# The Lamb Problem for an Inhomogeneous Elastic Half-Space

• A. G. Alenitsyn
Chapter
Part of the Topics in Mathematical Physics book series (TOMP, volume 1)

## Abstract

The following mixed boundary value problem for the system of equations of elasticity is considered: To find the displacement vector$$\vec u(x,y,z,t) = ({u_x},{u_y},{u_z})$$ satisfying in −∞<x<+∞,-∞<y<+∞, z>0, t>0 the equations of dynamics of an isotropic elastic medium
$${\overrightarrow {p\mu } _{tt}} = (\lambda + 2\mu )\nabla \left( {\nabla ,\vec u} \right) - \mu \left[ {\nabla ,\left[ {\nabla ,\vec u} \right]} \right] + \nabla \lambda \left( {\nabla ,\vec u} \right) + 2\left( {\nabla \mu ,\nabla } \right)\vec u + \left[ {\nabla \mu ,\left[ {\nabla ,\vec u} \right]} \right],$$
(1.2)
where the boundary conditions at z = 0 are
$${\tau _{zx}} \equiv \mu \left( {\frac{{\partial {u_X}}}{{\partial z}} + \frac{{\partial {u_z}}}{{\partial x}}} \right) = 0,{\tau _{zy}} \equiv \mu \left( {\frac{{\partial {u_y}}}{{\partial z}} + \frac{{\partial {u_z}}}{{\partial y}}} \right) = 0,{\tau _{zy}} \equiv 2\mu \frac{{\partial {u_z}}}{{\partial z}} + \lambda \left( {\nabla ,\vec u} \right) = \delta \left( x \right)\delta \left( y \right)\varepsilon \left( t \right),$$
(1.2)
and the initial conditions at t = 0 are
$$\vec u = {\vec u_t} = 0.$$
(1.3)

## Keywords

Transverse Wave Longitudinal Wave Rayleigh Wave Asymptotic Formula Reversal Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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