Early-time field response of a conducting cylinder excited by a line current

Considers the early-time field response of a perfectly conducting circular cylinder to cylindrical electromagnetic waves generated by a parallel filament carrying a unit-step current. The geometry of the problem is shown. The problem can be solved by applying the Watson transformation to obtain the solution for the shadow region and the physical optics approximation for the illuminated region. A double Laplace transform is applied to the wave equation of the vector potential with respect to time and the angular cylindrical coordinate. From the solution of the ordinary differential equation satisfied by the transformed vector potential, the transformed electromagnetic field quantities are derived. To obtain the time dependent solution the authors perform first the inversion of the angle transform for large values of the complex variable in the time transform, by using the theorem of residues for the shadow region and the saddle point technique for the illuminated region. A second inverse Laplace transform yields the time dependent solution