352672:1,.,1,(67(52:$1(,32:,(/$&=(
1$3, &,$



2*Ï/1$&+$5$.7(5<67<.$8.à$'Ï:

352672:1,&=<&+

=DVLODQLH XU]G]H HOHNWU\F]Q\FK Z HQHUJL  RGE\ZD VL  ]D

SRUHGQLFWZHP SUGX ]PLHQQHJR OXE SUGX VWDáHJR : W\P RVWDWQLP

SU]\SDGNXHQHUJLDPR*HE\üGRVWDUF]RQD]EDWHULL]DVLODMFHMOXEXNáDGX

SURVWRZQLF]HJRSU]HWZDU]DMFHJRHQHUJL SUGX]PLHQQHJRQDHQHUJL

SUGXVWDáHJRR*GDQHMZDUWRFLQDSL FLDLSUGX

3URFHV SURVWRZDQLD MDN ND*G\ SURFHV SU]HWZDU]DQLD HQHUJLL ]

SU]HPLDQ F] VWRWOLZRFL Z\PDJD X*\FLD HOHPHQWyZ QLHOLQLRZ\FK

VSHáQLDMF\FK IXQNFMH MHGQRNLHUXQNRZ\FK ]DZRUyZ SUGRZ\FK 3U]\

X*\FLXGLRGUHDOL]XMHVL XNáDG\SURVWRZQLF]HQLHVWHURZDQHZNWyU\FK

ZDUWRü VWDáHJR QDSL FLD OXE SUGX Z\MFLRZHJR MHVW XVWDORQD L QLH

SRGOHJD UHJXODFML =D SRPRF W\U\VWRUyZ PR*QD ]UHDOL]RZDü XNáDG\

SURVWRZQLF]H VWHURZDQH XPR*OLZLDMFH FLJá UHJXODFM  QDSL FLD OXE

SUGXZ\MFLRZHJRVWRVRZDQHQDMF] FLHMZXNáDGDFKUHJXODFMLPRF\

8NáDG\ SURVWRZQLF]H REHMPXM V]HURNL ]DNUHV ]DVWRVRZD Z ]DNUHVLH

PRF\RGNLONXZDWyZGRW\VL F\N:

8NáDG SURVWRZQLF]\ VNáDGD VL  ]D]Z\F]DM ] WUDQVIRUPDWRUD RUD]

GLRG SURVWRZQLF]\FK SHáQLF\FK URO  MHGQRNLHUXQNRZ\FK ]DZRUyZ

SUGRZ\FK ']L NL MHGQRNLHUXQNRZHPX SU]HZRG]HQLX SUGX SU]H]

