Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ofvp.phys.msu.ru/upload/iblock/c19/ekbtshjo_.pdf Äàòà èçìåíåíèÿ: Thu Sep 3 15:57:56 2009 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:34:15 2012 Êîäèðîâêà: ISO8859-5

. ..



IV

.. , ..


2007

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. ..


..

IV

.. , ..



2007

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3


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5. . . . . . . . . . . . . . . . . . . . . . . ?6. . . . . . . . . . . . . . . . . . . . . . ?7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?8. . . . . . . . . . . . . . . ?9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6 24 37 43 57 70 85 99 108 129

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4




, () . .. . (?Ë) (? Ë) 2006 . . , , , . , , . , . 150 . . . , , , . . . , 1 ( ) 3 ( ). " " . " " , , .

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5

. "" . , , . " " , , . . "" , , . . ". " . " . " , . "" , -. , . , . , . .. , .-. .. , .. .. , .. .. . , . .. .. , , .. . ..

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6

?1.

?1.
. . . , . - , . , . , . , . , , , , , , , . (r, t ) , . , , : 2 (r , t ) - c 2 (r , t ) = 0 , (1.1) t 2 c - . ( x, t ) : =0. (1.2) t 2 x 2 , , OX : (1.3) ( x, t ) = a 0 cos( t - kx + 0 ) 2 ( x, t ) -c
2

2 ( x, t )

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?1.

7

(1.4) ( x, t ) = Re{a0 e i ( t - kx + 0 ) } . a0 - , 0 - , Re . T : (1.5) = 2 / T . , (1.6) = 1 / T = / 2 . k : (1.7) k = 2 / . : (1.8) = cT . (1.5), (1.7) (1.8) : (1.9) k = / c = c/ . (1.3) (1.4) , OX. , OX, ( x, t ) : (1.10) ( x, t ) = Re{a e i ( t + kx + 0 ) } .
0

(1.3), (1.4) (1.10) ( x, t ) , . t = t* . (1.3), (1.4), (1.10) , , OX. , , , . (r , t ) : (1.11) a (r , t ) = 0 ei (t - kr + 0 ) , r r - . , , :

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8 ( r , t ) = a

?1.

0 i ( t - k + 0 ) e , (1.12) - . k . (1.13) w + (divS) = 0 , t (1.14) wd + Sd = 0 , t






w - , S - , - , . S k . w S : (1.15) S = cw . : [t] = , [] = , [] = /, [] = , [k] = -1, [c] = /c, [w] = /3, [S] = /2. p
2 p = c0 , : 2 2 - c0 = 0. (1.16) t 2 0 0 : (1.17) 1 2 0 = . 0

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?1. , : p 2 0 = 0 , 0 p0 - , - 1,4

9 . (1.18) .

( p 0 = 1,01 10 5 ; 0 = 1,3 /3) c0 330 /. Z, -, : (1.19) Z = 0 c 0 . V p , : (1.20) p = Z V . w : 1 1 w = 0 2 + p 2 2 2 S = pV w : w = 0V 2 ; w = p 2 ; S = 0 c 0V 2 ; S = S (1.21) (1.22) S

(1.23) p2 . 0 c0 (1.15) w S : (1.24) S = cw I , , (1.25) Z 1 2 2 I = V max I = p max , 2 2Z p max V max - .

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10

?1.

, , 16 20 . = 2 - 3 . , , , I 0 = 10-12 /2. () - I 0 A: (1.26) A = 10 lg(I / I 0 ) . , I = 1014 I 0 , A = 140 . : [] = /3; [] = 2/; [Z ] = /(2) = /, [ p ] = /2, [V ] = /. H : 2E - c 2 E = 0 . (1.27) 2 t : 1 , = (1.28) 0ÅÅ 0 0 = 8,85 10
-6 -12

()/()

-



,

Å 0 = 1,26 10 ()/() - , Å - . , = 1 Å = 1 , : (1.29) 0 = 2,998 108 /c. n c0/c,

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?1. n = Å . (1.30)

11

Z Å : Z


=

ÅÅ 0 . 0

(1.31)

Z 0 = 377 . E H : (1.32) E = Z H . , k , E, H , . = 0,39 , , = 0,76 , . - 0,55 . (, , , , ) Å = 1 , n = (1.30) (1.31) n Z : (1.33) 1 Å0 1 = Z = . n 0 nc 0 0 w ÅÅ w = 0 E 2 + 0 H 2 w 2 2 S = [EH ] (1.30)-(1.33) S = c 0 E 2 S S : (1.34) = 0 E 2 , w = ÅÅ0 H 2 , . S : = cÅÅ0 H 2 . (1.35) (1.36)

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12 (1.34) (1.36) w S , : S = cw . ) E0 : I=

?1. (1.37) I (

(1.38) c n 2 1 2 E 0 I = 0 0 E0 . 2 2Z p . , , . p : (1.39) p = w cos(e k , d) , e k - k , d - . RI : (1.40) p = (1 + RI ) w , w - . S , : (1.41) S C = 1,37 /2. 36% , . . , 46% , . Å , n . [E ] = /, - [H ] = /, [D ] = /2, [B ] = () = /2.

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?1.

13

1.1. = 15o. , l = 3 : I (/2) , a, Vmax, (V / t ) max, pmax. N = 30 , f = 50 . . . , , l . , . , , : = 2l2(1-cos(/2)). , I = N / = 6,2 10 -5 /2. (1.25) : V
max

=

2I , Z

Z = 0 0 - . I, (., , . , , .: , 1982) 0 = 103 /3 c0 1,5 103 /, = 9,2 10 -6 /. a , Vmax (V / t ) max: Vmax = 2fa (V / t ) max = 2f Vmax, V
max

: a = 2,9 10-11 ( V / t )max = 2,9 /2. , (1.20): p = ZV = 13 .

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14

?1.

1.2. 0,69 . , ( ) - . = 0,1 , W = 0,3 , - D = 5 . , ( ). . , , : () , () , () R = 0,9. . w S (1.37): w = S/c. S , W , , = D 2 / 4 . , , : 4W w= = 5,1 10 - 4 /3. 2 cD (1.36) S = c 0 E 0 2 , : S 4W E0 = = = 7,6 10 3 /. 2 c 0 c 0 D (1.40) p = w(1 + RI ) , RI - , . , R = 0; 0,9 1 p = 5,1 10 -4 ; 9,7 10-4 10,2 10 -4 , . 1.3. , d = 1 . = 0,5 . U = 10 . . . : dV m = -eE 0 cos( t ) , dt

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?1. m = 9,1 10 -31 e = 1,6 10-19 - , - , V - , eE sin(t ) V= 0 m (W ) =

15 , . .
2

mV 2 m eE 0 = . max 2 2 m , , eU. , , (W )max = eU . 2Um 2 . e (1.36): E
0 2

=

S = c 0 E 0 2 . , , , : 2 2 d 2 c 0 d Um = . P=S 4 2e , P 3,4 1012 .



1.4. = 109 /c , B 10 -6 . , E, , Smax. . (1.6) (1.9), = 2c / 1,88 . , () . (1.32), B H, , E = B c = 300 /.

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16

?1.

, (1.36) (1.38): Smax = 2 240 /2. 1.5. 30 . f = 2,5 , . . (1.26) , 30 , I = 10-9 /2. (1.19), : Z 430 /(2). (1.25) 2I 2,2 10 -6 /c. : V max = Z V , a 1,4 10 -10 . 1 1.6. (1) D = 10 N = 0,2 f = 600 . ; , , ; 2 , , 10 , . 1. . 2. . . . . . .: , 1976, . 1, . 418. 1.7. (1) , I, 1,4 106 , - 1,5 108 . , 1 ( ) 3 ( ).
1 max

= 2fa,

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?1.

17

E E. S, E E S E : 0,3 , = 10-4 , d = 5 , = 693,5 . . . , , .: , 1979, ? 4.4. 1.8. (1) N, l. I (/2), (), Vmax, a, ( V / t )max, pmax, max . () . N = 100 , l = 100 , f = 3 . () 30o, . N = 30 , l = 200 , f = 3 . 1.9. (2) , . , , , A = 0,36 (, = 0,75; = 0,24). 18%. E B , ? E B , ? ? 1.10. (1) - 3 10x10 2. S, E, w, p

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18

?1.

? , 10 100 , , . 1.11. (1) 1 = 1,06 2 = 1,09 . 20 10 /, . . .. . . .: , 1976, . 23, ? 135. .. . , . 3, ? 9,10. 1.12. (1) = 0,63 = 9 2. S = 107 /2. , 4 . , , R1 = 0,1, R2 = 1. . , , . . .. . . .: , 1976, . 34, ? 185,186. ... . .: , 1975. 1.13. (1) , , . , , 2 . , , - 20 . - , - . , = 50 /. . . .. . . .: , 1976, . 34, ? 185,186.

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?1.

19

1.14. (1) 1 = 1,06 2 = 1,09 , 30% . = 500 /. . .. . . .: , 1976, . 34, ? 185,186. .. . , . 3, ? 9,10. 1.15. (1) pmax = 2 10-5 ( ). , f = 10 ? /2. 1.16. (1) 1 5 /2. ( ) ? . .. . . .: , 1973. 1.17. (1) 1 , 10 -4 ? , ? 1.18. (1) = 100 1 . 1 , , . 1.19. (1) 500 . 60%. 1 , . . 1.20. (1) l = 100 100 . ,

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20

?1.

