Formal power series and iterative methods for wave vectors

In the case of a source-free plane-layered elastic medium, and for direct rays, the formal series ‘solution’ of the elastic equations in inverse powers of frequency can be obtained to any desired order by a process of integration by parts of a suitable iterate. The second iteration of the standard wave vector must be employed to ensure that all O(ω-1) terms are included in the iterative solution. For the recently introduced new wave vector, on the other hand, only the first iterate is needed. Under certain circumstances use of the new wave vector may prove to be a more efficient and accurate means of solving the elastic equations.