Tr&v

!"#$

s = s o + A cos(ωt + ϕ) = s o + Ae i ( ωt + ϕ) , %

ϕ=π/2

e ± iΦ = cos Φ ± i sin Φ &

!"

Φ'() )!Φ

, / Φ'ω01ϕ − 2 ϕ− 2 ω=3πν 4

[ω ] = rad ≡ s −1 , s

-

ν−

[ν ] =

!"Φ'*+ )!Φ# '.#

ω

y

A ϕ

5'67ν = 3π/ω

. . ≡ Hz s , 8'% ω01ϕ) 9'% !" ω01ϕ#

4

&

:' %)!Φ

%$

x

s = −ωA sin(ωt + ϕ) s = −ω2 A cos(ωt + ϕ) = −ω2s s + ω2s = 0, ;

&

2

4

-

<

< ,

<

,

-

>

4

= <

-

-

F = ma = my = − ky y+

k y=0 m

> -

&

2π m k , T= = 2π k m ω

<

ω01ϕ#

9'/ ?

ω=

4

-

-,

2

2

Ek

<

mv mω A mω A = sin 2 Φ = (1 − cos(2ωt + 2ϕ)) 2 2 2 2 ky 2 mω2 A 2 mω2 A 2 Ep = = cos 2 Φ = (1 + cos(2ωt + 2ϕ)) 2 2 2 2 mω2 A 2 ? E = Ek + Ep = = const 2 Ek =

2

2

2

2

Ep

E 4 3ω!

t

Iε = M Iα = −mgl sin α ≈ −mglα

α+ %

mgl α =0 I

ω=

I mgl , T = 2π mgl I

-

mg

2

I = ml 2

l

α

ω=

g l , T = 2π l g

<

,

α l m mg

@

A

Iε = M Iα = − Dα α+

D α=0 I

ω=

D I , T = 2π I D

α

4

- -

#

!

!

"!

u C = εL q = −Φ = −Lq C 1 q=0 q+ LC

C

ω=

1 , T = 2π LC LC

+ -

L

! "

#

# $

% &

-

$ !

%

2

ξ1 = A1 cos(ωt + ϕ1 )

ϕ

& B ,

ω

ζ

ξ 2 = A 2 cos(ωt + ϕ2 )

A

ξ = ξ1 + ξ 2 = ? C

$ ! !

ξ = A cos(ωt + ϕ)

!

:

tgϕ =

ϕ

!

A = A 12 + A 22 + 2A 1A 2 cos(ϕ 2 − ϕ1 ) A sin ϕ A1 sin ϕ1 + A 2 sin ϕ 2 = A cos ϕ A 1 cos ϕ1 + A 2 cos ϕ 2

<

$

A1

α

α = π − (ϕ 1 − ϕ 2)

A2 ξ

ξ = ξ1 + ξ 2 + ....... + ξ n = A cos(ωt + ϕ) A = A 12 + A 22 + ...... + A 2n + 2 tgϕ = -

j> i

A j A i cos(ϕ j − ϕ i )

A1 sin ϕ1 + A 2 sin ϕ 2 + ........ + A n sin ϕ n A1 cos ϕ1 + A 2 cos ϕ 2 + ........ + A n cos ϕ n "<

# $

% $

<

!

$

-

2

ζ

ξ1 = A1 cos(ω1t + ϕ1 )

! $

-

∆ω=ω2−ω1 A

ξ 2 = A 2 cos(ω2 t + ϕ2 ) ξ = ξ1 + ξ 2 = D ∆ω=ω2−ω1.

