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Fano resonances and bound states in the continuum in a simple phononic system

We derive expressions for the Fano parameter as an explicit function of known material parameters for a simple but realistic phononic system. Based on these expressions, the underlying mechanism is intuitively explained in connection with the resonant interaction of an incoming wave with bound states in the continuum. The derived expressions can serve as a guideline for the analysis of other complicated physical systems because of their analytical simplicity and the generality of the phenomena. Furthermore, they are useful for controlling the lineshape of Fano resonances.