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‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
Žªâï¡àì{¤¥ª ¡àì, 2001, ’®¬ 3, ‚ë¯ã᪠4

“„Š 532.5

…˜…ˆ… …‘’€–ˆŽ€Ž‰ Š€…‚Ž‰ ‡€„€—ˆ
‚“’…ˆ• Œ€ƒˆ’Žƒˆ„Ž„ˆ€Œˆ—…‘Šˆ• ‚Ž‹ €
Ž‚…•Ž‘’ˆ €‡„…‹€ ‘‹Ž…‚ Ž‚Ž„Ÿ™…‰ †ˆ„ŠŽ‘’ˆ
‚. ƒ. ‘®§ ­®¢, ˆ. „. Œã§ ¥¢, . ‘. ˜ã¬ ª®¢

‚ áâ âì¥ ¯®áâ ¢«¥­  ¨ à¥è¥­  ­¥áâ æ¨®­ à­ ï ªà ¥¢ ï § ¤ ç  ® ¢­ãâ७­¨å ¬ £­¨â®£¨¤à®¤¨­ ¬¨ç¥áª¨å ¢®«­ å ­  ¯®¢¥àå­®á⨠ࠧ¤¥«  á«®¥¢ ¯à®¢®¤ï饩 ¦¨¤ª®á⨠¢
áªà¥é¥­­ëå í«¥ªâà¨ç¥áª®¬ ¨ ¬ £­¨â­®¬ ¯®«ïå.

‡ ¤ ç  ¯®áâ ¢«¥­  ¢ ¡¥§¨­¤ãªæ¨®­-

­®¬ ¨ «¨­¥©­®¬ ¯à¨¡«¨¦¥­¨¨ ¤«ï ¨¤¥ «ì­®© ­¥á¦¨¬ ¥¬®© ¦¨¤ª®áâ¨.

®áâ ¢«¥­­ ï

­ ç «ì­®-ªà ¥¢ ï § ¤ ç  à¥è¥­   ­ «¨â¨ç¥áª¨ ¯ã⥬ ¯à¨¬¥­¥­¨ï ¬¥â®¤®¢ ®¯¥à æ¨®­­®£® ¨áç¨á«¥­¨ï ¨ ¨­â¥£à «ì­ëå ¯à¥®¡à §®¢ ­¨© ”ãàì¥. ‚ ®¬ ¢¨¤¥ ¯®«ã祭® ãà ¢­¥­¨¥ ¢®«­®¢®© ¯®¢¥àå­®á⨠ࠧ¤¥«  á«®¥¢, ¯®§¢®«ïî饥 ®¯à¥¤¥«¨âì ªà¨â¨ç¥áª®¥ ¯®«®¦¥­¨¥, ¯à¨ ª®â®à®¬ ­¥ ¯à®¨á室¨â § å¢ â  áâà â¨ä¨æ¨à®¢ ­­®© ¦¨¤ª®á⨠¨§ ¤à㣮£®
á«®ï.

«¥ªâ஬ £­¨â­ë¥ á¯®á®¡ë ®¡®£ é¥­¨© â¥á­® á¢ï§ ­ë á ¬ £­¨â®£¨¤à®¤¨­ ¬¨ç¥áª¨¬¨ § ¤ ç ¬¨ ® á«®¨á⮬ â¥ç¥­¨¨ ¯à®¢®¤ï饩 ¦¨¤ª®á⨠¢ áªà¥é¥­­ëå í«¥ªâà¨ç¥áª®¬ ¨ ¬ £­¨â­®¬ ¯®«ïå. à¨ í⮬ ­¥®¡å®¤¨¬® ¯à¥¦¤¥ ¢á¥£®
®¯à¥¤¥«¨âì ªà¨â¨ç¥áª®¥ ¯®«®¦¥­¨¥ ¯®¢¥àå­®á⨠ࠧ¤¥« , â. ¥. â ª®¥ ¯®«®¦¥­¨¥, ¯à¨ ª®â®à®¬ ­¥ ¯à®¨á室¨â § å¢ â  ¦¨¤ª®á⨠¨§ ¤à㣨å á«®¥¢ [1{3].
‚ á«ãç ¥, ª®£¤  ¦¨¤ª®áâì § ¡¨à ¥âáï ¨§ ­¨¦­¥£® á«®ï, ªà¨â¨ç¥áª®¥ ¯®«®-

¢¥àå­¨¬ ¯®«®¦¥­¨¥¬
­¨¦­¨¬ ¯®«®¦¥­¨¥¬ (à¨á. 2).