HOHPHQW\ SURVWRZQLF]H ZHMFLRZ\ SUG SU]HPLHQQ\ R VNáDGRZHM

UHGQLHM UyZQHM ]HUX MHVW Z XNáDG]LH SURVWRZQLF]\P SU]HWZDU]DQ\ QD

W WQLF\SUGMHGQRNLHUXQNRZ\RVNáDGRZHMUHGQLHMUy*QHMRG]HUD3R

RGILOWURZDQLXW WQLHZXNáDG]LHILOWUDF\MQ\PX]\VNXMHVL ZRGELRUQLNX

HQHUJLL*GDQZDUWRüSUGXLQDSL FLD

: ]DOH*QRFL RG OLF]E\ ID] ]PLHQQHJR QDSL FLD ]DVLODMFHJR

Z\Uy*QLDP\XNáDG\SURVWRZQLF]HMHGQRID]RZHLZLHORID]RZH

: XNáDG]LH MHGQRSRáyZNRZ\P QDSL FLH MHVW SURVWRZDQH Z FLJX

MHGQHJRSyáRNUHVXND*GHM]ID]QDWRPLDVWZXNáDG]LHGZXSRáyZNRZ\P

ZREXSyáRNUHVDFK=DWHPGZXSRáyZNRZ\VSRVyESURVWRZDQLDPR*QD

XZD*Dü ]D Z\QLN GZXNURWQHJR ]ZL NV]HQLD OLF]E\ ID] 'ODWHJR WH*

XNáDGMHGQRID]RZ\GZXSRáyZNRZ\E G]LHUR]ZD*DQ\MDNRGZXID]RZ\

: ]DOH*QRFL RG URG]DMX LPSHGDQFML REFL*HQLD ]PLHQLD VL  NW

SU]HSá\ZX SU]H] HOHPHQW SURVWRZQLF]\ : XNáDGDFK MHGQR L

GZXID]RZ\FK ] REFL*HQLHP UH]\VWDQF\MQ\P NW SU]HSá\ZX SUGX



Z\QRVL 2

Θ =

π

 ] REFL*HQLHP SRMHPQRFLRZ\P 2

Θ <

π



D]REFL*HQLHPLQGXNF\MQ\P2

Θ >

π



:áDVQRFLSURVWRZQLNyZRSLVXMQDVW SXMFHSDUDPHWU\

−

ZVSyáF]\QQLNZ\NRU]\VWDQLDQDSL FLRZHJRVSUDZQRüQDSL FLRZD

VNXWHF]QRüSURVWRZDQLD

η

X

2

VP

8

8

=



JG]LH

U

O

VNáDGRZDVWDáDQDSL FLDZ\MFLRZHJR



8

VP

DPSOLWXGD]PLHQQHJRQDSL FLDZHMFLRZHJR

:VSyáF]\QQLN Z\NRU]\VWDQLD QDSL FLRZHJR

η

u

 ]DOH*\ RG VNáDGRZHM

VWDáHMSUGXREFL*HQLDSRSU]H]]DOH*QRüQDSL FLDZ\MFLRZHJR

U

O

MDN

WRSRND]XMFKDUDNWHU\VW\NLREFL*HQLRZHSU]HGVWDZLRQHQDU\V

O

U

1

P

2

P

O

I

5\V&KDUDNWHU\VW\NLREFL*HQLRZH

SURVWRZQLNyZ

P

1

]REFL*HQLHPUH]\VWDQF\MQ\P

P

2

]REFL*HQLHPSRMHPQRFLRZ\P

:VSyáF]\QQLNZ\NRU]\VWDQLDQDSL FLRZHJR

η

u

]DOH*\RGURG]DMX

REFL*HQLDSU]\F]\PMHVWZL NV]\SU]\PDá\FKSUGDFKREFL*HQLD

−

Uy*QLF]NRZDUH]\VWDQFMDZ\MFLRZD

r

dU

dI

o

O

O

= −



5H]\VWDQFMDWDPR*HE\üZ\]QDF]RQDQDSRGVWDZLHFKDUDNWHU\VW\NL

REFL*HQLRZHM

−

ZVSyáF]\QQLNW WQLH

k

U

U

t

m

O

=

1



JG]LH

U

m

1

DPSOLWXGDVNáDGRZHMSRGVWDZRZHMW WQLHRSXOVDFMLm

ω



=DOH*QRü ZVSyáF]\QQLND W WQLH RG SUGX REFL*HQLD SU]HGVWDZLD

U\V



t

k

O

I

3



3



3



5\V=DOH*QRüZVSyáF]\QQLND

W WQLH k

t

RGSUGXREFL*HQLD I

O

SURVWRZQLND

P

1

]REFL*HQLHPUH]\VWDQF\MQ\P

P

2

]REFL*HQLHPSRMHPQRFLRZ\P

P

3

]REFL*HQLHPLQGXNF\MQ\P

3URVWRZQLN ] REFL*HQLHP SRMHPQRFLRZ\P FKDUDNWHU\]XMH VL

PDáZDUWRFLZVSyáF]\QQLNDW WQLHZ]DNUHVLHPDá\FKSUGyZ

REFL*HQLD  QDWRPLDVW ] REFL*HQLHP LQGXNF\MQ\P Z ]DNUHVLH GX*\FK

SUGyZ

−

ZVSyáF]\QQLNNV]WDáWXSUGX

k

m I

I

i

O

=

max





-HVW WR VWRVXQHN ZDUWRFL PDNV\PDOQHM

I

max

 GR VNáDGRZHM UHGQLHM

I

m

O

/

 SUGX Sá\QFHJR SU]H] ND*G ID]  :VSyáF]\QQLN WHQ RNUHOD

Z\NRU]\VWDQLH SUGRZH HOHPHQWyZ SURVWRZQLF]\FK L Z\PDJDQLD

RGQRQLHLFKPDNV\PDOQHJRSUGX

−

ZVSyáF]\QQLNZ\NRU]\VWDQLDSUGRZHJRVSUDZQRüSUGRZD

η

i

O

s

I

I

=



-HVWWRVWRVXQHNZDUWRFLUHGQLHM

I

m

O

/

SUGXID]RZHJRGRZDUWRFL

VNXWHF]QHM

I

s

WHJRSUGX

−

VSUDZQRüHQHUJHW\F]QD

η

η η

p

O

s

O

O

s

s

O

O

sm

s

u

i

P

m P

U I

m U I

U I

m U

I

=

=

=

=

2

2



-HVW WR VWRVXQHN PRF\ Z\MFLRZHM

P

U I

O

O

O

=

 SUGX VWDáHJR GR

PRF\ SR]RUQHM

m P

m U I

s

s

s

=

 REFL*DMFHM FDáH X]ZRMHQLH ZWyUQH

WUDQVIRUPDWRUD

−

QDSL FLH ]ZURWQH

U

R

  PDNV\PDOQH QDSL FLH QD HOHPHQFLH

SURVWRZQLF]\PZVWDQLHQLHSU]HZRG]HQLD

−

SUG XGDURZ\

I

u max

  PDNV\PDOQ\ SUG Sá\QF\ SU]H] HOHPHQW

SURVWRZQLF]\SRZáF]HQLXXNáDGX





8.à$'<=2%&,)(1,(05(=<67$1&<-1<0

 8NáDGMHGQRID]RZ\

1DMSURVWV]\PSU]\NáDGHPSURVWRZQLNDMHVWXNáDGMHGQRID]RZ\



m

=

1

]REFL*HQLHPUH]\VWDQF\MQ\PU\V

S

u

O

i

O

u

L

R

S

R

D

S

O

u

i ,

t

ω

max

I

Θ

2

π

π

2

O

i

8

VP

X

6

u

S

5\V-HGQRID]RZ\XNáDGSURVWRZQLF]\]REFL*HQLHPUH]\VWDQF\MQ\P

=DNáDGDMF*HVXPDUH]\VWDQFMLSU]HZRG]FHMGLRG\LUH]\VWDQFML

VWUDWX]ZRMHQLDZWyUQHJRZ\QRVL

R

S

RUD]SRPLMDMFQDSL FLHQD

SU]HZRG]F\P]áF]XSUGREFL*HQLDRSLVXMH]DOH*QRü

L

X

5

5

8

W

5

5

W

L

W

2

6

6

/

6P

6

/

2

≈

+

=

+

≤

≤

=

=

= ≤

≤








VLQ

ω

ω

π

π ω

π

GOD
GOD











Θ

Θ



5R]ZLMDMF]DOH*QRüZV]HUHJ)RXULHUDRWU]\PXMHP\

i

u

R

I

t

k

t

k

O

O

L

k

=

=

+

−

−











=

∞

∑

max

, , ,...

cos

1

1

2

2

1

2

2 4 6

π

ω

π

ω

VLQ



JG]LH

,

8

5

5

VP

6

/

PD[

=

+



.RU]\VWDMF]]DOH*QRFLPR*QDZ\]QDF]\üSDUDPHWU\

UR]ZD*DQHJRXNáDGX

−

VSUDZQRüQDSL FLRZD



(

)

η

π

π

π

X

2

6P

/

2

6P

/

VP

6

/

6

/

8

8

5 ,

8

5 ,

8

5 5

5

5

=

=

=

=

=

+

≈ ≈

<<

PD[













  

GOD



−

ZVSyáF]\QQLNW WQLH

k

U

U

I

I

t

m

O

m

O

=

=

= ≈

1

1

2

1 57

π

,



−

ZVSyáF]\QQLNNV]WDáWXSUGX

k

m I

I

I

I

i

O

O

=

=

=

max

max

π



−

VSUDZQRüSUGRZD

( )

