крутильные колебания штампа на слое сцепленном с

ufl
brsojxgrÀ¶ÃºÊ¾½Â¶ÈöÉÀoonbrlpdslb¾Ã¼
s¾ÂÊ»ÆÄÅÉÃÈ
lrutjm~o}glpmgcbojztbnqbobsmpg
sxgqmgoopnsqpmuqrpstrbostdpnspfgrhb{jnj
xjmjofrjygslu€qpmpst~
r»¶Á¾½Ä¸¶Ã¶Á¹ÄƾȺ¾Ã¶Â¾Í»ÇÀÄ¿½¶º¶Í¾r»¿Ç»Æ¶s¶¹Ä̾ºÁµÇÁĵÇÌ»Å
Á»ÃÃÄ¹Ä Ç ÅÄÁÉÅÆÄÇÈƶÃÇȸÄ Çĺ»Æ¼¶Ï¾Â¾ ̾Á¾ÃºÆ¾Í»ÇÀÉÔ ÅÄÁÄÇÈÒ º¶ÃÃÑ¿ ¸
>@ qƾ¸»º»ÃÑ Æ»½ÉÁÒȶÈÑ Í¾ÇÁ»ÃÃÄ¿ Æ»¶Á¾½¶Ì¾¾ ½¶º¶Í¾ qÆĸ»º»Ã ¶Ã¶Á¾½ ¸Á¾µ
þµ »˶þͻÇÀ¾Ë ¾ ¹»Ä»ÈƾͻÇÀ¾Ë Ŷƶ»ÈÆĸ ÇÁĵ ¾ ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ ö ¶Â
ÅÁ¾ÈɺÃÄͶÇÈÄÈÃÑ» ˶ƶÀȻƾÇȾÀ¾ ÀÄÁ»·¶Ã¾¿ ÎȶÂŶ qÆ»ºÁļ»ÃÑ ÅƶÀȾͻ
ÇÀ¾»ÅƾÁļ»Ã¾µÆ»½ÉÁÒȶÈĸºÁµÄ·»ÇŻͻþµÇ»¿ÇÂľ½ÄÁµÌ¾¾ÇÄÄÆɼ»Ã¾¿Ã¶Æ¶Ç
ÇÂÄÈÆ»ÃÃÄÂÄÇÃĸ¶Ã¾¾ÅƾöÁ¾Í¾¾ÀÆÉȾÁÒÃÑ˺¾Ã¶Â¾Í»ÇÀ¾Ë¸Ä½º»¿Çȸ¾¿
lÁ¶ÇǾͻÇÀ¶µ ½¶º¶Í¶ r»¿Çûƶs¶¹Ä̾ ¾ÇÇÁ»ºÄ¸¶Á¶ÇÒ ¸Å»Æ¸Ñ» ¸
>@ pºÃ¾Â ¾½ »» ķķϻþ¿ µ¸Áµ»Èǵ ƶÇÇÂÄÈÆ»Ãöµ ¸ >@ ½¶º¶Í¶ Ä
ÀÆÉȾÁÒÃÑË ÀÄÁ»·¶Ã¾µË ÎȶÂŶ ÇÌ»ÅÁ»ÃÃÄ¹Ä Ç ÂÃĹÄÇÁÄ¿ÃÑ ÉÅÆ É
¹¾ÂÄÇÃĸ¶Ã¾»ÂÇĺ»Æ¼¶Ï¾Â̾Á¾ÃºÆ¾Í»ÇÀÉÔÅÄÁÄÇÈÒdÓÈÄ¿½¶º¶Í»
ÇÁľ ÅÆ»ºÅÄÁ¶¹¶Á¾ÇÒ ÇÌ»ÅÁ»ÃÃѾ »¼ºÉ ÇÄ·Ä¿ ¶ ¹Æ¶Ã¾Ì¶ þ¼Ã»¹Ä
ÇÁĵ § Ç ¼»ÇÈÀ¾Â ÄÇÃĸ¶Ã¾»Â eƶþÌÑ Ì¾Á¾ÃºÆ¾Í»ÇÀÄ¿ ÅÄÁÄÇȾ ¾
¸»ÆËû¹ÄÇÁĵ¸Ã»ÎȶÂŶ §Ç¸Ä·ÄºÃÑÄÈöÅƵ¼»Ã¾¿lÄÁ»·¶Ã¾µÇÆ »
ºÑ ¸Ñ½¸¶ÃÑ ¸Ä½º»¿Çȸ¾»Â ö ÎȶÂÅ ¹¶ÆÂÄþͻÇÀÄ¹Ä ¸Ä ¸Æ»Â»Ã¾ ÀÆ É
ȾÁÒÃĹÄÂÄ»Ãȶ Re M 0 exp(iωt)ÄÈÃÄǾȻÁÒÃÄÄǾÅÄÁÄÇȾdÉÀ¶½¶ Ã
ÃÄ¿ ƶ·ÄÈ» ÅÄÁÉÍ»ÃÑ ¶Ã¶Á¾È¾Í»ÇÀ¾» ¸Ñƶ¼»Ã¾µ ºÁµ µºÆ¶ ¾ Åƶ¸Ä¿
ͶÇȾ ¾ÃÈ»¹Æ¶ÁÒÃÄ¹Ä Éƶ¸Ã»Ã¾µ vÆ»º¹ÄÁÒ¶ ¹Ä Æĺ¶ À ÀÄÈÄÆÄÂÉ
ǸĺµÈǵ ŶÆÃÑ» ¾ÃÈ»¹Æ¶ÁÒÃÑ» Éƶ¸Ã»Ã¾µ ÄÅÆ»º»ÁµÔϾ» Ç»ζÃÃÑ»
ÉÇÁĸ¾µÃ¶¹Æ¶Ã¾Ì»¸»ÆËû¹ÄÇÁĵy»Æ»½Æ»Î»Ã¾» ω (W) ¾ÃÈ»¹Æ¶ÁÒÃĹÄ
Éƶ¸Ã»Ã¾µ¸Ñƶ¼»ÃÑÈƶÃÇÊÄƶÃÈÑÅƻķƶ½Ä¸¶Ã¾¿d»·»Æ¶ÄÈÀÄ Ã
ȶÀÈÃÄ¹Ä Ã¶ÅƵ¼»Ã¾µ τ ( ) (ξ ) ¾ Ç»ϻþµ 8 ( ) (ξ ) ö ¸»ÆËû¿ ¹Æ¶Ã¾Ì»
ÂÃĹÄÇÁÄ¿ÃÄ¹Ä ÄÇÃĸ¶Ã¾µ o¶ ÄÇÃĸ» »Èĺ¶ öͶÁÒÃÑË Å¶Æ¶Â»ÈÆĸ
>@ÅÄÁÉÍ»ÃÑÆ»ÀÉÆ»ÃÈÃÑ»ÇÄÄÈÃÄλþµÅĽ¸ÄÁµÔϾ»¸Ñƶ½¾ÈÒ
˶ƶÀȻƾÇȾÀ¾ öÅƵ¼»ÃÃĺ»ÊÄƾÆĸ¶ÃÃÄ¹Ä ÇÄÇÈĵþµ § τ (ϕ) (U ]) M
]
8 ϕ( ) (U ] ) ÁԷĹÄÇÁĵ M §ÃÄ»ÆÇÁĵͻƻ½ τ ( ) (ξ ) M