¦¥­¨¥ ¯®¢¥àå­®á⨠ࠧ¤¥«  ­ §ë¢ ¥âáï
§ ¡®à¥ ¨§ ¢¥àå­¥£® á«®ï |

(à¨á. 1),   ¯à¨

x0z ç áâì ¯à®áâà ­á⢠, ®£à ­¨ç¥­­ ï ãá«®¢¨ï¬¨ 0 6 x 6 l, 0 6 z 6 H1 , ¯à¥¤áâ ¢«ï¥â ¢¥àå­¨© á«®© ­¥á¦¨¬ ¥¬®©
¯à®¢®¤ï饩 ¦¨¤ª®áâ¨, ¤à㣠ï ç áâì ¯à®áâà ­á⢠ | 0 6 x 6 l, ,H2 6 z 6 0 |
­¨¦­¨© á«®© (l | ¤«¨­  ¢ ­­ë, H1 ¨ H2 | £«ã¡¨­ë á«®¥¢, ®áì z | ­ ¯à ¢«¥­  ¢¢¥àå, ¯«®áª®áâì z = 0 ᮢ¬¥é¥­  á ¯®¢¥àå­®áâìî à §¤¥«  á«®¥¢). Ž¡  á«®ï
‚ ¯àאַ㣮«ì­®© á¨á⥬¥ ª®®à¤¨­ â

¦¨¤ª®á⨠¯®¬¥é¥­ë ¢ áªà¥é¥­­ëå ®¤­®à®¤­ëå í«¥ªâà¨ç¥áª®¬ ¨ ¬ £­¨â­®¬

¯®«ïå. ‘®§¤ ­­ ï í«¥ªâ஬ £­¨â­ë¬ ¯®«¥¬ ¯®­¤¥à®¬®â®à­ ï ᨫ  ­ ¯à ¢«¥­ 
¢¥à⨪ «ì­® ᢥàåã ¢­¨§ ¨ ᮧ¤ ¥â ¢®§¬®¦­®áâì £à ¢¨â æ¨®­­®£® ¢á¯«ë¢ ­¨ï
ç áâ¨æ ¯à¨¬¥á¨ ¨§ ­¨¦­¥£® á«®ï ¢ ¢¥àå­¨©. Žç¨é¥­­ë© ®â ¯à¨¬¥á¨ ­¨¦­¨©
á«®© ¦¨¤ª®á⨠ç¥à¥§ § ¡®à­®¥ ®ª­® ¢ë⥪ ¥â ¨§ ¢ ­­ë. ‡ ¡®à­®¥ ®ª­® ®£à ­¨ç¥­® ãá«®¢¨ï¬¨

x = 0, ,H2 6 z 6 ,H2 + h, £¤¥ h | ¢ëá®â  ®ª­ .

á®åà ­¥­¨ï ¯®áâ®ï­­ëå ã஢­¥© á«®¥¢ ¯®« £ ¥âáï, çâ® ¢ ­¨¦­¥¬ á«®¥ ¯à¨

„«ï

x=l

¯® ¢á¥© £«ã¡¨­¥ à §¬¥é¥­ë ¨áâ®ç­¨ª¨ á á㬬 à­®© ¬®é­®áâìî à ¢­®© à á室ã

c 2001 ‘®§ ­®¢ ‚. ƒ, Œã§ ¥¢ ˆ. „., ˜ã¬ ª®¢ . ‘.

