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Elastic Half-space Conditions

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. (gif) 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. (gif) we obtain the following boundary conditions,

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

Michael B. Porter
Tue Oct 28 13:27:38 PST 1997