5 /2 ? ? 1.21. (1) N, m = 4 10-9 . , . 1.22. (2) ( ), , W = 3/2 kT. . W0 I0 , m = 0,1 ? W0 W. I0 , 10 , ? , , . 1.23. (2) , ( f ). , V, , I 250 2000 . 2 10-3 32, 2000 16000 . 10 -5 ( ) 5 250 2000 . . . , 1982, . 263. 1.24. (2) GSM 1800 915 . . 90À. , 1 . , , Imin = 10 -10 /2.

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?1.

21

1) L, . E H . 2) , L, = 2 ( ), = 6 ( ) = 9 ( ). 3) , , - L-3. : = 10 lg(I 0 / I () ) , I0 - . 1.25. (2) - "Teramobile" , (Ti:sapphire laser). 0 = 0,8 . , 0,5 , t0,5 = 70 . De -2 , e -2 , De
-2

= 5 . W = 350 .

- , , , : x2 + y 2 2 exp - t . I ( x, y, z = o, t ) = I 0 exp - 2 2 a0 0 : a0 (); 0 (); I0 (/2); E0 (/). I ( x, y , t ) x , y t


,

I ( x , y , t ) , x y . De - 2 t 0,5 a0 0 .

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22

?1.

1.26. (2) "T4-" (Table-Top-Terawatt-Ti:sapphire laser) 0 = 0,8 . , 0,5 , t0 ,5 = 150 . I0 = 1011 /2. De - 2 , e -2 , De
-2

= 3 . f = 10 .

- , , , : x2 + y 2 2 exp - t . I ( x, y , z = o, t ) = I 0 exp - 2 2 a0 0 E ( x, y, z , t ) , OZ : le -1 , OZ, , e I0; , le -1 ( ). OZ E ( x = 0, y = 0, z, t = const ) : 0; le -1 , e-1 ; l .
0,5

, 0,5

1.27. (3) W. E0 E a 5 10 9 /. , , (Ti:Sapphire - laser).

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?1. t 0,5 , () ae -2 , ae
-2

23 0 = 0,8 . W = 5 , 0,5 I0, t0,5 = 150 . e-2 ,

= 5 . f = 10 .

I ( x, y, z = 0, t ) : x2 + y 2 2 exp - t . I ( x, y , z = o, t ) = I 0 exp - 2 2 a0 0 . , . , aF = 2,5 IF (/2); EF (/); ZF DF , Ea 5 109 /.

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24

?2.

?2.
Z1 Z 2 (.(1.19) (1.31)) , , , . , . ?1Ë ?2Ë . p1 p'1 , - p2: (2.1) p1 + p1 = p 2 . ?1Ë ?2Ë, : (2.2) V1 + V1 = V2 . , : (2.3) E1t + E1t = E 2 t H 1t + H 1t = H 2 t , E1 , H1 - , E1 , H1 - , E2 , H 2 - . (2.1) (2.2) (1.20) , (2.3) (1.32) , ( p1 / p1 ; E1 / E1 ) ( p 2 / p1 ; E 2 / E1 ) Z1 , Z 2 : (2.4) p E Z - Z1 p E 2Z 2 RP , E = 1 = 1 = 2 , TP , E = 2 = 2 = p1 E1 Z 2 + Z1 p1 E1 Z 2 + Z1

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?2.

25

(2.4) : (2.5) E n - n E 2n1 RE = 1 = 1 2 , TE = 2 = . E1 n1 + n2 E1 n1 + n2 (2.4) (2.5) , Z 2 < Z1 n2 > n1 ( ) , p1 E1 . : (2.6) T p,E - R p,E = 1 . RI TI (2.4) (1.25) (1.38) : I1 Z 2 - Z1 I 4 Z1Z 2 , TI = 2 = (2.7) = , I1 Z 2 + Z1 I1 (Z 2 + Z1 ) 2 I1 , I1 I 2 - , , . RI TI : RI = I1 n2 - n1 I 4n1n2 , TI = 2 = (2.8) = . I1 n2 + n1 I1 (n2 + n1 ) 2 RI TI , . , , . , . , , : (2.9) RI + TI = 1 . RI =
2 2

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26

?2.

2.1. h2 n2 = 0,5 10-6 (- ) n3 = 1,5. E H - - , E3 E1 . , h2 = /4n2 , n2 = n3 , . : () Å1 = Å 2 = Å3 = 1 ; () n1 = 1; () , - . . E1 H1, . , - . : E 2 H 2 , - E1 H1 . E1 + E , - 1 H1 + H . 1 " " , . , () . (2.3) E1 + E1 = E 2 H 1 - H1 = H 2 . - , k, E, H , . , (1.32):

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?2.

27

E =ZH . E1 + E1 = E 2 , E1 / Z 1 - E1 / Z 1 = E 2 / Z 2 . (2.4) ( RE = E1 / E1 ) ( TE = E2 / E1 ) , RE TE : 1 + RE = TE , 1 - RE = TE Z1 / Z 2 . Z -Z 2Z 2 . RE = 2 1 , TE = Z 2 + Z1 Z 2 + Z1 (1.33), n -n 2n1 , RE = 1 2 , TE = . n1 + n2 n1 + n2 , (2.8), RI = (n1 -n2 ) 2 4n1 n 2 , TI = . , 2 (n1 + n2 ) (n1 + n2 ) 2

: RI + TI = 1 . , : , 1, 2, 3. : - , - . , , , -,
( I 11) ; , , - - (

-I
I

( 2) 1

) .. I
(2) 1



:





(3 ) 1



=I

(1) 1

+I

+I

+ ... .

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28

?2.

: - -, . , , - , I, R12, T12 . - R23 . , : I
( 2 I1 2) = T12 R23 I1 , , (, R12 ( I1 3) = = R21 (1) 1

= R12 I 1 , ,

( R12 R23 I1 2 ) ... ..). ,

( ( n = 2, I1 n +1) = R12 R23 I1 n ) . ,

I



=I

(1) 1

+I

( 2) 1

+I

( 3) 1

+ ... = I

(1) 1

+I

( 2) 1


n =0



( R12 R 23 ) n .

R12 R23 < 1 , . : T2R 1 ( ( I = I 11) + I 1 2) = I 0 ( R12 + 12 23 ) 1 - R12 R 23 1 - R12 R 23 R I = I / I 1 ,
2 T12 R23 . , 1 - R12 R 23 - (n = 1.5) R I = 0,04 . , , , . , 2 R I R12 + T12 R23 . , : , -,

R I = R12 +

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?2.

29

, , - . .
( E11) , , : ( E11) = E1 R E 12

exp[i (t + k1 x)] .

RE TE , . ( .) , , :
( E1 2)

= E1T

E 12

R

E 23T E 21

exp[i (t + k1 x + )] .

, . = 2k 2 h2 (. ?Ë). , , , ,
( E11) E1( 2 ) , :

E



( ( E11) + E1

2)

= E1 exp[i (t + k1 x)]( R

E 12

+T

E 12

R

E 23 T E 21

exp(i)) .

, E . . , , , : RE12 , RE 23 << 1 . , (2.6) : TE12 , TE
21

1 . , TE12 TE


21

= 1.

, E R12 R12 n1 -n2 = n1 + n2



+ R23 exp(2ik 2 h2 ) = 0 . , , = R23 2k 2 h2 = . R12 = R23 , : n 2 -n3 , : n2 + n3

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30

?2.

n2 = n1 n3 . n1 , n2 n3 1,22 . h2 : h2 = / 2k 2 = 2 / 4 = / 4n2 . , , h2 0,1 . 2.2. h. . n = 1,5, n = 1. h >> , - . . . 2.1 , , . , (). , T2R 2.1 R = R12 + 12 12 . 2 1 - R12 , , R12 = 0,04 T12 = 0,96. , , , , 7,7% . , , 92,3%. 2.3. z . n(z) ( ). , . , . , : h 10 ; n ,

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?2.

31

n/ = 0,21 3/; (z) . n(z)/z, n(0)/z . . , (, z). . , z n, z + dz - n + dn. , , + d , n sin = (n + dn) sin( + d) . : dn n sin = ( n + dz)[sin cos( d) + sin( d) cos ] . dz d << 1 , dn n sin = n sin + sin dz + n cos d dz d 1 dn . = - tg dz n dz n( z ) z , k . - , k , . . , , r. , z. , : d 1 = . z / r = sin . , dz r cos z, 1 1 dn . = - tg r cos n dz

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32

?2.

, /2. tg 1/cos, 1 1 dn . =- r n dz n( z ) dn dn d , = . dz d dz , Ågz : = 0 exp - , Å - . RT , ( pV = RT ) :
gz = 0 exp - 0 p0 d dn , dz dz 1 / r , d 2 g = - 0 exp dz p0 0 gz - p 0 .

:
-1

dn p0 r = n d 2 g . 0 n 1 , r 3 10 7 ( : 6400 ). h , , . , . : r

(

R + h

)2

2 - R .

, h R, 2 R h . = r

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?2.

33

, 1,2 10 -2 ( : 9,3 10 -3 ). - ( , ). 2,7 . 1 2.4. (1) A = 70 . V p , : A , I /2, p , , f = 3 10 3 . . , , - . . , , 1974, . 5. . . ., 1979, . 4. 2.5. (1) = 0,63 n = 1,5. S = 107 /2. . . .. , , .: , 1976, . 23, ? 135, . 34, ? 185,186. .. , , . 3, ? 10. 2.6. (1)
1

.

, 1 ( ) 3 ( ).

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34

?2. . . 135.

, 30% Å = 1. .. , , .: , 1976, . 23, ? .. , , . 3, ? 10.

