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EIOOIHH.YE 34/2A 8-06-06 11:36 1

EIOOIHH.YE 34/2A 8-06-06 11:36 2

OYE TI YIKE EITHME

B

A

ATPA 2005

EIOOIHH.YE 34/2A 8-06-06 11:36 3

B

A

EYT.

, ,

IANNH K. IKPAMENO

/ 19972005

ISBN: 9605386291

K : YE 34/21

Copyright 2005

16 . , 26222 .: (2610) 367336, 367355, : (2610) 361420

. 2121/1993,

, .

EIOOIHH.YE 34/2A 8-06-06 11:36 4

M

.............................. 17

K 1

:

, , , .................................................................................................................................. 21

1.1 : .................................................................................................... 23

1.2 - .................................................. 27

1.2.1 .................................................................................................... 27

1.2.2 ..................................................................................................... 31

1.2 .................................................................................................................................. 35

1.3 ............................................... 36

1.3.1 ............................................................................................................................ 36

1.3.2 .................................................................................................................................... 40

1.4 ............................................................................................. 42

1.5 ..................................................................... 48

1.6 (1.27): ........................................................................................................................ 51

...................................................................................................................................................... 56

K 2

:

, , , .................................................................................................................................. 59

EIOOIHH.YE 34/2A 8-06-06 11:36 5

6 TA A N T E I K A I K Y M ATA

2.1 : ................................................................................... 61

2.2 ....... 64

2.3 ................................................................... 71

2.4 ........................................................................................... 78

2.5 ............ 83

2.6 ..................................................... 86

2.6.1 ....................... 87

2.6.2 ......... 90

2.6.3 ............. 95

2.6.4 .......................................................................................................... 98

2.7 .............................................................. 99

2.7.1 .................................. 100

2.7.2 ............................. 105

2.8 .............................................. 107

* 1

2.9 ................................................................................................................. 119

2.9.1 .............................................................................................................. 119

2.9.2 ............................................................................................................... 122

2.9.2.1 .............................................................. 122

2.9.2.2 ............................................................................. 124

2.9.3 ......................................... 127

2.9.4 .................................. 128

2.9.4.1 ................................................................................................................ 128

2.9.4.2 .......................................................................................................................... 133

.................................................................................................................................................. 139

K 3

, , , .............................................................................................................................. 143

3.1 : ....................... 145

3.2 .............................................................. 147

3.3 ...................................................................... 154

EIOOIHH.YE 34/2A 8-06-06 11:36 6

3.4 () ....................................................................................... 167

* 2

3.5 ............................................................................................................................... 173

3.5.1 ................................................................................................................. 173

3.5.2 ........................................................................................................... 173

3.5.3 .................................................................... 174

3.5.4 ................................................................................... 177

3.5.5 ................................................................................. 177

3.6 .............................................................................................. 179

3.6.1 RLC ........................................................................................ 179

3.6.2 ...................................................................................... 180

3.7 ( ) .................... 184

3.7.1 ............................................................................................ 184

3.7.2 ................................ 186

3.7.3 ................................... 187

* 3

3.8 .............................................................................................................. 190

.................................................................................................................................................. 193

K 4

, , , .............................................................................................................................. 197

4.1 ........................................................................................................................................................... 199

4.2 ............................................................. 199

4.3 ( ) ............................. 215

* 4

4.4 : - .......................................................................................... 222

4.5 ......................................... 234

.................................................................................................................................................. 237

7 E P I E X O M E N A A '

EIOOIHH.YE 34/2A 8-06-06 11:36 7

K 5

( )

, , , .............................................................................................................................. 239

5.1 ........................................................................................................................................................... 241

5.2 .................................................................................................................. 241

5.3 .................................................................................................................... 267

5.4 .............................................................. 272

5.4.1 .............................................................................................................................. 272

5.4.2 .............................................................................. 281

* 5

5.4.3 .................................................................................................................... 285

5.4.4 ................................................... 291

* 6

5.5 : Fourier ............................................................................................................................................... 295

5.5.1 ............................................................................................. 295

5.5.2 Fourier ...................... 296

5.5.3 ............................................... 303

5.5.4 ( ) ............................................................................................... 305

5.5.5 [0,L]. ........................................................................................................... 306

5.5.6 . .......................................................................................................................... 308

5.6 .............................................................................. 313

5.7 .................................................................................................................. 316

K .................................................................................................................................................. 320