η

π

ω

π

π

i

O

s

O

O

I

I

I

i d

t

=

=

= ≈

∫

1

2

2

0 64

2

0

,



−

VSUDZQRüHQHUJHW\F]QD

η

η η

π

π

p

O

s

u

i

S

L

S

L

P

P

R

R

R

R

=

=

=

+

≈

≈

≈

<<

2

2 2

1

1

2 2

0 29

2

2

/

,

GOD





−

QDSL FLH]ZURWQH

8

8

5

VP

=



−

SUGXGDURZ\

,

,

8

5

5

X

VP

6

/

PD[

PD[

=

=

+



=DWHP XNáDG MHGQRSRáyZNRZ\ ] REFL*HQLHP UH]\VWDQF\MQ\P

FKDUDNWHU\]XMH VL  PDá VSUDZQRFL QDSL FLRZ  SUGRZ

LHQHUJHW\F]Q RUD] GX*\PL W WQLHQLDPL R SXOVDFML UyZQHM SXOVDFML

SRGVWDZRZHM

( )

ω

 1LHNRU]\VWQH SDUDPHWU\ V NRQVHNZHQFM

MHGQRSRáyZNRZHJR VSRVREX SURVWRZDQLD RUD] EUDNX ILOWUDFML W WQLH

QDSL FLD]XZDJLQDUH]\VWDQF\MQ\FKDUDNWHUREFL*HQLD6NáDGRZDVWDáD

SUGXREFL*HQLD

I

O

Sá\QLHSU]H]WUDQVIRUPDWRU



 8NáDGGZXID]RZ\

6FKHPDWGZXID]RZHJRXNáDGXSURVWRZQLF]HJRMHGQRID]RZHJR

GZXSRáyZNRZHJR

m

=

2

SU]HGVWDZLRQRQDU\V

S

u

O

i

L

R

S

R

S

u

1

D

i

2

D

i

S

u

i ,

0

t

ω

max

I

Θ

2

π

π

2

O

u

1

D

i

2

D

i

i

O

i

1

D

2

D

8

VP

u

S

u

S

−

u

S

5\V'ZXID]RZ\XNáDGSURVWRZQLF]\]REFL*HQLHPUH]\VWDQF\MQ\P

: XNáDG]LH W\P GLRG\ SU]HZRG] QD SU]HPLDQ ND*GD SU]H]

SRáRZ  RNUHVX 'LRGD

D

1

 SU]HZRG]L JG\ QDSL FLH

u

S

 MHVW GRGDWQLH

]DGLRGD

D

2

 JG\ MHVW XMHPQH =DWHP SUG REFL*HQLD GDQ\ MHVW

]DOH*QRFL

L

L

8

W

5

5

L

L

8

W

5

5

2

VP

6

/

2

VP

6

/

= =

+

≤

≤

= = −

+

≤

≤



















VLQ

VLQ

ω

ω

π

ω

π ω

π

GOD

W

GOD

W





3RVW SXMFDQDORJLF]QLHMDNSRSU]HGQLRRWU]\PXMHP\

i

u

R

I

k

t

k

O

O

L

k

=

=

−

−











=

∞

∑

max

, , ,...

cos

2

4

1

2

2 4 6

π π

ω





6WG

η

π

π

X

6

/

6

/

5 5

5

5

=

+

≈

≈

<<









 





GOD



k

U

U

t

m

O

=

=

2

2

3



k

I

I

i

O

=

=

2

max

π





η

π

i

O

s

I

I

=

= ≈

2

2

0 64

,



η

π

π

p

S

L

S

L

R

R

R

R

=

+

≈

≈

<<

4 2

1

1

4 2

0 57

2

2

/

,

GOD



8

8

5

VP

=





-DN Z\QLND ] SU]HSURZDG]RQHM DQDOL]\ SDUDPHWUyZ

Z\NRU]\VWDQLH QDSL FLRZH XNáDGX GZXSRáyZNRZHJR MHVW GZXNURWQLH

ZL NV]HZSRUyZQDQLX]MHGQRSRáyZNRZ\PW WQLHQLDVQDGDOGX*H]H

Z]JO GX QD EUDN ILOWUDFML OHF] LFK F] VWRWOLZRü MHVW GZXNURWQLH

ZL NV]D :\NRU]\VWDQLH SUGRZH L VSUDZQRü XNáDGX V QDGDO PDáH

:DG XNáDGX MHVW NRQLHF]QRü VWRVRZDQLD V\PHWU\F]QHJR X]ZRMHQLD

ZWyUQHJR

 'ZXID]RZ\XNáDGPRVWNRZ\

8NáDG PRVWNRZ\*UDHW]D SU]HGVWDZLRQ\ QD U\V

SR]EDZLRQ\MHVWZDGMHGQRID]RZHJRXNáDGXGZXSRáyZNRZHJR

:XNáDG]LHW\PGODGRGDWQLFKZDUWRFLQDSL FLDZHMFLRZHJR

u

S

SU]HZRG] GLRG\

D

D

1

2

i

 D GOD XMHPQ\FK  GLRG\

D

D

3

4

i



]DSHZQLDMFSURVWRZDQLHGZXSRáyZNRZH=DWHPSUGMHVWSRELHUDQ\]