2.7. (2) n1 = 1,1 n2 = 2,0. . .. , , .: , 1976, . 5, ? 23, . 23, ? 135. .. , , . 3, ?? 10,11. 2.8. (1) . , . .. , , .: , 1976, . 23, ? 135. .. , , . 3, ? 10. 2.9. (3) , . z = 100 - 200 z. z . (). , , z : c ( z ) = c0 (1 + az ), z [0, z m ] . 1) , , , .

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?2.

35

a . 2) max ( , ) z. 3) max - , , 1450 /c z = 0 1480 /c z = 1000 . , . .. , 1974. 2.10. (2) n dn dh h , CO2 = 2,7 10 -23 3. r , . ? : 1) n = 1 + 4N . 2) N (h) h : P mg h N (h ) = 0 exp - , kT kT m - CO2, g = 8,76 /2, T = 740 . 3) , 1/r = (1/n) dn/dh. 2.11. (1) , 3À 1 ? , : / T = -1,95 10 -6 -1. 2.12. (1) . , . , , 3À. n / T = -0,98 10 -6 -1. h = 1,7 .

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36

?2.

2.13. (2) - : - = 60À. . - . , , . - . .. , , 1978, ? 2.4. .. , .. , , 1987, . 1, ? 74.

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?3.

37

?3.
, . ( ). E , k, . , E k, - . - . , , . - . - ( ). , , , ( ) . n1 n2 , , , . (3.1) tg = n2 / n1 .

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38

?3.

, , -, - 57o (n = 1,53). . , , , , . , . . . (ordinary). (, extraordinary) : , . , ( ). (no) (ne) . , , , . , . . , , . ( - - ). (3.2) = d , (3.3) = [ ] c d , - , [] - , c - d - . , - .

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?3.

39

3.1. , - . . OZ. XOZ, - YOZ. - : E1x ( z , t ) = E10 sin(t - kz), E1 y = E1z = 0 E 2 y ( z , t ) = E20 sin(t - kz + ), E
2x

=E

2z

=0.

, y- E y ( z, t ) x - E x ( z , t ) . y- E y ( z , t ) = E20 sin(t - kz) cos + E 20 cos( t - kz ) sin . sin(t - kz) = , : Ex E y = E20 20 1 - E sin . 10 E20, . E y ( z, t ) E x ( z , t ) : Ex cos + E E10
2 Ex = sin 2 . 2 E10 E20 E E10 cos = 0 (sin = Á1). 2 2 Ey Ex + 2 =1, 2 E20 E10 z = 0 E x = E10 sin t 2

E1x , E10

E

2 y 2 20

-2

Ex E

y

cos +

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40 E y = E20 sin(t +

?3.

+ n) = (-1) n +1 E20 cos t . 2 , n, z = 0, . (cos 0) E y ( z, t ) E x ( z , t ) , 2E10 2E20. 3.2. (n = 1,5) . . , . . , , , . , . ' (-) : sin = n sin ' . tg = n ( ). , : 1 . , sin ' = 1 + n2 tg ' = 1 / n . , (-) : tg = 1 / n . , . , = '+ = 2 . , tg( /2) = 1 / n , = 2arcctgn = - 2arctg n . , 67À.

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?3. 1

41

3.3. (1) , . . 3.4. (1) x y, , : Ex = Ecos(t-kz) Ey = Ecos(t-kz+). = /4, /3, /2, . , , . 3.5. (2) , , - /2. 3.6. (2) - . ( ), . 3.7. (2) ? 5 . (, ) , () ()

3.8. (2) : () () ?

, 1 ( ) 3 ( ).

1

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42

?3.

3.9. (2) , , . ( ) , 30 ( ). . 3.10. (2) , ; ; , , 10À. , .

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?4.

43

?4.
, - r , (r, t ) , , . (t ) , , S () . (t ) S () : (4.1) (t ) S () . (t ) , S () - . , . (t ) . (t ) S () , S () . (r, t ) , 0 , : (4.2) (t ) = a0 cos 0 t , a0 - . 0 - a0 : (4.3) S () = a0 ( - 0 ) . S () (4.3) 0 .

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44

?4.

1 , 2 a0 S () -: (4.4) S () = a0 ( - 1 ) + a0 ( - 2 ) . 1 2 , 1 - 2 << 1 , 2 , (t ) , : ~ (4.5) (t ) = 2a0 cos t cos 0t , ~ = - / 2 - , = ( + ) / 2 - . 1 2 0 1 2 ~ .
1 2 0

( = 600 - 1500 ), ( = 100 - 600 ) ( = 10 - 100 ) , . (t ) : (4.6) (t ) = a0 (1 + m cos mt ) cos 0t , m - , m - , 0 - , . S () (4.6) : (4.7) am am S () = a0 ( - 0 ) + 0 ( - m ) + 0 ( + m ) . 2 2 a0 S () . , S () : N -1 (4.8) ( - n ) , S () = a0


n =0

n = 1 + n , - n - ( n = 0, 1, ..., N - 1 ), - . , S () , 0 : (4.9) = ( N - 1) , 0 = 1 + / 2 .

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?4.

45

, : (4.10) << 0 , (t ) , N , : (4.11) (t ) = A(t ) cos 0t .1 A(t ) : (4.12) sin( Nt / 2) A(t ) = a0 . sin(t / 2) 2 (4.12) , t s = s ( s = 0, Á 1, Á 2, Á 3, ...) (4.13) A(t s ) = Na0 (-1) s . , , , A(t = t s ) N a0 . t s = 2s / 2s , . , : (4.14) 2 tq = q, N q - N, q = Á1, Á 2, Á 3, ..., Á ( N - 1), Á ( N + 1), ..., Á ( 2 N - 1), Á ( 2 N + 2), ... , : A(t q ) = 0 . ( N ) , ( 0 ), , S () . (4.14) : , A(t ) A(t ) 2 0 : << A(t ) . t 0
1

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46 tq =

?4.

(4.15) 2 q ( q = Á1, Á 2, Á 3, ... ). t = 0 , , . (4.15) . t , , A(t ) t +1 t -1 , (4.15): (4.16) t = 2 / t , : (4.17) t 2 . , t = 0 . (t ) . , t = 0 . (4.17) . , , . (t ) , : (4.18) (t ) = (t + T ) , T - . , , , - T . (t ) : (4.19) a (t ) = 0 + (an cos nt + bn sin n t ) . 2 n =1



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?4.

47

(t ) , S () ( ), n . , T : (4.20) = 2 / T . : (4.21) 2 n = n = n, n = 1, 2, 3, ... . T : an = 2 T
+T / 2 -T / 2 +



(t ) cos n t dt , bn =

2 T

+T / 2 -T / 2



(t ) sin n t dt .

(4.22)

: (t ) =


-

dn e

i n t

, d n =

1 T

+T / 2 -T / 2



(t ) e

- i n t

dt .

(4.23)

- : + (4.24) S () = d n ( - n ) .


-

d n ; an bn : (4.25) an = d n + d - n , bn = i (d n - d - n ) . (t ) . an bn - , d n :
2

dn = d

* -n

,







(4.26) an = 2 Re d n , bn = -2 Im d n . (t ) t = 0 , Im d n = 0
2

* .

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48 bn = 0 . (t ) , Re d n = 0 an = 0 an , bn (t ) .

?4. . d n

(t ) S () , . (t ) : + (4.27) i t . (t ) = S () e d
-



S () : S () = 1 2
+ -



(t ) e

- it

(4.28) dt .

S () . G () , , . G () 3: 2 (4.29) G ( ) = 2 S ( ) . W G () , , : + (4.30) W = G () d .
-



, W , (t ) :
2

3

(4.29) , .

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?4.
+

49 (4.31) W . (4.32)

W=

-



(t ) dt .

2

(4.30) (4.31) . (4.30) (4.31) tW tW - , :
+ t

- t



W

/2

(t ) dt = W .
/2

2

W

, W - , : + W (4.33) G () d = W .
- W



< 1 , = 0,8 - 0,9 . tW W , , : (4.34) tW W = C , C (t ) . , tW C , , W . () . S 3 () ( ) 1 (t ) 2 (t ) , , 1 (t ) S1 () , 2 (t ) S 2 () 3 (t ) = 11 (t ) + 2 2 (t ) , (4.35) S3 () = 1S1 () + 2 S2 () . () . (t ) S () , (4.36) (t - t ) S () e -it 0 .
0

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50 () . (t ) S () ,

?4.

(4.37) (t ) e i 0 t S ( - 0 ) . () . S 3 () 3 (t ) , 1 (t ) 2 (t ) , S1 () S 2 () , 1 (t ) S1 () , 2 (t ) S 2 () 3 (t ) = 1 (t ) 2 (t ) , (4.38) S () = S () S () ,
3 1 2

- , + (4.39) S3 () = S1 ( )S 2 ( - )d .
-



() . 1 (t ) S1 () , 2 (t ) S 2 () S3 () = S1 () S 2 () , (4.40) (t ) = (t ) (t ) .
3 1 2

4.1. , a. T. = 0,5T = 0,1T. . . (t), , : (t ) = a0 + 2


n =1



(an cos nt + bn sin nt ) ,

= 2/T T - . , (t) - . . a0 1 = 2T
T /2

-T / 2



(t )dt =

1 T

/ 2 - / 2



adt =

a , T

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?4. 2 T
T /2 -T / 2

51

an =



(t ) cos( nt )dt = =

2a T

/2 - / 2



cos( nt )dt =

2a sin(nt ) Tn

/2

=
- / 2

a n 2a n 2 sin sinc = , n T T T

sinc( x) sin( x) / x . , bn = 0 n, . , , n . T M n : T = M. , an = 0, = j T (j = 1, 2, 3, ...), n = Mj ( M). = 0,5T (M = 2) an. c = 0,1T (M = 10) , , . n a n = asinc 2 a n an = sinc . = 0,5T 5 10 n- n . (t), , :


(t ) = A() sin( t ) d + B() cos(t )d .
0 0







A() B(), ,
A() = 1


-



(t ) sin(t )dt

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52
1
-

?4.

