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Electrostatic Surface Wave Resonances for an Inhomogeneous Plasma Cylinder

Two solutions of the Laplace differential equation for an inhomogeneous cold plasma cylinder, whose radial electron density profile is considered parabolic and parametrically variable, are presented and discussed. In the first case (a smooth profile) the equation obtained is the confluent hypergeometric differential equation, also called the Kummer equation or the Pochhammer-Baines equation. The solution is expressed in terms of Kummer (or confluent hypergeometric) functions. In the second case, which is a more general case, the series solution to the same equation is achieved by the method of Frobenius. The surface resonance conditions are investigated and the profile-dependent frequency spectrum calculated when an harmonic electric field is incident perpendicularly to the cylinder axis of the plasma considered as collisionless and opaque. In particular, the effects of boundary different media in which the plasma cylinder is embedded are estimated.