Theory of the anomalous skin effect in a plasma pinch with a gradually falling-off electron concentration

The interaction of cylindrical electromagnetic waves in a plasma pinch is examined in the limiting case of strong anomalous skin effect. We obtain the integrodifferential equations describing a cylindrical pinch. Particular attention is paid to the examination of an electron concentration which falls off gradually with increasing distance from the axis of the pinch. We solve exactly an integrodifferential equation with a kernel, which does not reduce to the difference type. By use of a Mellin transform the integrodifferential equation reduces to a functional equation whose solution can be obtained in general form. We examine separately the absorption of waves of the electric and magnetic types. We obtain the coefficients of reflection from the pinch for each type of wave and the linear resistance of the plasma pinch.