B() =



(t ) cos(t ) dt .

- . A() = 0, B()
B() = 1
-



(t ) cos(t )dt =

a

/ 2 - / 2



cos(t ) dt =

a sin( t )

/ 2

=
- / 2

a sinc . 2

B() 2m/ (m - ). , , B(), (t). 4 4.2. (1) , , , z , . . .. , , .: - , 1983, . 3, ?? 2, 3, 14, 15. 4.3. (1) 0 , . , . a, t < / 2 4.4. (2) y (t ) = . 0, t > / 2 W = 1 = 1 . , -
4

, 1 ( ) 3 ( ).

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?4.

53

( ). . W fW . t a1 - , 0 t 4.5. (2) y (t ) = . 0, t < 0 t > = 1 = 1 . . T t y (t ) = a1 - . T . 4.6. (2) a exp(-t ), t > 0 y (t ) = . 0, t < 0 0. y(t) S().

4.7. (2) a exp(- t / 0 )sin t , t > 0 , (t ) = , 0, t < 0 0 >> 1. - , 0 - . S(). S() , 0 >> 1 , - << , . (t) S() 0 = 3, 10, 100. 0 . He-Ne , 0 10 -8 c .

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54

?4.

a sin(t + ), t1 t t 2 y (t ) = 0, t < t1 t > t 2 2 n , . = 0 = t 2 - t1 4.8. (2) n =1,2,10, n . S(). 4.9. (2) . T a exp i (t + ), 0 t (t ) = . , T >> 1. 0, < t < T = 0,5T = 0,1T. . 4.10. (1) 0 0+, << 0. , , , 0/. 4.11. (1) : a / 2, - T / 2 < t < 0 , 1(t) = 1(t+T); 1 (t ) = - a / 2, 0 < t < T / 2 T a 2 (t ) = 1 (t - ) ; 3 (t ) = 2 (t ) + . 4 2 4.12. (2) 2a (t + 2 ), - 2 < t < 0 2a (t ) = - (t - ), 0 < t < , (t) = (t+T) 2 2 0, - T / 2 < t < - / 2 / 2 < t < T / 2

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?4.

55

: = T/2, T/4, T/16. . 4.13. (2) a, 0 - 2 S () = 0, < 0 - 2 . , 0 + 2 ; << . 0 > 0 + 2 . ,

4.14. (2) : a , t < / 2 1 (t ) = , 2 (t ) = 1 (t - ) , 3 (t ) = 1 (t - t0 ) . 2 0, t > / 2 , . 4.15. (2) (t ) = a t t 2 exp(- 2 ) . t0 t0

(t ) t0,
- , S() t 0 1 , . t0 .

4.16. (2) , , . n = 600 . n = 1, 2, 3, 10. , , .

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56

?4.

4.17. (2) (t ) = a

t t 2 exp(- 2 ) . t1 t0

(t ) t0, S()
- t 0 1 , t1 = t 0 , 0,1t 0 , 10t 0 . t1 .

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?5.

57

?5.
(t ) , t , h ( j ) , h - , j = 1, 2, ..., N - , N - . h ( j ) , j , S h (n) ( n = 1, 2, ..., N - ). h ( j ) S h (n) : (5.1) h ( j ) S h (n), j , n = 1, 2, ..., N . h ( j ) , S h (n) . h ( j ) S h (n) N . h ( j ) h t (t ) , , h ( j ) S h () . = / 2 , , S h () : (5.2) S h () = S h ( + q / h), q = Á1, Á 2, ... . 1 / h , , h . S h (n) (5.2) : (5.3) S h (n) = S h (n + qN ), q = Á1, Á 2, ... , N - , . , , S h (n) ?4 (t ) = (t + sT ) ( s = Á1, Á 2, ...) , :

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58

?5.

(5.4) h ( j ) = h ( j + sN ), s = Á1, Á 2, ... . N , , , : (5.5) T = Nh . , (5.1) h ( j ) S h (n) . h ( j ) T S h (n) 1 / h N . . S h (n) h ( j ) - : S h ( n) = 1 N
N

j =1

N

h ( j) W

nj N

, n = 1, 2, ..., N .

(5.6)

h ( j ) S h (n) - : h ( j) = W
N


n =1

S h (n) W

- nj N

, j = 1, 2, ..., N ,

(5.7)

2 = exp i . N

nj WN , n n j , j j n :

j =1
N

N

W W

- nj N

W W

n j N

N n = n = , 0 n n

(5.8)

N j = j = . 0 j j n =1 , (4.35)-(4.40) . h ( j ) S h (n) (5.1), :
- nj N nj N

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?5. () .
(1)

59
(2)

h

( j) S

h

(1)

(n)

h

( j) S
h

h

(2)

(n) , (5.9) (5.10) (5.11)

(1) h (1) ( j ) + ( 2) h ( 2 ) ( j ) (1) S h (1) (n) + ( 2) S () . h ( j ) S h (n) ,
- h ( j - k ) WN nk S h (n) . () . h ( j ) S h (n) ,

( 2)

(n) .

h ( j)W
(1)

mj N h

S h ( n - m) . ( j) S
h (1) (1) h (1)

() .
h

(1)

(n)
h (2) (2)

h

(2)

( j) S

h

(2)

( j)
(1)

h

(2)

( j) S

(n) S

(n) (n) ,

(n) , (5.12) (5.13)


h (1)

h

( j)

h

(2) N

( j) S
h (1)

h

(n) S
( 2)

h

( j)

h

( 2)

( j) =

1 N


k =1

(k )

h

( j - k ) -

. S h () (t ) S h ( Á q / h) , q = Á1, Á 2, ... , 1 / h : S h ( ) =

S q = -

+

h

( + q / h) .

(5.14)

, (t ) , () ?Ë (5.14). , , N : (5.15) N = 1 / 2h , h - (t ) h ( j ) .

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60

?5.

max S () N , (t ) S h (n) , N . , , t , (t ) . S h (n) h ( j ) , (t ) -: sin (t - hj ) (5.16) h . (t ) = h ( j) (t - hj ) h max S () N , , S h (n) , (t ) t = hj , . , t hj , , , S h (n) , (t ) . , , . S () ( - < < + ), N : (5.17) S ( N ) << max S () .



(t + sT ) , s = Á1, Á 2, ... T : (t ) =

s = -



+

(t + sT ) .

(5.18) , .

, T , ?Ë . ,

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?5.

61

(t ) ( - < t < + ), T : (5.19) (ÁT / 2) << max (t ) . N h T (t ) , . , , , N = T / h . . 5.1. h T 2 (t ) = a exp - t 2 0 . . N . . (. ?4), , : 2 2 S () ~ exp - 2 0 4 S () ~ exp(- 2 2 0 ) , = 2 . , , , e e = 1 ( 0 ) . N . , () . . , , , . , S () , , S ( = 0) , = 0 . , , max . ,

(

)

(

)



:

exp(-2

2 2 max 0

) 10

-10

.

,



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62

?5.

2 2 ax 2 23 max 1.5 / 0 . N m 0 max , , : N = 2 / 0 . , (5.15) 1 / 2h 2 / 0 , , , h 0 / 4 . , h 0 , e , . . , T, t > T . , T, . . , . , , t > t max . 2 t ax (tmax ) = a exp(- m2 ) = a 10 -10 . , 0 t max 4.80 . ( T) , T / 2 = 5 0 T = 100 . , N = T / h = 40 . 1 5.2. (2) h T S h (n) . N , T . N .
1

, 1 ( ) 3 ( ).

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?5.

63

h ( ) ( ). (t) S() , , -1 , . () t2 (t ) = a exp - 2 sin t , / 2 = 105 , 0 = 10 -3 . 0 () t2 t2 (t ) = a exp - 2 - a exp - 2 , 0 = 10 -4 , 1 = 10 -6 . 0 1 () t (t ) = a exp - sin t , /2 = 1 , 0 = 10 -3 . 0 () t2 t (t ) = a exp - 2 , 0 = 10 -4 . 0 0 () t2 t2 (t ) = a 2 exp - 2 , 0 = 10 -3 , 1 = 10 -5 . 1 0 5.3. (2) (t), N . : , . , , . , . (t) , = 1 N = 10 100. 5.4. (3) S h (n) (t) . h = T/N. '(t) c h' = h/n.

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64

?5.

(t) '(t) , , -1 . N = 5, 11 n = 2, 4. N0 6N. () a, t < / 2 (t ) = . 0, t > / 2 () (t ) = () 2a (t + ), - / 2 < t < 0 2 2a (t - ), 0 < t < / 2 . - 2 0, t < - / 2 t > / 2 2a t + a, - / 2 < t < 0 2a t - a, 0 < t < / 2 . 0, t < - / 2 t > / 2

(t ) = ()

(t ) = a exp - (2t

(

)

2m

)

, m = 1, 2, 4, 8.

5.5. (2) (t) S(). , :


t =

-



(t ) dt

2

(max

(t )

)2

. ,

=

-



S () d

2

(max

S ( )

)2

.

, C : t = C. (t) S()

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?5.

