The potential and wave equations for a finite cylinder attached to one or two semi-infinite slabs

A detailed solution of Laplace's equation is obtained with boundary conditions appropriate for the study of classical transport through cylindrical constrictions. The potential, which is expressed in the spheroidal harmonic representation, can be used to describe the electrical and thermal spreading resistance problems and similar problems occurring in gaseous diffusion, magnetism, hydrodynamics and scalar diffraction theory (e.g. for acoustics) in the long-wavelength limit. The argument is based on standard variational and eigenfunction methods. A summary is given of the procedure required for adapting it to obtain solutions for the same configuration with different boundary conditions and also for the wave equation replacing Laplace's equation.