Shallow water propagation can be described using modal theory. For low frequency sources, propagated signals are composed of a few dispersive modes, each of them propagating with its own frequencydependent wavenumber. Modal estimation, and particularly wavenumber estimation, is of great interest in seabed characterization but classically requires a large and dense horizontal line array (HLA). The compressed sensing (CS) paradigm, which allows one to reduce the number of sensors, has been used to overcome this limitation. However, CS performance is directly linked to the discrete basis used in the process and is known to degrade with basis mismatch. To mitigate this issue, the current paper proposes a physics-based grid-free approach to perform wavenumber estimation using a HLA with a limited number of sensors and a single broadband source. The proposed method has three main features: it starts with a speed correction to prevent wavenumber aliasing (using water sound speed at the array location), it then embeds physical prior (the modal dispersion relation) at the core of the CS framework, followed by a CS grid-free approach. The performance of the method is quantified on simulated data using the Jaccard's distance. The method is then applied successfully on experimental data from the 2017 Seabed Characterization Experiment.