2 Excitation of the Waves with a Focused Source …
29
+
λ + 2μ − ρ D
2
q
2
−
μ + μl
2 k
2
− ρ D
2
B = 0,
(2.31)
which has nonzero solutions when:
l
2 k
2
− γ + β
2
1
l
2 k
2 q
6
+
l
2 k
2
1 + l
2 k
2
− β
2
1
+ γ
1 + l
2 k
2
1 − γβ
2
1
+ 2l
2 k
2
γ − l
2 k
2
q
4
+
1 + l
2 k
2
1 + l
2 k
2
− β
2
1
−
γ − β
2
1
2 +
γ − l
2 k
2
2
q
2
+
γ − β
2
1
1 + l
2 k
2
− β
2
1
= 0.
(2.32)
Here, β
2
1 = ρ D
2
/μ—the ratio of the square of the velocity of the source to the
square of the velocity of the shear wave; γ = λ + 2μ/μ—the ratio of the square of
the velocity of the longitudinal wave to the square of the velocity of the shear wave.
The roots of equations (2.32) q 1 , q 2 , q 3 should be sought when Re(q) > 0.
Longitudinal and transverse displacements will be expressed as sums:
u =
3
i=1
∞
0
A i e
kq i y sin kxdk, v =
3
i=1
∞
0
α i A i e
kq i y cos kxdk.
(2.33)
Here,
α i =
μl
2 k
2 q
4
i − μq
2
i
1 + l
2 k
2
+
λ + 2μ − ρ D
2
μl 2 k 2 q
3
i +
λ + μ − μl 2 k 2
q i
, B i = α i A i .
(2.34)
Let us substitute (2.33) into boundary conditions (2.27), we will get the system
for A 1 , A 2 , A 3 identification.
3
i=1
[λ + α i q i (λ + 2μ)]A i = −
P
π k
,
3
i=1
−α i
1 − l
2 k
2
+ q i
1 − lk
2
α i q i
+ lk
2 q i
1 − q
2
i
A i = 0,
3
i=1
q i (α + q i )A i = 0.
(2.35)
It is known that the effect of moment stresses is especially pronounced at short
waves (Erofeyev 2003). Therefore, we introduce a dimensionless small parameter
ε = 1/lk.
Accurate to values of order ε
2 from (2.32) we get: