ausers.ntua.gr/eglytsis/EM_B/Notes_6.pdf · 1. ITa npon()o:'l-"vo
Transcript of ausers.ntua.gr/eglytsis/EM_B/Notes_6.pdf · 1. ITa npon()o:'l-"vo
426a
lTopqltSiJ2jtCl Enl A~ns P~(ssC)n (And (31f3C T r PCIJPflIJltn8 JTC5Qshy
I~iOLl~) GE~ 5jO)
NQ (3P(88 Co 8UVoplU6 cent 6(0 E6WUpiLt6 lou 0POSUJvU00 XWfr oJ
Teu 6xn HQC05 nou n t PIltX ~ gtltwp ur nUAvd trt-cQ rpoPCfOV P(X) =pc
(c5 ( Tfx I 2 a)
x
~o j0
a2 d-tcentP (gtlt) cent -p(~12cent -J -Eo ax 2 + o~2 Ea
I OOOIJ
V24gto =- _ ec0 J E-ltJ
~p = iPp ()I) r--) d2 centp =- _ ~ cos(~)
dx 2 Eo Za
A5 OQuqJdc-lpc 1v ~~En
i~2 = _ 11 C p sln[~) dx 2 2a
2- I
Cp = (2() ) ~ IJ O-t () PEfl un rlU~n (fP I f ~ tOJ 11 Eo
6~V 4p (X) (~)2- ~ COS (~) 1T Eo 28
1 J
- ( ~)2 (E) CaS (~)b y CJli - - 1r bo 2a
a-8
H VC( 1(0 shy
=)
28 2 po (TiX)~ h (gtlt) u =b) = - ( - ) - co s shy
J 1T Eo 2~
~I = a ~ ~ ==-0
I f P IXpn61~OnOIWVLos -cn v ~t8-GS~ XWpI6)AOU -ew~ ~ELQ(I ~nt~v ()l~
rnv EIitHV1 L-alc1 CQ (rnllop~a(f0Jvtu~ lIEpla~~ -dlt)(ltd cJ lt~ b
x ( x) - A1 co S (k gtlt) + A~ sin (k gtlt )
ZA cos (~a) =0 -) A I Cos ft-a) 0
r--) 2 A2 SI n Ckd) a I A2 SIn (b) =0
~
T 01 E () I 0 V C U Q v c n 0 I U t Cl( ()
ItlIno (flv EJ (2) A2 a Z 6LE 010( VO( U (j vcshy
1lT ~ I D COJ 1 t S (1) COS Cka) = a -) k a = (Zn+ I ) n =J
2
nTr 111 )12- A1 =- 0 ) kr = - (n~f) l)rr) x) (x ) An sin (-r xja 2~- A2 = 0) 2d ) shykl) = (2Vl+I) (no)~) X1 (x ) 13 n co S ((2rl n) ~ )( )
Idltbl- ) 0 ~(lo 1J cvv d ~ j- I
( 2l1-t I ) h I 2Y +J )ltPI-) ( XI y=b ) - ~ti cos -- gtlt COS i - 1Tb ~ 2) 2Q
_ Po (~)2 CQS ( ~ 6 1 2a
Anol Tl1v TE-A~uicd~ ~01~r~ 11 =- 0
B 0 C(gt s~ 1f ta ) - ~ (~Q r gt
)
i26d
cosh (~) )
cosh (~)
I
poundfVCt(
I
A P (9 ~ nT Kn Lapllce -NtampoC--os
I
UQPC~ ( (cv [ (iulln
_ 0
E U) OQI (1 V n arpQ6CU (SO lnJ t(X )
f()lt~ 4x ) - (Xo)CFD) F or wov-d Dl fFeVnUl +l rXo ) ~
~X
f (Xc) - +(x~ - AX)-J((3)) 8ackvJltIlvd DffevelU ((x Q)
bX
f(xo +c-x) - +(~ -1 xl NtD) Ce( +vaL Oi ffevence = f (xo)
2 ~x
I OL amp(Q-flo~E~ 61 (Iv rtp~ce~60161 Tn ~p 06 WIS lt10 Inltgtpo0v yeJ cpOlV QIJ V
()(O UoL_W~ 6XYCl R) p B
I I
A I
I I
I I I I I
