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, D ( ),

Dt F ( ,

). ( ) , D ,

D V, ,

Dr. (“ ”,

<<l), ,

( ),

. , ,

Dm×c = F×Dt, Dm = rDV = {DV=SDx} = DrV = DrSDy = DrS×cDt ;

s =

= Ee – ( ), – ,

e =

(

) =

DVr

Dm /V

– ;

Dy Sr

m m /V

, FDt = SEeDt,

DrScDt×c = SEe×Dt

Dr×c2 = Ee= E

rc2 =

c =

(2)

§2.

x = Acos(wt -

, :

d2x

v =

= -Awsin(wt

y) ;

a =

= -Aw cos(wt -

y) .

(3)

dt2

= e = +

sin(wt -

y) ;

d2x

= -

Aw2

cos(wt -

y) .

(4)

dy2

e – ,

( )

( t).

(4) (3)

1 d2x		=	d2x	(5)
c2	dt2		dy2

– ).

( Z ),

3- :

151

1 d2x

d2x

d2 x

1 d2x

= x ,

dt 2

dX2

dy2

dZ2

c2 dt2

dy 2

§3. .

(

) x = Acos(wt - w y) . V c

, E = E + U:

mv2

m æ dx ö2

2 2

A w

sin

(wt -

y) ,

(6)

è dt ø

Dt<T i Dy<l. F

A =

FDx

= U ( .

1).

. 1

F=SeE, e =

U =

eESDx

eDx ×

e2Dy =

e2 ( – ).

Dy 2

(2) (4),

U =

c2rV

A2w2

sin

(wt

y) =

sin

(wt -

y) .

(7)

= +U = rVA w sin

(wt -

(8)

E ,U
1
1		E
2		E
2
λ	λ	y
2	λ	y
T	T	t
2
	. 2

( . 2), ,

, ,

t, ,

(“ ”)

(

), ,

– “ ” .

152

: (6),

(7) ,

, .

.2, sin2j 0 1,

1/2. ,

	=	1	rVw2 A2 .	(9)
E

2

(9) V – , –

“ ” –


w	=	E		=	1	rw2 A2	.	(10)
w	=			=		rw2 A2	.	(10)
	V		2

“ ”, “ ” –

, S,

( ):

1 r × DyS × w2 A2

2 Dy

1 2

rw A

S =

rw A

× c × S = w × c × S ,

(11)

“ ”:

j =

rw2 A2c =

×c

(12)

– ,

, ,

j =

× c

(12 )

1874

. ( ) (

), .

(12),

: (r× ) –

–

; (w2× 2)

, .

( l (D =l)

DV = S×Dy = S×c×T ) ( ),

: DV = Sl = Sc/v = ScT).

153

E = 0,5×rw2A2DV , w = /DV Þ	E = w DV;
= / = w ×DV/T = w×ScT/T = wcS;	j = /S = wc.

( ^

, ) ( , , ,

( )

S ^ ( ). ( – , ).

F, ,

. Dr, D – , S Dt D :

D = DrSDx = DrScDt, DK = Dm×c = FDt = DpSDt;

DrScDt×c = DpSDt Dr×c2 = Dp

c =

( e =

Dp = F/S = s = eE =

–

). – .

c|| = E / r . (2)

( )

c =	G	, G – .	(2 )
	r

c = , F – , 0 – .

, , ,

154

§4.

. ,

( )

( ) .

: 1) , 2)

, 3)

(

.3,

1 2

. 3

( 1= 2, =0), 1 3,

, ,

. :

-2 , Dj = 2 n;

-– + 2 , Dj = +2 n = (2n+1)× .

j1 j2 , (

), ,

( . . 3).

, ,

) , 1 2:


x		= A sin(wt -	2p		y ) = A sin j ,						j = wt -			2p		y		;

1		1		l 1				1	1		1				l 1
x		= A sin(wt -		2p		y		) = A sin j		,	j		= wt -		2p		y		.

	2	2		l			2	2	2			2			l			2

(13)

(13 )

sin j + sin j

= 2cos

j1 - j2

×sin

j1 + j2

= A sin

j1 + j2

( . 17):

+ y

x = x + x

2cosç

- y ) ÷

×sinçwt -

= A

sin(wt -

y) , (14)

1 ø

– .

( . 4)

“

” :

155

											19
A2	= A2	+ A2	− 2A A cos α = A2			+ A2	+ 2A A cos(ϕ − ϕ			2	) . (15)
p	1	2	1	2	1	2	1	2	1	2
	, (
	)							,
	:
	,								–
	,
	.
. 4	, ( )
. 4	,	d =					y2–y1	( ,
	,	d =					y2–y1	( ,

, . 5). , (13) (13 ),

ϕ − ϕ =

2π

(y − y

) =

2π

2πd

Δϕ = 2 n =

d = nλ=2n

( cos(2πn) = 0 )

= 1+ 2;

(15)

+2 n =

2π

d d =

(2n + 1) ,

. 5

= 1– 2. (cos(π+2πn) = 1)

– ,

( ,

) .