X]ZRMHQLD ZWyUQHJR SU]H] FDá\ RNUHV F]\OL WUDQVIRUPDWRU Z XNáDG]LH

*UDHW]DMHVWOHSLHMZ\NRU]\VWDQ\QL*ZXNáDG]LH]U\V

S

u

O

i

O

u

L

R

4

D

1

D

2

D

3

D

5\V'ZXID]RZ\XNáDG*UDHW]D

8NáDG *UDHW]D RSLVXM WH VDPH ]DOH*QRFL    FR XNáDG

GZXID]RZ\]WUy*QLF*HUH]\VWDQFMD

R

S

MHVWUyZQDVXPLHUH]\VWDQFML

VWUDW X]ZRMHQLD ZWyUQHJR L UH]\VWDQFML GZyFK GLRG SU]HZRG]F\FK

SRáF]RQ\FKV]HUHJRZR1DSL FLH]ZURWQHMHVWUyZQH

8

8

5

VP

=







352672:1,.,=),/75$0,2:(-&,832-(012&,2

:<0

3URVWRZQLNL ] REFL*HQLHP F]\VWR UH]\VWDQF\MQ\P QLH ]QDMGXM

SUDNW\F]QHJR ]DVWRVRZDQLD Z XNáDGDFK HOHNWURQLF]Q\FK ] SRZRGX

GX*HJR SR]LRPX W WQLH QDSL FLD EG( SUGX Z\MFLRZHJR

=PQLHMV]HQLHW WQLHQDSL FLDQDZ\MFLXXNáDGXSURVWRZQLF]HJRPR*QD

X]\VNDü SRSU]H] GRáF]HQLH ILOWUyZ Z\JáDG]DMF\FK 3RGVWDZRZ

F] FLWDNLHJRILOWUXVHOHPHQW\UHDNWDQF\MQHZSRVWDFLSRMHPQRFLOXE

LQGXNF\MQRFLNWyUHNRV]WHPZF]HQLHM]JURPDG]RQHMHQHUJLLPRJZ

RGSRZLHGQLFKPRPHQWDFKSRGWU]\PDüZDUWRüSU]HSá\ZDMFHJRSUGX

=H Z]JO GyZ DSOLNDF\MQ\FK SU]\ PRF\ SRELHUDQHM ] Z\MFLD

SURVWRZQLND QLH SU]HNUDF]DMFHM RN  : SRSXODUQLHMV]H V XNáDG\

SRMHPQRFLRZH :QDMSURVWV]\P SU]\SDGNX MHVW WR NRQGHQVDWRU

GRáF]RQ\UyZQROHJOHGRREFL*HQLDSURVWRZQLND

=DVDGDG]LDáDQLDSURVWRZQLND]ILOWUHPSRMHPQRFLRZ\P]RVWDQLH

SU]HGVWDZLRQD QD SU]\NáDG]LH XNáDGX MHGQRSRáyZNRZHJR 1D U\V

]DPLHV]F]RQRVFKHPDWREZRGXRUD]RGSRZLHGQLHSU]HELHJLF]DVRZH

S

u

O

i

O

u

L

R

S

R

D

D

i

S

O

u

u ,

t

ω

S

u

'

t

ω

''

t

ω

t

U

O

u

O

D

i

i ,

π

2

π

Θ

2

1

Θ

Θ

Θ

2

1

D

i

O

i

8

VP

t

ω

&

D

E

5\V-HGQRID]RZ\XNáDGSURVWRZQLF]\]ILOWUHPSRMHPQRFLRZ\PDVFKHPDW

XNáDGXESU]HELHJLQDSL üLSUGyZZXNáDG]LH

: VWDQLH XVWDORQ\P  NLHG\ QDSL FLH

u

S

 SU]HZ\*V]D QDSL FLH

u

O

(

)

u

u

S

O

>

  SUG GLRG\ Sá\QF\ Z SU]HG]LDOH F]DVX VSHáQLDMF\P

ZDUXQHN

Θ

Θ

Θ

1

1

2

≤

≤

+

ω

t

 NRQGHQVDWRU

C

 GRáDGRZXMH SUDZLH GR

PDNV\PDOQHMZDUWRFLQDSL FLD

8

VP

]HVWDáF]DVRZ

(

)

R R

C

R C

S

L

S

≈





2G FKZLOL

t'

 VSHáQLDMFHM ZDUXQHN

ω

t'

=

+

Θ

Θ

1

2

 Z NWyUHM QDSL FLD

u

u

S

O

i

 ]UyZQDM VL 

(

)

u

u

S

O

=

 GLRGD QLH SU]HZRG]L SUGX FR

SRZRGXMH*HNRQGHQVDWRU

C

UR]áDGRZXMHVL ]HVWDáF]DVRZ

R C

L



3URFHV WHQ WUZD GR PRPHQWX

(

)

t

t

''

''

ω

π

=

+

Θ

1

2

 SRQRZQHJR

]UyZQDQLD VL  QDSL ü

u

u

S

O

i

 SR F]\P NRQGHQVDWRU MHVW SRQRZQLH

GRáDGRZ\ZDQ\

:DQDORJLF]Q\VSRVyESUDFXMXNáDG\GZXID]RZHSU]HGVWDZLRQH

QDU\V=H Z]JO GX QD SURVWRZDQLH GZXSRáyZNRZH SURFHV

áDGRZDQLD L UR]áDGRZDQLD NRQGHQVDWRUD RGE\ZD VL  ] SXOVDFM

(

)

2

2

ω

m

=

 ]DWHP NRQLHF UR]áDGRZDQLD NRQGHQVDWRUD SU]\SDGD GOD

ω

π

t

m

''