dÇÁÉͶµËĺÃĹÄÇÁĵÈÄÁϾÃÑ K ¾ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ (K → ∞) ÀÄÈÄÆÑ» ¸ ͶÇÈÃÄÇȾ ÇÁ»ºÉÔÈ ¾½ ½¶º¶Í¾ ºÁµ ÂÃĹÄÇÁÄ¿ÃÄ¿ ÇÆ»ºÑ ¾
ƶÇÇÂÄÈÆ»ÃÃÑË Æ¶Ã»» ¸ >§@ ÅÄÁÉÍ»ÃÑ Í¾ÇÁ»ÃÃÑ» Æ»½ÉÁÒȶÈÑ ÅÆ» º
d º¶ÁÒû¿Î»Â ·Éº»Â ÅÄÁ¶¹¶ÈÒ
τ (ϕ) (U) = τ ( ) (U) ( )
8ϕ
M
]
M
M
( )
(U) = 8 (U)
M
Çȶ¸ÁµÔϾ» ÇÄ·Ä¿ ¶ÂÅÁ¾ÈɺÃÄͶÇÈÄÈÃÑ» ˶ƶÀȻƾÇȾÀ¾ byw À Ä
Á»·¶Ã¾¿ÎȶÂŶÀÄÃȶÀÈÃѻöÅƵ¼»Ã¾µÅĺÎȶÂÅľÇÆ»ºÃÔÔ½¶
Żƾĺ ÂÄÏÃÄÇÈÒ ¾½ÁÉͶ»ÂÉÔ ¸ ÇÆ»ºÉ d ÓÈ¾Ë ¼» ƶ·ÄÈ¶Ë ÄǸ»Ï»ÃÑ
»ÈĺÑǸ»º»Ã¾µ¸Ä½Ã¾À¶ÔϾËŶÆÃÑËÉƶ¸Ã»Ã¾¿Ç¸µ½¶ÃÃÑËǵºÆ¶Â¾
Åƻķƶ½Ä¸¶Ã¾¿d»·»Æ¶ÀÉƶ¸Ã»Ã¾ÔvÆ»º¹ÄÁÒ¶¶È¶À¼»ÀÄÃÈÉÆÃÑ»
Åƻķƶ½Ä¸¶Ã¾µ¸ÀÄÂÅÁ»ÀÇÃÄ¿ÅÁÄÇÀÄÇȾ¾ÃÈ»¹Æ¶ÁÒÃÑ˵º»ÆÄ·»ÇÅ »
;¸¶ÔϾ» ÓÊÊ»ÀȾ¸ÃÄÇÈÒ ¾Ë ¸Ñ;ÇÁ»Ã¾µ ¾ »º¾ÃÇȸ»ÃÃÄÇÈÒ Æ»Î»Ã¾µ
½¶º¶Í¾cÄÁ»»Í»ÂºÁµÄºÃĹÄÇÁĵÉÀ¶½¶Ãöµ½¶º¶Í¶ºÄ;ÇÁ»ÃÃÑËÆ »
½ÉÁÒȶÈĸ û ºÄ¸Äº¾Á¶ÇÒ t»Â û »û» ÅÆ»ºÇȶ¸Áµ»È ûÇÄÂûÃÃÑ¿
ÅƶÀȾͻÇÀ¾¿ ¾ÃÈ»Æ»Ç Æ»½ÉÁÒȶÈÑ Æ¶ÇÍ»Èĸ ȶÀÄ¿ ½¶º¶Í¾ ºÁµ º¸ÉË ¾
·ÄÁ»»ÇÁÄ»¸ÈÄǸµ½¶ÃÄÇÈ»ÂÍÈÄÀ¹»Ä»ÈƾͻÇÀ¾Â¾Ã»Æ̾ÄÃÃѾ
ÉÅÆɹ¾ÂǸĿÇȸ¶ÂÇÆ»ºÑÄÈÀÄÈÄÆÑ˽¶¸¾ÇµÈbywÀÄÁ»·¶Ã¾¿Îȶ Â
ŶÂÃĹÄÇÁÄ¿ÃÄÇÈÒ¸ÃÄǾȺÄÅÄÁþȻÁÒÃѻŶƶ»ÈÆѸÁ¾µÃ¾µ §Æ¶½
Á¾ÍÃÑ»ÀÄ·¾Ã¶Ì¾¾ÉÅÆɹ¾Ë¾¾Ã»Æ̾ÄÃÃÑËǸĿÇȸÇÁÄ»¸¶È¶À¼»¾Ë
ÈÄÁϾÃo¶ÄÇÃĸ¶Ã¾¾ÓÈĹÄÂļÃÄȶÀÅĺķƶÈÒ¸½¶¾ÂÃѻǸĿÇȸ¶
ÇÁÄ»¸ ÍÈÄ·Ñ Ä·»ÇŻ;ÈÒ ¸ ½¶º¶ÃÃÄ º¾¶Å¶½Äû ͶÇÈÄÈ ÄÈÇÉÈÇȸ¾» Æ »
½ÄöÃÇĸ ¾ ÇĽº¶ÈÒ È¶À¾Â ķƶ½Ä »ÇÈ»Çȸ»ÃÃÉÔ Ç¾ÇÈ»ÂÉ Ç»¿ÇÂľ½Ä
ÁµÌ¾¾
r¶ÇÇÂÄÈƾ ÇÁÉͶ¿ º¸ÉË ÇÁÄ»¸ Åƾͻ ÈÄÁϾÃÉ Ã¾¼Ã»¹Ä ÇÁĵ
ÉÇÈƻ¾ À ·»ÇÀÄûÍÃÄÇȾ ÈÄ ÇÄÄȸ»ÈÇÈ¸É»È ½¶º¶Í» Ä ÀÄÁ»·¶Ã¾µË
ÎȶÂŶÇÌ»ÅÁ»ÃÃĹÄÇÄÇÁÄ»ÂÁ»¼¶Ï»ÂöÅÄÁÉÅÆÄÇÈƶÃÇȸ»dÓÈÄÂ
ÇÁÉͶ» ¾ÃÈ»¹Æ¶ÁÒÃÄ» Éƶ¸Ã»Ã¾» vÆ»º¹ÄÁÒ¶ ¹Ä Æĺ¶ ¾ »¹Ä µºÆÄ ¾ ½
¸»ÇÈÃѾ¾Â»Ôȸ¾º >@
β
ω (W ) + ∫ ω (W ). (W V )GV = . (W β ) β = E − D ½º»ÇÒ E §Æ¶º¾ÉÇÎȶÂŶ D§Æ¶º¾ÉÇÅÄÁÄÇȾ
∞
. (W V ) = WV  ∫ ξ [) (ξ ) − ]- (ξ V )- (ξ W )Gξ ±


± Lπ ∑ 5HV ξ P [)P (ξ P ) − ]- (ξ P V ) - (ξ P W ) P

−
) (ξ ) = & ∞ (ξ )πDκ (ξ ) + ( ) (ξD )  

1 + ∆1WKη
C2\ ∞ (ξ ) =
∆1 WKη
µ κ (ξ )
η 1 = h1κ 1(ξ ) κ j(ξ ) = ξ 2 − kj2 ∆ =
µ κ (ξ )
?
