Again, I think mathematics is too much with us. The unwashed intro
student is asking about the "Newtonian" wave-particle distinction - what
Newton asked about light; what Crookes asked about cathode rays; etc - a
naive distinction about PHYSICAL reality, not about mathematical models:
Is the reality behind this observed phenomenon
1) a transfer of material stuff (ie., mass in the pre-relativity,
pre-quantum mechanical - ie., Newtonian - sense) from place to place, or
2) the propagation of changes in the properties of a space-filling medium?
In the modern post-Newtonian world, this is not a usefully definable
distinction.
P.S. The question of how we should restrict the appellation "wave" to
certain mathematical functions seems of little or no use to me.