30
A. M. Antonov et al.
q 1 = a 01 ε, q 2 =
(a 02 + a 12 ε)
√
ε
, q 3 =
(a 02 − a 12 ε)
√
ε
,
(2.36)
where
a 01 = −
γ
1 − β
2
0
√
2
, a 02 = −
1 + i
4
√
128
,
a 12 =
i(1 + i)
3 − β
2
0
√
2
2
4
√
8
, β
2
0 =
ρ D
2
λ + 2μ
.
(2.37)
From (2.34), (2.35), and (2.36), we get the following relations:
α 1 = α 01 + α 11 ε, α 2 = q 2 , α 3 = q 3 , A 1 = −
P
πλk
, A 2 = A 3 = 0,
α 01 =
1 + β
2
0
γ
2
− 2β
2
1
γ
1 − β
2
0
, α 11 = γ
1 − β
2
0
1 + β
2
0
γ
2
2
− β
2
1
(2.38)
which allow to write down displacements (2.33) in the following form:
u = −
P
πλ
e
−β 2 y
∞
k 0
sin kx
k
dk,
(2.39)
v = −
Pα 01
πλ
e
−β 2 y
∞
k 0
cos kx
k
dk −
Pα 11 k 0
πλ
e
−β 2 y
∞
k 0
cos kx
k 2 dk,
(2.40)
where k 0 = 1/l, β 2 = a 01 k 0 .
From the expressions (2.39) and (2.40), it can be seen that displacement amplitudes
will vary depending on the magnitude of the load of the moving source P and its
velocity D.
V 1 = −
vπλe
β 2 y
P
= α 01
∞
k 0
cos kx
k
dk + α 11 k 0
∞
k 0
cos kx
k 2 dk,
(2.41)
Figure 2.6 shows the dependence of the normalized amplitude of the transverse
displacement on the square of the dimensionless velocity of a moving source of
disturbances. It is clear from the graphic image that as the speed of a moving source
reaches the shear wave speed, V 1 increases unlimitedly.