¹º» kj = ω ρ j µ j M = § ÇÁÄ¿ M = § ÅÄÁÉÅÆÄÇÈƶÃÇÈ¸Ä ρ µ §
M
M
ÅÁÄÈÃÄÇÈÒ ¾ ÂĺÉÁÒ Çº¸¾¹¶ ÇÄÄȸ»ÈÇȸ»ÃÃÄ K § ÈÄÁϾö ÇÁĵ d - ([) + ( ) ([) §Ì¾Á¾ÃºÆ¾Í»ÇÀ¾»ÊÉÃÀ̾¾¹Ä¾¹ÄÆĺ¶ÃÉÁ»¸Ä¹Ä
¾ ¹Ä ÅÄƵºÀĸ d µºÆ» . (W V) § ¸ÈÄÆÄ» ÇÁ¶¹¶»ÂÄ» ÅƾÇÉÈÇÈ¸É»È ¸
ÇÁÉͶ»ÀĹº¶¸½¶¾ÂÃѻǸĿÇȸ¶ÇÁĵ¾ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶È¶ÀĸÑÍÈÄ
N < N ÈĹº¶ ö ÅÆÄ»¼ÉÈÀ» N < ξ < N ÊÉÃÀ̾µ ¾Â»»È ÀÄûÍÃÄ»
;ÇÁĸ»Ï»Çȸ»ÃÃÑËÅÄÁÔÇĸ ξ µ¸ÁµÔϾËǵ¸»Ï»Çȸ»ÃÃѾÃÉÁµÂ¾
ÈƶÃÇ̻ú»ÃÈÃĹÄÉƶ¸Ã»Ã¾µ WK η = −∆ ¸ÅÆÄȾ¸ÃÄÂÇÁÉͶ» N > N §
Éƶ¸Ã»Ã¾»¸»Ï»Çȸ»ÃÃÑËÀÄÆû¿Ã»¾Â»»È¾¸ÈÄÆÄ»ÇÁ¶¹¶»ÂÄ»¸¾ Ç
Í»½¶»ÈdŻƸÄÂÇÁÉͶ»¾ÃÈ»¹Æ¶ÁÅÄþ¶»Èǵ¸ÇÂÑÇÁ»lÄξ
nĺÉÁÒ ÀÄÂÅÁ»ÀÇÃÄ¿ ¶ÂÅÁ¾ÈÉºÑ É¹Á¶ ÅĸÄÆÄȶ ÎȶÂŶ ¶ÇÇÑ
P ¸Ñƶ¼¶»Èǵͻƻ½Æ»Î»Ã¾» ω (W) ¾ÃÈ»¹Æ¶ÁÒÃĹÄÉƶ¸Ã»Ã¾µÈ¶À¼»
P
À¶À¾¸Æ¶·ÄÈ¶Ë >§@¾¾Â»»È¸¾º
Φ =
(
Θ(Ω ) = M 0 πµ1b3
g
)
−1
Φ (Ω ) (1 − λ Re g )2 + (λ Im g )2
,
λ=
M Ω2
;
2π
¹º» 0 = P SE §Åƾ¸»º»ÃöµÂ¶ÇǶÎȶÂŶ S = E D Ω = ωE ρ µ §
Åƾ¸»º»Ãöµ ͶÇÈÄȶ ¸ÑÃɼº»ÃÃÑË ÀÄÁ»·¶Ã¾¿ ω § ̾ÀÁ¾Í»ÇÀ¶µ Ͷ Ç
ÈÄȶ
dÊÉÃÀ̾µ J = J(Ω) ¾Â»»ÈÇÂÑÇÁÀÄÂÅÁ»ÀÇÃÄ¿¶ÂÅÁ¾ÈɺÑÉ ¹
Á¶ÅĸÄÆÄȶ·»½ÑûÆ̾ÄÃÃĹÄÎȶÂŶ¾¸Ñƶ¼¶»Èǵ
β
2p3
g(Ω ) = 2
π
∫
sω (s)Y (s) ds − β Y (β )
0
β
∫
sω (s) ds − β
0
Y (s) =
∞
π
ξ −1Fm (ξ ) J0 (ξs) J0 (ξβ ) dξ ∫
20
y¾ÇÁ»ÃöµÆ»¶Á¾½¶Ì¾µ½¶º¶Í¾ jÃÈ»¹Æ¶ÁÒÃÄ»Éƶ¸Ã»Ã¾»½¶Â »
ÃÄ¿¾ÃÈ»¹Æ¶Á¶ÅÄÀ¸¶ºÆ¶ÈÉÆÃÄ¿ÊÄÆÂÉÁ»ÈƶŻ̾¿Ç¸Äº¾ÁÄÇÒÀÁ¾Ã» ¿
ÃÄ¿ ¶Á¹»·Æ¶¾Í»ÇÀÄ¿ ǾÇȻ» Éƶ¸Ã»Ã¾¿ s Ì»ÁÒÔ ÓÊÊ»ÀȾ¸ÃÄ¹Ä ¸ Ñ
;ÇÁ»Ã¾µ ¶ÈƾÌÑ Ç¾ÇÈ»ÂÑ ¾ ÊÉÃÀ̾¾ J(Ω) ¶ ȶÀ¼» Ä·»ÇŻͻþµ
»º¾ÃÇȸ»ÃÃÄÇȾ ƻλþµ ½¶º¶Í¾ ÍÈÄ Ç¸µ½¶ÃÄ Ç Ã»ÄºÃĽöÍÃÄÇÈÒÔ
ƶº¾À¶Áĸ κ j(ξ ) ¸ ÅÆĸĺ¾ÁÄÇÒ ÀÄÃÈÉÆÃÄ» Åƻķƶ½Ä¸¶Ã¾» ¾ÃÈ »
¹Æ¶Áĸ ¸ ÀÄÂÅÁ»ÀÇÃÄ¿ ÅÁÄÇÀÄÇȾ ζ cɺ»Â ƶÇǶÈƾ¸¶ÈÒ ÇÁ É
Ͷ¿ N > N ÈĹº¶¸¸ÑÍ»ÈÑÄÈÇÉÈÇȸÉÔÈr¶ÇÇÂÄÈƾ¾ÃÈ»¹Æ¶Á
qĺÑÃÈ»¹Æ¶ÁÒÃÄ» ¸Ñƶ¼»Ã¾» ÅÆ»ºÇȶ¸¾Â À¶À ÊÉÃÀÌ¾Ô ÀÄÂÅÁ»ÀÇÃÄ¿
Żƻ»ÃÃÄ¿ ζ = ξ + Lη ½¶Å¾Ç¶ÃÃÉÔ¸¸¾º»
5(ζ W V) = 5(ζ W V) + 5 (ζ W V)  J0 (ζ t) H 0(m ) (ζs) ,t ≤ s,
1
Rm (ζ ,t,s) = ζ [F (ζ ) − 1] 
(m = 1,2) 2
 J0 (ζ s) H 0(m ) (ζ t) ,t > s.