.................................................................. 323

8 TA A N T E I K A I K Y M ATA

EIOOIHH.YE 34/2A 8-06-06 11:36 8

M

K 6

, , , .................................................................................................................................. 15

6.1 ........................................................................................................ 19

6.1 .................................................................................................................................. 35

6.2 ...................................................................... 36

6.2.1 ................................................................................................................................................ 36

6.2.2 : .......... 37

6.2.3 : .............................................................................................................................. 41

6.3 .......................................... 44

6.3.1 : ...................................................... 44

6.3.2 : ................................................................. 45

* 7

6.4 ............................................ 47

6.4.1 : .......................................................... 47

6.4.2 : ......................................... 51

6.4.3 : ............................... 57

6.4.4 ............................................................ 60

6.4 .................................................................................................................................. 63

6.5 : () .................. 65

* 8

6.5.1 .............. 65

6.5.2 ( ) .................................. 72

6.5.3 ( ) .......... 74

6.5.4 ....................................................... 81

9 E P I E X O M E N A B '

EIOOIHH.YE 34/2A 8-06-06 11:36 9

1 0 TA A N T E I K A I K Y M ATA

* 9

6.5.5 .......................................................................... 84

6.5.5.1 .................... 84

6.5.5.2 ............................................................................................... 85

6.5.5.3 .............................................................. 87

6.5.5.4 ....................... 88

6.5.5.5 ( , ) ................................................................. 88

6.5.5.6 ( ): ................................... 89

6.6 ............................................................................................................................. 93

* 10

6.7 2: .......................................... 100

6.7.1 ............................ 100

6.7.2 ........................................... 104

6.8 Schrdinger ...................................................................................................... 105

.................................................................................................................................................. 107

K 7

, , , .............................................................................................................................. 109

7.1 .......................................................................................................................... 113

7.1.1 .............................................................................................................................. 113

7.1.2 ................... 119

7.1.3 . ............................................................................................................ 123

7.1.4 = ( ) ............................................................................................... 133

7.2 ........................................................................................................................................... 136

EIOOIHH.YE 34/2A 8-06-06 11:36 10

7.2.1 ..................................................................... 136

7.3 ........................... 144

7.3.1 . ............................................................................... 144

7.3.2 () ................................................................................ 147

7.4 . .............................................................................................................................................. 150

* 11

7.5 () . Fourier .......................................................................................................................................... 157

7.5.1 ............................................................................ 157

7.5.2 .................................................................... 160

7.5.3 ........................................ 178

7.5.4 .................................................................... 185

7.5.5 Fourier ................ 186

7.6 Huygens ............................................................................................................................... 190

7.6.1 ............................................................................................................................................. 190

7.6.2 .......................................................................................................................................... 190

7.6.3 ......................................................................................................................................... 192

7.6.4 .......................................................................................................................................... 193

7.6 ............................................................................................................................... 195

.................................................................................................................................................. 196

K 8

, , , .............................................................................................................................. 199

8.1 ...................................................................................................................... 202

8.2 ................... 206

8.3 ............................... 210

8.3.1 : , ......................... 210

8.3.2 () ........................................ 218

8.3.3 ............................................................... 220

1 1 E P I E X O M E N A B '

EIOOIHH.YE 34/2A 8-06-06 11:36 11


8.3.4 - : ....................... 221

8.4 .................................. 224

8.4.1 ...................................................................... 224

8.4.2 ................................................. 228

8.4.3 ............................................................................................................... 231

8.4.4 ..................................... 232

8.5 ........................................... 234

8.5.1 ......................................................................................................... 234

8.5.2 .................................................. 238

8.5.3 .......................................................................................................... 240

8.5.4 : Michelson ..................................................... 241

8.6 () ............... 245

8.6.1 ............................................................................................................................................. 245

8.6.2 ( ) ............................................................................................................... 247

8.6.3 ( ) ............................................................................................................... 248

* 12

8.6.4 ( ) : ......................... 253

* 13

8.6.5 ( ) : ............... 256

8.7 ................................................................................................... 262

8.7.1 : ............................................................................ 262

8.7.2 .................................................... 263

8.7.3 Fraunhofer ....................... 266

8.7.4 ........................................................................................................................ 273

EIOOIHH.YE 34/2A 8-06-06 11:36 12

* 14

8.7.5 Fourier ............................................................................................. 276

8.7.6 .............................................................. 277

8.7.7 ................................................................... 281

8.7.8 ......................................................................................................................... 286