A1

ϕ

'(

')

>

-

1

<

<

-

2

$

B -

,

4 $

<

ξ

4

-

E

A2

ω1 ω2

,

ω

i

ξ( t ) =

n i =1

A(ωi ) cos(ωi t + ϕi )

(ω) %

-

&

ω

ωi

$

ω2

ξ( t ) = A (ω) cos( ωt + ϕ(ω)) dω ω1

ω1

ω2

ω

[dB]

(ν) 50 40 30 20 10 0

A % -

0,1

1

A-

$

ν [κΗz]

6,4

! % C

4

,

<ω2 F

& < ' >

$

2 4-

-

< - B

,-$ * >

$

x = A cos(ω1 t + ϕ1 ), ρ = A

x = A cos(ω1 t + ϕ1 ), ρ = A

z = B cos(ω 2 t + ϕ 2 ), ω1 / ω 2 = 1

z = B cos(ω 2 t + ϕ 2 ), ω1 / ω 2 = p / q

B

G

H

ω1

& <-

-

! % I

& 4-

<

4

4

-

<

JK' KL$

ma = F = Fe + Fr k = ω02 , m 2 x + 2β x + ω0 x = 0

r = 2β m

mx = −kx − rx ,

%

β=0, -

< &

A

&

&

-

<

$

x = A cos(ωt + ϕ) = Ae i ( ωo t + ϕ) = Ae iΦ I -

<

x = Ae α t

-

A

x = αx , x = α 2 x

α 2 + 2βα + ω 02 = 0 6

&

α 1, 2 = −β ± β 2 − ωo2

ωο > β:

α1, 2 = −β ± i ωo2 − β 2 x = Ae −β.t e ±iωt = Ae −β.t cos ωt , ω = ωo2 − β 2 < ωo , T =

2π ω

&

$

#

%

-

2

<

β$

A( t ) = Ae−β.t , A( t + T ) = Ae−β.t e −βT A( t ) = eβT A( t + T) + $

λ = ln

!

Q=

,

? 4 $

2

-

λ

4

π π ω = = λ β T 2β

3π

mω2 A 2 ( t ) E 2 N= = = ∆E T ET 3

A( t ) = βT A( t + T)

#

1 1 π Q = = 2βT π 2π A( t ) 2 T A( t )

ωο M β$

α1, 2 = −β ± β 2 − ω2 < 0 x = A1e α1t + A 2 e α 2 t , N

-

ωο ' β

α 2 > β,

<

x $

3

t

x = 2A1e −β.t , α1, 2 = −β D

-

3

N

2

<

ω0 ' )!ω0 #

' (

%! )

%!

%

% J

A

O0

'J

ω0

ω$

&

ma = Fe + Fr + Fout mx = − kx − rx + Fo cos ωt x + 2β x + ωo2 x = f 0 cos ωt , f 0 = A

8'/

Fo m ω0 ϕ):

x = −ωA sin(ωt − ϕ), x = −ω2 A cos(ωt − ϕ)

β=0

(ωo2 − ω2 ) A cos(ωt − ϕ) − 2βωA sin(ωt − ϕ) = f o cos(ωt − ϕ + ϕ) = f o cos ϕ cos(ωt − ϕ) − f o sin ϕ sin(ωt − ϕ) (ωo2 − ω2 )A = f o cos ϕ A = ? ϕ=? 2βωA = f o sin ϕ A= ' P

fo (ωo2 − ω2 ) 2 + 4β 2 ω2

ϕ = arctg

,

%

-

2βω ω − ω2 2 o

!

! $

ωr = ωo2 − 2β 2 , A r =

fo 2β ω − β 2 o

2

, Ao =

fo ωo2

ωo2 ω Ar = ≈ o = Q, ako β << ωo 2 2 A o 2β ωo − β 2β . . .

% !

$ /%

β QQ ωο

ωr = ωo , v max = ωo A !

0

* ' 1 2 % 2 3 I >L

+! , !