r¶½Á¾Í¾» ¸ ÅÆ»ºÇȶ¸Á»Ã¾¾ 5 (ζ W V ) (P = ) Ǹµ½¶ÃÄ Ç ÅÄÇÁ»ºÉÔÏ»¿
ÄÌ»ÃÀÄ¿ ¾ÃÈ»¹Æ¶Á¶ ¸ ÀÄ Â
ÅÁ»ÀÇÃÄ¿ ÅÁÄÇÀÄÇȾ ÅÄ Ä À
ÆɼÃÄÇȵ ·ÄÁÒÎÄ¹Ä Æ¶º¾ É
Ƕ vÉÃÀ̾µ ÄÅÆ»º»Á»Ã¶
ö Í»ÈÑÆ»ËÁ¾ÇÈÃÄ¿ ƾ¶à Ä
¸Ä¿ Åĸ»ÆËÃÄÇȾ fÁµ »» Å Ä
ÇÈÆĻþµÅÆĸĺµÈǵƶ½Æ»½Ñ
¸ÑËĺµÏ¾» ¾½ ÈÄÍ»À ¸»È¸Á »
þµ ζ = ±N º¸É½Ã¶ÍÃÑË
ÊÉÃÀ̾¿ κ 1,2 (ξ ) d»È¸¾ ƶº¾
À¶Áĸ κ (ξ ) ÇÁ»ºÉ»È ¸Ñ·¾
ƶÈÒ È¶À¾Â ķƶ½Ä ÍÈÄ·Ñ Ç
ÉÍ»ÈÄ ¸Æ»Â»ÃÃÄ¹Ä ÂÃļ¾È »
Áµ H ω ƻλþ» 8 (U ]) Åƾ
] → ∞ ɺĸÁ»È¸ÄƵÁÄ ÉÇÁ Ä
¸¾Ô¾½ÁÉͻþµiÄ»ÆÊ»ÁÒº¶
>@»ÇÁ¾ < ξ < N ¾½¶ÈÉ˶
ÁÄ Ã¶ ·»ÇÀÄûÍÃÄÇȾ Åƾ
r¾Ç
ξ > k1,2 ÈÄ ¾Â»»È »ÇÈÄ Åƾ
Reκ 1,2 (ξ ) ≥ 0 Imκ j(ξ ) > 0 u;ÈѸ¶µÈ¶ÀÄ¿¸Ñ·ÄƸ»È¸»¿Æ¶½Æ»½ÑÌ »
Á»ÇÄķƶ½ÃÄÅÆĸĺ¾ÈÒ¸ºÄÁÒ Reκ 1,2 (ζ ) = 0 ¾¸Ñ·¾Æ¶ÈÒÁ¾ÇÈöÀÄÈ Ä
ÆÄ Reκ 1,2 (ζ ) > 0 fÁµÍ»ÈÀÄÇȾ¾Ã¶¹ÁµºÃÄÇȾÅÆĸ»º»Ã¾µÆ¶½Æ»½Ä¸¾
ÀÄÃÈÉÆĸ ¾ÇÅÄÁҽĸ¶Áǵ »Èĺ ÅÆ»º»ÁÒÃÄ¹Ä ÅĹÁÄϻþµ >@ È» N
½¶Â»ÃµÁ¾ÇÒö N = N − LN ′ ≤ N ′ << N P
L
M
M
W
M
r¶ÇÇÂÄÈƾÂÇÉÂÂɾÃÈ»¹Æ¶Áĸ
m =2
∑ ∫ Rm (ζ ) dζ
m =1Lm
m =2
= 2πi∑ (− 1)
m =1
5HV [Rm (ζ m )]
m −1
¹º»ÀÄÃÈÉÆÑ / ¾ÅÄÁÔǶ ζ = ζ ζ = ζ ÅÄÀ¶½¶ÃÑÃ¶Æ¾Ç ζ §ÀÄÂ
P
ÅÁ»ÀÇÃÑ¿ ÃÉÁÒ ÊÉÃÀ̾¾ H 2(2) (ζa) ¸ ŻƸÄ À¸¶ºÆ¶ÃÈ» qÄÇÁ» ÅÆ»º»Á Ò
ÃÑËŻƻËĺĸ ζ = ε → ζ = r → ∞ N ′ → ¾Åƻķƶ½Ä¸¶Ã¾¿ÅÄÁ É
;ºÁµµºÆ¶
k
∞
1  2
2i 1 2
K (t,s) ts=  ∫ ξ f1(ξ ,t,s) dξ − ∫ ξ f2 (ξ ,t,s) dξ +
a  0
π 0
k2

2i 2

+
ξ f3 (ξ ,t,s) dξ − π Re ζ 02f4 (ζ 0,t,s)  ; (s ≤ t) ,
∫
π k1

¹º»
I (ξ s)K 0 (ξ t) [π I2 (ξ a)C1(ξ ) + K 2 (ξ a)C2 (ξ )]
;
f1(ξ ,t,s) = 0
ξ ξ 2 + k12 K 2 (ξ a) π 2I22 (ξ a) + K 22 (ξ a)
[
f2 (ξ ,t,s) =
f3 (ξ ,t,s) =
]
J0 (ξ s)H 0(2) (ξ t)
ξ
k12
−ξ
H 2 (ξa)
(1)
2
J0 (ξ s)H 0(2) (ξ t)
ξκ 1(ξ ) H 2 (ξa)
(1)
2
[
2
]
C4 (ξ ) ;
C3 (ξ ) ;
J0 (ζ 0s)H 0(1) (ζ 0t)
f4 (ζ 0,t,s) =
C5 (ζ 0 ) ,
κ 1(ζ 0 )
½º»ÇÒ , ([) . ([) § Âĺ¾Ê¾Ì¾Æĸ¶ÃÃÑ» ÊÉÃÀ̾¾ c»ÇÇ»Áµ ÅÄƵºÀ¶ Q
Q
(Q = ) ¾
Q
(
)
(
)
~ (1 + WJ η~ ) (∆
~ + WJ η~ );
C (ξ ) = ∆
C (ξ ) = ∆ (WK η − 1) (∆ + WK η );
C (ξ ) = ∆ (1 + WJ η ) (∆ + WJ η );
~21 WJη~ ∆
~21 + WJ η~ ;
C1(ξ ) = 1 − ∆
2
1
3
1
4
1
2
1
1
C5 (ζ 0 ) = C2\ ∞ (ζ 0 ) ;
¹º»
2
1
2
1
2
1
a a a
∆ = δ δ η~1 = h1κ~1(ξ ) ∆ = δ δ η1 = h1κ 1(ξ )
