This paper addresses the physics of the oscillatory flow in the vicinity of a series of parallel plates forming geometrically identical channels. This type of flow is particularly relevant to thermoacoustic engines and refrigerators, where a reciprocating flow is responsible for the desirable energy transfer, but it is also of interest to general fluid mechanics of oscillatory flows past bluff bodies. In this paper, the physics of an acoustically induced flow past a series of plates in an isothermal condition is studied in detail using the data provided by PIV imaging. Particular attention is given to the analysis of the wake flow during the ejection part of the flow cycle, where either closed recirculating vortices or alternating vortex shedding can be observed. This is followed by a similarity analysis of the governing Navier-Stokes equations in order to derive the similarity criteria governing the wake flow behaviour. To this end, similarity numbers including two types of Reynolds number, the Keulegan-Carpenter number and a non-dimensional stack configuration parameter, d/h, are considered and their influence on the phenomena are discussed.