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M2
M3
d
dt
− ν∆ = −
~
∇p + ,
div = 0.
M4 :
d
dt
= −
1
ρ
~
∇p + , div = 0.
M4
=
~
∇ϕ
ϕ
∆ϕ = 0 ,
M0
(κ = 0, µ = λ = 0)
M5
dρ
dt
+ ρ div = 0, ρ
d
dt
= −
~
∇p + ρ ,
p = p(ρ, s ),
ds
dt
= 0.
M5
ρ = ρ
0
, s = s
0
, p
0
= p(ρ
0
, s
0
)
M5
M6 :
∂
∂t
+
1
ρ
0
~
∇p = ,
1
c
2
∂ρ
∂t
+ ρ
0
div = 0.
c
2
c
2
=
∂p
∂ρ
¯
¯
¯
¯
ρ=ρ
0
> 0.
c
M6
∂
2
p
∂t
2
− c
2
∆p = F, F = −c
2
ρ
0
div ,
c
p = Re[Φ(x)e
−ikct
], = Re[ (x)e
−ikct
], F = Re[ϕ(x)e
−ikct
].
Φ
∆Φ + k
2
Φ = g, g = −
ϕ
c
2
,
k = ω/c
ω
0
x = γ(ξ, t), ξ ∈ ω
0
, t ∈ (0, T )
T =
∂x
∂ξ
= x − ξ
T = I +
∂
∂ξ
.
T
|dx|
2
= | dξ|
2
+ dξ · (T
∗
T − I) hdξi
E =
1
2
(T
∗
T − I) ,
E =
1
2
µ
∂
∂ξ
+
∂
∗
∂ξ
¶
+
1
2
∂
∗
∂ξ
∂
∂ξ
.
d = kT − Ik
d
2
d
2
P = λ tr E ·I + 2µ · E.
M7 : ρ
0
∂
2
∂t
2
= ( λ + µ)
~
∇div + µ∆ + ρ
0
.
λ, µ
ω
0
µ∆ + (λ + µ)∇div + ρ
0
= 0 div P + ρ
0
= 0.
= 0 ∆ div = 0, ∆
2
= 0
ρ = 0 ρ = 0
p
=
p
0
µ
ρ
ρ
0
¶
γ
, γ
≥
1
ρ
0
h
α
P
M5
= 0 ρ = 0
M2
=
0 x
j
