Self intersections of wavefront evolution

In this paper we generalise results of V. I. Arnold on functions defined on discriminant sets of simple singularities to the case where we have a transverse intersection of such sets. This generalization is very easy, but the results are of some geometric importance since they describe (amongst other things) the quasi global behaviour of wavefront evolution that is the changes in the self intersections of the wavefronts as time progresses. We first prove the required generalisation and then discuss how to proceed when considering functions on the transverse intersection of bifurcation sets. Familiarity with the results and notation of Arnold's paper (1976, Wavefront evolution and equivariant Morse lemma, "Commun. pure appl. Math." 29, 557-82) is assumed.