In this paper we use equivalent circuits to analyze the wavelengths in a Fast and Slow wave Waffle-iron Ridge Guide (FS-WRG). An equivalent circuit for the transverse direction is employed and the transverse resonance method is used to determine the fast wave wavelength. Another equivalent circuit, for the inserted series reactance in the waveguide, is employed for the fast and slow wave wavelength. We also discuss the physical system that determines the wavelengths and the accuracy of this analysis by comparing the wavelengths with those calculated by EM-simulation. Furthermore, we demonstrate use of the results obtained in designing an array antenna.