Xo X
X~-lt )( I J h)I
~-----l-~ _
1----------------shy
ITa npon()ol-vo ltxpet n 1An TM 6POtihJ Ar wPpOj rnshy1
I I
i b8ckwovdJmiddotr~eYer)CQ ) n krI(n l~J PI3 Wfp~jL( Cnv ~VoQvdmiddot dttefenQ i
I
I I
pncfCI-gtA 110lt f3PO~I)E npaEf6(riff~
I I
fex o + 6X2 ) - f (X- A~2) Ceo ) ~)lt
-rCx )- -r ex -A)l)] _ llx
rex + Clx) - 2 fcxltraquo + fc Xo - ~)lt)
(hX) 2
A amppoi j o V TO-f
f(xc+6x) t- ~cxo-()x) -=- 2f=(xa) -t(t~if(x) + o (1x)4
Oro) 0(6gtlt)4 anl1ol)lt~ 6(0 6ltpoll~a ~TQ n 6Vpc 6o_1H~n((D)~
I I n e-ponQ VW ~A-flPClSVJ) ~xa01~
+(x a -+ 6 X ) - rex c - 6 x)
2 ~)(
+(gtlt0+ ClX) - 2 + (Xo) + f (Xc -6)lt)
(fJj )2
- l-
)IvJpO vl) eSy))
fllfl -I shyUUJ UUJ ex ncn gt0 gt0 ~N
I~ 3ol IN IN
)lj - J ~y
UCgt-( ltb(gtlti)Yj)= ltPj
-----------7gt X XL
2 1)CXc+6X) - 2 Sf) (XiYj) + p(x(-ax1_i) _d lt) C15iH JJ - 2 cD -+ P(_f~ a x 2
(6X)2 (6xF 1lt1) j
$)(X~)lj+pound)) - 2 ltP(XiJYj) + ~(xL~J-tiy) ~ ~Cj-+I - ZcentIJ + ~ l~j-I~l all- (~y)2(~yF
- x 1 ) J)
=
~ ( 1 2)I - +_ -I- q)l l + ~ - =())J l1x 2 ll )J + t
Y ~y2 Jj - a
dltP I -s - - vwctlln 01) 5
lAllXJ =lb]
Uvo 0 niv()(L(~ 1wV ~UVlr~CgtTwv
1 J
- ( ~)2 (E) CaS (~)b y CJli - - 1r bo 2a
a-8
H VC( 1(0 shy
=)
28 2 po (TiX)~ h (gtlt) u =b) = - ( - ) - co s shy
J 1T Eo 2~
~I = a ~ ~ ==-0
I f P IXpn61~OnOIWVLos -cn v ~t8-GS~ XWpI6)AOU -ew~ ~ELQ(I ~nt~v ()l~
rnv EIitHV1 L-alc1 CQ (rnllop~a(f0Jvtu~ lIEpla~~ -dlt)(ltd cJ lt~ b
x ( x) - A1 co S (k gtlt) + A~ sin (k gtlt )
ZA cos (~a) =0 -) A I Cos ft-a) 0
r--) 2 A2 SI n Ckd) a I A2 SIn (b) =0
~
T 01 E () I 0 V C U Q v c n 0 I U t Cl( ()
ItlIno (flv EJ (2) A2 a Z 6LE 010( VO( U (j vcshy
1lT ~ I D COJ 1 t S (1) COS Cka) = a -) k a = (Zn+ I ) n =J
2
nTr 111 )12- A1 =- 0 ) kr = - (n~f) l)rr) x) (x ) An sin (-r xja 2~- A2 = 0) 2d ) shykl) = (2Vl+I) (no)~) X1 (x ) 13 n co S ((2rl n) ~ )( )
Idltbl- ) 0 ~(lo 1J cvv d ~ j- I
( 2l1-t I ) h I 2Y +J )ltPI-) ( XI y=b ) - ~ti cos -- gtlt COS i - 1Tb ~ 2) 2Q
_ Po (~)2 CQS ( ~ 6 1 2a
Anol Tl1v TE-A~uicd~ ~01~r~ 11 =- 0
B 0 C(gt s~ 1f ta ) - ~ (~Q r gt
)
i26d
cosh (~) )
cosh (~)
I
poundfVCt(
I