65

, , -1 . , , , . . () a, t < / 2 (t ) = , 0, t > / 2 (t ) = 2a (t + / 2), - / 2 < t < 0 2a (t - / 2), 0 < t < / 2 ; - 0, t < - / 2 t > / 2
2

(t ) = a exp - t ()

(



2

)

.

a, t < / 2 (t ) = , 0, t > / 2 a cos (t ), t < / 2 "" (t ) = , 0, t > / 2 (t ) = a exp - t () a, t < / 2 (t ) = , 0, t > / 2 a (1 + cos (2t )), t < / 2 "" (t ) = 2 0, t > / 2 (t ) = a exp - t

(

2



2

)

.

(

2



2

)

.

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66 ()

?5.

a, t < / 2 ; (t ) = 0, t > / 2 (t ) = a exp - (2t () a, t < / 2 (t ) = , 0, t > / 2 a (1 + ) (1 + ) (1 - ) (1 + ) / 2 C 0 1. () a, t < / 2 (t ) = , 0, t > / 2 a, - / 2 < t < 0 (t ) = . - a, 0 < t < / 2 5.6. (3) (t) S(), . t ,
t / 2

(

)

2m

)

, m = 1, 2, 4, 8.

: =

- t / 2



(t ) dt

2

-



(t ) dt .

2

, :

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?5.
-

67

=

-



S () d

2



S () d . = 0,9

2

= 0,95. , C : t = C. (t) S() , , -1 . , . () a, (t ) = 0, 2a (t (t ) = - 2a 0, t < (t ) = a exp - t t < /2 , t > / 2 + / 2), - / 2 < t < 0 (t - / 2), 0 < t < / 2 , - / 2 t > / 2
2

(

2



)

.

() a, t < / 2 (t ) = , 0, t > / 2 (t ) = a exp - (2t () a, t < / 2 (t ) = 0, t > / 2

(

)

2m

)

, m = 1, 2, 4, 8.

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68

?5.

a (1 + ) (1 + ) (1 - ) (1 + ) / 2 C 0 1. 5.7. (3) h a, t < / 2 . (t ) = 0, t > / 2 , , -1 , h = 0,05, 0,1, 0,2, 0,4. (T 3) , N = T / h = 2 p , p - . . N . t a(1 - ), 0 t . y (t ) = 0, t < 0 t > , , -1 , h = 0,05; 0,1; 0,2; 0,4. (T 3) , N = T / h = 2 p , p - . . N . 5.9. (3) N i (i = 1, ..., N): 5.8. (3)

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?5.

69

a, 0 t 1 (t ) = ; i (t ) = 1 (t - (i - 1)T ) , i = 2, ..., N . 0, t < 0 t > - . G () . . G () N = 1, 2, 5, 10 , , -1 . T = 2 10. 5.10. (3) N i (i = 1, ..., N): 2a (t + / 2), - / 2 < t < 0 2a ; 1 (t ) = - (t - / 2), 0 < t < / 2 0, - T / 2 < t < - / 2 / 2 < t < T / 2 i (t ) = 1 (t - (i - 1)T ) , i = 2, ..., N . . . . G () N = 1, 2, 5, 10 , , -1 . T = 2 10.

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70

?6. .

?6. .
. - ( , ) , , . , , . , (1.9), V , k , : (6.1) (k ) = k V (k ) = (k ) . V k. V k , , n : (6.2) V = V (k ), V = V (), V = V (n) . , n : (6.3) n = n(), n = n() . V , k , , , n . , . (x,t) , , :
k 0 + k

( x, t ) =

k 0 - k



a(k ) exp(i ((k )t - kx) )dk .

(6.4)

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?6. .

71

k , , , , : (6.5) k , << 1 , k 0 0 0 - , k 0 - . (6.5) : (6.6) ( x, t ) = A( x, t ) exp(i (0 t - k 0 x )) , A( x, t ) , T : (6.7) A A T, << A . t x A( x, t ) (6.4), (6.1) 0 = (k 0 ) .
k0 + k

A( x, t ) =

k 0 - k



a(k ) exp i (k - k 0 ) (V t - x) dk .

(

)

(6.8)

(6.8) . a(k ) , A( x, t ) , OX V , : (6.9) d(k ) k = k 0 . dk , , V . , V = A( x, t ) : + 1 A( x, t ) = 0 . x V t (6.10) f (V t - x)

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72

?6. .

(x, ) , : (6.11) x = x, = t - x / V , A( x , ) : (6.12) A( x , ) =0. x , . (6.9) : (6.13) dV () V = V - d = 0 . (6.2). . , . , , . , 0 , 2 0, , /, : (6.14) S = 1 / 2 0. , . (4.17) 0 k , . : , , , , , - , . , , .

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?6. .

73

k , = (k ) . t , k0 - k k0 + k V (k 0 - k ) V (k 0 + k ) , . , t : t = l 2k 2 ,

0

(6.15)

l - . (4.16), (6.15) : t = 2l 2k
2 0 - 01 .

(6.16)

t + t , : 2k (6.17) - S = 2 + 4l 01 . 2 0 , , , . , = 1,55 , 2 k 2 = 0 = 1,3 . = 1,3 . A( x, t ) :
2 2 1 A( x, t ) + k A( x, t ) = 0 . 2i + x V t 2 t 2 -1

(6.18)

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74

?6. .

(6.11) A( x , ) : A( x, ) 2 k 2 A( x, ) 2i +2 =0. (6.19) x 2 (6.18) (6.19) , : (6.20) 2k 2 x 2
A( x , ) = A0
0

( x )

exp - isign exp - 2 2 2 ( x ) L 2 ( x )



,

( x ) = L


0 2 0

1 + (x L
2



)2

- x'

=



2 k

- , 2 .



6.1. , 1) 2) . . 1) (6.13) n = c / V , V 1 d 1 c = c - V = 1 + n d n n 2) 2c dn dn d = = . n d d d


dn . n d ,
-1

V k = n / c . V
-1

=

dk d

=

n dn + V c c d



=

c dn 1 + . n n d

-1

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?6. .

75

, V dn dn -1 = 1 + . , = 1 + V n d n d . 6.2. 1) ( V = a ), 2) ( V = b / ), 3) ( V = c ) (a, b c - ). . (6.13). , V = V . b b + = 2V . 2 V c V = c - = . 2 2 V


=

= 1 - ( / )2 , - . (), 2 = 2 + c 2 k 2 . . : V = V () = c 1 - (

6.3.

. n = n = c / V ,

[



/ )

2 -1 / 2

]

.

2 2 1 - / 2 = c 2 / V . V = / k

, : 1 - 2 / 2 = c 2 k 2 / 2 . , 2, : 2 = 2 + c 2 k 2 .

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76

?6. .

, = (k ) , k - d d 2 = 2c 2 k . V = , dk dk : V = V () = c 2k c 2 = = 1 - . V , c : V = c / n V = n .
2

, c, c. 6.4. , , 0 = 0,75 , . , (). .

2 n = 1 - ( )2 , - , . , , , n - , . , - . , , = arcsin(n). , , , >> n = 1. , . n , . , , ,

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?6. .

77

, , : > . , , , 0 , < 2c / 0 . Ne 2 , e m - m 0 , N - . , 42 2 m 0 N < . 2 e 2 0
2

=




-31



0 = 8,85 10
-19

-12

Ç/(Ç),



m = 9,11 10

e = 1,6 10

,

N < 2 109 -3. : 105 -3. 1 6.5. (2) , 1 11 = - V V c V V : dn() , d

c - , - , n - . c/V n(). . n() KCl, .. , , .: , 1976 . 28.3, V V = 680 = 600 .

1

, 1 ( ) 3 ( ).

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78

?6. .

6.6. (2) ka ka , k - a V = c sin 2 2 , c - . V. >> a ? ? V V = 0,4-0,7 a = 10-8 c = 5 103 /c. .. , , .: - , 1983, . 3, ? 9. . , , .: , 1974, . 6, ?? 6.1, 6.2. 6.7. (3) . = (k) k . . . , . . () l m, c, a . / , 2 = ca 2 / m . 0 = g / l . g - . () . ; T, m M, d. k / 2d . 6.8. (3) , LC , L

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?6. .

79

C . = (k) . . . , 0 = 1 / LC . , . n- . . () . L C . , - L C . () . C L . , - C L . 6.9. (3) , m M, l . , , . = (k) . . . . = 61 NaCl = 71 KCl, , = 15 /. . , , 1979, . 4. .. , .. , .. , , 1979.

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80

?6. .

6.10. (2) 3,6 . f1 = 150 f2 = 240 , t = 1,3 . , , f1 f 2 , . (1 3,09 1016 3,26 ). . = () , , . , N = 3 10-2 -3 . .. , . , .: , 1980, ? 87. 6.11. (2) . , . = (). . . , . f f. : 1) (f f) , , 2) (f >> f). . , , .: , 1974, ?? 3.5, 4.3, 6.1, 6.2. .. , . , .: , 1980, ? 87. 6.12. (3) H- TE ( E - , H ) X. YOZ Y a , Z - b . Hx.

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?6. .

81

k x = k x (, k y , k z ) , k y = m / a k z = n / b , m, n = 0, 1, 2, ... - . mn mn mn. Ey Ez Hy Hz . , X . . : 1) m = 0, n = 1; 2) m = 0, n = 2; 3) m = 1, n = 0. . , , .: , 1974, ? 7.2. .. , , 1983, ? 66. .. , .. , .. , ,1979, .10, ?3. 6.13. (3) E- TM ( H - , E ) X. YOZ Y a , Z - b . Ex. k x = k x (, k y , k z ) , k y = m / a k z = n / b ( m, n = 0, 1, 2, ... - ). mn mn Mmn. Ex . . . : 1) m = n = 1 2) m = 2, n = 1. . , , .: , 1974, ? 7.2. .. , , 1983, ? 66. .. ., , . 2, 1962, ? 38. 6.14. (3) , : 2( - 0 ) 2 2 0 a = a0 exp - , (2)2

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82

?6. .

