For a family of oscillatory Stokes layers, the spatio-temporal evolution of impulsively excited disturbances is investigated, using direct numerical simulations of the linearised Navier-Stokes equations. The semi-infinite planar Stokes layer is modified to incorporate a low-amplitude, highfrequency harmonic, which provides a simplified model of the external noise found in physical experiments. For the unmodified Stokes layer, impulsively excited disturbances are known to form family-tree-like structures, composed of multiple wavepackets. The long-term behaviour that is encompassed within these structures is studied, together with the effects upon them of the alterations to the base flow. In the absence of any base-flow modification, the disturbances are discovered to exhibit a subharmonic pattern of temporal growth, with a periodicity that is twice that of the basic state. It is also shown that when linear instability first arises, it takes an absolute rather than a convective form. Inclusion of a high-frequency harmonic into the basic state is found to have a strong and destabilising impact upon the impulse response. The development of the family-treelike disturbance structure can be disrupted, changing the character of the absolute instability and promoting its appearance at much reduced Reynolds numbers.