δ~j = µ jκ~j(ξ ) δ j = µ jκ j(ξ ) κ~j(ξ ) = ξ 2 + kj2 κ j(ξ ) =
ξ 2 − kj2 böÁĹ¾ÍÃÑ»ÀÄÃÈÉÆÃÑ»Åƻķƶ½Ä¸¶Ã¾µºÁµÊÉÃÀ̾¾º¶ÔÈ
i1
π ∞
Y (s) = −
f1(ξ ,β ,s) dξ − ∫ f2 (ξ ,β ,s) dξ +
2a ∫0
a0
k
i2
π2
Re f4 (ζ 0,β ,s) ;
+ ∫ f3 (ξ ,β ,s) dξ −
a k1
2a
k
(s ≤ β )
nļÃÄÅÄÀ¶½¶ÈÒÍÈÄÅĺÑÃÈ»¹Æ¶ÁÒÃÑ»ÊÉÃÀ̾¾¸ §Ä¹
ƶþͻÃÑ ¸ ÃÉÁ» ¶ ¸ ¾ÃÈ»¹Æ¶Á¶Ë ö ÅÄÁÉ·»ÇÀÄûÍÃÑË ÅÆÄ»¼ÉÈÀ¶Ë
ÓÀÇÅÄûÃ̾¶ÁÒÃÄ É·Ñ¸¶ÔÈ Ã¶ ·»ÇÀÄûÍÃÄÇȾ t¶À¾Â ķƶ½Ä ÅÄÁ É
Í»ÃÃÑ»ÅÆ»ºÇȶ¸Á»Ã¾µ¾ÃÈ»¹Æ¶ÁĸºÁµ . (W V) ¾ < (V) §µ¸ÁµÔÈǵÓÊÊ»À
Ⱦ¸ÃѾºÁµ¸Ñ;ÇÁ»Ã¾µÃ¶dn
fÁµÍ¾ÇÁ»ÃÃÄ¿Æ»¶Á¾½¶Ì¾¾½¶º¶Í¾ÄÈŻƻ»ÃÃÑË W V ¾ ξ §Å»
Æ»Ëĺ¾Á¾ À ·»½Æ¶½Â»ÆÃÑ Żƻ»ÃÃÑ τ σ ¾ [ Çĸ»Æζµ ½¶Â»ÃÑ
V = βσ W = βτ ≤ σ τ ≤ ξ = N [ d¸Äº¾Á¾ÇÒ È¶À¼» ·»½Æ¶½Â»ÆÃÑ»
¸»Á¾Í¾ÃÑ ε = N D β = β D + = K E P = µ µ α = N N (α ≤ ) pÈ»ȾÂÍÈÄÅƾ α = ¾ P = ¸Ñƶ¼»Ã¾µºÁµµºÆ¶ . (W V) §
¾ºÁµ < (V) §ÅÄÁÃÄÇÈÒÔÇĸŶº¶ÔÈÇ ¶Ã¶ÁĹ¾ÍÃѾ ¾ÃÈ»¹Æ¶Á¶Â¾
½¶º¶Í¾ºÁµÅÄÁÉÅÆÄÇÈƶÃÇȸ¶Ç̾Á¾ÃºÆ¾Í»ÇÀÄ¿ÅÄÁÄÇÈÒÔ >@
y¾ÇÁ»ÃöµÆ»¶Á¾½¶Ì¾µ½¶º¶Í¾ÅÆĸĺ¾Á¶ÇÒ Ã¶ dn gs ö
µ½ÑÀ» vÄÆÈƶà ,9 r»Î»Ã¾» ¶ÅÅÆÄÀǾ¾ÆÉÔÏ»¿ ǾÇÈ»ÂÑ Á¾Ã»¿ÃÑË
¶Á¹»·Æ¶¾Í»ÇÀ¾Ë Éƶ¸Ã»Ã¾¿ ÄÇÉÏ»ÇȸÁµÁÄÇÒ Åƾ ÅÄÂÄϾ Çȶú¶ÆÈÃÄ¿
ÅĺÅÆĹƶÂÂÑ &*$86>@ÅÆ»ºÃ¶½Ã¶Í»ÃÃÄ¿ºÁµÇ¾ÇÈ»ÂÇÀÄÂÅÁ»ÀÇÃÄ¿
¶Èƾ̻¿Á»Â»ÃÈѶÈƾÌÑÅÆ»ºÇȶ¸ÁµÔϾ»¾ÃÈ»¹Æ¶ÁÑ ¸ËĺµÏ¾»
¸¸Ñ;ÇÁµÁ¾ÇÒ¶º¶ÅȾ¸ÃÄ¿ÅĺÅÆĹƶÂÂÄ¿ 48$1&>@¸ÇÁÉͶ»
¸»Ï»Çȸ»ÃÃÑËÅĺÑÃÈ»¹Æ¶ÁÒÃÑËÊÉÃÀ̾¿¶ºÁµ¾ÃÈ»¹Æ¾Æĸ¶Ã¾µ ÀÄÂ
ÅÁ»ÀÇÃĽöÍÃÑË ÊÉÃÀ̾¿ À¶À ¾ ¸ >@ § ÇĽº¶¸¶Áǵ »» ÀÄÂÅÁ»ÀÇÃÑ¿
¶Ã¶ÁĹfÁµ¸Ñ;ÇÁ»Ã¾µÌ¾Á¾ÃºÆ¾Í»ÇÀ¾ËÊÉÃÀ̾¿Åƾ¸Á»À¶Á¾ÇÒÇȶ Ã
º¶ÆÈÃÑ» ÅĺÅÆĹƶÂÂÑ >@ ¶ ƻλþ» ¾ÃÈ»¹Æ¶ÁÒÃÄ¹Ä Éƶ¸Ã»Ã¾µ ¾
¸Ñ;ÇÁ»Ã¾»¶ÂÅÁ¾ÈɺÀÄÁ»·¶Ã¾¿ÎȶÂŶÄÇÃĸѸ¶ÁÄÇÒö·¾·Á¾ÄÈ »
À» ÂĺÉÁ»¿ ÇĽº¶ÃÃÄ¿ ƶû» ¶¸ÈÄÆÄ ƶ·ÄÈÑ >@ ¾ ÅÆ»ºÃ¶½Ã¶Í»ÃÃÄ¿
ºÁµÆ»Î»Ã¾µ¶Ã¶ÁĹ¾ÍÃÑ˽¶º¶ÍºÁµÅÄÁÉÅÆÄÇÈƶÃÇȸ¶¾ÇÁĵÇÅÄÁ Ä
ÇÈÒÔ
r»½ÉÁÒȶÈÑ Æ¶ÇÍ»Èĸ qƾ¸»º»Â ¸Ã¶Í¶Á» Æ»½ÉÁÒȶÈÑ Ë¶Æ¶ÀÈ»Æ ¾
½ÉÔϾ» ¸Á¾µÃ¾» º¸ÉËÇÁÄ¿ÃÄÇȾ ÄÇÃĸ¶Ã¾µ ö ÀÄÁ»·¶Ã¾µ ÎȶÂŶ ¸
Çƶ¸Ã»Ã¾¾ Ç ¶Ã¶ÁĹ¾ÍÃÄ¿ ½¶º¶Í»¿ ºÁµ ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ Ç Ì¾Á¾ÃºÆ ¾
Í»ÇÀÄ¿ÅÄÁÄÇÈÒÔfÁµÓÈĹÄöƵºÉǹ»Ä»ÈƾͻÇÀ¾Â¾Ë¶Æ¶ÀȻƾÇÈ ¾
À¶Â¾ § ÄÈÃÄǾȻÁÒÃѾ ƶº¾ÉÇÄ ÎȶÂŶ ¾ ÈÄÁϾÃÄ¿ ÇÁĵ ¸¸»º»Â
Ŷƶ»ÈÆÑ ÄÅÆ»º»ÁµÔϾ» ʾ½¾Í»ÇÀ¾» ǸĿÇȸ¶ ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ ¾