/

=

+

Θ

1

2

NWUR]áDGRZDQLD2

2

π

/ m

− Θ



3U]HGVWDZLRQHQDU\VESU]HELHJLF]DVRZHQDSL üLSUGyZV

UHSUH]HQWDW\ZQH GOD SURVWRZQLND R UH]\VWDQFML

R

S

 EDUG]R PDáHM

ZSRUyZQDQLX]UH]\VWDQFMDREFL*HQLD

R

L



O

u

t

ω

2

1

,

D

D

i

i

π

2

π

Θ

2

S

u

L

R

S

u

1

D

i

2

D

i

O

u

1

D

2

D

C

π

3

t

ω

-

àDGRZDQLH

-

5R]áDGRZDQLH

A

B

O

u

O

U

S

u

L

R

4

D

1

D

2

D

3

D

O

u

C

1

D

i

2

D

i

U

Sm

≈

−

8

W

5 &

VP

/

H[S

D

E

5\V'ZXID]RZHXNáDG\SURVWRZQLF]H]REFL*HQLHPSRMHPQRFLRZR

UH]\VWDQF\MQ\PDVFKHPDW\XNáDGyZESU]HELHJLSUGyZLQDSL üZSURVWRZQLNX

GZXSRáyZNRZ\P



3U]HELHJ QDSL FLD QD Z\MFLX SURVWRZQLND PR*QD REOLF]\ü

Uy*Q\PLPHWRGDPL3RQLHZD*]Z\NOHFHOHPDQDOL]\MHVWW\ONRRNUHOHQLH

ZDUWRFLUHGQLHMQDSL FLD

U

O

LZDUWRFLPL G]\V]F]\WRZHMW WQLH

U

t



DSU]\W\PQLHMHVWZ\PDJDQD]E\WGX*DGRNáDGQRüGODWHJRVWRVXMHVL

Uy*QHJRURG]DMXSU]\EOL*HQLD

1DMSURVWV]H SROHJD QD ]DVWSLHQLX U]HF]\ZLVWHJR SU]HELHJX

u

O

SU]HELHJLHP SLáRNV]WDáWQ\P RZDUWRFL PL G]\V]F]\WRZHM UyZQHM

ZDUWRFL PL G]\V]F]\WRZHM W WQLH

U

t

  QDáR*RQ\P QD VNáDGRZ VWDá

UyZQUHGQLHMZDUWRFLQDSL FLDZ\MFLRZHJR

U

O

MDNQDU\V

=DNáDGDMF *H

R

R

S

L

<<

 SURFHV áDGRZDQLD NRQGHQVDWRUD

SU]HELHJDZF]DVLHGX*RNUyWV]\PRGF]DVXUR]áDGRZDQLDDNRQGHQVDWRU

C

áDGXMHVL GRZDUWRFLV]F]\WRZHMQDSL FLD

8

VP

=DNáDGDMFWDN*H*H

ω

C R

L

>>

1

 SURFHV áDGRZDQLD PR*QD XZD*Dü ]D OLQLRZ\ FR SU]\

SU]\M W\FK ]DáR*HQLDFK R]QDF]D *H SU]HELHJ QDSL FLD Z\MFLRZHJR

u

O

MHVW]EOL*RQ\GRSLáRNV]WDáWQHJRSU]\VSHáQLHQLX]DOH *QRFL

8

8

8

2

VP

W

=

−







7UDNWXMF SURFHV UR]áDGRZDQLD MDNR OLQLRZ\ SUG UR]áDGRZXMF\

NRQGHQVDWRU PD Z SU]\EOL*HQLX ZDUWRü UyZQ I

U

R

R

O

L

L

=

/

 SU]\

U

U

t

O

<<



6WG

U

C

U

R

m

U

m

C R

t

O

L

O

L

=







 =

1

2

2

π

ω

π

ω

/



O

u

t

m

ω

2

π

2

π

π

3

O

u

O

U

t

U

0

≈

−

8

W

5 &

VP

/

H[S

8

VP

5\V3U]\EOL*RQ\SLáRNV]WDáWQ\SU]HELHJQDSL FLDQDZ\MFLX

GZXSRáyZNRZHJRXNáDGXSURVWRZQLF]HJR]REFL*HQLHP

SRMHPQRFLRZRUH]\VWDQF\MQ\P

.RU]\VWDMF]]DOH*QRFLLRWU]\PXMHP\



8

8

8

P & 5

2

VP

2

/

=

− π

ω



η

π

ω

X

2

VP

/

8

8

P & 5

=

≈

+







=DOH*QRü SU]HGVWDZLRQDZQLHFRLQQHMSRVWDFL

8

8

P &

,

2

VP

2

=

−

π

ω



SRND]XMHZSá\ZSUGXUH]\VWDQFMLREFL*HQLD

R

L

QDQDSL FLHZ\MFLRZH

XNáDGX

5R]NáDGDMFSU]HELHJSLáRNV]WDáWQ\ZV]HUHJ)RXULHUDDPSOLWXGD

VNáDGRZHMSRGVWDZRZHMZ\QRVL

U

U

m

t

1

2

=

π



=]DOH*QRFLPR*QDREOLF]\üZVSyáF]\QQLNW WQLH

k

t

k

U

U

m

C R

t

m

O

L

=

=

1

1

ω



1DSL FLH]ZURWQH GODXNáDGX]U\VZ\QRVL

8

8

8

8

5

VP

2

VP

=

+

≈





QDWRPLDVWSUGXGDURZ\WMPDNV\PDOQ\SUGGLRG\WX*SRZáF]HQLXZ

QDMEDUG]LHM QLHNRU]\VWQ\FK ZDUXQNDFK W]Q SU]\ UR]áDGRZDQ\P

NRQGHQVDWRU]HLZáF]HQLXQDSL FLDJG\

X

8

6

VP

=



,

8

5

X

VP

6

PD[



=



3DUDPHWU WHQ RNUHOD Z\PDJDQLD GRW\F]FH PDNV\PDOQHJR SUGX

X*\W\FKGLRG

3U]HGVWDZLRQD DQDOL]D SRND]XMH *H XNáDG\ SURVWRZQLF]H

]REFL*HQLHP SRMHPQRFLRZR  UH]\VWDQF\MQ\P FKDUDNWHU\]XM VL

GREU\P Z\NRU]\VWDQLHP QDSL FLRZ\P SURVWRZDQLH ]EOL*RQH GR

V]F]\WRZHJRLPDá\PLW WQLHQLDPLGODGX*\FKZDUWRFLVWDáHMF]DVRZHM

C R

L

 :\NRU]\VWDQLH SUGRZH MHVW ]QDF]QLH JRUV]H QL* Z XNáDGDFK

]REFL*HQLHPUH]\VWDQF\MQ\P

:áDFLZRFL

XNáDGyZ

SURVWRZQLF]\FK

]

REFL*HQLHP

SRMHPQRFLRZ\PVSUDZLDM*HVRQHQDMF] FLHMVWRVRZDQHZ]DNUHVLH

QLH]E\WGX*\FKPRF\!:SU]\VWRVXQNRZRGX*\FKZDUWRFLDFK

QDSL FLDLPDá\FKZDUWRFLDFKSUGX





352672:1,.,=),/75(02:(-&,8,1'8.&<-1<0

=DVWRVRZDQLH REFL*HQLD R FKDUDNWHU]H LQGXNF\MQ\P XPR*OLZLD

]ZL NV]HQLH NWD SU]HSá\ZX SUGX 2

Θ

 L RJUDQLF]HQLH W WQLH SUGX

3UDFD XNáDGX SURVWRZQLF]HJR ] REFL*HQLHP LQGXNF\MQ\P ]DOH*\ RG

FLJáRFLSU]HSá\ZXSUGXREFL*HQLD

 8NáDGMHGQRID]RZ\

1D U\V SU]HGVWDZLRQR VFKHPDW MHGQRID]RZHJR XNáDGX

SURVWRZQLF]HJR]REFL*HQLHPLQGXNF\MQ\P

: XNáDG]LH MHGQRID]RZ\P FLJáRü SUGX REFL*HQLD QLH MHVW

]DFKRZDQD'LRGD]DF]\QDSU]HZRG]LüJG\QDSL FLH]DVLODQLD

u

S

VWDMH

VL  GRGDWQLH 3UG

i

O

 MHVW RSy(QLRQ\ Z VWRVXQNX GR QDSL FLD

u

S

 QD

VNXWHNLQGXNF\MQHMUHDNWDQFMLREFL*HQLD

L 5 / GL

GW

X

8

W

L

2

2

6

VP

2

+

=

=

≤

≤

=

≤

≤









VLQ

ω

ω

ω

π

GOD

W

GOD

W









Θ

Θ



JG]LH

R

R

R

L

S

=

+

S

u

S

R

D

S

O

u

i ,

t

ω

max

'

