The solution for a homogeneous elastic medium is given in terms of P- and S-wave potentials, and respectively. For a bounded solution these potentials take the form:

where,

In terms of these potentials, the elastic displacements are given by,

So in terms of **u** and **w** we can write the most general form of the
half-space solution as:

Recall,

so that the most general solution in the lower half-space is

Taking the columns of the above matrix as two linearly independent
solutions and substituting into the the definitions of ** y**
in Eq.
() we obtain the following boundary conditions,

Note that the classical dispersion relation for Rayleigh waves is obtained by taking the free surface condition .

Tue Oct 28 13:27:38 PST 1997