¥è¥­¨¥ ­¥áâ æ¨®­ à­®© ªà ¥¢®© § ¤ ç¨

4{41

¦¨¤ª®á⨠ç¥à¥§ § ¡®à­®¥ ®ª­®. †¨¤ª®áâì áç¨â ¥âáï ¨¤¥ «ì­®©, ¤¢¨¦¥­¨¥ |
¡¥§¢¨åà¥¢ë¬ (¯®â¥­æ¨ «ì­ë¬).

H1

z
ρ1, σ 1

l

0

x

H2

ρ2, σ 2

h

Ðèñ. 1.
‚ ¡¥§¨­¤ãªæ¨®­­®¬ ¨ «¨­¥©­®¬ ¯à¨¡«¨¦¥­¨¨ áä®à¬ã«¨à®¢ ­­ ï ª®­â ªâ­ ï § ¤ ç  ¬ £­¨â­®© £¨¤à®¤¨­ ¬¨ª¨ ᢮¤¨âáï ª à¥è¥­¨î ¤¨ää¥à¥­æ¨ «ì­ëå
ãà ¢­¥­¨© ‹ ¯« á 

z

H1

ρ1, σ 1
0

x

H2

ρ2, σ 2

l
Ðèñ. 2.

@ 2 '1
@x2
@ 2 '2
@x2

@ 2 '1
@z 2
@ 2 '2
+
@z 2
+

6 z 6 H1;

(1)

, H2 6 z 6 0

(2)

=0

¯à¨ 0

=0

¯à¨

¯à¨ á«¥¤ãîé¨å ­ ç «ì­ëå ¨ £à ­¨ç­ëå ãá«®¢¨ïå:




@' 
@' 
'1 j =0 = 1  = 0; '2 j =0 = 2  = 0;
@t =0
@t =0



@'1 
@'1 
=
0
;
= 0;
@x  =0
@x  =


,V0 ¯à¨ , H2 6 z 6 ,H2 + h;
@'2 
= ,V (z ) =

@x =0
0
¯à¨ , H2 + h < z 6 0;

V0 h

@'2 
= ,V = ,
;

@x =
H2
t

t

t

x

x

l

x

l

(3)

t

x

(4)

l

(5)

4{42

‚. ƒ. ‘®§ ­®¢, ˆ. „. Œã§ ¥¢, . ‘. ˜ã¬ ª®¢

1



1

@ 2 '1
@t2

+

@'1
@t

1 2
B '1
1 0

+ 1B02 '1 + g11H1 = 0 ¯à¨ z = H1 ;
@'2
@'1
=

¯à¨ z = 0;
@z
@z

1
+ g1 @'
@z



= 2

@ 2 '2
@t2





@'2 
@z z =,H2

+

2 2
B '2
2 0

2
+ g2 @'
@z

(6)
(7)


= 0;

¯à¨ z = 0; (8)
(9)

£¤¥ ¯à¨­ïâë á«¥¤ãî騥 ®¡®§­ ç¥­¨ï: '1 (x; z; t) ¨ '2 (x; z; t) | ¯®â¥­æ¨ «ë
᪮à®á⥩ ¢ ¢¥àå­¥¬ ¨ ­¨¦­¥¬ á«®ïå ᮮ⢥âá⢥­­®, 1 ¨ 2 | ¯«®â­®áâ¨, 1
¨ 2 | í«¥ªâய஢®¤­®á⨠¢ ¢¥àå­¥¬ ¨ ­¨¦­¥¬ á«®ïå ¦¨¤ª®áâ¨,
g1

= g + 11 E0B0 ;

g2

= g + 22 E0B0;

(10)

= E | ­ ¯à殮­­®áâì í«¥ªâà¨ç¥áª®£® ¯®«ï, B0 = B | ¨­¤ãªæ¨ï ¬ £­¨â­®£® ¯®«ï.
Žâ­®á¨â¥«ì­® £à ­¨ç­®£® ãá«®¢¨ï (6) ®â¬¥â¨¬, çâ® ¢®«­®®¡à §®¢ ­¨¥ ­ 
᢮¡®¤­®© ¯®¢¥àå­®á⨠¢¥àå­¥£® á«®ï ­¥ ãç¨â뢠¥âáï.
‚®«­®¢ ï ¯®¢¥àå­®áâì à §¤¥«  á«®¥¢ ¯à¨ z = 0

