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A. M. Antonov et al.
μ x = 2μl
2 ∂
∂ x
∂v
∂ x −
∂u
∂ y
, μ y = 2μl
2 ∂
∂ y
∂v
∂ x −
∂u
∂ y
.
(2.26)
The principal difference in the formulation of the problem will be observed in the
process of setting the boundary conditions for solving the dynamic equation. In the
problem under consideration, the boundary conditions will have the following form:
σ y y = −Pδ(x), σ y x = 0, μ y = 0.
(2.27)
2.4.1 The Subsonic Case
Before proceeding to the solution of the problem with a moving source at supersonic speeds (c 1 > c 2 > D), let us consider the case of the motion of a source of
disturbances with a velocity less than the velocities of the longitudinal and shear
waves—the subsonic case.
To do this, we introduce a moving coordinate system (x, y), in which the source
of perturbations resides and which is connected with the fixed coordinate system by
the famous Galileo transformation:
x = x
− Dt, y = y
.
(2.28)
Relations (2.26), (2.28) allow to write down equations of dynamics in displacements:
λ + 2μ − ρ D
2
∂
2 u
∂ x 2 + λ
∂
2 v
∂ x∂ y
+ μ
∂
2 v
∂ x∂ y
+
∂
2 v
∂ y 2 + l
2
∂
2 v
∂ x∂ y
−
∂
2 u
∂ y 2
= 0,
(λ + 2μ)
∂
2 v
∂ y 2 + λ
∂
2 u
∂ x∂ y
+ μ
1 −
ρ D
2
μ
∂
2 v
∂ x 2 +
∂
2 u
∂ x∂ y
− l
2
∂
2 v
∂ x 2 −
∂
2 u
∂ x∂ y
= 0.
(2.29)
The solution to Eqs. (2.29) will look in the form:
u = Ae
kqy sin(kx), v = Be
kqy cos(kx).
(2.30)
After substitution (2.30) into (2.29) we get the system of algebraic equations with
respect to A and B
−μl
2 k
2 q
4
+ μq
2
1 + l
2 k
2
−
λ + 2μ − ρ D
2
+
μl
2 k
2 q
3
+ q
λ + μ − μl
2 k
2
B = 0,
−μl
2 k
2 q
3
− q
λ + μ − μl
2 k
2
A