ÇÁĵdÀ¶Í»Çȸ»Å¶Æ¶Â»ÈƶÄÅÆ»º»ÁµÔÏ»¹Äʾ½¾Í»ÇÀ¾»Ç¸Ä¿Çȸ¶ÇÆ»º
·Éº»Â ·Æ¶ÈÒ ¶ÀÉÇȾͻÇÀÉÔ ¼»ÇÈÀÄÇÈÒ § ρ µ ö½Ñ¸¶»ÂÉÔ È¶À¼»
M
M
¸ÄÁÃĸÑÂÇÄÅÆÄȾ ¸Á»Ã¾»ÂÇÆ»ºÑ>@
t¶·Á¾Ì¶ y¶ÇÈÄÈÃÑ»½¶¸¾Ç¾ÂÄÇȾÀÄÂÅÁ»ÀÇÃÄ¿¶ÂÅÁ¾ÈɺÑ
·»½ÑûÆ̾ÄÃÃĹÄÎȶÂŶ
+ =
S=
P = µ µ = α =
Ω
P = µ µ = 5H J π
,P J π
J π
5H J π
,P J π
J π
fÁµ¾½Éͻþµ¸Á¾µÃ¾µ¸½¶¾ÂÃÑËǸĿÇȸÇÁĵ¾ÅÄÁÉÅÆÄÇÈƶÃÇ È
¸¶ ö ÀÄÁ»·¶Ã¾µ ·Éº»Â ¾ÇËĺ¾ÈÒ ¾½ ƶ¸»ÃÇȸ¶ ¾Ë ¸ÄÁÃĸÑË Í¾Ç»Á §
(N = N ) ÍÈÄ Åƾ¸Äº¾È À ɺķÃÑ ºÁµ ƶÇÍ»Èĸ ¾ ÅÄÇÁ»ºÉÔÏ»¹Ä ¶Ã ¶
Á¾½¶ÇÄÄÈÃÄλþµÂ µ µ = ρ ρ = P r¶ÇÍ»ÈÑ Í¶ÇÈÄÈÃÑË ½¶¸¾Ç¾ÂÄÇÈ»¿ J(Ω) ¾ Φ(Ω) π ÅÆĸĺ¾Á¾ÇÒ
ºÁµ º¸ÉË ÇÁÉͶ»¸ ¶ P = µ µ = ÍÈÄ ÇÄÄȸ»ÈÇÈ¸É»È ÇÄÄÈÃÄλþÔ
ρµ < ρ µ È»ÇÁÄ¿¶ÀÉÇȾͻÇÀ¾·ÄÁ»»Âµ¹À¾¿Í»ÂÅÄÁÉÅÆÄÇÈƶ Ã
ÇÈ¸Ä ¾ · P = µ µ = ÈĹº¶ ρµ > ρ µ § ÇÁÄ¿ ¶ÀÉÇȾͻÇÀ¾
¼»ÇÈÍ»ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶dķľËÇÁÉͶµË¶¾·¹»Ä»ÈƾͻÇÀ¾»Å ¶
ƶ»ÈÆѷƶÁ¾ÇÒƶ¸ÃѾ + = S = dȶ·ÁÅƾ¸»º»Ãѽ¶¸¾Ç¾ÂÄÇȾº»¿Çȸ¾È»ÁÒÃÄ¿ 5H J ¾ÂþÂÄ¿ ,P J
ͶÇÈ»¿ÀÄÂÅÁ»ÀÇÃÄ¿¶ÂÅÁ¾ÈÉºÑ J(Ω) ɹÁ¶ÅĸÄÆÄȶ ·»½ÑûÆ̾ÄÃÃĹÄ
ÎȶÂŶÄÈÅƾ¸»º»ÃÃÄ¿¸ÑÃɼº»ÃÃĿͶÇÈÄÈÑÀÄÁ»·¶Ã¾¿¸º¾¶Å¶½Äû
< Ω < ºÁµÇÁÉͶ»¸¶¾·q»Æ»ËĺÄÈȶ·Á¾ÍÃÑ˽öͻþ¿ J(Ω)
À ûÅÆ»ÆѸÃÑ ºÁµ ÁÔ·Ä¿ ͶÇÈÄÈÑ ¾½ ƶÇǶÈƾ¸¶»ÂÄ¹Ä º¾¶Å¶½Äö
ÄÇÉÏ»ÇȸÁµÁǵ ÅÄÇÆ»ºÇȸÄ ¶ÅÅÆÄÀǾ¶̾¾ ÀÉ·¾Í»ÇÀ¾Â¾ ÇÅÁ¶¿Ã¶Â¾
sÄÄȸ»ÈÇȸÉÔϾ» ¹Æ¶Ê¾À¾ Åƾ¸»º»ÃÑ Ã¶ Æ¾Ç ¾½ ÀÄÈÄÆÄ¹Ä ¸¾ºÃÄ
ÍÈÄ 5H J ÄÇ̾ÁÁ¾ÆÉÔϾ» ÊÉÃÀ̾¾ Åƾͻ ÇÉÏ»ÇȸÉÔÈ ¾ÃȻƸ¶ÁÑ
ͶÇÈÄȺÁµÀÄÈÄÆÑËÊÉÃÀ̾¾ 5H J ¶Ã¶ËĺµÈǵ¸ÅÆÄȾ¸Äʶ½»·Ã ¶
ËĺµÈǵ¸Ê¶½»¸ÅÄÁÃÄÇÈÒÔÇĸŶº¶ÔÈ
pÈ»Ⱦ ÄÈÁ¾Í¾» Åĸ»º»Ã¾µ 5H J (Ω) ¸ º¶ÃÃÄ¿ ½¶º¶Í» Åƾ S = ÄÈ ¶Ã¶ÁĹ¾ÍÃÑË ½¶º¶Í ºÁµ ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ ¾ ĺÃÄ¹Ä ÇÁĵ Ç ÅÄÁ Ä
ÇÈÒÔ t¶À ¸ ½¶º¶Í» ºÁµ ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ >@ 5H J (Ω) µ¸Áµ»Èǵ ºÄÇȶ
ÈÄÍÃÄ ¹Á¶ºÀÄ¿ ÊÉÃÀ̾»¿ ɼ» ¸ ÇÁÉͶ» ÇÁĵ ö ûº»ÊÄƾÆÉ»ÂÄÂ Ä Ç
Ãĸ¶Ã¾¾>@Åĸ»º»Ã¾» 5H J (Ω) ½Ã¶Í¾È»ÁÒÃĽ¶¸¾Ç¾ÈÄÈÈÄÁϾÃÑÇÁĵ
Ãĸǻ¼»Ã»ÅƾÀ¶À¾Ë ÈÄÁϾöËÇÁĵû ÄÇ̾ÁÁ¾ÆÉ»ÈȶÀǾÁÒÃÄÀ¶À
¸ÇÁÉͶ»º¶ÃÃÄ¿½¶º¶Í¾
¸ º¶ÁÒû¿Î»Â ÇÁÉͶ¿ ¶ ·Éº»Â ö½Ñ¸¶ÈÒ § µ¹À¾¿ ÇÁÄ¿ ÇÁÉͶ¿ · § ¼»ÇÈÀ¾¿ ÇÁÄ¿
r¾Ç
vÉÃÀ̾¾ − ,P J(Ω) À¶À ÇÁ»ºÉ»È ¾½ Æ¾Ç ºÄÇȶÈÄÍÃÄ ¹Á¶ºÀ¾» ¾
ÂÄÃÄÈÄÃÃÄ ¸Ä½Æ¶ÇȶÔÈ t¶ÀÄ» ¼» Åĸ»º»Ã¾» ÓÈÄ¿ ÊÉÃÀ̾¾ ¾ ¸ ¶Ã¶Á Ä
¹¾ÍÃÑ˽¶º¶Í¶ËºÁµÅÄÁÉÅÆÄÇÈƶÃÇȸ¶¾ÇÁĵi¶Â»È¾ÂĺöÀÄÍÈĸ
½¶º¶Í»ºÁµÇÁĵöûº»ÊÄƾÆÉ»ÂÄÂÄÇÃĸ¶Ã¾¾ >@ÇÉÏ»ÇȸɻȽ¶Å ¾
ƶÔ϶µÍ¶ÇÈÄȶ Ω aZi þ¼»ÀÄÈÄÆÄ¿ ,P J (Ω) = ¾¸ÇÁĿû¾½ÁÉͶ» È
ǵÓûƹ¾µdöλÂÇÁÉͶ» ( S = + = ) ¾¸ÇÁÉͶ»ÅÄÁÉÅÆÄÇÈƶ Ã
Çȸ¶È¶ÀÄ»µ¸Á»Ã¾»Ã»¾Â»ÁÄ»Çȶ
o¶ Æ¾Ç Åƾ¸»º»ÃÑ byw ÀÄÁ»·¶Ã¾¿ ÎȶÂÅĸ ¶ÇÇÑ
0 = ÇÄÄȸ»ÈÇȸÉÔϾ»ÊÉÃÀ̾µÂ g(Ω ) öƾÇsƶ¸Ã ¾
¸¶µ ¾Ë Ç ¶Ã¶ÁĹ¾ÍÃѾ Æ»½ÉÁÒȶȶ¾ ½¶º¶Í¾ ºÁµ ÉÅÆÉ¹Ä¹Ä ÅÄÁÉÅÆ Ä
ÇÈƶÃÇȸ¶ Ç Ì¾Á¾ÃºÆ¾Í»ÇÀÄ¿ ÅÄÁÄÇÈÒÔ >@ ÂļÃÄ Çº»Á¶ÈÒ ÇÁ»ºÉÔϾ»
¸Ñ¸ÄºÑ ö ÇÆ»ºÃ¾Ë ͶÇÈÄÈ¶Ë Ω > ¼»ÇÈÀ¾¿ ÇÁÄ¿ ö ÅÄÁÉÅÆ Ä
ÇÈƶÃÇȸ» É»ÃÒζ»È Æ»½ÄöÃÇÃÑ» ¶ÂÅÁ¾ÈÉºÑ ÀÄÁ»·¶Ã¾¿ ÎȶÂŶ ¶
µ¹À¾¿§É¸»Á¾Í¾¸¶»Èo¶Ã¾½À¾ËͶÇÈÄÈ¶Ë §Ã¶Ä·ÄÆÄÈ
sÄÅÄÇȶ¸Áµµ¹Æ¶Ê¾À¾Ã¶Æ¾ÇºÁµÇÁÉͶ»¸¶¾·Â»¼ºÉÇÄ·Ä¿
Ä̻þ¸Á¾µÃ¾» ¶ÀÉÇȾͻÇÀ¾ËǸĿÇȸÇÁĵ¾ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶Ã¶º ¾
ö¾ͻÇÀ¾»Ë¶Æ¶ÀȻƾÇȾÀ¾ÀÄÁ»·¶Ã¾¿
o¶·ÄÁ»»¸ÑÇÄÀ¾ËͶÇÈÄÈ¶Ë Ω > µ¹ÀÄÂÉÇÁÄÔö¼»Ç È
ÀÄÂÅÄÁÉÅÆÄÇÈƶÃÇȸ»ÇÄÄȸ»ÈÇȸɻȷÄÁÒζµÍɸÇȸ¾È»ÁÒÃÄÇÈÒÈ»
ÇÅ»ÀÈƶÁÒÃÑ» Æ»½ÄöÃÇÃÑ» žÀ¾ ·ÄÁ»» ÄÇÈÆÑ» ¾ ¾Ë ·ÄÁÒλ ͻ ºÁµ
¼»ÇÈÀÄ¹Ä ÇÁĵ P = ÀÄÈÄÆÑ¿ ÉξƵ»È Á¾Ã¾¾ ÇÅ»ÀÈƶ ÉȈ Ò
ζµÅƾÓÈÄÂÆ»½ÄöÃÇÃÑ»¶ÂÅÁ¾ÈɺѸÈĸƻµÀ¶À¸ºÄ¸ÄÁÒÃÄÎ ¾
ÆÄÀÄ¿ÄÀÆ»ÇÃÄÇȾƻ½ÄöÃÇÃÑËͶÇÈÄȶÂÅÁ¾ÈɺÑɸ»Á ¾Í¾¸¶ÔÈǵ
o¶ þ½À¾Ë ͶÇÈÄÈ¶Ë ( < Ω < ) ¼»ÇÈÀ¾¿ ÇÁÄ¿ ö µ¹ÀÄÂ
ÅÄÁÉÅÆÄÇÈƶÃÇȸ» § ɸ»Á¾Í¾¸¶»È Æ»½ÄöÃÇÃÑ» ¶ÂÅÁ¾ÈÉºÑ ÍÈÄ ÇÄÄ È
¸»ÈÇÈ¸É»È ¸ ͶÇÈÃÄÇȾ ¾½¸»ÇÈÃÄÂÉ Æ»½ÉÁÒȶÈÉ ½¶º¶Í¾ Ä ÀÄÁ»·¶Ã¾µË
¸Ñ½¸¶ÃÃÑË º¾ÊƶÀ̾»¿ ºÁ¾ÃÃÑË ÉÅÆɹ¾Ë ¸ÄÁà Ǻ¸¾¹¶ ö ¶ÀÉÇȾͻÇÀ¾
¼»ÇÈÀĸÀÁÔͻþ¾¸Âµ¹ÀÄÂÅÄÁÉÅÆÄÇÈƶÃÇȸ» >@
r¾Ç
fÁµ ÄÌ»ÃÀ¾ ¸Á¾µÃ¾µ ¹»Ä»ÈƾͻÇÀ¾Ë ˶ƶÀȻƾÇȾÀ ÎȶÂŶ ¾
ÇÁĵ ö ÀÄÁ»·¶Ã¾µ ÅÄÁÉÍ»ÃÑ Í¶ÇÈÄÈÃÑ» ½¶¸¾Ç¾ÂÄÇȾ 5H J ,P J ºÁµ
ƶ½Á¾ÍÃÑË ÄÈÃÄǾȻÁÒÃÑË Æ¶º¾ÉÇĸ ÎȶÂŶ § S = ¾ S = + = α = P = Åƾ¸»º»ÃÃѻöƾÇsÄÄȸ»ÈÇȸÉÔϾ»bywºÁµÂ¶ÇÇ
ÎȶÂŶ 0 = ¾½Ä·Æ¶¼»ÃÑöƾǾ½ÀÄÈÄÆĹÄÄÍ»¸¾ºÃÄ
Ç É»ÃÒλþ»Â S Æ»½ÄöÃÇÃÑ» ͶÇÈÄÈÑ É»ÃÒζÔÈǵ ¶ ¶ÂÅÁ¾ÈɺÑ
¸Ä½Æ¶ÇȶÔÈÇÆÄÇÈĶÇÇÑÎȶÂŶƻ½ÄöÃÇÃÑ»¶ÂÅÁ¾ÈÉºÑ¸Ã¶Í ¶
Á» ¸Ä½Æ¶ÇȶÔÈ ¾ ÇȶÃĸµÈǵ ·ÄÁ»» ÄÇÈÆѾ ¶ ½¶È»Â ɷѸ¶ÔÈ Æ»½ Ä
öÃÇÃÑ» ͶÇÈÄÈÑ Åƾ ÓÈÄ ÂÄÃÄÈÄÃÃÄ É·Ñ¸¶ÔÈ i¶Â»È¾Â ÍÈÄ È¶ÀÄ¿
¼»À¶Í»Çȸ»ÃÃѿ˶ƶÀȻƾ»ÔȽ¶¸¾Ç¾ÂÄÇȾbywÄÈ S ¾ 0 ¸¶Ã¶
ÁĹ¾ÍÃÑ˽¶º¶Í¶ËºÁµÅÄÁÉÅÆÄÇÈƶÃÇȸ¶¾ÇÁĵÇÅÄÁÄÇÈÒÔ >@
o»·»½ÑÃȻƻÇÃÄ ÄÈ»ȾÈÒ ÍÈÄ Åƾ m = 0,5 H = 5 ÊÉÃÀ̾µ
§ ¹Á¶ºÀ¶µ ¾ ÅƶÀȾͻÇÀ¾ ÇĸŶº¶»È Ç ¶Ã¶ÁĹ¾ÍÃÄ¿