)

% 4

4

A

< 4-

&

ξ o = A cos ωt , τ =

x v

ξ = A cos(ωt − kx ),

v

O

ξ = A cos ω( t − τ) = A cos(ωt − k=

ω x) v

&

x

4

R ω

4 $

2π , λ= ω 2π λ= = k

P

ω v

D τ'87L S

T=

*

2π k 2π v = v.T ω

&

8

4

$

! n=const$

ξ = A cos(ωt − k.r ), k = kn T F

D

2 -

<

#-

<

B

4

U'VπK3

ξ= 7

&

<

W X VπK3/3 %X67K /'Y7K 4567K $

ao cos(ωt − kr ) r %

B

< 4

&

4

WX3πρZ /3 /X67ρ673 /'αο7ρ673 45ρ ρ/ρ:

I

ξ = [(ω0 8&6#

1

2 4

$

ξ=

αo cos(ωt − kρ) ρ

∂ξ = ωf ′(ωt − k .r ), ∂t

<

U'3πρZ

∂ 2ξ ω = ω 2 f ′′(ωt − k .r ), k = 2 v ∂t

∇ξ = −kf ′(ωt − k .r ), ∇ 2 ξ = ∆ξ = k 2 f ′′(ωt − k .r ) =

∆ξ −

1 ∂ 2ξ =0 v 2 ∂t 2

1 ∂ 2ξ v 2 ∂t 2

* 1 R 9

%

-

! $

vg v

! !

4 2

2

#! \

, 4

<

∂ξ ∂t .

< -

$

ξ = A o cos(ωt − kx ) + A o cos((ω + dω) t − (k + dk ) x ) 1 ξ ≈ 2A o cos (dω.t − dk.x ). cos(ωt − kx ) 2 1 ξ = A ( t , x ). cos(ωt − kx ), A ( t , x ) = 2A o cos (dω.t − dk.x ) 2 ]

Φ' ω0 S8#

2

-

#

Φ/' ^ω.0 ^S8#73$

#

Φ = const Φ = 0 = ω − k.x Φ = ω − k.v Φ vΦ = v = Φ A = const

ω k

Φ A = 0 = (dω − dk.x A ) / 2 = (dω − dk.v g ) / 2 dω 2π , ω = v.k , k = dk λ dv dv dv = v+ = v−λ vg = v + k dk d ln k dλ vg =

dv =0 dλ dv >0 dλ dv <0 dλ

1. 2. 3. E T-

vg = v vg < v vg > v 4 ,

4

"' 7L'" λ# -- _

. :

)

-

!

&

-

A

$

ξ1 =

a1 cos(ωt − k 1r1 + ϕ1 ) = A1 cos Φ1 r1

ξ2 =

a2 cos(ωt − k 2 r2 + ϕ 2 ) = A 2 cos Φ 2 r2

ξ = ξ1 + ξ 2 , A = A12 + A 22 + 2A1A 2 cos ∆Φ A ={

A1 + A 2 = max

2mπ , ako ∆Φ = { A1 − A 2 = min (2m + 1)π

1. ω1 = ω2 ∆Φ = const ( t ) = { } 2. ∆ϕ = ϕ 2 − ϕ1 = const

Ako ∆ϕ = 0, 2m

λ 2

∆Φ = k (r2 − r1 ) =

∆r = { λ (2m + 1) 2

max min

2mπ max 2π ∆r = { (2m + 1)π min λ

-

-

#

&

#

"

`

ξ

-

<

-

ξ P

,

x

ξ = ξ1 + ξ 2 = A cos(ωt − kx ) + A cos(ωt − k (2l − x ) + ϕ) ϕ ϕ ξ = 2A cos(k ( x − l) + ). cos(ωt − kl + ) = a cos Φ ( t ) 2 2 ϕ ϕ a = 2A cos(k ( x − l) + ) = 2A cos Φ A Φ( t ) = ωt − kl + 2 2 ] a

& , a , -

-

R )&

#

, -

ϕ = ± (2m + 1)π ; B

ϕ 2

cos

ϕ =0 2

%"

λ

λ/2&

;

!

a ( x = l) ≡ max = 2A cos ;

λ/2

D

3π %

ϕ = ±2mπ

-

ϕ π = ± (2m + 1) 2 2

-

-

7

ϕ 2

cos

π'λ/2). >

%

ϕ =1 2 -

!

