THEORY OF SYSTEMS WITH SMALL BOUNDARY ROUGHNESS IN APPLICATION TO ELECTRON STATES IN QUANTUM CHANNELS, ELECTRO- AND HYDRODYNAMICS

Аннотация

The solutions of Laplace and wave equations in the systems with small one-dimensional surface roughness are studied. The conformal mapping technique is used. This permits the exact solution of Laplace equation and approximate solution of the wave one, if the characteristic height of the roughnesses is smaller then the wavelength. It is shown that such a rough boundary can be replaced by a flat one, however, shifted with regard to the mean surface position. This is correct, if the roughnesses are small, but maybe not smooth. Different physical problems at such boundary are reduced to this formulation. Namely, the effective capacity of a flat capacitor, the resistivity of a conducting layer, reflection of the electromagnetic wave on the metal surface, the laminar hydrodynamic flow in the rough 2D tube, the edge effects of the electron states in a quantum layer, the wave resistance of a planar waveguide.