  A new approach was recently introduced by the authors for constructing
analytic solutions of the linear PDEs describing elastodynamics. Here, this
approach is applied to the case of a homogeneous isotropic half-space body
satisfying arbitrary initial conditions and Lamb's boundary conditions. A
particular case of this problem, namely the case of homogeneous initial
conditions and normal point load boundary conditions, was first solved by Lamb
using the Fourier-Laplace transform. The general problem solved here can also
be analysed via the Fourier transform, but in this case, the solution
representation involves transforms of \textit{unknown} boundary values; this
necessitates the formulation and solution of a cumbersome auxiliary problem,
which expresses the unknown boundary values in terms of the Laplace transform
of the given boundary data. The new approach, which is applicable to arbitrary
initial and boundary conditions, bypasses the above auxiliary problem and
expresses the solutions directly in terms of the given initial and boundary
conditions.
