(3.7)
, (. 3.1).
( ) x : u = dx/dt. , , . . (w t - k + j0) = const. , w dt kdx = 0,
u = w/ k. (3.8)
, . (3.2).
, .
(3.9)
(3.10)
: , . .
( ) , (. 3.9). .
u , . 3.9. ,
(3.11)
.
, p p u u
u = u -l×d u / dl. (3.12)
( ). .
, , , , . . d υ /dλ = 0. (3.12). , , , . d u /dl, , .
3.
x ,t = 0,5cos(p t /4 - p x /10) . : 1) ; 2) , = 5 .
. u = l/× . :
w = p/4 -1; k = p/10 -1;
= 2p/(p/4) = 8 ; l = 2p/(p/10) = 20 ;
u = 20/8 = 2,5 /.
, :
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x t = 0,5cos (p t /4 - p/2) .
: ut = -0,5(p/4) sin(p t /4 - p/2) /c. u 0 = 0,5 (p/4) 0, 4 /.
p
( ).
. , .
, Δ V
(3.13)
(3.14)
(3.13) (3.14) , , .. . , , .
Δ V Δ W =Δ W + Δ W ,
, (3.15)
.. .
. 3.10 , l .
( . 3.10),
. , . , ( . 3.10 1.3).
W 0 , , /3,
(3.16)
c .
(3.17)
, Δ S┴, , .. = Δ W/ Δ t, /.
u S. , W = W 0 υTS.
< > = r A 2w2 uS /2. (3.18)
U ( ) , , , .. U = Δ W / ( Δ S┴ Δ t). , /(2).
< U> = r A 2w2 u/ 2 =< W 0 >u. (3.19)
, (, ) .
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x1 = A cos(w t kx);
x2= A cos(w t + kx).
, :
x = x1 + x2 = cos w t. (3.20)
= ½2 × cos kx ½ = ½2 × cos(2p x/ l) ½, (3.21)
.
( ) . 3.11.
, .. .
, , .
: = ½2 cos(2p x /l)½ = 0, cos(2p x /l) = 0. , 2p x /l = (m + 1/2) p,
(m = 0, 1, 2, )(3.22)
, , .. = 2 , . : =½2 cos(2p x /l)½ = = 2 , cos(2p x /l) = 1. , 2p x /l = m p,
(m = 0, 1, 2, ). (3.23) (3.22) (3.23) , l/2. , .. l/ 4.
p , , , p , ( , ). .3.12 .
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1. , p p .
2. p p , , . cos(w t) . p p , ( ).
cos kx , .. . p.
3. p p p p ppp , p, p p , p. p p p p . , , .
, , , . l / 4.
. ( , .. r u).
( ), . . 3.13: , ( ), .. , . .
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( ), , , . , . . 3.14: , (). - , , .
(3.22) (3.23) . - (. 4).
. , , , , . .
4. x = 0,2cos ( p t/ 2 p x /3 ) . . : 1) : 2) , 3 ; 3) .
. 1. : x = 2 cos kx × cos w t (3.20). , = 0,2 , k = p / 3 -1, w = p / 2 -1. , x = 0,4cos(p / 3) cos (p t/ 2) .
2. :
½ ½ = ½0,4cos 3p / 3½ = 0,4 .
= 3 .
3. , p. : x = os(w t - kx) x = os(w t + kx + p). : x = 2 cos (kx + p/2) × cos (w t + p/2)
x = 2 sin (kx) × sin (w t). (3.24)
(3.24) ,
= m l/2; = (m +1/2) l/2, (3.25)
.. , = 3 .
5. , , (.3.15) 5 . , . 5100 /.
. - . ( ), (.3.15). , . l/2, , :
l = m l/2, (m = 1, 2, 3,..) (3.26)
n u l = u /n. (3.26),
n = mu /(2 l), w = m p u / l. (3.27)
(3.27) , , ( ) . m = 1 .
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m = 1, 2, 3 n , , 510, 1020, 1530. (. 3.15)
. 3.15
,
1. ?
?
2. : , , -
, .
3. : ?
4. , ? .
5. , ?
6. , , ?
7. ?
8. ? .
9. , : 1) l/2; 2) l/4; 3) l.
10. , ; .
11. t 4 (. 3. 16). . 3.16 l/2. , , . , . .
12. , .
13. . 3.17 . , . .
14. . ? ? .
15. , , ?
16. , ?
17. 0,8 . ?
18. : ?
19. , , : ) ; ) ?
20. : ) ; ) ?
21. ? ?
22. , ? ?
23.. ? ?
24. p?
25. , : ) ; ) .
26. . / 4; /2.
27. . 3.12 . , : ) ; ) .
28. , ? ?
29. ?
30. , , . ?
31. 3 . .
32. , . 3.13, 3.14. : ?
33. , . , . ?
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1. l = 62,5 .
2. T = l /u, T = 1,56.
3. n = 1/ T, n = 0,64.
4. w = 2pn, w = 1,28 p -1.
5. k = 2p/ l, k = 0,032p -1.
6. A = 95 .
7. , ( , ).
8. x = 95cos ( 1,28p t 0.032p x),.
9. 1 2 :
x1 =- 70, x2 = 70 .
10. H 1 2 p p , p ( w t - kx) t =0 x:24 - 1 55 - 2. - 0,77p p - 1,76p p 1 2 . 1 2 .
: 2p; = l 12,5 ; 2p/12,5.
11, 12. p :
x1 = 95×cos(1,28p t 0,77p) - 1,
x2 = 95×cos(1,28p t 1,76p)) - 2.
13. p p p p, Dj = j1-j2. : 6,25 , 2p/12,5, , p, .. .
2. . .
(.3.19) , ..
1. l = 12 .
2. : = 20 , = 40 .
. 3.19 .
3. = 0,04
4. w = 50p -1.
5. k = p/6 -1.
6. () , (3.24) x= 0,4sin(p x /6)sin(50p t) ;
7. :
, : 0 1 2 3 4 5 6
½ ½, : 0 20 34,8 40 34,8 20 0
8. x 32sin(50p t) c.
9. (3.25)
m 0 1 2 3 4
, 0 6 12 18 24
, 3 9 15 21 27
10. , D: Dj = 0, Dj CD = p.
11. , , .