E0

y

x

2
@'2
1,
( ) =  g ,1  g @'


@t
 g ,  g @t

 x; t

¨«¨

1 1
2 2 1 1
2
2
+ 2g 1,B0 1g '1 , 2 g 2,B0 1g '2
2
1
2
1

( )=

@ x; t
@t

2 2

(

)

@'1 x; z; t
@z






z

=0

=

(

)

@'2 x; z; t
@z






z

=0

(11)
:

(12)

„«ï ­¥¯à®¢®¤ï饩 ¦¨¤ª®á⨠íâ  § ¤ ç  ¯®áâ ¢«¥­  ¨ à¥è¥­  ¢ [4].
à¨áâã¯ ï ª à¥è¥­¨î ¯®áâ ¢«¥­­®© ­ ç «ì­®-ªà ¥¢®© § ¤ ç¨ (1){(9), ¯à¨¬¥­¨¬ ¨­â¥£à «ì­®¥ ¯à¥®¡à §®¢ ­¨¥ ‹ ¯« á  ®â­®á¨â¥«ì­® ¢à¥¬¥­¨ t.
~ (

)=

'1;2 x; z; p

+1

Z

0

'1;2 x; z; t e,pt dt:

(

)

(13)

‚ १ã«ìâ â¥ ¯à¥®¡à §®¢ ­¨ï (13) ¢ëà ¦¥­¨ï (1){(9) ¢ ¨§®¡à ¦¥­¨ïå § ¯¨èãâáï á«¥¤ãî騬 ®¡à §®¬

~'1 (x; z) = 0;
(14)

4{43
(15)

¥è¥­¨¥ ­¥áâ æ¨®­ à­®© ªà ¥¢®© § ¤ ç¨


~'2 (x; z) = 0;
@ '~1 
= 0;
@x x=0

V (z )
@ '~2 
=
,
;

@x x=0
p


1 p'~1 + 1 B02 '~1 +

@ '~

p + 1 B02 p '~1 + g1 1
1
@z



1



2



= 2

@ '~1 
= 0;
@x x=l

Vl
@ '~2 
=
,
;

@x x=l
p


g1 1 H1
=0
p
@ '~1 @ '~2
= @z
@z



(18)

¯à¨ z = 0;

(19)



@ '~2 
@z z=,H2


(17)

¯à¨ z = H1;


@ '~
p + 2 B02 p '~2 + g2 2
2
@z
2

(16)



¯à¨ z = 0; (20)

= 0:

(21)

à¨¬¥­¨¬ ª®­¥ç­®¥ ª®á¨­ãá-¯à¥®¡à §®¢ ­¨¥ ”ãàì¥ ®â­®á¨â¥«ì­® ¯¥à¥¬¥­®© x:
l
n
(22)
'~1;2;n (z ) = '~1;2 (x; z )cos x dx;
l
Z

d2 '~2;n
dz 2


'~1;0 (z )

d2 '~1;n
dz 2

0

, a2n '~1;n = 0 ¯à¨ 0 6 z 6 H1;

(23)

, a2n '~2;n = Vl (,p 1) , V p(z) ¯à¨ , H2 6 z 6 0;

(24)

=
,
g1
z =H1

n

H1 L

p(p + 1B1 0 )
2



'~1;n 

;

z =H1

= 0 (n = 1; 2; 3 : : : );

d'~1;n d'~2;n
= dz ¯à¨ z = 0;
dz


1 2 
d'~1;n

2
1 p + B0 p '~1;n +g1
1
dz


d'~2;n
2 2 

2
=2 p +  B0 p '~2;n + g2 dz
2
d'~2;n 
dz z=,H2


= 0:

(25)
(26)

¯à¨ z = 0;

(27)
(28)

4{44

‚. ƒ. ‘®§ ­®¢, ˆ. „. Œã§ ¥¢, . ‘. ˜ã¬ ª®¢

¥è¥­¨ï ¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨© (23) ¨ (24) á £à ­¨ç­ë¬¨ ãá«®¢¨ï¬¨ (25) ¨ (28) ¨¬¥îâ á«¥¤ãî騩 ¢¨¤:

'~1;n (z ) = c~1;n sh (an (z , H1 ));
'~2;n (z ) = c~2;n ch (an (z + H2 )) +
®áâ®ï­­ë¥

c

c

Zz 

1

an

H1

Vl (,1)n
p

(29)

, V p( )



a z ,  )) d:

sh ( n (

(30)

c1 ¨ c2 ®¯à¥¤¥«ïîâáï ¨§ £à ­¨ç­ëå ãá«®¢¨© (26) ¨ (27).

