The modeling of source a distributions of finite spatial extent in ultrasound and medical imaging applications is a problem of longstanding interest. In time-domain methods, such as the finite-difference time-domain or pseudospectral approaches, one requirement is the representation of such distributions over a grid, normally Cartesian. Various artifacts, including staircasing errors, can arise. In this short contribution, the problem of the representation of distribution over a grid is framed as an optimization problem in the Fourier domain over a preselected set of grid points, thus maintaining control over computational cost and allowing the fine-tuning of the optimization to the wavenumber range of interest for a particular numerical method. Numerical results are presented in the important special case of the spherical cap or bowl source.