#

%

2 $

& 2

-

λ. ;

a ( x = l) ≡ 0 = 2A cos

% (&

l

ϕ = ± mπ 2 %"

&

#

& 4 &

! \

4

,

$ ξ 8 0#

ξ 81^8 0#

P-

x

δl σ = l E b

W`

l'^8

σ

-

ma = F, ρSdx.

∂ 2ξ 1 ∂ 2ξ − =0 ∂x 2 E ρ ∂t 2 v=

E ρ

<

δl = δ(dx ) = ξ( x + dx , t ) − ξ( x , t ) =

δl δ(dx ) ∂ξ σ = = = ∂x E l dx =

dx

σ(x , t)

σ(x + dx,t)

,

∂ξ dx ∂x

∂ 2 ξ 1 ∂σ = ∂x 2 E ∂x

∂σ ∂ 2ξ = [σ( x + dx , t ) − σ( x , t )].S = S.dx 2 ∂x ∂t -

ρ

∂ 2 ξ ∂σ ∂ 2ξ = = E ∂t 2 ∂x ∂x 2 ∆ξ −

1 ∂ 2ξ = 0 #$ v 2 ∂t 2

#!

%

v =

µ=

E 1− µ ≈ ρ (1 + µ)(1 − 2µ)

l

E ρ

∆a / a <1 ∆l / l

b

∆l ∆a ∆b ~ = l a b

B

#!

v⊥ =

E ρ.2(1 + µ)

v II = v⊥

2(1 − µ) >1 1 − 2µ

∆t = t ⊥ − t

D

2 -

=

B

$

l l l v − = − 1) ( v⊥ v v v⊥

4

#

-

l=

v .∆t v − 1) ( v⊥ 2

c'.

v⊥ = 0

2

%

T

v=

v= α

?

!

#!

=

ρRT µ

EV

2π λ

F.l F = ρS.l ρl

4

ρl

J

%

v=

EV = ρ

-

dV dp =− V κp

g k +α $ ρ k

4 4

!

dV dp =− V EV p=

g k

k ρ

v=

<

v=

%

κRT ≈ 340 m / s µ =

v=

EV = ρ

κp = ρ

,

κRT ∴ µ

pV κ = const

0'3. d

κ=

cp cV

/ F δl ∂ξ σ ES = , F= = = δl = δl, l ∂x E E.S l εk =

∆E k ∆mv 2 ρ ∂ξ = = ∆V 2∆V 2 ∂t ∆E p

δl 2 E δl εp = = = ∆V 2S.l 2 l

2

2

=

ρω2 A 2 sin 2 (ωt − kx ) = ε p , 2

E ∂ξ = 2 ∂x

2

!

k=

Ek 2 A 2 ρω 2 A 2 sin 2 (ωt − kx ) = = sin 2 (ωt − kx ) 2 2

ε = ε + ε p = ρω 2 A 2 sin 2 (ωt − k.r ), ε =

ρω 2 A 2 2

Φ = ε S.v g = εS ⊥ v g = JS ⊥

ε, ε k

=

εp

Ek 2 = E

-

:

I≡

ES l

2π ω = λ v

ω2 = ρω 2 2 v

000

Φ = εv g = J S⊥

!

=

∂B ∂t ∂D ∇×H = j+ ∂t

∇×E = −

'!

j=0

∇.D = ρ ∇.B = 0

D = εE B = µH

ρ = 0$

∂B ∇×E = − ∂t ∂E ∇ × B = εµ ∂t

∇.E = 0 ∇.B = 0

∂ 2B ∇ × (∇ × B) = −εµ 2 ∂t ∂ 2E − ∇ × (∇ × E) = εµ 2 ∂t 2 1 ∂ E ∆E − =0 1 / εµ ∂t 2 1 ∂ B ∆B − =0 1 / εµ ∂t 2 2

∂E ∂ 2B ∇× =− 2 ∂t ∂t ∂B ∂ 2E ∇× = εµ 2 ∂t ∂t

a × (b × c) = b(a.c) − c(a.b)

E

B

∆ ≡ ∇2

-

$

F

E = E 0 e i ( ωt −k .r ) , B = B 0 e i ( ωt −k .r )

. =

4

!

