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,
and from Eq. (
) we obtain,
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 
.