2 Excitation of the Waves with a Focused Source …
27
Fig. 2.5 Dependence of the amplitudes of the stresses in the surface wave on the depth in the
classical case (when L = 0)
2.4 The Statement and Solution to the Problem
of a Gradient-Elastic Medium with a Moving Source
Generating Surface Waves
Let us denote the fixed coordinate system as x
, y
.
Along with the above problem, there is an interest in the question of the generation
of surface waves by a source moving along the border of a gradient-elastic half-space
with a constant velocity D which is greater the velocity of shear c 2 =
μ
ρ
and
longitudinal c 1 =
λ+2μ
ρ
waves.
The equations of dynamics equivalent to the vector equation in the twodimensional case are written as:
∂σ x x
∂ x +
∂τ y x
∂ y = ρ
∂
2 u
∂t 2 ,
∂τ x y
∂ x +
∂σ y y
∂ y = ρ
∂
2 v
∂t 2 .
(2.25)
Components of stress tensor and moment stresses μ x i μ y expressed through
displacements u and v:
σ x x = λ
∂u
∂ x +
∂v
∂ y
+ 2μ
∂u
∂ x , σ y y = λ
∂u
∂ x +
∂v
∂ y
+ 2μ
∂v
∂ y ,
σ x y = μ
∂v
∂ x +
∂u
∂ y − l
2
∂
2
∂ x 2 +
∂
2
∂ y 2
∂v
∂ x −
∂u
∂ y
,
σ y x = μ
∂v
∂ x +
∂u
∂ y + l
2
∂
2
∂ x 2 +
∂
2
∂ y 2
∂v
∂ x −
∂u
∂ y
,