Uniform leaky-wave antennas (LWAs) with limited length have suffered from the presence of high-level side lobes, which could lead to the reception of unwanted noise especially when the LWA is short. This communication examines how the phase constant of a short LWA can be manipulated to achieve side lobe suppression. The tapering of the phase constant is shown to result in higher order 'image patterns,' which can be used advantageously for side lobe suppression. Validation of this principle is given by a cosine-shaped width-tapering profile design of a half-mode substrate-integrated waveguide (HMSIW) LWA. The measured results show that the side lobe level (SLL) of a short (1.4 λ0) HMSIW LWA has been suppressed to below-16dB in the entire space wave leakage region using this tapering profile.