On in‐out splitting of incident fields and the far‐field behaviour of Herglotz wavefunctions.

It is easy to write down entire solutions of the Helmholtz equation: Examples are plane waves and Herglotz wavefunctions. We are interested in the far‐field behaviour of these solutions motivated by the following question: When is it legitimate to split the far field of such an entire solution into the sum of an incoming spherical wave and an outgoing spherical wave? We review the relevant literature (there are disjoint physical and mathematical threads), and then we answer the question for Herglotz wavefunctions, using a combination of the 2‐dimensional method of stationary phase and some explicit examples. [ABSTRACT FROM AUTHOR]