  We introduce a new approach to constructing analytic solutions of the linear
PDEs describing elastodynamics. This approach is illustrated for the case of a
homogeneous isotropic half-plane body satisfying arbitrary initial conditions
and Lamb's boundary conditions. A particular case of this problem, namely the
case of homogeneous initial conditions, was first solved by Lamb using the
Fourier-Laplace transform. The solution of the general problem can also be
expressed in terms of the Fourier transform, but this representation involves
transforms of 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 data.
