  Resonant accelerating cavities are key components in modern particles
accelerating facilities. These take advantage of electromagnetic fields
resonating at microwave frequencies to accelerate charged particles. Particles
gain finite energy at each passage through a cavity if in phase with the
resonating field, reaching energies even of the order of $TeV$ when a cascade
of accelerating resonators are present. In order to understand how a resonant
accelerating cavity transfers energy to charged particles, it is important to
determine how the electromagnetic modes are exited into such resonators. In
this paper we present a complete analytical calculation of the resonating
fields for a simple cylindrical-shaped cavity.
