The dispersion characteristics of periodically loaded waveguides are examined. As is well known, a loaded waveguide has the properties of a filter having infinitely many stop bands, which occur when the phase change per section for the wave is a multiple of π. For linear accelerator and other applications it is sometimes desirable to operate the structure in the π-mode but the resulting zero group velocity raises considerable difficulties.In the first part of the paper it is shown how a finite group velocity at the π-mode frequency can be obtained in a waveguide loaded with solid dielectric discs. This is accomplished by making the discs reflectionless at the required frequency; the stop band being thus eliminated, the structure can be regarded as a confluent band-pass filter.To permit the passage of an electron beam it is necessary for the discs to have central holes. Evanescent and other propagating modes are set up, but it is shown in the latter part of the paper that it is still possible to operate the structure in the π-mode with a finite group velocity. This result cannot be obtained if the guide is loaded by thin metal discs, as used in most present-day linear accelerators.