This document analyzes wave propagation in a multiple groove rectangular waveguide using Fourier transforms. It presents: 1) A rigorous dispersion relation for a multiple groove guide derived from enforcing boundary conditions on the electric and magnetic fields. 2) Numerical results showing good agreement with prior work on double groove guides and demonstrating that a dominant mode approximation is accurate. 3) Plots of the magnetic field distributions for the first few modes of a quadruple groove guide, confirming the validity of the dominant mode approximation.