Surface gravity waves play a major role in the exchange of momentum, heat, energy, and gases between the ocean and the atmosphere. The interaction between currents and waves can lead to variations in the wave direction, frequency, and amplitude. In the present work, we use an ensemble of synthetic currents to force the wave model WAVEWATCH III and assess the relative impact of current divergence and vorticity in modifying several properties of the waves, including direction, period, directional spreading, and significant wave height Hs. We find that the spatial variability of Hs is highly sensitive to the nature of the underlying current and that refraction is the main mechanism leading to gradients of Hs. The results obtained using synthetic currents were used to interpret the response of surface waves to realistic currents by running an additional set of simulations using the llc4320 MITgcm output in the California Current region. Our findings suggest that wave parameters could be used to detect and characterize strong gradients in the velocity field, which is particularly relevant for the Surface Water and Ocean Topography (SWOT) satellite as well as several proposed satellite missions.