν
(39)
 (39)  (35), .
( )
0
0
=rR
, ,
( )
rR
. 
( ) ( )
rJrR
k
ν=
0
,

−xJ
m
. 
 (39) 
( )
rN
k
ν
, 

0
=
r
.

k
, 
() ( )
0
00
== rJrR
k
ν
.
, 
0
rk

xJ
k
( )
( )
( )
,....2,1,0,....,3,2,1,0,
0
==== knJ
r
k
nk
k
n
µ
µ
ν
,
( )
( )
,.....2,1,0,....,3,2,1,
0
0
==
= knr
r
JrR
k
n
k
µ
 (37), 
:
( )
( )
( )
×
+=
00
,
sincos;;
r
at
B
r
at
Atru
k
n
k
n
kn
µµ
ϕ
( )
( )
+× r
r
JkCk
k
n
k
0
21
sincos
µ
ϕϕ
(40)
            c                                           ,                                           Φ(ϕ )
             2π −                   .                                                       ,
= −k ,
    2
             k = 0, 1, 2,.... .
                         Φ(ϕ ) = 1 cos kϕ + C2 sin kϕ ,
     1 , C2 −                             .
                    R(r )
                      ′        
               2 (rR ′)
           −r           + ν 2  = −k 2 ,
                 rR           

                                1       2 k2 
                           R′′ + R′ + ν − 2  R = 0                                              (39)
                                r         r 
                               (39)                                             (35),   .
                                      R (r0 ) = 0
,                , R(r )                                                    .
                                      R(r0 ) = J k (ν r ) ,
    J m (x) −                                 .
           (39)                                                  N k (ν r ) ,
      r=0                                                                                       .
                           k                                            ,
                                      R(r0 ) = J k (ν r0 ) = 0 .
                       ,        k r0                                                                 J k (x )
        µ n(k )
     ν=
         r0
                           ( )
                , J k µ n(k ) = 0, n = 1, 2, 3,...., k = 0,1, 2,....

                   ,
                               µ n(k ) 
                R(r0 ) = J k         r , n = 1, 2, 3,...., k = 0,1, 2,.....
                               0 
                                 r
                                                         (37),
                                                                                   :
                                            (k )                 (k )
                                  µ at              µ at 
      un ,k (r ;ϕ ; t ) =  A cos      + B sin
                                            n
                                                          ×    n

                                   r0                r0   
                                         µ n(k ) 
      × ( 1 cos kϕ + C2 sin kϕ )J k            r                                                (40)
                                         0 
                                           r

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