O

I

Θ

2

π

π

2

S

u

O

i

'

O

i

O

u

L

R

L

O

i

''

O

i

''

'

L

R

L

ω

=

5

=

20

=

U

Sm

5\V-HGQRID]RZ\XNáDGSURVWRZQLF]\]REFL*HQLHPUH]\VWDQF\MQR

LQGXNF\MQ\P

5R]ZL]DQLHUyZQDQLDPDSRVWDü

( )

(

)

{

}

(

)

[

]

i t

U

R

t

t ctg

O

Sm

=

−

+

−

cos

sin

sin

exp

ϕ

ω

ϕ

ϕ

ω

ϕ



JG]LH

tg

L

R

R

S

L

ϕ ω

=

+

/ (

)



.WSU]HSá\ZX 2

Θ

RNUHODZDUXQHN

( )

i 2

0

Θ =





6WG

(

)

(

)

[

]

sin

sin

exp

2

2

0

Θ

Θ

−

+

=

ϕ

ϕ

ϕ

ctg



.W SU]HSá\ZX 2

Θ

 ]PLHQLD VL  Z ]DOH*QRFL RG LQGXNF\MQRFL Z

]DNUHVLH RG 2

Θ =

π

  GOD

ω

L

R

R

S

L

/ (

)

+

=

0

 GR 2

2

Θ =

π

  GOD

ω

L

R

R

S

L

/ (

)

+

→ ∞

%UDNFLJáRFLSUGXZREFL*HQLXSRZRGXMH*H]H

Z]URVWHPNWDSU]HSá\ZX 2

Θ

PDOHMHZDUWRüPDNV\PDOQDSUGX

I

O max

RUD] MHJR VNáDGRZD

I

O

 FR SRZRGXMH *H ZVSyáF]\QQLN Z\NRU]\VWDQLD

QDSL FLRZHJR

η

u

LVSUDZQRüHQHUJHW\F]QD

η

p

PDOHMDW WQLHQLDURVQ

=DWHP VWRVRZDQLH REFL*HQLD LQGXNF\MQHJR Z XNáDG]LH MHGQRID]RZ\P

MHVWQLHFHORZHSRQLHZD*SUGZREFL*HQLXMHVWQLHFLJá\

 8NáDG\GZXID]RZH

:XNáDGDFKGZXID]RZ\FKU\V]DFKRZDQ\MHVWZDUXQHN

FLJáRFLSU]HSá\ZXSUGX

O

i

t

ω

π

2

π

S

u

S

u

1

D

i

2

D

i

1

D

2

D

S

u

L

R

4

D

1

D

2

D

3

D

0

u

L

1

D

i

2

D

i

L

R

O

u

O

i

L

5

,

0

=

1

=

0

=

L

R

L

ω

10

=

∞

=

5\V'ZXID]RZHXNáDG\SURVWRZQLF]H]REFL*HQLHPUH]\VWDQF\MQR

LQGXNF\MQ\PGODUy*Q\FKZDUWRFL

ω

/5

/

3UG

i

D1

GLRG\

D

1

Sá\QLHZF]DVLHXMHPQHJRSyáRNUHVXQDSL FLD

u

S

 -HVW RQ RSy(QLRQ\ Z]JO GHP QDSL FLD

u

S

 R NW

ϕ

 ]JRGQLH ]

]DOH*QRFL  : PRPHQWDFK SU]HMFLD QDSL FLD

u

S

 SU]H] ]HUR

SUG REFL*HQLD

i

O

 MHVW SU]HMPRZDQ\ QDSU]HPLHQQLH SU]H] GLRG\



D

D

1

2

i

 FR ]DSHZQLD MHJR FLJáRü L NW SU]HSá\ZX SUGX 2

2

Θ =

π



=H Z]URVWHP LQGXNF\MQRFL RWU]\PXMHP\ VWDá ZDUWRü SUGX

REFL*HQLD SU]\ QLHZLHONLFK DPSOLWXGDFK VNáDGRZ\FK ]PLHQQ\FK

:PLDU Z]URVWXLQGXNF\MQRFLSU]HELHJLSUGyZREXGLRG]EOL*DMVL

GRSU]HELHJyZSURVWRNWQ\FK

8NáDG\ ] REFL*HQLHP UH]\VWDQF\MQR  LQGXNF\MQ\P Z\ND]XM 

PDáH W WQLHQLD L EDUG]R GREUH Z\NRU]\VWDQLH SUGRZH

η

i

 SU]\ QLHFR

JRUV]\P Z\NRU]\VWDQLX QDSL FLRZ\P :áDVQRFL ILOWUDF\MQH XNáDGyZ

SROHSV]DM VL  ]H Z]URVWHP SUGX REFL*HQLD L GODWHJR V QDMF] FLHM

VWRVRZDQHZ]DVLODF]DFKGX*HMPRF\



8.à$'<352672:1,&=(=),/75(0,1'8.&<-12

32-(012&,2:<0

3ROHSV]HQLH ILOWUDFML XNáDGX ] REFL*HQLHP UH]\VWDQF\MQR 

SRMHPQRFLRZ\PX]\VNXMHVL SU]\]PQLHMV]HQLXSUGXREFL*HQLD

I

O



Z SU]HFLZLHVWZLH GR XNáDGX ] REFL*HQLHP UH]\VWDQF\MQR 

LQGXNF\MQ\PZNWyU\PSROHSV]HQLHILOWUDFMLX]\VNXMHVL GODZL NV]HJR

SUGX REFL*HQLD =DVWRVRZDQLH Z XNáDGDFK SURVWRZQLNyZ ILOWUyZ

LQGXNF\MQR  SRMHPQRFLRZ\FK QLHNLHG\ Z SRVWDFL NLONX RJQLZ ILOWUX

/& VNXWNXMH ]QDF]Q\P SROHSV]HQLHP ILOWUDFML W WQLH 3U]HELHJL

F]DVRZH QDSL ü L SUGyZ Z GZXID]RZ\P XNáDG]LH SURVWRZQLF]\P ]

