When subjected to rapid acceleration, a metal plate that is not perfectly flat displays a type of Rayleigh-Taylor instability, which is affected by shear strength. We investigate the initial stage of this instability assuming that the deviation from flatness is small and the pressure producing the acceleration is moderate. Under these assumptions, the plate can be modeled as elastic and incompressible, and the linearized form of the governing are valid. We derive a linear initial/boundary-value problem that models the flow and obtain analytical formulae for the solutions. Our solutions exhibit vorticity inside the plate, an important feature caused by shear strength that was omitted in previous solutions. The theoretical relationship between the acceleration and the critical perturbation wave length, beyond which the flow is unstable, agrees quantitatively with results of numerical simulations and experiments.