Acoustic wave devices with shear horizontal displacements, such as quartz crystal microbalances (QCM) and shear horizontally polarized surface acoustic wave devices, provide sensitive probes of changes at solid-solid and solid-liquid interfaces. Increasingly the surfaces of acoustic wave devices are being chemically or physically modified to alter surface adhesion or coated with one or more layers to amplify their response to any change of mass or material properties. In this work, we describe a model that provides a unified view of the modification in the shear motion in acoustic wave systems by multiple finite thickness loadings of viscoelastic fluids. This model encompasses QCM and other classes of acoustic wave devices based on a shear motion of the substrate surface and is also valid whether the coating film has a liquid or solid character. As a specific example, the transition of a coating from liquid to solid is modeled using a single relaxation time Maxwell model. The correspondence between parameters from this physical model and parameters from alternative acoustic impedance models is explicitly given. The characteristic changes in QCM frequency and attenuation as a function of thickness are illustrated for a single layer device as the coating is varied from liquid like to that of an amorphous solid. Results for a double layer structure are explicitly given and the extension of the physical model to multiple layers is described. An advantage of this physical approach to modeling the response of acoustic wave devices to multilayer films is that it provides a basis for considering how interfacial slip boundary conditions might be incorporated into the acoustic impedance used within circuit models of acoustic wave devices. Explicit results are derived for interfacial slip occurring at the substrate-first layer interface using a single real slip parameter, s, which has inverse dimensions of impedance. In terms of acoustic impedance, such interfacial slip acts as a single-loop negative feedback. It is suggested that these results can also be viewed as arising from a double-layer model with an infinitesimally thin slip layer which gives rise to a modified acoustic load of the second layer. Finally, the difficulties with defining appropriate slip boundary conditions between any two successive layers in a multilayer device are outlined from a physical point of view.