0 - , 0 - . , x = 0,
2 2( - 0 ) 2 0 exp(it )d . exp - 2 (2) - . , ,

( x = 0, t ) = a0



k () = k 0 +

1 1 2k ( - 0 ) + V 2 2

( - 0 )

0

2

, V

2k 2

.

0

I() = ()*() = t - x / V x. x.


W =

-



I ()d .

x. .. , .. , , 1970, . 8, ? 1. .. , .. , .. , , 1988, ? 1.1 1.3. 6.15. (1) , f = 108 n = 0,9. . . f. f = 108 . . , , .: , 1974, ?? 3.5, 4.3. .. , . , .: , 1980, ? 87. 6.16. (2) l = 10 .

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?6. .

83

= 1 , n = 1,5. , (0) = 1 . T (l) . ( /), , n/ = 4 10-3 -1. , . , 2n 1 n << . 2 n
2

.. , .. , , . 1, 1987, ? 85. 6.17. (2) . T (l) . ( /), , l . , f. . . () l = 600 . = 3 . (0) = 3 . f = 30 . () l = 1000 . f0 = 3 . (0) = 50 . f = 2,6 . .. , . , .: , 1980, ? 87. .. , .. , , . 1, 1987, ? 85. 6.18. (2) f0 . mn ,

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84

?6. .

H - , E . ( m, n = 0, 1, 2, ... - .) (0) = 5 . z = 100 . T (l) . ( /), , a = 1 b = 3 . , :
2 2 2 k z = 2 / c 0 - ( k x + k y ) ,



k x = m / a



k y = n / b .

. () 11 f0 = 20 . () 33 f0 = 60 . .. ., , . 2, 1962, ? 38. .. , , 1983, ? 66. .. , .. , .. , ,1979, .10, ?3.

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?7. .

85

?7. .
. . , . I : (7.1) I = I + I + 2 I I cos ,
1 2 12

I1 I2 - . : (7.2) = 2 / . (), : , . , n 1, . : (7.3) I = I + I + 2 I I cos ,
1 2 12

I1 I2 - . , " ", : (7.4) = 2m, m = 0, 1, 2, ... . , : (7.4) = 2m , m = 0, 1, 2, ... . 2 , : (7.5) = (2m+1), m = 0, 1, 2, ... .

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86

?7. .

: (7.5) = (2m + 1) , m = 0, 1, 2, ... . 2 I max : (7.6) I = I +I +2 I I ,
max 1 2 12

(7.7) I min = I1 + I 2 - 2 I1 I 2 . I1 = I2 = I0 (7.3) : (7.8) I = 2 I0 (1 + cos ) . - I max = 4 I 0 , - I min = 0 . ( ) , x ( ) : (7.9) x = . ( 2) 2 sin ( << 1 ), , (7.9) x . , , d L , . d << L () x , (7.8): L (7.10) x . d , , , . , , , . ,

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?7. .

87

() . , ( ) 12 ( ) : 1 (t + ) (t ) 2 12 () , () = = I1 I 2 I1 I 2 (7.11)

, 1 (t + ) 2 (t ) - , , . = / c , c - . . : (7.12) I = I + I + 2 I I Re () .
1 2 12

, a , 1, 2 (t ) = a1, 2 (t ) exp(it ) , a T << a , t I = I1 + I 2 + 2 I1 I
2

() Re(exp(i )) .

(7.13)

Re(exp(i)) = Á1 , () - : (7.14) I = I + I Á 2 I I ( ) .
max, min 1 2 12

: II I max - I min = 2 1 2 () . I max + I min I1 + I 2 I1 = I 2 = I V () = V () : V () = () . (7.15)
0



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88 -

?7. .
2










() = (t + ) (t )

G ( ) = 2 S () (. (4.29)):


G () =

-



() exp(i )d ; () =

1 2

-



G () exp(-i )d .

(7.16)

7.1. . , . , x x /. , , ? . , ( 2). , . , , , x . . , , . , () B A d = x sin( / 2) x / 2 . AB A : AB = k 2d 2x / x , AB = 2 . . , 2x / = 2 x = / , . , , , , .

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?7. .

89

, d1 = x sin( + / 2) , - d 2 = x sin( - / 2) . A : 2 2x AB = k (d1 - d 2 ) = x sin + - sin - cos 2 2 , , . , : x = , cos , . 7.2. , L = 2 . , f = 40 , ? f. . . , y , . : d1 = L2 + ( y + d / 2)2 d 2 = L2 + ( y - d / 2)2 d - . d << L, : 1 2y + d 2 1 2 y - d 2 yd = (d1 - d 2 ) L1 + . - L1 + = 2 2L 2 2L L y , k = 2, . , : yd 2yd k = = 2 y = L / d . L L

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90

?7. .

( = 500 ). , , d = 5 y = 0,2 . . , , , . , , , d tg = , d << f, 2 2f = d / f . , , , , y = f / d . , y / y = L / f = 5 . 7.3. d. l = 1 , . n = 1,5. ( ) , , ( = 500 ). , , ? . , , = 2nl . = 2l . , n . (n - 1)l = 0.5 , 1000 = 500 . , , 2. 7.2. y

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?7. . , 1 =

91

2yd . L << ( ). , 2yd + 2 = . L( + ) 2 - 2 , . , + , , , . 2yd L2 . , 2 yd L2 . , , L = 2 d = 5 y = 5 . 1 . , . . + , 2 2l (n - 1) = l (n - 1) , 1 - = 0 - 0 ( + ) ( ) 0 = 2000 . 0 / 2000 = 0.25 . 7.4. = 500 a = 1 . ( d = 2 ) , ?

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92

?7. .

. , , . L, , l. . - x ( -a / 2 x a / 2 ). , , . , , y . x x, y, d << l , L : d x d y + L l (. 7.2). , , , : I = 2 I 0 (1 + cos k( x) ) I0 - . , , . , y , . : ( x ) = 1 + cos 2d x + y dx . L l -a / 2 -a / 2 , , . , I ( y) ~
a/2



(1

+ cos k( x) )dx =

a/2



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?7. .
a/ 2

93 2 sin(a 2) a = a1 + cos y sinc 2

I ( y ) ~ a + cos y

-a / 2



cos xdx = a + cos y

2d 2d = = . l L , : I -I a da V = max min = sinc = sinc . I max + I min 2 2 L , V = 1. , d, a , L . . , - . , . , V 1, : da da << 1 L >> . 2 L , L , : L >> 4 . . . . , : I1 = 2 I 0 (1 + cos k1 ) I 2 = 2I 0 (1 + cos k 2 ) , d a d y d a d y 1 = + 2 = - + . 2L l 2L l ( L ), 1 2 , I1 I2 , . , L ,

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94

?7. .

. , L, 1 - 2 = / 2 , , . , . , , : d a 1 - 2 << / 2 << . L 2 L , . 1 7.5. (1) , S = [EH ] , . .. , , .: - , 1983, . 3, ? 11. 7.6. (1) w = 0 E 2 w = ÅÅ0 H 2 , . .. , , .: - , 1983, . 3, ? 11. 7.7. (1) x V1 V2 ( 1 2, ). : 1) , , , 1 ( ) 3 ( ).
1

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?7. .

95

2) U , . 7.8. (2) , , . , : 1) , [0, 2], 2) , [0, ], 3) , , ? .. , , .: , 1976, . 4, ?? 11-14. . , , .: , 1974, . 9, ?? 9.1-9.3. 7.9. (1) = 0,5 L1 = 1 L2 = 10 . , . .. , , .: - , 1983, . 3, ? 15. .. , , .: , 1976, . 4, ?? 11-13. 7.10. (2) f = 50 d. . ( = 600 ). , . x = 0,5 . d . .. , , .: , 1976, . 4, ?? 11-16. . , , .: , 1974, . 9, ?? 9.1-9.3. 7.11. (2) , f = 12 .

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96

?7. .

, , d = 1,1 . , a = 20 . L = 480 . , = 610 . .. , , .: , 1976, . 4, ?? 11-16. . , , .: , 1974, . 9, ?? 9.1-9.3. 7.12. (2) D , (. 7.11), f = 12 , ? d = 1,1 . a = 50 . L = 480 . = 610 . .. , , .: , 1976, . 4, ?? 11-17. . , , .: , 1974, . 9, ?? 9.1-9.4. 7.13. (2) ( d = 0,5 ) - , , = 632,8 . l , L = 5 ? , = 100 , , , I = 0,5I 0 (1 + cos( 2t )) ? .. , , .: , 1976, . 4, ?? 11-16. . , , .: , 1974, . 9, ?? 9.1-9.3. 7.14. (2) - . , . , , , - .

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?7. .

97

. , h. m 1 = 589,0 2 = 589,6 , h = 20 m = 2, 10, 20. .. , , .: , 1976, . 4, ?? 11-15, . 7, ?? 30. 7.15. (1) , h . , . . 7.16. (2) h = 0,1 n = 1,4. L = 1 , 60À. , , . . 7.17. (2) - . l = 7,36 . , = 540 2 ? 7.18. (1) L = 480 . , . , . , 10%

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98

?7. .