A P (9 ~ nT Kn Lapllce -NtampoC--os
I
UQPC~ ( (cv [ (iulln
_ 0
E U) OQI (1 V n arpQ6CU (SO lnJ t(X )
f()lt~ 4x ) - (Xo)CFD) F or wov-d Dl fFeVnUl +l rXo ) ~
~X
f (Xc) - +(x~ - AX)-J((3)) 8ackvJltIlvd DffevelU ((x Q)
bX
f(xo +c-x) - +(~ -1 xl NtD) Ce( +vaL Oi ffevence = f (xo)
2 ~x
I OL amp(Q-flo~E~ 61 (Iv rtp~ce~60161 Tn ~p 06 WIS lt10 Inltgtpo0v yeJ cpOlV QIJ V
()(O UoL_W~ 6XYCl R) p B
I I
A I
I I
I I I I I
Xo X
X~-lt )( I J h)I
~-----l-~ _
1----------------shy
ITa npon()ol-vo ltxpet n 1An TM 6POtihJ Ar wPpOj rnshy1
I I
i b8ckwovdJmiddotr~eYer)CQ ) n krI(n l~J PI3 Wfp~jL( Cnv ~VoQvdmiddot dttefenQ i
I
I I
pncfCI-gtA 110lt f3PO~I)E npaEf6(riff~
I I
fex o + 6X2 ) - f (X- A~2) Ceo ) ~)lt
-rCx )- -r ex -A)l)] _ llx
rex + Clx) - 2 fcxltraquo + fc Xo - ~)lt)
(hX) 2
A amppoi j o V TO-f
f(xc+6x) t- ~cxo-()x) -=- 2f=(xa) -t(t~if(x) + o (1x)4
Oro) 0(6gtlt)4 anl1ol)lt~ 6(0 6ltpoll~a ~TQ n 6Vpc 6o_1H~n((D)~
I I n e-ponQ VW ~A-flPClSVJ) ~xa01~
+(x a -+ 6 X ) - rex c - 6 x)
2 ~)(
+(gtlt0+ ClX) - 2 + (Xo) + f (Xc -6)lt)
(fJj )2
- l-
)IvJpO vl) eSy))
fllfl -I shyUUJ UUJ ex ncn gt0 gt0 ~N
I~ 3ol IN IN
)lj - J ~y
UCgt-( ltb(gtlti)Yj)= ltPj
-----------7gt X XL
2 1)CXc+6X) - 2 Sf) (XiYj) + p(x(-ax1_i) _d lt) C15iH JJ - 2 cD -+ P(_f~ a x 2
(6X)2 (6xF 1lt1) j
$)(X~)lj+pound)) - 2 ltP(XiJYj) + ~(xL~J-tiy) ~ ~Cj-+I - ZcentIJ + ~ l~j-I~l all- (~y)2(~yF
- x 1 ) J)
=
~ ( 1 2)I - +_ -I- q)l l + ~ - =())J l1x 2 ll )J + t
Y ~y2 Jj - a
dltP I -s - - vwctlln 01) 5
lAllXJ =lb]
Uvo 0 niv()(L(~ 1wV ~UVlr~CgtTwv
ZA cos (~a) =0 -) A I Cos ft-a) 0
r--) 2 A2 SI n Ckd) a I A2 SIn (b) =0
~
T 01 E () I 0 V C U Q v c n 0 I U t Cl( ()
ItlIno (flv EJ (2) A2 a Z 6LE 010( VO( U (j vcshy
1lT ~ I D COJ 1 t S (1) COS Cka) = a -) k a = (Zn+ I ) n =J
2
nTr 111 )12- A1 =- 0 ) kr = - (n~f) l)rr) x) (x ) An sin (-r xja 2~- A2 = 0) 2d ) shykl) = (2Vl+I) (no)~) X1 (x ) 13 n co S ((2rl n) ~ )( )
Idltbl- ) 0 ~(lo 1J cvv d ~ j- I
( 2l1-t I ) h I 2Y +J )ltPI-) ( XI y=b ) - ~ti cos -- gtlt COS i - 1Tb ~ 2) 2Q
_ Po (~)2 CQS ( ~ 6 1 2a
Anol Tl1v TE-A~uicd~ ~01~r~ 11 =- 0