8.8 ............................................................................... 290

8.9 : ( Fermat) ..................................................................................................................................... 294

8.9.1 ............................................................................................................................................. 294

* 15

............ 298

8.9.2 () Fermat ......................................................................................................... 300

8.9.3 , Fermat .......................................................................................................................................... 303

* 16

8.9.4 Fermat ........................................... 307

.................................................................................................................................................. 313

.................................................................................................... 315

........................................................................ 321

1 3 E P I E X O M E N A B '

EIOOIHH.YE 34/2A 8-06-06 11:36 13

EIOOIHH.YE 34/2A 8-06-06 11:36 14

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EIOOIHH.YE 34/2A 8-06-06 11:36 23

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EIOOIHH.YE 34/2A 8-06-06 11:36 26

1.2

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EIOOIHH.YE 34/2A 8-06-06 11:36 27

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EIOOIHH.YE 34/2A 8-06-06 11:36 28

, 1.5.

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x.

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EIOOIHH.YE 34/2A 8-06-06 11:36 29

1.7

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EIOOIHH.YE 34/2A 8-06-06 11:36 30

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x xa

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u

EIOOIHH.YE 34/2A 8-06-06 11:36 31

3 2 1 : :

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()

x=x0, .

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x

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T

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T

T

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t = 0

t = T

t =T4

t =2T4

t =3T4

EIOOIHH.YE 34/2A 8-06-06 11:36 32

3 31 . 2

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= (1.1)

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EIOOIHH.YE 34/2A 8-06-06 11:36 34

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EIOOIHH.YE 34/2A 8-06-06 11:36 35

3 6 1 : :

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2pl

EIOOIHH.YE 34/2A 8-06-06 11:36 36

.

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(x,t) = (x,t + mT), m = 1, 2, 3,

3 71 . 3

EIOOIHH.YE 34/2A 8-06-06 11:36 37

3 8 1 : :

(1.5)

f(x) f(x), >0. ( 1.5) ,

x f(x) x=x, -

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x. =t, (1.5)

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pn= =

22

T

Y ( , ) sin[ ( )]x t A k x t= - u

k 2p

l

f x f x t( ) ( ).= - u

EIOOIHH.YE 34/2A 8-06-06 11:36 38

(1.7) :

(1.9)

(1.7) (1.9):

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EIOOIHH.YE 34/2A 8-06-06 11:36 39

4 0 1 : :

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x t

Tcos

EIOOIHH.YE 34/2A 8-06-06 11:36 40

. x

(, ) ,

:

x t = (1.14)

() (+), ,

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).

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EIOOIHH.YE 34/2A 8-06-06 11:36 41

4 2 1 : :

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=

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x

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x

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=

= -

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x u

x u= -x t

EIOOIHH.YE 34/2A 8-06-06 11:36 42

(1.16) (1.17), :

(1.18)

(1.18)

(

) , (x,t), -

x. (1.18)

(1.5) :

(x,t) = Cf(xt) (1.5)

C ,

.

, , ( )

(1.2) (1.9) (1.18). ,

(1.2) (1.9) (1.18), ,

() :

(1.19)

(1.19) (1.1).

( )

x ( ). [7]

x ( -

) :

(x,t) = g(x+t) (1.20)

( 1.11)

(1.20) :

(1.21)

x}

uY Y( , ) ( , )x t

t

x t

x=

x}

u

l w

= =

T k

x}

uY Y( , ) ( , )x t

t

x t

x= -

4 31 . 4

[7] g(x+a) x ( -

) a ( a=t) g(x).

EIOOIHH.YE 34/2A 8-06-06 11:36 43

4 4 1 : :

(1.9) -

(1.21) :

(1.22)

(1.18) (1.21), -

, -

, 1.4, 1.5,

.

-

, x, y ( -

x) -

. , :

(1.23)

(1.18) (1.19) :

(1.24)

(1.24), :

(1.25)

, (1.24) x, :

(1.26)

(1.23).

(1.25) (1.26),

d F

d

2

2

( )x

x

=

x

x1,

=

x

u

t

x x

x

x

x

2

2

2

2

2

21

Y ( , )( )

( ) ( )x t

x x

d F

d

d F

d= +

=

x

u

x

x

u

x

x

u

x

x

2

2

2

22

2

22

2

2

Y Y( , ) ( , ) ( )( )

( ) ( )x t

t t

x t

t t

d F

d

d F

d

d F

d=

=

= =

x x

x

u

x

x

Y ( , ) ( ) ( )x t

t t

dF

d

dF

d=

=

u

Y Y( , ) ( , )x t

t

x t

x=

x u= x t

x} Y ( , ) sin( )x t A kx t= + w

EIOOIHH.YE 34/2A 8-06-06 11:36 44

(1.27) [8] , -

(1.5) (1.20) . , ,

, (1.27),

. , (1.27) :

(x,t) = c1 f (x t) + c2 g(x + t) (1.28)

c1, c2 / .

