In a free electron laser where the resonator has a wave-guide structure the phase velocity of the light has a significant impact on the wavelength of the output radiation. This effect is well known in Raman-type FEL's. It is however less known that this effect can also play an important role in Compton-type FEL's. This effect has been experimentally verified in our 6 MeV Compton-type FEL, having a wave-guide resonator with a hole in the outcoupling mirror. Successive transverse modes, which have distinct phase velocities, show up at different wavelengths in the output spectrum. The difference appeared to be as large as 5% of the resonance wavelength. In normal lasers this difference is not more than about 0.1%.