B 0 C(gt s~ 1f ta ) - ~ (~Q r gt
)
i26d
cosh (~) )
cosh (~)
I
poundfVCt(
I
A P (9 ~ nT Kn Lapllce -NtampoC--os
I
UQPC~ ( (cv [ (iulln
_ 0
E U) OQI (1 V n arpQ6CU (SO lnJ t(X )
f()lt~ 4x ) - (Xo)CFD) F or wov-d Dl fFeVnUl +l rXo ) ~
~X
f (Xc) - +(x~ - AX)-J((3)) 8ackvJltIlvd DffevelU ((x Q)
bX
f(xo +c-x) - +(~ -1 xl NtD) Ce( +vaL Oi ffevence = f (xo)
2 ~x
I OL amp(Q-flo~E~ 61 (Iv rtp~ce~60161 Tn ~p 06 WIS lt10 Inltgtpo0v yeJ cpOlV QIJ V
()(O UoL_W~ 6XYCl R) p B
I I
A I
I I
I I I I I
Xo X
X~-lt )( I J h)I
~-----l-~ _
1----------------shy
ITa npon()ol-vo ltxpet n 1An TM 6POtihJ Ar wPpOj rnshy1
I I
i b8ckwovdJmiddotr~eYer)CQ ) n krI(n l~J PI3 Wfp~jL( Cnv ~VoQvdmiddot dttefenQ i
I
I I
pncfCI-gtA 110lt f3PO~I)E npaEf6(riff~
I I
fex o + 6X2 ) - f (X- A~2) Ceo ) ~)lt
-rCx )- -r ex -A)l)] _ llx
rex + Clx) - 2 fcxltraquo + fc Xo - ~)lt)
(hX) 2
A amppoi j o V TO-f
f(xc+6x) t- ~cxo-()x) -=- 2f=(xa) -t(t~if(x) + o (1x)4
Oro) 0(6gtlt)4 anl1ol)lt~ 6(0 6ltpoll~a ~TQ n 6Vpc 6o_1H~n((D)~
I I n e-ponQ VW ~A-flPClSVJ) ~xa01~
+(x a -+ 6 X ) - rex c - 6 x)
2 ~)(
+(gtlt0+ ClX) - 2 + (Xo) + f (Xc -6)lt)
(fJj )2
- l-
)IvJpO vl) eSy))
fllfl -I shyUUJ UUJ ex ncn gt0 gt0 ~N
I~ 3ol IN IN
)lj - J ~y
UCgt-( ltb(gtlti)Yj)= ltPj
-----------7gt X XL
2 1)CXc+6X) - 2 Sf) (XiYj) + p(x(-ax1_i) _d lt) C15iH JJ - 2 cD -+ P(_f~ a x 2
(6X)2 (6xF 1lt1) j
$)(X~)lj+pound)) - 2 ltP(XiJYj) + ~(xL~J-tiy) ~ ~Cj-+I - ZcentIJ + ~ l~j-I~l all- (~y)2(~yF
- x 1 ) J)
=
~ ( 1 2)I - +_ -I- q)l l + ~ - =())J l1x 2 ll )J + t
Y ~y2 Jj - a
dltP I -s - - vwctlln 01) 5
lAllXJ =lb]
Uvo 0 niv()(L(~ 1wV ~UVlr~CgtTwv
i26d
cosh (~) )
cosh (~)
I
poundfVCt(
I
A P (9 ~ nT Kn Lapllce -NtampoC--os
I
UQPC~ ( (cv [ (iulln
_ 0
E U) OQI (1 V n arpQ6CU (SO lnJ t(X )
f()lt~ 4x ) - (Xo)CFD) F or wov-d Dl fFeVnUl +l rXo ) ~
~X
f (Xc) - +(x~ - AX)-J((3)) 8ackvJltIlvd DffevelU ((x Q)
bX
f(xo +c-x) - +(~ -1 xl NtD) Ce( +vaL Oi ffevence = f (xo)
2 ~x
I OL amp(Q-flo~E~ 61 (Iv rtp~ce~60161 Tn ~p 06 WIS lt10 Inltgtpo0v yeJ cpOlV QIJ V
()(O UoL_W~ 6XYCl R) p B
I I
A I
I I
I I I I I
Xo X
X~-lt )( I J h)I