(1.28) , 1.3, -

.

, , -

(1.27), n () , 1, 2, n, -

, , , -

() :

(1.29)

[

(1.18), (1.20), (1.27) ],

, -

, :

Y ( , ) ( , )x t c x tn n

i

n

=

=

y

1

(1.27)

}

4 51 . 4

: (1.27)

u

2

2 2

2

2

10

Y Y( , ) ( , )x t

x

x t

t- =

[8] , -

, .

1.6: (1.27), -

, (1.28);

: (1.27) -

-

. ,

, -

EIOOIHH.YE 34/2A 8-06-06 11:36 45

4 6 1 : :

. , -

, ,

.

, -

(x) (t) -

. -

[9]

1.7: (1.27) (1.28);

: (1.27) ,

. (1.28)

(1.27). (1.28), -

, f g,

(),

. (1.27) -

. (1.27) / , -

,

(1.28). , ,

.

1.8: (1.27) , , ,

;

: , (1.27) -

, , . .

, ,

Maxwell, -

, (1.27),

-

. , , -

1.14.

1.9: (1.18)

(1.21). . (1.18) (1.21)

(1.27);

[9] ,

, .

EIOOIHH.YE 34/2A 8-06-06 11:36 46

: ( )

( ) ,

.

: , (1.18) (1.21) , (1.27) -

. (1.18) (1.21)

( ,

) (1.27).

4 71 . 4

-

. f g

(1.28) (1.9) (1.22) ,

, :

(1.30)

(x,t) -

, y(x)=2Ainkx, -

, (t)=cost. (1.30)

1.5 (t). ,

, (-

) , ,

.

(1.29), , -

. , -

Beethoven

, -

.

( -

). ,

( -

), .

,

.

Y ( , ) sin( ) sin( ) sin cos ( ) ( )x t A kx t A kx t A kx t y x t= - + + = w w w j2

EIOOIHH.YE 34/2A 8-06-06 11:36 47

4 8 1 : :

* 1.5

-

( ). -

, . -

, .

. ,

1.11, ,

, . -

, ,

( , 1.11 ,

, ). , -

. , -

, . ,

, .

, (.. -

). -

x x+x. x,

S , , Sx,

Sx, .

x

x + T F + FS

S()

x

x

x + x

x + T x + x- + + T

() F 1.11

.

,

.

(, , ) -

EIOOIHH.YE 34/2A 8-06-06 11:36 48

.

x F, , x+x,

F+F. F. F

. -

x x+x. F,

x x+, -

x+x x+x++, x.

, x x+

, . , ,

Hooke, F/S ( F

S) . -

, :

(1.31)

(Y) Young.

( x) , :

(1.32)

,

(1.32) x, :

(1.33)

(F)

Sx

. :

(1.34) F S x

t=

r

x

2

2

g

x

=

2

2t

F s x= r g ,

=

FSY

x x

2

2

x

F =

s y

x

x

F

SY

x=

x

F

SY Y

x= =e

x

x

x.

e =

l

l,

4 91 . 5

EIOOIHH.YE 34/2A 8-06-06 11:36 49

5 0 1 : :

, :

(1.35)

Taylor F x=0,

[ x (x)n, n 2, -].

(1.35), (1.33) (1.34), :

(1.36)

, :

(1.37)

, (1.37) -

() :

(1.38)

( -

Young) () .

(1.27),

6.

u

r

=

Y

u

r

2=

Y

=

2

2

2

2

x x x

x Y t

SY

xS

t

=

2

2

2

2

x

r

x

F

F

xx=

EIOOIHH.YE 34/2A 8-06-06 11:36 50

* 1.6 (1.27):

, , , -

, (

) . : ,

, ,

. , -

, ( ).

-

. , 6, -

(1.27).

( ), . -

, ( )

(1.39)

. -

() . , -

1.1.

, ,

,

(1.40)

6.