ILOWUHPZ\JáDG]DMF\P/&SU]HGVWDZLRQRQDU\V

O

u

1

π

/

2

0

kr

L

L

u

η

t

ω

t

ω

t

ω

L

i

u

Θ

2

kr

L

L

<

u

u

L

i

L

i

kr

L

L

>

ω

3

L

kr

R

L

L

=

=

O

I

S

u

S

u

1

D

i

2

D

i

1

D

2

D

L

R

C

L

L

i

u

5\V'ZXID]RZ\XNáDGSURVWRZQLF]\]ILOWUHP/&



'OD PDáHM LQGXNF\MQRFL XNáDG SUDFXMH MDN SURVWRZQLN ]REFL*HQLHP

SRMHPQRFLRZ\PSU]\PDá\PNFLHSU]HSá\ZX 2

Θ



:PLDU Z]URVWXLQGXNF\MQRFLNWSU]HSá\ZXURQLHLGOD

L

L

kr

=

SUG

REFL*HQLD

i

O

VWDMHVL FLJá\2

2

Θ =

π

'DOV]\Z]URVWLQGXNF\MQRFL

SRZRGXMH *H ZáDVQRFL XNáDGX ]EOL*DM VL  GR SURVWRZQLND ]

REFL*HQLHP LQGXNF\MQ\P R OHSV]HM ILOWUDFML W WQLH G]L NL G]LDáDQLX

SRMHPQRFL



325Ï:1$1,(:à$612&,8.à$'Ï:

352672:1,&=<&+

1D U\V GRNRQDQR JUDILF]QHJR SRUyZQDQLD SDUDPHWUyZ

GZXID]RZ\FK XNáDGyZ SURVWRZQLF]\FK ] Uy *Q\PL ILOWUDPL

Z\JáDG]DMF\PL

8NáDG\ SURVWRZQLF]H ] ILOWUHP R ZHMFLX SRMHPQRFLRZ\P

Z\ND]XM GREUH Z\NRU]\VWDQLH QDSL FLRZH Z ]DNUHVLH PDá\FK PRF\

LPDá\FKSUGyZSURVWRZDQLHEOLVNLHV]F]\WRZHPXRUD]PDáHW WQLHQLD

,FK SDUDPHWU\ SUGRZH ZVSyáF]\QQLN Z\NRU]\VWDQLD SUGRZHJR

ZVSyáF]\QQLN NV]WDáWX SUGX L SUG XGDURZ\ V QLHNRU]\VWQH FR

SRZRGXMH VáDEH Z\NRU]\VWDQLH WUDQVIRUPDWRUD VLHFLRZHJR L GLRG

3UDNW\F]QLHXNáDG\WHVWRVXMHVL MDNRXNáDG\PDáHMPRF\RVWRVXQNRZR

GX*\PQDSL FLXLPDá\PSUG]LH

8NáDG\ SURVWRZQLF]H ] ILOWUHP R ZHMFLX LQGXNF\MQ\P PDM

ZáDVQRFL SU]HFLZQH :\NRU]\VWDQLH QDSL FLRZH W\FK XNáDGyZ MHVW

PQLHMV]HQL*ZXNáDGDFK]REFL*HQLHPSRMHPQRFLRZ\PSURVWRZDQLH

UHGQLHLQLH]DOH*\RGSUGXREFL*HQLD

1

π

/

2

0

o

I

u

η

1

5

,

0

3

/

2

uc

η

2

=

m

uL

uR

η

η

,

tR

k

tC

k

tL

k

tLC

k

t

k

1

π

/

2

0

u

η

π

5

,

0

iL

η

2

=

m

iR

η

i

k

i

k

100

iC

η

}

R

L

C

2

/

1

1

o

I

5\V3RUyZQDQLHFKDUDNWHU\VW\NUy*Q\FKXNáDGyZSURVWRZQLF]\FK

8NáDG\ WH Z\ND]XM PDáH W WQLHQLD L GREUH Z\NRU]\VWDQLH SUGRZH Z

]DNUHVLHGX*\FKSUGyZREFL*HQLDD]DWHPQDGDMVL V]F]HJyOQLHGR



]DVWRVRZDQLD Z XNáDGDFK GX*HM PRF\  SU]\ GX*\FK ZDUWRFLDFK SUGX

REFL*HQLD,FKVáDEHZáDVQRFLILOWUDF\MQHZ]DNUHVLHPDá\FKZDUWRFL

SUGXREFL*HQLDPR*QDSRSUDZLüVWRVXMFILOWU\PLHV]DQH/&:yF]DV

ZVSyáF]\QQLN W WQLH MHVW PDá\ L Z ]DVDG]LH QLH ]DOH*\ RG SUGX

REFL*HQLD



32:,(/$&=(1$3, &,$

3RZLHODQLH QDSL FLD MHVW PR*OLZH SU]\ ]DVWRVRZDQLX XNáDGyZ

SURVWRZQLF]\FK ] REFL*HQLHP SRMHPQRFLRZ\P 2GG]LHOQH XNáDG\

SURVWRZQLF]HREFL*RQHSRMHPQRFLRZRZ\WZDU]DMQDNRQGHQVDWRUDFK

QDSL FLD NWyUH VXPXM VL  Z REZRG]LH REFL*HQLD SRZRGXMF

]ZLHORNURWQLHQLHQDSL FLDZ\MFLRZHJR
D6\PHWU\F]Q\SRGZDMDF]QDSL FLD

1DMSURVWV]\PSRZLHODF]HPQDSL FLDMHVWV\PHWU\F]Q\SRGZDMDF]