~-----l-~ _
1----------------shy
ITa npon()ol-vo ltxpet n 1An TM 6POtihJ Ar wPpOj rnshy1
I I
i b8ckwovdJmiddotr~eYer)CQ ) n krI(n l~J PI3 Wfp~jL( Cnv ~VoQvdmiddot dttefenQ i
I
I I
pncfCI-gtA 110lt f3PO~I)E npaEf6(riff~
I I
fex o + 6X2 ) - f (X- A~2) Ceo ) ~)lt
-rCx )- -r ex -A)l)] _ llx
rex + Clx) - 2 fcxltraquo + fc Xo - ~)lt)
(hX) 2
A amppoi j o V TO-f
f(xc+6x) t- ~cxo-()x) -=- 2f=(xa) -t(t~if(x) + o (1x)4
Oro) 0(6gtlt)4 anl1ol)lt~ 6(0 6ltpoll~a ~TQ n 6Vpc 6o_1H~n((D)~
I I n e-ponQ VW ~A-flPClSVJ) ~xa01~
+(x a -+ 6 X ) - rex c - 6 x)
2 ~)(
+(gtlt0+ ClX) - 2 + (Xo) + f (Xc -6)lt)
(fJj )2
- l-
)IvJpO vl) eSy))
fllfl -I shyUUJ UUJ ex ncn gt0 gt0 ~N
I~ 3ol IN IN
)lj - J ~y
UCgt-( ltb(gtlti)Yj)= ltPj
-----------7gt X XL
2 1)CXc+6X) - 2 Sf) (XiYj) + p(x(-ax1_i) _d lt) C15iH JJ - 2 cD -+ P(_f~ a x 2
(6X)2 (6xF 1lt1) j
$)(X~)lj+pound)) - 2 ltP(XiJYj) + ~(xL~J-tiy) ~ ~Cj-+I - ZcentIJ + ~ l~j-I~l all- (~y)2(~yF
- x 1 ) J)
=
~ ( 1 2)I - +_ -I- q)l l + ~ - =())J l1x 2 ll )J + t
Y ~y2 Jj - a
dltP I -s - - vwctlln 01) 5
lAllXJ =lb]
Uvo 0 niv()(L(~ 1wV ~UVlr~CgtTwv
I
A P (9 ~ nT Kn Lapllce -NtampoC--os
I
UQPC~ ( (cv [ (iulln
_ 0
E U) OQI (1 V n arpQ6CU (SO lnJ t(X )
f()lt~ 4x ) - (Xo)CFD) F or wov-d Dl fFeVnUl +l rXo ) ~
~X
f (Xc) - +(x~ - AX)-J((3)) 8ackvJltIlvd DffevelU ((x Q)
bX
f(xo +c-x) - +(~ -1 xl NtD) Ce( +vaL Oi ffevence = f (xo)
2 ~x
I OL amp(Q-flo~E~ 61 (Iv rtp~ce~60161 Tn ~p 06 WIS lt10 Inltgtpo0v yeJ cpOlV QIJ V
()(O UoL_W~ 6XYCl R) p B
I I
A I
I I
I I I I I
Xo X
X~-lt )( I J h)I
~-----l-~ _
1----------------shy
ITa npon()ol-vo ltxpet n 1An TM 6POtihJ Ar wPpOj rnshy1
I I
i b8ckwovdJmiddotr~eYer)CQ ) n krI(n l~J PI3 Wfp~jL( Cnv ~VoQvdmiddot dttefenQ i
I
I I
pncfCI-gtA 110lt f3PO~I)E npaEf6(riff~
I I
fex o + 6X2 ) - f (X- A~2) Ceo ) ~)lt
-rCx )- -r ex -A)l)] _ llx
rex + Clx) - 2 fcxltraquo + fc Xo - ~)lt)
(hX) 2
A amppoi j o V TO-f
f(xc+6x) t- ~cxo-()x) -=- 2f=(xa) -t(t~if(x) + o (1x)4
Oro) 0(6gtlt)4 anl1ol)lt~ 6(0 6ltpoll~a ~TQ n 6Vpc 6o_1H~n((D)~
I I n e-ponQ VW ~A-flPClSVJ) ~xa01~
+(x a -+ 6 X ) - rex c - 6 x)
2 ~)(