. -

4 5. , ,

( , ) -

, ,

. -

, , 6.

u

lr

e

p

=

2 T

u

l

b

p

=

g

2

5 11 . 6 ( 1 . 2 7 ) :

EIOOIHH.YE 34/2A 8-06-06 11:36 51

5 2 1 : :

.

x t,

, =

, .

,

-

-

. -

: ,

-

1.4

(1.2) x

t = 0, ,

1.9. -

.

4

4

T

T T T

4

2

4

3

4, ,

1.5

.

Y ( , ) sin .x t A

x t

T= +

2pl

1.6

EIOOIHH.YE 34/2A 8-06-06 11:36 52

5 31 . 6 ( 1 . 2 7 ) :

; ( )

) (1.27)

.

)

.

)

.

)

.

) (1.27)

,

.

)

(1.16) (1.19).

)

.

) (1.27)

1.4.

- -

22

p

p

l

A

T

x t

Tcos

1.7

EIOOIHH.YE 34/2A 8-06-06 11:36 53

5 4 1 : :

.

(x,t) = f(x t)

(x,t) = g(x+t)

(x+n,t) = (x,t), n = 1, 2, 3,

u

y x t

t

y x t

x

( , ) ( , )= -

Y ( , ) sin( )x t A kx t= - w

- -

22

p

p

l

A

T

x t

Tcos

Y x t B kx t( , ) sin cos= w

j u w - = -k( )x t kx t

2

22

2

2

E x t

xA

E x t

t

( , ) ( , )=

w

k

u

Y Y( , ) ( , )x t

t

x t

x=

Y x t A

x t

T( , ) sin= +

2pl

1.8

, CGS -

=1.0 sin (3x2t) cm.

: () , () , () ,

() , () ()

.

1.9

EIOOIHH.YE 34/2A 8-06-06 11:36 54

5 51 . 6 ( 1 . 2 7 ) :

-

x t, x=1/3 cm t=0.5 sec.

1.10

(1.2) x

t . (1.3) (1.4).

=10 m/sec, (x,t)

xt=, xt=10 m. -

(x,t) (1.2) ; ;

1.11

-

() x

;

u

Y Y( , ) ( , ).

x t

t

x t

x=

1.12

() (),

(1.27).

: (1.14).

1.13

Maxwell, Faraday

Ampere, ( )[10] -

(=0, J=0): (1.29)

m e

E

x

B

tB

x

E

t

= -

= - 0 0

1.14

[10] , , 10 ., , ,

(2001).

EIOOIHH.YE 34/2A 8-06-06 11:36 55

5 6 1 : :

y

z. 0 0

.

S ( SA) :

0=8.8541012C2/Nt.m2 0 = 410

7 Nt/A2

-

. :

(1.30) (1.31)

(1.32)

(1.33)

. -

. ;

c m= =

12 998 10

0 0

8

m e

. / sec

2

2 2

2

2

1B

x c

B

t=

2

2 2

2

2

1E

x c

E

t=

m e

2

2 0 0

2

2

E

x

E

t=

1

48 988 10

0

9 2 2

pe

. . /x Nt m C

,

.

;

1.15

1) ( ) -

. ,

() .

2) ,

, .

3) ,

EIOOIHH.YE 34/2A 8-06-06 11:36 56

. .

, ,

,

.

4) :

) (.. , , ) ) -

(.. ) )

, () () (..

, , ).

5)

(x,t), -

, (

) -

.

6) ( ) :

(1.27)

(-

, ) (), .

,

, .

(

) .

7) (1.27), -

,

(x,t)=f(xt), - . ,

( n) , , .

,

. .

8) (1.27), -

Y x t c x tn n

i

n

( , ) ( , ).==

y

1

y n x t( , )

u

2

2 2

2

2

10

Y x t

x

Y x t

t

( , ) ( , )- =

5 7 Y N O H K E A A I O Y

EIOOIHH.YE 34/2A 8-06-06 11:36 57

5 8 1 : :

, (x,t)=c1 f(xt)+c2g(x+t).

-

f g.

c1 c2 .

c1=c2 f=g, , -

. -

, , .

9) -

.

( )

, x -

( ) t (

). , -

,

. .

.

() (+).

10) () , -

(1.27), =

=/k.

()

,

(=kxt). () -

-

.

w

p

pn= =

22

T k 2p

l

Y ( , ) sin( )x t A kx t= - w

Y ( , ) sin .x t A

x t

T= -

2pl

EIOOIHH.YE 34/2A 8-06-06 11:36 58

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