1

v=

1

=

4

εµ !

v=

∇.E = −ik.E = 0 ∇.B = −ik.B = 0

1 ε 0µ 0

= c ≈ 3.10 8 m / s

k⊥E

B

!

∂B ∂t = ∂E ∇ × B = −ik × B = εµ.iωE = εµ ∂t ∇ × E = −ik × E = −iωB = −

0

k × E = ωB

ω ω n×B = − 2 E v v

k × B = −εµωE E

E = −( v × B ) = B × v , B = n × E / v

J

B % a ,

E !

E

4

> 4

/ =

\, -

-

a

F -

p = j.E = E.(∇ × H ) − E

-

4

∂D , om ∇.(E × H ) = (∇ × E).H − E.(∇ × H ) ∂t

p = −∇.(E × H ) + (∇ × E).H − E p=−

∂D ∂B ∂D = −∇.(E × H ) − H −E ∂t ∂t ∂t

∂ 1 [E.D + H.B] − ∇.(E × H ) ∂t 2

∂ε = − p − ∇.(E × H ) ∂t

ε ≡ εe + εm =

E.D H.B ε E 2 µ H 2 + = + 2 2 2 2

4 4

J = E × H = −( v × B) × H = H × ( v × B) = (H.B) v = ε v

:

ε E2 2

=

ε (− v × B)2 2

=

ε v2 B2

I= J =εv

2

=

4

44

εe =

p = j.E ∂D ∂D ∇×H = j+ , j = ∇×H − $ ∂t ∂t

&

B2 µ H 2 ε = = εm = 2µ 2 2

$ !

>

4

4

1 1 1 ε 2 1 µ 2 I = J = ε v = (ε E 2 + µ H 2 ) = E + H 2 2 ε εµ 2 µ

.

)

0 B

2 -

!

2π 2π 2π = = n= n k= λ v.T c.T λ0

$

λ0

-

∆Φ = k 2 r2 − k 1 r1 + ∆ϕ =

2m

λ0

, 4 ` , -

δ = 2d m +

2

,

2

R>>dm#

2m

λ0 2

={ λ (2m + 1) 0 2

max min

R-dm

rm

min

λ0 2

_ _ _

-A &

λ0

min

<

2/ <-

max

max

4-

∆ = r2 − r1

--

2mπ (n 2 r2 − n 1 r1 ) + ∆ϕ = { (2m + 1)π λ0

2 ={ λ ± λ0 / 2 (2m + 1) 0 2

δ = n 2 r2 − n 1 r1 + {

,

2π

0

δ

<

2π

"' 7L

B

+1

rm2 = d m (2R − d m ) ≈ 2d m R

2d m =

rm2 R

λ

(2m − 1) 0 max rm2 2 ={ λ0 R 2m min 2

λ0 =

rm2 mR

- )

'

2 + -

!

- ) D

4

, B

2

&

^? 6

67K

A 2 4

#

4 , , - ^? 6

4

]

4

P

]

<

λ/2. F

&

4

%

$

A P = A1 − A 2 + A 3 − A 4 + ......... + (−1) n +1 A n =

A1 A A A A + ( 1 − A 2 + 3 ) + ( 3 − A 4 + 5 ) + ... 2 2 2 2 2

A1 = A 0 , I P ~ A 02 = I 0 2 ]

$

# <

,

dS dS.r cos α = 2 , ξ P = dξ P , I P =< ξ P2 > r r

2

%

2 & 2

]

B B

=

/

2 ,

&

4

3/ >4

B ]

!)