. = 0,5 . . 7.19. (2) , 106 . = 693 . 7.20. (2) -, ( ) , t2 2 ( x = 0, t ) = a0 exp - 2 + it ; << . .. , . , .: , 1980, ? 31. 7.21. (2) . , 0. a , 0 - 2 0 + 2 S () = , 0, < 0 - 2 > 0 + 2
0

<< 0 a = I

(I0 - ).

7.22. (2) D- , d = 0,15 . << 0, 0 = 0,59 . .. , , 1978, ? 5.6. .. , , .: , 1976, ? 25.

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?8. .

99

?8. .
, . , . (7.8) I min = 0 I max = 4 I 0 . . , . . () . , , . , , . , . , , , . , , . () N , d . r, , D = ( N - 1)d . , , , . , .

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100

?8. .

, ,
E = A(r ) exp(i(t - kr ))

exp(- n=0

N -1

inkd sin ) = A(r , ) exp(i(t - kr ))

(8.1)

, - . , A(r , ) r, . i, i +1 i- (i+1)- . , i- (i+1)- dsin , i, i +1 = kd sin . , I, N , 2 Nkd sin 2 kd sin I ~ I 0 sin 2 sin ; I0 ~ A . (8.2) 2 2 , I ~ N 2 I 0 , , : (8.3) d sin max = n , n = 0, Á 1, Á 2, ... . min , : (8.4) Nd sin min = p , p = Á1, Á 2, ... , p Nq q = 1, 2, ... . (8.4) : (8.5) d sin min = (n Á s / N ) , n = 0, Á 1, Á 2, ... , s = Á1, Á 2, ..., Á ( N - 1) , n- ( n = 0, Á 1, Á 2, ... ), - , n d sin - = (n - 1 / N ) , + , n n
+ d sin n = (n + 1 / N ) .



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?8. . , : = ( + n ,

101

(8.6) << 1 : (8.7) (8.8) c (N >> 1)

- - ) 2 . n N >> 1 Nd : / D , D Nd - .

8.1. N = 32 , d = 7 = 21 . . N d, ? . , , , . , , , , , . , , , , i- , (i-1)- kdsin (k = 2/). ,
N -1



exp(-
n=0

inkd sin ) .

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102

?8. .

, , 1 - exp(iNkd sin ) . : 1 - exp(ikd sin ) Nkd sin sin 1 - exp(iNkd sin ) 2 i ( N - 1)kd sin . = exp 1 - exp(ikd sin ) 2 sin kd sin 2 , : Nkd sin 2 kd sin I ~ sin 2 sin . 2 2 . , kd sin , = n , n = 0, Á 1, Á 2, ... - 2 2n . , sin max = = n . << d, kd d n : max = / d . , max = / d = 0,03 1,8À. , : Nkd sin = Á . 2 2 : + = Nkd Nd - - . Nd 2 : = + - - . , . Nd , , :

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?8. .

103

2 0,21 0,002 0,1À. 32 7 , Nd . , , , , , . , , . 8.2. 3M , . , 1) 2) ? . . , 8.1. ( ) , , , . , , max = . 2d . , 8.1, : 1 + exp(- ikd sin ) + exp(- i 3kd sin ) + exp(- i 4kd sin ) + + exp(- i 6kd sin ) + exp(- i 7kd sin ) + ... =
M -1 l =0 M -1 m=0

=



exp(- i 3mkd sin ) + exp(- ikd sin ) =

exp(-

i3lkd sin ) =

1 - exp(- i3Mkd sin ) (1 + exp(- ikd sin )). 1 - exp(- i 3kd sin )

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104

?8. .

, , , kd sin 2 3Mkd sin 2 3kd sin I ~ cos 2 sin sin , 2 2 2 - . 3Md 3d. , ( n = 0, Á 1, Á 2, ... ). sin max = n 3d n max = . , 3d . 2 3Md . kd sin cos 2 . 2 , sin = j d ( j = 0, Á 1, Á 2, ... ). , , .. . ( 4-, 5-, 7-, 8- ..) , 1 kd 2 kd 2 2 cos 2 = cos = cos = . 2 3d 2 3d 3 4 1 8.3. (2) 256 , , 1 ( ) 3 ( ).
1

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?8. .

105

, . , , 622 . , . = 0,5 . , , : 1) , , , 2) , "" , , 45À , . 8.4. (3) GSM-1800 925 . , L = 4,5 . , , P = 5 . , , , , 10 -14 /2. . , : 1) , 2 - , 2) , 6 - , 3) , 9 . 8.5. (2) , , . 2N , N - . - d, - l. . , , 1) d 0,1 2 l = const;

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106

?8. .

2) l 0,1 2 d = const ( - ). d = l = N = 4. . , , .: , 1974, . 9, ? 9.6. .. , , .: , 1976, . 9, ? 46. 8.6. (2) , , . 2N (N ). - d, - l. . , , l = 0,5; ; 1,5; 2 ( - ). d > d < . . , , .: , 1974, . 9, ? 9.6. .. , , .: , 1976, . 9, ? 46. 8.7. (2) L a << L. , , R >> L , . .. , , .: , 1976, . 9, ? 39. 8.8. (2) . /2 ( - ). 3/4. . , ? . 2 4, . .. , , .: , 1976, . 4, ? 13. . , , .: , 1974, . 9, ? 9.2.

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?8. .

107

8.9. (3) , , . 2N (N ). - /2, - /4, - . . , /2. , , . 8.10. (3) GSM 1800 0,34 . , L = 4,5 . D = 50 . 5 . , , Z = 10 I = 10 -14 /2. , 10 .

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108

?9.

?9.
, . (1.1) (r, t ) , , . (r, t ) :
+

(r , t ) =

-



E (r ) e

it

d .

(9.1)

(1.1) (9.1) E (r ) : (9.2) E (r ) + k 2 E (r ) = 0 . (9.2) (.., .., ), - E ( P ) P 1: (9.3) 1 G E E ( P) = -G E d , 4 n n




1 G = e - ikr - , n - r , P . (9.3) E , P , , .
1

.

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?9.

109

. . P , , (9.3) , P , . (9.3) : (9.4) = + . , , . (9.3) , : (9.5) 1 G ( M ) E (M ) E ( P) = - G(M ) S E S (M ) d , 4 n n





E S (M ) - M , , G ( M ) - . , r0 r , : (9.6) << r0 , r . - (9.3) , -: E ( P) = E S (M ) = A e ik 4
- ikr0



ES ( M )

e

- ikr

r

(cos - cos )d



,

(9.7)



- , r0 S M , - n - r0 , S M , - n - r , P M .

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110

?9.

- - E ( P ) , (9.7). , P , , / 2 . , Rn , : Rn = n r0 , + r0 (9.8)

n = 1, 2, ..., N - , N - , - P n = 1 , r0 - , S , . - P En , : E ( P) =


n =1

N

En .

(9.9)

En n - : K n (-1) n . (9.10) + r0 n. En = A E
n +1

e

- ik ( + r0 )

K

n

En < En . (9.11)
I (P) = E (P)
2

E ( P ) .

, b , ,

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?9.

111

RP P : (9.12) b, RP << r0 , r , z , z - P . 0 , (9.7) cos 1, cos -1 . , (9.12) , r z . , E ( x, y, z ) P , x, y z , :
E ( x, y , z ) = ik e 2z
-i kz




ik (x - x )2 + ( y - y E ( x, y ) exp - 2z

(

)2

)

dx dy

(9.13)

(9.12) , , OZ . (). E ( x, y, z ) : (9.14) E ( x, y, z ) = A( x, y , z )e -ikz . A( x, y, z ) (9.13) : (9.15) ik ik (x - x)2 + ( y - y )2 dxdy A( x, y, z ) = A( x, y) exp - 2z 2z



(

)



: 2ik

A( x, y, z )



A 2 A 2 A = + . (9.16) z x 2 y 2 : R1 . , ( ), : (9.17) b z , b - , z - .

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112

?9.

, , ( ) . A( x, y, z ) . OX A( x, z ) , OZ, :
A( x, z ) =
n =+ n =-



An ( z )e

ikn z

,

(9.18)

kn = 2n/a - , , a - A( x, z ) . An ( z ) z . z = 0 (4.23) : An (0) = 1 A( x, z = 0) e a


0

a

-ik n x

dx ,

(9.19)

A( x, z = 0) - , (9.16)

. A( x, z ) (9.18)

e ikn x An ( z ) : dA ( z ) 2ik n + k n 2 An ( z ) = 0 . (9.20) dz k = 2 - . (9.20) : (9.21) A ( z ) = A (0)e in ( z ) .
n n

n ( z ) : n ( z) = kn 2 1 2 z n ( z ) = n z . 2k 2k a
2

(9.22)

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?9.

113

n ( z ) , n- , n z. z T , , An ( z ) , 2 . (9.22) , z T , 1 ( z T ) = 2 , - n ( z T ) = 2n 2 , : (9.23) z T = 2a 2 . zT An ( z = z T ) z = 0 . , ( x, z T , t ) . ( x, z T , t ) ( x, z = 0, t ) , . , (9.18) (9.14) (9.1), , : ( x, z T , t ) =



An (0) exp(i n z

T

)

exp(i (t - kz T + k n x) ) .

(9.24) I ( x, z T ) ,


2



( x, z T ) , I ( x, z = 0) , . , , z T : (9.25) z ( s ) = s z , s = 1, 2, 3, ... .
T T

, , , b, :

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114

?9.

