Wave / Particle contradiction

"Everyone knows" there's a contradiction between the wave and particle
concepts - a wave can't be a particle and vice versa. I never
understood it. Can someone please explain?

I understand wave to be any physical phenomena that satisfies the wave
equation. For a particle I didn't find any definition, but intuitively
it means something with energy, momentum and maybe velocity, defined
in some region of space/time. Why can't such a thing also satisfy the
wave equation? The only problem is that it may be somewhat hard to
imagine.

.

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