+(gtlt0+ ClX) - 2 + (Xo) + f (Xc -6)lt)
(fJj )2
- l-
)IvJpO vl) eSy))
fllfl -I shyUUJ UUJ ex ncn gt0 gt0 ~N
I~ 3ol IN IN
)lj - J ~y
UCgt-( ltb(gtlti)Yj)= ltPj
-----------7gt X XL
2 1)CXc+6X) - 2 Sf) (XiYj) + p(x(-ax1_i) _d lt) C15iH JJ - 2 cD -+ P(_f~ a x 2
(6X)2 (6xF 1lt1) j
$)(X~)lj+pound)) - 2 ltP(XiJYj) + ~(xL~J-tiy) ~ ~Cj-+I - ZcentIJ + ~ l~j-I~l all- (~y)2(~yF
- x 1 ) J)
=
~ ( 1 2)I - +_ -I- q)l l + ~ - =())J l1x 2 ll )J + t
Y ~y2 Jj - a
dltP I -s - - vwctlln 01) 5
lAllXJ =lb]
Uvo 0 niv()(L(~ 1wV ~UVlr~CgtTwv
1----------------shy
ITa npon()ol-vo ltxpet n 1An TM 6POtihJ Ar wPpOj rnshy1
I I
i b8ckwovdJmiddotr~eYer)CQ ) n krI(n l~J PI3 Wfp~jL( Cnv ~VoQvdmiddot dttefenQ i
I
I I
pncfCI-gtA 110lt f3PO~I)E npaEf6(riff~
I I
fex o + 6X2 ) - f (X- A~2) Ceo ) ~)lt
-rCx )- -r ex -A)l)] _ llx
rex + Clx) - 2 fcxltraquo + fc Xo - ~)lt)
(hX) 2
A amppoi j o V TO-f
f(xc+6x) t- ~cxo-()x) -=- 2f=(xa) -t(t~if(x) + o (1x)4
Oro) 0(6gtlt)4 anl1ol)lt~ 6(0 6ltpoll~a ~TQ n 6Vpc 6o_1H~n((D)~
I I n e-ponQ VW ~A-flPClSVJ) ~xa01~
+(x a -+ 6 X ) - rex c - 6 x)
2 ~)(
+(gtlt0+ ClX) - 2 + (Xo) + f (Xc -6)lt)
(fJj )2
- l-
)IvJpO vl) eSy))
fllfl -I shyUUJ UUJ ex ncn gt0 gt0 ~N
I~ 3ol IN IN
)lj - J ~y
UCgt-( ltb(gtlti)Yj)= ltPj
-----------7gt X XL
2 1)CXc+6X) - 2 Sf) (XiYj) + p(x(-ax1_i) _d lt) C15iH JJ - 2 cD -+ P(_f~ a x 2
(6X)2 (6xF 1lt1) j
$)(X~)lj+pound)) - 2 ltP(XiJYj) + ~(xL~J-tiy) ~ ~Cj-+I - ZcentIJ + ~ l~j-I~l all- (~y)2(~yF
- x 1 ) J)
=
~ ( 1 2)I - +_ -I- q)l l + ~ - =())J l1x 2 ll )J + t
Y ~y2 Jj - a
dltP I -s - - vwctlln 01) 5
lAllXJ =lb]
Uvo 0 niv()(L(~ 1wV ~UVlr~CgtTwv
)IvJpO vl) eSy))
fllfl -I shyUUJ UUJ ex ncn gt0 gt0 ~N
I~ 3ol IN IN
)lj - J ~y
UCgt-( ltb(gtlti)Yj)= ltPj
-----------7gt X XL
2 1)CXc+6X) - 2 Sf) (XiYj) + p(x(-ax1_i) _d lt) C15iH JJ - 2 cD -+ P(_f~ a x 2
(6X)2 (6xF 1lt1) j
$)(X~)lj+pound)) - 2 ltP(XiJYj) + ~(xL~J-tiy) ~ ~Cj-+I - ZcentIJ + ~ l~j-I~l all- (~y)2(~yF
- x 1 ) J)
=
~ ( 1 2)I - +_ -I- q)l l + ~ - =())J l1x 2 ll )J + t
Y ~y2 Jj - a
dltP I -s - - vwctlln 01) 5
lAllXJ =lb]
Uvo 0 niv()(L(~ 1wV ~UVlr~CgtTwv