This paper presents a global exponential stability (GES) proof for a signal-based nonlinear wave encounter frequency estimator. The estimator under consideration is a second-order nonlinear observer designed to estimate the frequency of a sinusoid with unknown frequency, amplitude and phase. The GES proof extends previous results that only guarantee global stability. Typical applications are control and decision-support systems for marine craft, where it is important to know the sea state and wave frequency. The theoretical results are verified experimentally by analyzing data from towing tank experiments using a container ship scale model. The estimates for both regular and irregular waves confirm the results. Finally, the estimator is applied to full-scale data gathered from a container ship operating in the Atlantic Ocean during a storm. Again the theoretical results are confirmed.