A comparison of parabolic wave theories for linearly elastic solids

The Schrödinger equation describes a theory for propagating scalar waves which is frequently termed a parabolic theory. This theory has been demonstrated to provide a paraxial, or narrow-angled, approximation to the theory of acoustic wave propagation, described by the Helmholtz equation, by a variety of seemingly different procedures. Several authors have considered the question of an approximation to the time harmonic equations of linear elastodynamics, which is parabolic in the above described sense. Since none of the deivation procedures employed can be termed rigorous, and since the results of these procedures are different, the validity of each of the theories is suspect and all should be considered further. In this paper we consider three parabolic theories of elastodynamics; by Hudson, by Landers and Claerbout, and by McCoy; and apply them in turn to a computational experiment that can be solved in the perturbation limit using the exact equations of elastodynamics. The principal conclusion achieved is that an approximate theory for propagating vector waves must be based on a representation of the vector wave field, that explicitly incorporates different wave speeds for the dilitational and rotational components, if predictions of the approximate theory are to approach those of the exact formulation for narrow angles.