If a particle can also be a wave, is that wave spread out and frozen in a "time slice" a cross space time?
so, to handle your question one must leaf through the history of investigations in 'dual' nature of particles- are they waves or particles?
The de Broglie proposition that a particle of certain momentum can show a 'wavy' nature was confirmed by several experimenters looking at scattering of particles like 'electron', 'neutron' diffraction experiments.
the diffraction peaks were very much identical to X-ray diffraction peaks as if the the waves scattered were interfering and using Bragg's Law could explain the pattern.
So , that is history of looking at particles but getting a wavy nature.
The wavelength of those particles of momentum p came out to be exactly as
wavelength = h / momentum(p) ; where h is the Planck's constant.
So, all confusions could be set aside as physical world gives its 'signature' through different designed experiments and observations are to be taken in even if with a pinch of salt.
But your picture that the 'waves' may be 'frozen' in time slice did not come out as in the experiments in laboratories had never did a time slice measurement.the time was flowing as continuous river through out the observations.
I'm seeing a wave as if I dropped a rock in a pond. I have 3 measure points spread out from the first wave and .....
Your later observation of looking at a purely classical wave and trying to locate the particle nature is not in line with the dual picture observed in the described above diffraction experiments .
looking for a particle Einstein looked at the light waves of different frequencies falling on a metal plate and found some electrons coming out from it...the ejection of electrons from the metal surface followed an
energetics that came down to 'picturing' light waves as stream of balls/particles hitting the surface electrons and exchanging its energy such
that the incident frequency( energy =h.f) just matched the energy of electrons coming out added to the work function of individual metals (which represented a binding energy of surface electrons of metals).
the energetics of these 'photo electrons' told a story of light waves behaving like a bunch of balls called photons and the phenomena could be described as photons hitting the individual electrons and sharing energy such that
Photon Energy = work function + Kinetic energy of electron
these E.M. waves do not need medium to pass through, whereas your observation on ripples on water is actually water/medium particles oscillating up and down
and you perhaps tried to freeze it for looking at a particle representation of waves.
However , the people trying to reconcile the two observations (described above) of " particles diffracting like waves" and "waves moving like particles"
have tried their best to theoretically construct representation of particles by superposition of waves of varying momentum and getting a packet of waves whose
movements may represent a particle picture and this wave packet analogy fits the quantum picture.
and particles moving with a wavy pattern associated with it fits the other side...but it will depend on design of your experiment whether you wish the wave character to be expressed or a particle picture.
thereby one will have to look for "proper" measurements in the quantum world.