5H J(Ω) ºÁµ S = ºÁµ ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ Ç ÅÄÁÄÇÈÒÔ >@ ¸ ÈÄ ¸Æ»Âµ À¶À ºÁµ S = 5H J(Ω) § ǾÁÒÃÄ ÄÇ̾ÁÁ¾ÆÉ»È Æ¾Ç ÈÄ ÂļÃÄ Ä·ÐµÇþÈÒ ÅÄ
¸¾º¾ÂÄÂÉÈ»ÂÍÈÄÅƾŶƶ»ÈÆ» p ¶ÁĸÇƶ¸Ã»Ã¾¾ÇÈÄÁϾÃÄ¿
ÇÁĵ H p = 1,1 H = 5 Àƶ»¸Ä¿ ÓÊÊ»ÀÈ ÀÄÈÄÆÑ¿ À¶À ¾½¸»ÇÈÃÄ ÄÇ Ä
·»ÃÃÄ ÅÆĵ¸Áµ»Èǵ Åƾ ƶº¾ÉÇ» ÎȶÂŶ ·Á¾½ÀÄ À ƶº¾ÉÇÉ ÅÄÁÄÇȾ
p → 1 >@ Å嶸Áµ»È ¸Á¾µÃ¾» ÇÁĵ ÇÌ»ÅÁ»ÃÃÄ¹Ä Ç ÅÄÁÉÅÆÄÇÈƶÃÇ È
¸Ä d ÇÁÉͶ» ¼» ÀĹº¶ Ŷƶ»ÈÆ p Çƶ¸Ã¾Â Ç ÈÄÁϾÃÄ¿ ÇÁĵ H
p = 2 H = 5 Àƶ»¸Ä¿ ÓÊÊ»ÀÈ ÄÇÁ¶·»¸¶»È ¾ ÅƻķÁ¶º¶»È ¸Á¾µÃ¾»
ÇÁĵ ÊÉÃÀ̾µ 5H J(Ω) Åƾ ÓÈÄ À¶Í»Çȸ»ÃÃÄ ¾½Â»Ãµ»Èǵ
ÄÇ̾ÁÁ¾ÆÉ»È
sÀ¶¼»Â ¸ ½¶ÀÁÔͻþ» ÍÈÄ ºÁµ ½¶Æ¶Ã»» ¾½¸»ÇÈÃÑË Å¶Æ¶Â»ÈÆĸ
ÇÄÄÆɼ»Ã¾µÎȶÂŶ¸½¶º¶ÃÃÄÂͶÇÈÄÈÃĺ¾¶Å¶½ÄûöÄÇÃĸ¶Ã¾¾
ÅĺķÃÑËƶÇÍ»ÈĸÂļÃÄÇÅÆÄ»ÀȾÆĸ¶ÈÒº¸ÉËÇÁÄ¿ÃÑ¿ÊÉú¶Â»ÃÈÇ
ʾ½¾Í»ÇÀ¾Â¾ ˶ƶÀȻƾÇȾÀ¶Â¾ ρ j,µ j Ä·»ÇŻ;¸¶ÔϾ¾ »¹Ä Ç»¿ Ç
¾ͻÇÀÉÔ¾Á¾¸¾·Æ¶Ì¾ÄÃÃÉÔ·»½ÄŶÇÃÄÇÈÒ
sžÇÄÀ¾ÇÅÄÁҽĸ¶ÃÃÄ¿Á¾È»Æ¶ÈÉÆÑ
iÄ»ÆÊ»ÁÒº b f¾ÊʻƻÃ̾¶ÁÒÃÑ» Éƶ¸Ã»Ã¾µ ¸ ͶÇÈÃÑË ÅÆľ½¸ÄºÃÑË
ʾ½¾À¾§n§ Ç n¶Á¾Ìqsþ̻Æbr s»¿Ç¾ͻÇÀÄ» ÅÄÁ» ¸Ñ
½¸¶ÃÃÄ»¸ÄÁÃĿǺ¸¾¹¶¸ÅÄÁÉÅÆÄÇÈƶÃÇȸ»Ç¾ÃÄÆĺÃѸÀÁÔͻþ»Âö¹Æ¶Ã¾Ì»
f¾Ã¶Â¾Í»ÇÀ¾»Ç¾ÇÈ»Âѧ§dÑŧs n¶Á¾Ìqsþ̻Æbr
z»¸ÁµÀĸ € b p ƻλþ¾ ½¶º¶Í¾ r»¿Çûƶs¶¹Ä̾ ºÁµ ÅÄÁÉÅÆÄÇÈƶÃÇȸ¶ Ç Ì¾
Á¾ÃºÆ¾Í»ÇÀÄ¿ÅÄÁÄÇÈÒÔqƾÀÁ»˶þÀ¶§§t²§s§n¶
Á¾Ìqsþ̻ÆbrlÆÉȾÁÒÃÑ»ÀÄÁ»·¶Ã¾µÀÆɹĸĹÄÎȶÂŶöÉÅÆɹÄÂÅÄ
ÁÉÅÆÄÇÈƶÃÇȸ» ¾ ÇÁÄ» Ç Ì¾Á¾ÃºÆ¾Í»ÇÀÄ¿ ÅÄÁÄÇÈÒÔ s¾ÂÊeu § s¾ÂÊ»ÆÄÅÄÁÒ
§Ç§f»Å¸uÀÆojjotj²uÀj˼»lÆÉȾÁÒÃÑ»
ÀÄÁ»·¶Ã¾µ ÀÆÉ¹Ä¸Ä¹Ä ÎȶÂŶ ö ÉÅÆɹÄ ÇÁÄ» Ç Ì¾Á¾ÃºÆ¾Í»ÇÀÄ¿ ÅÄÁÄÇÈÒÔ f¾
ö¾ͻÇÀ¾»Ç¾ÇÈ»Âѧ§dÑŧs n¾Èƶrm¾s böÁ¾È¾Í»
ÇÀ¾»Â»ÈĺÑÈ»Äƾ¾¸ÄÁÃĸĺĸ§nn¾Æ§Çrµ·É˾ötbr»Î»
þ»Ç¾ÇÈ»ÂÁ¾Ã»¿ÃÑ˶Á¹»·Æ¶¾Í»ÇÀ¾ËÉƶ¸Ã»Ã¾¿ÇÀÄÂÅÁ»ÀÇÃѾÀÄÓÊʾ̾»Ãȶ¾
ÇNÅƶ¸Ñ¾ ͶÇȵ¾ »ÈĺÄ e¶ÉÇǶ n¶È»Â¶È¾Í»ÇÀÄ» Ä·»ÇŻͻþ» gs dn§
n¾ÃÇÀ§dÑŧss¶¸¶Æ»ÃÇÀ¾¿gvs»¿Ç¾ͻÇÀ¾»¸ÄÁÃѧn
o»ºÆ¶§Ç sþ̻Æbr i¶º¶Í¶r»¿Çûƶs¶¹Ä̾ ºÁµ ÂÃĹÄÇÁÄ¿ÃĹÄ
ÄÇÃĸ¶Ã¾µÇ̾Á¾ÃºÆ¾Í»ÇÀÄ¿ÅÄÁÄÇÈÒÔf¾Ã¶Â¾Í»ÇÀ¾»Ç¾ÇÈ»Âѧ§dÑÅ
§svÄÆǶ¿Èf¼n¶ÁÒÀÄÁÒÂnnÄÉÁ»Æln¶Î¾ÃÃÑ» »ÈÄºÑ Â¶È»
¶ȾͻÇÀ¾Ë¸Ñ;ÇÁ»Ã¾¿§nn¾Æ§Çz»¸ÁµÀĸ€bn¶ÈƾÍÃÑ»
¶Á¹ÄƾÈÂѸȻÄƾ¾ÉÅÆɹÄÇȾûĺÃÄÆĺÃÑËÇÆ»ºlpº»ÇǶÇn¶
Ȼ¶ȾͻÇÀĻķ»ÇŻͻþ»gsdndÑŧn¾ÃÇÀ5HLVV
QHU(6DJRFL+))RUFHGWRUVLQDORVFLOODWLRQRIHODVWLFKDOIVSDFH-$SSO3K\V
9RO²3 qÄÇÈÉžÁ¶¸Æ»ºÀÄÁÁ»¹¾Ô