QDSL FLDU\V

S

u

L

R

D

1

D

i

1

1

,

C

D

u

i

t

ω

2

2

,

C

D

u

i

π

Θ

2

1

C

u

2

C

u

C

C

=

1

C

C

=

2

2

D

i

S

u

1

D

i

S

u

2

D

i

1

C

u

2

C

u

t

ω

O

u

t

ω

Θ

2

π

2

r

τ

r

τ

π

π

U

Sm

U

Sm

i

O

u

u

u

O

C

C

=

+

1

2

u

O

5\V6\PHWU\F]Q\SRGZDMDF]QDSL FLD

: VWDQLH XVWDORQ\P JG\ QDSL FLH ]DVLODMFH

u

S

 MHVW GRGDWQLH

SRMHPQRü

C

1

GRáDGRZXMHVL ]HVWDáF]DVRZ

τ

s

S

R C

=

1

SU]H]GLRG

D

1

 ZVNXWHN SU]HSá\ZX SUGX

i

D1

 1DVW SQLH JG\ SUG

i

D1

 ]DQLND



SRMHPQRü

C

1

 UR]áDGRZXMH VL  SU]H] UH]\VWDQFM  REFL*HQLD

R

L

 L

SRMHPQRüGUXJLHJRXNáDGXSURVWRZQLF]HJR

C

2

]HVWDáF]DVRZ

τ

U

/

/

5 & &

& &

5 &

& &

&

=

+

=

=

=















GOD



: SyáRNUHVLH XMHPQHJR QDSL FLD ]DVLODMFHJR

u

S

 DQDORJLF]QLH MDN

SRSU]HGQLR GRáDGRZXMH VL  SRMHPQRü

C

2

 ZVNXWHN SU]HSá\ZX SUGX

i

D2

  D QDVW SQLH UR]áDGRZXMH VL  SU]H] UH]\VWDQFM  REFL*HQLD

R

L

 L

SRMHPQRüSLHUZV]HJRXNáDGXSURVWRZQLF]HJR

C

1

]HVWDáF]DVRZ

τ

r



1DSL FLH Z\MFLRZH

u

O

 MHVW VXP QDSL ü

u

u

C

C

1

2

i

  MHVW ZL F EOLVNLH

 8

VP

L]DZLHUDW WQLHQLDRSXOVDFML 2

ω



E1LHV\PHWU\F]Q\SRGZDMDF]QDSL FLD

8NáDGQLHV\PHWU\F]QHJRSRGZDMDF]DQDSL FLDSU]HGVWDZLRQRQD

U\V

S

u

0

2

u

u

C

=

2

D

1

D

i

1

C

u

2

D

i

1

D

1

C

2

C

u

i,

sm

U

t

ω

1

2

Θ

S

u

1

D

i

1

C

u

2

2

Θ

S

u

2

D

i

1

2

Θ

2

2

Θ

1

D

i

2

D

i

S

u

−

u

u

C

O

2

=

5

/

5\V1LHV\PHWU\F]Q\SRGZDMDF]QDSL FLD

:VWDQLHXVWDORQ\PZSyáRNUHVLHXMHPQHJRQDSL FLD]DVLODMüHJR

u

S

SRMHPQRü

C

1

áDGXMHVL ]HVWDáF]DVRZ

τ

s

S

C R

1

1

=



SUGHP

i

D1

GRZDUWRFLEOLVNLHMV]F]\WRZHM

8

VP

*G\SUG

i

D1

]DQLNDSR

]DNRF]HQLX áDGRZDQLD SRMHPQRFL

C

1

 GLRGD

D

1

 XOHJD ]DWNDQLX

LSRMHPQRü

C

1

QLHPR*HVL UR]áDGRZDüGRSyNLQLH]DF]QLHSU]HZRG]Lü

GLRGD

D

2

SRZRGXMFGRáDGRZDQLHNRQGHQVDWRUD

C

2

GRZDUWRFLEOLVNLHM

8

8

8

&

VP

VP





+

≈

]HVWDáF]DVRZ

(

)

τ

s

S

L

S

S

L

R R

C C

C

C

C R

C

C

C

R

R

2

1

2

1

2

1

2

1

2

=

+

≈

=

=

<<

dla

,



*G\SURFHVáDGRZDQLDSRMHPQRFL

C

2

LUR]áDGRZDQLDSRMHPQRFL

C

1

τ

τ

r

s

1

2

=

NRF]\VL GLRGD

D

2

SU]HVWDMHSU]HZRG]LüLUR]SRF]\QDVL

UR]áDGRZDQLHSRMHPQRFL

C

2

]HVWDáF]DVRZ

τ

r 2





τ

r

L

C R

2

2

=





1LHV\PHWU\F]Q\ SRGZDMDF] QDSL FLD Z SRUyZQDQLX ] XNáDGHP

V\PHWU\F]Q\PFKDUDNWHU\]XMHVL ZL NV]\PLW WQLHQLDPL R GZXNURWQLH

ZL NV]HM SXOVDFML SRGVWDZRZHM ZL NV] ]DOH*QRFL QDSL FLD

Z\MFLRZHJR RG SUGX REFL*HQLD L EDUG]LHM QLHUyZQRPLHUQ\P

REFL*HQLHP GLRG LSRMHPQRFL 1DWRPLDVW MHJR ]DOHW MHVW LVWQLHQLH

ZVSyOQHJR ]DFLVNX QDSL FLD ]DVLODQLD L REFL*HQLD

R

L

 : REX

SU]\SDGNDFK SRZLHODF]\ QDSL FLD SUG REFL*HQLD SUDNW\F]QLH QLH

SU]HNUDF]DG]LHVLWNyZP$
F3RZLHODF]HZLHORNURWQH

S

u

S

u

5\V:LHORNURWQHSRZLHODF]HQDSL FLDDQLHV\PHWU\F]QHEV\PHWU\F]QH

'RGDQLH GDOV]\FK VHNFML XPR*OLZLD ]ZL NV]HQLH NURWQRFL

SRZLHODQLDSU]\F]\PXNáDG\WDNLHPRJ PLHüVWUXNWXU QLHV\PHWU\F]Q

OXEV\PHWU\F]QU\V

: SUDNW\FH NURWQRü SRZLHODF]D MHVW RJUDQLF]RQD GR  GOD

PDá\FKSUGyZLGRNLONXG]LHVL FLXGODSUGyZREFL*HQLDU] GX

µ

$