At the free surface we have to consider different boundary conditions. If an upcoming wave hits the free surface it will be measured as a particle velocity vector at the surface. Reflected downgoing waves will also be produced. The free surface boundary condition is that there are no normal stresses on the free surface. For a wave hitting the free surface we have,

This gives the solutions,

where,

For a traction applied to the free surface, there is no upgoing wave. To solve for the downgoing waves produced by a source traction, we have

This gives the solutions,

12/18/1997