Considering low-frequency acoustic sources, shallow-water environments act as modal dispersive waveguides. In this context, the signal can be described as a sum of a few modal components, each of them propagating with its own wavenumber. When dealing with broadband sources, wavenumber-frequency (f-k) diagrams constitute popular representations naturally enabling modal separation. Based on a Fourier transform, they require however a large number of sensors to resolve wavenumbers with a high-resolution. This limitation can be overcame by adding some physical priors to the processing method. In the continuation of previous works, we propose here a new grid-free algorithm allowing a super-resolution of the (f-k) diagram by benifiting from the sparse nature of the wavenumber spectrum and embedding the broadband behavior of the wavenumbers within the algorithm. The method is validated on simulated data.