Realistic density profiles in the very periphery of tokamak discharges have been considered in the study of lower hybrid wave coupling by a waveguide array (grill). The density profile is assumed to be exponential in the scrape-off region up to the limiter radius and then linear. The effect of a sharp density gradient or of a density discontinuity at the limiter radius is discussed. It is shown that these effects lead to a strong modification of the transmission coefficient which can be optimized by adjusting the grill mouth position.