(9.26) b << z . (9.15) A( x, y, z ) :
A( x , y , z ) = ik ik ik 2 exp - ( x + y 2 ) A( x , y ) exp ( xx + yy ) dx dy . 2z 2z z



(9.27)

. x , y , x, y , , : (9.28) x = x z , y = y z . , x, y , : (9.29) k x = k x, k y = k y . (9.28) (9.29) (9.27) x, y k x , k y :
+ -

A(k x , k y ) = const A( x , y ) exp i k x x + k y y dx dy .



(

)

(9.30)

A(k x , k y ) . A( x, y )

I () : kb sin I () sin 2 2 b - , - . kb sin (9.31) , 2 (9.31) ,
2

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?9.

115

I () m , : (9.32) b sin m = Á m , m = Á1, Á 2, ... - , . ( m << 1) : (9.33) m = Ám b . , ( m = Á1 ), : (9.34) = / b . . D (9.35) = 1,22 / D . . , , I () : kb sin 2 Nkd sin sin 2 sin (9.36) 2 2 , I () = I 0 2 kd sin kb sin sin 2 2 2 b - , d - , N - . (9.36) (9.31) b (8.2) N , d . (9.36) , .

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116

?9.

d sin n = n , n = 0, Á 1, Á 2, ... - . b sin m = Á m , m = 0, Á 1, Á 2, ... - . Nd sin p = p , p = Á1, Á 2, ..., Á ( N - 1), Á ( N + 1), Á ( N + 2), ... , p qN , q - . b < d , . N >> 1 , . R , , , (9.37) R = = nN . 9.1. = 600 D = 1,2 . b = 10 . b , , ? . n- R n = nz , z - . , D D2 n= . , , n = 6. 4z , z + z . , . , , , n' = 4. ,

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?9.

117

. , z + z = 1,5z z = 5 . 9.2. n- , N ( d, b, d > b). , = 500 d = 4 ? . b . - , E ( y) = E0 b
b/2

-b / 2



exp(-

ikyy' )dy ' , l

E0 - . y' , y y' , . l . tg = y / l . ( ) tg sin , , , E E = 0 b
b/2

-b / 2



exp(-iky ' sin )dy ' .

, ky' sin , y', , . , . kb sin E = E0 sinc() , = 2 sinc( z ) sin( z ) z . , , ,

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118

?9.

. ( ) kb sin = = n (n = 1, 2, ...) 2 sin = n / b . . , . , , , , kdsin . . , E0sinc() , .


n= 0

N -1

exp(-inkd sin ) . ,

- iNkd sin Nkd sin exp sin 2 2 . - ikd sin kd sin exp sin 2 2 , : 2 kb sin sin 2 Nkd sin sin 2 . 2 I () ~ kb sin kd sin 2 sin 2 2 kd sin : = n, (n = 1, 2, ...) sin = n . , 2 d n- , / b = n / d d = nb. 1 = nmax / d . 1 - exp(-iNkd sin ) 1 - exp(-ikd sin )

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?9. d 4 nmax = = =8. 0.5

119

9.3. , . , l = 50 . 0,2 . ( - 400-700 ). . . 9.2 E ( x, y ) = E b
b/2 0 2

-b / 2



ikxx' exp - dx' l

b/2

-b / 2



ikyy' exp - dy ' , l

E0 - . x' y' , x y x' y', , . l . 1 2 tg1 = x / l tg 2 = y / l . ( ) tg sin , E= E b
b/ 2 0 2

-b / 2



b/2

exp(-ikx' sin 1 )dx'

-b / 2



exp(-iky' sin 2 )dy .

E = E 0 sinc()sinc() , kb sin 1 kb sin 2 = = . 2 2 E 2 . , , , . kb sin 3 = sin = 3 / 2b . l 2 2

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120

?9.

D = ltg lsin = 3l / 2b . , D . = 700 . , 2D 5 . 9.4. . , l = 1,2 ; D = 0,5 ; = 500 . L , , , ? . . (9.27), = 1,22 / D . , , . () l/L. , , : / 2 . , L lD 19,7 . h 0.61


L

2 Rh , R -

. R = 6,371 106 h = 1,7 , L 4,7 . , , . 9.5. n. d ( x ) = d 0 + d1 sin(x) .

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?9.

121

, . n = 1,5, = 500 , d1 = 0,1 , = 2,1 10 4 -1. . (9.22), .
+

A() = const

-



A( x ) exp(ikx sin )dx

A(x'), . , , x' : 2 (d1 + d1 (n - 1)sin x ) , A(x') 2 A( x ) = a exp i d1 (n - 1)sin x , a - . , . d1 , : 2 2 A( x ) = a exp i d1 (n - 1) sin x a1 + i d1 (n - 1)sin x . , 2 d1 (n - 1) sin x exp i (kx sin )dx . 1 + i - , . , L = 2/ 3 . sin = e ia - e -i 2i , : A() a
+



(

)

+

A() ~

-



exp (ikx sin )dx + C exp(ix (k sin + ))dx - C exp (ix (k sin - ))dx ,
-

+



+

-



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122 C =

?9.

d1 (n - 1) . -. , , : sin 0 = 0, sin -1 = - k = - L sin +1 = k = L . , , ( ) Á1 Á9,6 o . , C 2 , . . , . 2 9.6. (2) ( = 500 ) I0 2 . 1) z1, z2, ..., zn P1, P2, ..., Pn, , 1, 2, ..., n . 2) . 3) , P1, , , P1. .. . . .: , 1976, . 8, ?? 33-35. .. . , .: - , 1983, . 3, ? 22.

, 1 ( ) 3 ( ).

2

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?9.

123

9.7. (2) f = 1 ( = 530 ). . P , ; P - 0,5 . ? : , . 9.8. (2) , n h r1 = 2 , r2 = 4 r3 = 6 . fmax = 500 . , h . Imax , I0? .. . . .: , 1976, . 8, ?? 34, 35. 9.9. (3) , , h b, , . n h , . ? n = 1,5, h = 0,1 , = 500 b = 2 10 -4 . .. . . .: , 1976, . 9, ? 46. . . . .: , 1974, ? 9.6. .. . . . .: , 1980, ?? 52, 53. 9.10. (3) = 108 I = 10- = 500 .
4

/ 2.

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124

?9.

, h = 10 . . ?1. . , , . .. . . ? 33. .. . . . .: , 1980, ? 52. 9.11. (2) T ( x) T(x) = a + bcospx. . = 500 , a = b = 0.5, p = 2 102 -1 . .. . . . .: , 1980, ?? 52, 53. 9.12. (1) D = 5 , 384000 . = 600 . S , N = 10 ? . ? ? . 9.13. (1) f = 2,5 D . = 630 . , , . D = 2 2 . 9.14. (3) z = 0 :

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?9. x2 + y E (x, y , z = 0, t ) = E0 exp - 2 2a0 ( z >>
2 ka0 2

125 exp(i (t - kz )) , k = 2/.
2 2



2 ( z ka0 )

). I(r) ( r = x 2 + y
2 ka0



) z = 0 z = . a z, a(0) = a0 . .. . . .: , 1976, . 9, ?? 43. 9.15. (2) d = 0,1 . , - d, , = 0,05 . ( = 500 ). , ( ). n, . 9.16. (2) a . a/2, h n. T1 T2. . ? .. . . .: , 1976, . 9, ?? 39, 40. 9.17. (2) , n, . " " - N. " " - a h. " " a . h - " ", a >> h. , . .. . . . .: , 1980, ?? 52, 53.

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126

?9.

9.18. (3) A T(x) = 1 + 0,5sinpx. Á0,06 . x = 0 . d = 10-3 , = 600 . . .. . . . .: , 1980, ?? 52, 53. 9.19. (1) f = 10 D = 5 . , R = 100 -1. , , h = 10 . 9.20. (1) P = 10 D = 30 . , = 630 , , P = 10 -14 .

, . 90% . d = 20 . .. , . .: , 1976, ?? 42, 96. 9.21. (2) D. I(r) , z (z >> D2/). . . .. , . .: , 1985, ? 33. . , -, .: , 1970. 9.22. (3) b l.

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?9.

127

(d - b) . , . . - . 9.23. (3) L, l (l < L). , . . . 9.24 (3) , . T(x',y') d D f (d < f). . a < Dd f . . . -, .: , 1970. 9.25. (3) , . T(x',y') d f. . , . , . d = f. . . -, .: , 1970. 9.26. (3) b . a . z = 0,5 z T 1,5 z T ,

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128

?9.

z T = 2a 2 - . . 9.27. (3) . OX a x , OY - a y , bx b y , . , , . z T z .
(s) T

,

9.28. (3) a . b = a / 2 . z = 0,25z T ; 0,5 z T ; 0,75 z T z T , z T = 2a 2 - . .

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129


, c0 = 2,998 108 / ( ), e = 1,602 10 , m = 9,110 10
-31 -19




-27

, 1 ... = 1,661 10 , 0 = 8,854 10
-6



-12

()/()
1

, Å 0 = 1,257 10 ()/() () , N A = 6,022 10 23 - , k = 1,381 10-23 / , R = 8,314 /() , h = 6,626 10
-34



, c = 331,5 / 25À, c = 1497 / , R = 6371 n ( ): 1,0003 (); 1,33 (); 1,5-1,7 ( ); 2,42 () 10-1 (, d) 101 (, da) 10-2 (, s) 10-3 (, m) 10-6 (, Å) 10-9 (, n) 10 10 10
-12 -15 -18

102 (, h) 103 (, k) 106 (, M) 109 (, G) 1012 (, T) 1015 (, P) 1018 (, E)

(, p) (, f) (, )

.

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