Hydrofoil sections operating near the free surface are highly prone to marine cavitation, which is the appearance of vapor cavities inside an initially homogeneous liquid medium. Advances in computational and experimental research have led to a better understanding of this multiphase phenomenon, thus enabling researchers to predict and alleviate its adverse effects on hydrofoil performance. In this work, a new BEM-based adjoint optimization prediction method, where the attached cavity is modeled as a free-streamline, is proposed for the analysis of partially cavitating hydrofoils that move with a constant speed beneath the free surface. Following an inverse problem approach, this potential method poses an alternative to higher-fidelity multiphase flow models for sheet cavitation with free surface effects for hydrofoils at moderate submergence depths and angles of attack. The proposed numerical scheme predicts well the cavity shape and cavitation number, under the given cavity length assumption, when compared against experimental data and other methods. The developed numerical tool could facilitate the preliminary design of partially cavitating lifting surfaces with, or without, free surface effects.