§5.

, ,

, ( ).

, , +

ξ	= Asin(ωt −	ω	x) = Asin ϕ , –

		c	1

. “ ”

, , - ,

: ξ	= Asin(ωt +	ω	x) = Asin ϕ2 .

		c

: sin ϕ1 + sin ϕ2 = 2 cos ϕ1 − ϕ2 sin ϕ1 + ϕ2 ):

2 2

156

					19
x = x	+ x	= 2A cos	w	x ×sin wt = A sin wt ,	(16)
x = x	+ x	= 2A cos		x ×sin wt = A sin wt ,	(16)
			c	p
			c

A = 2A cos( w x) = 2A cos(2p x) – ,
p	c	l

, . ,

), .

, –

. ,

cos , , ±1 0.

1) cos \|	2p	x \|= 0 , = 0 – ,

	l

( , ). cos(p + np) = 0 (n = 0,1,2,3, ...),

x =

+ np Þ

x = ±

(n +

) = ±(

n) = ±

(2n +1) ,

(17)

x = ±	l	; ±	3	l; ±	5	l; ... – ( , D, F

4		4		4

. 6). l/4 ,

l/2.

					2)	cos \| 2p x \|= ±1,				= 2 – .
ξ	A	C	E			l
							cos(np)=1,
0	B	D	F	x
	B	D	F	x
	λ		λ		( , , )
	4		λ		( , , )					l
						2p				l
		. 6				l x = ±np		Þ	x	= ± 2 n .	(18)

§6.

, ,

. “ ” .

– .

x = 2Acos

x ×sin wt ,

= 2Awcos

x ×cos wt ;

mv2

2 2

2 2p

E =

(2Awcos

x ×cos wt)

= 2rA w V × cos

x ×cos

wt . (19)

157

( =f( )),

U=k 2/2 .

F=k ,

( )

F(x),

k(x).

”

dx,

d /dx (

s e).

U =

Fdx

1 F dx

dxS =

seV =

Ee2

2 S dx

V ,

– ( )

( ),

( 2 2 )

= r 2.

e = dx = -2A 2p sin 2p x ×sin wt = -2A w ×sin 2p x ×sin wt ;

| w=2 , c=l

rc2

2 w2

2 2p

2 2

2 2p

U = 2

V ×4A

sin

x ×sin

wt = 2rA

w V

×sin

x ×sin

wt .

(20)

(19) (20), ,

U /4,

l/4 ( . 7).

sinwt ( coswt)

( –

. 7).

. 7

( . 16 .6) ( )),

, :

U(x = λ ) =

2rA2w2V ×sin2 wt .

( :

(x = λ ) =

2rA2w2V ×cos2 wt )

, ,

. ,

158

20.

. .

. ,

. . .

§1.

– ,

(20-20000 ).

20 ,

20000 – . ,

1), (1 – 1 );

2)J, /( × 2)= 2,

j = wc º J = 1 rcA2w2 , rc = Ra – ;

1)( ) –

;

600-2000 ,

2)L.

) ( , ),

. – , .

– J0,

, .

æ J ö

L( ) = lgç ÷ ,

çè J0 ÷ø

)	-2					100
2
(	-4					80
0							L ( )
J	-6					60
	-8					40
	-10					20
	-12					0
	10	100	1000		10000
				( )
		. 1

æ	J	ö
L( ) =10 lgç	J	÷	,	(1)
L( ) =10 lgç		÷	,	(1)
ç	J0	÷
è	J0	ø

J0 – ( )

( ),

J0=10-12 2. , ,

2 20 ,

100 .

. J0

159

(800-3000 ), . 1,

(lg) ,

, ,

, .

J = J0 e−αx ,

(2)

L21 ( ) =10lg(J2 / J1

, 20 , 100 ).

3)– ( ) :

( ),

(“ ” ).

4)– .

§2.

( ) x = Asin w(t - y / c) .
V ( .
2), F S
D	D ,	. 2
	Dp.

, F=SDp=D ,

, , ), .

dp =

dm ×a

dm dv

dy = -r

d2 x

dy ;

(3)

dV dt

dt2

“ ” , : dp>0, dy<0.

(a=d2x/dt2):

v =

= Awcos w(t -

) ,

a =

= -Aw2 sin(wt - w

) .

(4)

(4) (3):

dp = rAw2 sin(wt -

y)dy

òdp = rAw2 òsin(wt -

y)dy

Dp = -rAw2 (-

) cos(wt -

y) = rAwc × cos w(t -

) = Dp

cos w(t -

) .

D 0 = r w

( ).

160

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