A travelling-wave long sinusoidal slot in the narrow face of a millimeter-wave (Ka-Band) rectangular waveguide is investigated. To the best of our knowledge, a slot of this shape is applied to the narrow face of a rectangular waveguide for the first time at this frequency range. The analytical study is based on the magnetic current model. Experimentally, far-field and S-parameters are measured and a good match is obtained between analytical and measured patterns. Frequency scan rates and velocity ratios of the structure are calculated. The effect of the sinusoidal slot's geometrical parameters (amplitude, period and number of periods) on the radiation patterns is also investigated. The drawback of the sinusoidal radiator is its cross-polarized beam that is not convenient for antenna applications. These co and cross-polarized beams scan in two different angle regions. To filter/suppress the cross-polarized beam, a grid polarizer is mounted to the sinusoidal slot and a 15 dB suppression level is reached. S-parameters measurement shows that the antenna efficiency increases remarkably after the application of the grid polarizer to the slot.