  In this paper we demonstrate that using a mathematical physics approach
(focusing the attention to the physics and using mathematics as a tool) it is
possible to visualize the formation of the transverse modes inside a
cylindrical waveguide. In opposition, the physical mathematics solutions
(looking at the mathematical problem and then trying to impose a physical
interpretation), when studying cylindrical waveguides yields to the Bessel
differential equation and then it is argued that in the core are only the
Bessel functions of the first kind those who describe the transverse modes. And
the Neumann functions are deemed non physical due to its singularity at the
origin and eliminated from the final description of the solution. In this paper
we show, using a geometrical-wave optics approach, that the inclusion of this
function is physically necessary to describe fully and properly the formation
of the propagating transverse modes. Also, the field in the outside of a
dielectric waveguide arises in a natural way.
