Exact analogue of the Hatano-Nelson model in 1D continuous nonreciprocal systems

We propose a general framework that enables the exact mapping of continuous nonreciprocal 1D periodic systems to the Hatano-Nelson (HN) model. Our approach, based on the two-port transfer matrix, is broadband and is applicable across various physical systems and, as an illustration, we consider the implementation of our model in acoustic waveguides. Through theoretical analysis and experimental demonstrations, we successfully achieve the mapping to the HN model by utilizing active acoustic elements, thereby observing the renowned skin effect. Moreover, our experimental setup enables the exploration of the transition from periodic to open boundary conditions by employing diaphragms of varying radii. Our experimental results, unveil the exponential sensitivity of the system to changes in boundary conditions. By establishing a profound connection between continuous systems and the fundamental discrete HN model, our results significantly broaden the potential application of nonreciprocal wave systems and the underlying phenomena.