„«ï ~1;n ¨ ~2;n ¯®«ãç îâáï á«¥¤ãî騥 ¢ëà ¦¥­¨ï:

c~1;n =

2 p(p



+ 2
2

B

2 1
0 ) an

R0



,H2

,

Vl ( 1)n
p

,



V ()
p

a H2 +  )) d

ch ( n (



2 ch (an H1 ) ch (an H2 ) + 1 sh (an H2 ) sh (an H1 ) (p

2

+

sp + q )

;

2 B02 ch (an H1 ) ch (an H2 ) + 1 B02 sh (an H2 ) sh (an H1 )
;
2 ch (an H1 ) ch (an H2 ) + 1 sh (an H1 ) sh (an H2 )
(2 g2 , 1 g1 )an sh (an H2 ) ch (an H1 )
;
q=
2 ch (an H1 ) ch (an H2 ) + 1 sh (an H1 ) sh (an H2 )

s=

c~2;n =



a H1 )

ch ( n




, 2p p + 2 B02 a1
2
n


Z0


Vl (,1)n

+ 2 g2
p
,H2

, gan ch (an H1 )
1





1

an

Z0 

,H2

V ( ) 
, p ch (an  ) d
Z0 

H2

Vl
p



Vl (,1)n
p

, V p( )



(31)

(32)

a  d

sh ( n )

 

1 2 
+ 1 p p +
B sh (an H1 )
1 0


, V p( ) ch (an  ) d



1



2 ch (an H1 ) ch (an H2 ) + 1 sh (an H1 ) sh (an H2 ) (p2 + sp + q )

:
(33)

‚ १ã«ìâ â¥ ¯à¨¬¥­¨¬ëå ¨­â¥£à «ì­ëå ¯à¥®¡à §®¢ ­¨© (13) ¨ (22) ¢ëà ¦¥­¨¥ (12) ¯à¨­¨¬ ¥â ¢¨¤



p~n =

d'~1;n 
:
dz z=0

(34)

®¤áâ ¢¨¢ ¢ëà ¦¥­¨ï (29) ¨ (31) ¢ (34), ¯®«ã稬



~n = 

a H1 )2 (p + 22 B02 )n

;
2 ch (an H1 ) ch (an H2 ) + 1 sh (an H1 ) sh (an H2 ) p(p2 + sp + q )
ch ( n

(35)

¥è¥­¨¥ ­¥áâ æ¨®­ à­®© ªà ¥¢®© § ¤ ç¨

4{45

£¤¥

0 = 0; n =

, V

( 1)n l

an

V0
sh (an h)
an

a H2 ) ,

sh ( n

(

n = 1; 2; 3 : : : ):

(36)



„«ï ~n ®¡à â­®¥ ª®á¨­ãá ¨­â¥£à «ì­®¥ ¯à¥®¡à §®¢ ­¨¥ ”ãàì¥ ¨¬¥¥â á«¥¤ãî騩 ¢¨¤

~(x; p) =

1
X

2

l

n=1

n (p)cosan x:

„«ï ­ å®¦¤¥­¨ï ®à¨£¨­ «  ä㭪樨

(37)

 (x; t) ¤®áâ â®ç­® ¨á¯®«ì§®¢ âì â ¡-

«¨æë ®¯¥à æ¨®­­®£® ¨áç¨á«¥­¨ï.
“à ¢­¥­¨¥ ¢®«­®¢®© ¯®¢¥àå­®á⨠ࠧ¤¥«  á«®¥¢ ¯®«ãç ¥âáï ¢ á«¥¤ãî饬
¢¨¤¥

