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:
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:
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 .