Abstract: I will formally discuss the dispersion of waves arising from charge density oscillations near a fixed plane in 3D at zero temperature from a Partial-Differential-Equation (PDE) perspective. The goal is to describe the interplay of microscopic scales that include a binding length in the emergence of the surface plasmon (SP), a collective low-energy charge excitation in the vicinity of the plane. The starting model is a time-dependent Hartree-type PDE in 3D that aims to provide a mean-field description of a confined interacting-particle quantum system. Two main ingredients are the repulsive Coulomb interaction, and an idealized binding external potential in the vertical (z-) direction. The linearization of this equation around the ground state yields a homogeneous integral equation for a suitably defined scattering amplitude. The existence of nontrivial solutions implies a dispersion relation, which non-linearly connects the temporal frequency and the wave number of charge oscillations near the plane. This relation is obtained exactly in closed form by a transform technique. If time permits, I will discuss the effect of a periodic potential, and aspects of the semiclassical limit.