 (x; t) =
£¤¥

n (t) =

1
X

2

l

n=1

n (t)cos

n
x;
l

(38)

r

2
qa H1s)2 2n e, s2 tsin q , s4 t
dn q , 4

ch ( n

s sin

,e, 2s t 2

q

q,

s2 t +

q

q,
q

4

s2 cos

q

4

q,

s2 t
4

q
+

1
q,
A;
s2

q

(39)

4

dn = 2 ch (an H1 ) ch (an H2 ) + 1 sh (an H1 ) sh (an H2 ):

(40)

‘ â®çª¨ §à¥­¨ï ॠ«¨§ æ¨¨ ­  ‚Œ 楫¥á®®¡à §­® ¢ëà ¦¥­¨î (39) ¯à¨¤ âì á«¥¤ãîéãî ä®à¬ã:

 (x; t) =

r
1 (,1)n h th (an H2 ) , sh (an h)
s
V0 X
n
H2
ch (an H2 )
, 2 t sin qn , s2n t
q
e
2
l n=1
4
d q , sn a

2

,e, s2n t
£¤¥

q

sn sin
2

n

n

q

4

n

q

qn , s4n t + qn , s4n cos qn , s4n t
+
qn
2

dn = 1 +
qn = g2 an

2

2

q

1

qn , s4n
A;
qn
2

(41)

1
th (an H1 ) th (an H2 );
2



1



, 21 gg21 th (an H2)


1 + 1 th ( n
2



a H1 ) th (an H2 )

;

(42)

4{46

‚. ƒ. ‘®§ ­®¢, ˆ. „. Œã§ ¥¢, . ‘. ˜ã¬ ª®¢

sn

=

th (an H1 ) =
sh (an h)

ch (an H2 )

2 B02
2
1






1 + 1 th (an H1 ) th (an H2 )
2
;

1 + 1 th (an H1 ) th (an H2 )
2

, e,2an H1 ;

1 + e,2an H1
=

th (an H2 ) =

1

, e,2an H2 ;

1 + e,2an H2

e,an (H2 ,h) , e,an (H2 +h)
;
1 + e,2an H2

h < H2 :

‹¨â¥à âãà 
1. ®¢å ˆ. ‹. ’¥å­¨ç¥áª ï £¨¤à®¬¥å ­¨ª .|‹.: Œ è¨­®áâ஥­¨¥, 1976.|501 á.
2. ®¢å ˆ. ‹, Š ¯ãáâ  €. , —¥ª¨­ . ‚. Œ £­¨â­ ï £¨¤à®¤¨­ ¬¨ª  ¢ ¬¥â ««ãࣨ¨.|Œ.:
Œ¥â ««ãࣨï, 1974.|240 á.
3.

‘¯à ¢®ç­¨ª ¯® £¨¤à ¢«¨ª¥ ¯®¤ । ªæ¨¥© ‚. €. ®«ìè ª®¢ .|Š¨¥¢:

‚¨é  誮« ,

1977.|278 á.
4. ‘®§ ­®¢ ‚. ƒ, Œã§ ¥¢ ˆ. „, ’ã ¥¢  †. „.,

Œã§ ¥¢  ’. ‚.

®áâ ­®¢ª  ¨ à¥è¥­¨¥

­ ç «ì­®-ªà ¥¢®© § ¤ ç¨ ¢­ãâ७­¨å ¢®«­ ¯à¨ ᥫ¥ªâ¨¢­®¬ ¢®¤®§ ¡®à¥ ¨§ áâà â¨ä¨æ¨à®¢ ­­®£® ¢®¤®¥¬  // ˆ§¢.

¢ã§®¢.

‘¥¢¥à®-Š ¢ª §áª¨© ॣ¨®­, ¥áâ¥á⢥­­ë¥ ­ ãª¨.

®á⮢-­ -„®­ã.|2001.|ü 1.|‘. 104{106.

£. ‚« ¤¨ª ¢ª §

‘â âìï ¯®áâ㯨«  26 ᥭâï¡àï 2001