` 3/

dξ P =

1

A P = A1 = 2

A1 , I P ~ A 12 = 4I 0 2

%

]

%

6

+ -

N

! -

e

]

$

$

A1 , I P ~ 16A 02 = 16I 0 2 A A P = A 1 + A 3 + A 5 ≈ 3A 1 = 6 1 , I P ~ 36A 02 = 36I 0 2

R

A P = A 1 + A 3 ≈ 2A 1 = 4

] ]

` *+f *. 7 3+16#

3

δ

,

I ]

l >> a

δ = ∆ = a sin θ m = {

2m

λ 2

(2m + 1)

+ 2

!

% A

!

λ73

δ

=

!

I A

;

-

min

λ 2

max < 4

,

λ/2. B

λ

d sin θ m = 2m

2

= mλ

N.d sin θ m′ = 2m' .

=

λ 2

max

= m' λ

"

min

$

% -

!

=

!

`

,

('λ/∆λ. b

< λ:

-

sin θ' m = (m ±

<

λ+∆λ

-

1 λ λ + ∆λ ) = sin θ m = m N d d

R=

λ = mN ∆λ

. R ?

) λ = 6.

(X –

4 A @

. θm

2.

6. 2

1

;

+ a , A

-

#

7 6.

6.

+

+Y8

!

λ

63

.

-

∆ = δ = 2d sin θ m = mλ 1.

g

-

A

+

.

!

N ; >

+ > - # &

A

λ +

`

A 2

G" !

34

)

)

%

-

βMM '6#

, h

\

4

b

- #

ν0 ν ν.

# & b

&

-<# λ0 %

,

b) -<# L)

L

-

LK

D

λ

< $

6

λ L5

6

λ

B

L 5.

ν 0 675.

,

L , ,

LK λ.

L)# 5.

i

, +

L7 λ.

L) 5. ' L &

#

- !

$

L)'. λ=λο

L )M. λQλο

L )Q. λMλο

v − vr 1 v − vr = ν0 v − v e T0 v − ve

2.

ν=

2a.

λ = λ0

v − ve v − vr L

>

LK

3

3 # B& % L

,

LK

b

%

L

L K MM L

%

L) ' L

$

2

ν = ν0

< ,

4

%

C

<

3.

sin

-

2

, 4

a <

4

, $

<

k4

$

α v 1 = = 2 ve M B

<

4

!

L) Q L

4 #

!

? -

L) Q L

2

4 , 2

-

<-

v − vr 1− vr / v ≈ ν 0 (1 + e ) v 1 − ve / v

λ '.

3 b

B j

a`

a <

<

-

< - !

E

# !

, ,

D$

5 )

-

B

!

4 2

#

4 2

4 →

k ' = ω' =

ω = ω0

k − β.k x 1 − β2

=

2 →

k − β.k cos θ 1 − β2

=ω

1 − β cos θ 1 − β2

, ω' = ω 0

1 − β2 1 − β cos θ

#%

θ'.

$

5a.

ω = ω0

1+ β > ω0 1− β

#%

θ=π

$

5 .

ω = ω0

1− β < ω0 1+ β

n

P A e % b

--

Sm 'S

| k |= k 2 − k 2 = 0, k 0 = k

k = (k , k ) = (ω, ωn),

5.

#

l2

B

4.

'6

θ=π/2 \

2 2

#

$

5 .

ω = ω0 1 − β 2 %

>

-

6oNg4

βMM '6 #$

'$

k'x =

6. a <

4

k x − β.k

ω' cos θ' = ω

1 − β2 cos θ' =

<

cos θ − β , 1 − β cos θ 4

cos θ − β

cos θ =

1 − β2

4

= ω'

#

$

1 − β 2 cos β − β 1 − β cos θ 1 − β 2

cos θ'+β 1 + β cos θ' ,

,

4

,