10 The Effect of Preheating on the Thermoelastic …
143
x
i =
x i · ω∗
V
l
, t
= ω ∗ t, V
(n)
P =
c
(n)
11 /ρ
(n)
0 ,
u
(n)
i
=
u
(n)
i
· ρ
(n)
0 ω ∗ V
(n)
P
β (n)
1
θ 0
, i = 1 − 3,
E
(n)
=
θ 0 β
(n)2
1
ρ
(n)
0 c
(n)
ε c
(0)
11
, u
(n)
4
=
u
(n)
4
θ 0
, c
(n)
i jkl =
c
(n)
i jkl
c
(0)
11
,
(10.11)
β
(n)
i
=
β
(n)
i
β
(0)
1
, λ
(n)
i =
λ
(n)
i
λ
(0)
1
, ω
=
ω
ω∗
, ω∗ =
c
(0)
ε c
(0)
11
λ
(0)
1
10.4 The Plane Surface Waves
The propagation of harmonic plane oscillations is considered. For the purpose of
this, we present all the functions in the form:
f (x 1 , x 2 , t) = f (x 1 )e
−iω t
Then, the defining expressions (10.1)–(10.3) and the boundary conditions (10.6)–
(10.10) are rewritten:
c
(n)∗
11 u
(n)
1,11 + c
(n)∗
3113 u
(n)
1,33 + ω
2 u
(n)
1 + c
(n)
2 u
(n)
3,13 − β
(n)∗
1 u
(n)
4,1 = 0,
c
(n)
2 u
(n)
1,13 + c
(n)∗
1331 u
(n)
3,11 + ω
2 u
(n)
3 + c
(n)∗
33 u
(n)
3,33 − β
(n)∗
3 u
(n)
4,3 = 0,
(10.12)
u
(n)
4,11 + λ
(n)
3 u
(n)
4,33 + iωθ 1 u
(n)
4 + iωθ
(n)
1 E
β
∗(n)
1 u
(n)
1,1 + β
∗(n)
3 u
(n)
3,3
= 0.
x 3 = 0,
⎧
⎪ ⎨
⎪ ⎩
c
(1)∗
3113 u
(1)
1,3 + c
(1)∗
1313 u
(1)
3,1 = q
0
1 ,
c
(1)∗
33 u
(1)
3,3 + c
(1)∗
13 u
(1)
1,1 − β
(1)∗
3
u
(1)
4 = q
0
3 ,
−λ
(1)∗
33 u
(1)
4,3 = q
0
4 ,
(10.13)
x 3 = −h,
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u
(1)
1 = u
(0)
1 ,
u
(1)
3 = u
(0)
3 ,
c
(1)∗
3113 u
(1)
1,3 + c
(1)∗
1313 u
(1)
3,1 −
−c
(0)∗
3113 u
(0)
1,3 − c
(0)∗
1313 u
(0)
3,1 = 0,
c
(1)∗
33 u
(1)
3,3 + c
(1)∗
13 u
(1)
1,1 − β
(1)∗
3
u
(1)
4 −
−c
(0)∗
33 u
(0)
3,3 − c
(0)∗
13 u
(0)
1,1 + β
(0)∗
3
u
(0)
4 = 0,
(10.14)