2
, , , , .
. , . : , , - . , , : , . , , , , . , . , , , , .
. , , . , , , , . , . , . , , . , , , , , , .
, . , . , , u x t, y z. u (x, t).
? , , . , u (x, t). , ό , , , . , . , , , , . , , . , . , .
|
|
, , , , , . , , . , ρ, x y. , , .
. .
1. , , :
1. , .
2. .
3. .
(2). , . 0 ρ 0. = f (ρ), 0 = f (ρ 0). . (1 = 105 /2). 1 (1 = 1,0133 ). , , , . I :
(3.1)
,
. = 103 = 2 10-7 60 . , , , 1 . . , , ; 1 . , . 100 . 120 .
= 0 + , ρ = ρ 0 + ρ (3.2)
, 0, ρ ρ 0.
0 + = f (ρ 0 + ρ ) = f (ρ 0) + ρ f ' (ρ 0), (3.3)
0 = f (ρ 0) f ' (ρ 0) f (ρ 0), ρ = ρ 0. , ρ . , , ρ ; κ:
|
|
= κρ , κ = f ' (ρ 0) = . (3.4)
(2).
(1). , , , x, t u (x, t), x + u (x, t), . 3.1.
u (x, t) |
u (x+ Δ x, t) |
x |
x+ Δ x |
x + u (x, t) |
(x+ Δ x) + u (x+ Δ x, t) + χ(x, t), |
. 3.1. x u (x, t), x+ Δ x u (x+ Δ x, t) , , Δ x, Δ x + u (x+ Δ x, t) u (x, t) |
, x+ Δ x, x+ Δ x + u (x+ Δ x, t). . , , x, .. . , Δ x, ρ 0Δ x, ρ 0 , , . , , x + u (x, t) x+ Δ x + u (x+ Δ x, t), , Δ x . ρ ,
ρ 0Δ x = ρ [ x + Δ x + u (x+ Δ x, t) x u (x, t)] (3.5)
Δ x , u (x+ Δ x, t) u (x, t) = (du / dx) Δ x. , u x, .
(3.6)
(3.7)
, ρ , u du / dx . , ,
(3.8)
(du / dx) (du / dx). , (1):
. (3.9)
. x, . : u x, , .
, . , . Δ x , x, ρ 0Δ x, d 2 u / dt 2, , , ρ 0Δ x (d 2 u / dt 2). ( Δ x , , - ). , , x, ρ 0Δ x (d 2 u / dt 2). x P (x, t), + x, x+ Δ x , P (x+ Δ x, t) (. 3.2):
(3.10)
Δ x |
P (x, t) |
P (x+ Δ x,t) |
x |
. 3.2. , x, , x, (dP / dx)Δ x |
, Δ x x. , (3)
|
|
(3.11)
, , x. (3.11) (3.4):
, (3.12)
ρ (3.9). ρ 0
(3.13)
(3.14)
, .
2.
, . , t , , 0 ( ).
u t :
,
.
u x :
,
.
ω 2 k 2 ( ω 2 k 2 ). :
.
,
k = 2π/ λ, ω = 2π ν = 2π / λ,
:
(3.15)
, (3.14), , 0 . (2.9) (. 1). , 0 . : - ( u) t x , (3.15), .