Screened sinusoidal surface-charge mode ($p=\sqrt{k^2+{\kappa }_D^2}$)

A lateral sinusoidal surface charge density with wave number $k$ in the Debye–Hückel approximation generates vertical exponential or hyperbolic functions with inverse length $p=\sqrt{k^2+{\kappa }_D^2}$. In a homogeneous electrolyte filling all space, an isolated sheet at $z_s$ produces $\varphi ={\sigma }_k(x)e^{-p|z-z_s|}/(2\epsilon p)$. Boundary conditions determine the normalization when the electrolyte occupies only a gap.

Two equally modulated charge sheets in identical full-space electrolyte have charging energy per area ${\alpha }^2[1+e^{-pd}\cos \theta ]/(4\epsilon p)$, where $p=\sqrt{k^2+{\kappa }_D^2}$. Its phase-dependent interaction favors a half-wavelength offset, with opposite charge patches facing each other. For an electrolyte gap with negligible exterior displacement response, the energy is instead ${\alpha }^2[\coth (pd)+\cos \theta /\sinh (pd)]/(2\epsilon p)$, with the same minimizing phase. Absolute energies require specified exterior electrostatic interface boundary conditions.