Replacement of a summation by an integration in structural acoustics

An analytical formulation of a transfer function pertaining to a simple structural model is used both to examine the result of replacing a summation by an integration and to interpret this result. The model consists of a coated plate immersed in fluid media on either side. The first fluid interfaces the coating atop and is semi-infinite in extent. The second fluid interfaces the bottom of the plate and terminates at a baffle that defines a reflection coefficient. An external drive is placed on a plane that lies in the fluid between the plate and the baffle. Focus is centered on the transfer function between a spectral component in the external drive and a spectral component on the interface with the top fluid. The presence of the baffle generates resonances and anti-resonances in this transfer function. These are discernible in those regions of the frequency domain in which the modal overlap parameters are less than unity. In those regions of the frequency domain in which the modal overlap parameters approach and exceed unity the resonances and the anti-resonances merge into values that are the same as those in the absence of the cavity. It transpires that replacing a summation by an integration is tantamount to artificially merging the resonances and the anti-resonances also in those regions in which the modal overlap parameters are less than unity. The merging, again, are into values that are the same as those in the absence of the cavity. It follows then that in those regions of the frequency domain where resonances and anti-resonances are present under summations, these resonances and anti-resonances are suppressed when the summations are replaced by integrations.