We take into account two further physical models which play an utmost importance in the framework of Analogue Gravity. We first consider Bose--Einstein condensates (BEC) and then surface gravity waves in water. Our approach is based on the use of the master equation we introduced in a previous work. A more complete analysis of the singular perturbation problem involved, with particular reference to the behavior in the neighbourhood of the (real) turning point and its connection with the WKB approximation, allows us to verify the thermal character of the particle production process. Furthermore, we can provide a simple scheme apt to calculate explicitly the greybody factors in the case of BEC and surface waves. This corroborates the improved approach we proposed for studying the analogue Hawking effect in the usual limit of small dispersive effects.
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