The Analysis of a Model for Wave Motion in a Liquid Semiconductor: Boundary Interaction and Variable Conductivity

Abstract

The theory of conducting fluids in relative motion with small conductivity is studied with a model including the Maxwell displacement current. The model is linearized, and the interaction of waves with a plane boundary in three space is studied for two orientations of the external magnetic field. It is found that two families of boundary conditions preserve energy in one orientation (external field orthogonal to the boundary), while in the other (external field parallel to the boundary) only one condition exists which preserves energy. It is shown that generalized Fourier transforms exist, generated from the generalized eigenfunction expansions. Further, it is shown that surface waves are not supported by this model, indicating that their presence is unstable when relative motion of the fluid is allowed (surface waves exist in the still fluid case). Finally, the problem of variable conductivity (decaying to zero at infinity) is studied and steady-state and time dependent solutions are shown to exist for c...